Modeling Immune Control Effects on Viral Dynamics during Primary, Chronic and Treated Phases of Viral Infection.

David Burg Faculty of Life Sciences

PhD Thesis

Submitted to the Senate of Bar Ilan University Ramat Gan, Israel

January 2006

This research was conducted under the supervision of Prof. Avidan Neumann The Mina and Everard Goodman Faculty of Life Sciences Bar-Ilan University

Acknowledgements

My highest appreciation goes to Prof. Avidan Neumann for being my mentor for more than six years. These last years of training years with one of the leading experts in viral dynamics have been outstanding. His encouragement has made this time at Bar Ilan University a wonderful scientific voyage. Thanks to my friends and colleagues from Bar-Ilan University namely, Asher Uziel, Rachel Levy-Drummer, Yonit Homburger, Ronen Tal, Ester Hagai, Menachem Skarltz, Baruch Elbaz, Yafit Maayan and Maya Shamilov. The teamwork, brainstorming and feedback from them have been invaluable. This thesis could not have been written without the active participation of many scientists all over the world. I would particularly like to thank all the many participants and researchers for their collaboration which have made this thesis possible. Among the most notable: Prof. Brent Korba, for providing the WHV data; and Gilead Sciences for the collaboration in the analysis of HBV infection. I would like to express my gratitude to Dr. Ramit Mehr and Dr. Ronit Sarid for their ongoing encouragement and outstanding advice. Thanks to Prof. Moshe Kaveh, along with the entire administration staff, in his support of not only my research through President's Fellowship for Excellence, but of the other students who benefit from his vision to advance the leading edge of science. I also thank the members and staff of the Faculty of Life Sciences here at Bar Ilan University, particularly Prof. Zvy Dubinsky and Prof Yossef Steinberger among many others who have enlightened my path through their guidance and counsel. Having the good fortune to exchange with these people and researchers was a privilege I value deeply and have made these last four years memorable. Finally, thanks to my wife Rachel along with Efrat, Avital and Michael for their perseverance, understanding, love and support through times of tribulation.

Table of Contents

Abstract......................................................................................................................... i Chapter 1 ......................................................................................................................1 Introduction and Aims of the Study 1.1 Introduction .........................................................................................................2 1.1.1 Structure of the Study................................................................................2 1.1.2 Aims of the Thesis ....................................................................................2 1.1.3 Mathematical Models in Biology..............................................................3 1.1.4 The Simplest Viral Dynamics Model........................................................5 1.1.5 Infected cells in viral dynamics models ....................................................6 1.1.6 Target-Cell Limiting Viral Infection Model .............................................7 1.1.7 Immune Control Model.............................................................................8 1.2 Methods .............................................................................................................11 1.2.1 Phase Plane Representation.....................................................................11 1.2.2 Steady States ...........................................................................................11 1.2.3 Model Stability........................................................................................11 1.2.4 Parameter Estimation ..............................................................................13 1.3 Results ...............................................................................................................14 1.3.1 Model Simplification ..............................................................................14 1.3.2 Nullclines ................................................................................................14 1.3.3 Steady States ...........................................................................................15 1.3.4 Model Stability........................................................................................16 1.3.5 Viral Parameters Estimations..................................................................16 1.3.6 Phase Plane Analysis ..............................................................................17 1.3.7 Immune Control Dynamics .....................................................................19 1.4 Summary............................................................................................................20 Chapter 2 ....................................................................................................................21 Primary Human Immunodeficiency Viral Dynamics 2.1 Introduction .......................................................................................................22 2.1.1 Virion Genome and Structure .................................................................24 2.1.2 Viral Life Cycle.......................................................................................25 2.1.3 Virus Entry ..............................................................................................26 2.1.4 DNA Provirus Synthesis .........................................................................27 2.1.5 Nuclear Transport, Integration and Gene Expression .............................28 2.1.6 Virion Assembly .....................................................................................28 2.1.7 Mathematical Modeling of HIV..............................................................29

Chapter 2 -Primary Human Immunodeficiency Viral Dynamics (continued)

2.2 Methods .............................................................................................................34 2.2.1 Patient Data .............................................................................................34 2.2.2 Modeling, Parameter Estimation and Fitting ..........................................34 2.3 Results ...............................................................................................................36 2.3.1 Parameter Estimates ................................................................................36 2.3.2 Simulations..............................................................................................37 2.3.3 Tables and Graphs...................................................................................38 2.4 Summary............................................................................................................43 Chapter 3 ....................................................................................................................46 Hepatitis B Viral Chronic Infection Dynamics 3.1 Introduction .......................................................................................................47 3.1.1 Virion Structure.......................................................................................47 3.1.2 HBV Genome..........................................................................................48 3.1.3 HBV Life Cycle ......................................................................................49 3.1.4 Chronic HBV Characteristics..................................................................52 3.1.5 Chronic HBV Kinetics ............................................................................52 3.2 Methods .............................................................................................................54 3.2.1 Inclusion Criteria.....................................................................................54 3.2.2 Pattern Recognition Algorithm ...............................................................54 3.2.3 DNA Measurements................................................................................56 3.2.4 Model ......................................................................................................56 3.2.5 Nonlinear Data Fitting.............................................................................57 3.2.6 Statistical Analysis ..................................................................................57 3.3 Results ...............................................................................................................59 3.3.1 Individual Viral Kinetics Patterns...........................................................59 3.3.2 No “Placebo Effect”................................................................................60 3.3.3 Definition of Spontaneous Declines .......................................................60 3.3.4 Kinetic profiles........................................................................................61 3.3.5 Baseline correlations ...............................................................................62 3.3.6 Modeling of SPDs ...................................................................................63 3.3.7 Tables and Graphs...................................................................................65 3.4 Summary............................................................................................................71 Chapter 4 ....................................................................................................................73 Woodchuck Hepatitis Viral Chronic Infection during Therapy 4.1 Introduction .......................................................................................................74 4.1.1 Woodchuck Hepatitis Virus (WHV).......................................................74 4.1.2 Clevudine (L-FMAU) .............................................................................75 4.1.3 The Relevance of the Amimal Model .....................................................75

Chapter 4 -Woodchuck Hepatitis Viral Chronic Infection during Therapy (continued)

4.2 Methods .............................................................................................................77 4.2.1 Animals ...................................................................................................77 4.2.2 Treatment Regime...................................................................................77 4.2.3 Sampling .................................................................................................77 4.2.4 DNA Measurements................................................................................78 4.2.5 Mathematical Model ...............................................................................79 4.2.6 Immune Control Model...........................................................................80 4.2.7 Parameter Estimations.............................................................................81 4.2.8 Nonlinear Fitting .....................................................................................81 4.2.9 Statistical Analyisis.................................................................................82 4.3 Results ...............................................................................................................83 4.3.1 WHV Kinetic Profiles .............................................................................83 4.3.2 Parameter Value Estimations ..................................................................84 4.3.3 Simulations..............................................................................................87 4.3.4 Tables and Graphs...................................................................................88 4.4 Summary............................................................................................................94 Chapter 5 ....................................................................................................................96 Discussion References .................................................................................................................102 ‫תקציר‬..........................................................................................................................‫א‬

Figures and Tables Figure 1.1 Basic biological model of viral infection dynamics ...........................................7 Figure 1.2: Nullcline analysis of Equation 1.5 ..................................................................15 Figure 1.3: Phase plane analysis. .......................................................................................18 Figure 1.4: Graph of simulation of the immune control effect. .........................................18 Figure 2.1: The dynamics of HIV and CD4+ T cells during HIV illness and AIDS. ........31 Figure 2.2: HIV genome organization. ..............................................................................31 Figure 2.3: HIV life cycle ..................................................................................................31 Figure 2.4: Reverse Transcription......................................................................................32 Figure 2.5: Patient kinetics and simulation results ............................................................41 Figure 2.6: Excerpt from Stafford et al. 2000. ...................................................................42 Figure 3.1: Outcomes of HBV infection and their incidence in the population. ...............50 Figure 3.2 Hepatitis B virus schematic diagram................................................................50 Figure 3.2 Hepatitis B virus schematic diagram and micrograph......................................50 Figure 3.3: Hepatitis B viral genome. ................................................................................51 Figure 3.4: Hepatitis B viral life cycle...............................................................................51 Figure 3.5: Pattern recognition algorithm flowchart..........................................................55 Figure 3.6: Mean viral load per month of study with placebo...........................................65 Figure 3.7: Patient viral load maximum and minimum values..........................................65 Figure 3.8: Percentage of patients to have SPD per month. ..............................................65 Figure 3.9: Baseline parameters correlation with SPD......................................................66 Figure 3.10: Individual chronic HBV viral load and ALT kinetic patterns.......................68 Figure 3.11: Viral load and ALT data and theoretical curves SPD patients......................69 Figure 3.12: Second phase Rx slope correlation with Pre-Rx slope. ..................................70 Figure 4.1: Individual viral load during Clevudine and Lamivudine therapy. ..................88 Figure 4.3: Results of fitting the model to the WHV kinetic data. ....................................90 Figure 4.4: Gender vs. infected cell loss rate.....................................................................92 Figure 4.5: Viral suppression of Clevudine and Lamivudine therapy in woodchucks......92

Table 1.1: Model outcomes as a function of k and θ .........................................................19 Table 2.1: HIV proteins. ....................................................................................................33 Table 2.3. HIV kinetic parameters. ....................................................................................38 Table 2.4: HIV model parameter approximations. ............................................................39 Table 2.5: HIV fitted parameter values..............................................................................40 Table 3.1: Prevalence of RVD and concomitant ALT elevations. ....................................66 Table 3.2: Prevalence of kinetic patterns and HBeAg. ......................................................66 Table 3.3: HBV model parameter approximations. ...........................................................67 Table 4.1: WHV kinetic characteristics. ............................................................................89 Table 4.2: WHV therapy parameters. ................................................................................91 Table 4.4: Estimated values for k and θ.............................................................................93

Abbreviations

AIDS – Acquired Immunodeficiency Syndrome ALT – Alanine Aminotransferase AST – Asparate Aminotransferase BMI – Body Mass Index cccDNA – covalently closed circular DNA CLV – Clevudine (L-FMAU) CTL – Cytotoxic T Lymphocytes DNA – Deoxyribonucleic acid DWN – Patients exhibiting slow viral decline kinetic profiles FLT – Patients exhibiting no viral decline kinetic profiles ge – Genomic equivalents HAART – Highly Active AntiRetroviral Therapy HBeAg – Hepatitis B Virus envelope antigen HBsAg – Hepatitis B surface antigen HBV – Hepatitis B Virus HCC – Hepatocarcinoma HIV – Human Immunodeficiency Virus IFN-α – Interferon-α IU – International units LAM – Lamivudine (3TC)

Abbreviations (continued)

MHC – Major Histocompatibility Complex PCR – Polymerase chain reaction RNA – Ribonucleic acid RVD – A rapid decline in viral load Rx – Treatment SPD – Patients exhibiting RVD of >1 log or between 1 log and 0.5 log with ALT flares VL – Viral Load WHV – Woodchuck Hepatitis Virus

Abstract

Viral infection dynamics represents a complex non-linear interaction between target cells, infected cells and the virus, with elements from both predator-prey and epidemics models. It can be described as naturally reproducing susceptible target cells are hunted and infected by virus, consequently enabling the virus to reproduce using the infected cell compartment. Viral control is established by target cell limitation and/or loss of infected cells to the immune system reaction. Earlier studies showed the importance of simple non-linear models, in which the dynamical parameters were assumed to be constant in time, to increasing our understanding of viral infections. However, because of their simplicity, they fail to capture the full scope of viral infection behavior. For example, the assumption that target cell dynamics or the immune response is constant over time cannot adequately account for the intricacies of the relationship between host and virus. Interaction of the immune system with the infected cell population must be considered as a dynamic process, in contrast to earlier studies. The model proposed here includes a saturation immune control term in which its effect on loss of infected cells depends on the number of infected cells. This process, while being highly abstract, demonstrates how the combination of simple rules and their interaction in a non-linear system can generate a variety of immune response patterns observed during experimentation. Three possible outcomes of particular interest are eradication of the infection after exposure to a virus, the death of the host due to the virus overwhelming the host or the establishment of chronic infection whereby host and virus survive for long periods of time. The model successfully recreates all three scenarios, depending on initial conditions and parameter values. Moreover, the model can explain the existence of long-term low-level viremia

i

during anti-viral therapeutic perturbation seen in many chronic infections. It also confers the seemingly complex multi-phasic behavior exhibited during the different phases of viral infection and also during therapy. I apply the saturation immune control model in 3 different viral systems, each in a different phase of infection: primary infection with human immunodeficiency virus (HIV); the natural dynamics of Hepatitis B virus (HBV) during chronic phase of infection; the decline of the woodchuck hepatitis virus (WHV) during treatment with antivirals. During primary HIV infection the plasma viral load increases, reaches a peak which then declines leveling off at steady state value. Previous papers have suggested that the decline is due to a limitation in the number of cells susceptible to HIV infection, while other authors have suggested that the decline in viremia is due to an immune response. Here I address this issue by implementing the immune control model on primary HIV-1 infection data from ten anti-retroviral and drug-naïve infected patients. Stafford et al. (2000) showed that the data from all ten patients are consistent with a target-cell-limited model from the time of initial infection until shortly after the peak in viremia, while the kinetics of the subsequent fall and recovery in virus concentration in some patients are not consistent with the predictions of the target-cell-limited model. Although they modeled the data with two different immune system responses, the resulting simulations were not convincing. I illustrate that a simple immune control mechanism, can account for kinetics in viral load data not predicted by the original target-cell-limited model or by other more complicated immune system models. Applying nonlinear least-squares estimation, I find that relatively small variations in parameters are capable of mimicking the highly diverse patterns found in patient viral load data. The half-life of productively infected cells during

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primary infection ranged between 0.2-11.6 days, values consistent with results obtained by drug perturbation experiments. Opportunities also exist for modeling to provide insights into the dynamics of other infectious diseases. Hepatitis, which currently infects more than 9% of the world’s population, is an important target for modeling, and work in this direction is ongoing. Models that incorporate immune responses and deal with the issue of drug resistance that can arise during treatment are of great importance and can yield insights into treatment strategies. Moreover, understanding viral host interactions are paramount in designing efficient antiviral treatment. The analysis of viral dynamics assumes that, in general, viremia in patients without antiviral treatment is in a steady state over time scale of days to months, as is indeed the case for HIV and HCV. However, in chronic Hepatitis B (HBV) patients it is common to observe sporadic ALT flares, changes in HBV- DNA levels and HBeAg loss. This study characterizes in detail the different viral kinetic patterns of HBV-DNA in placebo-treated patients and correlates it with baseline parameters, ALT kinetics and HBeAg loss. 170 HBe Antigen-positive (HBeAg) chronic HBV patients randomized to receive placebo were followed for 48 weeks during a phase 3 study of adefovir dipivoxil. Viral kinetics patterns were blindly categorized in 164 patients with HBV-DNA determined every 4 weeks using the Roche Amplicor PCR assay (LD 400 cp/mL). An HBV-DNA decline was defined as minimum 2 consecutive samples with viremia lower by a threshold of 0.5 or 1.0 log copies/mL than previous levels. An ALT elevation was defined by an increase of 1.5-fold more than pre-decline steady state levels. A spontaneous HBV-DNA decline of more than 1 log (mean 2.8, range 1-5 logs) was observed in 44% of patients during the 1 year period of observation. ALT elevations were temporally correlated (P<0.01) with the start of viral decline (79% of patients with iii

decline 1.0
iv

importance of these spontaneous declines for optimization of the management of antiHBV therapy needs further study. Woodchucks chronically infected with Woodchuck Hepatitis virus (WHV) are an animal model for treatment of humans with chronic Hepatitis B viral (cHBV) infection. It is well established that WHV, like chronic HBV infection, is a dynamic interaction between virus, hepatocyte "target cells" and the host's ability to fight viral infection. Following antiviral therapy two phases of viral load decay are observed: one corresponding to clearance of free virions, and a second slower phase corresponding to the loss of infected cells. To date, no dynamic analysis of Clevudine (L-FMAU) has been undertaken, nor has there been a quantitative comparison to show the differences among the different treatment regimes in the woodchuck/WHV model, and between the model and human cHBV. I also characterize the in vivo dynamic parameters of anti-WHV therapy in chronically infected woodchucks, using a mathematical model of viral infection. Sixteen chronically WHV infected woodchucks were treated with 0.3, 1.0, 3.0 and 10mg/kg Clevudine. Serum was collected at days 0, ½, 1, 2, 3, 5, 7, 14, 21 and 30, and WHV DNA was amplified with an in-house PCR reaction (detection level: >30pgDNA/mL). Values for the antiviral efficacy along with first and second phase decline slopes were generated using kinetics analysis and then using nonlinear fitting of the viral dynamic model on serum WHV DNA measurements during days 0-28. The viral infection model assumes that: 1) target cell dynamics are unchanged during the first 21 days of treatment, and 2) free virions and infected cells are in a quasi-steady state. The same analysis was performed on the viral kinetic data of 5 woodchucks treated with Lamivudine (3TC). The viral load decline during therapy displayed a biphasic kinetic profile. Viral production was dose-dependent (P<0.007) and suppressed viral reproduction with a mean v

efficacy (ε) of 99.8% (range 97.5-99.9993%), indicating that very low levels of viral production persisted during therapy. Viral half-lives, derived from the first phase decline, ranged between 5±2 hours, with no correlation to Clevudine doses. Mean infected cell half-life, represented by the second phase slope ranged between 1 and more than 70 days. Interestingly, even though small doses Clevudine are highly potent, antiviral efficacy and the pre-decline delay showed strong dose-dependency (P<0.001 and P<0.045). Furthermore, at higher doses, post-treatment viral load rebound steady-state values were considerably less than the pretreatment baseline values (P<0.041). However, woodchucks treated with Lamivudine had lower antiviral efficacy (85±10%)). These results of Lamivudine treatment are in agreement with previous findings. Wolters et al (2002) calculated for cHBV patients treated with 3TC the antiviral efficacy, viral and infected cell half-life values of 93%, 17hr and 7d, respectively. In contrast, L-FMAU with its stronger antiviral effects, characteristic rebound delay and failure to reach pretreatment values might be a more effective antiviral than 3TC. The combination of short pharmacokinetic delay, high efficacy and distinctive post-treatment rebound properties, deliver a highly potent antiviral therapeutic. Even though high doses of Clevudine have similar antiviral potential, increased dosing of Clevudine could intensify its other attributes, not only the direct antiviral efficacy; and further study is required to determine if this is so. Our understanding of the pharmacokinetics of Clevudine needs to be expanded in order to understand significance of the delay for end-of-treatment rebound and the slow rebound slope. This "lingering effect", not seen in other treatments, hints that the virus/host parameters have been altered. Also, studies following viral kinetics made need to take into account the very early viral kinetics, since the first phase decline only lasts 2-3 days.

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Chapter 1

Introduction and Aims of the Study

1

1.1 Introduction

1.1.1

Structure of the Study

The main thread of this thesis is the formulization of an immune control viral dynamics mathematical model and its implementation in three different clinical and experimental settings. To accomplish this task: •

I introduce the model and perform analysis and explain its advantages.

•

Each viral model will be dealt with in its own chapter.

•

I will give sufficient introduction.

•

I then individually analyze the viral kinetics of each dataset.

•

I implement the mathematical model.

•

I summarize the main points for each chapter.

•

I end this study with a discussion of the importance the immune control hypothesis has on each viral model and scientific and clinical ramifications where appropriate.

1.1.2

Aims of the Thesis


Explore the dynamic features of viral infection at three junctures: •

Beginning of infection (primary infection phase). The characteristic kinetics of this stage are critical for the determination of the final outcome for the infected individual.

•

During the chronic phase. HBeAg-positive placebo-controlled patients show sporadic, spontaneous and rapid declines in viral load with temporally significant concomitant changes in serum Alanine Aminotranferase (ALT) levels. This is the first time chronic viral infection show spontaneous perturbations in the virus-host steady state.

•

Under anti-viral treatment. Although antiviral dynamics have played an important role in recent years, the multi-phasic kinetics of chronic Hepatitis B virus/Woodchuck Hepatitis virus observed during treatment have yet to modeled successfully.

2


Theorize that control is paramount in viral dynamics behavior which has yet to be explained to satisfaction and deduced quantitatively by current hypotheses as part of a modeling strategy.
Implementation an immune control mechanism integrated into the basic viral dynamics mathematical model. This will offer an analytical description of the central forces at play during viral infection.
Show the importance of the immune response to viral control in vivo. The collated varied viral kinetics data from HIV primary infection, Hepatitis B viral chronic infection and treatment of chronic Woodchuck Hepatitis viral infection will be analyzed using mathematical modeling of viral infection under immune control.
Explain the difference between standard models of viral infection, their drawbacks and the advantage of including viral immune control in mathematical models.
Draw conclusions from observations and analysis, as well as highlight implications this study may have in guiding research into novel therapeutics and optimization of standard treatment regimes.

1.1.3

Mathematical Models in Biology

Modern mathematical theory of the population biology of infectious diseases dates back a least as far as Daniel Bernoulli’s mathematical analysis of smallpox control in 1760. Significant applications of mathematics to biology have occurred for nearly a century, starting from the early work of Vito Volterra and Alfred Lotka on interacting populations, and maturing through fundamental work in population genetics, epidemiology , development and neurobiology. Much of this research stimulated important contributions by other mathematicians; however, until the mid-1990’s, communication between mathematicians and biologists remained problematical; much work in mathematical

3

biology was relatively sterile, uncontaminated by contact with data, while experimental work suffered from a lack of theoretical generality. The situation has changed dramatically in the past decade. Biologists are becoming aware of the power of mathematical modeling, and mathematicians have learned the importance of becoming immersed in data. The spectrum collaboration is providing a continuum of highly mathematical work to experimentalists and clinicians. New and exciting areas (e.g. molecular biology, epidemiology and immunology) have opened up to mathematical investigations. A century of research has elucidated fundamental mechanisms in evolution, collective phenomena and pattern formation, and laid the foundations for more specialized modeling; and the development of new computational tools has greatly expanded the potential both for fundamental studies and for communications. The main impetus for this highly successful field has been the great impact of disease on human health, both historically and in facing the threat of acquired immunodeficiency syndrome (AIDS) among other emerging diseases. Simple models have been remarkably successful in capturing many features of host/parasite dynamics and control. Only until recently, mathematical and computational methods have played a large role in immunology and virology. Impressive advances have come from the use of simple models applied to the interpretation of quantitative data. The best example is in AIDS research. AIDS develops slowly - the average time from HIV infection to the development of full-blown AIDS is about 10 years. Modeling of the progression to AIDS has received considerable attention. The suggestion that progression to AIDS involves a diversity threshold has generated debate, new theory and new experimentation. The role of the immune response in determining the pace of disease progression has yet to be clarified, but mathematical modeling has helped focus attention on the role of cytotoxic T cells. Other key areas in which modeling has played and will continue to play an 4

important role is the understanding of how HIV evolves resistance to antiretroviral drugs and the design of treatment strategies. Mathematical modeling is a theoretical tool that helps in understanding biological systems, and is therefore constrained by biological reasoning. One of the first population models was formulated by Verhulst in 1836 (Murray, 1993). He proposed that populations have initial exponential growth, but their numbers are limited to the environment, and introduced a "logistic term" into the exponential growth equation formulated by Malthus (1798). The term, 1-(N/K), decreases as the number of individuals N approaches the carrying capacity K, thereby reducing the growth rate r by an equal amount for each addition of an individual to the population. The equation dN  N = r 1 −  N dt  K became known as the "logistic equation" and it still serves as a way of describing the process of population growth. Volterra (1925) derived a simple model to describe the dynamics of two-species interaction systems. The model consists of two differential equations, expressing the interdependencies and the change of the populations over time. He then sought to determine the model parameters from data obtained from a fishing community in Italy. Since then, studies have made use of this model and others to interpret biological systems (Elton and Nicholson 1942 and others). As biology has developed into a quantitative science the application of mathematics becomes possible.

1.1.4

The Simplest Viral Dynamics Model

Data obtained from in vivo perturbation experiments using potent antivirals in humans infected with HIV-1, showed an exponential fall in the amount of virus measured in blood plasma (Ho et al. 1994). This seemed to agree with a simplistic model which tracks viral dynamics as a function of time: 5

dV = P − cV dt P an unknown function representing the rate of virus production, c being the clearance

rate constant and V is the virus concentration. If the antiviral drug completely blocks viral production (i.e., P = 0), then the model predicts that V will fall exponentially. Plotting lnV versus time and using linear regression to fit the data to the solution V(t)=V0e-ct (where t=0 is the time therapy) allows the estimation of the half-life of virus in the plasma, a function of c.

1.1.5

Infected cells in viral dynamics models

Since viruses by nature are required to infect cells in order to replicate, it became obvious that more complex models were needed. Models incorporating infected cell infection in the dynamics were needed to explain viral kinetic data obtained during in vivo antiviral perturbation experiments especially during antiviral treatment of Hepatitis C Virus (HCV), which showed complex bi-phasic decreases in viral concentrations. Neumann et al. (1998) explored the mechanistic effects these perturbation experiments were demonstrating. The model proposed that, the action of the antivirals was either to block infection (1-η) or to block production (1-ε) with an efficacy between 0-100%.

dI = (1 − η ) β V − δI dt dV = (1 − ε ) pI − cV dt The apparent mechanism, when using non-linear least-squares fitting of the model to clinical data, seemed to be that antivirals against HCV were blocking production of viral RNA. Other studies went on to ratify this finding.

6

1.1.6

Target-Cell Limiting Viral Infection Model

Phillips (1996) showed that a simple mathematical model - the target-cell limiting viral infection model could explain the control of viral concentrations in HIV primary infection dynamics independent of the immune response. This model used in the research of viral dynamics is one that describes the interactions among target cell, infected cell and virus populations. This model has been the focus of numerous studies of viral infection in vivo. Influx of target cells is assumed to be from a constant source, or from proliferating undifferentiated cells. Target cell death is assumed to proportionate to the number of cells. Virus infects target cells, and infected cells are lost at a constant rate. Infected cells produce and release virions to the extra-cellular region, and virus is cleared from the body. s Target cell population

Infected cell population

d

δ β p

Viral load c

Figure 1.1 The basic biological model of viral infection dynamic assumes constant rates for all parameters and linear effects between the compartments.

The basic model can be translated into a system of differential equations, which describe the interaction that occurs among populations of target cells, infected cells and free virions.

 dT  dt = s − dT − βVT   dI  = β VT − δI  dt  dV  dt = pI − cV 

7

The system of differential equation expresses the kinetics of target cells where T is the concentration of target cells, s and d are the production rate and rate of loss constants of target cells. β is the infection rate constant of target cells by virus; therefore, βVT is the number target cells that become infected cells; and is also the chance of a productive collision between a virion and a target cell. It also expresses the kinetics of infected cells where I is the concentration of infected cells; δ is the loss rate of infected cells. The dynamics of the virus are expressed by V the concentration of virions; p is the production rate constant of virions by infected cells and c is the virus clearance rate constant. The target-cell limiting model is accurate in certain applications. This model has only two steady states - pre-infection and chronic states. Its major drawback lay in the assumption that infected cell loss is constant. This leads to highly oscillatory and inflexible kinetics which are not observed in most reliable viral load data. Further, the simple steady state expressions are inappropriate for complex biologic systems. These problems are seen in published works, but not solved in an adequate or elegant manner. For example, Stafford et al. (2000) attempted to model primary HIV infection data using the target-cell limiting model. They show that, while the model is sufficient for the very early data it cannot be fitted to the data. Therefore they attempt to incorporate an immune response into the model. Even so, the results derived from their model are highly speculative and do not correctly reproduce the data, in the opinion of this writer.

1.1.7

Immune Control Model

It became clear in my earlier work in primary Simian Immunodeficiency virus (SIV) infection, that while the target-cell limiting model was sufficient to explain viral dynamics during treatment, it was incapable of the more complex and robust kinetics needed to model primary infection. Further preliminary work showed that this model

8

could be applied in a number of different and varied datasets to accurately simulate their kinetics. Increasingly complicated models have been proposed to address immune response dynamics. Most use high-dimensional differential systems (one dimension for each population being modeled) increasing the number of variables and parameters. Because of their complexity there remains incomplete understanding of the mechanisms of the role of immune system in controlling viral infection. Furthermore, problems arise in highdimensional models, such as a lack of analytical solutions, parameter estimations and insight into the model itself. I present a model with a different strategy. Motivated by nonlinear dynamics theory, it demonstrates that considerable robustness can result in simple but interacting mechanisms. This model attempts to deal with the viral dynamics by a simple immune response against viral infection. The model assumes that the immune response is a function of viral stimulus, time and which is limited in intensity. These are the hallmarks of the immune system’s response to viral infection. This model is a system of differential equations, describing the temporal changes and interactions among populations of target cells (T), infected cells (I) and free virions (V). dT = s − dt − β VT dt dI = βVT − δI dt dV = pI − cV dt I δ = d + k (t ) I +θ

(1.1) (1.2) (1.3) (1.4)

This model is similar to the basic viral infection model (Equations 1.1-1.3), in target cell and viral compartment dynamics. The essential feature of this model is the expression

9

(see Equation 1.4), a function of viral concentration and time, for immune control which has dramatic effects. The virus invades target cells turning them into infected cells. The infected cells are then lost due to viral-induced (δ) and immune-mediated cytoxicity, the velocity of which is determined by κ (t)

I . This is reasonable, since the immune I +θ

system has an increasing but finite ability to control infection, similar to a saturation function. Furthermore, the immune system directly recognizes infected cells and not free viral particles. The advantages of this model include complex steady state solutions and multi-faceted kinetics. The possible multiple outcomes and dynamic robust behavior, dependent upon parameter values, contribute to this immune control model’s superiority over existing models in the area of viral dynamics research.

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1.2 Methods

1.2.1

Phase Plane Representation

Phase plane representations are the graph of one variable as a function of another. In this case, viral load as a function of target cell availability. These graphs give insight into the important stages of viral infection.

1.2.2

Steady States

Steady states of the model are conditions where there is no change in T, I and V; defined by the equilibria of the system when in mathematical terms, dT/dt=0, dI/dt=0 and dV/dt=0. Since Equations 1.1-1.4 model a biological system, this study is interested only in non-negative equilibria, and all parameters are positive (>0). Formalization of the steady states as functions of viral parameters gives the pre-infection and post-infection steady state values for T, I and V.

1.2.3

Model Stability

While designating steady states and their values is paramount in solving the system of equations, the stability characteristic of the entire system or of each steady state is unknown. A general of stability can be obtained through nullclines. The definition of a nullcline is when the derivative equals zero. Plotting the nullcline functions on a phase plane gives information on the vectors around the steady states (Murray, 1993). If the vectors converge on the steady state then it is stable, whereas the equilibria are unstable if the vectors diverge from it. This is a geometric method for finding the equilibria and extending our understanding of the solutions of the differential equations. In the case of nonlinear systems global solutions not possible and are derived via nullclines. The Jacobian matrix analysis approximates a nonlinear system linearization of 11

the system one around the equilibrium point. The assumption is that the behavior of the solutions of the linear system approximates the nonlinear one. The Jacobian matrix yields the general vectors around the steady states and this determines the specific stability of each steady state.  ∂f x1   ∂t  ∂f x  2 J ei =  ∂t  M  ∂f  xi  ∂t 

∂f x1 x1

∂t

x1

∂t M ∂f xi ∂t

∂f x1

L

∂f x 2

x2

∂f x 2 x1

L

x1

x2

∂t

∂t O M ∂f xi L ∂t

  xi    xi      x1 

Substitution of the steady state values (ei) will give the analytical solution for the stability around ei by the determinant and trace of the matrix  a11 a12  , which are defined by    a 21 a 22 

a11·a22-a12·a21 and a11+a22, respectively. The steady state will be stable when DetJ>0 and TraceJ<0 (Keshet, 1993). The eigenvalues of the Jacobian matrix characterize the fate of the solutions around the equilibrium point from the eigenvalues. Negative or complex eigenvalues with a negative real part, then the equilibrium point is a sink. If the eigenvalues are positive or complex with positive real part, then the equilibrium point is a source (that is all the solutions will move away from the equilibrium point. If they are complex, then the solutions will spiral around the equilibrium point (for the sink) or will spiral away (for the source). When the eigenvalue is a real number with different sign (one positive and one negative) then the equilibrium point is a saddle. The eigenvalues are derived from: λ2+(TraceJ)λ+DetJ=0 Eigenvalues will have imaginary components when ∆=(DetJ)2-4TraceJ<0 (Perko, 1991).

12

1.2.4

Parameter Estimation

Deriving analytical solutions of the model’s differential equations using approximations based on biological assumptions allow mathematical formulation of model parameters. These estimates are tools in understanding the relationships in the biological system. They also aid in fitting the model to clinical data, since the number of “free” parameters are smaller, thereby decreasing the multidimensional search space and simplifying the gradient search topology used generated by nonlinear fitting algorithms.

13

1.3 Results

1.3.1

Model Simplification

Viral dynamics have been shown to be very rapid for HIV (Ho et al., 1995) and Perelson et al. 1996, and for HCV by Neumann et al. (1998). Consequently, V is constrained only by the growth of I and a quasi-steady state can be assumed and therefore V≈pI/c. This assumption is sufficient for ratios of p/c that are large, denoting a rapid viral turnover rate. Therefore Equations 1.1-1.4 are simplified to:  dT  dt = s − (d + β I )T   dI =  β T − δ − k I  I  dt  I + θ  where β =

β' p c

(1.5)

, d is the target-cell death, δ is the infected-cell loss and β are the target-

cell infection rate constants. Iv More complicated saturation terms are possible, such as v , however the assumption I +θu v=u=1 is sufficient to describe viral kinetic behavior (see Results in Chapters 2, 3 and 4) and would only introduce more variability into the model.

1.3.2

Nullclines

Global stability for complex nonlinear systems can be obtained by nullcline analysis. The immune control model has three steady state points. The first steady state, the pre-infection state, is defined as (T>0, I=0). The second is the post-infection chronic steady state, which is the most relevant to this study. The third has negative values for infected cells, which while is only theoretical in biologically relevant systems, may be important in viral infections which are cleared from the host.

14

I

I1 I2

I2

I3 I3

T

Figure 1.2: Nullcline analysis of Equation 1.5. This representaiotn allows the global characterization of the model dynamics.

This representation is mutable, depending on parameter values. A different picture is obtained when the post-infection steady state intersects (T«Tmin, I>0). At these parameter values the infection will be fatal for the host as there are no more target cells, usually cells necessary for host survival. Furthermore, other conditions may allow the post-infection steady state to be only locally attractive thereby allowing to the system to return to the pre-infection steady state.

1.3.3

Steady States

The pre-infection steady state (e1) is defined by I=0, therefore: e1: T1 = s , I1 = 0 d However, the post-infection steady state (e2) is defined by I>0, and is the positive solution of the quadratic function:

a = β (k + δ ) b = d (k + δ − β T0 ) + βδθ

− b ± b 2 − 4ac I2 = 2a

c = dθ (δ − β T0 )

Because dδ and βδ products are very small, the coordinates for e2 can be approximated by: e2: T2 = s

d +I2

, I2 =

d ( β T0 − k ) +

[d (βT0 − k )]2

2( k + δ ) 15

1.3.4

Model Stability

The general Jacobian for Equation 1.5 is: − (d + βI )  − βT  J= I  βI βT − δ − k   I +θ   The Jacobian approximation around e1 is given by:

− d J e1 =   0 

− βs  d  βs − δ  d 

e1 will be a saddle node when

βs

−δ − d d DetJ e1 = dδ − β s

TraceJ e1 =

s δ s δ > , and stable when < . d β d β

The Jacobian approximation around the positive post-infection stead state (e2) is:

J e2

− (d + β I ) − βT  θδI  = − βI   (θ + I ) 2  

TraceJ e1 = −(d + βI ) − DetJ e1 =

θδI (θ + I ) 2

<0

θδI (d + βI ) >0 (θ + I ) 2

Since the trace and determinant are always positive, assuming biologically relevant positive values for parameters and variables, e2 is stable.

1.3.5

Viral Parameters Estimations

Assuming a pre-infection steady state, I derive the target cell influx constant: s = dT0

(1.6)

Estimation of the viral production rate constant may be defined as (Dahari et al., 2005):

p=

cVmax T0

(1.7)

where Vmax is the maximal viremia measured during primary infection and T0 is the concentration of target cells before infection, both of which can be found from data.

16

Equation (1.7) shows the dependence of the viral burst size on pre-infection target cell value. That means, any change in T0 will be compensated for by p/c. I derive an estimate for the target cell loss rate constant by the post-infection viral steady state V =

sp d − , substituting δ , p, and s with Equations 1.4, 1.6 and 1.7: δc β  kθVmax 1  d ≈V  −  β  3

−1

(1.8)

The parameter θ is around the level of I at which the increase or decrease of I tapers off, which can be measured from data. In addition, in viral infections known to be noncytotoxic, δ can be assumed to equal d. Further parameter estimates can be made as data allow, as shown in my earlier work (Burg, 2000).

1.3.6

Phase Plane Analysis

The kinetics of viral infection is determined by Equations 1.1-1.5. Each solution depends on initial conditions and parameter values. Although analytic expression of the model as t=>0 is not available due to its non-linearity, the qualitative behavior of this twodimensional (2D) system can be carried out by phase-plane portrait. The number of target cells (T=T0) is assumed constant, and no immune control is present (I0=0 therefore k=0) prior to exposure to the pathogen. This is initial steady state. Figure 1.3 shows the trajectories of four outcomes of the model. The post-infection outcome can be characterized by the attraction properties of the two steady states. When the preexposure steady state is attractive then the dynamics characteristic of an acute infection are observed and the virus is cleared from the host returning the system to its “virgin” state (Figure 1.3C). However, if it is locally repelling three post-infection possibilities are possible: 1) Death of the host sine the steady state value of target cells is very low (Figure 1.3D); 17

2) establishment of chronic infection with no clearance of the infection (Figure 1.3.B) or 3) chronic infection with damped oscillatory kinetics (Figure 1.3.A). A log[Infected Cell Concentration]

log[Infected Cell Concentration]

B

log[Target Cell Concetration] D log[Infected Cell Concentration]

log[Infected Cell Concentration]

log[Target Cell Concetration] C

log[Target Cell Concetration]

log[Target Cell Concetration]

Figure 1.3: Phase plane analysis of 4 main infection outcomes of viral infection.

Time

C

δ (t )

B

δ (t )

δ (t )

A

Time

Figure 1.4: Graph of simulation of the immune control effect.

18

Time

1.3.7

Immune Control Dynamics

The diverse dynamics and outcomes are possible in significant part to the immune control effect in the model (Equation 1.4). Depending on parameter values, three different shapes of the control function exist, which dictate the outcome of viral infection. The solutions are bounded due to the fact that the target cell population is limited in size. Host death occurs when almost no control is generated against the pathogen and it has a high replicating capacity, i.e., k is very high and θ is high. Pathogen clearance is accomplished when there is an immediate, strong and continuous build up of control. This can occur when k is high and θ is low (Figure 1.4A). The semi-sigmoid increases control to allow the establishment of a chronic infection, and is possible when k and θ are high (Figure 1.4B and 1.4C).

Table 1.1: Model outcomes as a function of k and θ

Outcome

k

θ

Clearance

High

Low

Death

Low

High

Chronicity

Medium

Medium

19

1.4 Summary Interaction of the immune system with the infected cell population, along with target cell and viral compartments is considered to be a dynamic process. I describe this process with 3 ordinary differential equations. Although the immune control term is highly idealized it demonstrates how the combination of simple rules for immune control generates the expected complexity and robustness which are observed in experimental and clinical data in viral infections. The model successfully replicates the acute and chronic infection dynamics exhibited by many different viral/host models. I also find that this model has damped and undamped oscillatory dynamics, along with simpler behavior known to exist in different viral infection data. There are a number of advantages to this model, which endeavors to simplify the immune system’s control of viral infection. First, the variety of different types of outcomes among different disease and even within a single disease can be obtained by the combination of simple mechanisms and interactions. Second, phase plane analysis gives an almost complete understanding of the system, due to its 2-dimensional space. While this is possible in higher-dimensional models, it is difficult at best. Lastly, quantitative parameter estimations are possible and highly informative of the interactions between host and virus. Beyond reproducing viral infection dynamics and outcomes, this model gives a high understanding of the mechanisms at play during immune infection and may aid in discovery of strategies in order to cure or control disease by maintaining the pathogen at sub-clinical levels.

20

Chapter 2

Primary Human Immunodeficiency Viral Dynamics

21

2.1 Introduction Acquired Immunodeficiency Syndrome (AIDS) is a disease that has spread throughout the world. The focal points of the pandemia are areas in which the numbers of infected individuals is still increasing, especially in the poor areas of the world, in particular Africa. Currently, more than 40.3 million people carry the virus in their bodies worldwide. During the year 2005, 4.9 million people were newly infected, 70% of them in Africa. As a result of increasing awareness and advanced medical treatment, in developed countries the prevalence of the disease is decreasing (UNAIDS, 2005). Human Immunodeficiency Virus (HIV) belongs to the family of Retroviridae, characterized by the enzyme RNA dependent DNA polymerase, better known as reverse transcriptase. This unique viral enzyme creates a DNA sequence from an RNA template and nucleosides. HIV also integrates the proviral DNA into the host genome, utilizing the viral enzyme Integrase, in order to complete its life cycle. Another characteristic of this group of viruses is the ability to induce a chronic long-term asymptomatic infection (Fields ed). HIV illness begins with exposure to the virus and productive infection. Primary infection is characterized by the exponential increase in the number of viral particles in peripheral blood, which attains a peak (Vmax) followed by a spontaneous decrease to steady state levels ( V ). Since the main target cells are CD4+ T lymphocytes, there is a decrease in the number of these cells from the initial steady state values (CD40), leading to a minimum (CD4min), and increasing to a new equilibrium ( CD4 ) which is lower than the pre-infection values. This chronic infection can persist without symptoms for a long period of time, with high inter-patient variability. There is also a high probability of transmission during viremia through sexual contact, blood and related products. After the incubation term and through an undetermined mechanism, there is an uncontrolled 22

increase of viral replication, and destruction of the immune system at which time the patient succumbs to opportunistic infections. This stage is termed Acquired Immunodeficiency Syndrome (AIDS) (Hirsch MS and Curran J., 1990). Although mechanisms of AIDS pathogenesis in humans remain unknown, much is understood about the patterns of progression from the initial HIV infection to AIDS (Coffin, 1995 and Mittler et al., 1995). During primary infection, , viral titer rises to high levels within weeks after exposure to HIV, causing a brief influenza-like illness, and the density of circulating HIV decreases. After 3 to 6 weeks this time the infection can be diagnosed by the presence of antibodies to HIV, known as seroconversion. The immune system is unable to entirely clear the virus and the infection enters an asymptomatic phase. During this period the CD4+ T-cell population in the circulating blood gradually declines as a direct or indirect result of HIV infection (Coffin, 1995). Apparently, the immune system is capable of spending many years coping with the burden of HIV infection but after about 10 years on average, the immune system is in total collapse leaving the host unable to combat other threatening microorganisms. At this end stage of infection viral load increases dramatically and immunodeficiency becomes clinically overt. Some scientists have suggested that HIV is not the causative agent of AIDS, arguing that explanations such as an individual's lifestyle can account for immunodeficiency (Duesberg, 1995). However, the ‘lifestyle hypothesis’ can not explain the hemophiliacs, blood transfusion recipients, infants and partners of heterosexual infected individuals that succumbed to AIDS infection. Irrefutable scientific evidence exists proving that HIV is the infectious agent responsible for AIDS (Ho et al., 1995 and Wei et al., 1995). Many of the studies identifying the significance of HIV as a disease progenitor were coupled to advances in molecular biology technology (Mittler et al., 1995). 23

Until the middle of the 1990’s, the principal hypothesis explaining the pattern of primary viral infection in HIV was that the immune system was playing a prominent role in controlling viral replication, was regarded to be in a state of latency during the asymptomatic chronic infection (Temin and Bolognesi 1993). After inception of more sensitive and direct molecular essaying of genomic material, by polymerase chain reaction, and of cell populations, these hypotheses were tested. By using the mathematical modeling, attempts have been made to give explicit estimations of viral parameters to many viruses.

2.1.1

Virion Genome and Structure

Retroviruses are enveloped, positive-strand RNA that rely on a unique enzyme, reverse transcriptase, to convert their RNA genome into a DNA provirus, and are then integrated into the cellular genome. The viral envelope is a lipid bi-layer produced by the cellular plasma membrane and contains the protruding viral env glycoprotein. The composition of the core viral particle includes the p24 capsid protein and contains the viral RNA and enzymes. All retroviruses have in common the three main encoding regions: the capsid proteins (gag), the viral enzymes necessary for replication (pol), and the external glycoprotein (env) responsible for the infectivity of the viral particle. The viral enzymes encoded by pol are reverse transcriptase (RT), integrase (IN), and protease (PR). Figure 2.2 shows the

genomic organization of HIV. Retroviruses have one promoter and one polyadenylation site within the long terminal repeats (LTRs), and express one primary transcript. The location of the polyadenylation signal and processing site allow efficient polyadenylation only at the 3' LTR. In HIV-1

24

U3 and U5 region elements have been proposed responsible for polyadenylation at the 3' LTR. To produce many proteins from a single primary transcript, the retroviruses use different strategies: 1) generation and proteolytic processing of precursor polyproteins, 2) ribosomal frame shifting or suppression of translation termination, 3) alternative splicing of the primary transcript and 4) bicistronic mRNAs producing two proteins. The additional proteins expressed by HIV (Table 2.1) are either part of the viral particle, regulate directly viral gene expression, or interact with the cellular machinery to promote virus propagation. The additional proteins increase the complexity of the organization and expression of HIV and the other lentiviruses. Another distinguishing characteristic of these viruses is the ability to regulate their own expression via virally encoded protein factors. Some scientists propose this property to be essential for the long-term association of lentiviruses with the host and the generation of chronic active infections.

2.1.2

Viral Life Cycle

After the virus attaches to the cell and penetrates the plasma membrane, reverse transcriptase converts the viral RNA to DNA. The produced DNA is transported to the nucleus and integrated into the cellular DNA by the viral integrase. Due to the replication characteristics of the retroviruses, the proviral DNA is flanked by tandem repeated sequences with important regulatory functions, termed the long terminal repeats (LTR). After integration, the retroviral provirus uses the cellular transcription machinery to express the viral RNA that has two essential roles. First, it serves as the genomic RNA that is incorporated in the virion and second, as the messenger RNA that produces all the viral proteins. The genomic RNA and the viral proteins are assembled into the viral

25

particle, which buds out of the cell and infects new cells by attaching to specific cellular receptors. The retroviral life cycle is depicted in Figure 2.3.

2.1.3

Virus Entry

Env binds to the surface receptor CD4, which is a transmembrane glycoprotein of 58kd,

and a member of the Ig superfamily (McDougal et al., 1986). CD4 is found on T lymphocytes, monocytes, B lymphocytes and other cells. It contributes to T cell recognition of foreign antigens in the context of major histocompatibility complex (MHC) class II determinants. CD4 interacts with MHC-II molecules and also with the T cell receptor (TCR). CD4 is the receptor for HIV (Sattentau et al., 1988). Although CD4 is the primary receptor for these viruses, additional factors are necessary for infection, such as CCR5. The env-binding site on the CD4 glycoprotein has been mapped to amino acids 40-82. Specific activation of T cells by antigen presentation induces the CD4-TCR cointernalization. CD4 interaction with env is of paramount importance for HIV-1 and leads to infection and CD4 dysregulation. This affects the function of T cells and eventually leads to depletion of the CD4+ subset of T cells and to immunodeficiency in pathogenic infections. The importance of the env-CD4 interaction for HIV is further underscored by the multiple mechanisms that lead to CD4 modulation; for example, env, vpu and nef interact with and affect CD4 at various stages.

Virus entry requires the fusion of the virus and cell membrane which is mediated by the N-terminal hydrophobic region of the env transmembrane subunit (Freed et al., 1990). HIV env promotes fusion at the near neutral pH of the extracellular milieu. It is also responsible for the formation of syncytia resulting in membrane fusion among many cells.

26

2.1.4

DNA Provirus Synthesis

After virus entry, disruption of the viral capsid occurs and the viral RT is fully activated. A ribonucleoprotein (RNP) complex forms in the cytoplasm of the infected cell and triggers reverse transcription and transport to the nucleus. This RNP complex contains the genomic RNA together with the NC and MA protein, and the viral enzymes RT and IN. During reverse transcription, the two RNA molecules in the virion are converted to a linear double-stranded DNA. The process of reverse transcription requires priming provided by the tRNAlys3 selectively incorporated in the HIV virion, (Figure 2.4) which anneals to the primer-binding site (PBS) at the 5' part of the viral genome during particle formation. The virion contains, on average, eight molecules of tRNAlys3 per two copies of genomic RNA. Elongation at the PBS results in the generation of a nascent DNA molecule of approximately 630nt spanning the region from the PBS to the CAP site at the 5' end of the viral RNA (step 1). The Ribonuclease H (RNase H) activity specifically degrades the RNA part of the RNA-DNA hybrid of RT (step 2). This leaves the 3' end of the strong-stop DNA free to anneal to the R region of the second RNA molecule (step 3). This template 'jump' requires components of the viral capsid and depends on the affinity of RT for its template. After this, elongation by RT results in a nearly complete DNA copy terminating at the PBS, since the R and U5 regions have been removed by the RNase H (step 4). RNase H degrades the second RNA template except the polypurine tract (PPT) (step 5). Synthesis of the complementary (plus-strand) DNA is initiated at the junction between the polypurine tract (PPT) and the U5 region of the LTR. Specific cleavage of the RNA-DNA hybrid at this site initiates the plus-strand DNA synthesis (step 6), which terminates within the PBS. Copying of the PBS and removal of the RNA by RNaseH generates a DNA molecule that can anneal to the 3' end of the opposite DNA strand at the minus-strand PBS (step 7). This second template jump allows elongation of

27

the plus-strand DNA to the end of a complete LTR (step 8). The completion of DNA synthesis results in a linear molecule with one complete LTR at each end (step 9). This molecule is the subject of integration after transfer to the nucleus. Side products of this process are circular molecules containing one or two LTRs, which can be detected after acute infection by HIV-1. These molecules are not able to integrate but may express proteins if transported into the nucleus.

2.1.5

Nuclear Transport, Integration and Gene Expression

The double-stranded proviral DNA, complexed with proteins (pre-integration complex), is transported to the nucleus of the infected cell. The integration of the proviral DNA occurs at random throughout the cellular genome by the action of viral Integrase. The integrated provirus flanked by the tandem LTRs is organized as a eukaryotic transcriptional unit. Although the 5'- and 3' LTR sequences are similar in structure the 5' LTR acts as a strong enhancer/promoter while the 3' LTR's function is a polyadenylation site. Transcription of the provirus by the cellular RNA polymerase II results in a primary transcript that has two important functions: incorporation of the genomic RNA into the virion and also providing all the mRNAs encoding the viral proteins. Both viral and cellular factors regulate the viral promoter. Its activity varies greatly depending on the cell status. In many infected cells in HIV-positive individuals, virus expression remains undetectable. Thus, a state of viral latency exists in individual cells, although the infection is chronically active due to continuous expression of HIV in a fraction of the cells.

2.1.6

Virion Assembly

The expressed structural proteins accumulate inside the plasma membrane. gag also interacts with env, and Pr55gag multimerization results in the initiation of particle formation. Together with the Pr55gag, some Pr160gag-pol is also incorporated into the 28

virion. Two molecules of genomic RNA are also encapsidated together with tRNA molecules, primarily tRNAlys3. The integration of Pr160gag-pol into the particle leads to protease dimerization and activation. The orderly cleavage of Pr160gag-pol and Pr55gag leads to particle maturation and budding. This is an essential step for production of infectious virions, since immature viral particles containing the precursor molecules are non-infectious. Cleavage also initiates the maturation and activation of the other viral enzymes. RT associates with the tRNA-viral RNA complex and initiates reverse transcription if nucleotide triphosphates are available. The accessory proteins vif and vpr, and possibly nef are also incorporated into the virion, along with cellular proteins.

2.1.7

Mathematical Modeling of HIV

Much of the 10-year period until AIDS develops has been characterized as a period of clinical latency, with low but constant levels of virus and infected cells in circulation. Giving HIV-1–infected patients potent antiretroviral drugs and using simple dynamical models to analyze the ensuing decline in viral load has led to important insights into the in vivo processes involved in HIV infection. This analysis established that HIV is rapidly replicating and cleared from the body (Ho et al., 1995) and revealed that the average rate of HIV production was greater than 10 billion virus particles per day, that free virus particles were cleared with a half-life that is probably 6 hours or less and that productively infected T cells had a life-span of about 1.5 days (Ho et al., 1994). These results, which are derived from mathematical modeling, firmly put to rest the view of AIDS as a slow disease in which little happens for years after infection, and replaced it with a new paradigm in which rapid viral dynamics was the centerpiece. Most important, uncovering the rapid replication of HIV led to a new understanding of the observed rapid evolution of the virus and the seemingly inevitable emergence of drug-resistant forms of HIV-1. In part as a result of this increased understanding, treatment protocols using a 29

single drug are being replaced by protocols using combinations of antiretroviral drugs, which have a greater antiretroviral effect and which increase the number of mutations needed for resistance. During primary HIV infection the viral load in plasma increases, reaches a peak, and then declines. While Phillips (1996) has suggested that the decline is due to a limitation in the number of cells susceptible to HIV infection it only holds for declines of 1 log magnitude or less. Other authors (e.g., Kaur et al., 2000) have suggested that the decline in viremia is due to an immune response. Here I address this issue by implementing a novel immune control model to primary HIV-1 infection, and by comparing predictions from these models with data from ten anti-retroviral, drug-naive, infected patients. Applying nonlinear least squares estimation, I and that relatively small variations in parameters are capable of mimicking the highly diverse patterns found in patient viral load data. This approach yields lifespan values of productively infected cells during primary infection consistent with results obtained by drug perturbation experiments. The kinetics of the subsequent fall and recovery in virus concentration in some patients are not consistent with the predictions of the target-cell limiting mode when patient data are examined over periods longer than 30 days after infection. Thus, the target-cell-limited model fails to explain the data in many patients, with the viral load either falling below the prediction or not reaching down to steady state values. However, by including immune response mechanisms the data can be explained. I propose that the immune control model could account for declines in viral load data not predicted by the basic viral infection model.

30

Figure 2.1: The kinetics of HIV, anti-HIV antibodies and CD4+ T cells during HIV illness and progression to AIDS. (Perelson et al. 2000)

Figure 2.2: HIV genome organization. (Frontiers in Bioscience 2005.10:2064-2081)

Figure 2.3: HIV life cycle (Russell Kightley Media)

31

Figure 2.4: Reverse Transcription (http://www-micro.msb.le.ac.uk/3035/Retroviruses.html)

32

Table 2.1: HIV proteins. Encodes the capsid proteins. The precursor is the p55 myristylated protein, which is

gag processed to p17 (MAtrix), p24 (CApsid), p7 (NucleoCapsid), and p6 proteins, by the viral protease. Gag associates with the plasma membrane where the virus assembly takes place. The genomic region encoding the viral enzymes protease, reverse transcriptase and pol integrase. These enzymes are produced as a gag-pol precursor polyprotein, which is processed by the viral protease; the gag-pol precursor is produced by ribosome frameshifting at the C-terminus of gag. env Viral glycoproteins produced as a precursor (gp160) and processed to the external glycoprotein gp120 and the transmembrane glycoprotein gp41. Transactivator of HIV gene expression. Needed for HIV gene expression. Low levels of

tat are found in persistently infected cells. Tat has been localized primarily in the

rev

vif

vpr

vpu

nef

tev

nucleolus/nucleus by immunofluorescence. Extracellular tat can be found and can be taken up by cells in culture. Necessary regulatory for HIV expression. Localized primarily in the nucleolus/nucleus, rev acts by binding to RRE and promoting the nuclear export, stabilization and utilization of the viral mRNAs containing RRE. Rev is considered the most functionally conserved regulatory protein of lentiviruses. Rev cycles rapidly between the nucleus and the cytoplasm. Viral infectivity factor, promotes the infectivity but not the production of viral particles. In the absence of vif the produced viral particles are defective, while the cell-to-cell transmission of virus is not affected significantly. Found in almost all lentiviruses. vpr detected in the cell is localized to the nucleus. It induces nuclear import of preintegration complexes, cell growth arrest, transactivation of cellular genes, and induction of cellular differentiation. Viral protein U is unique to HIV-1, is a type I integral membrane protein with at least two different biological functions: (a) degradation of CD4 in the endoplasmic reticulum, and (b) enhancement of virion release from the plasma membrane. Vpu is involved in env maturation. Nef has been identified in the nucleus and found associated with the cytoskeleton. Its association with the virion is suspected but not proven. One of the first HIV proteins to be produced in infected cells, it is the most immunogenic of the accessory proteins. Initially thought to be a negative factor. The nef genes are dispensable in vitro, but are essential for efficient viral spread and disease progression in vivo. It is necessary for the maintenance of high virus loads and for the development of AIDS in macaques. Nef downregulates CD4, the primary viral receptor, and is also proposed to increase viral infectivity Found primarily in the nucleus. tev contains the first exon of tat, a small part of env and the second exon of rev. It exhibits both tat and rev functions and can functionally replace both these essential regulatory proteins. It is produced very early in infection.

33

2.2 Methods

2.2.1

Patient Data

Data was obtained for patients who remained untreated from a published study by Stafford et al. (2000). They collected viral load data for patients 1 and 2 from Schacker et al. (1996), data for patients 3-9 were provided by the Aaron Diamond AIDS Research Center, and patient 10 data were provided by the Cedars-Sinai Medical Center in Los Angeles, CA. Virus concentration measurements in peripheral blood from ten patients used in parameter estimation. Since specimens were taken only after patients presented with acute infection symptoms, except for patient 10, time of exposure to HIV is not precisely known.

2.2.2

Modeling, Parameter Estimation and Fitting

To analyze acute HIV infection, I use the mathematical model from Chapter 1. Initial target cell population (T0) is estimated by normal CD4+ T cell count in humans which is on the order of 103 cells/mL. This is a maximal estimate since all cells bearing the CD4 molecule, such as T cells, dendritic cells and macrophages are susceptible to infection (Zhang et al., 1999). Assuming a steady state in CD4+ T cell population prior to infection, the value of s can be expressed as s=dT0. Also I assume a quasi-steady state between

viral

and

infected

cell

concentrations

in

peripheral

blood

the

production:clearance ratio for HIV in humans is large (Perelson et al., 1996), and initial values of infected cells are expressed by I0=cV0/p. The HIV-RNA production rate constant is proportional to the maximal viral load which can be measured directly from data, and initial target cell values (see Chapter 1, Methods).

34

Since there is no early viral load data, I set V0 as 10-1 copies HIV-RNA per mL. Even though the exact value is critical for the overall dynamics of the system, the “free” parameter values compensate for shifts in initial viral load concentrations. Although I accept the estimate from earlier studies (Mittler et al., 1999; Ramratnam et al., 1999) of the clearance rate constant of 4 day-1, values of up to 30-fold have no significant impact on the kinetics of the model. The parameter θ, which slows the immune control expression, is given the value of viral load steady state after the spontaneous decline from peak concentrations. The parameters b, k and d are given initial values and ranges according to literature (Table 2.2) and fit to

the individual datasets using nonlinear least squares regression (Oster and Macey, 2000). Since patients’ viral load data collection did not start at time within a short timeframe, time of infection was not known, except in one patient. I therefore used a bootstrap method to find the time shift needed to accurately fit the data.

35

2.3 Results

2.3.1

Parameter Estimates

Bootstrapping the initial time of exposure of the patients to HIV generated mean time shifts of 8.6±5.0 days (range: 0-14 days) allowing the best fit of the model parameters to patients’ datasets. The slope of viral spontaneous decline after peak viremia ranges between 0.01-1.0 log/week, along with the large inter-patient variability of viral half-life values, are consistent with the infected cells decay rate during HAART (Ho et al., 1994), I rule out acceleration of free virion clearance and increased blocking of virion production, as both would necessarily give faster declines. Additionally, inhibition of viral infection of target cells would not allow viral load to reach peak values as high as observed in the data. Table 2.4 shows the parameter estimations of viral infection rate of target cells, target cell- and infected cell half life values derived by the non-linear least-squares fit. The maximum infected cell half-life values due to accelerated cell loss (k) range between 0.411.6 days. Further, the rate acceleration necessary to obtain the observed decline slopes, the model with a constant k can give rise to steady states significantly lower than observed; and when constant cell loss rate acceleration is assumed, a new steady state is obtained only following significant oscillations in viremia, which are not observed in the data. The target cell half life due to intrinsic cell loss rate (d) before and during primary infection is on the order of 30-200 days. Average target cell infection rates (β) was 1.3·10-6±1.7·10-6 µl·virion-1·day-1. These values are on par with those estimated by Stafford et al. (2000).

36

2.3.2

Simulations

Viral load declines substantially after the peak all patients observed (see Figure 2.6), However the simple target-cell-limited model predicts solutions that are highly oscillatory during their approach to the steady-state (Burg, et al 2001, Stafford et al 2000).. Insufficient data exists in two cases (CMO and JSW) to verify or refute the model's predictions; however the oscillatory behavior is not typically seen in data (see Figure2.6 ). The viral load does not rebound as the basic model would predict following steep declines, and the it's predictions clearly do not agree with patient data in the region just beyond the local minima (Stafford et al, 2000). By Incorporating an immune response term the model can mimic patients' data between beyond 200 days, without oscillations. In comparison with Stafford et al. (2000) (see Figure 2.6), the simulations in Figure 3.5 are a better fit to the data with much simpler assumptions in the immune model.

37

2.3.3

Tables and Graphs

Table 2.3. HIV kinetic parameters. log[Vmax]

day at Vmax

log[ V ]

Max down slope r1

log[copeis RNA·mL-1]

day

log[copies RNA·mL-1]

log/wk

1 2 3 4 5 6 7 8 9 10

5.5 7.1 6.7 5.6 6.8 7.0 6.6 6.7 6.2 5.6

40 10 12 10 7 4 3 7 8 9

5.0 4.5 5.2 4.0 5.0 4.6 3.3 3.8 4.7 4.1

0.01 0.22 0.09 0.07 1.00 0.08 0.03 0.17 0.33 0.05

Mean

7.71

11

5.21

±

0.53

10.6

0.44

0.01 0.22

38

Table 2.4: HIV model parameter approximations.

Parameter Vmax

Peak viral load transient

θ

Value or range

Units

Source

105-107

RNA·mL-1

From data

105-108

RNA·mL-1

From data

cells·mL-1·d-1

Preinfection steady state assumption

cV0 p

RNA·mL-1

Quasi-steady state assumption

cVmax T0

d-1

Computed

4

d-1

From literature

0.1

RNA·mL serum-1

Set

10-2

d-1

Fit

10-3

µl·V-1·d-1

Fit

0.2-1.0

d-1

Fit

s

Target cell generation constant

s = dT0

I0

Initial Infected cell concentration

I0 ≈

p

Virion production rate

p=

c V0 d

β k(I,t)

Virion clearance rate Viral RNA concentration at exposure Target cell loss rate constant Target cell Infection rate Infected cell loss rate function

39

Table 2.5: HIV fitted parameter values. Time Shift

β

days

10 µL·V ·day

1 2 3 4 5 6 7 8 9 10

7 14 10 10 7 5 21 10 14 0

Mean ±

8.6 5.0

ID

-3

Target cell half life

k

Infected cell half life

10 day

days

day-1

days

1.57 0.10 0.53 5.20 0.03 0.83 0.43 0.41 0.93 1.31

8.3 5.3 7.5 2.2 3.3 7.3 23.0 1.7 1.3 6.5

84 131 92 315 210 95 30 408 533 107

0.06 1.61 0.34 0.80 0.84 0.69 0.52 0.90 1.55 0.38

11.6 0.4 2.0 0.9 0.8 1.0 1.3 0.8 0.4 1.8

1.31 1.77

6.5 7.7

200 165

0.77 0.50

2.1 3.4

d -1

-1

-3

-1

40

BORI

1113 7

6

6

log copies RNA/mL

HIV RNA (log copies/ml)

7

5

4

4

3

3

2

2

1

1

0

0

-1

-1 0

60

120 1019 180 240 Days Post-Treatment

300

360 HIV RNA (log copies/ml)

log copies RNA/mL

6 5 4 3

240

300

0

60

360

0

60

360

120 180 240 Dr Days Post-Treatment

300

360

300

360

7 HIV RNA (log copies/ml)

7

300

3

1 180 SUMA Days

240

4

1 120

120 IMME 180 Days

5

2

60

60

6

2

0

0 7

7

log copies RNA/mL

ALT (IU/ml)

5

6 5 4 3 2 1

6 5 4 3 2 1

0

60

120

180 Days

240

300

360

120

180 Days

240

Figure 2.5: Individual numerical simulations of 6 out of 10 patients' data after fitting said data to free parameters (see Section 2.2.2). Notice the difference in fit quality between this figure and Figure 2.6 Patient kinetics (black dots) and simulation results (grey curves). 6 of 10 representative

41

A

B

Figure 2.6: Excerpt from Stafford et al. 2000. The representative fits are circumspect, especially for patient B. Also notice the secondary peak viremia which is not confirmed by the data in both patients.

42

2.4 Summary Although complex processes occur during the rise and fall in viral load following initial HIV infection, I have shown that temporal changes in virus concentration observed early after infection are not consistent with the assumptions embodied in the simple target-cell limited model (Equations 1.1-1.3). Therefore, the modified immune control model was implemented. Relatively small variations in parameters determined by the host (d) and parameters determined by interaction between host and virus (k, θ) can be used to account for the diverse viral concentration transients observed during following primary infection. Mean values of 0.8±0.5day-1 for k over our ten patients. Although these values are slightly higher than the estimates of 0.39 day-1 (Stafford et al. 2000), 0.47day-1 (Wu et al. 1999), 0.5day-1 (Perelson et al 1996) and 0.38 day-1 (Little et al. 1999) obtained for the first phase decay rate obtained by drug perturbation studies, they are nonetheless of the same magnitude and within the standard deviations from these studies. These slightly elevated values are reasonable since they are the maximum immune control rate attainable and are lowered as infected cells concentrations fluctuate. Target cell infection rates are comparable to those estimated elsewhere (Stafford et al 2000, Little et al 1999), but are not unique. As the slope of initial viral increase to peak is mostly dependent this parameter any modification of the slope inclination will adjust the target cell infection rate constant value. Other studies go on to evaluate other parameter values, however, the assumptions put forth here as to parameter inter-dependencies are sufficient to accurately simulate the data presented. On the other hand, using these techniques I am unable to validate them with clinical data and experimentation. Also, the dependencies between the parameters imply that my estimates may not be unique.

43

The results do not support the hypothesis that the initial decline in viral load is due solely to target-cell limitation, as suggested by Phillips (1996); rather an immune response is at play to control replication in vivo. In order to support the hypothesis of immune system control during primary infection there is evidence of a temporal correlation between an increase in CTL precursor frequency and decline in viral load (Koup et al., 1994). Further, anti-human CD8+ monoclonal antibody was used to deplete CD8+ T cells in macaques inoculated with simian immunodeficiency virus (SIV) (Matano et al., 1998). Measurements of SIV concentration showed marked increases in peak viremia as compared to controls, suggesting a cytotoxic control of the viral load. These studies may provide direct evidence that CTLs are decreasing the HIV-1 viral load from its early peak level, as predicted by the immune control model. It may also be that cytotoxic cells are generated as viral load increases thereby decreasing the viral stimulus, and as the viral load declines the antigenic stimulus diminishes and immune control also declines leading to an establishment of chronic viral infection also predicted by the model. It has been shown in vitro that the late HIV-1 protein nef causes down regulation of MHC class I molecules on HIV-infected cells making. These cells are thought to be poor targets for cytolytic killing (Collins et al., 1998) which has the effect of causing them to be invisible to the immune control mechanism. This may effectively cause the immune control from reaching it maximum potential and contributing to the longer infected cell half life values exhibited during chronic infection. I show here that the frequently used target-cell limited model cannot mimic the highly diverse temporal changes in viral load of ten patients during primary HIV-1 infection. These results are consistent with, and provide evidence for, immune control being the principle mechanism of the initial decline in viral load following the initial viremia peak while target cell limitation may be secondary. The immune control model accurately

44

predicts viral load in all patients well beyond the initial transient, suggesting that one or more unmodeled processes in the basic viral dynamic model lower the viral load are present. This model exploits a simple control processes which increases the rate of infected cell loss as a function of viral stimulus. Although estimated parameter values are similar between my work and that of Stafford et al. (2000), the model proposed here has improved dynamic properties and can recreate primary HIV infection kinetics with enhanced success.

45

Chapter 3

Hepatitis B Viral Chronic Infection Dynamics

46

3.1 Introduction Hepatitis B is a viral disease with a high incidence and prevalence worldwide. Whereas 80% of infected individuals clear the virus, there are an estimated 350 million HBV carriers in the world, representing nearly 6% of the global population, of whom approximately one million die annually from HBV-related liver disease. Hepadnaviridae (Hepa = liver; dna = deoxyribonucleic acid) includes one virus that is pathogenic to man: Hepatitis B virus (HBV). As their names imply, all of the known hepadnaviruses are hepatotropic, infecting liver cells, and all can cause hepatitis in their known host. Hepatitis is characterized by inflammation of the liver. Although 90% of patients clear HBV infection in an acute response, the remaining 10% will develop chronic HBV (cHBV) which may lead to fulminant hepatitis (1.4%) cirrhosis (2%), Primary Hepatocellular Carcinoma (HCC or PHC, 3.2%) and death in approximately 2% of infected individuals (see Figure 3.1).

3.1.1

Virion Structure

The hepatitis B virion, also known as the Dane particle, is the one infectious particle found within the body of an infected patient. This virion has a diameter of 42nm and its outer envelope contains a high quantity of hepatitis b surface proteins. The envelope surrounds the inner nucleocapsid which is made up of 180 hepatitis B core proteins arranged in an icosahedral arrangement. The nucleocapsid also contains at least one hepatitis B polymerase protein (P) along with the HBV genome. In infected people, virions actually compose a small minority of HBV-derived particles. Large numbers of smaller subviral particles are also present, that unusually outnumber the virions by a ratio of 100:1. These two other subviral particles, the hepatitis B filament and the hepatitis B sphere, are often referred to as a group named surface antigen (HBsAg) particles. They are both 22nm in diameter and are totally composed of hepatitis B surface proteins. The 47

sphere contains both middle and small hepatitis surface proteins whereas the filament also includes large hepatitis B surface protein. The absence of the hepatitis B core, polymerase, and genome causes these particles to have a non-infectious nature. High levels of these non-infectious particles can be found during the acute phase of the infection. Since the non-infectious particles present the same epitopes as the virion, they induce a significant immune response and are thought to be non-advantageous for the virus. However, it is also believed that the presence of high levels of non-infectious particles may allow the infectious viral particles to travel undetected by antibodies through the blood stream (Ganem and Schneider, 2001).

3.1.2

HBV Genome

The HBV virion genome is circular and approximately 3.2 kb in size and consists of DNA that is mostly double stranded. It has compact organization, with four overlapping reading frames running in one direction and no noncoding regions. The minus strand is unit length and has a protein covalently attached to the 5' end. The other strand, the plus strand, is variable in length, but has less than unit length, and has an RNA oligonulceotide at its 5' end. Thus, neither DNA strand is closed and the circularity is maintained by cohesive ends. The four overlapping open reading frames (ORFs) in the genome are responsible for the transcription and expression of seven different hepatitis B proteins. The transcription and translation of these proteins is through the used of multiple in-frame start codons. The HBV genome also contains parts that regulate transcription, determine the site of polyadenylation and a specific transcript for encapsidation into the nucleocapsid. The genomic arrangement of the hepatitis B virus family makes it unique among viruses.

48

3.1.3

HBV Life Cycle

In order to reproduce, the hepatitis B virus must first attach onto a cell which is capable of supporting its replication. Although hepatocytes are known to be the most effective cell type for replicating HBV, other types of cells in the human body have be found to be able to support replication to a lesser degree. The initial steps following HBV entry are not clearly defined although it is known that the virion initially attaches to a susceptible hepatocyte through recognition of cell surface receptor that has yet to be identified. The DNA then enters into the nucleus, where it is known to form a covalently closed circular form called cccDNA. The (-) strand of such cccDNA is the template for transcription by cellular RNA polymerase II of a longer-than-genome-length RNA called the pregenome and shorter, subgenomic transcripts, all of which serve as mRNAs. The shorter viral mRNAs are translated by ribosomes attached to the cell's endoplasmic reticulum and the proteins that are destined to become HBV surface antigens in the viral envelope are assembled. The pregenome RNA is translated to produce a polymerase protein, P, which then binds to a specific site at the 3' end of its own transcript, where viral DNA synthesis eventually occurs. Occurring at the same time as capsid formation, the RNA-P protein complex is packaged and reverse transcription begins. Soon after infection, the DNA is re-circulated to the nucleus, where the process is repeated, resulting in the accumulation of 10 to 30 molecules of cccDNA and an increase in viral mRNA concentrations

49

Figure 3.1: Outcomes of HBV infection and their incidence in the population. (Based on http://www.gsk.com.my/gsk_pharmaceutical_hepatitis.asp?cat=07)

Figure 3.2 Hepatitis B virus schematic diagram. (Based on numerous sources)

Figure 3.2 Hepatitis B virus schematic diagram (right) and micrograph (left). (http://web.uct.ac.za/depts/mmi/stannard/hepb.html)

50

Figure 3.3: Hepatitis B viral genome. (Kidd-Ljunggren K, et al. 2002)

Figure 3.4: Hepatitis B viral life cycle. (Ganem and Prince, 2004)

51

3.1.4

Chronic HBV Characteristics

Most people with chronic hepatitis B are asymptotic carriers and do not usually develop severe liver problems. However, some people have "flare ups" during which they exhibit the symptoms of acute hepatitis, including flares of aminotransferases. In some patients, these flares are followed by HBeAg/anti HBe seroconversion (Liaw et al., 1983). Furthermore, spontaneous HBeAg seroconversion occurs in 50-70% of patients with elevated aminotransferases. HBeAg to anti-HBe seroconversion can be observed at annual rates of 2.7%-25%, with female gender being predicative of HBeAg seroconversion. (Violo et al., 1981 and Hoofnagle et al., 1981). Transition from replicative to non-replicative infection may be rapid, smooth and clinically silent or prolonged, fluctuating and marked by recurrent exacerbations (Lok, 1992). Transition of viral DNA to RNA during the life cycle of hepadnaviruses has similarities to that of retroviruses (Ganem and Schneider, 2001). Integration of viral DNA into the host genome is not, however, essential for replication of hepadnaviruses as is the case with retroviruses. The integration of HBV DNA sequences into host cell genomic DNA may have an important role in hepatocarcinogenesis.

3.1.5

Chronic HBV Kinetics

During chronic HCV and HIV viral infection, replication kinetics is at a steady state. This means that virus production by infected cells is compensated by peripheral virion degradation, and de novo infection of non-infected cells is compensated by the infected cell loss. Fluctuations in steady state viremia are relatively insignificant in untreated patients with chronic HCV infection (Nguyen et al. 1996) and chronic HIV infection (Jurriaans and Goudsmit et al., 1996). In contrast to these, chronic HBV viral infection is highly dynamic. In this study, I show that in a placebo-controlled phase III trial, many patients showed marked declines in viral load and ALT flares over 52 weeks. I analyze 52

these changes in viral and ALT kinetics and explain their biological relevance in clinical settings.

53

3.2 Methods Data presented here were received in collaboration with Gilead Pharmaceuticals.

3.2.1

Inclusion Criteria

170 patients, from 78 centers, enrolled in the 437 phase III Adefovir clinical trial to receive placebo. They were both female (29%, pregnancy-negative and were using effective contraception) and male (71%); ages 16 to 65 years of age (median: 40 years). They all were HBV treatment naïve for at least 12 weeks, and seronegative for HIV, HCV or HDV. Furthermore, they were HBsAg-positive, had HBV-DNA titers in excess of 106 copies/mL (measured with the Roche Amplicor Monitor polymerase-chain-reaction [PCR] assay), aminotransferase levels higher than 1.2 to 10 times the upper limit of the normal range for six month prior to trial initiation; and had an adequate blood count. Exclusion criteria included coexisting serious medical or psychiatric illness; immune globulin, interferon, or other immune- or cytokine-based therapies with possible activity against HBV disease within 6 months before screening; organ or bone marrow transplantation; recent treatment with systemic corticosteroids, immunosuppressants, or chemotherapeutic agents; liver disease that was not due to hepatitis B. Of these 170 patients, 6 dropped out after 6 months. Asians constituted 61% of the cohort, 36% were Caucasians and 3% were Black or of different ethnicity. Patients with high or low BMI (>23kg/m2 or >25kg/m2, Asians or others respectively) were distributed evenly among the cohort.

3.2.2

Pattern Recognition Algorithm

To find changes in kinetics of the data I formalized a pattern recognition algorithm which was used to detect both spontaneous rapid declines in serum viral load (RVD), as well as flares in ALT value, over three consecutive datapoints (see Figure 3.5). Two helping 54

variables were used. LMAX and COUNTER representing local maximum and the number of consecutive datapoints in each phase, respectively. The formalized algorithm is characterized by the following: STEP 1: For each ID I initialize LMAX with the first VL and COUNTER with 1. STEP 2: If (VL-LMAX)<θ2 decrease COUNTER by 1, otherwise, increase COUNTER by 1. Once COUNTER decreases from a positive value, it receives the value (-1). Once COUNTER increases from a negative value, it receives the value (+2) and LMAX receives VL. If COUNTER equals θ1, this patient is RVD and STOP. STEP 3: If not updated at STEP 2, LMAX receives VL if the current VL>LMAX. STEP 4: Proceed to the next VL point and go to STEP 2.

Figure 3.5: Pattern recognition algorithm flowchart. 55

Patients with fluctuations in viral load less than <0.5 log were defined as FLT. Patients with at least one spontaneous decline in viral load of more than 1 log over three datapoints were defined as "RVD". Initially, I was faced with three categories of viral kinetics: FLT (fluctuations in VL<0.5 log), SPD fluctuations in VL>1.0 log) and sSPD (VL fluctuations between 0.5-1.0 log). In order to separate the sSPD patients into FLT and SPD I integrated the ALT flare analysis to determine the distribution of ALT flares which coincide with VL declines in SPD and FLT patients. The viral and ALT kinetic patterns showed a temporal correlation between viral decline and ALT flare in SPD patients but not in FLT patients (see Results). Therefore, sSPD patients with a coinciding ALT flare were categorized as SPD. Alternately sSPD patients without a coinciding ALT flare were categorized as FLT. Patients which had a slow viral load decline over most of the 12-month followup period were defined as DWN. HBeAg loss was compared with serum viral concentration. I compared HBeAg loss with suppression of viral load, and as a function of RVD onset.

3.2.3

DNA Measurements

Blood samples were collected from patients every 4 weeks. Blood serum DNA content was analyzed by Roche Amplicor Monitor polymerase-chain-reaction (PCR) assay, an in vitro nucleic acid amplification test for the quantification of HBV DNA.

3.2.4

Model

I tested several models with changes in all possible parameters, although I only report here the final model and results for parameter changes that yielded a good fit with the empiric data. To analyze rapid viral declines and ALT flares during chronic HBV infection, I developed a new mathematical model, generalizing our immune control model for HBV 56

dynamics under treatment, by including additional terms for liver cell proliferation, increased death of infected cells at certain timepoints, and an equation to describe the ALT kinetics. The differential equations describing the model are: dT T   = s 1 −  − dT − β VT dt  T max 

(3.1

dI kV   = β VT −  d + I dt V + θ  

(3.2

dV = pI − cV dt

(3.3

k0 (t ) = 0  − γ ( t − t1 ) k1 (t ) = κ 1 − e

(

)

, t1 ≤ t ≤ t2 , t1 > t > t2

dA = φdT + ϕδI − σA dt

κ1 »κ 0

(3.4 (3.5

This model is derived from the model put forth in Chapter 1. I incorporated into the immune control model a description of ALT dynamics known to play an important role in viral Hepatitis (Equation 3.5) that expresses the ALT dynamics parameters of φ and ϕ which are the target and infected cell ALT release constants, respectively; and σ is the ALT clearance rate constant. s(T) in Equation (3.1) is the logistic proliferation rate function of hepatocytes. Equation (3.4) is an immune control perturbation term.

3.2.5

Nonlinear Data Fitting

Each individual dataset used included viral load and ALT data. log[V(t)] was fit to the measured log viral load (copies HBV-DNA/mL) while log[A(t)/A(t0)] was fit simultaneously to the measured log ALT enzymatic values (IU/L). Fitting was done by a nonlinear least squares regression method using Berkeley Madonna software (Oster and Macey, 2000). I estimated the “free” parameters as described in Results.

3.2.6

Statistical Analysis

The kinetic pattern groups were broken up according different baseline parameters: VL and ALT, ethnicity, age, BMI and gender. For the baseline analysis, “DWN” patients 57

were considered as “SPD”. I employed Fisher's Exact Test for statistical analysis of the groups. Multivariate analysis was not possible due to the small numbers of patients in each group generated by the analysis. Statistical analyses were performed using SPSS (SPSS Inc., Chicago, IL). Fisher exact test (2x2 tables) was used to determine the statistical significance of the distribution of categorical variables between groups. Mean results are given with standard deviations. Multivariate analysis was not possible due to the small numbers of patients in each group.

58

3.3 Results

3.3.1

Individual Viral Kinetics Patterns

The mean HBV DNA of all patients at each visit is almost constant over time; starting at 8.1 log copies/mL and ending at 7.1 after 48 weeks of Placebo (see Figure 3.6). Thus, seemingly an HBV steady-state exists for HBeAg-positive chronic infection. However, the large standard deviations that are observed at each time point, along with the 1 log drop in HBV-DNA over time, indicate that the mean description may be faulty. Moreover, when verifying the viral load per patient, one can observe that in a large fraction of patients the HBV-DNA level spans a large magnitude between minimum and maximum during the whole period rather than being in steady state (Figure 3.6). The patients' maximum viral load values are relatively the same in all patients, but their minimum values are highly variable among patients. Indeed, when looking at the individual patterns of HBV DNA kinetics I observe that while a number of patients have a steady state in viral load (e.g. Figures 3.9A and 3.9B), many other patients show large and rapid viral declines of a significant magnitude up to 4.7 logs (e.g. Figures 3.9C-3.9G). In order to analyze this phenomenon more formally, I devised a pattern recognition algorithm that identifies rapid viral declines (RVD); namely a rapid drop in HBV-DNA level from one visit to the next that is sustained for at least 2 measurements (about 8 weeks). I found that 45% of patients have at least one RVD with a magnitude higher than 1 log copies/mL during the 52 weeks of study, 32% have RVD with a magnitude between 0.5 and 1 log and 23% have an apparent steady state with RVD less than 0.5 log (see Table 3.1). The population mean of the minimum-maximum viral loads over time per patient was 6.0-8.5 log respectively for RVD and 7.6-8.7 for FLT patients, in concurrence with our original observation.

59

3.3.2

No “Placebo Effect”

A placebo effect is ruled out since I observe RVDs starting at any time during the study. The fraction of patients starting RVD during the first month of placebo (6%) is not larger than the fraction of patients starting RVD during the 6 coming months (7%-17%). Moreover, 17% of patients start an RVD already before placebo initiation. Most patients with an RVD exhibit a rebound in viral load after a mean duration of 19±12 weeks (range 8-52). A fraction of patients (35%) do not show a rebound for a mean period of 21±13 weeks, possibly because their RVD started towards the end of the study. Other 18% of patients, with short RVD, exhibit multiple (2-4) RVDs and rebounds during the study.

3.3.3

Definition of Spontaneous Declines

In order to evaluate the biological relevance of these RVDs, I implemented the pattern recognition algorithm to identify ALT elevations; namely ALT levels that are 1.5-fold higher then the 2 previous ALT measurements. 43% of patients had at least one ALT elevation (see Table 3.1 and Figure 3.2). Significantly (P<0.001) more patients with RVD larger than 1.0 log had ALT elevation (80%) as compared to patients with no RVD (19%). An ALT elevation is found in 56% of the patients with an intermediate magnitude of RVD (0.5-1.0 log). Moreover, ALT elevations and RVDs are correlated in time (P<0.001), with most ALT elevations occurring during a time frame of 4 weeks before or after the initiation of RVD (see Figure 3.9D-3.9F). I assume that a biological process giving rise to a spontaneous decline (SPD) in viral load is initiated in a large fraction of patients without any treatment. The markers of this process are primarily a rapid viral decline and secondly an ALT elevation. I consider patients with RVD larger than 1.0 log (clearly larger than measurement error) and spanning at least 2 time points (usually 8 weeks, thus excluding miscellaneous errors) as having an SPD. On the other hand, patients with no apparent RVD (less than 0.5 log) are 60

not considered here as having an SPD even if they have an ALT elevation. Among the patients with an intermediate magnitude of RVD (0.5-1.0 log), I used the ALT elevation as a marker for SPD (see gray zones and double lines in Table 3.1). At least one significant spontaneous decline (SPD) in HBV-DNA is found in 63% of the HBeAgpositive chronic HBV-infected patients studied here over 48 weeks of placebo (see Table 3.2). The mean magnitude of RVD is 1.4±1.0 log copies/mL (range: 0.5-4.7) bringing HBV-DNA down to an average of 6.7±1.2 log copies/mL (range 4.0-9.2) as the minimum of the RVD. A large, but slow, decline (DWN) in HBV-DNA over 48 weeks is nevertheless observed (see figure 3.9H) in 17% of patients without an SPD (6% of all). Only 31% of the patients have a flat kinetic pattern (FLT) in HBV-DNA (see Figure 3.9A-3.9B). At least one HBeAg-negative visit during the study was observed in 20% of the patients (see Figure 3.2). There is a significant (P<0.002) difference in the fraction of patients with HBeAg-loss between SPD (26%) and FLT (4%) patients (Table 3.2). Note that 30% of DWN patients show an HBeAg-loss during the study. HBeAg becomes negative when HBV-DNA falls below 5.4±0.9 (range: 3.6-7.8) log copies/mL, and becomes HBeAgpositive again with the increase in HBV-DNA. Also the ALT levels follow in general the decline and rise concomitant with HBV-DNA during SPDs, with the exception of the ALT elevation that occurs at the beginning of the SPD.

3.3.4

Kinetic profiles

Individual kinetic profiles showed 6 distinct patterns using three serum markers: HBV serum DNA concentration, serum ALT levels and HBeAg seroconversion. While the chronic flat pattern is characterized by small changes in the viral load and no HBeAg or seroconversion (3.10A, 27%), some patients had elevations in their baseline ALT values (3.10B, 4% of all). Another group had one or more viral load declines of more than 1061

fold spanning 3 datapoints or more. Some of these patients had no change in ALT serum values (3.10C, 14%, while others had ALT elevations of more than 150% from baseline values (3.10E, 25%), were usually much higher than the upper normal limit. Some patients with RVDs rebound to pre-RVD viral load values. Another group had multiple RVDs, rebounds and ALT elevations (3.10D, 10%) and their viral load kinetics appeared to have oscillatory properties. Another group had two or more consecutive RVDs of more than 1 log magnitude, with a steady-state phase between them. They always had only one ALT elevation, coinciding with their first RVD (3.10G, 2%). The last group of patients had a slow and continuous decline in viral load, with no ALT elevations (3.10F, 6%). HBeAg seroconversion seems to be a threshold function of the serum viral load; when it decreases beneath a certain patient-based value.

3.3.5

Baseline correlations

The occurrence of SPD was correlated with several baseline factors (see Figure 3.9). 89% of patients with high baseline ALT (>180 IU/mL) have SPDs, while only 57% (P<0.001) of patients with lower ALT concentrations have SPD. Moreover, all high ALT patients with baseline viremia lower than 108 copies/mL had SPD. Low baseline viral load was also associated with more SPDs (58%) among patients with low baseline ALT, in comparison to patients with high viral load (73%). SPDs are significantly (P<0.002) more common (78%) among Asians in comparison to Caucasians (53%), independently of HBV genotype. There are small differences in occurrence of SPDs among the HBV genotypes, but those are not significant. Moreover, note that among the few Asian patients infected with non-Asian HBV genotypes (A and D), as well as the few Caucasian patients infected with Asian HBV genotypes (B and C), the frequency of SPDs is more influenced by the race rather then by the genotype. Other baseline factors that were correlated with higher frequency of SPDs were: females (88%) versus males (60%, 62

P<0.0005), younger than 40 years (74%) versus older (57%, P<0.022), and lower BMI (72%) versus higher BMI (55%, P<0.03).

3.3.6

Modeling of SPDs

In order to investigate the mechanism behind the occurrence of SPDs I use a viral dynamics model (Equations 4.1-4.5) based on Neumann et al (1998) and verify which parameter changes could give rise to a SPD-like phenomenon. Since the half-life of viral decline during SPDs is of the order of 2-18 days I rule out acceleration of free virion clearance and increased blocking of virion production, as both would necessarily give faster declines. The lack of faster decline is not due to infrequent sampling, since in many patients the same viral decline rate is observed over three data-points (8 weeks) or more (see Figure 3.11). The range of viral decline half-lives, and its large inter-patient variability, is consistent with the loss rate of infected cells in HBV-DNA (Hadziyannis et al., 1990). I, therefore, propose that SPDs are the result of an acceleration in infected cell loss rate (from κ=0 to κ >0 in Equation 4.5). However, given the rate acceleration necessary to obtain the observed decline slopes, the model with a constant κ can only give rise to steady states significantly lower than observed. Moreover, when a constant cell loss rate acceleration is assumed, a new steady state is obtained only following significant oscillations in viremia, which are not observed in the data. Therefore, I assume that the increased loss rate of infected cells is also dependent on the amount of infected cells (see saturation function in Equation 3.4), such that when the number of infected cells declines their loss rate also decreases. In addition, I assume that the loss rate of infected cells is also a factor giving rise to increased ALT levels (see Equation 3.4). I use this model (Equations 3.1-3.5) to qualitatively fit the viral load and ALT data from all 79 patients with RVDs exceeding 1 log copies HBV-DNA/month to the model. Figure 3.11 shows the theoretical curves for eight representative patients after 63

fits increased infected cell loss from d to κ1. The cell half life due to intrinsic cell loss rate (d), before and during SPDs, is order of 68.8±52.2 days (range: 8-180 days), while the maximum half life values during SPDs (due to the accelerated loss rate κ1) are of 2.5±1.6 days (range: 0.6-5 days). Moreover, also the rebounds after SPD were simulated in 17 patients by returning κ1 to d values. Other model parameter estimates obtained were within excepted value ranges but are not reported since the sampling in this study is too sparse to obtain reliable estimates.

64

Mean HBV DNA (log copies/ml

Tables and Graphs

10 9 8 7 6 5 4 3 2 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Months

Figure 3.6: Mean viral load per month of study with placebo.

HBV DNA (log copies/ml)

10 9 8 7 6 5 4 3 2 0

50 100 Patient (sorted by SD)

150

Figure 3.7: Patient viral load maximum and minimum values. SPD Patients (Cummalative %)

3.3.7

20%

15%

10%

5%

0% Pre

1

2

3

4 5 Months

6

7

8

Figure 3.8: Percentage of patients to have SPD per month.

65

9

Table 3.1: Prevalence of RVD and concomitant ALT elevations.

Rapid Viral Decline (RVD) Magnitude

N (% of total)

With ALT elevation (% of RVD)

No ALT elevation (% of RVD)

RVD >1 log

74 (45%)

59 (80%)

15(20%)

0.5< RVD <1 log

53 (32%)

30 (56%)

23 (44%)

RVD <0.5 log

37 (23%)

7 (19%)

30 (81%)

Table 3.2: Prevalence of kinetic patterns and HBeAg.

SPD

N (% of total) 104 (63%)

HBeAg Negative (% of pattern) 27 (26%)

FLT

50 (31%)

2 (4%)

DWN

10 (6%)

3 (30%)

Kinetic pattern

100%

94

100 55%

42

100 112%

82

100 82%

93

100% 61

80%

80%

80%

80%

80%

60%

60%

60%

60%

60%

40%

40%

40%

40%

40%

20%

20%

20%

20%

20%

0%

0% Asian Cauc. Ethnicity

0% Female Male Gender

0% High Low BMI

0% <40yrs >40yrs Age

129

35

SPD FLAT

<180 >180 baseline ALT IU/mL

Figure 3.9: Baseline parameters correlation with SPD. Y-axis is percentage of patients with SPD or FLT kinetic profiles. Values in each column designate the number of patients in each group.

66

Table 3.3: HBV model parameter approximations.

Parameter

Value or range

Units

Source

105-109

DNA·mL-1

From data.

10-400

IU·mL-1

From data.

Cells·d-1

Steady state assumption

µL·V-1·d-1

Steady state assumption

ALT·d-1·mL-1

Steady state assumption

cells·mL-1

Quasi-steady state assumption

d

d-1

Non-cytopathacy assumption

3

d-1

From literature

107.3

cells· mL serum-1

Hepatocyte number/intercellular volume. From literature

105

DNA·d-1·ml-1

Fit

Infected cell ALT generation constant ALT clearance rate constant

10-3

ALT·d-1·mL-1

Fit

0.1

ALT·d-1·mL-1

Fit

p

Virion production rate

10-100

d-1

Fit

d

Target cell loss

2·103

d-1

Fit

κ1

RVD infected cell loss

λ *d

d-1

Fit

V0 A0

Steady state viral load concentration Steady state ALT concentration

s

Target cell generation

β

Target cell infection

φ

Target cell ALT generation

I0

Initial Infected cell concentration

k0 c

T0

Pre-RVD infected cell loss rate constant Virion clearance rate Initial Target cell concentration

θ ψ σ

s=

d + βV0 1 − T0 / Tmax 2

β =d+ φ=

σA0 dT0

k 0V0 T0 (V0 + θ )

− ϕd +

I0 ≈

cV0 p

67

k 0V 0 V0 + θ

3036

A

800

9 8

6

400

5 4

200

3

600

eAg+

6

eAg-

5

ALT Data

400

4

200

3

0 60

120

10

180 240 300 1001 Days Post-treatment

360

2

420

0 0 D

60

120

180 1047

9 8

240

300

360

420

Days

10 800

800

9

HBV DNA (log copies/ml). ALT (IU/ml)

6

400

5 4

200

3

VL Data

7

600

eAg+

6

eAg-

5

ALT Data

400

4

ALT (IU/ml)

8

600

7

200

3

2

0

E0

60

120

180

10

240 1023 Days

300

360

2

420

0 0 F

60

120

180 1138

8

360

420 800

9

HBV DNA (log copies/ml). ALT (IU/ml)

9

300

Days

10 800

240

6

400

5 4

200

VL Data

7

600

eAg+

6

eAg-

5

ALT Data

400

4

ALT (IU/ml)

8

600

7

200

3

0

2

FG

60

120

180 HBV DNA (log copies/ml).

10 9

240 3013 Days

300

360

0

H

3036 800

120

0

180 4006

60

120

600 240

180

9

7

300

6 Days Post-treatment 400 5

6 5 4

200

240

300

360

420

Days

8

2

7

60

10

7

8

0

2

420

HBV DNA (log copies/ml). ALT (IU/ml)

0

VL Data 360 420 eAg+

800 600 400 200 0

800

600

eAg-

400

ALT Data

4

200

3

3 2

0 0

60

120

180

240

300

360

420 420

2

0 0

60

120

180

240

300

Days

Days VL Data

eAg+

eAg-

ALT Data

Figure 3.10: Representative individual chronic HBV viral load (black dots) and ALT (grey dots) data corresponding to different kinetic patterns. Notice the rapid drops in viral load in the absence of antiviral treatment.

68

360

420

ALT (IU/ml)

3

ALT (IU/ml)

HBV DNA (log copies/ml).

VL Data

7

ALT (IU/ml)

8

600

7

C0

HBV DNA (log copies/ml).

800

9

HBV DNA (log copies/ml). ALT (IU/ml)

HBV DNA (log copies/ml).

10

2

HBV DNA (log copies/ml).

6065

B

10

9

10 1000 9

8

800 8

7

7

600

6 5

400 5

4

4

600 400

60

120

180 A240 300 1138 Days Post-Treatment

360

2

D

420

8

800 8

60

120

180 A240 300 3020 Days Post-Treatment

360

420 1000

HBV DNA (log copies/ml) ALT (IU/ml)

9

10 1000 9

0 0

7

800

7

600

6 5

600

6

400 5

4

400

ALT (IU/ml)

0 0

4

200

200

3

2

0

E0

60

120

10

180 A240 300 4059 Days Post-Treatment

360

2

0

F 0

420

60

120

10

8

800

360

420 1000

HBV DNA (log copies/ml) ALT (IU/ml)

9

1000 9

180 A240 300 6013 Days Post-Treatment

8

800

7

7

600

6 5

600

6 5

400

400

ALT (IU/ml)

3

4

4

200 2

2

0 60

120 180A 240 7024 Days Post-Treatment

300

360 1000

-60

6

0

7 120 180 240 300 360 600 Days Post-Treatment 6 60

180 A240 300 7037 Days Post-Treatment

360

420 1000

HBV DNA (log copies/ml) ALT (IU/ml)

HBV DNA (log copies/ml)

7

120

9

1023 800 8

10 9 8 7 6 5 4 3 2

60

10

9 8

0

H0

1000 800 600 400 200 0

ALT (IU/ml)

G0 10

200

3

3

800 600

400 5

5 4

400

4

200

200

3

3 2

0 0

60

120

180 240 Days

300

360

420

2

0 0

60

120

VL Data

VL simulation

ALT Data

ALT simulation

180 240 Days

300

360

Figure 3.11: Viral load and ALT data (dots) and theoretical curves for 10 representative SPD patients. Vertical line denote the time k was increased. Second vertical lines in patients B,C and G are the times when k was returned to baseline value; and in patient H it is when k was increased further. Notice the good fit to both ALT and viral load data. 69

420

ALT (IU/ml)

C

200

3

10 HBV DNA (log copies/ml).

800

200 2

HBV DNA (log copies/ml).

1000

6

3

HBV DNA (log copies/ml).

A 1037

HBV DNA (log copies/ml). ALT (IU/ml)

HBV DNA (log copies/ml).

B

A 1023

ALT (IU/ml)

A 10

HBV DNA 2nd phase slope (d-1)

0

-0.05

-0.10

-0.15

-0.20

0.60

-0.40

-0.20

0

0.20

0.40

HBV DNA Pre-Rx decline slope (d-1) 5mg (N=2)

10mg (N=14)

30mg (N=15)

60mg (N=6)

Figure 3.12: Second phase Rx slope correlation with Pre-Rx slope. (Shmailov and Neumann, 2005)

70

0.60

3.4 Summary The advantage of individual kinetics analysis allowed for the observation of complex kinetics not seen heretofore. This permitted the observation of rapid declines during placebo treatment. These declines are unusual in comparison with other chronic infections in humans which have only minor fluctuations in viral load. Further study pointed to the possibility of profiling the patients according to their kinetic data. It could be argued a ‘placebo effect’ may be accountable for the aforementioned declines. I prove this argument to be false. The majority of patients are SPD which would preclude the marginal impact expected from a ‘placebo effect’. Also the RVDs would necessarily begin at the onset of placebo treatment; however, I show that the fraction of patients that experienced a decline in viral load is relatively constant during the course of followup. In fact, the highest fraction of patients who experienced an RVD before placebo is initiated. The pattern recognition algorithm developed formalized definitions of the declines and allowed the separation of the patients into two groups: RVD and FLT. This led to the finding that almost 70% of patients had at least one RVD. Therefore, this phenomenon is not aberrational but is an important process during chronic HBV infection. Baseline parameters such as ethnicity, gender, age, BMI and pre-placebo ALT levels are correlated with the occurrence of SPD. ALT is known as a marker for infected cell apoptosis or necrosis, are processes linked to immune function against those cells. It is widely accepted that immune system activity is higher in young people; and females in addition have immune enhancing progesterone and estrogen. Some scientists have considered that Asians as well as people with low BMI also share a similar heightened immune activity. A biological process and/or processes that occur during an SPD are characterized by three markers. First, a rapid viral decline of a magnitude higher than 1 log drop; second, an 71

ALT flare coinciding in time with the RVD; and third, the correlation between HBeAg seroconversion and a drop in viral load. These observations show that there are ongoing underlying processes driving the SPDs during chronic HBV infection; the well documented HBeAg-to-anti-HBe switch, an increase in endogenous IFN-α production known to increase spontaneous antiviral reduction or development of mutant viral forms evoking a more intense immune response by virtue of the antigenic difference. The observation that RVDs have slopes with magnitudes similar to infected cell loss rate derived by previous studies along with the successful implementation of the immune control model and may indicate an increase in specific anti-HBV activity by the immune system may instigate the occurrence of SPDs. To further test the validity of this hypothesis, application of nonlinear regression in the fitting the equations of the immune control model (see Chapter 1 and Section 3.2.4) to the clinical HBV chronic infection data was highly successful. Taken together, these results imply that spontaneous and sporadic changes in viral load, ALT measurements and HBeAg seroconversion are the consequence of the alteration of the immune/viral balance established during chronic viral infection.

72

Chapter 4

Woodchuck Hepatitis Viral Chronic Infection during Therapy

73

4.1 Introduction The woodchuck hepatitis virus (WHV) was the first of the mammalian and avian hepadnaviruses described after discovery of the hepatitis B virus (HBV). Woodchucks chronically infected with WHV develop progressively severe hepatitis and hepatocellular carcinoma, which present as lesions that are remarkably similar to those associated with HBV infection in humans. It is possible to infect neonatal woodchucks born in the laboratory with standardized inocula and produce a high rate of chronic WHV carriers that are useful for controlled investigations. WHV has been shown experimentally to cause hepatocellular carcinoma, supporting conclusions based on epidemiological and molecular virologic studies that HBV is an important etiological factor in human hepatocarcinogenesis. Chronic WHV carrier woodchucks have become a valuable animal model for the preclinical evaluation of antiviral therapy for HBV infection, providing useful pharmacokinetic and pharmacodynamic results, in a relevant animal disease model. It also has been shown that the pattern of toxicity and hepatic injury observed in woodchucks treated with certain fluorinated pyrimidines is remarkably similar to that observed in humans that were treated with the same drugs, suggesting the woodchuck has significant potential for the pre-clinical assessment of antiviral drug toxicity.

4.1.1

Woodchuck Hepatitis Virus (WHV)

WHV is classified as a member of the genus Orthohepadnavirus, family Hepadnaviridae. The genetic organization of WHV is similar to HBV and to other mammalian hepadnaviruses. Numerous 22-nm filaments and spherical particles are found in the serum of infected woodchucks and are composed of the envelope protein of the virus. Complete virions are 42 to 45nm in diameter and are composed of the exterior envelope (WHsAg), the nucleocapsid or core protein (WHcAg), and, within the nucleocapsid, the DNA

74

genome. The replicative cycle of WHV appears to be identical to that of HBV (Ganem and Schneider, 2001).

4.1.2

Clevudine (L-FMAU)

The nucleoside analogue L-FMAU [1-(2-fluoro-5-methylb,L-arabinofuranosyl) uracil] also known as Clevudine has been shown to have significant antiviral activity against hepatitis B virus (HBV) in humans (Marcellin et al., 2004) and in cell culture replication (Ma et al., 1997). Cell-culture studies have demonstrated that HBV variants that have been associated with resistance to current treatments most notably Lamivudine therapy in chronically infected patients are sensitive to Clevudine (Brunelle et al., 2005). L-FMAU has markedly less toxicity than its D-enantiomer, D-FMAU (Kleiner et al., 1997), and the related nucleoside analogue, D-FIAU (fialuridine) (Fourel et al., 1990). Fialuridine demonstrated potent anti-HBV activity in clinical trials for chronic HBV infection, but also induced severe delayed toxicity associated with lactic acidosis and hepatic failure that resulted in the death of several patients (McKenzie et al., 1995). D-FMAU, D-FIAU, and D-FEAU have been shown to be potent inhibitors of woodchuck hepatitis virus (WHV) replication in chronically infected woodchucks, but all of these nucleoside analogues also demonstrated severe toxicity at daily doses greater than 2 mg/kg body weight (Tennant et al., 1998). L-FMAU has been shown to have a favorable pharmacokinetic profile and sufficient oral bioavailability in woodchucks that makes it suitable for once-daily administration in antiviral perturbation experiments.

4.1.3

The Relevance of the Amimal Model

WHV and its natural host, the Eastern woodchuck (M. monax), constitute a useful model of HBV-induced disease, including hepatocellular carcinoma (HCC) (Kroba et al., 1989). A variety of published studies from several laboratories have reported the use of chronic 75

WHV infection in woodchucks to investigate potential antiviral therapies for chronic HBV infection (Korba et al., 2000). Most of the antiviral agents that have been used to treat chronic HBV infection: FTC, Famciclovir, Fialuridine, Ganciclovir, Lamivudine, Ribavirin, Thymosin, Vidarabine and AZT, along with combination therapy with either Lamivudine/Interferon-α or Lamivudine/Famciclovir have shown similar antiviral activities against WHV and similar toxicity profiles in chronically infected woodchucks. These studies demonstrate that the WHV/woodchuck model of chronic HBV infection can be considered to be a predictive model for new, potential therapeutic applications against HBV infection in humans, especially nucleosides.

76

4.2 Methods Data presented here were received in collaboration with Korba et al. (2005).

4.2.1

Animals

The woodchucks used in these studies were born to WHV-negative females in a breeding colony maintained at Cornell University. Animals were inoculated at 3 days of age with 5 million woodchuck infectious doses of a standardized WHV inoculum pool (WHV7p1) that characteristically produces a 65% to 70% chronic WHV carrier rate. The diet consisted of laboratory animal chow formulated for rabbits (Agway Red Rabbit Food, Syracuse, NY), specially pelleted in blocks for woodchucks. Both food and water were provided ad libitum. Six groups of 4 adult (approximately 2 years old) chronic carrier woodchucks were treated with Clevudine at doses of: 0.03, 1.0, 3.0, or 10 mg/kg body weight. An additional group of 8 age-matched chronic WHV carriers served as a placebo control.

4.2.2

Treatment Regime

L-FMAU was dissolved in isotonic saline solution and administered orally in a liquid diet once daily for 28 days. A placebo liquid diet was also administered daily to the control animals.

4.2.3

Sampling

Serum samples were taken for analysis on the first day of treatment before administration of the initial dose of drug (“day 0”), after 0.5, 1, 2, 3, 5, 7, 14, 21, and 28 days of treatment. Followup sampling was done at 1, 2, 3, 4, 6, 8, and 12 weeks. The general health of the woodchucks was assessed by daily observations at the time they received food and water, at the time of drug or placebo administration, and at the times they were 77

anesthetized. Any abnormalities in behavior, appearance, or intake of food or water were recorded. Body weights were determined when serum samples were taken.

4.2.4

DNA Measurements

Viremia in serum samples was assessed by different quantitative methods depending on the concentration of WHV DNA. Serum containing concentrations of WHV virion DNA above 107 WHV genome equivalents per milliliter of serum (WHVge/mL) were analyzed by dot-blot hybridization (four 10-mL replicates per sample). Samples containing WHV DNA below the dot-blot sensitivity cutoff and above 1,000 WHVge/mL were analyzed using a quantitative polymerase chain reaction (PCR)-based method (Korba et al., 2000). Serum that contained WHV-DNA levels below 1,000 WHVge/mL was processed in duplicate 100-mL aliquots. Nucleic acids from these serum samples were extracted by a proteinase K–sodium dodecyl sulfate (SDS)/phenol/chloroform procedure, followed by ethanol precipitation. WHV-DNA samples then were resuspended in 10 mL of nucleasefree water and subjected to PCR amplification and blot hybridization using standards representing 30 to 1,000 WHVge/ mL (Cote et al., 2000). The sensitivity of this detection procedure was approximately 30 WHVge/mL serum. Assessments of the level of WHV genome equivalents for the PCR analyses were determined by direct comparisons with parallel PCR amplifications of a dilution series (30 to 1,000,000 WHVge/mL) of the standardized serum pool (WHV7p1). This pool was used to infect the experimental animals and has a known WHV genome content. The dilution standards were included, in duplicate, with each run of the thermocycler and were blotted, in duplicate, on each hybridization membrane containing the test samples amplified in the same thermocycler run. Two negative controls also were included, in duplicate, in the standards: uninfected

78

woodchuck serum used for the dilution of the standards and the water/buffers used as PCR components.

4.2.5

Mathematical Model

A standard viral infection model was used for modeling the viral kinetics during therapy (Neumann et al., 1998.). Over periods longer than a few days, the loss of infected cells can be neglected, since uninfected hepatocytes turn over slowly it is reasonable to assume that target cell dynamics remain at baseline values. The model is described by the differential equations: dI = (1 − η ) β V − δI dt dV = (1 − ε ) pI − cV dt

(4.1) (4.2)

t < t pk ε = 0  0 < ε < 1 t ≥ t pk Equation (1) expresses the dynamics of infected cells where I is the concentration of infected cells, δ is the infected cell loss rate. Equation (2) expresses the dynamics of the virus where V is the virions' concentration, p is the production rate constant of virions by infected cells and c is the virus clearance rate constant. I assume that both the viral load and the number of infected cells are in quasi-steady state before therapy initiation (Perelson et al., 1994). This assumption is supported in so far as the maximum variation in viral load is only 2-fold over timescales of months in placebo-treated animals (data not shown). Since therapy profiles exhibit biphasic kinetics (see Figure 4.1), the first-phase decline is 10-fold more rapid than observed during potent antiretroviral therapy for HIV, along with its strong dose dependence - findings deduce that CLV's mode of action is similar to HCV- and HBV-inhibitors. It follows that the major effect of CLV is to block viral production or release. Therefore, this model sets ε>0 and η=0, and refers to ε as the 79

antiviral efficacy. If productively infected cells live longer than 2 days the model assumes that their number remains relatively constant during that period and the solution of

(

)

Equation 4.2 is V (t ) = V0 I − ε + εe c ( t −t0 ) . Time point t0 is the observed pharmacokinetic delay. This solution was previously derived for Interfereon-a therapy of HCV (Neumann et al., 1998) and successfully applied in different contexts. The subsequent study derives analytical solutions for the initial exponential decay slope as c and the secondary decay slope as δ as well as ε the antiviral efficacy of the drug.

4.2.6

Immune Control Model

While this model (Equations 4.1 and 4.2) is sufficient to describe the data for the first week or so, it is deficient in modeling the long-term and complex kinetics exhibited in therapy of WHV. I therefore include a derivative of the immune control model (see Chapter 1) where target cells are assumed to be constant, the infected cell loss constant is replaced by the immune control term (see Chapter 1, Equation 1.4) and tpk is the pharmacokinetic of therapy intervention: dI = (1 − η ) β V − δI dt dV = (1 − ε ) pI − cV dt I δ =d +k I +θ

(4.3) (4.4) (4.5)

t < t pk ε = 0  0 < ε < 1 t ≥ t pk

80

4.2.7

Parameter Estimations

Before therapy initiation I assume that viral load is at a steady state. Initial viral load values (V0) are set as the pretreatment serum WHV-DNA concentration. The infection rate constant is derived from the steady state assumption: β =

δc p

Virion production values between 101-109 copies WHV-DNA·day-1·mL-1 had no effect on model kinetics. “Free” parameters in the fit were the pharmacological delay (t0), the infected cell loss (δ) and virion clearance rate (c) constants. The observed pharmacological delay (t0) was fit with ranges between the first timepoints with a decline in viremia of at least 1 log/wk. The most rapid exponential decay slopes between days 1-2 or 2-3 (depending on pharmacological delay) and between days 5-21 were used as minimal estimates for the virion clearance rate constant (c) and the infected cell loss rate constant (δ).

4.2.8

Nonlinear Fitting

Fitting was done by a nonlinear least squares regression method using Berkeley Madonna software (Oster and Macey, 2000). log[V(t)] was fitted to the measured log viral load (copies DNA/mL) for each woodchuck individually. First I fit the data using Equations 4.1-4.2 to and analytical estimations of 1st and 2nd decay slopes as minimal estimates for c and δ, thereby determining optimal values for c, ε and δ. To derive estimates for k and

θ, I again fit the data using Equations 4.3-4.5 while setting values generated for c, ε and using δ as a minimal estimate for k. AST

(aspartate

aminotransferase),

the

maker

for

liver

disfunction

used

in

WHV/woodchuck kinetic studies, was measured and quantified. However the sampling

81

rate was insufficiently frequent to derive accurate fits, and was therefore not included here.

4.2.9

Statistical Analyisis

Statistical analyses were performed using SPSS (SPSS Inc., Chicago, IL). Mean results are given with standard deviations. Wilcoxon (independent samples T-test) was used to find the statistical significance among paired data. Mean results are given with standard deviations.

82

4.3 Results

4.3.1

WHV Kinetic Profiles

After treatment initiation there was no effect on viremia during the first 12-43 hours. This pharmacokinetics delay was dose-dependent. Initiation of Clevudivne (CLV) therapy induced a dose-dependent decline in viremia in chronically infected woodchucks (Figure 4.1). The reduction in viremia exhibited a classic biphase kinetic in 8 animals (50%). The other 8 animals had complex multi-phasic kinetics uncharacteristic of other viral infections during treatment. The first phase decline rate reduction of viremia was rapid. Initial rate of clearance of viremia were virtually the same for all subjects from 0.5 to 2 days of treatment. Second phase slopes were highly variable among the cohort. The average level of viremia in woodchucks treated with 0.3 mg/kg L-FMAU was reduced approximately 104-fold by the end of the treatment period, while higher dosages were able reduce viral concentrations more than 108-fold. There was a high degree of variation in the antiviral suppression among all animals. The average level of viremia in woodchucks treated with 10 mg/kg CLV was reduced approximately 106-fold after 7 days of therapy and more than 100 million-fold by the end of the treatment period. Termination of therapy resulted in a rebound in viremia. These rebounds were generally dose-related (Table 4.1). Viremia in all animals treated with 0.3 mg/kg L-FMAU rose significantly within 1 week following the end of therapy and had returned to pretreatment levels in all animals by 2 weeks post-treatment. Viremia in the animals treated with higher doses of CLV remained suppressed for various lengths of time following the withdrawal of treatment. The observed rebound slope was progressively slower as a function of the dosage. This slow rebound did not allow viral load to return to

83

pretreatment levels until 8 to 12 weeks after the end of therapy in the 1.0 and 3.0 mg/kg regimes, respectively. Viremia in all of the animals treated with 10 mg/kg remained suppressed 6 weeks following the end of therapy. Viremia then rebounded at the slowest rate observed among treatments and did return to pre-treatment levels.

4.3.2

Parameter Value Estimations

First, I used the model given by Equations 4.1 and 4.2 to quantitatively fit the empiric data of each woodchuck from anti viral therapy initiation to day 3 of treatment. This allows the estimation of the ‘free’ parameters c, ε and δ. Minimal parameter values for c and δ were determined by the fastest decay rate between days 0.5-3 and 3-14, respectively. I then quantitatively fit the data which exhibited tri-phasic kinetic profiles using Equations 4.3-4.4 and parameter values derived previously. I set the minimal value for k with the preceding estimate for δ. This method permits the evaluation of the parameters k and θ for datasets without deviation from early viral kinetics. The other parameters do not change during the period of treatment and were either fixed or computed from other parameters (see Table 4.3) for each animal independently. After initiation of therapy, all animals exhibited a delay (t0) during which the viral load remained approximately at baseline (see Figure 4.1). The average CLV pharmacokinetic delay was 18±12 hours (range: 8-42 hrs), and was highly dose-dependent (P<0.01). Thereafter, serum viral load declined rapidly, in characteristic biphasic kinetics; although 8 subjects (50%) had seemingly tri-phasic kinetics. The first phase slopes were characterized by an average virion clearance half-life of 3.9±1.3 hours (range: 2-7 hours) irrespective of treatment regime and agrees with previous studies (Dahari et al., 2005).

84

Mean antiviral suppression values for the 0.3-, 1-, 3- and 10-mg/kg regimens were 2.1-, 2.4-, 3.5- and 4.4-log copies WHV-DNA/mL, with suppression values in excess of 6.5 log copies WHV-DNA/mL. These values translate to 99.3%, 99.6%, 99.7% and 99.996% effective blockage (ε) for each dosage. These efficacies show a strong dose dependency (P <0.01). After 24 to 36 hours of CLV treatment, the viral decline slowed and stabilized at 82% and 57% of baseline values for the 0.3-1 mg/kg and 3-10 mg/kg dose regimens, respectively (see Table 4.1). The second phase slopes have half life ranging from 1 to mare than 70 days with a high inter-animal variability and is consistent with previous reports of infected cell loss rates in HBV perturbation experiments (Krogsgaard et al., 1994). The mean estimates of the new parameter k have values of 0.65±0.49 day-1 (see Table 4.3). These values are only slightly higher than the mean estimate for δ (0.29±0.28 day-1). However, five animals have significantly higher differences between k and δ. This is due to the insufficiency of the basic model in representing the long-term viral kinetics. The immune control in Equation 4.3-4.5 represented by the saturation function has a dynamic behavior, where the value of k is the maximum immune activity potential (see Chapter 1, Figure 1.4). This is in contrast to the parameter δ which is constant over time. I used the model given by Equations 4.1 and 4.2 to quantitatively fit the empiric data of five woodchucks which received Lamuvidine (LAM) antiviral therapy initiation for 84 days until end of treatment. The ‘free’ parameters c, ε and δ were given minimal estimations, with parameter values for c and δ as determined by the fastest measured decay rate between days 0.5-3 and 3-14, respectively. This method permits the optimization of the ‘free’ parameters. The other parameters do not change during the period of treatment as for fitting CLV data.

85

gAfter initiation of therapy, all animals exhibited a delay (t0) during which the viral load remained approximately at baseline (see Figure 4.1A and Table 4.1). The average LAM pharmaco-kinetic delay was 204±233 hours (range: 24-494 hrs), and was highly dependent upon gender (P<0.01). Thereafter, serum viral load declined rapidly, in characteristic biphasic kinetics. The first phase slopes were characterized by an average virion clearance half-life of 6.7±4.6 hours (range: 2.5-15 hours) irrespective of treatment regime. Mean antiviral suppression averaged 1.3-log copies WHV-DNA/mL which is indicative of approximately 95% effective viral production blockage (ε), with maximum suppression values less than 103 copies WHV-DNA/mL. The second phase slopes have half life ranging 5-17 days with a high inter-animal variability and is consistent with previous reports of infected cell loss rates in HBV therapy with Lamivudine (Hadziyannis et al., 2000; Wolters et al., 2000 and others). The mean baseline viral load in males is higher than in female woodchucks, in so far as 77% of female subjects had lower viremia values prior to treatment vis-à-vis the male woodchucks (Figure 4.6). This gender disparity, while not statistically significant could be the due to the small number of animals, may be important due to the virus-host interplay. Figure 4.4 shows that female woodchucks have higher values for infected cell loss parameter δ than males. These two variables are significantly correlated (P<0.04). This may be an important since sustained viral response in other viral infections is highly correlated with this parameter. Viral suppression of CLV is represented here as the decline magnitude. The mean decline magnitudes for 0.3-, 1.0-, 3.0- and 10 mg/kg dosages were 2.1, 2.4, 3.5 and 4.4-log. Viral suppression under CLV treatment is highly correlated with dosage (P<0.01). The high 86

dosage of LAM treatment (10 mg/kg) shows a decrease in serum viral concentration at the end of the first phase slope is not as effective as the lowest dosage of CLV (Figure 4.5).

4.3.3

Simulations

The numerical integration of the model (see Figure 4.3, black dashed line) after fitting it to the data (black dots) reveals that the basic infection model, while correctly simulating the data until approximately day 7, is insufficient to explain the long term viral kinetics. Whereas, the immune control model (see Equations 4.3-4.5) can fit early data similarly to the basic model, it can continue to do so for the long term kinetic data (see Figure 4.3, grey curve).

87

4.3.4

Tables and Graphs

Figure 4.1: Individual viral load during Clevudine (LFMAU) and Lamivudine (LAM) therapy. Y-axis denotes the change in viral load relative to the baseline. Therapy began on day 0 (x-axis) and cessation is denoted by the vertical line. Note the change in scale after 21 or 28 days.

88

Table 4.1: WHV kinetic characteristics. ID 5247 5254 5304 5175

Gender Female Female Female Male

Drug CLV CLV CLV CLV

Dosage mg /kg 0.3 0.3 0.3 0.3

Mean SD 5280 5303 5140 5202

Female Female Male Male

CLV CLV CLV CLV

1.0 1.0 1.0 1.0

Mean SD 5196 5238 5160 5191

Female Female Male Male

CLV CLV CLV CLV

3.0 3.0 3.0 3.0

Mean SD 5196 5273 5279 3702

Female Female Female Male

CLV CLV CLV CLV

10 10 10 10

Mean SD 5764 5784 5796 5858 5857 Mean SD

Male Female Female Female Male

LAM LAM LAM LAM LAM

10 10 10 10 10

Initial viral load

Viral load

Viral load

log copies/mL

end 1st phase % of baseline

before end RX % of baseline

% of baseline

Decline Phases number

10.9 10.6 10.6 11.2

73 74 50 75

75 77 47 76

101 97 99 102

2 2 2 3

10.8 0.3

68 12

69 15

100 2

2.5 0.5

11.1 10.7 11.2 10.9

59 43 37 44

46 48 39 47

101 101 101 101

3 3 3 2

11.0 0.2

46 9

45 4

101 0

2.8 0.5

11.6 11.0 11.1 11.0

40 51 38 55

45 10 39 11

95 103 98 98

2 3 3 2

11.2 0.3

46 8

26 18

99 3

2.5 0.6

11.4 10.8 11.6 10.5

28 51 52 71

10 24 36 12

99 39 86 79

3 2 2 3

11.1 0.5

51 18

21 12

76 26

2.5 0.6

11.1 10.2 10.2 10.3 10.9

no 1st phase

no 1st phase

88 58 61 60 88

98 95 100 95 100

2 2 2 2 2

10.4 0.3

85 80

71 16

98 3

2.0 0.0

89

76 88 90

Viral Rebound

Figure 4.3: Results of fitting the model to the WHV kinetic data. The data points are represented by symbols, and the lines show the best fit of the data to the basic (black dashed line) and immune control (grey line) mathematical model. Detection limit is below 100 copies DNA/mL.

90

Table 4.2: WHV therapy parameters. ID

Gender

Drug

5247 5254 5304 5175

Female Female Female Male

CLV CLV CLV CLV

Dosage mg /kg 0.3 0.3 0.3 0.3

Mean SD 5280 5303 5140 5202

Female Female Male Male

CLV CLV CLV CLV

1.0 1.0 1.0 1.0

Mean SD 5196 5238 5160 5191

Female Female Male Male

CLV CLV CLV CLV

3.0 3.0 3.0 3.0

Mean SD 5242 5273 5279 3702

Female Female Female Male

CLV CLV CLV CLV

10 10 10 10

Mean SD 5764 5784 5796 5858 5857 Mean SD

Male Female Female Female Male

LAM LAM LAM LAM LAM

10 10 10 10 10

Pharmacologic WHV-DNA WHV DNA Infected cell Delay Blocking ½-life ½-life days log hours days 1.5 2.21 5.4 2.9 1.0 2.39 6.7 >70 1.8 1.75 3.0 1.8 1.4 2.22 5.9 9.4 1.4 0.3

2.1 0.3

5.2 1.6

4.7 4.1

0.5 0.5 0.5 1.5

3.66 2.40 1.80 1.60

4.3 3.8 7.1 7.6

2.3 1.1 0.8 1.1

0.8 0.5

2.4 0.9

5.7 1.9

1.3 0.7

0.3 0.8 0.3 0.3

6.52 3.40 2.16 1.94

5.0 4.8 4.1 3.6

13.9 1.7 0.8 0.5

0.5 0.2

3.5 2.1

4.4 0.6

4.2 6.4

0.2 0.5 0.6 0.3

4.77 5.16 4.96 2.70

3.8 3.3 4.2 2.4

1.1 1.9 2.2 1.0

0.4 0.2

4.4 1.1

3.4 0.8

1.6 0.6

16.5 2.0 1 2.7 20.6

0.23 2.30 0.82 2.68 0.41

4.5 6.5 2.5 14.47 5.6

4.8 6.6 6.9 17.3 9.9

8.5 9.3

1.29 1.13

6.7 4.6

9.1 4.9

91

1.2 1.0 0.8

δ

0.6 0.4 0.2 0.0 -0.2 Female

Male

7

7

6

6

5

5

4 3

۞

۞

۞

2

۞

۞

۞

۞

-log(1-ε )

Efficacy

Figure 4.4: Gender vs. infected cell loss rate. (P<0.04, T-test/independent samples)

4 3 2

1

1

0

0 0.3 0.3 0.3 0.3 1

1

1 1 3 3 CLV (mg/kg)

3

3

10 10 10 10

۞

۞

10 10 10 10 10 LAM (mg/kg)

Figure 4.5: Viral suppression of Clevudine and Lamivudine therapy in woodchucks. Dose dependency of viral suppression is statistically significant (P<0.01) Horizontal lines are the mean values per regime. Males are shown with ۞. Note that all the males in high dosage regimes have significantly (P<0.04) lower efficacy values. 12

logV0

11

10

9 F F F F F F F F F F F F F MMMMMMMM Gender

Figure 4.6: WHV-DNA values prior to Clevudine therapy in woodchucks. The horizontal line is the mean value for males. The females (77%) have somewhat lower baseline values than the mean value for males. 92

Table 4.3: WHV model parameter approximations. Parameter V0

Value or range

Units

Source

105-109

copies·mL-1

From data

104-108

DNA·mL-1

From data

Steady state viral load concentration

θ β

Infected cell generation β = kc V 0 rate p V0 + θ

cV0 p

Steady state assumption DNA·mL-1

Quasi-steady state assumption

10 1-7

DNA·d-1·mL-1 d-1

Set Fit

Infected cell loss rate

0.01-1

d-1

Fit

Infected cell loss rate function

0.01-1

d-1

Fit

I0

Initial infected cell concentration

p c

Virion production rate Virion clearance rate

δ k(I,t)

I0 ≈

Table 4.4: Estimated values for k and θ. ID

Gender

δ

k day-1 0.31 0.06 0.79 0.08

θ

5247 5254 5304 5175

Female Female Female Male

day-1 0.31 0.06 0.79 0.08

5280 5303 5140 5202

Female Female Male Male

0.24 0.56 0.86 0.24

0.33 0.56 0.86 0.24

5.2 5.1 4.8

5196 5238 5160 5191

Female Female Male Male

0.15 0.24 0.10 1.47

0.15 0.24 0.10 1.47

5.1 5.2 4.9 4.4

5242 5273 5279 3702

Female Female Female Male

0.64 0.06 0.19 1.26

0.64 1.05 0.22 1.26

4.0 5.4 3.7 2.1

0.29 0.28

0.65 0.49

5.4 1.9

Mean SD

93

log copies/mL

8.2 8.9 5.6 8.7

4.4 Summary The viral production rate must equal the viral clearance rate to assure relatively constant baseline viral load values before treatment (Neumann et al., 1998, and others). Exploiting the perturbation in the viral steady-state allows an estimation of the viral production rate in each animal by the product cV0 times a factor equal to the extra-cellular fluid volume. This enables a calculation of the mean daily virion production rate of 5·1011 virions per day. A slower viral decline, characteristic of second phase, occurred between 3-5 and 28 days of treatment (Figure 4.1). These exponential decay slopes exhibited a large inter-animal variation, ranging between 1-53 days, with no statistical difference among the regimens. The considerable variation in productively infected cell half-life could be indicative of the particular cellular immunity against WHV of the individual woodchucks (Ho et al., 1995.). These results imply that CLV has significantly better antiviral efficacy than LAM and, based on other findings, higher efficacy than current standard treatments. Studies with more animals and therapeutic trials in clinical settings are necessary to understand the importance of this experimental drug manifested as a greater than 1,000-fold drop in viral load after only 2-4 days of therapy. Studying the rapid dynamics of various WHV treatments has implications for the possible use of new therapeutic agents such as Clevudine; and, by inference, assessing the viability of viral eradication and for managing chronic HBV treatment. It is important to note that in light of this fact most clinical studies which track data at intervals of 1 week miss this very instense first phase altogether. Results of Lamivudine treatment are in agreement with previous findings. Wolters et al. (2002) calculated the antiviral efficacy, viral and infected cell half-life values of 0.93, 94

17hr and 7d, respectively, in cHBV patients treated with 3TC. In contrast, L-FMAU with its stronger antiviral effects, characteristic rebound delay and failure to reach pretreatment values might be a more effective antiviral than 3TC. The combination of short pharmacokinetic delay, high efficacy and distinctive post-treatment rebound properties, deliver a highly potent antiviral therapeutic. Even though very small doses of Clevudine have potent antiviral potential, increased dosing of Clevudine could intensify its other attributes not only the direct antiviral efficacy; and further study is required to determine if this is accurate. Our understanding of the pharmacodynamics of Clevudine needs to be expanded in order to understand significance of the delay for end-of-treatment rebound and the slow rebound slope. This “lingering effect”, not seen in other treatments, hints that the virus/host parameters have been altered. Studies following viral kinetics made need to take into account the very early viral kinetics, since the first phase decline only lasts 2-3 days. Although the basic infection model (see Chapter 1, Section 1.1.6) can accurately predict the early viral kinetics, it cannot account for long term multi-phasic behavior seen in the data for Clevudine therapy of all dosage regimes. However, the incorporating immune control into the basic model does in fact recreate the data in a very detailed manner. This gives evidence for the theory that the immune system is paramount in viral control and may be a factor in therapy innovation and optimization.

95

Chapter 5

Discussion

96

Mathematical modeling in virology is a fast growing, well recognized field and is an exciting modern application of mathematics. Due to the vast improvements in viral quantification in the infected individual, the increased use of mathematics in virology has become feasible. In general, mathematical models must be biologically relevant to if they are to augment and complement laboratory techniques and biologic theory. The best mathematical models of viral kinetics can suggest what the governing mechanisms are, quantitatively evaluate dynamical host and viral parameters, and predict what may follow. The aim of this thesis is to explore the dynamic features of viral infection at three junctures: the beginning of infection (the primary infection phase), during the chronic phase and under anti-viral treatment. To understand viral dynamics during primary HIV infection, chronic HBV infection and treatment, I developed a new mathematical model. This model along with accurately measured viral dynamics has brought a new understanding into the elements of the nature of the steady-state between the virus and the immune system. However, they do not allow research into the underlying mechanisms of the pathology of chronic viral infection; and reveal nothing about the parameter shifts occurring slowly over years from first exposure to pathogenesis. This model does not include, for example, whether antibody mediated immune responses contribute to the rapid decline of plasma virions after seroconversion, and what the exact immune response modes that lead to spontaneous viral clearance are unknown at this time. Nor does it illuminate the difficult task of modeling viral infection from onset to outcome trough the various phases. Even so, this thesis shows the important inter-play between immune response and the virus, and accurately projected viral kinetics for more than the first weeks of infection or treatment. 97

To understand the primary infection phase I developed more complex model to explain the viral kinetics displayed by HIV. To better characterize host-viral interplay I added an enhanced term for dynamics of infected cell loss. This extension of the target cell-limiting model assumes an immune control of infected cells as the primary mechanism of cell loss. This led the way to modeling the complex dynamics in HBV chronic infection, with the addition of Alanine aminotransferase (ALT) dynamics, necessary for modeling viral liver disease. A hybrid of the basic therapy model and the immune control model was used in WHV to accurately describe drug efficacies, first viral decay slope along with the relatively long-term multi-phasic second phase decline. The implementation of this model in a number of clinical settings allow evaluation of host and viral dynamics and suggest the governing host mechanisms for explaining viral kinetics during viral infection. In order to ascertain the model’s validity, I analyzed 203 individual datasets from primary, chronic and treatment of viral infections. This analysis allowed us to point to the infected cell control as the primary mechanism in the complex multi-phasic dynamics. Early HIV kinetics HIV infected patients could not be explained by conventional modeling. Previous models either lacked simplicity or failed to reproduce the clinical results in an elegant fashion. I hypothesize that the multi-phasic viral kinetics shown in data from diverse sources could be explained by incorporating infected cell control into the basic model. Model simulations predict that this infected cell control to have highly diverse dynamics. The results of this model give accurate predictions of viral kinetics during long periods of time (Chapter 1). Moreover, model results suggest what the governing mechanisms are, and raise several new hypotheses of viral and host interplay that can be tested in the laboratory. Although many complex processes occur during the rise and fall in viral load following initial HIV infection, I show that temporal changes in 98

virus concentration observed early after infection are consistent with the assumptions embodied in the extension of simple target-cell limited model (Equations 1.1-1.5). HBV chronic infection shows highly dynamic kinetics manifested as rapid declines in viral load in the absence of antiviral intervention. These rapid declines have not previously been documented. This is significant as other viral infections exhibit only inconsequential fluctuations. A pattern recognition algorithm was developed and categorized the patients into those with biologically relevant viral decreases (SPD) and those with flat kinetic profiles (FLT). The RVDs have magnitudes similar to those demonstrated during the second phase of anti-HBV treatment. ALT flares and HBe antigen seroconversion are concomitant with RVDs. Interestingly, patients who are assumed to have more effective immune responses, e.g., females, had a higher incidence of SPDs. Preceding dynamic models were unsuccessful at fitting the data collected from 104 SPD patients out of 163 who received placebo. I successfully implemented the immune control model to qualitatively fit the SPD patients’ data. All of the above discussed parameters evident that the major component of this phenomenon indicates the immune system’s activity during SPDs. The immune control model was extended in the context of chronic HBV infection. I hypothesized the existence of homeostasis of only non-infected hepatocytes, which was determined after testing various mechanisms of hepatocyte proliferation, namely: a) no proliferation of hepatocytes at all, b) proliferation only of non-infected cells, c) blind homeostasis proliferation of both infected and non-infected cells. The decision to focus on non-infected hepatocytes homeostasis mechanism was based on the following results: I) infected cells are being cleared faster than they can proliferate; II) simulation results show that this assumption is sufficient to model the data; III) without the homeostasis 99

mechanism the liver total mass might decrease 2-fold or more during therapy. Such evidence was provided by Summers et al (2003) in the study of woodchuck hepatitis infection. Another hypothesis was the existence of hepatocyte proliferation in HBV infection. Even though constant influx rate of de-novo hepatocytes to the liver has been put forth as a source of target cells for HBV, and probably holds for HIV, this study shows that hepatocyte proliferation may be the main mechanism of liver regeneration, consistent with liver regeneration after liver transplantation procedures. This hypothesis is important in the explanation of both constant liver size and elevated ALT before followed by an ALT flare. In addition, Summers et al (2003) suggest that hepatocyte proliferation plays a significant role (together with killing of infected cells) in the direct elimination of infected cells in WHV. Research conducted by Shmailov and Neumann (2005) showed a correlation between the pre-treatment and the magnitude of the second slopes during Adefovir therapy (see Figure 3.12). Pre-treatment slopes with values more or equal to zero exhibited very slow second slopes; while negative values of pre-treatment slopes had exceptionally fast second slopes (see adjacent table). Yafit Maayan (2006) calculated the minimal sampling rate to determine if a patient will be SPD. Taken together, this suggests a clinically relevant strategy to begin treatment for chronic HBV during an SPD.

Pre-Rx slope

Second slope

≥0 <0

slow fast

100

Woodchuck Hepatitis virus exhibits complex viral kinetics during antiviral treatment with Clevudine. Although half of the woodchucks had bi-phasic decay slopes similar to those seen in HCV treatment, the remaining animals had tri-phasic kinetic profiles. Slow viral rebound slopes are another interesting phenomenon seen in after CLV treatment cessation. These rebound slopes are dose-dependent. The basic model used to describe therapy dynamics (Neumann et al., 1998) is inadequate to encompass the behavior displayed by the data for the full 28 days of study, and beyond. I apply the immune control model in conjunction with the basic model. This approach allows the capture of the long-term viral kinetics utilizing the immune control model along with the clearance rate and the drug efficacy employing the basic model. Although the use of viral dynamics has improved our understanding of viral infection, a range of critical questions about remain unanswered and are not within the scope of this thesis. These topics include: mechanisms responsible for complex antiviral effects, mechanisms describing dose-dependent rebound slopes observed during Clevudine therapy and formulation of a general theory of viral infection. The viral kinetics of both HBV and WHV show complex and diverse patterns not seen in other chronic viral infections, and can be explained by the immune control model. Understanding the immune mechanisms hinted at by the immune control model may be fundamental for the development of new antiviral drugs and therapy regimes. The mathematical model proposed here offers new biological hypotheses. It represents an immune control response against infected cells. It allows the accurate simulation of many different and complex kinetics, and explains the various outcomes of viral disease. I hope that this new mathematical model outlined here and its implications will stimulate virologists to empirically confirm immune control processes in these and other viral infections. 101

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‫תקציר‬ ‫דינאמיקה נגיפית מהווה וריאנט מורכב ביחסי הגומלין של טורף‪-‬נטרף הבלתי‪-‬ליניאריים‬ ‫שבין תאי המטרה ובין הנגיף‪ .‬ניתן לתאר יחסי גומלין אלו על ידי מאגר מתחדש של תאי‬ ‫מטרה אשר מודבקים בנגיף המאפשרת התרבות הנגיף‪ .‬השליטה והבקרה על הנגיף‬ ‫מושגים בעזרת הגבלה במספר תאי המטרה ו‪/‬או בעזרת איבוד התאים המודבקים כתוצאה‬ ‫מתגובה חיסונית‪ .‬מחקרים קודמים הוכיחו את חשיבותם ביישום מודלים פשוטים בלתי‪-‬‬ ‫ליניאריים ככלי להבנה ההדבקות הנגיפיות‪ .‬במודלים אלה הנחת היסוד היא כי הפרמטרים‬ ‫אינם משתנים; אך מפאת פשטותם של מודלים אלה מורכבותה המלאה של המערכת‬ ‫הביולוגית אינה באה לידי ביטוי‪ .‬דוגמא לכך‪ ,‬שההנחה שהדינאמיקה של תאי מטרה או‬ ‫ַבּויֹות של‬ ‫רּכ‬ ‫מערכת החיסון קבועים על ציר הזמן‪ ,‬אינה מסוגלים לקחת בחשבון את המּוְ‬ ‫התנהגות ההדבקה הנגיפית‪ .‬אינטראקציה של מערכת החיסון עם אוכלוסיית התאים‬ ‫המודבקים חייבת להיחשב כאל תהליך דינאמי‪ .‬המודל המתואר כאן מציע מנגנון פשוט‪,‬‬ ‫שעל אף המופשטות שלו‪ ,‬מראה כיצד צירוף כללים פשוטים‪ ,‬המשפיעים והמושפעים‬ ‫במערכת הבלתי‪-‬ליניארית מחוללת מגוון תבניות של תגובה חיסונית הנצפות בנתונים‬ ‫קליניים וניסיוניים‪ .‬שלושה מצבים בעלי עניין רב הם הדבקה אקוטית אשר הנגיף מפונה‬ ‫מהגוף לאחר חשיפה‪ ,‬מוות של המאכסן בשל התקפת המחץ של הנגיף המכריע אותו או‬ ‫כינון הדבקה כרונית שבה המאכסן והנגיף ׂשורדים יחדיו במשך תקופה ארוכה‪ .‬המודל‬ ‫מסוגל לשחזר בהצלחה רבה את שלושת המצבים הללו‪ .‬יתר על כן‪ ,‬המודל מסביר את‬ ‫קיומה המתמשכת של רמת נגיף נמוכה במשך טיפול אנטי‪-‬נגיפי המופיע בהדבקות כרוניות‬ ‫רבות‪ .‬הוא גם מאפשר התנהגות מורכבת ורב‪-‬פאזית )‪ (multi-phasic‬במהלך שלבים שונים‬ ‫של הדבקה נגיפית וטיפול‪.‬‬ ‫אני משתמש במודל המתאר בקרה אימונית כפונקצית סטוראציה )‪ (saturation‬בשלוש‬ ‫מערכות נגיפיות שונות‪ ,‬בשלבים שונים של הדבקה‪ :‬הדבקה ראשונית בנגיף ה‪human -‬‬ ‫‪ ;(HIV) immunodeficiency virus‬דינאמיקה טבעית של צהבת נגיפית מסוג ‪ B‬במהלך‬ ‫הדבקה כרונית; טיפול אנטי‪-‬ויראלי של מרמיטות ב‪.(WHV) woodchuck hepatitis virus-‬‬

‫א‬

‫במשך הדבקה ראשונית ב‪ ,HIV-‬ריכוז הנגיף בדם‬

‫עולה‪ ,‬מגיע לשיא ולאחר מכן יורד‬

‫ומתייצב‪ .‬מסקנה העולה ממחקרים קודמים היא שירידה זאת נובעת משני גורמים‪:‬‬ ‫מההגבלה במספר התאים הרגישים להדבקה בנגיף ‪ HIV‬או מהתגובה החיסונית‪ .‬כאן אני‬ ‫מיישם מודל של בקרה אימונית על נתונים של הדבקה ראשונית ב‪ HIV-1-‬בעשרה חולים‪.‬‬ ‫‪ (2000) Stafford et al‬הראה כי הנתונים מעשרת החולים מתאימים למודל המגביל את‬ ‫מספר תאי המטרה עד מעט לאחר התייצבות הנגיף בירידה מהשיא‪ ,‬כאשר הקינטיקה‬ ‫ממחולים מסוימים אינה מתאימה למודל זה‪ .‬אף על פי שהם הפעילו מודל המדמה שתי‬ ‫תגובות חיסוניות שונות‪ ,‬תוצאות הסימולציות לא השביעו רצון‪ .‬אני מתאר שמודל פשוט של‬ ‫מנגנון מערכת חיסון מסביר את הירידות בריכוז הנגיף שאינן מצופות על ידי המודל הבסיסי‬ ‫או על ידי מודלים מורכבים של מערכת חיסון‪ .‬בעזרת ‪nonlinear least-squares estimation‬‬ ‫אני מראה כי המודל המוצע מסוגל לתאר את התבניות השונות הנצפות בנתונים קליניים‪.‬‬ ‫זמני מחצית החיים של תאים מודבקים במהלך הדבקה ראשונית נעו בין ‪ 0.2‬ל‪ 12-‬ימים‪,‬‬ ‫ערכים שהם עקביים לתוצאות שהושגו בניסויים תרופתיים המדכאים את הנגיף‪.‬‬ ‫מידבקות נוספות‪.‬‬ ‫מודלים נותנים לנו הזדמנויות לספק תובנות לגבי הדינאמיקה של מחלות ִ‬ ‫צהבת נגיפית‪ ,‬אשר גורמת להדבקה ביותר מ‪ 9%-‬מאוכלוסיית העולם‪ ,‬מהווה מטרה בעלת‬ ‫חשיבות גבוהה למחקר בתחום המודלים המתמטיים‪ .‬מודלים‪ ,‬שבהם משולבות תגובות‬ ‫חיסוניות ואשר מטפלים בנושא חסינות הנגיפים לטיפול תרופתי‪ ,‬בעלי חשיבות רבה‬ ‫ויכולים להניב תובנות לאסטרטגיות טיפוליות‪ .‬יתר על כן‪ ,‬להבנה ביחסי הגומלין בין נגיף‬ ‫למאכסן ישי חשיבות עליונה בעיצוב טיפולים תרופתיים יעילים כנגד נגיפים‪.‬‬ ‫אנליזת דינאמיקה נגיפית מניחה כי ריכוז הנגיף בדם בחולים שאינם מטופלים נמצא במצב‬ ‫יציב לאורך ימים וחודשים‪ .‬המצב אכן כך בנגיפים כמו ‪ HIV‬וצהבת נגיפית מסוג ‪.(HCV) C‬‬ ‫על אף זאת‪ ,‬בחולי צהבת נגיפית מסוג ‪ (HBV) B‬נצפים שינויים ספוראדיים ברמות‬ ‫‪ ,HBV-DNA‬איבוד אנטיגנים מסוג ”‪ (HBeAg) “e‬ועליות חדות ברמות ה‪ .ALT-‬מחקר זה מאפיין‬ ‫בפרוטרוט את התבניות הקינטיות של הנגיף בחולים אשר קבלו טיפול "‪ "placebo‬ואת‬ ‫המתאם שיש בין ירידות ספונטאניות אלה לבין פרמטרים בסיסיים )כמו גיל‪ ,‬מוצא וכו'‪(...‬‬ ‫לבין קינטיקה של ‪ ALT‬ואיבוד ‪.HBeAg‬‬ ‫ב‬

‫‪ 170‬חולי ‪ HBV‬כרוניים קבלו ‪ placebo‬והיו במעקב במשך ‪ 48‬שבועות במחקר קליני בשלב ‪3‬‬ ‫של התרופה ‪ .adefovir dipivoxil‬תבניות נגיפיות סווגו עבור ‪ 164‬חולים ורמות ‪HBV-DNA‬‬ ‫דוגמו כל ארבעה שבועות‪ .‬ירידה נגיפית מהירה )‪ (RVD‬בתמונה הקינטית הוגדרה במצב‬ ‫שבו היה שינוי ברמת ה‪ DNA-‬אל מתחת לסף של ‪ 1.0 log‬או ‪ 0.5 log‬עותקים לסמ"ק‪ .‬בנוסף‬ ‫שינוי ברמת ה‪ ALT-‬בדם העולה על פי ‪ 1.5‬מהמצב היציב הוגדרה כעלייה ב‪ .ALT-‬גודל ‪RVD‬‬ ‫ממוצע היה ‪) 2.8 log‬טווח‪ (1-5 log :‬ונצפה ב‪ 44%-‬מהחולים במעקב שנמשך כשנה אחת‪.‬‬ ‫ישנו מתאם בזמן בין העליות ב‪ ALT-‬לבין התחלת ה‪) RVD-‬יותר מ‪ 79%-‬מהחולים היו בעלי‬ ‫ירידה של יותר מ‪ (1.0 log-‬בהשוואה לחולים עם ירידה פחות מ‪ .(21%) 0.5 log-‬לכן החולים‬ ‫חולקו על פי ‪ 3‬תבניות קינטיות‪ :‬מצב יציב )‪ (31% :FLT‬בעלי ירידות בנגיף פחות מ‪0.5 log-‬‬ ‫או בעלי ירידה שבין ‪ 0.5-1.0 log‬ללא שינוי ברמת ה‪ ;ALT-‬ירידות ספונטאניות מהירות עם‬ ‫ירידות בנגיף גדולות מ‪ ,1.0 log-‬או בעלי ירידה שבין ‪ 0.5-1.0 log‬עם עלייה ברמת ה‪ALT-‬‬ ‫)‪ ;(63% :SPD‬ירידה איטית אך ממושכת בנגיף של יותר מ‪ .(6% :DWN) 0.5 log-‬רמת ה‪-‬‬ ‫‪ DNA‬בחולים בעלי ‪ SPD‬הגיע למצב יציב נמוך יותר )ממוצע ‪ 4.5 log‬עותקים לסמ"ק‪ ,‬טווח‬ ‫‪ .(3.0-6.5 log‬כמוכ ן‪ ,‬רמות הנגיף חזרו לרמות הנגיף שנמדדו קודם לירידהב‪45%-‬‬ ‫מהחולים האלה בתוך ‪ 2-9‬חודשים מתחילת הירידה‪ .‬לא נצפה אובדן ‪ HBeAg‬באף אחד‬ ‫מהחולים בקבוצת ה‪ ,FLT-‬לעומת ‪ 27%‬מקבוצת ה‪.(P<0.001) SPD-‬‬ ‫התפלגות הופעת ה‪ SPD-‬הייתה שווה במשך השנה בתדירות של ‪ 8-15%‬בכל חודש ולא‬ ‫היה מתאם בין תחילות הירידות לבין תחילת טיפול ב‪.placebo-‬‬

‫הופעת ה‪ SPD-‬הייתה‬

‫צא אירופאי‪ .‬יש גם‬ ‫ממֹו ָ‬ ‫צא אסיאתי לעומת חולים ִ‬ ‫ממֹו ָ‬ ‫גבוהה בצורה משמעותית בחולים ִ‬ ‫מתאם בין ‪ SPD‬לבין גיל צעיר‪ ,‬מסת גוף )‪ (BMI‬נמוכה‪ ,‬רמת בסיסית נמוכה של ‪HBV-DNA‬‬ ‫וכן רמה בסיסית גבוהה של ‪.ALT‬‬ ‫צה ספוראדית של איבוד תאים מודבקים‬ ‫ה‪ָ j‬‬ ‫מודל דינאמיקה נגיפית המניחה כי ישנה ֵ‬ ‫מתאים הן לקינטיקת הנגיף וכן לקינטיקת ה‪ ALT-‬במשך ‪ .SPD‬החזרה לרמות הנגיף שהיו‬ ‫לפני ה‪ SPD-‬מדומה על ידי שיחזור קצב איבוד התאים המודבקים לערכו מלפני ה‪.SPD-‬‬ ‫נפילה ברמת הנגיף )זמן מחצית חיים )½‪ (t‬ממוצע‪ 7.8 :‬ימים( וקצב חזרה רמה שלפני ‪SPD‬‬

‫ג‬

‫)זמן הכפלה )‪ (t2‬ממוצע‪ 7.3 :‬ימים( הן מתאימות לקצב איבוד תאים מודבקים אשר נקבע‬ ‫בזמן טיפול אנטי‪-‬ויראלי‪.‬‬ ‫‪ HBV‬בחולים ללא טיפול מראה קינטיקה נגיפית מורכבת )אשר נמצא במתאם עם עליות ב‪-‬‬ ‫‪ ALT‬ואיבוד ‪ (HBeAg‬בניגוד להדבקות נגיפיות כרוניות אחרות‪ .‬הירידות הספונטאניות אשר‬ ‫נצפו מתאימות להאצה ספוראדית של איבוד תאים מודבקים על ידי מערכת חיסון וחזרה‬ ‫ִּטּוב‬ ‫בעקבות בריחת הנגיף מבקרה חיסונית‪ .‬החשיבות בירידות ספונטאניות אלה עבור מ‬ ‫ניהול טיפול כנגד ‪ HBV‬דורשת מחקר נוסף‪.‬‬ ‫מרמיטות )‪ (woodchucks‬אשר הודבקו באופן כרוני ב‪(WHV) Woodchuck Hepatitis virus-‬‬ ‫מהוֹות מודל אנימלי עבור טיפול בבני אדם הסובלים מ‪ HBV-‬כרוני‪ .‬מחקרים קודמים ביססו‬ ‫ש‪ ,WHV-‬כמו גם הדבקה ‪ HBV‬כרונית‪ ,‬הינה מערכת דינאמית של יחסי גומלין שבין הנגיף‬ ‫לבין תאי המטרה ‪ -‬תאי כבד‪ ,‬וכן יכולת המאכסן להילחם בהדבקה נגיפית‪ .‬ישנן שתי‬ ‫פאזות בירידה ברמת הנגיף בעקבות הטיפול כנגדו‪ :‬הראשון מקביל לפינוי ויריונים )‪(virions‬‬ ‫חופשיים‪ ,‬והשני האיטי יותר המתאים לאיבוד תאים מודבקים‪ .‬נכון לעכשיו‪ ,‬לא בוצעה‬ ‫אנליזה דינאמית ל‪ ,(L-FMAU) Clevudine-‬וכן לא נעשתה השוואה כמותית להראות את‬ ‫ההבדלים בין שיטות טיפול שונות במודל מרמיטה‪ WHV/‬ובין המודל לצהבת נגיפית מסוג ‪B‬‬ ‫כרונית בבני אדם‪.‬‬ ‫כמו כן‪ ,‬אני מאפיין את הפרמטרים הדינאמיים בטיפול כנגד ‪ WHV‬במרמיטות מודבקות‬ ‫באופן כרוני‪ ,‬בעזרת מודל מתמטי של הדבקה נגיפית‪ .‬מ‪ 16-‬מרמיטות אשר הודבקו ב‪WHV-‬‬ ‫באופן כרוני ואשר טופלו ב‪ Clevudine-‬בריכוזים ‪ 3 ,1 ,0.3‬ו‪ 10-‬מ"ג‪/‬ק"ג נלקחו בדיקות דם‬ ‫בימים ‪ 21 ,14 ,7 ,5 ,3 ,2 ,1 ,½ ,0‬ו‪ ;30-‬וה‪ WHV-DNA-‬הוגבר ונמדד על ידי תגובת ‪PCR‬‬ ‫)‪ .(detection level: >30pgDNA/mL‬בהתאמה בלתי ליניארית )‪ (nonlinear fitting‬של הנתונים‬ ‫למודל הופקו ערכים עבור הפרמטרים של יעילות התרופה והשיפועים הראשון והשני‬ ‫בירידה הבי‪-‬פאזית )‪ ,(bi-phasic‬במהלך ימים ‪ .0-21‬המודל מניח כי‪ :‬דינאמיקה של תאי‬ ‫מטרה אינה משתנית במשך הימים הראשונים של הטיפול‪ ,‬ונגיפים חופשיים ותאי מודבקים‬ ‫הם ב‪ .quasi-steady state-‬אנליזה דומה בוצעה על נתונים קינטיים של ‪ 5‬מרמיטות שטופלו‬ ‫ב‪.(3TC) Lamivudine-‬‬ ‫ד‬

‫ממוצע יעילות התרופה הייתה ‪) 99.8%‬טווח ‪ ,(97.5-99.9993%‬דבר המצביע על כך שהנגיף‬ ‫מתרבה ברמות מאוד נמוכות בזמן הטיפול‪ .‬ממוצע זמני מחצית החיים של הנגיף אשר‬ ‫הופקו מירידת הפאזה הראשונה הייתה בטווח ‪ 5±2‬שעות‪ ,‬ללא קשר להריכוזים ‪.0‬של‬ ‫‪ .Clevudine‬זמן מחצית החיים הממוצע של תאים מודבקים‪ ,‬המיוצגים על ידי פאזת הירידה‬ ‫השניה‪ ,‬היה ‪ 2.5‬ימים )טווח‪ 1-70 :‬יום(‪ .‬על אף שריכוזים נמוכים של ‪ Clevudine‬הם בעלי‬ ‫עוצמה רבה‪ ,‬היעילות האנטי‪-‬ויראלית והעיכוב הפרמקוקינטי )‪(pharmacokinetic delay‬‬ ‫מראים מתאם לריכוז הניתן לחיות ) ‪ P<0.001‬ו‪ .(P<0.045-‬יתר על כן המצב היציב שהושג‬ ‫לאחר סיום הטיפול בריכוזים גבוהים יותר של ‪ Clevudine‬היו נמוכות בצורה משמעותית‬ ‫מאשר לפני הטיפול )‪ .(P<0.041‬למרות זאת‪ ,‬היעילות התרופתית של ‪ Lamivudine‬אשר‬ ‫ניתן למרמיטות הייתה נמוכה בצורה משמעותית )‪ (85±10%‬מאשר הטיפול ב‪. Clevudine-‬‬ ‫תוצאה זו של ‪ Lamivudine‬נמצאת בהתאמה לממצאים קודמים‪ (2002) Wolters et al .‬חישבו‬ ‫את היעילות האנטי‪-‬נגיפית‪ ,‬זמני מחצית החיים של הנגיף ותאים מודבקים של ‪17 ,93%‬‬ ‫שעות ו‪ 7-‬ימים בהתאמה‪ ,‬בחולי ‪ HBV‬כרוני אשר טופלו ב‪ .Lamivudine-‬בהשוואה‪ ,‬ייתכן ו‪-‬‬ ‫‪ L-FMAU‬יעיל יותר מאשר ‪ 3TC‬בשל השפעה אנטי‪-‬ויראלית עוצמתית יותר‪ ,‬שהייה אופיינית‬ ‫בעלייה בחזרה של הנגיף למצב קדם‪-‬טיפול וחוסר היכולת של המערכת לחזור לערכים‬ ‫קדם‪-‬טיפוליים של הנגיף בדם‪ .‬בנוסף לעובדה כי ריכוזים גבוהים של ‪ Clevudine‬בעלי‬ ‫השפעה אנטי‪-‬ויראלית חזקה‪ ,‬ייתכן ועלייה בריכוז של התרופה תגביר את המאפיינים‬ ‫הנוספים שלו‪ .‬נדרשת הרחבה בהבנה שלנו בפרמקוקינטיקה של ‪ Clevudine‬על מנת‬ ‫להסביר את חשיבותם של השהייה והחזרה האיטית של הנגיף בסיום טיפול‪ .‬תופעה זו‪,‬‬ ‫אשר לא נצפתה בטיפולים קודמים‪ ,‬מרמז כי הפרמטרים ביחסי הגומלין בין הנגיף למאכסן‬ ‫עברו שינוי‪ .‬כמו כן מחקרים העוקבים אחרי קינטיקה נגיפית צריכים לקחת בחשבון את‬ ‫הקינטיקה המאוד מוקדמת‪ ,‬משום שהירידה בפאזה הראשונה נמשכת רק ‪ 2-3‬ימים‪.‬‬

‫ה‬

‫תוכן עניינים‬

‫תקציר ‪ .............................................................................................................‬א‬

‫פרק ‪1............................................................................................................... 1‬‬ ‫מבוא ומטרות העבודה‬ ‫‪ 1.1‬מבוא‪2....................................................................................................‬‬

‫‪ 1.1.1‬מבנה המחקר ‪2............................................................................‬‬ ‫‪ 1.1.2‬מטרות התזה ‪2..............................................................................‬‬ ‫‪ 1.1.3‬מודלים מתמטיים בביולוגיה ‪3.........................................................‬‬ ‫‪ 1.1.4‬מודל דינאמי נגיפי פשוט ‪5..............................................................‬‬ ‫‪ 1.1.5‬תאים מודבקים במודלים נגיפים דינאמיים ‪6......................................‬‬ ‫‪ 1.1.6‬הגבלה בתאי מטרה במולדים דינאמיים נגיפיים ‪7..............................‬‬ ‫‪ 1.1.7‬מודל בקרה חיסונית‪8....................................................................‬‬ ‫‪ 1.2‬שיטות ‪11 ................................................................................................‬‬ ‫‪ 1.2.1‬הצגת מרחב פאזי ‪11 ....................................................................‬‬ ‫‪ 1.2.2‬מצבים יציבים ‪11 ...........................................................................‬‬ ‫‪ 1.2.3‬יציבות המודל ‪11 ...........................................................................‬‬ ‫‪ 1.2.4‬הערכת פרמטרים ‪13 .....................................................................‬‬ ‫‪ 1.3‬תוצאות ‪14 ..............................................................................................‬‬ ‫‪ 1.3.1‬הפשטת המודל ‪14 ........................................................................‬‬ ‫‪14 ................................................................................. Nullclines 1.3.2‬‬ ‫‪ 1.3.3‬מצבים יציבים ‪15 ...........................................................................‬‬ ‫‪ 1.3.4‬יציבות המודל ‪16 ...........................................................................‬‬ ‫‪ 1.3.5‬הערכת פרמטרים נגיפים ‪16 ...........................................................‬‬ ‫‪ 1.3.6‬אנליזת המרחב הפאזי ‪17 ...............................................................‬‬ ‫‪ 1.3.7‬דינאמיקת בקרה חיסונית ‪19 ...........................................................‬‬ ‫‪ 1.4‬סיכום ‪20 ..............................................................................................‬‬ ‫פרק ‪21 ............................................................................................................. 2‬‬ ‫דינאמיקה נגיפית בהדבקה ראשונית ב‪HIV-‬‬ ‫‪ 2.1‬מבוא‪22 ..................................................................................................‬‬

‫‪ 2.1.1‬מבנה וגנום הנגיף ‪24 .....................................................................‬‬ ‫‪ 2.1.2‬מחזור חיים‪25 ...............................................................................‬‬ ‫‪ 2.1.3‬כניסת הנגיף לתאים ‪26 .................................................................‬‬ ‫‪ 2.1.4‬סינתזת ‪ DNA‬עבור ה‪27 .................................................... Provirus-‬‬ ‫‪ 2.1.5‬מעבר לגרעין‪ ,‬השתלבות בגנום וביטוי גנים ‪28 .................................‬‬ ‫‪ 2.1.6‬הרכבת הנגיף ‪28 ..........................................................................‬‬ ‫‪ 2.1.7‬מודלים מתמטיים ב‪29 ............................................................ HIV-‬‬

‫פרק ‪ - 2‬דינאמיקה נגיפית בהדבקה ראשונית ב‪) HIV-‬המשך(‬

‫‪ 2.2‬שיטות ‪34 ................................................................................................‬‬ ‫‪ 2.2.1‬נתונים מחולים‪34 ..........................................................................‬‬ ‫‪ 2.2.2‬מידול‪ ,‬הערכת פרמטרים והתאמה )‪34 .................................. (fitting‬‬ ‫‪ 2.3‬תוצאות ‪36 ..............................................................................................‬‬ ‫‪ 2.3.1‬ערכי פרמטרים ‪36 .........................................................................‬‬ ‫‪ 2.3.2‬טבלאות גרפים ‪37 .........................................................................‬‬ ‫‪ 2.3.8‬הדמיות‪38 ....................................................................................‬‬ ‫‪ 2.4‬סיכום‪43 .................................................................................................‬‬ ‫פרק ‪46 ............................................................................................................. 3‬‬ ‫דינאמיקה של הדבקה כרונית בצהבת נגיפית מסוג ‪B‬‬ ‫‪ 3.1‬מבוא‪47 ..................................................................................................‬‬

‫‪ 3.1.1‬מבנה הנגיף ‪47 .............................................................................‬‬ ‫‪ 3.1.2‬הגנום של ‪48 ......................................................................... HBV‬‬ ‫‪ 3.1.3‬מחזור החיים ‪49 ............................................................................‬‬ ‫‪ 3.1.4‬מאפיינים של הדבקה כרונית ב‪52 ........................................... HBV-‬‬ ‫‪ 3.1.5‬קינטיקה של הדבקה כרונית ב‪52 ............................................ HBV-‬‬ ‫‪ 3.2‬שיטות ‪54 ................................................................................................‬‬ ‫‪ 3.2.1‬קריטריונים להשתתפות במחקר ‪54 .................................................‬‬ ‫‪ 3.2.2‬אלגוריתם זיהוי תבנית ‪54 ...............................................................‬‬ ‫‪ 3.2.3‬מידידת ‪56 ............................................................................ DNA‬‬ ‫‪ 3.2.4‬המודל המתמטי‪56 ........................................................................‬‬ ‫‪ 3.2.5‬התאמה בלתי‪-‬ליניארית )‪57 ................................... (nonlinear fitting‬‬ ‫‪ 3.2.6‬אנליזה סטטיסטית ‪57 ....................................................................‬‬ ‫‪ 3.3‬תוצאות ‪59 ..............................................................................................‬‬ ‫‪ 3.3.1‬תבניות בקינטיקה הנגיפית ‪59 .........................................................‬‬ ‫‪ 3.3.2‬ביטול טענת קיום השפעה של ה‪60 .................................... placebo-‬‬ ‫‪ 3.3.3‬הגדרה של "ירידות ספונטאניות" ‪60 ................................................‬‬ ‫‪ 3.3.4‬תבניות קינטיות ‪61 ........................................................................‬‬ ‫‪ 3.3.5‬מתאמים בפרמטרים בסיסיים )‪62 ................... (baseline correlations‬‬ ‫‪ 3.3.6‬מידול של ‪63 ..........................................................................SPD‬‬ ‫‪ 3.3.7‬טבלאות גרפים ‪65 .........................................................................‬‬ ‫‪ 3.4‬סיכום‪71 .................................................................................................‬‬ ‫פרק ‪73 ............................................................................................................. 4‬‬ ‫טיפול במהלך הדבקה נגיפית כרונית של ‪Woodchuck Hepatitis Virus‬‬ ‫‪ 4.1‬מבוא‪74 ..................................................................................................‬‬

‫‪74 ............................................. Woodchuck Hepatitis Virus (WHV) 4.1.1‬‬ ‫‪75 ................................................................. Clevudine (L-FMAU) 4.1.2‬‬ ‫‪ 4.1.3‬הרלוונטיות של המודל האנימלי‪75 ..................................................‬‬

‫פרק ‪ - 4‬טיפול במהלך הדבקה נגיפית כרונית של ‪) Woodchuck Hepatitis Virus‬המשך(‬

‫‪ 4.2‬שיטות ‪77 ................................................................................................‬‬ ‫‪ 4.2.1‬בעלי החיים ‪77 .............................................................................‬‬ ‫‪ 4.2.2‬שיטת הטיפול ‪77 ..........................................................................‬‬ ‫‪ 4.2.4‬דיגום ‪77 .......................................................................................‬‬ ‫‪ 4.2.5‬מדידת ‪78 ............................................................................. DNA‬‬ ‫‪ 4.2.6‬מודל מתמטי ‪79 ............................................................................‬‬ ‫‪ 4.2.7‬מודל בקרה חיסונית‪80 ..................................................................‬‬ ‫‪ 4.2.8‬הערכת פרמטרים ‪81 .....................................................................‬‬ ‫‪ 4.2.7‬התאמה בלתי‪-‬ליניארית )‪81 ................................... (nonlinear fitting‬‬ ‫‪ 4.2.9‬אנליזה סטטיסטית ‪82 ....................................................................‬‬ ‫‪ 4.3‬תוצאות ‪83 ..............................................................................................‬‬ ‫‪ 4.3.1‬תבניות קינטיות של ‪83 ...........................................................WHV‬‬ ‫‪ 4.3.2‬ערכי הפרמטרים ‪84 ......................................................................‬‬ ‫‪ 4.3.3‬הדמיות‪87 ....................................................................................‬‬ ‫‪ 4.3.4‬טבלאות גרפים ‪88 .........................................................................‬‬ ‫‪ 4.3‬סיכום‪94 .................................................................................................‬‬ ‫פרק ‪96 ............................................................................................................. 5‬‬ ‫דיון‬

‫מקורות ‪102 .........................................................................................................‬‬ ‫תקציר באנגלית )‪i............................................................................... (Abstract‬‬

‫תודות‬

‫אבקש להודות בראש ובראשונה לפרופ' אבידן נוימן‪ ,‬אשר היווה עבורי מורה דרך במשך‬ ‫למעלה משש שנים‪ .‬שנים אלה‪ ,‬בהן זכיתי לחסות תחת כנפיו של אחד המומחים המובילים‬ ‫בדינאמיקה נגיפית‪ ,‬היו יוצאות מן הכלל‪ .‬עידודו וחיזוקו תרמו רבות להפיכת זמן העבודה‬ ‫באוניברסיטת בר‪-‬אילן למסע מדעי מרתק ביותר‪.‬‬ ‫תודותיי הרבות נתונות לשותפיי לעבודה באוניברסיטת בר‪-‬אילן‪ :‬דר' אשר עוזיאל‪ ,‬רחל‬ ‫לוי‪-‬דרומר‪ ,‬יונית הומבורגר‪ ,‬רונן טל‪ ,‬אסתר חגי‪ ,‬מנחם סקרלץ‪ ,‬ברוך אלבז‪ ,‬יפית‬ ‫מעיין ומאיה שמאילוב‪ .‬סיעור המוחות וההיזון החוזר אשר קיבלתי מהם התקבלו על ידי‬ ‫באהדה רבה‪.‬‬ ‫לא יכלה עבודה זו להיכתב ללא שיתוף הפעולה המלא של מדענים רבים מכל קצוות תבל‪.‬‬ ‫אני מבקש להודות במיוחד למשתתפים והחוקרים הרבים‪ ,‬אשר חברו יחדיו וסייעו ביצירה זו‪.‬‬ ‫תרומתו הגדולה במיוחד התקבלה מידי פרופ' ברנט קורבה בכך שסיפק את הנתונים על‬ ‫‪ WHV‬וכן ל‪ Gilead Sciences-‬אשר סיפקו את הנתונים אודות הדבקה בנגיף ה‪.HBV-‬‬ ‫מר ולד"ר רונית שריד על עידודן הבלתי פוסק‬ ‫אבקש להביע הערכתי לד"ר רמית ֲ‬ ‫והעצות היעילות שקיבלתי מהן‪.‬‬ ‫תודה לפרופ' משה קוה‪ ,‬ואיתו כל צוות האדמיניסטרציה‪ ,‬על תמיכתו המוחלטת בי‪ ,‬על‬ ‫שהעניק את מלגת הנשיא‪ ,‬לי ולסטודנטים רבים הנהנים מראייתו את החשיבות בקידום‬ ‫המדע‪.‬‬ ‫כמו כן אבקש להודות לחברי הפקולטה למדעי החיים וצוותה ובמיוחד לפרופ' צבי‬ ‫דובינסקי ופרופ' יוסף שטיינברגר‪ ,‬בתוך רבים אחרים אשר האירו את דרכי בהדרכתם‬ ‫וייעוצם‪ .‬יתרון גדול ראיתי בכך שהיה לי הכבוד להכיר אנשים וחוקרים אלה‪ ,‬אשר הותירו‬ ‫בי זכרון חזק וטוב מארבע השנים האחרונות‪.‬‬ ‫ובסוף‪ ,‬תודתי העמוקה ביותר נתונה לרעייתי רחל יחד עם אפרת‪ ,‬אביטל ומיכאל על‬ ‫גילוי ההבנה‪ ,‬האהבה והתמיכה‪.‬‬

‫עבודה זו בוצעה בהנחייתו של‬ ‫פרופ' אבידן נוימן‬ ‫הפקולטה למדעי החיים‬ ‫אוניברסיטת בר אילן‬

‫מידול השפעת הבקרה החיסונית על‬ ‫הדינמיקה הנגיפית במהלך הדבקה‬ ‫ראשונית‪ ,‬כרונית וטיפול‬

‫מאת‬ ‫דוד בורג‬ ‫הפקולטה למדעי החיים‬

‫מוגש לסנט של אוניברסיטת בר אילן‬ ‫רמת גן‪ ,‬ישראל‬

‫טבת תשס"ו‬