July 30-August 2, 2008
Society for Mathematical Biology Conference

hosted by the Centre for Mathematical Medicine, Fields Institute
held at University of Toronto, Medical Sciences Bldg


1) Viral dynamics - Mathematical models of viral infection and treatment I and II (2 minisymposia)
Principal organiser: Dr. Helen Moore,
Modeling & Simulation Group, Genentech
Organized by Mark Dresser and Helen Moore

Although significant progress has been made in the treatment of viral infections, many infections remain difficult to treat. Determining optimal doses and regimens is a key component in success of existing and future therapies. Differential equation models of dynamics (in a single patient or in a population) can be used to better understand the life cycle of viruses, as well as host immune response. Such models also hold great promise in determining optimal doses, regimens, and strategies. This session is intended to provide an update on some of the latest and best mathematical models, with an emphasis on models tied to data, ideas that could be applied to other viral infections, and remaining unresolved modeling issues.

Audience: This mini-symposium is aimed at researchers and students in the field of viral dynamics, as well as those who would like an introduction to the area. The emphasis on models tied to data is intended to appeal to industry researchers.

Session I, 10:00 am - 12:00 noon, Thursday, July 31st

Alan Perelson, Los Alamos National Laboratories,
Modeling hepatitis C virus kinetics: PK/PD versus time-varying drug effectiveness models

The current approved therapy for hepatitis C virus (HCV) infection involves giving pegylated interferon (PEG-IFN) once weekly in combination with daily ribavirin. We have shown that the concentration of PEG-IFN 2b decreases towards the end of its weekly dosing interval and as a consequence the serum levels of HCV tend to increase in some patients. To account for this rebound in HCV levels we have developed a pharmacokinetic/pharmacodynamics (PK/PD model in which the effectiveness of PEG-IFN treatment is related to it serum concentration. In one set of patients in which simultaneous measurements of PEG-IFN and HCV concentrations were available this model was validated. However, in most clinical studies drug levels are not available at each time point. To handle cases in which PK is not known in each patient, we have developed a class of models called decreasing effectiveness models where we simply assume that drug effectiveness declines accordingly to a known function towards the end of the dosing interval. We shall show how models with either exponential or linear decreases in drug effectiveness perform as compared with a full PK/PD model on this one set of patients.

Robert Nachbar, Merck,
Modeling HCV drug pharmacokinetics and pharmacodynamics

Hepatitis C virus infection results in chronic disease of the liver, and as hepatocytes have a different life cycle than T-lymphocytes our model for HCV infection and treatment has important differences from earlier HCV models derived from those for HIV. We will briefly review our model and its relationship to a similar model published recently by Perelson, and then describe how our model has been coupled in a natural way to the typical compartmental pharmacokinetics model for drug treatment. We have also extended the model for use with combination therapy, and demonstrate its utility with data from in vitro and in vivo experiments.

Shasha Jumbe, Roche,
Population modelling of hepatitis C virus dynamics in 1773 CHC patients after long-term treatment with PEG-IFN alfa 2a and ribavirin

Mathematical models for hepatitis C viral (HCV) RNA kinetics have provided important insights into the life cycle of HCV and have increased the understanding of the mechanism of action of the current standard treatment of care: i.e. combination therapy of pegylated interferon (PEG-IFN) and ribavirin [1]. However, these models are unable to explain all of the observed long-term HCV RNA profiles under long-term treatment and after cessation of therapy [2]

1) To develop a HCV viral kinetic model describing the individual HCV RNA profiles in chronic hepatitis C (CHC) patients after a long-term treatment with PEG-IFN alfa 2a (Pegasys®) and ribavirin (Copegus®). 2) To undertake exploratory mechanistic simulations explaining phenomena such as break-through during therapy and relapse after discontinuation of therapy.

A total of 18937 HCV RNA concentration-time data were available from 1773 CHC patients who participated in clinical trials evaluating different 24 or 48-week dosing schemes of PEG-IFN alfa-2a as monotherapy or in combination with different doses of ribavirin. The original model of HCV infection and treatment based on the Lotka-Volterra principle [3], including three differential equations representing the populations of target cells (T), productively infected cells (I) and virus (V), was modified and extended (e.g. liver regeneration and viral extinction) to allow fitting long-term viral load data. The MATLAB® version of MONOLIX 2.3 was used in combination with user-defined functions in C++ solving the ODE's describing the kinetics of T, I and V. Finally, an extension of the SAEM algorithm was used to handle left censoring due to the lower limits of quantification of the HCV RNA levels [4].

The individual long-term HCV RNA versus time profiles were well described by the extended HCV viral kinetic model, with estimated free virus clearance rates and infected cells death rates similar to those previously found in the literature. The estimated effect of PEG-IFN alfa-2a was confirmed to be higher in HCV genotype non 1 patients as compared to patients infected with HCV genotype 1. The model provided a convincing picture of how ribavirin enhances the long-term outcome of interferon-based therapy. Analogous to HIV [5], exploratory mechanistic simulations revealed that the concept of the basic reproductive ratio is playing a major role in predicting the individual outcome in CHC patients.

Long term hepatitis C virus dynamics in 1773 CHC patients after a 24-48 week treatment with PEG-IFN alfa 2a and ribavirin was successfully modelled. Mechanistic simulations have provided additional insights into the understanding of the possible synergy between ribavirin and PEG-IFN and the factors explaining long-term individual outcomes in CHC patients which could assist in treatment decisions. The effect of other hepatitis C drugs with a new mechanism of action can be incorporated into the existing model allowing predictions for these other drugs or drug combinations to aid in optimizing the design of future clinical trials.

[1] Layden-Almer JE, Cotler SJ, Layden TJ. Viral kinetics in the treatment of chronic hepatitis C. Journal of Viral Hepatitis, 2006; 13: 499-504.
[2] Layden JE, Layden TJ. Viral kinetics of hepatitis C: new insights and remaining limitations. Hepatology, 2002; 35: 967-970.
[3] Neumann AU, LAM NP, Dahari H, Gretch DR, Wiley TE, Layden TJ, Perelson AS. Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy. Science, 1998; 282: 103-107.
[4] Samson A, Lavielle M, Mentré F. Extension of the SAEM algorithm to left-censored data in non-linear mixed-effects model: application to HIV dynamics model. Computational Statistics and Data Analysis, 2006; 51: 1562-1574.
[5] Jacqmin P, McFadyen L, Wade JR. Basic PK/PD principles of proliferative and circular systems. PAGE, 2007; Abstr 1194.

David Schley, Institute for Animal Health (UK)
Mathematical modelling of foot-and-mouth disease virus - epithelium dynamics to identify the determinants of lytic behaviour.
Coauthors: John Ward, Loughborough University (UK) Zhidong Zhang, Institute for Animal Health (UK)

Foot-and-mouth disease virus (FMDV) causes an economically important disease of cloven-hoofed livestock. The virus primarily infects epithelium: on the skin around the feet and tongue the virus rapidly replicates, killing the cell and resulting in growing lesions. Eventually the immune response tends to clear the virus from the system and these symptoms gradually disappear [1]. In the soft palate, however, lesions do not occur and the virus can persist inside cells long after the animal has recovered [2]: this has implications for the control of the disease, especially if vaccination is used in an outbreak, while a better understanding of why this occurs would also contribute to the fight against foot-and-mouth disease (FMD) more generally. To help identify which of the differences between oral and pharyngeal epithelia are responsible for such dramatically divergent FMDV dynamics a simple models [3] has been developed. Virus concentration is made explicit to allow the lytic behaviour of cells to be fully considered. Initial results suggest that localised structuring of what are fundamentally similar cells may induce bifurcation, although analysis is complicated by the fact that quantitative changes (prior to the immune response) alone may be sufficient to generate qualitative changes in outcome. Results for the biologically relevant parameter space of the system are presented, based upon clinical and experimental data, together with an indication of novel experiments which the modelling work has instigated. The extension of the system to an age-structured model of cells, considered necessary to identify more subtle factors (that could then be tested experimentally), is work in progress.

[1] Alexandersen et al. (2003) J Comp Pathol 129, 1-36.
[2] Ward et al. (2007) in Final Report of the 7th Mathematics in Medicine Study Group, Southampton (UK).
[3] Zhang & Alexandersen (2004) J Gen Virol 85, 2567-2575.

Session II, 10:00 am - 12:00 noon, Friday, August 1st

Bambang Adiwijaya, Vertex Pharmaceuticals,
A novel, multi-variant, viral dynamic model of genotype 1 HCV to assess the evolution of protease-inhibitor resistant variants

Hepatitis C virus (HCV) genotype 1 variants resistant to protease inhibitors have been observed in clinical trials. The in vivo fitness of these variants was estimated from clinical trial data of subjects dosed in monotherapy with the protease inhibitor telaprevir.

Avidan Neumann, Bar-Ilan University,
Multi-level models of viral dynamics and evolution combining intra-cellular replication with cell infection - modeling the response to the novel generation of direct anti-HCV therapy
Coauthor: J. Guedj

The current paradigm for modeling viral dynamics and the development of viral resistance, based on the HIV experience, considers the dynamics of the circulating virus and the cellular infection levels only (CI model). While this may be accurate enough approximation for HIV, a retrovirus RNA virus for which mutation occurs mainly at the RT step, it is known that for HCV all processes of resistance evolution - mutation, selection and amplification - can occur on a faster time-scale of RNA synthesis at the intra-cellular level. Here we explore, with a novel mathematical model (IC+CI model) that considers a multi-level dynamical processes, on both intra-cellular level replication and evolution dynamics and cellular infection level viral dynamics, the clinical implication of intra-cellular resistance evolution for direct anti-HCV drugs.
In the model, intra-cellular RNA (ICR) is used to form replication units (RU), which in turn synthesize more ICR that is partially packaged and secreted as virions. Direct anti-HCV drugs can have an anti-viral effect through blocking of RU formation, ICR synthesis and/or virion export. The development of resistance is modeled in the intra-cellular level by the evolution and dynamics of different strains of RU and ICR with different drug-sensitivity and different relative-fitness. Resistance evolution also impacts the cellular infection level as result of the exported virus of different strains.

We have already shown that the IC+CI model gives rise to more rich viral dynamics scenarios than the CI model. In particular, a critical threshold of anti-viral effectiveness is predicted above which intra-cellular clearance of RU becomes the dominant viral dynamics process and viral decline is then governed by the rate of RU clearance (gamma). On the other, below that critical effectiveness threshold, as is probably the case for the standard IFN-a based therapy, viral decline is governed by the slower loss rate of infected cells (delta).
Furthermore, evolution of resistant virus is faster when it occurs at the intra-cellular level as compared to occurring only at the cellular infection level, assuming similar rates of mutation, and similar distribution of sensitivity to drug and relative fitness of the different strains in both models. The combination of resistance evolution on the intra-cellular level with cellular infection viral dynamics level allows for more complex viral kinetic patterns than possible with resistance evolution on the cellular infection level only. In general, a complete rebound, a tri-phasic decline, a bi-phasic decline with a shoulder, a decline at the delta mode (with the rate of infected cell loss) or a decline at the gamma mode (with the faster rate of intra-cellular RU loss) are all possible as function of the mutation rate and the distribution of fitness and sensitivity.

More rapid and more complex patterns of viral resistance evolution are predicted when considering HCV resistance evolution at the intra-cellular level together with cellular level viral dynamics. Some of the patterns predicted by the model were already observed in data publicly released for different direct anti-HCV drugs. The model makes several predictions with important clinical implications, such as the possibility for decline with either the rapid gamma-rate or the slower delta-rate as a function of the effectiveness, drug sensitivity and relative fitness parameters.

Abba Gumel, University of Manitoba,
Mathematical study of the transmission dynamics and control of HIV/TB co-infection

The talk addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic mathematical model, which incorporates many of the essential biological and epidemiological features of the two diseases. In addition to analysing the qualitative dynamics of the model, various targeted treatment strategies for the two diseases (e.g., treating only HIV or TB or both) will be explored.
It will be shown, for instance, that the effectiveness of the targeted use of antiretroviral drugs (to treat those with or without AIDS symptoms) is dependent on whether or not it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold.

Grace Kepler, North Carolina State University,
Modeling CMV infection in transplant patients

HCMV infection is a signifcant health threat to immunosuppressed patients. Patient health outcome could be improved with suitable mathematical modeling that could predict the disease course in individuals and one that could suggest optimized treatment strategies. We illustrate our approach to within-patient modeling and prediction with similar HIV-1 modeling work by this group. We then present an initial model for HCMV infection in healthy and immunosuppressed patients and show how this model can begin to provide a window into the dynamics of HCMV infection in transplant patients.



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