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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