Mathematical model of antiviral immune response. II. Parameters identification for acute viral hepatitis B

Examining the cooperativity mode of antibody and CD8+ T cell immune responses for vaccinology

A fundamental unsolved issue in vaccine design is how neutralizing antibodies and cytotoxic CD8⁺ T cells cooperate numerically in controlling virus infections. We hypothesize on a viewpoint for the multiplicative cooperativity between neutralizing antibodies and CD8⁺ T cells and propose how this might be exploited for improving vaccine-induced protective immunity.

MATHEMATICAL MODELLING OF IMMUNE PROCESSES AND ITS APPLICATION

The aim of the study was to develop a mathematical model to research hypoxic states in case of simulation of an organism infectious lesions. The model is based on the methods of mathematical modeling and the theory of optimal control of moving objects. The processes of organism damage are simulated with the mathematical model of immune response developed by G.I. Marchuk and the members of his scientific school, adapted to current conditions. This model is based on Burnet’s clone selection theory of the determining role of antigen. Simulation results using the model are presented. The dependencies of infectious courses on the volumetric velocity of systemic blood flow is analyzed on the complex mathematical model of immune response, respiratory and blood circulation systems. The immune system is shown to be rather sensitive to the changes in blood flow via capillaries. Thus, the organ blood flows can be used as parameters for the model by which the respiratory, immune response, and blood circulation systems interact and interplay.

Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation

We study the global stability of a class of models for in-vivo virus dynamics that take into account the Cytotoxic T Lymphocyte immune response and display antigenic variation. This class includes a number of models that have been extensively used to model HIV dynamics. We show that models in this class are globally asymptotically stable, under mild hypothesis, by using appropriate Lyapunov functions. We also characterise the stable equilibrium points for the entire biologically relevant parameter range. As a by-product, we are able to determine what is the diversity of the persistent strains.

A probability cellular automaton model for hepatitis B viral infections

The existing models of hepatitis B virus (HBV) infection dynamics are based on the assumption that the populations of viruses and cells are uniformly mixed. However, the real virus infection system is actually not homogeneous and some spatial factors might play a nontrivial role in governing the development of HBV infection and its outcome. For instance, the localized populations of dead cells might adversely affect the spread of infection. To consider this kind of inhomogeneous feature, a simple 2D (dimensional) probability Cellular Automaton model was introduced to study the dynamic process of HBV infection. The model took into account the existence of different types of HBV infectious and non-infectious particles. The simulation results thus obtained showed that the Cellular Automaton model could successfully account for some important features of the disease, such as its wide variety in manifestation and its age dependency. Meanwhile, the effects of the model's parameters on the dynamical process of the infection were also investigated. It is anticipated that the Cellular Automaton model may be extended to serve as a useful vehicle for studying, among many other complicated dynamic biological systems, various persistent infections with replicating parasites.

A kinetic model of telomere shortening in infants and adults

We have previously demonstrated that telomeres shorten more rapidly in peripheral mononuclear cells (PBMC) of infants than in adults (Zeichner et al., Blood 93 (1999) 2824). Here we describe a mathematical model that allows quantification of telomere dynamics both in infants and in adults. In this model the dependence of the telomere dynamics on age is accounted by assuming proportionality between the body growth, as approximated by the Gompertz equation, and the increase in the number of PBMCs. The model also assumes the existence of two subpopulations of PBMC with significantly different rates of division. This assumption is based on the results from a previous analysis of in vitro data for telomere dynamics in presence of telomerase inhibitors and our recent data obtained by measurements of BrdU incorporation in T lymphocytes in humans (Kovacs et al., J. Exp. Med. 194 (2001) 1731). The average telomere length of PBMC was calculated as the average length of these two subpopulations. The model fitted our experimental data well and allowed to derive a characteristic time of conversion of the rapidly proliferating cells to slowly proliferating cells on the order of 20 days. The half-life of the slowly proliferating cells was estimated to be about 6 months, which is in good agreement with data obtained by independent methodologies. Comparison of the one-population and two-subpopulations models demonstrated that one population model cannot explain the observed parameters of the terminal restriction fragment (TRF) dynamics while two-subpopulations model does. These results suggest that the rapid telomere shortening in infants is largely determined by the faster PBMC turnover compared to adults. This may have major implications for elucidation of the HIV pathogenesis in infants. One can speculate that the more rapid course of the HIV disease in infants is due to the existence of rapidly dividing cells, which are susceptible to HIV infection. In addition, these results could have implications for understanding of mechanisms of aging.

Dynamics of hepatitis B virus infection

Mathematical models of the dynamics of HIV and hepatitis C virus infection have proven to be of great utility in understanding pathogenesis and designing better treatments. Here, we review the state of the art in modeling and interpreting data obtained from hepatitis B virus infected patients treated with antiviral agents.

Cancer cell dynamics in presence of telomerase inhibitors: analysis of in vitro data

The inhibition of telomerase activity in actively dividing cells leads to suppression of cell growth after a time delay (inhibitory delay) required to reach a threshold telomeric DNA size. We developed a mathematical model of the dynamics of telomere size distribution and cell growth in the presence of telomere inhibitors that allowed quantification of the inhibitory delay. The model based on the solution of a system of differential equations described quantitatively recent experimental data on dynamics of cultured cells in presence of telomerase inhibitors. The analysis of the data by this model suggested the existence of at least two distinct subpopulations of cells with different proliferative activity. Size distribution of telomeres, fraction of proliferating cells, and tumor doubling times are of critical importance for the dynamics of cancer cells growth in presence of telomerase inhibitors. Rapidly growing cells with large telomeric DNA heterogeneity and small proliferating fractions as well as those with very short homogeneous telomeres would be the most sensitive to telomerase inhibitors.

Mathematical modeling of citric acid production by repeated batch culture

A mathematical model has been created for the process of citric acid biosynthesis by yeast (mutant strain Yarrowia lipolytica) cultivated by the repeated batch (RB) method on ethanol under conditions of nitrogen limitation. The model accounts for cell growth as a function of nitrogen concentration in the culture liquid; nitrogen uptake by growing cells; citric acid production; pH control in the fermentor by means of NaOH addition; and changes in system volume. The model represents a system of five nonlinear differential equations. Experimental measurements of cell concentration, citric acid concentration, and cultivation broth volume were used with the least squares method to determine the values of eight model parameters. The parameter values obtained were consistent with literature data and general concepts of cell growth and citric acid biosynthesis. The model has been used to predict optimum RB culture conditions.

Resource competition as a mechanism for B cell homeostasis

Cellular competition for survival signals offers a cogent and appealing mechanism for the maintenance of cellular homeostasis [Raff, M. C. (1992) Nature (London) 356, 397-400]. We present a theoretical and experimental investigation of the role of competition for resources in the regulation of peripheral B cell numbers. We use formal ecological competition theory, mathematical models of interspecific competition, and competitive repopulation experiments to show that B cells must compete to persist in the periphery and that antigen forms a part of the resources over which B cells compete.

Mathematical modeling of T-cell proliferation

A mathematical model of the T-lymphocyte proliferation process (in vivo and in vitro) is presented. This model takes into account cell-cycle progression and the regulation by lymphokines (lymphocyte activating factor interleukin 1 and T-cell growth factor interleukin 2). Using data on the generalized picture of the short-term course of viral hepatitis B, the parameter estimation procedure is carried out. The possibility of immunocorrection (by means of injection of a pharmacologic dose of IL-2) during the immune response to viral hepatitis B with T-lymphocyte deficiency is shown.

Analysis of a cellular model to account for the natural history of infection by the hepatitis B virus and its role in the development of primary hepatocellular carcinoma

Infection with the hepatitis B virus (HBV) can have many different outcomes. Transient infection may result in acute hepatitis or may remain subclinical. Persistent infection may also be subclinical, or may involve chronic active hepatitis, and can finally lead to the development of primary hepatocellular carcinoma. A mathematical model is given to account for the many different outcomes of HBV pathogenesis. The model is based on the assumption that the liver contains two cell populations with differing abilities to support active HBV replication and/or viral integration into the genome. The model helps account for the relationship of the different clinical courses of HBV infection to the age when the disease is acquired, together with the state of the immune system of the patient.

Mathematical model of antiviral immune response. I. Data analysis, generalized picture construction and parameters evaluation for hepatitis B

The present approach to the mathematical modelling of infectious diseases is based upon the idea that specific immune mechanisms play a leading role in development, course, and outcome of infectious disease. The model describing the reaction of the immune system to infectious agent invasion is constructed on the bases of Burnet's clonal selection theory and the co-recognition principle. The mathematical model of antiviral immune response is formulated by a system of ten non-linear delay-differential equations. The delayed argument terms in the right-hand part are used for the description of lymphocyte division, multiplication and differentiation processes into effector cells. The analysis of clinical and experimental data allows one to construct the generalized picture of the acute form of viral hepatitis B. The concept of the generalized picture includes a quantitative description of dynamics of the principal immunological, virological and clinical characteristics of the disease. Data of immunological experiments in vitro and experiments on animals are used to obtain estimates of permissible values of model parameters. This analysis forms the bases for the solution of the parameter identification problem for the mathematical model of antiviral immune response which will be the topic of the following paper (Marchuk et al., 1991, J. theor. Biol. 15).