An introduction to lymphocyte and viral dynamics: the power and limitations of mathematical analysis

Modeling inoculum dose dependent patterns of acute virus infections

Inoculum dose, i.e. the number of pathogens at the beginning of an infection, often affects key aspects of pathogen and immune response dynamics. These in turn determine clinically relevant outcomes, such as morbidity and mortality. Despite the general recognition that inoculum dose is an important component of infection outcomes, we currently do not understand its impact in much detail. This study is intended to start filling this knowledge gap by analyzing inoculum dependent patterns of viral load dynamics in acute infections. Using experimental data for adenovirus and infectious bronchitis virus infections as examples, we demonstrate inoculum dose dependent patterns of virus dynamics. We analyze the data with the help of mathematical models to investigate what mechanisms can reproduce the patterns observed in experimental data. We find that models including components of both the innate and adaptive immune response are needed to reproduce the patterns found in the data. We further analyze which types of innate or adaptive immune response models agree with observed data. One interesting finding is that only models for the adaptive immune response that contain growth terms partially independent of viral load can properly reproduce observed patterns. This agrees with the idea that an antigen-independent, programmed response is part of the adaptive response. Our analysis provides useful insights into the types of model structures that are required to properly reproduce observed virus dynamics for varying inoculum doses. We suggest that such models should be taken as basis for future models of acute viral infections.

The cycle of EBV infection explains persistence, the sizes of the infected cell populations and which come under CTL regulation

Previous analysis of Epstein-Barr virus (EBV) persistent infection has involved biological and immunological studies to identify and quantify infected cell populations and the immune response to them. This led to a biological model whereby EBV infects and activates naive B-cells, which then transit through the germinal center to become resting memory B-cells where the virus resides quiescently. Occasionally the virus reactivates from these memory cells to produce infectious virions. Some of this virus infects new naive B-cells, completing a cycle of infection. What has been lacking is an understanding of the dynamic interactions between these components and how their regulation by the immune response produces the observed pattern of viral persistence. We have recently provided a mathematical analysis of a pathogen which, like EBV, has a cycle of infected stages. In this paper we have developed biologically credible values for all of the parameters governing this model and show that with these values, it successfully recapitulates persistent EBV infection with remarkable accuracy. This includes correctly predicting the observed patterns of cytotoxic T-cell regulation (which and by how much each infected population is regulated by the immune response) and the size of the infected germinal center and memory populations. Furthermore, we find that viral quiescence in the memory compartment dictates the pattern of regulation but is not required for persistence; it is the cycle of infection that explains persistence and provides the stability that allows EBV to persist at extremely low levels. This shifts the focus away from a single infected stage, the memory B-cell, to the whole cycle of infection. We conclude that the mathematical description of the biological model of EBV persistence provides a sound basis for quantitative analysis of viral persistence and provides testable predictions about the nature of EBV-associated diseases and how to curb or prevent them.

Epstein-Barr virus (EBV) is a herpesvirus that establishes a lifelong persistent infection in virtually all human beings. This infection is a risk factor for the subsequent development of certain tumors and possibly also autoimmune diseases. In order to understand the origin of these diseases, it is necessary to first understand how EBV maintains persistent infection. We have used mathematical analysis to study this question. We find that the characteristic cycle of infected stages that EBV establishes in vivo allows it to persist stably at extremely low levels. This represents a consistent mathematical description of EBV infection and allows us to describe what must change to convert benign infection into pathogenic infection, as well as what kind of efficacy drugs and vaccines must have in order to be useful.

Quantifying T lymphocyte turnover

Peripheral T cell populations are maintained by production of naive T cells in the thymus, clonal expansion of activated cells, cellular self-renewal (or homeostatic proliferation), and density dependent cell life spans. A variety of experimental techniques have been employed to quantify the relative contributions of these processes. In modern studies lymphocytes are typically labeled with 5-bromo-2'-deoxyuridine (BrdU), deuterium, or the fluorescent dye carboxy-fluorescein diacetate succinimidyl ester (CFSE), their division history has been studied by monitoring telomere shortening and the dilution of T cell receptor excision circles (TRECs) or the dye CFSE, and clonal expansion has been documented by recording changes in the population densities of antigen specific cells. Proper interpretation of such data in terms of the underlying rates of T cell production, division, and death has proven to be notoriously difficult and involves mathematical modeling. We review the various models that have been developed for each of these techniques, discuss which models seem most appropriate for what type of data, reveal open problems that require better models, and pinpoint how the assumptions underlying a mathematical model may influence the interpretation of data. Elaborating various successful cases where modeling has delivered new insights in T cell population dynamics, this review provides quantitative estimates of several processes involved in the maintenance of naive and memory, CD4(+) and CD8(+) T cell pools in mice and men.

Numerical solutions for a model of tissue invasion and migration of tumour cells

The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points. Our numerical experiments illustrate the metastatic ability of tumour cells.

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.

How does HTLV-I persist despite a strong cell-mediated immune response?

Human T-lymphotropic virus type 1 (HTLV-1) is a pathogenic retrovirus that infects human CD4(+) T lymphocytes. Despite its presence in T cells, HTLV-1 causes little overt immunosuppression. This host-virus relationship has therefore been exploited as an excellent model system for studying the dynamic interaction between a persistent retrovirus and the normal human immune system. We use a combination of mathematical and experimental techniques to identify key factors on both sides of the in vivo host-virus interaction that significantly determine HTLV-I proviral load and disease risk. We develop a model to describe how these factors interact to enable viral persistence.

The evolution of mathematical immunology

The types of mathematical models used in immunology and their scope have changed drastically in the past 10 years. Classical models were based on ordinary differential equations (ODEs), difference equations, and cellular automata. These models focused on the 'simple' dynamics obtained between a small number of reagent types (e.g. one type of receptor and one type of antigen or two T-cell populations). With the advent of high-throughput methods, genomic data, and unlimited computing power, immunological modeling shifted toward the informatics side. Many current applications of mathematical models in immunology are now focused around the concepts of high-throughput measurements and system immunology (immunomics), as well as the bioinformatics analysis of molecular immunology. The types of models have shifted from mainly ODEs of simple systems to the extensive use of Monte Carlo simulations. The transition to a more molecular and more computer-based attitude is similar to the one occurring over all the fields of complex systems analysis. An interesting additional aspect in theoretical immunology is the transition from an extreme focus on the adaptive immune system (that was considered more interesting from a theoretical point of view) to a more balanced focus taking into account the innate immune system also. We here review the origin and evolution of mathematical modeling in immunology and the contribution of such models to many important immunological concepts.

Determining thymic output quantitatively: using models to interpret experimental T-cell receptor excision circle (TREC) data

T cells develop in the thymus and then are exported to the periphery. As one ages, the lymphoid mass of the thymus decreases, and a concomitant decrease in the ability to produce new T cells results. Human immunodeficiency virus (HIV) infects CD4(+) T cells and, hence, can also affect thymic function. Here we discuss experimental techniques and mathematical models that aim to quantify the rate of thymic export. We focus on a recent technique involving the quantification of T-cell receptor excision circles (TRECs). We discuss how proper interpretation of TREC data necessitates the critical development of appropriate mathematical models. We review the theory for interpretation of TREC data during aging, HIV infection, and anti-retroviral treatment. Also, we show how TRECs can be used to accurately quantify thymic output in the context of thymectomy experiments. We show that mathematical models are not only useful but absolutely necessary for these analyses. As such, they should be taken as just another tool in the immunologist's arsenal.

Human T-lymphotropic virus 1: recent knowledge about an ancient infection

Human T-lymphotropic virus 1 (HTLV-1) has infected human beings for thousands of years, but knowledge about the infection and its pathogenesis is only recently emerging. The virus can be transmitted from mother to child, through sexual contact, and through contaminated blood products. There are areas in Japan, sub-Saharan Africa, the Caribbean, and South America where more than 1% of the general population is infected. Although the majority of HTLV-1 carriers remain asymptomatic, the virus is associated with severe diseases that can be subdivided into three categories: neoplastic diseases (adult T-cell leukaemia/lymphoma), inflammatory syndromes (HTLV-1-associated myelopathy/tropical spastic paraparesis and uveitis among others), and opportunistic infections (including Strongyloides stercoralis hyperinfection and others). The understanding of the interaction between virus and host response has improved markedly, but there are still no clear surrogate markers for prognosis and there are few treatment options.

Quantification of T-cell dynamics: from telomeres to DNA labeling

Immunology has traditionally been a qualitative science describing the cellular and molecular components of the immune system and their functions. Only quite recently have new experimental techniques paved the way for a more quantitative approach of immunology. Lymphocyte telomere lengths have been measured to get insights into the proliferation rate of different lymphocyte subsets, T-cell receptor excision circles have been used to quantify the daily output of new T cells from the thymus, and bromodeoxyuridine and stable isotope labeling have been applied to measure proliferation and death rates of naive and memory lymphocytes. A common problem of the above techniques is the translation of the resulting data into relevant parameters, such as the typical division and death rate of the different lymphocyte populations. Theoretical immunology has contributed significantly to the interpretation of such quantitative experimental data, thereby resolving diverse controversies and, most importantly, has suggested novel experiments, allowing for more conclusive and quantitative interpretations. In this article, we review a variety of different models that have been used to interpret data on lymphocyte kinetics in healthy human subjects and discuss their contributions and limitations.

A dynamical model of human immune response to influenza A virus infection

We present a simplified dynamical model of immune response to uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Innate immunity is represented by interferon-induced resistance to infection of respiratory epithelial cells and by removal of infected cells by effector cells (cytotoxic T-cells and natural killer cells). Adaptive immunity is represented by virus-specific antibodies. Similar in spirit to the recent model of Bocharov and Romanyukha [1994. Mathematical model of antiviral immune response. III. Influenza A virus infection. J. Theor. Biol. 167, 323-360], the model is constructed as a system of 10 ordinary differential equations with 27 parameters characterizing the rates of various processes contributing to the course of disease. The parameters are derived from published experimental data or estimated so as to reproduce available data about the time course of IAV infection in a naïve host. We explore the effect of initial viral load on the severity and duration of the disease, construct a phase diagram that sheds insight into the dynamics of the disease, and perform sensitivity analysis on the model parameters to explore which ones influence the most the onset, duration and severity of infection. To account for the variability and speed of adaptation of the adaptive response to a particular virus strain, we introduce a variable that quantifies the antigenic compatibility between the virus and the antibodies currently produced by the organism. We find that for small initial viral load the disease progresses through an asymptomatic course, for intermediate value it takes a typical course with constant duration and severity of infection but variable onset, and for large initial viral load the disease becomes severe. This behavior is robust to a wide range of parameter values. The absence of antibody response leads to recurrence of disease and appearance of a chronic state with nontrivial constant viral load.

Cellular immune response to HTLV-1

There is strong evidence at the individual level and the population level that an efficient cytotoxic T lymphocyte (CTL) response to HTLV-1 limits the proviral load and the risk of associated inflammatory diseases such as HAM/TSP. This evidence comes from host population genetics, viral genetics, DNA expression microarrays and assays of lymphocyte function. However, until now there has been no satisfactory and rigorous means to define or to measure the efficiency of an antiviral CTL response. Recently, methods have been developed to quantify lymphocyte turnover rates in vivo and the efficiency of anti-HTLV-1 CTLs ex vivo. Data from these new techniques appear to substantiate the conclusion that variation between individual hosts in the rate at which a single CTL kills HTLV-1-infected lymphocytes is an important determinant, perhaps the decisive determinant, of the proviral load and the risk of HAM/TSP. With these experimental data, it is becoming possible to refine, parameterize and test mathematical models of the immune control of HTLV-1, which are a necessary part of an understanding of this complex dynamic system.

Cell-mediated immune response to human T-lymphotropic virus type I

Human T-lymphotropic virus type I (HTLV-I) is a retrovirus that causes persistent infection in many populations in tropical and subtropical regions. HTLV-I chronically activates the cell-mediated arm of the host adaptive immune response. There has been much debate about the role of the immune response in determining the outcome of HTLV-I infection: most seropositive individuals remain lifelong asymptomatic carriers of the virus, whereas a small proportion-usually those with higher equilibrium proviral loads-develop an inflammatory disease of the central nervous system known as HAM/TSP. Here we discuss the cell-mediated immune response to HTLV-I infection. We summarize recent data on the HTLV-I-specific CD4(+) cell response and explore its potential role in HAM/TSP pathogenesis. We also explore the controversy surrounding the role of the CD8(+) cell response in controlling HTLV-I infection and/or contributing to HAM/TSP disease, highlighting recent studies of T cell gene expression profiles and a newly developed assay of CD8(+) cell functional efficiency. Finally, we introduce a possible role for cellular innate immune effectors in HTLV-I infection.

Quantifying cell turnover using CFSE data

The CFSE dye dilution assay is widely used to determine the number of divisions a given CFSE labelled cell has undergone in vitro and in vivo. In this paper, we consider how the data obtained with the use of CFSE (CFSE data) can be used to estimate the parameters determining cell division and death. For a homogeneous cell population (i.e., a population with the parameters for cell division and death being independent of time and the number of divisions cells have undergone), we consider a specific biologically based "Smith-Martin" model of cell turnover and analyze three different techniques for estimation of its parameters: direct fitting, indirect fitting and rescaling method. We find that using only CFSE data, the duration of the division phase (i.e., approximately the S+G2+M phase of the cell cycle) can be estimated with the use of either technique. In some cases, the average division or cell cycle time can be estimated using the direct fitting of the model solution to the data or by using the Gett-Hodgkin method [Gett A. and Hodgkin, P. 2000. A cellular calculus for signal integration by T cells. Nat. Immunol. 1:239-244]. Estimation of the death rates during commitment to division (i.e., approximately the G1 phase of the cell cycle) and during the division phase may not be feasible with the use of only CFSE data. We propose that measuring an additional parameter, the fraction of cells in division, may allow estimation of all model parameters including the death rates during different stages of the cell cycle.

Human T-lymphotropic virus type 1 (HTLV-1): persistence and immune control

The human retrovirus human T-lymphotropic virus type 1 (HTLV-1) is associated with two distinct types of disease: the malignancy known as adult T-cell leukemia and a range of chronic inflammatory conditions including the central nervous system disease HTLV-1-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Until recently, it was believed that HTLV-1 was largely latent in vivo. However, evidence from a number of types of experiments shows that HTLV-1 persistently expresses its genes, and that the "set point" of an individual's proviral load of HTLV-1 is mainly determined by the efficiency of that individual's cellular immune response to the virus. These conclusions have two main consequences. First, HTLV-1 may be vulnerable to antiretroviral drug therapy or immunotherapy. Second, HTLV-1 infection has become a useful system to analyze the determinants of the efficiency of the antiviral immune response.