Models of Immune Systems: The Use of Differential Equations

(see also
Burrascano's Guidelines and Immune Response Modeling,
Mathematical Immune Response Models)


A Medline Literature Survey
Date: 13. April 2000

Joachim Gruber

Table of Contents
(Note: only some of the papers in this file have been linked to the Table of Contents)


 Health Phys 2000 Mar;78(3):289-94

A stochastic model of radiation-induced bone marrow damage.

Cotlet G, Blue TE

The Ohio State University, Nuclear Engineering Program, Columbus 43210, USA.

A stochastic model, based on consensus principles from radiation biology, is used to estimate bone-marrow stem cell pool survival (CFU-S and stroma cells) after irradiation. The dose response model consists of three coupled first order linear differential equations which quantitatively describe time dependent cellular damage, repair, and killing of red bone marrow cells. This system of differential equations is solved analytically through the use of a matrix approach for continuous and fractionated irradiations. The analytic solutions are confirmed through the dynamical solution of the model equations using SIMULINK. Rate coefficients describing the cellular processes of radiation damage and repair, extrapolated to humans from animal data sets and adjusted for neutron-gamma mixed fields, are employed in a SIMULINK analysis of criticality accidents. The results show that, for the time structures which may occur in criticality accidents, cell survival is established mainly by the average dose and dose rate.

PMID: 10688451, UI: 20151159


IEEE Trans Med Imaging 1999 Nov;18(11):1108-14

Microwave imaging using the finite-element method and a sensitivity analysis approach.

Rekanos IT, Panas SM, Tsiboukis TD

A method for reconstructing the constitutive parameters of two-dimensional (2-D) penetrable scatterers from scattered field measurements is presented. This method is based on the differential formulation of the forward scattering problem, which is solved by applying the finite-element method (FEM). Given a set of scattered field measurements, the objective is to minimize a cost function which consists of two terms. The first is the standard error term, which is related to the measurements and their estimates, while the second term, which is related to the Tikhonov regularization, is used to heal the ill posedness of the inverse problem. The iterative Polak-Ribiere nonlinear conjugate gradient algorithm is applied to the minimization of the cost function. During each iteration of the algorithm, the direction of correction is computed by using a sensitivity analysis approach, which is carried out by an elaborate finite-element scheme. The adoption of the finite-element method results in sparse systems of equations, while the computational burden is further reduced by applying the adjoint state vector methodology. Finally, a microwave medical imaging application, which is related to the detection of proliferated bone marrow, is examined, while the robustness of the proposed technique in the presence of noise and for different regularization levels is investigated.

Publication Types: Letter

PMID: 10661328, UI: 20124944


Theor Popul Biol 1999 Feb;55(1):94-109

Dynamics of co-infection with M. Tuberculosis and HIV-1.

Kirschner D

Department of Microbiology and Immunology, The University of Michigan Medical School, 6730 Medical Science Building II, Ann Arbor, Michigan, 48109-0620, USA.

Since 1985, there has been a renewed epidemic of tuberculosis (TB) that was previously thought to be in check. There is evidence to believe the main factor for this resurgence has been the human immunodeficiency virus (HIV). Co-infection with HIV and M. Tuberculosis has profound implications for the course of both diseases. This study represents a first attempt to understand how the introduction of an opportunistic infection, namely Mycobacterium tuberculosis, the bacteria that causes TB, affects the dynamic interaction of HIV-1 and the immune system. We create a mathematical model using ordinary differential equations to describe the interaction of HIV and TB with the immune system. It is known that infection with TB can decrease the CD4(+) T cell counts-a key marker of AIDS progression; thus, it shortens survival in HIV infected individuals. Another main marker for HIV progression is the viral load. If this load is increased due to the presence of opportunistic infections, the disease progression is much more rapid. We also explore the effects of drug treatment on the TB infection in the doubly-infected patient. Copyright 1999 Academic Press.

PMID: 9925811, UI: 99126627


Mech Ageing Dev 1998 Nov 16;105(3):241-64

Dynamic phenotypic restructuring of the CD4 and CD8 T-cell subsets with age in healthy humans: a compartmental model analysis.

Jackola DR, Hallgren HM

Department of Medicine, University of Minnesota Medical School, Minneapolis 55455, USA.

In healthy humans, phenotypic restructuring occurs with age within the CD3+ T-lymphocyte complement. This is characterized by a non-linear decrease of the percentage of 'naive' (CD45RA+) cells and a corresponding non-linear increase of the percentage of 'memory' (CD45R0+) cells among both the CD4+ and CD8+ T-cell subsets. We devised a simple compartmental model to study the age-dependent kinetics of phenotypic restructuring. We also derived differential equations whose parameters determined yearly gains minus losses of the percentage and absolute numbers of circulating naive cells, yearly gains minus losses of the percentage and absolute numbers of circulating memory cells, and the yearly rate of conversion of naive to memory cells. Solutions of these evaluative differential equations demonstrate the following: the memory cell complement 'resides' within its compartment for a longer time than the naive cell complement within its compartment for both CD4 and CD8 cells; (2) the average, annual 'turnover rate' is the same for CD4 and CD8 naive cells. In contrast, the average, annual 'turnover rate' for memory CD8 cells is 1.5 times that of memory CD4 cells; (3) the average, annual conversion rate of CD4 naive cells to memory cells is twice that of the CD8 conversion rate; (4) a transition in dynamic restructuring occurs during the third decade of life that is due to these differences in turnover and conversion rates, between and from naive to memory cells.

PMID: 9862233, UI: 99077049


J Theor Biol 1998 Nov 7;195(1):41-52

Mathematical model of a virus-neutralizing immunglobulin response.

Funk GA, Barbour AD, Hengartner H, Kalinke U

Mathematical Institute, University of Zurich, Winterthurerstrasse 190, Zurich, 8057, Switzerland.

We present a mathematical model to simulate the kinetics of B-cell activation and the virus-neutralizing immunoglobulin response in the spleen of mice after infection with vesicular stomatitis virus (VSV). Our model combines data from in vitro experiments and in vivo kinetic observations.

RESULTS: Copyright 1998 Academic Press.

PMID: 9802949, UI: 99021701


IMA J Math Appl Med Biol 1998 Sep;15(3):235-56

The effect of chemotaxis and chemokinesis on leukocyte locomotion: a new interpretation of experimental results.

Byrne HM, Cave G, McElwain DL

Department of Mathematics, University of Manchester Institute of Science and Technology, UK.

A mathematical model is developed to describe the motion of leukocytes through a Boyden chamber. The model is based on the Keller-Segel model of cell motion and comprises three partial differential equations which describe the evolution of the neutrophils, the chemoattractant, and a neutrophil-derived chemokinetic factor. Where other authors have concentrated on chemotaxis, here attention is focused on the manner in which the chemokinetic factor influences neutrophil locomotion. Numerical simulations show how the number of neutrophils initially placed on top of the chamber affects cellular motion through the system and reproduce the qualitative behaviour observed by Takeuchi Persellin (Am. J. Physiol. 236, C22-C29). In particular, the simulations show how dense packing of the neutrophils increases the levels of the chemokinetic factor. This enhances random cell motion and increases the speed with which the neutrophils reach the source of chemoattractant. For a particular asymptotic limit of the system parameters, the model reduces to a nonlinear partial differential equation for the neutrophils. Similarity solutions of this caricature model yield algebraic expressions relating the speed with which the neutrophil front penetrates into the chamber to the number of neutrophils initially placed on top of it. The implications of the results are also discussed.

PMID: 9773518, UI: 98446695


Ultrasound Med Biol 1998 Jun;24(5):621-9

Ultrasonographic evaluation of small cervical lymph nodes in head and neck cancer.

Yoshida H, Yusa H, Ueno E, Tohno E, Tsunoda-Shimizu H

Department of Oral and Maxillofacial Surgery, University of Tsukuba, Japan.

To establish sonographic criteria for differentiating metastasis and nonmetastasis in small cervical lymph nodes, correlations between sonographic parameters and histological diagnosis were statistically examined in 117 lymph nodes with maximal diameter of up to 10 mm in the sonographic findings, consisting of 26 metastatic and 91 nonmetastatic nodes. The equations obtained with logistic regression analysis showed lambda predictive values of -1.5 and 0.5 as effective cutoff-point criteria, and were considered to be a reliable indicator for differentiating small nodes with predictive values outside of -1.5 < lambda < 0.5. The sensitivity, specificity and accuracy with predictive values outside of -1.5 < lambda < 0.5 were 83%, 97% and 95%, respectively.

PMID: 9695264, UI: 98360377


J Theor Biol 1998 Jun 7;192(3):283-308

Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses.

Bocharov GA

Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia.

Lymphocytic choriomeningitis virus (LCMV) infection in mice provides an example of an extraordinarily dynamic process with an extreme sensitivity of phenotype of infection to parameters of virus/host interaction.

A mathematical model is developed to examine the dynamics of virus-specific cytotoxic T lymphocyte (CTL) response for LCMV infection in mice.

The model, formulated by a system of nonlinear delay-differential equations, considers the interacting

Clonal elimination of virus-specific cytotoxic T cells in high-dose LCMV-Docile infection represents an example of the classical phenomenon--high zone tolerance.

To describe both conventional and exhaustive CTL responses in the acute phase of LCMV-D infection 2 mechanisms are invoked:

  1. the high virus antigen load inhibition of T-cells proliferation via energy induction and
  2. the activation-induced cell death by apoptosis.
Parameters of the model, characterizing the rates of virus and CTL production and elimination in spleen, are estimated by assimilating with the model data on the LCMV-D infection in C57BL/6 mice for low-, moderate- and high-dose infections.

It is suggested that

  1. not only the clonal expansions have to be described in mathematical models as being virus regulated
  2. but also the later phases of primary immune response.
  3. Down-regulation of the primary CTL response is controlled by a network of mechanisms inducing anergy and apoptosis in activated T cells.
The model is used to investigate the effect of variations in virus and CTL response parameters on LCMV infection outcome and suggest predictions for experimental studies, in particular the phenotype of LCMV-WE infection in C57BL/6 as a function of initial virus doses.

PMID: 9650288, UI: 98313974


J Theor Biol 1997 Sep 7;188(1):127-40

The dynamics of Plasmodium falciparum blood-stage infection.

McKenzie FE, Bossert WH

Department of Organismic and Evolutionary Biology and the Division of Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA 02138, USA.

We develop a system of ordinary differential equations to model the dynamics of the blood-stages of the malaria parasite, Plasmodium falciparum.

The values of several parameters of our models can be estimated from previous empirical work.

Although the dynamics of the variants differ somewhat, in each variant some set of values of the 3 unconstrained parameters, different from one variant to the next, produces a range of behaviours quantitatively consistent with those reported from clinical studies.

  1. Some parameter values produce infections which quickly terminate,
  2. while others approach a chronic equilibrium level or
  3. produce oscillations, with repeated severe peaks separated by periods of undetectable parasitemia.
We examine these and several other distinctions that might be used to assess model variants and focus further empirical research. Copyright 1997 Academic Press Limited.

PMID: 9299316, UI: 97446222


J Math Biol 1997 Aug;35(7):775-92

Optimal control of the chemotherapy of HIV.

Kirschner D, Lenhart S, Serbin S

Department of Mathematics, Texas A and M University, College Station 77843, USA.

Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we

Using an objective function based on a combination of maximizing benefits based on T cell counts and minimizing the systemic cost of chemotherapy (based on high drug dose/strength), we solve for the optimal control in the optimality system composed of 4 ordinary differential equations and 4 adjoint ordinary differential equations.

PMID: 9269736, UI: 97415141


Antimicrob Agents Chemother 1997 Feb;41(2):449-53

Modeling of the change in CD4 lymphocyte counts in patients before and after administration of the human immunodeficiency virus protease inhibitor indinavir.

Stein DS, Drusano GL

Department of Medicine, Albany Medical College, New York 12208, USA.

We investigated the relationships between changes in CD4 lymphocytes counts over 24 weeks after the initiation of therapy with indinavir at dosages of > or = 2.4 g/day (n = 15) in human immunodeficiency virus-positive patients and compared them to the baseline values. Starting CD4 count were linked to the time-weighted average CD4 cell count (return) through a nonlinear effect model. The diminution of destruction of CD4 cells after the initiation of indinavir therapy was estimated by fitting simultaneous differential equations to the data by using a linked lymph node (LN)-blood (BL) (two-compartment) system in which there is a constant rate of generation (R), first-order transfer rate constants (KLN-BL and KBL-LN) of compartment exchange, and first-order rate constants of CD4 destruction in the absence and presence of indinavir (KLN-OUT1 and KLN-OUT2). The half-life of CD4 lymphocytes was calculated from the rate constants by standard two-compartment methods. The CD4 lymphocyte counts at the start and return were linked in a sigmoid-Emax model were the maximal effect (Emax) was at 574.6 cells/microliters and 50% of the effect occurred at 157.1 cells/microliters (r2 = 0.94; P < 0.001). The mean +/- standard deviation (median) KLN-OUT2 was 0.574 +/- 0.202 (0.589), indicating that indinavir decrease the destruction of CD4 cells by circa 41 to 42%. The mean (median) CD4 half-life was 11.5 +/- 5.72 day (10.3 days). In multivariate analysis, KLN-OUT2 was significantly correlated with starting the CD4 cells count and the change in the CD4 cell count on therapy. The relationship between CD4 lymphocyte half-life and the starting CD4 lymphocyte count was hyperbolic, with a rapid increase in half-life as the CD4 count decreased. On the basis of the calculated half-life, the average production (destruction) of CD4 lymphocytes was approximately 3 x 10(9) cells/day, with an individual variation of 44-fold. These findings suggest that (i) the CD4 lymphocyte cell count at the start is significantly correlated to both the decrease in the destruction rate of CD4 cells and the degree of change in the CD4 lymphocytes on therapy, (ii) the lower the initial CD4 lymphocyte count, the higher the amount of CD4 lymphocyte turnover and the lower the ability of the immune system to increase absolute CD4 lymphocyte levels after viral suppression, consistent with a decreased regenerative capacity with progression of disease; and (iii) the increase in CD4 lymphocytes is likely secondary to the expansion of proliferating pool of cells since our determinations are based on 24 weeks of effect.

Publication Types: Clinical trial Clinical trial, phase i Clinical trial, phase ii

PMID: 9021206, UI: 97173300


J Theor Biol 1996 Dec 7;183(3):285-305

Toxicity and neuroendocrine regulation of the immune response: a model analysis.

Muraille E, Thieffry D, Leo O, Kaufman M

Laboratoire de Physiologie Animale, Universite Libre de Bruxelles, Belgium. emura@

Various models have been proposed for the regulation of the primary immune response. Most of the models focus on the ability of the immune system to control a multiplying pathogen, and take into account the cross-regulations between different immune components.

In the present study, we integrate the immune system in the general physiology of the host and consider the interaction between the immune and neuroendocrine systems.

From a biological point of view, our model accounts for four stable regimes which can be described as
  1. "pathogen elimination/organism healthy",
  2. "pathogen elimination/ organism death",
  3. "pathogen growth/organism death" and
  4. "chronic infection".
The size of the basins of attraction of these different regimes varies as a function of some crucial parameters.

Our model allows moreover to interpret

  1. the interplay between pathogen immunogenicity and neuro-hormonal feedback,
  2. the effects of stress on immunity and
  3. the toxic shock syndrome,
in terms of transitions among the steady states.

PMID: 9015451, UI: 97167834


Environ Health Perspect 1996 Dec;104 Suppl 6:1293-301

Modeling marrow damage from response data: evolution from radiation biology to benzene toxicity.

Jones DT, Morris MD, Hasan JS

Chemical and Biological Physics Section, Oak Ridge National Laboratory, TN 37831-6101, USA.

Consensus principles from radiation biology were used to describe a generic set of nonlinear, first-order differential equations for modeling toxicity-induced compensatory cell kinetics in terms of sublethal injury, repair, direct killing, killing of cells with unrepaired sublethal injury, and repopulation. This cellular model was linked to a probit model of hematopoietic mortality that describes death from infection and/or hemorrhage between 5 and 30 days. Mortality data from 27 experiments with 851 dose-response groups, in which doses were protracted by rate and/or fractionation, were used to simultaneously estimate all rate constants by maximum-likelihood methods. Data used represented 18,940 test animals:12,827 mice, 2925 rats, 1676 sheep, 829 swine, 479 dogs, and 204 burros. Although a long-term, repopulating hematopoietic stem cell is ancestral to all lineages needed to restore normal homeostasis, the dose-response data from the protracted irradiations indicate clearly that the particular lineage that is critical to hematopoietic recovery does not resemble stemlike cells with regard to radiosensitivity and repopulation rates. Instead, the weakest link in the chain of hematopoiesis was found to have an intrinsic radioresistance equal to or greater than stromal cells and to repopulate at the same rates. Model validation has been achieved by predicting the LD50 and/or fractional group mortality in 38 protracted-dose experiments (rats and mice) that were not used in fitting of model coefficients.

PMID: 9118909, UI: 97147057


Proc R Soc Lond B Biol Sci 1996 Nov 22;263(1376):1487-93

On the role of angiogenesis in wound healing.

Pettet G, Chaplain MA, McElwain DL, Byrne HM

School of Mathematics, Queensland University of Technology, Brisbane, Australia.

Angiogenesis, the formation of blood vessels, may be described as a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then develop and organize themselves into a dendritic structure. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we present a mathematical model which describes the role of angiogenesis as observed during (soft-tissue) wound healing. We focus attention on certain principal players involved in this complex process, namely capillary tips, capillary sprouts, fibroblasts, macrophage-derived chemical attractants, oxygen and extracellular matrix. The model consists of a system of nonlinear partial differential equations describing the interactions in space and time of the above substances. Numerical simulations are presented which are in very good qualitative agreement with experimental observations.

PMID: 8952092, UI: 97109835


Math Biosci 1996 Nov;138(1):1-22

HIV-1 infection kinetics in tissue cultures.

Spouge JI, Shrager RI, Dimitrov DS

National Center for Biotechnology Information, National Library of Medicine, Bethesda, Maryland, USA.

Despite intensive experimental work on HIV-1, very little theoretical work has focused on HIV-1 spread in tissue culture. This article uses 2 systems of ordinary differential equations to model 2 modes of viral spread, 

  • cell-free virus and
  • cell-to-cell contact. The 2 models produce remarkably similar qualitative results.
  • Some previous models of in vivo HIV-1 infection oscillate, but only in unrealistic parameter regimes. Experimental tissue infections sometimes display several sequential cycles of oscillation, however, so our models can at least mimic them qualitatively. Significantly, the models show that infective oscillations can be explained by infection dynamics; biological heterogeneity is not required.

    The models also display proportionality between infected cells and cell-free virus, which is reassuringly consistent with assumptions about the equivalence of several measures of viral load, except that the proportionality requires a relatively constant total cell concentration. Tissue culture parameter values can be determined from accurate, controlled experiments.

    Therefore, if verified, our models should make interpreting experimental data and extrapolating it to in vivo conditions sharper and more reliable.

    PMID: 8942173, UI: 97097568


    J Theor Biol 1996 Oct 21;182(4):513-29

    Published erratum appears in J Theor Biol 1996 Nov 7;183(1):119 A model of the immune network with B-T cell co-operation. I--Prototypical structures and dynamics.

    Carneiro J, Coutinho A, Faro J, Stewart J

    Unite d'Immubiologie, CNRS URA 1961, Institut Pasteur, Paris, France,

    Hitherto, "second generation" network models of the immune system have all been restricted to B-lymphocytes and the Ig molecules they produce. These models have not so far been able to provide a convincing mechanism for the distinction between a "Central Immune System" (CIS) composed of a connected network of lymphocyte clones which couple with "self" antigens in a tolerant mode, and a "Peripheral Immune System" (PIS) composed of clones with little or no supra-clonal organization and which produce classical immune responses when interacting with "non-self" antigens. Here, we present a new network model which explicitly incorporates B-T cell co-operation. In this model, B-cell activation is dependent on T-cell help, and activated T-cells are down-regulated by engagement of their TCRs by soluble Ig. We discuss the underlying biology on which we base the system of ordinary differential equations which defines the present network model. We then illustrate some basic features of the model by examining several prototypical situations with a small number of clones. Depending on the idiotypic connectivity structure, the model exhibits two distinct modes of coupling with antigens: "immune response" mode in which T- and B-cell clones grow exponentially; and a "tolerant" mode in which T-cell clones are controlled by inclusion of all TCRs in the repertoire of an idiotypic B-cell network. Finally, we discuss the simplifying assumptions of the present model and argue that its range of validity is indeed the region of the state-space of the system where the discrimination between the CIS and the PIS take place.

    PMID: 8944897, UI: 97100336


    Cancer Res 1996 Aug 15;56(16):3771-81

    Physiologically based kinetic model of effector cell biodistribution in mammals: implications for adoptive immunotherapy.

    Zhu H, Melder RJ, Baxter LT, Jain RK

    Steele Laboratory, Department of Radiation Oncology, Massachusetts General Hospital, Boston 02114, USA.

    The goal of the present investigation was to develop a physiologically based kinetic model to describe the biodistribution of immunologically active effector cells in normal and neoplastic tissues of mammals based on the current understanding of lymphocyte trafficking pathways and signals.

    In the model, The model was used to simulate the following biodistribution data:
    1. nonactivated T lymphocytes in rats;
    2. interleukin 2-activated tumor-infiltrating lymphocytes in humans;
    3. nonactivated natural killer (NK) cells in rats; and
    4. interleukin 2-activated adherent NK cells in mice.
    Comparisons between simulations and data demonstrated the feasibility of the model and the scaling scheme. PMID: 8706023, UI: 96328102


    Artif Organs 1996 Aug;20(8):866-77

    Analysis of nonlinear properties of immune network reactions.

    Hirayama H, Nishimura T, Fukuyama Y

    Department of Public Health, Asahikawa Medical College, Japan.

    On the basis of biochemical reaction dynamics, the temporal behavior of the immune network system was analyzed theoretically to promote the analysis of quantitative changes in the reactions of immune disorders and organ substitution. The idiotype immune network reaction system was expressed by 64 nonlinear differential equations that comprised four kinds of antibodies and B-cell subpopulations. All four kinds of antibodies decreased rapidly. With the progress of the reactions, they have increased gradually. The single and double bound antibodies increased rapidly from the onset of the reaction. The single-bound antibodies did not show a definite increase after the rapid increasing phase. The antibody-antibody complex increased parallel with the double bound antibodies. The effects of rate constant expand to all the immune complexes in the network system. The double bound antibodies and antibody-antibody complexes were oscillatory functions of a given antibody. Therefore, the idiotypic immune network system must be a chaotic one. The present theoretical method is available to evaluate the total ability of immune reaction system that operates as a network system.

    PMID: 8853798, UI: 97006505


    Math Biosci 1996 Aug;136(1):35-63

    A model of wound-healing angiogenesis in soft tissue.

    Pettet GJ, Byrne HM, McElwain DL, Norbury J

    School of Mathematics, Queensland University of Technology, Brisbane, Australia.

    Angiogenesis, or blood vessel growth, is a critical step in the wound-healing process, involving the chemotactic response of blood vessel endothelial cells to macrophage-derived factors produced in the wound space. In this article, we formulate a system of partial differential equations that model the evolution of the capillary-tip endothelial cells, macrophage-derived chemoattractants, and the new blood vessels during the tissue repair process. Chemotaxis is incorporated as a dominant feature of the model, driving the wave-like ingrowth of the wound-healing unit. The resulting model admits traveling wave solutions that exhibit many of the features characteristic of wound healing in soft tissue. The steady propagation of the healing unit through the wound space, the development of a dense band of fine, tipped capillaries near the leading edge of the wound-healing unit (the brush-border effect), and an elevated vessel density associated with newly healed wounds, prior to vascular remodeling, are all discernible from numerical simulations of the full model. Numerical simulations mimic not only the normal progression of wound healing but also the potential for some wounds to fail to heal. Through the development and analysis of a simplified model, insight is gained into how the balance between chemotaxis, tip proliferation, and tip death affects the structure and speed of propagation of the healing unit. Further, expressions defining the healed vessel density and the wavespeed in terms of known parameters lead naturally to the identification of a maximum wavespeed for the wound-healing process and to bounds on the healed vessel density. The implications of these results for wound-healing management are also discussed.

    PMID: 8755336, UI: 96334198


    Health Phys 1996 Jun;70(6):787-97

    Stem cell responses after radiation exposure: A key to the evaluation and prediction of its effects.

    Fliedner TM, Tibken B, Hofer EP, Paul W

    Department of Clinical Physiology, Occupational and Social Medicine, University of Ulm, Federal Republic of Germany.

    A biomathematical model of granulocytopoiesis is described and used to analyze the blood granulocyte changes seen in the blood of dogs and humans after continuous and after acute external radiation exposure. This allows to relate the cell change pattern seen to the extent of stem cell damage in the hematopoietic bone marrow distributed as semi-autonomous units throughout the skeletal bones.

    The model is described briefly and consists of 8 cellular and 2 regulatory compartments and is described by 37 differential equations. With the help of this model, it can be shown that the chronic radiation exposure of dogs at a rate of between 0.003 and 0.12 Gy per day results in a system failure with subsequent death of the animal, if the stem cell pool decreases below 2.5% of its normal content. In human beings exposed to a single radiation exposure (as seen in radiation accidents) the simulation of the granulocyte pattern results in the finding that a reduction of the stem cell pool to 5-10% of normal is compatible with the assumption of its "reversible" damage (to be treated by conventional replacement therapy including cytokines), whereas the reduction of blood granulocytes to levels of less than 200-300 per mm3 on day 5-6 after exposure indicates that no stem cells remain from which a spontaneous regeneration could occur and hence would require a substitution therapy by stem cell transplantation. In order to test the approach, the same model was used to correlate the changing granulocyte pattern seen after autologous blood stem cell transfusion in patients treated with a supralethal radiochemo conditioning regimen.

    The results indicate a proportionality of progenitor cells in the transfusate with the calculated stem cell number of the modeling exercise. It is proposed to use the pattern of granulocyte changes in the blood as a principal indicator to predict the outcome of a radiation exposure and to select appropriate therapeutic strategies.

    PMID: 8635902, UI: 96216259


    Bull Math Biol 1996 Mar;58(2):376-90

    A model for treatment strategy in the chemotherapy of AIDS.

    Kirschner D, Webb GF

    Department of Mathematics, Texas AM University, College Station 77845, USA. dek/

    Mathematical models are developed for the chemotherapy of AIDS. The models are systems of differential equations describing the interaction of the HIV infected immune system with AZT chemotherapy. The models produce the three types of qualitative clinical behavior: an uninfected steady state, an infected steady state (latency) and a progression to AIDS state. The effect of treatment is to perturb the system from progression to AIDS back to latency. Simulation of treatment schedules is provided for the consideration of treatment regimes. The following issues of chemotherapy are addressed: (i) daily frequency of treatment, (ii) early versus late initiation of treatment and (iii) intermittent treatment with intervals of no treatment. The simulations suggest the following properties of AZT chemotherapy: (i) the daily period of treatment does not affect the outcome of the treatment, (ii) treatment should not begin until after the final decline of T cells begins (not until the T cell population falls below approximately 300 mm-3) and then, it should be administered immediately and (iii) a possible strategy for treatment which may cope with side effects and/or resistance, is to treat intermittently with chemotherapy followed by interruptions in the treatment during which either a different drug or no treatment is administered. These properties are revealed in the simulations, as the model equations incorporate AZT chemotherapy as a weakly effective treatment process. We incorporate into the model the fact that AZT treatment does not eliminate HIV, but only restrains its progress. The mathematical model, although greatly simplified as a description of an extremely complex process, offers a means to pose hypotheses concerning treatment protocols, simulate alternative strategies and guide the qualitative understanding of AIDS chemotherapy.

    PMID: 8713663, UI: 96342043


    J Math Biol 1996;34(4):361-412

    Stochastic model of receptor-mediated cytomechanics and dynamic morphology of leukocytes.

    Tranquillo RT, Alt W

    Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis 55455, USA.

    The proposed mathematical model investigates the simplified cytomechanics of cell shape change driven by stochastic stimulation from chemosensory receptors. The cytomechanical component of our model describes the dynamical distribution of F-actin and associated forces in an idealized cortical actin network around the cell periphery. The chemosensory component describes the distribution of chemotactic receptors in the cell membrane surrounding the cortex, where bound receptors give rise to an intracellular signal which modulates some property of the cortical network. As in our earlier models, an account is made for (1) the reactive, contractive properties of cortical actin, but here also for a stress induced by curvature of the cortex-membrane complex which carries an effective surface tension, and (2) statistical fluctuations in receptor binding, but generalized here to include statistical fluctuations in the spatial distribution of receptors, entirely determined by the additional prescription of membrane diffusion coefficients along with total receptor number, receptor binding rate constants and the local concentration field of chemotactic factor. We simplify the analysis by restricting the model to a prototype in which viscous stresses in the cortical network are negligible and the radial extension of the cell cortex is a prescribed function of the cortical actin concentration. We assume in particular that the assembly rate of cortical actin depends on the local density of bound receptors. These assumptions lead to a 4th-order parabolic differential equation on the unit circle coupled to a system of stochastic differential equations. We characterize via bifurcation analysis, stochastic simulations, and analytical correlation functions the spatial-temporal pattern of cell morphology under the influence of fluctuations in the bound receptor distribution for the case of a uniform concentration field of chemotactic factor. In addition to addressing the biological significance of our model, we remark on its relevance to the generic problem of the influence of correlated stochastic perturbations on spatial patterns in morphogenetic media.

    PMID: 8867995, UI: 97021635


    Stem Cells 1995 May;13 Suppl 1:290-300

    An approach to a biomathematical model of lymphocytopoiesis.

    Hofer EP, Brucher S, Mehr K, Tibken B

    Department of Measurement, Control and Microtechnology, University of Ulm, Germany.

    Fundamental principles for the development of a biomathematical model of lymphocytopoiesis are presented in this paper. The first step in this modeling approach is the definition of appropriate anatomical compartments in order to identify dominant locations of lymphocytes in the human body, and the definition of functional compartments in order to model different maturation stages. In the second step these compartment structures are combined, and thus form the basis of a dynamical model consisting of linear differential equations. Cell balance equations are used to derive the biomathematical dynamical model which is presented using the tools of modern systems theory. As a result of intensive discussions between engineers and medical doctors, our model of lymphocytopoiesis consists of six anatomical and six functional compartments. Virtual marking technique plays a key role for the interpretation of the simulation results which are in solid agreement with biological observations. Future work is directed towards implementation of radiation damages in order to meet the final goal, namely, the evaluation of our model against the data derived from a group of chronically irradiated uranium miners.

    PMID: 7488959, UI: 96003182


    Medinfo 1995;8 Pt 2:1091

    A non-linear mathematical model for the in vivo evaluation of the RES phagocytic function.

    Bondareva IB, Parfenov AS

    Dept. of Radiology, The Research Institute of Physico-Chemical Medicine, Moscow, Russia.

    A new non-linear mathematical model was constructed in order to perform in vivo quantification of the RES phagocytic function. This method is based on the same technical facilities as used for the routine liver-spleen scintigraphy with radiocolloids [1, 2]. But kinetic modeling of dynamic Tc-99m-sulfur colloid data produced estimations of the functional RE-parameters: the clearance rate of the colloidal particles, the rate of phagocytosis, and the RES functional volume, which can not be obtained by classical approaches.

    This non-linear model was designed on the basis of the principal characteristics of particulate material interaction with macrophages (attachment, phagocytosis, digestion) [3, 4, 5]. The theoretically examined behavior of this in vivo mathematical model corresponds with the experimental behavior of the RES. The mathematical expression of the dynamics is the system of non-linear differential equations with constant coefficients that have no analytical solution. Fitting of the normalized heart blood time-activity curve was obtained to identify the unknown model parameters via non-linear regression. For this purpose general interactive PASCAL procedure IDPAR for a PDP-11/34 computer was used (an IBM PC version is also available). Two to three iterations were needed to estimate the set of unknown parameters for any patient study (1-1.5 min). A very good fitting was obtained between experimental and model curves in every case of different pathologies (error of the approximation is about 2-3%). Studies were performed using an in vivo bolus injection of 3.6 mg/80 kg commercially available colloid KOREN labeled with 3m-Ci 99m-Tc (analog of TCK-1).

    Our method was used to determine the RES functional parameters for patient groups with different levels of the RES dysfunction. Obtained results illustrate the possibilities of our technique to quantitatively estimate not only great pathology (portal cirrhosis), but also small changes of the RE-function (case of hyperlipidemia and ulcer gaster). In all patient groups marked changes of Tc-99m-sulfur colloid turnover were observed. In general, tracer clearance from the circulation was decreased, and the rate of phagocytosis and the RES volume were diminished compared with controls. The effect of a reduction of phagocytosis increases when the RES dysfunction becomes stronger. It can be shown that a non-parametric Wilcoxon-Mann-Whitney test gives a significant difference (P95%) for these patient groups. Further, we represent the possibility of using the model for monitoring changes of the RES-function parameters during and after therapy. The quantitative test of the RES function can significantly enhance the diagnosis and management of different diseases. Serial colloidal studies may document changes in the RES-function for the tumors, cirrhosis, hyperlipidemia, reticulosis, hepatitis, thrombosis, infection, AIDS, burn injury, shock and trauma patients. The technique may be useful for the different RES investigations with laboratory animals. Created computer software can be used as a tool for kinetic models, simulation, and unknown parameters identification.

    PMID: 8591376, UI: 96174198


    Int J Biomed Comput 1994 Aug;36(4):293-8

    A non-linear compartmental model of human basophil activation.

    Nkobetchou F, Cherruault Y, Sainte-Laudy J

    Laboratoire Medimat, Universite Pierre et Marie Curie, Paris, France.

    In this paper human basophil activation is modelled by means of a system of differential equations. The resolution of this system allows the justification of the incubation period and the unwedging period existing between the human basophil degranulation and the liberation of histamine.

    PMID: 7528175, UI: 95095399


    Bull Math Biol 1994 Jul;56(4):687-721

    Reverse engineering: a model for T-cell vaccination.

    Segel LA, Jager E

    Department of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel.

    A class of minimal models is constructed that can exhibit several salient phenomena associated with T-cell inoculations that prevent and cure autoimmune disease. The models consist of differential equations for the magnitude of two populations, the effectors E (which cause the disease), and an interacting regulator population R. In these models, normality, vaccination and disease are identified with stable steady-states of the differential equations. Thereby accommodated by the models are a variety of findings such as the induction of vaccination or disease, depending on the size of the effector inoculant. Features such as spontaneous acquisition of disease and spontaneous cure require that the models be expanded to permit slow variation of their coefficients and hence slow shifts in the number of steady-states. Other extensions of the basic models permit them to be relevant to vaccination by killed cells or by antigen, or to the interaction of a larger number of cell types. The discussion includes an indication of how the highly simplified approach taken here can serve as a first step in a modeling program that takes increasing cognizance of relevant aspects of known immunological physiology. Even at its present stage, the theory leads to several suggestions for experiments.

    PMID: 8054891, UI: 94332045


    Proc Natl Acad Sci U S A 1994 Jan 18;91(2):544-8

    Diversity and virulence thresholds in AIDS.

    de Boer RJ, Boerlijst MC

    Utrecht University, The Netherlands.

    We propose a model for the interaction between human immunodeficiency virus and the immune system. Two differential equations describe the interactions between one strain of virus and one clone of T lymphocytes. We use the model to generalize earlier results pertaining to the AIDS diversity threshold [Nowak, M. A., Anderson, R. M., McLean, A. R., Wolfs, T. F. W., Goudsmit, J. and May, R. M. (1991) Science 254, 963-969]. Our model has (i) a stable steady state corresponding to the "controlled" persistence of the virus and (ii) a region corresponding to AIDS. The separatrix between the two regimes is formed by the stable manifold of a saddle point. We define a dimensionless "virulence" parameter which combines the infectivity and antigenicity of a virus strain. We derive analytically two parameter conditions involving virulence. The first corresponds to a saddle-node bifurcation which causes AIDS due to the loss of the stable equilibrium. The second corresponds to a global bifurcation which causes AIDS due to a change in the basins of attraction. Incorporating diversity into the model, we derive a diversity threshold corresponding to the saddle-node bifurcation. In this threshold condition diversity and virulence have an equivalent effect. By studying the effect of diversity on the critical virulence that is required for a new mutant to cause AIDS, we again establish that diversity and virulence are equivalent parameters. Because in our model increasing diversity decreases the critical virulence, the strain that eventually causes AIDS need not be a virulent one.

    PMID: 7904755, UI: 94119923


    IMA J Math Appl Med Biol 1994;11(2):107-47

    Modelling and simulation of Rosenberg-type adoptive cellular immunotherapy.

    Nani FK, Oguztoreli MN

    Department of Mathematics, University of Alberta, Edmonton, Canada.

    A mathematical model is developed to describe the process of adoptive cellular immunotherapy (ACI) using the scheme of Rosenberg and other investigators. The model exhibits the dynamics of tumour cells as well as the time evolution of the tumoricidal immunocytes, such as

    The model is described mathematically by a system of nonlinear functional-differential equations. Computer simulations based on the model equations are performed using parametric configurations analogous to the protocols used in the clinical trials.

    The model elucidates explicitly the effects of

    PMID: 8089590, UI: 94375994


    J Theor Biol 1993 Oct 7;164(3):271-90

    Studies on a recent class of network models of the immune system.

    Faro J, Velasco S

    Departamento de Fisica Aplicada, Universidad de Salamanca, Spain.

    It is argued that the realism of computer simulations of network models of the immune system depends basically on the coherence of these models with the essentials of the known physiology of the cells and molecules selected to be modelled and on the incorporation in them of the different compartments of activated B cells. Focusing on these two aspects, here we analyse the simplifications and assumptions that go implicit in the formulation of a recently developed new class of network models that distinguish between immunoglobulins and B cells. This is approached by first building a general model which incorporates explicitly the kinetics of different B-cell compartments as well as a splenic compartment and a peripheric one for immunoglobulins, and then formally studying the simplifications on this model that are necessary to recover the initial simpler models. Following this procedure, it is shown that the effective coefficients of the different rate terms in the simpler models are particular combinations of the elementary rates obtained empirically. These relations reflect the particular assumptions associated with each simplification step. Also, it is shown that the usual biological interpretation of some of the coefficients in the ordinary differential equations of the simpler models is inconsistent with the more exact general model, unless one makes certain unreasonable assumptions about B-cell physiology. The relevance of this approach in providing variables with a biologically identifiable reality and for realistic, testable, computer simulations is discussed.

    PMID: 8246520, UI: 94066471


    Exp Hematol 1993 Jun;21(6):816-22

    A cell-kinetics model for radiation-induced myelopoiesis.

    Jones TD, Morris MD, Young RW, Kehlet RA

    Health and Safety Research Division, Oak Ridge National Laboratory, TN 37831.

    A mathematical model of time-dependent cellular damage, repair, killing and repopulation of bone marrow following treatments with ionizing radiations is described. Effects from variable dose rates, multiple exposures, different radiation sources and arbitrary intervals between treatments can be modeled by ordinary differential equations. Of several unique features, the most unusual is that rate constants for injury, repair, killing and proliferation of cells are evaluated by likelihood analysis of animal mortality data. Results indicate that a relatively radioresistant pool of bone marrow cells mediates the proliferation of the hematopoietic stem cells. Applications include modeling of 1) myelopoietic integrity as a function of time and dose rate, 2) the whole-body survival curve (at any point in the treatment protocol) for cells critical to myelopoiesis, 3) a prompt dose equivalence from any completed portion of a therapeutic schedule and 4) potential gain from schedule changes during the course of the treatment.

    Comments: Comment in: Exp Hematol 1994 Jan;22(1):2; discussion 3-4 Comment in: Exp Hematol 1994 Jul;22(7):535-8

    PMID: 8500579, UI: 93272905


    J Theor Biol 1993 May 7;162(1):23-40

    Chemical control of eukaryotic cell movement: a new model.

    Sherratt JA, Sage EH, Murray JD

    Centre for Mathematical Biology, Mathematical Institute, St Giles', Oxford, U.K.

    Cellular chemotaxis and chemokinesis play important roles in many biological processes. Most continuum mathematical models for these regulatory mechanisms are based on the model of Keller Segel (1971 a, b), in which cells respond directly to the local concentration of extracellular chemical. We have developed a new model which reflects the receptor-based mechanisms underlying chemical control of cell motion. Our model consists of three coupled partial differential equations, and we use the Boyden chamber (millipore) assay to compare it with a simpler model based on the Keller-Segel approach. The predictions of our model capture the key qualitative features of the experimental data, whereas the simpler model only does so when appropriate functional forms are chosen for the dependence of the transport coefficients on chemical concentration. Using experimental data on the variation of receptor kinetic parameters with temperature, we use our model to predict the effect of decreasing the temperature on both the "leading front" and "migrated cell" measurements taken from Boyden chamber assays. Our results show that changes in the kinetic parameters play a key role in controlling the temperature dependence of cell chemotaxis and chemokinesis.

    PMID: 8412219, UI: 94017856


    Math Biosci 1993 Mar;114(1):81-125

    Dynamics of HIV infection of CD4+ T cells.

    Perelson AS, Kirschner DE, De Boer R

    Theoretical Division, Los Alamos National Laboratory, New Mexico.

    We examine a model for the interaction of HIV with CD4+ T cells that considers four populations: uninfected T cells, latently infected T cells, actively infected T cells, and free virus. Using this model we show that many of the puzzling quantitative features of HIV infection can be explained simply. We also consider effects of AZT on viral growth and T-cell population dynamics. The model exhibits two steady states, an uninfected state in which no virus is present and an endemically infected state, in which virus and infected T cells are present. We show that if N, the number of infectious virions produced per actively infected T cell, is less a critical value, Ncrit, then the uninfected state is the only steady state in the nonnegative orthant, and this state is stable. For N > Ncrit, the uninfected state is unstable, and the endemically infected state can be either stable, or unstable and surrounded by a stable limit cycle. Using numerical bifurcation techniques we map out the parameter regimes of these various behaviors. oscillatory behavior seems to lie outside the region of biologically realistic parameter values. When the endemically infected state is stable, it is characterized by a reduced number of T cells compared with the uninfected state. Thus T-cell depletion occurs through the establishment of a new steady state. The dynamics of the establishment of this new steady state are examined both numerically and via the quasi-steady-state approximation. We develop approximations for the dynamics at early times in which the free virus rapidly binds to T cells, during an intermediate time scale in which the virus grows exponentially, and a third time scale on which viral growth slows and the endemically infected steady state is approached. Using the quasi-steady-state approximation the model can be simplified to two ordinary differential equations the summarize much of the dynamical behavior. We compute the level of T cells in the endemically infected state and show how that level varies with the parameters in the model. The model predicts that different viral strains, characterized by generating differing numbers of infective virions within infected T cells, can cause different amounts of T-cell depletion and generate depletion at different rates. Two versions of the model are studied. In one the source of T cells from precursors is constant, whereas in the other the source of T cells decreases with viral load, mimicking the infection and killing of T-cell precursors.

    PMID: 8096155, UI: 93208399


    Bull Math Biol 1993;55(4):745-80

    Immune network behavior--I. From stationary states to limit cycle oscillations.

    De Boer RJ, Perelson AS, Kevrekidis IG

    Utrecht University, The Netherlands.

    We develop a model for the idiotypic interaction between 2 B cell clones. This model takes into account

    Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior.

    After appropriate nondimensionalization, the model consists of 8 ordinary differential equations and a number of parameters.

    1. We estimate the parameters from experimental sources.
    2. Using a coordinate system that exploits the pairwise symmetry of the interactions between 2 clones, we analyse 2 simplified forms of the model and obtain bifurcation diagrams showing how their 5 equilibrium states are related.
    PMID: 8318929, UI: 93306242


    Arch Immunol Ther Exp (Warsz) 1993;41(1):21-31

    Kinetic approach and estimation of the parameters of cellular interaction between the immunity system and a tumor.

    Kuznetsov VA, Zhivoglyadov VP, Stepanova LA

    Department of Chemical and Biological Processes, Russian Academy of Sciences, Moscow.

    A method is suggested to estimate multi component dynamic systems, which permits, with the help of the computer-calculated kinetic curves, to obtain information about the possible mechanisms of the system component interaction. The method is based on the structural and parametrical identification of mathematical models presented in the form of a system of nonlinear differential equations, using a multi-criterial approach. Using experimental data of studies on growth kinetics and regression of multicellular tumor EMT6 line spheroids in the mouse allogenic system and the immune system cell accumulation in spheroids a mathematical model has been developed of the cellular interaction process in a spheroid. It has been stated that the rate of macrophage and neutrophil accumulation in a spheroid depends on the amount of tumor cells and is determined by the hyperbolic law (as analogous to the Michaelis-Menthen kinetics), while the accumulation of immune lymphocytes in a tumor is determined besides that by the three-cellular cooperation of lymphocytes, macrophages and tumor cells. According to the model, the inhibition of the process of neutrophil and lymphocyte (but not of macrophages) accumulation is realized through the auto-suppression mechanism. The numerical values of the process parameters, which characterise the rates of accumulation, cellular death in a tumor and of local cellular interactions intensity are obtained.

    PMID: 8239905, UI: 94058555


    J Pharmacokinet Biopharm 1992 Aug;20(4):319-31

    Pharmacoimmunodynamics of methylprednisolone: trafficking of helper T lymphocytes.

    Fisher LE, Ludwig EA, Jusko WJ

    Department of Pharmaceutics, School of Pharmacy, State University of New York, Buffalo.

    A two-compartment closed model was used to characterize the cell trafficking behavior of helper T cells in response to various single doses of methylprednisolone. Steroids are assumed to inhibit the circadian-determined cell return from extravascular sites to blood in a classic inhibitory pattern reflected by an IC50. The rate of cell efflux from tissues is modeled with a cosine function having a period of 24 hr and a maximum at about 1 AM. Nonlinear least-squares regression employing differential equations was used to analyze helper T-cell data from three human studies from our laboratory. The IC50 value of methylprednisolone of 12-19 ng/ml approximates receptor KD values. Simulations were performed to demonstrate the log-linear role of steroid dose or AUC on the integral of effect of helper T cells over a wide range of methylprednisolone doses. This pharmacodynamic model allows flexibility for characterizing any type of steroid dosing regimen and is relevant in describing complex response data for corticosteroid immunosuppressive effects in man.

    PMID: 1479558, UI: 93124381


    Biosystems 1992;26(3):177-83

    A kinetic model of CD4+ lymphocytes with the human immunodeficiency virus (HIV).

    Bailey JJ, Fletcher JE, Chuck ET, Shrager RI

    Laboratory of Applied Studies, National Institutes of Health, Bethesda, Maryland 20892.

    This report describes a kinetic model of in vitro cytopathology involving interactions of human immunodeficiency virus (HIV) with CD4+ helper T lymphocytes. The model uses nonlinearly coupled, ordinary differential equations to simulate the dynamics of infected and uninfected cells and free virions. It is assumed that resting cells are more readily infected than activated cells, but once infected, only activated cells produce more virus. Resting cells can be activated by some appropriate stimulus (e.g. phytohemagglutinin, soluble antigen). The model predicts that the initial inoculum of virus is taken up by resting cells and without stimulation the system comes to a steady state of two populations, namely infected and uninfected cells. Stimulation of this system produces two additional populations, namely infected and uninfected activated cells which, along with the previous populations, exhibit cyclic behavior of growth, viral expression/release, and death. Additional stimuli enhance or diminish the cyclic behavior depending upon their occurrence in time. These simulations suggest a similar dynamics in human HIV infection and may explain a major factor responsible for the widely varying depletion rate of (CD4+) helper T cells in AIDS patients.

    PMID: 1348962, UI: 92232930


    Kosm Biol Aviakosm Med 1991 Sep-Oct;25(5):53-6

    [Study of cyclic kinetics of immunity by mathematical modeling methods].

    [Article in Russian]

    Smirnova OA

    The mathematical model of the dynamics of humoral immune responses to soluble antigens has been developed. This is a system of nonlinear differential equations describing concentrations of immunocompetent B-lymphocytes, plasma cells, antibodies and antigen. The model reproduces cyclic kinetics of the immune reaction to slowly catabolizing antigens which is observed experimentally. Within the framework of the model a description of the mechanism of origin of oscillatory modes of the dynamics of the immune response is presented. It has been shown that the feedback in the control of the antibody synthesis by antibodies is due to neutralization of the main stimulus of the immune system, i.e., free molecules of the antigen--by circulating antibodies.

    PMID: 8577146, UI: 96164137


    J Theor Biol 1991 Jul 7;151(1):1-40

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

    Marchuk GI, Petrov RV, Romanyukha AA, Bocharov GA

    Department of Numerical Mathematics U.S.S.R. Academy of Sciences, Moscow.

    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).

    PMID: 1943135, UI: 92047484


    Health Phys 1991 Jul;61(1):87-95

    An experimental and mathematical analysis of lymphopoiesis dynamics under continuous irradiation.

    Zukhbaya TM, Smirnova OA

    Institute of Medicobiological Problems of USSR, Ministry of Health, Moscow.

    A mathematical model describing the dynamics of lymphopoiesis in mammals continuously exposed to ionizing radiation has been developed. It is based on the theory of chalone regulation of hematopoiesis. The model comprises a system of nine differential equations. Results from the model were compared with our experimental data for bone marrow and blood lymphocytes of rats continuously exposed to gamma radiation in a wide range of dose rates. The model reproduces the lymphopoiesis dynamics that we observed in our experiment, in particular, the radiation hormesis at low dose rates, the reduction of lymphopoiesis at intermediate dose rates, and extinction of lymphopoiesis at high dose rates of continuous radiation. The possible explanation of the hormesis is suggested by the framework of the model. The model can be used for predicting the lymphopoiesis dynamics in mammals under continuous irradiation.

    PMID: 1829440, UI: 91285858


    Kosm Biol Aviakosm Med 1991 May-Jun;25(3):40-2

    [Stimulating effect of long-term low-dose radiation on granulocytopoiesis].

    [Article in Russian]

    Smirnova OA, Zukhbaia TM

    A mathematical model simulating variations in granulocytopoiesis of mammals exposed to chronic irradiation has been built. The model has been developed to take into account the chalone mechanism of hemopoiesis regulation. The model represents a system of 12 nonlinear differential equations. The modelling results have been found to be in agreement with experimental observations of variations in granulocytopoiesis of rats within a wide range of radiation dose rates. This model has been used for the first time to simulate a stimulating effect of prolonged irradiation with a low dose rate on granulocytopoiesis. Manifestations of this effect are explained. This model can be used to predict the effect of prolonged irradiation with different dose rates on variations of granulocytopoiesis in mammals.

    PMID: 1770766, UI: 92122707


    Folia Biol (Praha) 1991;37(1):10-20

    Mathematical model of antiviral immune response regulation. II. Mathematical formalization of the modelled processes. Imitation of acute course of hepatitis B.

    Chuykov VV, Bazhan SI, Kulichkov VA

    All-Union Research Institute of Molecular Biology, Novosibirsk Region.

    The mathematical formalization of the conceptual model for antiviral immune response regulation described in the preceding report was carried out. The mathematical model is presented as a system of 30 ordinary nonlinear differential equations with delays. The algorithm for numerical integration of the mathematical model is based on Gear's methods of variable step and variable order. Initial conditions and parameters, as well as intervals of plausible values for them, were chosen for adaptation of the model for description of acute hepatitis B.

    PMID: 1830011, UI: 91301285


    J Theor Biol 1990 May 10;144(1):93-101

    Dynamics of a class of immune networks. I. Global stability of idiotype interactions.

    Varela FJ, Stewart J

    CREA, Ecole Polytechnique, Paris, France.

    This paper establishes the conditions under which a class of differential equations which appear in the study of immune systems (Varela et al., 1988a, In: Theoretical Immunology Part II. New Jersey: Addison Wesley), are globally stable. This is proved by adapting a Liapunov functional originally proposed by Cohen Grossberg (1983, IEEE Transac SMC 13, 815-826) for competitive systems. The global stability thus obtained is valid on the fast time scale where only idiotypic interactions are relevant, thus excluding both lymphocyte proliferation processes and repertoire change via recruitment from immature bone marrow B cells.

    PMID: 2385112, UI: 90348210


    Biochemistry 1990 Mar 13;29(10):2464-71

    Phase behavior of cholesteryl ester dispersions which model the inclusions of foam cells.

    Snow J, Phillips MC

    Chemistry Department, Philadelphia College of Pharmacy and Science, Pennsylvania 19104.

    In order to understand the phase behavior of the approximately 1-micron-diameter droplets which occur in the cytoplasm of cholesterol-enriched cells, differential scanning calorimetry has been utilized to elucidate the factors controlling the rate of crystallization of cholesteryl esters. The kinetics of the thermotropic transitions between liquid, liquid-crystal, and crystal states which occur in mixtures of cholesteryl oleate and cholesteryl palmitate present in monodisperse, phospholipid-stabilized, emulsion droplets have been determined and are compared to the characteristics of these transitions in bulk mixtures. Cholesteryl palmitate is observed to crystallize in undercooled phospholipid-stabilized dispersions of cholesteryl palmitate/cholesteryl oleate (50/50 w/w) at temperatures up to 50 degrees C lower than it does in bulk mixtures of the same cholesteryl ester composition. It is postulated that this difference between crystallization temperatures is due primarily to the presence of impurities present in bulk mixtures which act as catalysts that promote crystallization. It is suggested that phospholipid-stabilized dispersions of cholesteryl palmitate/cholesteryl oleate are more appropriate models than bulk mixtures of these cholesteryl esters for studying the kinetic and thermodynamic basis of the phase behavior in cholesteryl ester rich inclusions characteristic of foam cells and atherosclerotic plaque. The thermotropic phase behavior of these dispersions can be satisfactorily analyzed by using the equations of homogeneous nucleation theory. The interfacial tension between the crystal nucleus and the surrounding fluid cholesteryl ester is about 10 erg/cm2.

    PMID: 2334676, UI: 90241863


    Cell Biophys 1989 Apr;14(2):139-73

    A dynamical model for receptor-mediated cell adhesion to surfaces in viscous shear flow.

    Hammer DA, Lauffenburger DA

    School of Chemical Engineering, Cornell University, Ithaca, NY 14853.

    We present a dynamical model for receptor-mediated cell adhesion to surfaces in viscous shear flow when the surfaces are coated with ligand molecules complementary to receptors in the cell membrane. This model considers the contact area between the cell and the surface to be a small, homogeneous region that mediates the initial attachment of the cell to the surface. Using a phase plane analysis for a system of nonlinear ordinary differential equations that govern the changes in free receptor density and bond density within the contact area with time, we can predict the conditions for which adhesion between the cell and the surface will take place. Whether adhesion occurs depends on values of dimensionless quantities that characterize the interaction of the cell and its receptors with the surface and its ligand, such as the bond formation rate, the receptor-ligand affinity, the fluid mechanical force, the receptor mobility, and the contact area. A key result is that there are two regimes in which different chemical and physical forces dominate: a rate-controlled high affinity regime and an affinity-controlled low affinity regime. Many experimental observations, including the effects of temperature and receptor mobility on adhesiveness, can be explained by understanding which of these regimes is appropriate. We also provide simple approximate analytical solutions, relating adhesiveness to cell and surface properties as well as fluid forces, which allow convenient testing of model predictions by experiment.

    PMID: 2472206, UI: 89288209


    Kosm Biol Aviakosm Med 1989 Jan-Feb;23(1):47-51

    [Stimulating effect of long-term small-dose radiation on lymphopoiesis in mammals].

    [Article in Russian]

    Zukhbaia TM, Smirnova OA

    Mathematical models of lymphopoietic variations in mammals exposed to acute and chronic irradiation have been developed. They are based on the cheilone theory of hemopoiesis regulation. The models are systems of 9 nonlinear differential equations. They give a qualitative and quantitative description of lymphocytes in peripheral blood and their precursors in bone marrow of irradiated mammals. These models have for the first time simulated the stimulating effect of prolonged irradiation with small dose rates on recovery processes in the lymphopoietic system.

    PMID: 2709752, UI: 89217728


    Radiobiologiia 1988 Nov-Dec;28(6):796-802

    [A mathematical model of the dynamics of granulocytopoiesis in mammals].

    [Article in Russian]

    Zukhbaia TB, Smirnova OA

    A mathematical model has been developed for the dynamics of granulocytopoiesis in mammals subjected to chronic irradiation. The model involves a chalones mechanism of haemopoiesis regulation and comprises 12 nonlinear differential equations. The simulation results agree with the experimental data concerning the dynamics of granulocytopoiesis in rats affected by radiation within a wide range of dose rates.

    PMID: 2975393, UI: 89113671


    Antibiot Khimioter 1988 Oct;33(10):767-71

    [Study of the combined action of an antibiotic and an immunostimulator using mathematical modeling].

    [Article in Russian]

    Churnosov EV, Iurkevich IuV, Tsyplenkov PV

    A mathematical model of antibiotic and immunostimulator (IMS) combined effect on various elements of the immune system and general state of patients with infectious diseases is described.

    The model was constructed as a system including 6 usual differential equations of the 1st order. With the use of this model and a computer many diverse variants of infection development under conditions of treatment with IMS at the background of antibiotic therapy were modeled.

    The results of the mathematical modeling corresponded to the data on protective effect of salmozan (IMS) and doxycycline (antibiotic) combination in animals (albino mice).

    It was concluded that the described mathematical model was adequate for validation and optimization of schemes for combined use of IMS and antibacterial agents.

    PMID: 3214212, UI: 89104542


    Radiobiologiia 1988 Sep-Oct;28(5):626-31

    [Experimental and theoretical study of the dynamics of lymphopoiesis during prolonged irradiation].

    [Article in Russian]

    Zukhbaia TM, Smirnova OA

    A mathematical model has been developed describing the dynamics of lymphopoiesis in mammals chronically exposed to ionizing radiation: it is based on the chalone mechanism of regulation of the rate of bone marrow blast cell reproduction and represents a system of 9 nonlinear differential equations. The results of the mathematical simulation and experimental evidence have been found to agree within a wide range of dose rates.

    PMID: 3194493, UI: 89058074


    J Theor Biol 1987 Nov 21;129(2):141-62

    Model analysis of the bases of multistationarity in the humoral immune response.

    Kaufman M, Thomas R

    Universite Libre de Bruxelles, Service de Chimie-Physique II, Belgium.

    A formal analysis of the regulation of antibody production has been developed. It comprises two complementary approaches: a logical analysis in terms of discrete (boolean) variables and functions and a more classical analysis in terms of differential equations. A first paper dealt mostly with the logical description which provided global information on how complex the network needs to be in order to account for some main aspects of the immune response, without having to specify the details of the cellular interactions or to introduce a great number of parameters. Here we present the continuous approach and, in particular, a detailed study of the steady states and a discussion of their role in the dynamics of the immune response. The model subject to this analysis is a minimal one, which takes into account a small number of well-established facts concerning lymphocyte interactions and some reasonable assumptions. The core of the model is a negative feedback loop between the helper (TH) and suppressor (TS) T lymphocytes on which autocatalytic loops of the TH and TS populations on themselves are grafted. The salient feature of this minimal scheme is the prediction, for given environmental and parametrical conditions, of a multiplicity of steady states. This multistationarity occurs both in the absence of antigen or for constant antigen levels. Variations in the external constraints provoke switches among the steady states which might be related to the various modes of the humoral immune response, and depend on the doses of antigen injected and on the previous antigenic history of the system. In particular, high and low dose paralysis appear to be associated with two distinct steady state branches.

    PMID: 2458507, UI: 88333839


    Vopr Pitan 1987 Nov-Dec;(6):39-42

    [Mathematical modelling of perorally induced immunological tolerance].

    [Article in Russian]

    Sharmanov TSh, Kurmangalinova IA, Nikitin SA, Chernov GIa, Kurmangalinov SM

    A mathematical model of the mechanism of development of orally induced immunologic tolerance has been suggested. The model presents a system of differential non-linear equations, and it is realized as a program in FORTRAN. The model describes primary and secondary immune responses, reflects the main features of the immune system response to antigen intake with food. The immune system model response to varying doses and frequency of the antigen intake with food has been studied. It has been established that repeated administration of small doses of the food antigen leads to a deeper tolerance due to lower stimulation of the immune system. The existence of optimal tolerogenic doses of the food antigen has been proved. Qualitative changes in the immune system response to the food antigen have been recorded in case of increased permeability of the intestinal wall.

    PMID: 3439085, UI: 88146667


    J Math Biol 1987;24(6):691-722

    Transient behavior of a chemotaxis system modelling certain types of tissue inflammation.

    Alt W, Lauffenburger DA

    A spatially-distributed mathematical model for the inflammatory response to bacterial invasion of tissue is proposed which includes leukocyte motility and chemotaxis behavior and chemical mediator properties explicitly. This system involves three coupled nonlinear partial differential equations and so is not amenable to analysis. Using scaling arguments and singular perturbation techniques, an approximating system of two coupled nonlinear ordinary differential equations is developed. This system now permits analysis by phase plane methods. Using the approximating model, the dependence of the dynamic behavior of the inflammatory response upon key process parameters, including leukocyte chemotaxis, is studied.

    PMID: 3572263, UI: 87196990


    J Theor Biol 1986 Sep 7;122(1):33-67

    A neural network model based on the analogy with the immune system.

    Hoffmann GW

    The similarities between the immune system and the central nervous system lead to the formulation of an unorthodox neural network model. The similarities between the two systems are strong at the system level, but do not seem to be so striking at the level of the components. A new model of a neuron is therefore formulated, in order that the analogy can be used. The essential feature of the hypothetical neuron is that it exhibits hysteresis at the single neuron level. A network of N such neurons is modelled by an N-dimensional system of ordinary differential equations, which exhibits almost 2N attractors. The model has a property that resembles free will. A conjecture concerning how the network might learn stimulus-response behaviour is described. According to the conjecture, learning does not involve modifications of the strengths of synaptic connections. Instead, stimuli ("questions") selectively applied to the network by a "teacher" can be used to take the system to a region of the N-dimensional phase space where the network gives the desired stimulus-response behaviour. A key role for sleep in the learning process is suggested. The model for sleep leads to prediction that the variance in the rates of firing of the neurons associated with memory should increase during waking hours, and decrease during sleep.

    PMID: 3796008, UI: 87088312


    Cancer Immunol Immunother 1986;23(3):159-64

    A survey of some formal models in tumor immunology.

    Dullens HF, Van der Tol MW, De Weger RA, Den Otter W

    Computer technology has acquired an important role in structuring a variety of biological systems. The availability of modern powerful computers has stimulated the development of good and accurate models of biological systems. Biological systems, such as the immune response against cancer, are complex and it is difficult to experimentally control all the interacting elements constituting the immune response of a host to cancer. Complex biosystems do not always behave or act as expected during experimental investigation. In these cases computer models can be helpful in understanding the behavior of such complex systems. The purpose of this review is to consider the use of mathematical models to study the immune response against cancer. The logic and design of some operable models relevant for tumor immunology will be discussed. Special attention is given to the conceptualization of a model based upon a new hypothesis of tumor rejection presented by De Weger et al. [10]. Technical details concerning the mathematical aspects, differential equations, details on hardware and software package etc. are not included in this survey. These details are contained to in the original papers.

    PMID: 3491679, UI: 87078138


    J Theor Biol 1985 Jun 21;114(4):615-40

    Cluster formation in a symmetrical network: a dynamical system for the description of the suppression among non-immune T lymphocytes and its application to the effects of immunization.

    Fey K, Eichmann K

    A mathematical model has been developed for the description of the suppressive regulation between polyclonally activated normal and immune T cells. The model assumes reversible cell-cell interactions to interpret results from limiting dilution experiments performed to determine the frequencies of precursor cells for antigen-specific T effector lymphocytes and to analyse mechanisms regulating the maturation of precursor into effector T cells. In particular, the model deals with the changes induced in the T lymphocytes population following immunization with antigens. In these limiting dilution experiments, T cells are placed in cultures at varying cell numbers with all other essential culture constituents kept in excess. After polyclonal activation of the T cells in culture they are supplied with growth and maturation factors so that they form daughter clones of functionally active T effector cells. The typical result observed was that effector T cells develop in cultures at low cell input but that this development is totally suppressed at high cell numbers. This result suggested that, at high cell numbers, the effector T cells are exposed to a sufficient number of other T cells of appropriate specificity to permit suppressive interactions. Whereas this is the case for non-immune T cells, T cells after immunization develop into effector cells both at high as well as at low cell concentrations, though with efficiencies less than proportional to their number of precursors. Our mathematical model is made up of a set of first order autonomous ordinary differential equations in many variables permitting the calculations of numbers of free cells and of cells engaged in cellular clusters of varying sizes. Free cells can develop into effector cells whereas cells engaged in clusters cannot. We calculate the consequences of several reasonable hypotheses concerning the effects of immunization. We consider the possibility that immunization modifies the growth behavior of the antigen-specific cells to permit an increased or accelerated clonal expansion in culture. Alternatively, we consider the possibility that immunization changes the interaction strength between cells specific for the immunizing antigen and other cells. Thirdly, we have connected both behaviors by calculating the case of an inverse relationship between growth rates and intensities of interaction between cells. Our model has been inspired by the symmetrical network model and can be interpreted in this framework. It proposes that immune regulation is a consequence of idiotype-anti-idiotype interactions.

    PMID: 3875001, UI: 85266287


    J Theor Biol 1985 Jun 21;114(4):527-61

    Towards a logical analysis of the immune response.

    Kaufman M, Urbain J, Thomas R

    We present a new way to conceive, formalize and analyse models of the immune network. The models proposed are minimal ones, based essentially on the well-established negative feedback loop between helper and suppressor T cells. The occurrence of T-T interactions in both helper and suppressor circuits. These T-T interactions are represented here by autocatalytic feedback loops on TH and TS. The fact that immature B cells are sensitive to negative signaling, as was originally suggested by Lederberg (1959). There is a functional inactivation of immature B cells encountering antigen or anti-idiotypic antibody. This prevents further differentiation to a stage where the B cells become fully responsive. We describe the role of a logical method in the generation and analysis of the models, and the complementarity between this logical method and the more classical description by continuous differential equations. Logical analysis and numerical simulations of the differential equations show that the emerging model accounts for, the occurrence of multiple steady states (a virgin state, a memory state and a non-responsive state) in the absence of antigen, the kinetics of primary and secondary responses, high dose paralysis, low dose of paralysis. Its fit with real situations is surprisingly good for a model of this simplicity. Nevertheless, we give it as an example of what can now be done in the field rather than as a stable model.

    PMID: 3875000, UI: 85266281


    Kosm Biol Aviakosm Med 1985 Jan-Feb;19(1):77-80

    [Mathematical model of cyclic kinetics of granulocytopoiesis].

    [Article in Russian]

    Smirnova OA

    A model of time-course variations of granulocytopoiesis which is a system of three non-linear differential equations has been developed. The model describes the basic stages of granulocyte development and includes the chalone mechanism regulating the proliferation of granulocyte precursors in bone marrow. Theoretical investigations applying the vibration theory and computer-aided calculations have shown that the model presents aperiodic and vibrational kinetics of reduction processes in the system of granulocytopoiesis as well as steady-state vibrations of concentrations of mature granulocytes and their precursors (limiting cycles). The variations of the model parameters within which the above dynamic modes occur have been identified. The conditions under which the limiting cycles arise have been examined. The fact that the model simulates various experimentally observed situations suggests that it can be used to predict changes in granulocytopoiesis induced by adverse effects responsible for hemopoietic abnormalities.

    PMID: 3974189, UI: 85135799


    Biorheology 1983;20(1):41-56

    Theoretical modeling of filtration of blood cell suspensions.

    Skalak R, Impelluso T, Schmalzer EA, Chien S

    A theoretical model of filtration of suspensions containing red blood cells (RBCs) and white blood cells (WBCs) has been developed. Equations are written for the pressure drop, the filtration flow and the fractions of filter pores containing RBCs (alpha) and WBCs (alpha*). Because the relative resistances (ratios of resistance of cell to resistance of suspending fluid) of RBCs (beta) and WBCs (beta*) through the filter pore are greater than one, the transit of these cells (especially WBCs) through the filter is slower than that of suspending fluid; this leads to values of alpha and alpha* higher than those simply expected from the hematocrit and leukocrit, respectively, in the entering and exiting suspensions. In the absence of pore plugging by the cells (steady flow), the pressure drop can be computed from alpha, alpha*, beta and beta*. In order to model unsteady flow, differential equations are written to include pore plugging and the subsequent unplugging by the rising filtration pressure at a constant flow. By specifying the fractions of entering RBCs (epsilon) and WBCs (epsilon*) which would plug the pores and the rate at which the plugged pores would unplug in response to pressure rise (epsilon u), as well as the fractions of entering RBCs (epsilon p) and WBCs (epsilon p*) that would plug the pores permanently, theoretical pressure-time curves can be generated by numerical integration, and the results fit the experimental data well. From such fitting of theoretical curve to experimental data, information can be deduced for epsilon, epsilon*, epsilon u, epsilon p and epsilon* p.

    PMID: 6871425, UI: 83257696


    J Math 1982;15(2):173-201

    A mathematical model of iron metabolism.

    Franzone PC, Paganuzzi A, Stefanelli M

    A mathematical model of iron metabolism is presented. It comprises the following iron pools within the body: transferrin-bound iron in the plasma, iron in circulating red cells and their bone marrow precursors, iron in mucosal, parenchymal and reticuloendothelial cells. The control exerted by a hormone, called erythropoietin, on bone marrow utilization of iron for hemoglobin synthesis is taken into account. The model so obtained consists of a system of functional differential equations of retarded type. Most model parameters can be estimated from radiotracer experiments, others can be measured and numerical values can be assigned to the remaining ones making few reasonable assumptions according to the available physiological knowledge. Iron metabolism behavior under different therapeutical treatments was stimulated. Model predictions were compared to experimental data collected in clinical routine.

    PMID: 7153668, UI: 83110616


    J Immunol 1981 Jun;126(6):2443-9

    A kinetic analysis of immune-mediated clearance of erythrocytes.

    Meryhew NL, Runquist OA

    A mathematical expression has been derived that successfully correlates the kinetic data for the immune-mediated clearance of red blood cells. The expression resulted from the solution of differential equations arising from a clearance mechanism that was, essentially, consistent with that described by Schreiber and Frank. The mathematical expression correlated data for both IgG-and IgM-mediated reactions. Four different rate constants appear in the final kinetic equation; these constants, which measure the rates of the various steps in the clearance process, were evaluated by an iterative curve-matching process. The values of the rate constants were found to be dependent upon type of sensitizing immunoglobulin, number of C1-fixing sites, and several known immune system modifiers. Correlation of the derived rate expression with the experimental data provided a critical test for the Schreiber-Frank mechanism and the values of the rate constants provided additional insights into the immune clearance process.

    PMID: 7229383, UI: 81193120


    J Math Biol 1981;13(1):67-86

    A mathematical model of B lymphocyte differentiation: control by antigen.

    Klein P, Sterzl J, Dolezal J

    A mathematical model of B lymphocyte differentiation, based on experimental results, has been developed. The model focuses on the role of antigen in initiating and regulating B cell differentiation while other mechanisms, acting in concert with antigen but the functioning of which can be circumvented under appropriate conditions, are not considered. The importance of presence of antigen at individual stages of B cell differentiation was studied in experiments with an easily metabolizable antigen. Immunocompetent cells (ICC), arising by antigen-independent differentiation of stem cells, are activated by antigen (they become immunologically activated cells--IAC). Excess of antigen drives IAC into the terminal stage (antibody-forming cells--AFC) thereby restricting proliferation. Exhaustive terminal differentiation results in tolerance. A low primary dose permits IAC to escape antigen; IAC proliferate and later give rise to resting memory cells (MC) which are amenable to reactivation. MC have higher avidity for antigen (due to higher affinity, number and density of receptors) and the effect of different doses of antigen on MC is diverse. A very low secondary dose induces tolerance, a medium dose secondary response, and the administration of a high dose of antigen also brings about tolerance. The model suggests that the fate of memory cells is controlled by the ratio R:Ag, of the number of immunoglobulin receptors on B cells (R) to the number of available antigenic molecules (Ag), low values R:Ag favouring stimulation to differentiation while high values of R:Ag favouring inactivation. A nonlinear system of ordinary differential equations, describing the development of the populations involved in antigen-driven B cell differentiation, was used to simulate experiments and good qualitative agreement was achieved.

    PMID: 6977606, UI: 82144167


    J Math Biol 1980 Aug;10(1):1-12

    A mathematical model of canine granulocytopoiesis.

    Steinbach KH, Raffler H, Pabst G, Fliedner TM

    The granulocyte cell renewal system of the dog is represented by a mathematical model consisting of the following compartments: The pool of pluripotential stem cells, the committed stem cell pool, divided into a blood and a bone marrow compartment, the proliferation pool, the maturation pool, the reserve pool and the blood pool of functional granulocytes. This chain of compartments is described by a system of non-linear differential equations. Cell losses anyplace in the system provoke increased production in all pools containing cells capable to divide. A reduced number of granulocytes in the blood pool stimulates production of a "granulocyte releasing factor" which mobilizes a rising number of cells to transit from the marrow reserve into the blood pool. The model was simulated on a digital computer. It was found to be capable to reproduce the steady state conditions and it also fits the data of two distinct experimental perturbations of the system both equally well. These perturbations are a loss of proliferating cells as it occurs after the administration of cytostatic drugs and losses of functional cells as they are induced by leukapheresis experiments of differing leukapheresis rates.

    PMID: 7205074, UI: 81143739


    Cell Tissue Kinet 1975 Mar;8(2):153-69

    A compartmental analysis of circulatory lymphocytes in the spleen.

    Hammond BJ

    A tentative model describing the passage of circulatory lymphocytes through the spleen is formulated in accord with known anatomical features. In order to preserve isomorphism between the model and the splenic system, the model is formulated in compartmental form and its design allows alternative routes and modes of lymphocyte transit to be considered. The simultaneous differential equations arising from the model are solved using an analogue computer which also provides the means whereby the performance of the model may be compared with suitable dynamic data drawn from literature. This not only allows the selection of a particular configuration of the model in preference to its alternatives, but also allows the numerical determination of certain unknown parameters. In the case of the rat spleen, best agreement between model and experimental data is obtained when between 10 and 25% of the total lymphocyte flux in the model spleen passes through the marginal zone where the average dwell time of the lymphocytes is about 50 min. The white pulp receives a lymphocyte flux from the marginal zone amounting to about 10% of the total splenic flux and the white pulp lymphocytes are sequestered for a period of 4-6 hr before release to the venous circulation. The red pulp receives 90% of the total splenic flux but the majority of lymphocytes find transit through the red pulp in less than 5 min. The remaining flux of lymphocytes, amounting to 10% of the splenic input, is delayed in transit through the red pulp by 2-3 hr before release to the venous circulation.

    PMID: 1168543, UI: 75148287

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