Health Phys 2000 Mar;78(3):289-94
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
2:
IEEE Trans Med Imaging 1999 Nov;18(11):1108-14
Publication Types: Letter
PMID: 10661328, UI: 20124944
3:
Theor Popul Biol 1999 Feb;55(1):94-109
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
4:
Mech Ageing Dev 1998 Nov 16;105(3):241-64
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
5:
J Theor Biol 1998 Nov 7;195(1):41-52
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.
PMID: 9802949, UI: 99021701
6:
IMA J Math Appl Med Biol 1998 Sep;15(3):235-56
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
7:
Ultrasound Med Biol 1998 Jun;24(5):621-9
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
8:
J Theor Biol 1998 Jun 7;192(3):283-308
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
To describe both conventional and exhaustive CTL responses in the acute phase of LCMV-D infection 2 mechanisms are invoked:
It is suggested that
PMID: 9650288, UI: 98313974
9:
J Theor Biol 1997 Sep 7;188(1):127-40
We develop a system of ordinary differential equations to model the dynamics of the blood-stages of the malaria parasite, Plasmodium falciparum.
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.
PMID: 9299316, UI: 97446222
10:
J Math Biol 1997 Aug;35(7):775-92
Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we
PMID: 9269736, UI: 97415141
11:
Antimicrob Agents Chemother 1997 Feb;41(2):449-53
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
12:
J Theor Biol 1996 Dec 7;183(3):285-305
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.
Our model allows moreover to interpret
PMID: 9015451, UI: 97167834
13:
Environ Health Perspect 1996 Dec;104 Suppl 6:1293-301
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
14:
Proc R Soc Lond B Biol Sci 1996 Nov 22;263(1376):1487-93
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
15:
Math Biosci 1996 Nov;138(1):1-22
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, 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
16:
J Theor Biol 1996 Oct 21;182(4):513-29
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
17:
Cancer Res 1996 Aug 15;56(16):3771-81
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.
18:
Artif Organs 1996 Aug;20(8):866-77
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
19:
Math Biosci 1996 Aug;136(1):35-63
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
20:
Health Phys 1996 Jun;70(6):787-97
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
21:
Bull Math Biol 1996 Mar;58(2):376-90
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
22:
J Math Biol 1996;34(4):361-412
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
23:
Stem Cells 1995 May;13 Suppl 1:290-300
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
24:
Medinfo 1995;8 Pt 2:1091
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
25:
Int J Biomed Comput 1994 Aug;36(4):293-8
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
26:
Bull Math Biol 1994 Jul;56(4):687-721
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
27:
Proc Natl Acad Sci U S A 1994 Jan 18;91(2):544-8
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
28:
IMA J Math Appl Med Biol 1994;11(2):107-47
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 elucidates explicitly the effects of
29:
J Theor Biol 1993 Oct 7;164(3):271-90
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
30:
Exp Hematol 1993 Jun;21(6):816-22
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
31:
J Theor Biol 1993 May 7;162(1):23-40
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
32:
Math Biosci 1993 Mar;114(1):81-125
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
33:
Bull Math Biol 1993;55(4):745-80
We develop a model for the idiotypic interaction between 2 B cell clones.
This model takes into account
After appropriate nondimensionalization, the model consists of 8 ordinary
differential equations and a number of parameters.
34:
Arch Immunol Ther Exp (Warsz) 1993;41(1):21-31
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
35:
J Pharmacokinet Biopharm 1992 Aug;20(4):319-31
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
36:
Biosystems 1992;26(3):177-83
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
37:
Kosm Biol Aviakosm Med 1991 Sep-Oct;25(5):53-6
PMID: 8577146, UI: 96164137
38:
J Theor Biol 1991 Jul 7;151(1):1-40
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
39:
Health Phys 1991 Jul;61(1):87-95
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
40:
Kosm Biol Aviakosm Med 1991 May-Jun;25(3):40-2
PMID: 1770766, UI: 92122707
41:
Folia Biol (Praha) 1991;37(1):10-20
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
42:
J Theor Biol 1990 May 10;144(1):93-101
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
43:
Biochemistry 1990 Mar 13;29(10):2464-71
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
44:
Cell Biophys 1989 Apr;14(2):139-73
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
45:
Kosm Biol Aviakosm Med 1989 Jan-Feb;23(1):47-51
PMID: 2709752, UI: 89217728
46:
Radiobiologiia 1988 Nov-Dec;28(6):796-802
PMID: 2975393, UI: 89113671
47:
Antibiot Khimioter 1988 Oct;33(10):767-71
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.
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
48:
Radiobiologiia 1988 Sep-Oct;28(5):626-31
PMID: 3194493, UI: 89058074
49:
J Theor Biol 1987 Nov 21;129(2):141-62
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
50:
Vopr Pitan 1987 Nov-Dec;(6):39-42
PMID: 3439085, UI: 88146667
51:
J Math Biol 1987;24(6):691-722
PMID: 3572263, UI: 87196990
52:
J Theor Biol 1986 Sep 7;122(1):33-67
PMID: 3796008, UI: 87088312
53:
Cancer Immunol Immunother 1986;23(3):159-64
PMID: 3491679, UI: 87078138
54:
J Theor Biol 1985 Jun 21;114(4):615-40
PMID: 3875001, UI: 85266287
55:
J Theor Biol 1985 Jun 21;114(4):527-61
PMID: 3875000, UI: 85266281
56:
Kosm Biol Aviakosm Med 1985 Jan-Feb;19(1):77-80
PMID: 3974189, UI: 85135799
57:
Biorheology 1983;20(1):41-56
PMID: 6871425, UI: 83257696
58:
J Math 1982;15(2):173-201
PMID: 7153668, UI: 83110616
59:
J Immunol 1981 Jun;126(6):2443-9
PMID: 7229383, UI: 81193120
60:
J Math Biol 1981;13(1):67-86
PMID: 6977606, UI: 82144167
61:
J Math Biol 1980 Aug;10(1):1-12
PMID: 7205074, UI: 81143739
62:
Cell Tissue Kinet 1975 Mar;8(2):153-69
PMID: 1168543, UI: 75148287
Version: 26. November 2003.
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.
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, carneiro@pasteur.fr
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.
In the model,
The model was used to simulate the following biodistribution data:
Comparisons between simulations and data demonstrated the feasibility of
the model and the scaling scheme.
PMID: 8706023, UI: 96328102
Analysis of nonlinear properties of immune network reactions.
Hirayama H, Nishimura T, Fukuyama Y
Department of Public Health, Asahikawa Medical College, Japan.
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.
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 model for treatment strategy in the chemotherapy of AIDS.
Kirschner D, Webb GF
Department of Mathematics, Texas AM University, College Station 77845,
USA. dek/math.tamu.edu
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.
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.
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 non-linear compartmental model of human basophil activation.
Nkobetchou F, Cherruault Y, Sainte-Laudy J
Laboratoire Medimat, Universite Pierre et Marie Curie, Paris, France.
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.
Diversity and virulence thresholds in AIDS.
de Boer RJ, Boerlijst MC
Utrecht University, The Netherlands.
Modelling and simulation of Rosenberg-type adoptive cellular immunotherapy.
Nani FK, Oguztoreli MN
Department of Mathematics, University of Alberta, Edmonton, Canada.
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.
PMID: 8089590, UI: 94375994
Studies on a recent class of network models of the immune system.
Faro J, Velasco S
Departamento de Fisica Aplicada, Universidad de Salamanca, Spain.
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.
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.
Dynamics of HIV infection of CD4+ T cells.
Perelson AS, Kirschner DE, De Boer R
Theoretical Division, Los Alamos National Laboratory, New Mexico.
Immune network behavior--I. From stationary states to limit cycle oscillations.
De Boer RJ, Perelson AS, Kevrekidis IG
Utrecht University, The Netherlands.
Here we investigate, by means of stability and bifurcation analysis, how
each of the processes influences the model's behavior.
PMID: 8318929, UI: 93306242
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.
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 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.
[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.
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.
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.
[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.
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.
Dynamics of a class of immune networks. I. Global stability of idiotype
interactions.
Varela FJ, Stewart J
CREA, Ecole Polytechnique, Paris, France.
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.
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.
[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.
[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.
[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 results of the mathematical modeling corresponded to the data on protective
effect of salmozan (IMS) and doxycycline (antibiotic) combination in animals
(albino mice).
[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.
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.
[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.
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.
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.
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.
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.
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.
[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.
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.
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.
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.
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.
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.
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.
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