Classic multidrug resistance (MDR) is a phenomenon by which cells nonspecifically extrude noxious agents from the cutoplasm before lethal concentrations buils up. Some chemotherapeutically treated tumors exhibit these same dynamics. In tumor systems, the most common mechanism of facilitating MDR is the upregulation of the P-glycoprotein pump. This protein forms a transmembrance channel, and agter binding the chemotherapeutic agent and 2ATP molecules, forces the noxius agent through the channel. Hydrolysis of ATP to ADP provides the energy component of this reaction. General mathematical models describing drug resistamce are reviewed in this article. One model describing the molecular function of the P-glycoprotein pump in MDR cell lines is developed and presented in detail. The pump is modeled as an energy-dependent facilitated diffusion process. A partial differential equation is linked to a pair of ordinary differential equations to form the core of the model. To describe MDR reversal, the model is extended by additing an inhibitor to the equation system. Equations for competitive, one-site non-competitive, and allosteric non-competitive inhibition are then derived. Numerical simulations have been run to describe P-glycoprotein dynamics both in the presence and absence of inhibition, and these results are briefly reviewed. The character of the pump and its response to inhibition are discussed within the comtext of the models.

Seth Michelson

Multidrug Resistance and Its Reversal: Mathenatical Models

The propagation of electrical activity in cardiac tissue can be modelled by reaction diffusion equations, where a tensor of diffusion coefficients represents anisotropy due to fiber orintation, and excitation is represented by high-order, stiff differential systems. The effects of external electrical stimulation, as in artifical pacemakers, or in defibrillators, requires bidomain models, in wich intra- and extracellular currents are treated separately.simplified approaches are taken to this problem to illustrate two methods of defibrillation: by a sinhle large pulse, that eliminates all propagating activity, and by a series of smaller amplitude perturbations, that drive out re-entrant sources of excitation.

Arun V. Holden

Defibrillation in Models of Cardiac Muscle

Electrical stimulation of excitable cardiac tissue results in a variety of rhythms similar to those clinically recorded. Simplified momlinear equations and ionic models give complementary descriptions of the dynamics in cardiac preparations. Nonlinear midels often capture the essential dynamical features of the experimental observations but do not properly describe the underlying ionic mechanisms. Ionic midels are descriptive, complex, and despite their general sucess in reproducing the cardiac action potential, seldom undergo extensive testing against experimentally determined rhythms. We review work from our group devoted to the modeling of cardiac rhythms, including nonlinear models and the Shrier-Clay ionic model of electrical activity in embryonic chick heart cell aggregates. Since sustained stimulation may have long lasting consequences on the intrinsic excitability of the tissue, we also discuss recent advances in modeling the stimulation history-dependent aspects of pacemaking.

Arkady Kunysz

Alvin Shrier

Leon Glass

The Challenge of Complex Dynamics for Theoretical Models of Cardiac Activity

Bayes' theorem is a vehicle for incorporating prior knowledge in updating the degree of belief in light of data. For example, the state of tomorrow's weather can be predicted using belief or likelihood of tomorrow's weather given today's weather data. We give a brief review of the recent advances in the area with emphasis on high-level Bayesian image analysis. It has been gradually recognised that knowledge-based algorithms based on Bayesian analysis are more widely applicable and reliable than ad hoc algorithms. Advantages include the use of explicit and realistic statistic models making it easier to understand the working behind such algorithms and allowing confidence statements to be made about conclusions. These systems are not necessarily as time consuming as might be expected. However, more care is required in using the knowledge effectively for a given specific problem; this is very much an art rather than a science.

K. V. Mardia

Bayesian Image Analysis

Some different types of models of cell proliferation are discussed. The importance of basing the models on experimental data is emphasised, but warnings are given about some of the pitfalls in fitting models of data. The importance of investigating alternative models which might lead to similar experimental findings is stressed. The use of simulation to assess the ability of an analytical method to extract correct information from experimental data is advocated. In this instance, the modelling process takes place in advance of the data collection. Models described relate to cell proliferation in a transplantable tumor, the prostate of the castrate mouse stimulated with testosterone, and stratified squamous epithelium. Experimental techiques include measurment of tumor size, calculation of labelling and mitotic indices over time, and the fraction labelled mitoses method.

David R. Appleton

An Overview of Models of Cell Proliferation

A Survey of cell-population models is presented. The so-called cell cycle is discussed, including some models which are not population models but which contribute to the better understanding proliferation of cell dynamics. Classical linear models of structured cell-population models are described with a review of classical results and a comparison between different approaches. Finally, some non-linear models are also developed.

O. Arino

E. S&#225;nchez

A Survey of Cell Population Dynamics

A Mathematical model of the HIV infected human immune system is presented. The model consists of a system of ordinary differential equations for the populations of uninfected CD4+T cells, infected CD4+T cells and free virus in the blood. The model is constructed, and parameters chosen, so that turnover rates and life spans for these populations agree with clinical data. The model is used to simulate chemotherapy treatment of HIV infection. The simulations are based upon preliminary clinical reports of treatment with combinations of antiviral drugs involving standard reverse transcriptase inhibitors and newly developed protease inhibitors. The models incorporate the apperance of drug-resistant viral strains, which is the key limiting factor in the effectiveness of HIV Chemotherapy. The simulations focus upon the timing of treatment initiation.

Denis E. Kirschner

G. F. Webb

A Mathematical Model of Combined Drug Therapy of HIV Infection

Corneal epithelium has a highly specialided wound-healing response. The biological aspects of this repair process are reviewed, and methods of modelling it mathematically are described. A model which focuses on the source of epidermal growth factor (EGF) within a healing wound id described. By considering mathematical representations of a number of possible source terms, it is shown that the EGF presents in the tear film is insufficient to explain the observed rate of healing, and experimental approaches are suggested for distinguishing between other sources. Also, the simulation of exogenous addition of EGF using the miodel is described. An issue that has been the subject of considerable debate in the literature is the role of eyeball curvature. The model is used to show that this curvature is not significant for either the speed or form of healing in the epithelium. In conclusion, a comparison is made between wound healing in the corneal epithelium with that in the epidermis of the skin. Possible diredtions for future modelling work are considered.

Eamonn A. Gaffney

Philip K. Maini

Jonathan A. Sherratt

Paul D. Dale

Wound Healing in the Corneal Epithelium: Biological Mechanisms and Mathematical Models

Experimental evidence indicates that tumour metastases can exist for long periods in a dormant state, with cell proliferation balancing cell death. However, this balance can be upset, by removing the primary tumour for instance, which causes the metastasis to grow, or by adminstering a substances inhabiting angiogenesis which causes the metastasis toi regress. A mathematical model is presented for the growth of a tumour metastasis, which by postulating the possibility of a local imbalance between cell proliferation and cell death though apoptosis, is able to explain some of these observations. A prediction of the model is that at any position within the metastasis there will be a radial movement of cells, even in the document state.

Malcolm I. G. Bloor

Michael J. Wilson

A Mathematical Model of a Micrometastasis

mathematical modelling of tumour response in breast cancer offers the potential for further understanding of the mechanisms involved in a tumour&#39;s imperfect response to chemotherapy. Three different models of assessing response are studied; the simplest consisting of fitting a regression line to the logarithm of the tumour volumes; a study using exponential growth and an S-shaped growth response curve; and one that assumes log cell-kill and the possibilitu of primary tumour resistance to therapy. All thre can explain some facets of tumour biology, but it is the introduction of the possibility of resistance that appears to result in correlations with clinical outcome. The issue of Gompertz growth is discussed, since it is considered, although not without controversy, to best describe not only xenograft but also clinical tumour growth, and yet has not been used in any of the three models discussed. It appears that much of the data used to clinically validate Gompertz growth is before the period of maximum deceleratin, and thus the true relevance of this function to clinical tumour growth remains uncertain.

D. A. Cameron

Mathematical Modelling of the Response of Breast Cancer to Drug Therapy

The human body is a complex structure, but our understanding of its movements is greately enhanced by extremely simple mathematical models. A model of walking shows why we have to break into a run at speeds above 3 meters per second, and why the critical speed is lower for children and on the moon. A model of running jumps show why long jumpers run up at a fast sprinting speed, but high jumpers run up much more slowly. Finally, a model od standing jumps explains why athletes can jump higher by using a countermovement than from a static squatting position. The body is represented in these models as a small number of rigid segments connected by hinge joints, powered by muscles with realistic physiological properties.

R. McNeill Alexander

Simple Models of Human Locomotion

Cell proloferation and differentiation phenomena are key issues in immunology, tumour growth and cell biology. We study the kinetics of cell growth in the immune system using mathematical models formulated in terms of ordinary and delay differential equations. We study how the suitability of the mathematical models depends on the nature of the cell growth data and the types of differential equations by minimizing an objective function to give a best-fit parameterized solution. We show that mathematical models that incorporate a time-lag in the cell division phase are more consistent with certain reported data. They also allow various cell proliferation characteristics to be estimated directly, such as the average cell-boubling time and the rate of commitment of cells to cell division. Specifically, we study the interleukin-2-dependent cell division of phytohemagglutinin stimulated T-cells — the model of whic can be considered to be a general model of cell growth. We also review the numerical techniques available for solving delau differential equations and calculating the least-squares best-fit parameterized solution.

C. T. H. Baker

G. A. Bocharov

C. A. H. Paul

Mathematical Modelling of the Interleukin-2 T-Cell System: A Comparative Study of Approaches Based on Ordinary and Delay Differential Equation

The development of scarring following accidental injury or surgery is an important clinical problem, often resulting in adverse effects on frowth and function, as well as an undersirable cosmetic appearance. Adult wound healing and scar formation are characterised by acute inflammation, wound contraction and disordered collagen deposition. Similar processes may also be involved in the progression of fibrotic disease states such as pulmonary fibrosis and hepatic cirrhosis. A major clinical objective is therefore, the reduction, and ideally the prevention, of scarring. The embryo and early fetus respond to injury in a way that is fundamentally different from the adult. In general, the early fetus heals rapidly, without scar formation. Extracellular matrix deposition in the fetal wound is rapid and organised in its structure, much more similar in appearance to unwounded skin, whilst, in the adult, matrix deposition is disordered. The inflammatory response, charasterisitic of the early phase of adult wound healing, is absent, or highly limited, in the fetus and the levels of cytokines are generally greatly reduced. Research into fundamental cellular and molecular differences between adult and fetal wound healing have revealed a number o targets in the adult wound which can be manipulated to more closely resemble the fetal wound environment and hence result in the reduction of adult scarring.

R. L. McCallion

M. W. J. Ferguson

Scar-Free Embryonic Wound Healing and the Prevention of Scarring Following Wound Healing in the Adult

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Computational and Mathematical Methods in Medicine publishes research and review articles focused on the application of mathematics to problems arising from the biomedical sciences.

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Computational and Mathematical Methods in Medicine - Table of contents 1997

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Computational and Mathematical Methods in Medicine publishes research and review articles focused on the application of mathematics to problems arising from the biomedical sciences.

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Computational and Mathematical Methods in Medicine