This paper presents a mathematical model for the growth of a cancer micrometastasis in the form of a vascular cuff. The model postulates the possibility of a local imbalance between the rate of cell proliferation and the rate of cell death through apoptosis which is taken as dependent on the concentration of an angiogenesis-inhibitor such as angiostatin. This imbalance produces non-zero cell velocities within the micrometastasis. The local cell velocity is related to an interstitial pressure gradient through a Darcy's Law type of equation, and the spatio-temporal development of the micrometastasis in an environment with a non-uniform nutrient concentration is followed by treating its outer boundary as an advancing front.

Malcolm I. G. Bloor

Michael J. Wilson

The Non-Uniform Spatial Development of a Micrometastasis

The potential distribution in a three dimensional model of the abdomen during pregnancy was studied numerically. Magnetic Resonance Imaging scans were used to construct a three dimensional model with eight different compartments of the abdomen: fetal body, vernix caseosa layer, placenta, uterus fluid, soft tissue, spinal cord, fat and muscles. Each compartment was assigned with its own conductivity and the integral form of the conservation volume conductor equation was solved by a finite volume method. The surface potential distribution of the maternal abdomen was studied for different compartment conductivity and for varying dipole source locations and orientations in the fetal heart. The present study supports the hypothesis that the reduction in the amplitude of the abdominal fetal electrocardiogram around the 30th week of gestation is related to the appearance of a low conductive isolating layer around the fetus. The study shows that an opening in the vernix caseosa layer increases the amplitude of the abdominal fetal ECG with the creation of a preferred signal pathway. It is also shown that the surface potential distribution depends on the fetal posture. Thus, choosing the correct bipolar recording and the appropriate sites of the recording electrodes may result in an increase in the peak to peak amplitude of the fetal ECG.

Shimon Abboud

Michael Arbel

Moshe Rosenfeld

Izuka Udom-Rice

Dan Skupski

Deborah Friedman

Frank A. Chervenak

The Potential Distribution due to a Dipole Source in the Fetal Heart: A Numerical Model of the Maternal Abdomen

Modeling a hormone system requires a number of simplifying assumptions. Often, the final conceptual model incorporates a number of aggregated processes that have no correspondence with a single enzyme-catalyzed reaction. In such cases, it is discussible using models based on classical biochemical kinetics rate laws that are valid only under specific conditions. The power-law formalism provides an alternative framework for building up a mathematical model in such cases. The resulting model is a set of ordinary differential equations with a special structure that allows efficient symbolic and numerical analysis of the system's properties. In these equations, the underlying rate-laws of each of the component processes are represented by a power-law that is an exact representation of the actual rate-law at the operating point. The particular form of these equations allows representation of a wide range of kinetic features without changing the basic power-law form. Moreover, its parameters have an immediate interpretation as apparent kinetic-orders and rate-constants. This is especially helpful for incorporating both quantitative and qualitative information in the process of model definition. This is particularly useful when detailed kinetic information concerning system's components is not available. In this paper we show the utility of the power-law approach in this context by deriving an illustrative model of a complex physiological system: the hypothalamus-anterior pituitary-thyroid network. First, we derive a conceptual model that incorporates the key features of this system. Then, we derive an S-system model, one of the preferred variants within the power-law formalism, and we show its utility in exploring the system properties. The model qualitatively reproduces the response of normal, hyperthyroid, and hypothyroid patients to a clinical test involving a thyrotropin releasing hormone injection. Finally, we illustrate the utility of this modeling strategy for studying the system's response to different dynamic patterns of regulatory signals, and for exploring how altered dynamic patterns of stimulatory signals can cause pathological states.

Albert Sorribas

Angel Gonz&#xe1;lez

The Power-Law Formalism as a Tool for Modeling Hormonal Systems

A dynamical network model of insulin-glucose interactions in subjects with Type I Diabetes was developed and applied to data sets for 40 subjects. Each data set contained the amount of dextrose + insulin infused and blood glucose (BG) determinations, sampled every 5 minutes during a one-hour standardized euglycemic hyperinsulinemic clamp and a subsequent one-hour BG reduction to moderate hypoglycemic levels. The model approximated the temporal pattern of BG and on that basis predicted the counterregulatory response of each subject. The nonlinear fits explained more than 95% of the variance of subjects' BG fluctuations, with a median coefficient of determination 97.7%. For all subjects the model-predicted counterregulatory responses correlated with measured plasma epinephrine concentrations. The observed nadirs of BG during the tests correlated negatively with the model-predicted insulin utilization coefficient (r = -0.51, p &lt; 0.001) and counterregulation rates (r = -0.63, p &lt; 0.001). Subjects with a history of multiple severe hypoglycemic episodes demonstrated slower onset of counterregulation compared to subjects with no such history (p &lt; 0.03).

Boris P. Kovatchev

Leon S. Farhy

Daniel J. Cox

Martin Straume

Vladimir I. Yankov

Linda A. Gonder-Frederick

William L. Clarke

Modeling Insulin-Glucose Dynamics during Insulin Induced Hypoglycemia. Evaluation of Glucose Counterregulation

In this paper we build on the mathematical model of Ward and King (1998) to study the effects of high molecular mass mitotic inhibitors released at cell death. The model assumes a continuum of living cells which, depending on the concentration of a generic nutrient, generate movement (described by a velocity field) due to the changes in volumes caused by cell birth and death. The necrotic material is assumed to consist of two diffusible materials: 1) basic cellular material which is used by living cells as raw material for mitosis; 2) a generic non-utilisable material which may inhibit mitosis. Numerical solutions of the resulting system of partial differential equations show all the main features of tumour growth and heterogeneity. Material 2) is found to act in an inhibitive fashion in two ways: i) directly, by reducing the mitotic rate and ii) indirectly, by occupying space, thereby reducing the availability of the basic cellular material. For large time the solutions to the model tend either to a steady-state, reflecting growth saturation, or to a travelling wave, indicating continual linear growth. The steady-state and travelling wave limits of the model are derived and studied, the regions of existence of these two types of long-time solution being explored in parameter space using numerical methods.

J. P. Ward

J. R. King

Mathematical Modelling of the Effects of Mitotic Inhibitors on Avascular Tumour Growth

In this paper we present a simple mathematical model to describe the initial phase of placental development during which trophoblast cells invade the uterine tissue as a cells invade the uterine tissue as a continuous mass of cells.The key physical variables involved in this crucial stage of mammalian development are assumed to be the invading trophoblast cells, the uterine tissue trophoblast derived proteases that degrade the uterine tissue, and protease inhibitors that neutralise the action of the protease.Numerical simulations presented here are in good qualitative agreement with experimental observations and show how changes in the system parameters influence the rate and degree of trophoblast invasion.In particular we suggest that chemotactic migration is a key feature of trophoblast invasion and that the rate at which proteases are produced is crucial to the successful implantation of the embryo.For example both insufficient and excess production of the proteases may result in premature halting of the trophoblasts. Such behaviour may represent the pathological condition of failed trophoblast implantation and subsequent spontaneous abortion.

M. H. Byrne

M. A. J. Chaplain

G. J. Pettet

D. L. S. McElwain

A Mathematical Model of Trophoblast Invasion

The distribution of the stem villi arteries and arterioles diameter was evaluated on histological sections from 6 normal (control) and 8 Intrauterine Growth Restricted (IUGR) placentas. No significant difference between the diameter distributions was found. This result can be due either to a similar pattern in vascularization or to the inadequacy of the diameter distribution in showing existing differences. Further investigation is needed to make a distinction between these two hypotheses. A multiparametric model is proposed, which predicts the diameter distribution that is expected when a few parameters (describing the vessel geometry and the tree branching structure) are varied in their normal range. The diameter distribution is shown to slightly change or not to change at all when important parameters, such as the vessel asymmetry, are changed. We conclude that the diameter distribution is insensitive to variation in certain placental structural parameters. A reliable analysis of the stem villi vasculature in normal and TUGR placentas cannot be based on the comparison of the respective diameter distributions alone.

C. Guiot

T. Todros

P. G. Piantà

A. Sciarrone

G. Kosanke

P. Kaufmann

The Diameter Distribution of the Stem Villi Arteries Does Not Discriminate between Normal and Intra Uterine Growth Restricted Placentas

We show that the mechanical properties of the outer covering (or capsule) of pig spleen can be modelled as a nonlinear elastic membrane with an exponential stress-strain Law. Knowledge of the caspule's elastic properties is important for the development of a virtual reality training package for keyhole surgery. Bccause capsule tissue is very fragile and tears when clamped and pulled it is not possible to obtain experimental data for which the usual uniform stretch data-fitting approach can be adoptcd.Instcad we perform experiments in which the capsule deforms non-uniformly and describe how numerical solutions of the corresponding boundary value problem can be fitted to the experimerimental data in order to validate the biomechanical model.

Penny J. Davies

Fiona J. Carter

David G. Roxburgh

Alfred Cuschieri

Mathematical Modelling for Keyhole Surgery Simulations: Spleen Capsule as an Elastic Membrabe

Early detection of the self-limiting or chronic tendency towards disease at the beginning of the hepatitis process is a canonical element of the analysis and synthesis to formulate the diagnosis and to decide the most useful and economic therapeutic strategy. Early treatment with antiviral drugs before liver functions undergo deterioration may reduce death and disability especially in young patients.A formula suggested by a Max Planck equation provides mathematical curves that can be obtained very early during the course of hepatitis using just a few initial serum measurements of the alanine-aminotransferase (ALT) enzyme. The clinical ALT data forming the unpredictable early natural curve of ALT can be transformed into mathematical values capable of providing a deterministic curve that intersects the normal ALT value at what is known as the G point, and allows some predictions to be made concerning the acute or chronic evolution of the disease.The G point seems to be a good predictor of whether or not a patient with clinically evident first episode of viral hepatitis will develop chronic liver disease as it revealed the initial phase of the first cytolytic episode in 56 untreated patients of our 59 cases; the three wrong predictions involved patients being treated with interferon.

Nicola Dioguardi

Antonio Barbieri

Renato Dacc&#xf2;

Fabio Grizzi

A Simple Mathematical Method to Predict the Course of a First Episode of Liver Viral Cytolysis

Simple modeling is proposed to represent the screening of gene polymorphisms for association with a progressive disease of insidious onset such as Alzheimer's disease. The modeling demonstrates that when a polymorphism affects the rate of progression as well as the risk of disease, the correct interpretation of DNA data requires an accurate sampling of the living, diseased population. Furthermore, in this population, the effect of the polymorphism on disease risk cannot be distinguished from a corresponding effect on the rate of progression of the disease, and a null result does not preclude a significant effect of the gene on the disease. By contrast, when the population is sampled either at time of diagnosis or at autopsy, the effect of the polymorphism on disease frequency can be directly related to the frequency of the polymorphism in the sample, but evaluating the rate of disease progression requires additional data. When the only available data are obtained from a live patient population, substantial differences in interpretation can result from subtle differences in the patient selection protocol. When existing DNA databases are used in which this protocol is not well characterized, there is a corresponding uncertainty introduced into the deduced effect of the polymorphism on disease risk and rate of progression.

Larry Baum

Eric Baum

Progressive Diseases: Interpretation of Genetic Data

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Computational and Mathematical Methods in Medicine - Table of contents 1999

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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