Biomathematics

The following sections are included:

• THE EDITOR

• PREFACE

• CONTENTS

This article first provides a concise introduction to the statistical approach to phylogenetics. It then describes a new method for detecting mosaic structures in DNA sequence alignments, which is based on combining two probabilistic graphical models: (1) a taxon graph (phylogenetic tree) representing the relationships among the taxa, and (2) a site graph (hidden Markov model) representing spatial correlations between nucleotides.

The common scientific approaches to the reasoning of problems are mathematical reasoning or statistical reasoning. Mathematical or formal reasoning is mostly deductive, in that, one reasons from general assumptions to specifics using mathematical logic and axioms for multi-criteria decision-making. The purpose of this chapter is to relate statistical methodology and model design to planning and policy making for a viable solution to the ravaging Human Immune Deficiency Virus (HIV)/Acquired Immune Deficiency Syndrome (AIDS). Demonstrating the benefits that can be derived from adapting the concept and model building approach in planning and decision-making of public health and urban development. Discussion in this chapter is presented the following sequence: (a) introduction, (b) deductive/inductive approach, (c) statistical methodology and model design, (d) adaptation of “Seldom Do” models to human behaviour, (e) the discrete choice modelling and its application to the socio-behaviour of HIV transmission. The chapter concludes that the “Seldom Do” model approach offers potential for addressing the development of planning and multi-criteria decision processes associated with health and urban development problems in our society.

During recent years the medical community has been aggressively searching for a cure of the HIV disease, but so far a cure has not been found. Consequently, the goal of HIV/AIDS treatments has been to impact the health of infected individual and to extend their life-expectancy with the hope that in time the medical community may find a cure. Much success has been reported in improving the life-span of infected individuals. How this increased life-span for infected individuals effects the overall impact of the spread of the disease remains unknown. Thus, while success of HIV treatments, such as the HAART therapy, is beneficial in the infected population, consequences must be considered for the susceptible audience. The investigation described in this article will focus on the heterosexual population; therefore, a thorough investigation must consider the multiple facets present in the heterosexual population. A stochastic model for the heterosexual population with sufficient parameters to model the multiple facets impacting the spread of the disease in this audience is considered. Proximity to threshold conditions specify when the disease will spread if a small number of infected people are introduced into a susceptible population. Determination of distance to the threshold condition for a population participating in HIV treatments and a population without HIV treatments is investigated. Simulations suggest that from the perspective of only successful implementation of HIV treatments to this audience, a more rapid spread of the disease throughout the susceptible sector occurs. The merits of HIV treatments is not in question but, in this era of more advances in HIV treatments, HIV treatments must be coupled with attention to general awareness and further education and prevention efforts, for proper control of the epidemics spread.

This work presents two mathematical models explicitly reflecting the magnetic-field-induced transitions in biologically significant processes: oxidation of n-hexane and linolenic acid, and describes the methodology used in obtaining the models. For the n-hexane oxidation, the range of the magnetic field strength is found (0.05–0.3 T) with the trend indicating a significant magnetic-field-induced change in the reaction rates (up to 50% at 0.2 T). For the linolenic acid oxidation, a pronounced magnetic-field-induced change in the rate of malonaldehyde (MDA) production is found (at 0.1 T). The equations describing the effects of the magnetic field on the photoinduced free radical reaction of oxidation involving a lipid-modeling substance, hexane, and a fatty acid, linolenic acid, are obtained on the basis of chemical kinetics and data from batch experiments. The magnetic-field-induced changes in n-hexane and linolenic acid oxidation are validated (the latter only for diene conjugates) using the identification technique based on the real time input-output data in separately conducted flow-through experiments.

In this paper we describe supercomputer simulations for the self reorganization of tissue which has been separated into endoderm, mesoderm, and ectoderm cells.

Models based on the Murray–Oster continuum framework have been applied to a variety of biological settings in order to investigate the morphogenesis of living tissues. Collagen-matrix contracted by fibroblast model is an important example and a suitable way to study reciprocal geometric and mechanical interactions that regulate wound contraction of connective tissue cells. This contraction, which is due to cell traction forces, is essential in wound healing and pathological contractures. In the present contribution, where thin disk sample geometry is considered, an attempt is made to investigate the effect of initial stress upon the kinematics of contraction. This aspect is probably source of novel insight into the roles of key biological parameters in determining the biomechanical properties of contracted biological gel. Our hope is that this contribution will find a logical sound and contribute to gain a greater understanding of wound contraction mechanism.

The following sections are included:

• Phenomena Associated with Peristalsis

• Physiological Systems Associated with Peristalsis
  • Digestive system
  • Oesophagus
  • Stomach
  • Small intestine
  • Ureter
  • Vas deferens
  • Experimental investigations on peristalsis

• Theoretical Studies on Peristaltic Transport
  • Newtonian flows
  • Non-Newtonian flows
  • Non-stationary initial flows
  • Two-phase flows
  • Two-layer flows

• Flows through Tubes of Non-Uniform Cross-Section

• Numerical Investigations

• References

The following sections are included:

• Introduction

• Mathematical Background
  • Topological tools for DNA analysis
  • Definitions
  • 2-string tangles
  • Rational tangle
  • 4-plats
  • Classification of 4-plats

• Biological Statement and Assumptions

• Tangle Model Assumptions
  • Other substrates

• Site-Specific Recombination

• Processive Recombination

• Useful Facts and Theorems About Tangles

• Model for the Tn3 Resolvase

• Model for the Xer Recombinase and Topoisomerases III and IV
  • Biological model for recombinases and topoisomerases
  • Biological model (unknotted substrates)
  • Biological model (catenated substrates)
  • Tangle equations for unknotted substrates
  • Tangle equations for catenated substrates
  • Problems
  • Results

• Modelling Conclusions

• Acknowledgement

• References

Within the last decade computer models of cardiac excitation spread have become increasingly more realistic due to the interdisciplinary merging of techniques from biophysics, mathematics, cardiology and computer sciences. Computer models are considered as an indispensable complement to experimental and clinical studies. Both experimental as well as clinical methods for the determination of the cardiac activation sequence rely in most instances on the measurement of potentials outside the myocardium. For the interpretation of such measurement a profound understanding of the relationship between electrical processes in the tissue and the electric field caused by them outside the tissue is essential. Monodomain computer models are one of the most frequently used tools for the investigation of this relationship since they represent a balanced trade-off between level of detail and computational tractability. This paper summarizes theoretical basics necessary for the implementation of monodomain computer models for the simulation of the cardiac excitation spread and the concomitant electric field and reviews numerical techniques used for this purpose.

The following sections are included:

• Introduction
  • Wall shear stress and atherosclerosis
  • Arterial stenosis
  • Entry flows
  • Influence of curvature
  • Artefacts of catheters

• Mathematical Formulation
  • Flow geometry
  • Co-ordinate system
  • Governing equations of motion
  • Boundary conditions

• Methods of Solution
  • Perturbation analysis
  • Numerical approach

• Discussions
  • Pressure drop and impedance
  • Wall shear stress
  • Flow behavior

• Concluding Remarks

• References

Biomechanistic and theoretical models can be applied to investigate complex reproductive processes. The application of Laplace equation to follicle rupture helps in obtaining insight into the simultaneous effect of a number of factors pertaining to bursting of the follicle. Gamete transport dynamics via peristaltic analysis for cilia beat has been outlined for the oviduct. The metachronal wave generated in the wall of the vas deferens contradicts peristalsis to be the dominating factor in spermatozoa transport. Biomechanical characterization of the forces involved in the mechanics of sperm-egg interactions are outlined. Generalized Hooks law to obtain displacements for specific load conditions during fetal head moulding and formulation of a fertilization index based on Von Foerster's equation accounting for both epididymal spermatozoa reserves and spermatozoa numbers for a specific ejaculation frequency are described.

Investigation of bioelectricity phenomena has gained recently a steadily increasing interest in medical and engineering applications. This chapter deals with the computational aspects of bioelectromagnetic interactions and their related bioelectric processes, aiming to provide a better physical understanding of the effect of functional electrical stimulation (FES) on biological tissue and to set-up models that can provide quantitative insights into this bioelectromagnetic phenomenon. These goals are achieved here by an explicit image series construction of the macroscopic electromagnetic field within the multilayer tissue. The novel image series expansion scheme, outlined here for quasistatic Green's function in multilayer media, utilizes a unique and explicit recursive representation for Green's function. Our recursive construction convergences under rather general constraints on the media parameters. The usefulness and effectiveness of the proposed analysis is demonstrated through an hybrid scheme, combining image series and moment method procedures, that are capable of handling effectively layered medium problems excited by an electrode array. The inclusion of a collective image term, expressed in a closed form asymptotic evaluation of the series remainder integral, significantly accelerated the image series convergence and the overall algorithm speed and accuracy. This proposed computational procedure can be used as a simple tool for producing analytical data for testing numerical subroutines applied to simulate direct (FES) and inverse (bioelectromagnetic imaging) problems in biomedical applications.

Humanoid robots are human-like, anthropomorphic mechanisms with biodynamics that resembles human musculo-skeletal dynamics. The present chapter enlightens the underlying unique global mathematical structure beneath the general humanoid dynamics (HD, for short). It presents a parallel development of Hamiltonian and Lagrangian formulations of HD, proves both differential-geometrical and algebraic-topological dualities between these two formulations, and finally establishes a unique functorial relation between HD-geometry and HD-topology.

The following sections are included:

• Introduction

• Modeling Human Daily Energy Expenditure
  • Equilibrium models
  • Setpoint analysis and modeling nonequilibrium energy needs
  • Comparing models

• Energy from Fat/Nonfat Body Mass
  • The personnel fat ratio
  • The ratio of nonfat loss to total weight loss
  • The energy density and the energy density ratio

• The Setpoint/Body Composition Adjusted Energy Rate Equation

• Analysis
  • The Minnesota experiment
  • The characteristic time and rate of weight loss
  • Comparing the models in dynamics
  • Comparing the models in equilibrium

• Discussion and Conclusions

• References

We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals.

The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short- and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of the McKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed.

This chapter gives a brief account of various mathematical models developed by different investigators in connection with various studies in Biomechanics of Bones. A mention of some relevant experimental investigations has also been made. For the convenience of readers some basic concepts and some associated topics useful for a better comprehension of this interesting interdisciplinary area of study are also included.

The following sections are included:

• INDEX

• CONTRIBUTORS