A survey is presented of some mathematical problems encountered in biological studies. A brief description of the problems to be discussed are as follows. 1. Blood flow: The mean flow of blood through arteries and veins does not behave in accordance with Poiseuille's law all the time. 2. Tracer analysis: The inferences to be drawn from radioactive tracer studies of physiological systems, called compartment analysis, is an inverse problem. 3. Cell populations: The growth of cell populations is conveniently described by differential equations which utilize either age and time or maturity and time as independent variables. I would like to take this opportunity to tell you about some of the mathematical problems I have encountered in studying biological phenomena during the past ten years or so. These phenomena are all describable by differential equations. However, I think you will see that the problems with which they are associated have novel features when compared to those usually encountered in engineering and the physical sciences. My descriptions will be unsatisfactory in the sense that they will be very brief. However, brevity will permit me to present you with an overview of several biomathematical problems deriving from biological experience. Steady blood flow. I will describe first a problem in biomechanics, because the latter is one area of biology in which the underlying mathematical theory is very well understood. When the mean steady volume flux of blood through the vena cava of a dog is measured as a function of the pressure difference between the upstream end and the downstream end, a curve is determined [2], as illustrated in Figure 1. Similar observations have been made of viscous fluid flow through a rubber tube [2], as shown in Figure 2. In fact, the observed blood flux through the entire circulatory system of the dog displays similar properties [2]-[5]. In determining the curves of Figures 1 and 2, the upstream pressure was held fixed while the downstream pressure was decreased. The noteworthy feature of these experiments is that when the pressure difference increases beyond a certain value, the flux does not increase, unlike the flow through a rigid tube. A simple theory of viscous fluid flow through a tube can account for these observations provided the collapsibility of the tube is accounted for [6]. In the This article is the text of an address presented to the American Mathematical Society at its meeting in New York on April 10-13, 1974. It is based on some of the material presented in the author's monograph [1]; received by the editors October 16, 1974. AMS (MOS) subject classifications (1970). Primary 92A05.

/pdf/some-mathematical-problems-in-biology-47a70i1ayy.pdf

Some mathematical problems in biology

Identifiability of Model Parameter

Structures are unidentifiable if there exist several parameter values which lead to the same input-output relations. This paper presents a method, called exhaustive modeling, to obtain the whole set of output indistinguishable linear compartmental models. The method uses standard decomposition of a minimal representation into four mutually exclusive parts, namely: observed-controlled, observed-uncontrolled, unobserved-controlled, unobserved- uncontrolled. The connection of this standard decomposition with Kalman's canonical form is developed. Several simple examples are included to illustrate the method.

Unidentifiable compartmental models: what to do?

This paper analyzes the reasons for the current marked interest shown by workers in the biosciences to applications of compartmental structures, both in regard to the special analytic properties of compartmental structures and the physiological properties to be attached to the compartments and intercompartmental flows. This paper presents the various classes of compartmental structure (perturbation and tracer, time varying, nonlinear, and stochastic), examines their fundamental properties (realizability, stability, observability and controllability, identifiability, and decomposability), considers practical aspects of identification, including computer programs, and reviews the state of the art in the areas of metabolism, pharmacokinetics, ecology, and chemical kinetics. The object of the review is to highlight the main streams of work via key references.

Compartmental System Analysis: State of the Art

A number of inconsistencies, misunderstandings and ambiguities have arisen in the recent literature regarding identifiability concepts, evidenced in part by the number of "point-counterpoint" notes and comments published on this subject. It appears that the problem is due fundamentally to a lack of a unified set of definitions. A partially new, simple and hopefully complete set of definitions is proposed here for deterministic models. They are based on an extended model, termed the "constrained structure"-appropriate for complete analysis of identifiability properties of a system. They rely minimally on special jargon and are consistent with reported definitions for notions of stochastic identifiability. Relationships with other definitions are discussed and it is also shown that an important class of identifiable models are not necessarily uniquely observable/reconstructible. The latter problem has important implications in applications.

On parameter and structural identifiability: Nonunique observability/reconstructibility for identifiable systems, other ambiguities, and new definitions

A review is presented of the central mathematical problem of compartment analysis: What is the set of coefficient matrices appearing in the compartment equations that are compatible with the part of the solution that is observed? The matrix elements are shown to depend on a generally underdetermined set of nonlinear algebraic equations. Some general properties of the coefficient matrix for an n -compartment system are deduced. The matrix may be made “minimal” by the imposition of null constraints on the matrix elements. The compatibility of such arbitrary constraints with the algebraic equations is investigated. The algebraic simplicity of two-compartment and three-compartment systems permits a complete characterization of the set of minimal matrices in parametric form. Properties of the graphs representing some minimal matrices of a four-compartment system are obtained.

Alice Winzer

Papers

Compartment analysis: An inverse problem