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Mathematical modelling in biomedicine: optimal control of biomedical systems.
BOOK
REVIEW
Y. Cherruault, Mathematical Modelling in Biomedicine: Optimal Control of Biomedical Systems. D. Reidel, Dordrecht, Holland, 1986, xviii+258 pp,, $49.50,
Dfl 130.00, E36.25.
This monograph on biomedical modelling more closely resembles a collection of the author’s work and that of his colleagues at Medimat Laboratory, primarily the numerical solution of optimization (in particular, optimal control) problems arising in biomedical systems. Reading it requires a willingness to sacrifice the desire to see models carefully developed based on biological assumptions and principles in favor of discussing techniques for attaining approximate solutions of various systems of equations. In the Introduction the author states, “The aim of this book is to present mathematical methods for building models of biomedical systems.” The book delivers only half what it promises. There is virtually no building of models. On the other hand, the main mathematical methods presented are data analysis; analytical or numerical solution of differential, partial differential, and integral equations; parameter identification; optimization of functions; and optimal control for finding optimal policies according to one or more criteria. All of these ideas are treated using concrete models and examples. General remarks on modelling are brought out in the first chapter. The author considers compartmental modelling using linear models with constant rate coefficients to be the most popular method in biomathematics and hence devotes an appropriate amount of space to developing these models. More conventional notation in the rate coefficients of the mass-balance equations and the diagrams would have helped so that the usual matrix-vector format could be used. The novice will be dismayed by frequent typographical errors that must be corrected to understand fully many basic equations in tbis chapter. Chapter 2 is concerned with identification and control in linear compartmental models. Initially there is discussion of identification for particular two and three compartment examples, where the standard approach of fitting data to sums of exponentials is used. Cherruault could have made more up-to-date comments on the current state of research on the general identifiability problem: reference is made only to a 1975 paper. One interesting feature of this chapter is a single algebraic system expressing the mathematical relations between all the unknowns of the linear constant coefficient model (p. 25). This system is used to define a sum-of-squares functional which leads to an optimal set of parameters when minimized. When the number of compartments is greater than three, several different numerical techniques are suggested for identification of model MATHEMATICAL OElsevier
Science
52 Vanderbilt
BIOSCIENCES Publishing
87:233-235
233
(1987)
Co., Inc., 1987
Ave., New York, NY 10017
0025-5564/87/$3.50
234
BOOK REVIEW
parameters, the primary one being numerical integration of the linear differential equation model in combination with minimization of a functional involving the observed states of the model. Standard optimization topics make up the remaining part of the chapter. A key technique, introduced here and mentioned throughout the text, is the Alienor method. Optimal control in compartmental analysis is the subject of Chapter 3. Input to a compartmental system is the control variable which is chosen to optimize some measure of the system’s efficiency, usually an integral functional. Also examined is the influence of certain model parameters on the observed components of the solution. Penalty functions and dynamic programming are applied to biomedical problems. The logical line of the previous chapter is continued in Chapter 4, which centers on the relation between dose and effect. Two and three compartment pharmacokinetics models are utilized to study the action of a drug and the optimization of this action. The treatment shows that modelling and maximization methods allow the proposal of more rational therapeutics. The rest of this chapter deals with optimal control in a compartmental model with time lag. Once the parameter estimation of the transfer coefficients and the time lag is handled using the Alienor method, the optimal control problem is analyzed. General modelling in medicine is the title of Chapter 5. The chapter has rather limited scope because it only studies the imbalance and interaction of the two hormones, cortisone and vasopressin, by a simulation model. Parameter estimation is by minimization of functionals involving linear combinations of exponentials. Again penalty functions, gradient techniques, and the Alienor method are employed to gain numerical solutions. As soon as the model parameters are computed, optimal control is possible and this is carried out by the calculus of variations and the Pontryagin principle. Chapter 6 deals with blood glucose regulation and diabetes. Identification and control are done as in previous chapters. Featured in this section is a proof of the existence and uniqueness of a minimum point for a functional designed to yield optimal therapeutics. Chapters 7 and 8 treat linear and nonlinear integral equations in biomedicine and their numerical solution. Linear combinations of known functions forming a basis in the space of control functions are used to reduce the associated control problem to a linear algebraic system. Some applications of integral equations are then presented (population growth, biomechanics, oxygen diffusion). The introduction in Chapter 9 to partial differential equation (PDE) models is by way of enzyme kinetics. The equations form the standard enzyme-substrate complex model. Little motivation is used in setting up the equations; mass balance is mentioned, but there is no reference to the law of mass action. The Michaelis-Menten hypothesis and the Briggs-Haldane quasi-steady-state hypothesis are quickly examined to arrive at approximations to the rate of formation of product. Using this reaction velocity approximation, the model is now allowed to depend on time and space and thus a PDE model is established. Dirichlet conditions are mentioned as being satisfied, but are they shown to be used? This space-time model is the study of a porous, enzyme containing
REVIEW
235
membrane which separates two compartments into which substrate is introduced. The equations are given solely to demonstrate how PDE models arise in biomedicine; the model is not again discussed. The remainder of this chapter treats different techniques for the solution of PDE’s, parameter identification, and optimal control. The final portion of the book covers optimization in human physiology, sensitivity analysis, and some open problems in biomathematics, as well as an appendix containing a BASIC code for the Alienor method. The principal value of this monograph, in the opinion of the reviewer, is as an illustrated guide to techniques of model parameter estimation and optimal control for the bioscientist. There is contact with practical work in the field, an appreciation of the requirements of fitting models to available data, and numerical procedures which can be adapted to a variety of specific situations. The book is recommended for the person working in biology and medicine, but not as a textbook from which mathematical model building, fundamentals of numerical analysis, and principles of optimal control can be learned. DAVID H. ANDERSON Department of Southern Methodist Dallas, TX
REVIEW
Y. Cherruault, Mathematical Modelling in Biomedicine: Optimal Control of Biomedical Systems. D. Reidel, Dordrecht, Holland, 1986, xviii+258 pp,, $49.50,
Dfl 130.00, E36.25.
This monograph on biomedical modelling more closely resembles a collection of the author’s work and that of his colleagues at Medimat Laboratory, primarily the numerical solution of optimization (in particular, optimal control) problems arising in biomedical systems. Reading it requires a willingness to sacrifice the desire to see models carefully developed based on biological assumptions and principles in favor of discussing techniques for attaining approximate solutions of various systems of equations. In the Introduction the author states, “The aim of this book is to present mathematical methods for building models of biomedical systems.” The book delivers only half what it promises. There is virtually no building of models. On the other hand, the main mathematical methods presented are data analysis; analytical or numerical solution of differential, partial differential, and integral equations; parameter identification; optimization of functions; and optimal control for finding optimal policies according to one or more criteria. All of these ideas are treated using concrete models and examples. General remarks on modelling are brought out in the first chapter. The author considers compartmental modelling using linear models with constant rate coefficients to be the most popular method in biomathematics and hence devotes an appropriate amount of space to developing these models. More conventional notation in the rate coefficients of the mass-balance equations and the diagrams would have helped so that the usual matrix-vector format could be used. The novice will be dismayed by frequent typographical errors that must be corrected to understand fully many basic equations in tbis chapter. Chapter 2 is concerned with identification and control in linear compartmental models. Initially there is discussion of identification for particular two and three compartment examples, where the standard approach of fitting data to sums of exponentials is used. Cherruault could have made more up-to-date comments on the current state of research on the general identifiability problem: reference is made only to a 1975 paper. One interesting feature of this chapter is a single algebraic system expressing the mathematical relations between all the unknowns of the linear constant coefficient model (p. 25). This system is used to define a sum-of-squares functional which leads to an optimal set of parameters when minimized. When the number of compartments is greater than three, several different numerical techniques are suggested for identification of model MATHEMATICAL OElsevier
Science
52 Vanderbilt
BIOSCIENCES Publishing
87:233-235
233
(1987)
Co., Inc., 1987
Ave., New York, NY 10017
0025-5564/87/$3.50
234
BOOK REVIEW
parameters, the primary one being numerical integration of the linear differential equation model in combination with minimization of a functional involving the observed states of the model. Standard optimization topics make up the remaining part of the chapter. A key technique, introduced here and mentioned throughout the text, is the Alienor method. Optimal control in compartmental analysis is the subject of Chapter 3. Input to a compartmental system is the control variable which is chosen to optimize some measure of the system’s efficiency, usually an integral functional. Also examined is the influence of certain model parameters on the observed components of the solution. Penalty functions and dynamic programming are applied to biomedical problems. The logical line of the previous chapter is continued in Chapter 4, which centers on the relation between dose and effect. Two and three compartment pharmacokinetics models are utilized to study the action of a drug and the optimization of this action. The treatment shows that modelling and maximization methods allow the proposal of more rational therapeutics. The rest of this chapter deals with optimal control in a compartmental model with time lag. Once the parameter estimation of the transfer coefficients and the time lag is handled using the Alienor method, the optimal control problem is analyzed. General modelling in medicine is the title of Chapter 5. The chapter has rather limited scope because it only studies the imbalance and interaction of the two hormones, cortisone and vasopressin, by a simulation model. Parameter estimation is by minimization of functionals involving linear combinations of exponentials. Again penalty functions, gradient techniques, and the Alienor method are employed to gain numerical solutions. As soon as the model parameters are computed, optimal control is possible and this is carried out by the calculus of variations and the Pontryagin principle. Chapter 6 deals with blood glucose regulation and diabetes. Identification and control are done as in previous chapters. Featured in this section is a proof of the existence and uniqueness of a minimum point for a functional designed to yield optimal therapeutics. Chapters 7 and 8 treat linear and nonlinear integral equations in biomedicine and their numerical solution. Linear combinations of known functions forming a basis in the space of control functions are used to reduce the associated control problem to a linear algebraic system. Some applications of integral equations are then presented (population growth, biomechanics, oxygen diffusion). The introduction in Chapter 9 to partial differential equation (PDE) models is by way of enzyme kinetics. The equations form the standard enzyme-substrate complex model. Little motivation is used in setting up the equations; mass balance is mentioned, but there is no reference to the law of mass action. The Michaelis-Menten hypothesis and the Briggs-Haldane quasi-steady-state hypothesis are quickly examined to arrive at approximations to the rate of formation of product. Using this reaction velocity approximation, the model is now allowed to depend on time and space and thus a PDE model is established. Dirichlet conditions are mentioned as being satisfied, but are they shown to be used? This space-time model is the study of a porous, enzyme containing
REVIEW
235
membrane which separates two compartments into which substrate is introduced. The equations are given solely to demonstrate how PDE models arise in biomedicine; the model is not again discussed. The remainder of this chapter treats different techniques for the solution of PDE’s, parameter identification, and optimal control. The final portion of the book covers optimization in human physiology, sensitivity analysis, and some open problems in biomathematics, as well as an appendix containing a BASIC code for the Alienor method. The principal value of this monograph, in the opinion of the reviewer, is as an illustrated guide to techniques of model parameter estimation and optimal control for the bioscientist. There is contact with practical work in the field, an appreciation of the requirements of fitting models to available data, and numerical procedures which can be adapted to a variety of specific situations. The book is recommended for the person working in biology and medicine, but not as a textbook from which mathematical model building, fundamentals of numerical analysis, and principles of optimal control can be learned. DAVID H. ANDERSON Department of Southern Methodist Dallas, TX