Mathematical Modelling in Biomedicine

Approach your problems from the right It isn't that they can't see the solution. It end and begin with the answers. Then is that they can't see the problem. one day, perhaps you will find the final question. G.K. Chesterton. The Scandal of Father Brown 'The point of a Pin'. 'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, cod­ ing theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical pro­ gramming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Contenu

0 Introduction.- 1 General Remarks on Modelling.- 1.1 Definitions.- 1. 2 The main techniques for modeling.- 1.2.1 Compartmental analysis.- 1.2.2 Systems with diffusion-convection reactions.- 1.2.3 Simulation models.- 1.3 Difficulties in modeling.- 2 Identification and Control in Linear Compartmental Analysis.- 2.1 The identification problem.- 2.2 The uniqueness problem.- 2.3 Numerical methods for identification.- 2.4 About the non-linear case.- 2.5 Optimization techniques.- 2.5.1 General considerations.- 2.5.2 Numerical methods.- 2.5.3 Descent methods.- 2.5.4 A global optimization technique.- 3 Optimal Control in Compartmental Analysis.- 3.1 General considerations.- 3.2 A first explicit approach.- 3.3 The general solution.- 3.4 Numerical method.- 3.5 Optimal control in non-linear cases.- 3.5.1 A general technique.- 3.5.2 A method for non-linear compartmental systems.- 3.5.3 Another practical approach.- 3.5.3 A variant of dynamic programming technique.- 3.5.4 A simple idea applied to optimal control problems.- 4 Relations Between dose and Effect.- 4.1 General considerations.- 4.2 The non-linear approach.- 4.3 Simple functional model.- 4.4 Optimal therapeutics.- 4.5 Numerical results.- 4.6 Non-linear compartment approach.- 4.7 Optimal therapeutics using a linear approach.- 4.8 Optimal control in a compartmental model with time lag.- 5 General Modelling in Medicine.- 5.1 The problem and the corresponding model.- 5.2 The identification problem.- 5.3 A simple method for defining optimal therapeutics.- 5.4 The Pontryagin method.- 5.5 A simplified technique giving a sub-optimum.- 5.6 A naive but useful method.- 6 Blood Glucose Regulation.- 6.1 Identification of parameters in dogs.- 6.2 The human case.- 6.3 Optimal control for optimal therapeutics.- 6.4 Optimal control problem involving several criteria.- 7 Integral Equations in Biomedicine.- 7.1 Compartmental analysis.- 7.2 Integral equations from biomechanics.- 7.3 Other applications of integral equations.- 8 Numerical Solution of Integral Equations.- 8.1 Linear integral equations.- 8.2 Numerical techniques for non-linear integral equations.- 8.2.1 Numerical solution using a sequence of linear integral equations.- 8.2.2 A discretised technique.- 8.2.3 An iterative diagram with regularity constraints.- 8.3 Identification and optimal control using integral equations.- 8.4 Optimal control and non-linear integral equations.- 9 Problems Related to Partial Differential Equations.- 9.1 General remarks.- 9.2 Numerical resolution of partial differential equations.- 9.2.1 Semi-discretization technique.- 9.2.2 Optimization method for solving partial differential equations.- 9.2.3 Solution of partial differential equations using a complete discretization.- 9.3 Identification in partial differential equations.- 9.4 Optimal control with partial differential equations.- 9.5 Other approaches for optimal control.- 9.6 Other partial differential equations.- 10 Optimality in Human Physiology.- 10.1 General remarks.- 10.2 A mathematical model for thermo-regulation.- 10.3 Optimization of pulmonary mechanics.- 10.4 Conclusions.- 11 Errors in Modelling.- 11.1 Compartmental modeling.- 11.2 Sensitivity analysis.- 12 Open Problems in Biomathematics.- 12.1 Biological systems with internal delay.- 12.2 Biological systems involving retroaction.- 12.3 Action of two (or more) drugs in the human organism.- 12.4 Numerical techniques for global optimization.- 12.5 Biofeedback and systems theory.- 12.6 Optimization of industrial processes.- 12.7 Optimality in physiology.- 13 CONCLUSIONS.- Appendix - The Alienor program.- References.