Mathematical Modelling in Biomedicine

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.