Mathematical Modelling in Biomedicine: Optimal Control of Biomedical Systems

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Springer Science & Business Media, Feb 28, 1986 - Mathematics - 258 pages
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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.
 

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Contents

GENERAL REMARKS OH MODELLING
3
12 The main techniques for modelling
4
122 SYSTEMS WITH DIFFUSIONCONVECTION REACTIONS
8
123 SIMULATION MODELS
10
13 Difficulties in modelling
12
IDENTIFICATION AND CONTROL 18 LINEAR COMPARtMEHTAL ANALYSIS
15
22 The uniqueness problem
28
23 Numerical methods for identification
31
63 Optimal control for optimal therapeutics
143
64 Optimal control problem involving several criteria
146
INTEGRAL EQUATIONS IN BICHEDICINE
152
72 Integral equations from biomechanics
154
73 Other applications of Integral equations
157
NUMERICAL SOLUTION OF INTEGRAL EQUATIONS
162
82 Numerical techniques for nonlinear integral equations
164
821 NUMERICAL SOLUTION USING A SEQUENCE OF LINEAR
165

the nonlinear case
35
25 Optimization techniques
37
252 NUMERICAL METHODS 46
42
253 DESCENT METHODS 20
44
254 A GLOBAL OPTIMIZATION TECHNIQUE 11 14
46
OPTIMAL CONTROL IN COMPARTMENTAL ANALYSIS
53
32 A first explicit approach
56
33 The general solution 13
62
34 Numerical method 36
63
35 Optlanl control in nonlinear cases 3
76
352 A METHOD FOR NONLINEAR COMPARTMENTAL SYSTEMS
80
353 ANOTHER PRACTICAL APPROACH
82
353 A VARIANT OF DYNAMIC PROGRAMMING TECHNIQUE 5 11
85
354 A simple idea applied to optimal control probl
94
RELATIONS BKTWEES DOSE AND EFFECT
96
42 The nonlinear approach
98
43 Simple functional Model
102
44 Optimal therapeutics
105
45 Numerical results
110
46 Nonlinear compartment approach Formula 424 or 425 allows the calculation of the
111
47 Optimal therapeutics using a linear approach
115
48 Optical control in a compartmental model with time lag
117
GENERAL MODELLING IN MEDICINE
122
52 The indentification problem
123
53 A simple method for defining optimal therapeutics
128
54 The Pontryagin Method 3 11
129
55 A simplified technique giving a suboptium
133
56 A naive but useful Method 74
135
BLOOD GLOCOSE REGULATION
137
62 The huotan case
140
822 A DISCRETISED TECHNIQUE
166
823 AN ITERATIVE DIAGRAM WITH REGULARITY CONSTRAINTS 9
169
83 Identification and optimal control using integral equations
170
84 Optimal control and nonlinear integral equations
171
PROBLEMS RELATED TO PARTIAL DIFFERENTIAL EQUATIONS
173
92 Numerical resolution of partial differential equations 56 23
177
922 OPTIMIZATION METHOD FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS
179
923 SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS USING A COMPLETE DISCRETIZATION 56 57
181
93 Identification in partial differential equations
183
94 Optimal control with partial differential equations
184
95 Other approaches for optical control
189
96 Other partial differential equations
190
OPTIMALITY IH HOMAH PHYSIOLOGY
193
102 A mathematical model for thermoregulation 30
194
103 Optimization of pulmonary mechanics
202
104 Conclusions
208
ERRORS IN MODELLING
209
112 Sensitivity analysis
212
OPEN PROBLEMS IN BIOMATHEMATICS
217
122 Biological systems involving retroaction
225
123 Action of two or more drugs In the huvan organSB
228
124 Numerical techniques for global optimization
231
125 Biofeedback and systeas theory 32 33 34
234
126 Optimization of industrial processes
236
127 Optiwality in physiology
237
CONCLUSIONS
238
THE ALIENOR PROGRAM IMPLEMENTED BY A GUILLKZ
240
REFERENCES
251
INDEX
257
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Page ix - 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, coding theory and the structure of water meet one another in packing and covering theory; quantum...
Page ix - Brown The point of a Pin'. 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...
Page ix - SERIES EDITOR'S PREFACE Approach your problems from the right end and begin with the answers. Then one day, perhaps you will find the final question. The Hermit Clad in Crane Feathers
Page xi - As long as algebra and geometry proceeded ics in science ... along separate paths, their advance was slow and their applications limited. Eugene Wigner But when these sciences joined company they drew from each other fresh vitality and Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace towards perfection. Bruce Bairnsfather Joseph Louis Lagrange. What is now proved was once only imagined.
Page x - With such books, which are stimulating rather than définitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields. Because of the wealth of scholarly research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA...

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