## Mathematical Modelling in Biomedicine: Optimal Control of Biomedical SystemsApproach 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 |

251 | |

257 | |

### Other editions - View all

Mathematical Modelling in Biomedicine: Optimal Control of Biomedical Systems Y. Cherruault No preview available - 2014 |

### Common terms and phrases

algorithm Alienor alveolus approach approximation Archimedean spiral associated biofeedback biological system biomathematics blood calculated Chapter chosen classical coefficients compartment compartmental analysis compartmental model compartmental system concentration constant control variables convergence convex corresponding criterion defined depends developed differential system diffusion discretization doses drug dynamic programming example experimental data explicit exponential f(xi,ui fixed formula Furthermore given gives global optimization technique glucose Guillez hormonal hypothesis identification problem identified from experimental initial conditions injection insulin integral equations interval introduced involves iterative known Laplace transform Let us consider linear algebraic system linear system local minimum mathematical model matrix measured minimised minimization minimum necessary non-linear numerical methods numerical results obtained oocyte optimal control problem optimal solution optimal therapeutics optimization method optimization problem optimum partial differential equations partial pressures possible Sandoz laboratories single variable solve Tertatolol theorem tube uniqueness values variation vector xi(t

### Popular passages

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...

### References to this book

Metaheuristics for Hard Optimization: Methods and Case Studies Johann Dréo No preview available - 2006 |