Lecture Notes in Biomathematics

1. Transformation Systems.- 1.1 Introduction.- 1.2 Formalism.- 1.3 An example: nonlinear chemical kinetics.- 1.4 Specific problems of transformation system modelling.- 1.5 Conclusion.- 2. Structural Properties and Main Approaches to Checking Them.- 2.1 Introduction.- 2.2 Definitions.- 2.2.1 Structural properties and genericity.- 2.2.2 Connectability.- 2.2.3 Structural observabi1ity and structural controllability.- 2.2.4 Structural identifiability.- 2.2.5 Relations between these notions.- 2.3 Practical methods for checking structural observability and structural controllability of linear models.- 2.3.1 All nonzero entries are free.- 2.3.1.1 Graph theoretic approach.- 2.3.1.2 Algebraic approach.- 2.3.1.3 Conclusion.- 2.3.2 Nonzero entries are dependent.- 2.4 Main approaches to structural identifiability.- 2.4.1 Identifiable canonical representations.- 2.4.2 Global optimization.- 2.4.3 Berman and Schoonfold's approach.- 2.4.4 Transfer function approach.- 2.4.5 Minimal representation approach.- 2.4.6 Local approaches.- 2.4.7 Power series approach.- 2.4.8 Identifiability of large-scale linear models.- 2.5 Conclusion.- 3. Local Identifiability.- 3.1 Introduction.- 3.2 Methods.- 3.2.1 Use of the implicit function theorem.- 3.2.2 Local stability of identification algorithms.- 3.2.2.1 Newton and Gauss-Newton algorithms.- 3.2.2.2 Gauss-Seidel algorithm.- 3.2.2.3 Quasilinearization algorithm.- 3.2.3 Observability of the extended state.- 3.2.4 Information matrix.- 3.3 Linear models.- 3.4 Computer aided design of models.- 3.5 Implementation for linear transformation systems.- 3.5.1 Method A.- 3.5.2 Structural nature or the result obtained.- 3.5.3 Method B.- 3.5.4 Examples.- 3.6 Conclusion.- 4. Global Identifiability of Linear Models.- 4.1 Introduction.- 4.2 Properties of the transition matrix.- 4.3 Parametrization of the transition matrix.- 4.3.1 All the eigenvalues of A are real.- 4.3.2 Some eigenvalues of A are complex conjugates.- 4.3.3 Connection with Lagrange-Sylvester polynomials.- 4.4 Application to checking s.g. identifiability.- 4.4.1 The experimental data are entries of ?.- 4.4.1.1 No constraintexists on A.- 4.4.1.2 General procedure.- 4.4.1.3 Example: two-class transformation systems.- 4.4.2 Method for any B and C.- 4.4.3 Problems raised by inequality constraints.- 4.5 Conclusion.- 5. Exhaustive Modelling for Linear Models.- 5.1 Introduction.- 5.2 Class of the studied models.- 5.3 The matrices B and C are known.- 5. 3.1 The matrices B and C are standard.- 5.3.2 The matrices B and C are known, but non-standard.- 5.3.2.1 Standardization of CB.- 5.3.2.2 Standardization of B and C.- 5.4 The matrices B and C are partially unknown.- 5.5 Connections with Kalman's canonical form.- 5.6 Applications of exhaustive modelling.- 5.7 Conclusion.- 6. Examples.- 6.1 Introduction.- 6.2 Chemotherapeutic model.- 6.2.1 First experimental set-up.- 6.2.1.1 Connectabi1ity, structural observability and structural controllability.- 6.2.1.2 Structural local Identifiability.- 6.2.1.3 Exhaustive modelling.- 6.2.2 Second experimental set-up.- 6.2.2.1 Structural local identifiability.- 6.2.2.2 Exhaustive modelling.- 6.3 Hepatobiliary kinetics of B.S.P..- 6.3.1 Connectability, structural observability and structural controllability.- 6.3.2 Structural local identifiability.- 6.3.3 Exhaustive modelling.- 6.4 Metabolism of iodine.- 6.4.1 Structural local identifiability.- 6.4.2 Structural global identifiability.- 6.4.2.1 Input-output transformation.- 6.4.2.2 Standardization.- 6.4.2.3 Determination of AS.- 6.4.2.4 Introduction of the structural constraints on AS.- 6.5 Systemic distribution of Vincamine.- 6.6 Conclusion.- 7. Global Identifiability of Nonlinear Models.- 7.1 Introduction.- 7.2 Series expansion approach.- 7.2.1 Time-power series.- 7.2.2 Generating series.- 7.3 Linearization approach.- 7.3.1 Principle.- 7.3.2 Application to nonlinear transformation systems.- 7.3.2.1 Physical linearization by tracer inclusion.- 7.3.2.2 Mathematical linearization.- 7.3.2.3 What is the best linearization?.- 7.3.3 Generalization.- 7.4 Conclusion.- Conclusion.- References.