Mathematical Modelling and Computers in Endocrinology

1 Modelling in Biology.- 1.1 The Nature of Scientific Models.- 1.1.1 Different Kinds of Models.- 1.1.2 Reductionism in Biology.- 1.1.3 The Scope of Modelling.- 1.2 Clarity from Complexity.- 1.2.1 Experimental Frames.- 1.2.2 Variables and Parameters.- 1.2.3 Diagrams.- 1.2.4 Lumping.- 1.2.5 Hierarchical Levels.- 1.2.6 Stochastic and Deterministic Behaviours.- 1.2.7 Problems of Individuality.- 1.3 Experimental Data.- 1.4 Predictions from Models - Simulation.- 1.5 A Model of Complexity Producing Organized Simplicity.- 1.6 Subjectivity in Modelling.- 1.7 Mathematics in Modelling.- 1.8 Computers and Models.- 1.9 Description of Models.- 1.10 Modelling in Perspective.- 1.11 Advantages in Modelling.- 2 Mathematical Descriptions of Biological Models.- 2.1 Theoretical Modelling - Analysis of Mechanism.- 2.1.1 Differentials.- 2.1.1.1 Modelling Dynamic Systems with Differential Equations.- 2.1.1.2 Making Differential Equations and Data Comparable.- 2.1.1.3 Requirements for the Use of Differential Equations..- 2.1.2 Transient and Steady States.- 2.1.3 Defining Model Equations.- 2.1.4 Systems to Which Theoretical Modelling Can Be Applied.- 2.1.5 Compartmental Analysis and Inhomogeneity.- 2.2 Linearity and Non-Linearity.- 2.3 Empirical Modelling - a Description of System Response.- 2.3.1 Empirical Equations and Curve-Fitting.- 2.3.1.1 Spline Functions.- 2.3.1.2 Self-Modelling Non-Linear Regression.- 2.3.2 Transfer Functions.- 2.3.3 Convolution Integrals.- 2.3.4 Combining Subsystems.- 2.3.5 Frequency Domain Analysis.- 2.4 Point Stability in Models.- 2.4.1 Tests for Stability of Linear Models.- 2.4.2 Stability of Non-Linear Models.- 2.5 Concepts of Feedback.- 2.5.1 Biological Homeostasis and the Concept of Feedback.- 2.5.2 Feedback Control.- 2.5.2.1 Point Stability in Models of Negative Feedback.- 2.5.2.2 Effectiveness of Feedback Loops.- 2.5.2.3 Negative Feedback Loops and Their Mechanisms in Endocrinology.- 2.6 Biological Development and Mathematics Beyond Instability.- 2.6.1 An Example of Parameter-Dependent Changes in System Stability.- 2.6.2 Limit Cycles.- 2.6.3 Stability Behaviour and Spatial Inhomogeneity.- 2.6.4 Examples of Non-Linear Equations Showing Instabilities.- 2.6.5 Generalized Descriptions of Instabilities in Biology.- 2.7 Finite Level Modelling.- 2.8 The Need for Statistics.- 2.8.1 Transformation and Weighting.- 2.8.2 Parameter Uncertainty.- 2.8.2.1 Monte Carlo Simulation.- 2.8.3 Testing Hypotheses.- 2.8.4 Normal Distributions in Biology.- 2.8.5 Statistics and Experimental Design.- 3 Comparing Models with Experimental Results.- 3.1 Analogue Simulation.- 3.2 Approximate Simulation by Digital Computer.- 3.3 Suitable Digital Computers.- 3.4 Estimation of Parameters.- 3.4.1 Linearity and Non-Linearity in Parameter Estimation.- 3.4.2 Defining the "Best" Fit of a Model to Data.- 3.4.3 Methods of Parameter Search in Non-Linear Models.- 3.4.3.1 Direct Search Methods.- 3.4.3.2 Gradient Search Methods.- 3.5 Practical Details of Fitting Non-Linear Models to Data.- 3.5.1 Weighting by Variance in the Data.- 3.5.2 Measurement Error Estimated from the Fitting Process.- 3.5.3 Constraining the Search.- 3.5.4 Terminating the Search and Examining the Residuals.- 3.5.5 Outliers.- 3.5.6 Interpretation of Parameter Estimates.- 3.5.7 Goodness of Fit of the Model.- 3.5.8 Difficulties in Parameter Optimization.- 3.5.8.1 Parameter Interaction and Sums of Exponentials.- 3.6 Models Containing Differential Equations.- 3.7 Using the Computer to Fit Models to Data.- 3.7.1 Exponential Decay: Clearance of PMSG.- 3.7.1.1 Application of Monte Carlo Simulation.- 3.7.2 Growth of Elephants.- 3.7.3 How Thick is the Wall of an Ovarian Follicle?.- 4 Design of Analytical Experiments.- 4.1 Principles of Design.- 4.1.1 Randomization.- 4.1.2 Replication.- 4.1.3 Reduction of Random Variation.- 4.1.4 Factorial Experiments.- 4.2 Sequential Design.- 4.2.1 Design Criterion for Parameter Estimation.- 4.2.2 Design Criterion for Model Discrimination.- 4.2.3 Combined Model Discrimination and Parameter Estimation.- 4.2.4 Termination Criteria.- 4.2.5 Implementation.- 4.2.6 A Computer Algorithm for Combined Design Criteria.- 4.2.7 Does Sequential Experimentation Work?.- 4.2.8 Testing the Design Criteria by Monte Carlo Simulation.- 5 Dynamic Systems: Clearance and Compartmental Analysis.- 5.1 Clearance.- 5.1.1 Clearance of PMSG from the Blood.- 5.2 Compartmental Analysis.- 5.2.1 Writing Equations to Describe Dynamic Systems.- 5.2.2 Hormonal Influence on Zinc Transport in Rabbit Tissues.- 5.2.2.1 Analytical Integration: Two Compartments.- 5.2.2.2 Analytical Integration: Three Compartments.- 5.2.2.3 Other Applications.- 5.2.3 Numerical Integration of Dynamic Model F-quations.- 5.2.3.1 A Two-Compartment System.- 5.2.3.2 A Three-Compartment System.- 5.2.3.3 Comparison of Series and Parallel Models.- 5.2.4 A Generalized Mammilary System.- 5.2.4.1 Transport of Albumin.- 5.2.4.2 Transport of PMSG.- 5.2.5 Extension of the Generalized Mammillary System.- 6 Ligand-Protein Interaction and Competitive Displacement Assays.- 6.1 Interactions Between Ligands and Macromolecules.- 6.1.1 The Binding of Ligands to Non-Interacting, Independent Sites.- 6.1.1.1 The Binding Properties of Human Pregnancy Plasma.- 6.1.1.2 Binding Expressed as a Molar Ratio.- 6.1.2 Sequential Binding and Co-operativity.- 6.1.2.1 Use of the General Binding Model.- 6.2 Competitive Protein-Binding Assays.- 6.2.1 Theoretical Models.- 6.2.1.1 Effects of Labelled Ligand and Cross-Reactants.- 6.2.1.2 Analysis of a Radio-Immunoassay for Testosterone.- 6.2.2 Empirical Models.- 6.2.2.1 Logit-Log and Non-Linear Regression Models.- 6.2.2.2 Simulation of the Testosterone Radio-Immunoassay.- 6.2.2.3 Spline Functions.- 6.2.3 Weighting.- 6.2.4 Sensitivity and Precision.- 6.2.5 Optimization of Assays.- 7 Mathematical Modelling of Biological Rhythms.- 7.1 Biological Rhythms: Experimental Evidence.- 7.2 The Contribution of Mathematics.- 7.2.1 Description of Rhythms.- 7.2.2 Mathematical Models of Response to Stimuli.- 7.3 Response of Rhythms to Stimuli - a Description of System Response.- 2.3.1 Empirical Equations and Curve-Fitting.- 2.3.1.1 Spline Functions.- 2.3.1.2 Self-Modelling Non-Linear Regression.- 2.3.2 Transfer Functions.- 2.3.3 Convolution Integrals.- 2.3.4 Combining Subsystems.- 2.3.5 Frequency Domain Analysis.- 2.4 Point Stability in Models.- 2.4.1 Tests for Stability of Linear Models.- 2.4.2 Stability of Non-Linear Models.- 2.5 Concepts of Feedback.- 2.5.1 Biological Homeostasis and the Concept of Feedback.- 2.5.2 Feedback Control.- 2.5.2.1 Point Stability in Models of Negative Feedback.- 2.5.2.2 Effectiveness of Feedback Loops.- 2.5.2.3 Negative Feedback Loops and Their Mechanisms in Endocrinology.- 2.6 Biological Development and Mathematics Beyond Instability.- 2.6.1 An Example of Parameter-Dependent Changes in System Stability.- 2.6.2 Limit Cycles.- 2.6.3 Stability Behaviour and Spatial Inhomogeneity.- 2.6.4 Examples of Non-Linear Equations Showing Instabilities.- 2.6.5 Generalized Descriptions of Instabilities in Biology.- 2.7 Finite Level Modelling.- 2.8 The Need for Statistics.- 2.8.1 Transformation and Weighting.- 2.8.2 Parameter Uncertainty.- 2.8.2.1 Monte Carlo Simulation.- 2.8.3 Testing Hypotheses.- 2.8.4 Normal Distributions in Biology.- 2.8.5 Statistics and Experimental Design.- 3 Comparing Models with Experimental Results.- 3.1 Analogue Simulation.- 3.2 Approximate Simulation by Digital Computer.- 3.3 Suitable Digital Computers.- 3.4 Estimation of Parameters.- 3.4.1 Linearity and Non-Linearity in Parameter Estimation.- 3.4.2 Defining the "Best" Fit of a Model to Data.- 3.4.3 Methods of Parameter Search in Non-Linear Models.- 3.4.3.1 Direct Search Methods.- 3.4.3.2 Gradient Search Methods.- 3.5 Practical Details of Fitting Non-Linear Models to Data.- 3.5.1 Weighting by Variance in the Data.- 3.5.2 Measurement Error Estimated from the Fitting Process.- 3.5.3 Constraining the Search.- 3.5.4 Terminating the Search and Examining the Residuals.- 3.5.5 Outliers.- 3.5.6 Interpretation of Parameter Estimates.- 3.5.7 Goodness of Fit of the Model.- 3.5.8 Difficulties in Parameter Optimization.- 3.5.8.1 Parameter Interaction and Sums of Exponentials.- 3.6 Models Containing Differential Equations.- 3.7 Using the Computer to Fit Models to Data.- 3.7.1 Exponential Decay: Clearance of PMSG.- 3.7.1.1 Application of Monte Carlo Simulation.- 3.7.2 Growth of Elephants.- 3.7.3 How Thick is the Wall of an Ovarian Follicle?.- 4 Design of Analytical Experiments.- 4.1 Principles of Design.- 4.1.1 Randomization.- 4.1.2 Replication.- 4.1.3 Reduction of Random Variation.- 4.1.4 Factorial Experiments.- 4.2 Sequential Design.- 4.2.1 Design Criterion for Parameter Estimation.- 4.2.2 Design Criterion for Model Discrimination.- 4.2.3 Combined Model Discrimination and Parameter Estimation.- 4.2.4 Termination Criteria.- 4.2.5 Implementation.- 4.2.6 A Computer Algorithm for Combined Design Criteria.- 4.2.7 Does Sequential Experimentation Work?.- 4.2.8 Testing the Design Criteria by Monte Carlo Simulation.- 5 Dynamic Systems: Clearance and Compartmental Analysis.- 5.1 Clearance.- 5.1.1 Clearance of PMSG from the Blood.- 5.2 Compartmental Analysis.- 5.2.1 Writing Equations to Describe Dynamic Systems.- 5.2.2 Hormonal Influence on Zinc Transport in Rabbit Tissues.- 5.2.2.1 Analytical Integration: Two Compartments.- 5.2.2.2 Analytical Integration: Three Compartments.- 5.2.2.3 Other Applications.- 5.2.3 Numerical Integration of Dynamic Model F-quations.- 5.2.3.1 A Two-Compartment System.- 5.2.3.2 A Three-Compartment System.- 5.2.3.3 Comparison of Series and Parallel Models.- 5.2.4 A Generalized Mammilary System.- 5.2.4.1 Transport of Albumin.- 5.2.4.2 Transport of PMSG.- 5.2.5 Extension of the Generalized Mammillary System.- 6 Ligand-Protein Interaction and Competitive Displacement Assays.- 6.1 Interactions Between Ligands and Macromolecules.- 6.1.1 The Binding of Ligands to Non-Interacting, Independent Sites.- 6.1.1.1 The Binding Properties of Human Pregnancy Plasma.- 6.1.1.2 Binding Expressed as a Molar Ratio.- 6.1.2 Sequential Binding and Co-operativity.- 6.1.2.1 Use of the General Binding Model.- 6.2 Competitive Protein-Binding Assays.- 6.2.1 Theoretical Models.- 6.2.1.1 Effects of Labelled Ligand and Cross-Reactants.- 6.2.1.2 Analysis of a Radio-Immunoassay for Testosterone.- 6.2.2 Empirical Models.- 6.2.2.1 Logit-Log and Non-Linear Regression Models.- 6.2.2.2 Simulation of the Testosterone Radio-Immunoassay.- 6.2.2.3 Spline Functions.- 6.2.3 Weighting.- 6.2.4 Sensitivity and Precision.- 6.2.5 Optimization of Assays.- 7 Mathematical Modelling of Biological Rhythms.- 7.1 Biological Rhythms: Experimental Evidence.- 7.2 The Contribution of Mathematics.- 7.2.1 Description of Rhythms.- 7.2.2 Mathematical Models of Response to Stimuli.- 7.3 Response of Rhythms to Stimuli - Rhythm Coupling.- 7.3.1 Phases Sensitive to Stimulation.- 7.3.2 Disturbances in Models Involving Limit Cycles.- 7.3.2.1 Pulse Stimuli Causing Phase Changes.- 7.3.2.2 Limit Cycle Interpretation of Phase Changes.- 7.3.2.3 Application of a Limit Cycle Model.- 7.3.3 Entrainment of Oscillators by Rhythmic Forcing Functions.- 7.3.4 Interactive Coupling of Limit Cycles.- 7.4 Rhythms, Endocrinology and Biological Control.- 7.5 Empirical Characterization of Rhythms from Data.- 7.5.1 The Sine Wave Model.- 7.5.1.1 Temperature Variation During the Menstrual Cycle.- 7.5.2 Auto-Correlation: Analysis in the Time Domain.- 7.5.3 Frequency Analysis of Rhythms.- 7.5.3.1 Auto-Correlation and Frequency Analysis of Temperature Data.- 7.5.4 Bivariate Processes; Cross-Correlation and Cross-Spectra.- 8 Large Systems: Modelling Ovulatory Cycles.- 8.1 Modelling the Ovulatory Cycle.- 8.2 A Description of the Ovulatory Cycle.- 8.3 Use of Differential Equations with Cyclic Solutions.- 8.4 Use of a Threshold Discontinuity to Produce Cyclicity.- 8.5 A Physiologically Based Model of the Rat Oestrous Cycle.- 8.6 An Empirical Model of the Rat Oestrous Cycle Controlled by Time.- 8.7 An Attempt to Include More Variables.- 8.8 Eliminating the Differential Equations: A Finite Level Model.- 8.9 A "Complete" Description.- 8.9.1 Testing the Model.- 8.9.1.1 The Effect of Short-Term Random Oscillations.- 8.9.1.2 Response to Infusions of Oestradiol and GnRH.- 8.9.2 Further Modifications.- 8.10 Conclusions.- 9 Stochastic Models.- 9.1 Non-Parametric Statistical Models.- 9.1.1 Comparing Two Independent Samples.- 9.1.2 Comparing Two Samples Related by Pairs.- 9.1.3 Comparing More than Two Samples with Related Individuals.- 9.1.4 Identifying the Difference: Critical Range Tests.- 9.1.5 Comparing More than Two Independent Samples.- 9.1.6 Conclusions.- 9.2 Multivariate Analysis.- 9.2.1 Principal Component and Factor Analysis.- 9.2.1.1 Factor Analysis: Steroid Production by Ovarian Follicles.- 9.2.2 Cluster Analysis.- 9.2.3 Discriminant Analysis.- 9.2.3.1 Application to Steroid Production by Ovarian Follicles.- 10 Appendix A: A Summary of Relevant Statistics.- 10.1 Variance, Standard Deviation and Weight.- 10.2 The Propagation of Variance.- 10.3 Covariance and Correlation.- 10.4 z-Scores, or Standardized Measures.- 10.5 Testing Hypotheses.- 10.6 Runs-Test.- 10.7 Chi-Square Test.- 10.8 Tests of Normality of Distribution.- 10.9 Comparing Two Parameters: The r-Test.- 10.10 Comparing Any Number of Parameters: Analysis of Variance.- 10.11 The Variance Ratio or F-Test.- 10.12 Confidence Intervals.- 10.13 Control Charts.- 11 Appendix B: Computer Programs.- 11.1 MODFIT: A General Model-Fitting Program.- 11.2 SIMUL: A Program for Monte Carlo Simulation.- 11.3 FUNCTN Subprogram RESERVR: Exponential Decay.- 11.4 FUNCTN Subprogram EXPCUBE: Growth of Organisms.- 11.5 FUNCTN Subprogram POLYNOM: General Polynomial.- 11.6 FUNCTN Subprogram FOLL: Growth of Ovarian Follicles.- 11.7 DESIGN: A Program for Efficient Experimental Design.- 11.8 FUNCTN Subprogram CAI: Compartmental Analysis.- 11.9 FUNCTN Subprogram CA2: Compartmental Analysis.- 11.10 FUNCTN Subprogram CA2A: Compartmental Analysis.- 11.11 FUNCTN Subprogram CA3: Compartmental Analysis.- 11.12 FUNCTN Subprogram MASSACT: Ligand Binding.- 11.13 FUNCTN Subprogram GENBIND: Ligand Binding.- 11.14 FUNCTN Subprogram RIA: Competitive Protein-Binding.- 11.15 FUNCTN Subprogram RIAH: Competitive Protein-Binding.- 11.16 FUNCTN Subprogram SIN: Rhythmical Data.- 11.17 FUNCTN Subprogram LIMIT: Limit Cycles.- 11.18 Other Programs for Fitting Non-Linear Models to Data.- 11.18.1 SPSS NONLINEAR.- 11.18.2 MLAB: An Online Modelling Laboratory.- 11.18.3 SAAM: Simulation, Analysis and Modelling.- 11.18.4 Other Programs.- 12 Appendix C: Analytical Integration by Laplace Transform.- References.