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Mathematical Modelling and Computers in Endocrinology: (16 Monographs on Endocrinology)

Mathematical Modelling and Computers in Endocrinology: (16 Monographs on Endocrinology)

          
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About the Book

The building of conceptual models is an inherent part of our interaction with the world, and the foundation of scientific investigation. Scientists often perform the processes of modelling subconsciously, unaware of the scope and significance of this activity, and the techniques available to assist in the description and testing of their ideas. Mathematics has three important contributions to make in biological modelling: (1) it provides unambiguous languages for expressing relationships at both qualitative and quantitative levels of observation; (2) it allows effective analysis and prediction of model behaviour, and can thereby organize experimental effort productively; (3) it offers rigorous methods of testing hypotheses by comparing models with experimental data; by providing a means of objectively excluding unsuitable concepts, the development of ideas is given a sound experimental basis. Many modern mathematical techniques can be exploited only with the aid of computers. These machines not only provide increased speed and accuracy in determining the consequences of model assumptions, but also greatly extend the range of problems which can be explored. The impact of computers in the biological sciences has been widespread and revolutionary, and will continue to be so.

Table of Contents:
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.


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Product Details
  • ISBN-13: 9783540096931
  • Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
  • Publisher Imprint: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Language: English
  • Returnable: N
  • Weight: 750 gr
  • ISBN-10: 3540096930
  • Publisher Date: 01 Feb 1980
  • Binding: Hardback
  • No of Pages: 337
  • Series Title: 16 Monographs on Endocrinology


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