Application of Network Thermodynamics to Problems in Biomedical Engineering

D. C. Mikulecky

Table of Contents:

Chapter 1: Introduction to Network Thermodynamics

What Is Network Thermodynamics?

Electronic Network Theory

Complex Systems Theory and the Characteristics of Modern Biology.

The Modeling Relation and Rosen's Distinction

The Explicit Recognition of Topological Contributions to System's Behavior

Physical Systems Theory

A Review of Classical Thermodynamics

Chapter 2: Equilibrium Thermodynamics:Review and Vocabulary

What Is "Thermodynamic" Reasoning?

Callen's First Postulate

The Extremum Principle and the Remainder of Callen's Postulates

The Chemical Potential as a Constitutive Rrelation

The Electrochemical Potential

Examples of the Use of the Chemical Potential to Calculate Relations for Constrained Equilibria

The Gibbs-Donnan Equilibrium Obeys the Nernst Equation as an Extremely Good Approximation

An aside: Application to Resting Cells

Chapter 3: Non-equilibrium Thermodynamics: From Onsager toPrigogine

The Nature of Stationary States away from Equilibrium (Steady States)

Entropy Production During an Irreversible Process and the Direction of Heat Flow

The Isothermal Dissipation Function

The Practical Phenomenological Equations

Chemical Reactions, Active Transport and Curie's Principle

Chapter 4: The Basic Ideas of Network Thermodynamics: I. The Constitutive Laws and Dynamic Systems

From Thermodynamics to Dynamic Systems: an Overview

Through and Across Variables

The Building Blocks: Constitutive Relations

Simple Independent Storage Events and the Generalization of Capacitance

Some Independent Inertial Processes

Chapter 5: The Basic Ideas of Network Thermodynamics: II. Graph Theoretical Methods for the Encoding of System Topology .

A Brief History of Graph Theory

Introduction to the Study of Networks and Graphs

An Example: The Network Approach to Some Rate Processes

Chapter 6: Network Thermodynamic Solutions to Steady State Linear Systems: Nodal and Mesh Analysis and Duality

The Systematic Approach to Networks

The Intimate Relation Between the Network's Topology and Kirchhoff's Laws

Nodal Analysis of a Network

The Generalized Dissipative Branch

Duality in Networks: Loop-Mesh Analysis is Equivalent to Node Analysis on the Dual

Dual Networks

Loop and Mesh Analysis

Chapter 7Dynamic Linear Systems and Cutsets: The Network Thermodynamic Generation of State Vector Equations

Time Dependent or "Dynamic" Linear Networks

Cut Set Analysis of Time Dependent Linear Networks

Cut Set Analysis: A Method Which Extends to Networks with Nonlinear Resistors, Capacitors and Inductors

Generalized Cut Set Analysis and the Analysis of RC Networks

State Vector Equations for Continuous-time Dynamic Networks

Nonlinear Dynamic Systems

The Use of the Proper Tree to Organize the Analysis

Using This Approach on Nonlinear Networks

Chapter 8: N-ports and Tellegen's Theorem: Energy Conversion, Degree of Coupling, Efficiency and Metrics . . . . . . .

Non-equilibrium Thermodynamics Provides the Phenomenological Description of N-ports

Network Thermodynamics Provides a Holistic View of the N-port

The Degree of Coupling and Efficiency of Energy Conversion in N-ports

The Degree of Coupling's Geometric Significance: the Metric and Its Demonstration of the Cannonical Nature of the Network Representation

The Phenomenological Approach

Phenomenology in Hierarchical Systems

Tellegen's Theorem and the Holistic View of Networks

Simplification of Computation in Linear Networks

Chapter 9: Applications of the N-port Concept: Topological Contributions to System Behavior, Multiport Current Dividers, and Other Innovations

Toward a Biological Circuit Theory

The Current Divider Principle in 1-port Networks

The Node-branch Incidence Matrix for the Current Divider

The Extension of the Current Divider Principle to N-ports

Mathematical Description of a Solute-volume Flow 2-port

Chapter 10: Making Nonlinearity Look Linear: The Reference State, Kinetics, and Non-equilibrium thermodynamics Get Married

The Relationship Between Kinetics and Onsager's Thermodynamics

Onsager's Triangle Reaction as a Network

Nonlinear Systems as Networks and the Reference State

The Need for Reference States: A Simple Kinetic Example

The Reference State

The Simple Carrier Model: Hill's Method of Analysis

Peusner's "Thermokinetic" Networks

The Creation of Peusner Networks for Thermokinetic Systems

The Principal of Detailed Balance: First Order vs. Pseudo-first Order Steps in the Network

The Choice of Reference State

An Example with Two Degrees of Freedom: Linear Description and Onsager Coefficients for a Given Reference State

Chapter 11: Simulation of Network Models Using SPICE: Making the Difficult Seem Easy

SPICE as a General Purpose Simulator

Using Resistors, Capacitors and Constant Sources

Some Examples

Controlled Sources, the "Magic" of SPICE

Chapter 12: Applications and Models: Compartmental Systems, Pharmacokinetics, Reaction Networks and Cancer Chemotherapy

Applications of Kinetics to Systems

The Compartmental Model of Frog Skin

The Energetics of Coupled Processes and Its Relationship to Compartmental Analysis

A Study of the Energetics of Sodium Transport in Frog Skin

Pharmacokinetic Models

Reaction Networks and Cancer Chemotherapy

Cardiovascular Models

Ecological Models

Structural Identifiability: What Network Thermodynamics Has to Offer

Appendix: A Short Review of Laplace Transforms

Chapter 13: Applications and Models: Epithelial Membranes in Kidney, Gut, and the Lingual Epithelium

The Epithelial Membrane Revisited

The Model of Coupled Solute/Volume Flow Across Loose Epithelia

The Tight Epithelium Model for Ion Transport and Electrophysiological Responses

Chapter 14: Epilogue: Towards a Theory of Complex Systems

Networks, Organization, Complexity and Life

Appendix: Some Topological Considerations in Thermodynamics: An Application of Network Thermodynamics

Onsager Reciprocity as a Topological Property

Reciprocity Defined as a Class of Simultaneous Properties

Reciprocity in Equilibrium Systems and the Existence of Potential Functions

Caratheodory's Proof and the Second Law of Thermodynamics

The Accessibility Theorem of Caratheodory

Proof of the Accessibility Theorem

Forms and the Exterior Calculus

Reciprocity in Classical Thermodynamics: Maxwell's Relations. . .

The Meaning of Receprocity in a Non-Equilibrium (Dynamic) System

The Topological Implications of Reciprocity

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