Modeling Systems and Processes Anthony McGoron, PhD Associate Professor Department of Biomedical Engineering Florida International University

Mathematical Modeling A model is any representation of a real system. –May deal with structure or function –May involve words, diagrams, mathematical notation, physical structure –May have the same meaning as “hypothesis” –Must always involve simplification of the real system –A mathematical model may be as simple as a single equation relating a single dependent variable (y) to another independent variable (x) such as: y = ax + b –May be multi-component involving the interaction of many equations having several mutually dependent variables

Building Models Stepwise replacement of a system component with a model equation. 1.Conceptual model of the real system. Without an understanding of the real system and the interaction of the system with its environment, no model can be developed. 2.Design experiments and collect “good” data that accurately represents the real system. 3.Examine the data to determine the parameter set that defines the system f(x,y,t,a,b,c…). 4.Define an equation based on the data (empirical) and/or based on the characteristics of the system (theory based). For example, y = ax + b. y and x are variables. a and b are parameters. 5.Find the optimal (most correct) values for the parameters a and b. 6.Implement the model to “experiment” with new concepts.

Building Models: An Example Food Chain/Ecosystem/Photosynthesis

Conceptual components of a hypothetical system are replaced by equations to form a multi-component model of a system (Keen and Spain, 1992)

The role of quantitative modeling and simulation within the process of research (Keen and Spain, 1992)

Modeling Application - Transport mass, energy, momentum Hemodialysis Heart Lung Bypass Machine

Pharmacology – The history, source, physical and chemical properties, biochemical and physiological effect, mechanisms of action, absorption, distribution, biotransformation and excretion, and therapeutic and other uses of drugs. Pharmacokinetics – Absorption, Distribution, Metabolism (biotransformation) and Excretion of drugs (ADME). Pharmacodynamics – Biochemical and physiological effects and their mechanisms of action An Example: Drug Distribution Mass Transport

Concentration of drug in the body as a function of time for two types of drug dosage forms (Rowland and Beckett, 1964)

Physiochemical factors in transfer of drugs across membranes: absorption, distribution, biotransformation, and excretion of a drug involve its passage across cell membranes. Systemic Circulation AbsorptionExcretion Metabolites Biotransformation Free Drug Bound Drug Locus of Action “receptors” Bound Free Tissue Reservoirs Bound Free

General compartment model for the human body (Bischoff and Brown, 1966)

Numerical details of a specific pharmacokinetic model of the body. There will be 36 equations (Bischoff and Brown, 1966).

Model for a local tissue region (Bischoff and brown, 1966)

Body Absorption Elimination k 1 Simple Compartmental Model (lumped) 1 st order absorption: R or k 0 E A B IC’s: A(o)=A 0 B(0)=0 E(o)=0 Solution: E A B

P T k0k0 k1k1 k 12 k 21 Absorption Simple Compartmental Model (lumped) dA/dt=-k o *A dP/dt=k 0 *A-k 1 *P-k 12 *P+k 21 *T dT/dt=k 12 *P-k 21 T dE/dt=k 1 P Elimination Plasma Tissue

© Crump Institute for Molecular Imaging UCLA School of Medicine Plasma time activity curve and Tissue time activity curve Medical Application Nuclear Medicine Imaging

© Crump Institute for Molecular Imaging UCLA School of Medicine Three compartment FDG model

© Crump Institute for Molecular Imaging UCLA School of Medicine Building the TTAC from the ROI

© Crump Institute for Molecular Imaging UCLA School of Medicine Building the TTAC from the ROI

© Crump Institute for Molecular Imaging UCLA School of Medicine Building the TTAC from the ROI

© Crump Institute for Molecular Imaging UCLA School of Medicine Model Simulation and optimization

© Crump Institute for Molecular Imaging UCLA School of Medicine Model Simulation and optimization

© Crump Institute for Molecular Imaging UCLA School of Medicine Model Simulation and optimization