Modeling Biosystems

1. Modeling Biosystems • Mathematical models are tools that biomedical engineers use to predict the behavior of the system. • Three different states are modeled • Steady-state behavior • Behavior over a finite period of time • Transient behavior

2. Modeling Biosystems • Modeling in BME needs an interdisciplinary approach. • Electrical Engineering: circuits and systems; imaging and image processing; instrumentation and measurements; sensors. • Mechanical Engineering: fluid and solid mechanics; heat transfer; robotics and automation; thermodynamics. • Chemical Engineering: transport phenomena; polymers and materials; biotechnology; drug design; pharmaceutical manufacturing • Medicine and biology: biological concepts of anatomy and physiology at the system, cellular, and molecular levels.

3. Modeling Biosystems • A framework for modeling in BME • Step one: Identify the system to be analyzed. • Step two: Determine the extensive property to be accounted for. • Step three: Determine the time period to be analyzed. • Step four: Formulate a mathematical expression of the conservation law.

4. Modeling Biosystems • Step one: Identify the system to be analyzed • SYSTEM: Any region in space or quantity of matter set side for analysis • ENVIRONMENT: Everything not inside the system • BOUNDARY: An infinitesimally thin surface that separates the system from its environment.

5. Modeling Biosystems • Step two: Determine the extensive property to be accounted for. • An extensive property doe not have a value at a point • Its value depends on the size of the system (e.g., proportional to the mass of the system) • The amount of extensive property can be determined by summing the amount of extensive property for each subsystem comprising the system. • The value of an extensive property for a system is a function of time (e.g., mass and volume) • Conserved property: the property that can neither be created nor destroyed (e.g. charge, linear momentum, angular momentum) • Mass and energy are conserved under some restrictions • The speed of the system << the speed of light • The time interval > the time interval of quantum mechanics • No nuclear reactions

6. Modeling Biosystems • Step three: Determine the time period to be analyzed. • Process: A system undergoes a change in state • The goal of engineering analysis: predict the behavior of a system, i.e., the path of states when the system undergoes a specified process • Process classification based on the time intervals involved • steady-state • finite-time • transient process

7. Modeling Biosystems • Step four: Formulate a mathematical expression of the conservation law. • The accumulation form (steady state or finite-time processes) • The rate form (transient processes)

8. Net amount Accumulated Inside the system Net amount transported Into the system Net amount generated Inside the system = + Modeling Biosystems • The accumulation form of conservation • The time period is finite • Mathematical expression: algebraic or integral equations • It is not always possible to determine the amount of the property of interest entering or exiting the system.

9. Rate of change inside the system at t Transport rate into the system at t Generation rate Into the system at t = + Modeling Biosystems • The rate form of conservation • The time period is infinitesimally small • Mathematical expression: differential equations

10. Modeling Biosystems Example: How to derive Nernst equation? Background: Nernst equation is used to describe resting potential of a membrane The flow of K+ due to (1) diffusion (2) drift in an electrical field

11. Modeling Biosystems Example: How to derive Nernst equation? Diffusion: Fick’s law • J: flow due to diffusion • D: diffusive constant (m2/S) • I: the ion concentration • : the concentration gradient

12. Modeling Biosystems Example: How to derive Nernst equation? Drift: Ohm’s law • J: flow due to drift •  : mobility (m2/SV) • I: the ion concentration • Z : ionic valence • v: the voltage across the membrane

13. Modeling Biosystems Example: How to derive Nernst equation? Einstein relationship: the relationship between diffusivity and mobility • K: Boltzmann’s constant (1.38x10-23J/K) • T : the absolute temperature in degrees Kelvin • q: the magnitude of the electric charge (1.60186x10-19C)

14. Modeling Biosystems Example: How to derive Nernst equation? K+

15. Concepts of Numerical Analysis • Errors: absolution and relative (given a quantity u and its approximation) • The absolute error: |u - v| • The relative error: |u – v|/|u| • When u 1, no much difference between two errors • When |u|>>1, the relative error is a better reflection of the difference between u and v.

16. Concepts of Numerical Analysis • Errors: where do they come from? • Model errors: approximation of the real-world • Measurement errors: the errors in the input data (Measurement system is never perfect!) • Numerical approximation errors: approximate formula is used in place of the actual function • Truncation errors: sampling a continuous process (interpolation, differentiation, and integration) • Convergence errors: In iterative methods, finite steps are used in place of infinitely many iterations (optimization) • Roundoff errors: Real numbers cannot be represented exactly in computer!

17. Concepts of Numerical Analysis • Taylor series: the key to connecting continuous and discrete versions of a formula • The infinite Taylor series • The finite Taylor formula

18. Concepts of Numerical Analysis • h=10.^(-20:0.5:0); • dif_f=[sin(0.5+h)-sin(0.5)]./h; % numerical derivative for sin(0.5) • delta=abs(dif_f-cos(0.5)); % absolute errors • loglog(h,delta,'-*') • h>10-8, truncation errors dominate roundoff errors • h<10-8, roundoff errors dominate truncation errors

19. Concepts of Numerical Analysis Floating point representation in computer Not saved! • IEEE 754 standard, used in MATLAB • di = 0 or 1 • 64 bits of storage (double precision) • 1bit: sign s; 11 bits: exponent (e); 52 bits: fraction (t) • A bias 1023 is added to e to represent both negative and positive exponents. (e.g., a stored value of 1023 indicates e=0)

20. Concepts of Numerical Analysis Floating point representation in computer • Overflow: A number is too large to fit into the floating-point system in use. FATAL! • Underflow: The exponent is less than the smallest possible (-1023 in IEEE 754). Nonfatal: sets the number to 0. • Machine precision (eps): 0.5*2^(1-t)

21. Concepts of Numerical Analysis Floating point representation in computer • How to avoid roundoff error accumulation and cancellation error • If x and y have markedly different magnitudes, then x+y has a large absolute error • If |y|<<1, then x/y has large relative and absolute errors. The same is true for xy if |y|>>1 • If x y, then x-y has a large relative error (cancellation error)

22. Concepts of Numerical Analysis The ill-posed problem: The problem is sensitive to small error Example: Consider evaluating the integrals n=0,1,2,…25 n=1,2,3,…25

23. Concepts of Numerical Analysis The ill-posed problem: The problem is sensitive to small error y=zeros(1,26); %allocate memory for y y(1)=log(11)-log(10); %y0 for n=2:26,y(n)=1/(n-1)-10*y(n-1);end plot(0:25,y)