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
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.
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.
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.
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
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
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)
Modeling Biosystems
The accumulation form of conservation
• The time period is finite
Net amountgenerated
Inside the system
Net amountAccumulated
Inside the system
Net amounttransported
Into the system = +
• )()( consumedgeneratedoutputinput PPPPinside
initialP
inside
finalP
• 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.
Modeling Biosystems
The rate form of conservation
• The time period is infinitesimally small
Generation rateInto the system
at t
Rate of changeinside the system
at t
Transport rateinto the system
at t = +
• Mathematical expression: differential equations
• )()(
cgoi PPPPdt
dP
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
Modeling Biosystems
Example: How to derive Nernst equation?
Diffusion: Fick’s law
dx
dIDJ diffusion
• J: flow due to diffusion
•D: diffusive constant (m2/S)
• I: the ion concentration
• : the concentration gradientdx
dI
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
dx
dvZIJ drift
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)
q
KTD
Modeling Biosystems
Example: How to derive Nernst equation?
0 driftdiffusion JJdt
dJ
K+
ioi K
K
q
KTvv
][
][ln 0
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.
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!
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
)(!
)()(''
2
)()(')()()( 0
)(00
20
000 xfk
xxxf
xxxfxxxfxf k
k
10
)(00
20
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)()(''
2
)()(')()()(
kk
k
Rxfk
xxxf
xxxfxxxfxf
xxfk
xxR k
kk
0)1(
101 ),()!1(
)(
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
10-20
10-15
10-10
10-5
100
10-10
10-8
10-6
10-4
10-2
100
Concepts of Numerical Analysis
Floating point representation in computer
ettdddd
xfl 2)2842
1()( 321
• 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)s; 11 bits: exponent (e); 52 bits: fraction (t)
• A bias 1023 is added to e to represent both negative and A bias 1023 is added to e to represent both negative and positive exponents. (e.g., a stored value of 1023 indicates e=0) positive exponents. (e.g., a stored value of 1023 indicates e=0)
Not saved!
Concepts of Numerical Analysis
Floating point representation in computer
• OverflowOverflow: : A number is too large to fit into the floating-point system in use. FATALFATAL!
• UnderflowUnderflow: 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)
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)
Concepts of Numerical Analysis
The ill-posed problem: The problem is sensitive to small error
Example: Consider evaluating the integrals
dxx
xy
n
n
1
0 10n=0,1,2,…25
10ln11ln0 y
1101
nn yn
y n=1,2,3,…25
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); %y0for n=2:26,y(n)=1/(n-1)-10*y(n-1);endplot(0:25,y)
0 5 10 15 20 25-2
0
2
4
6
8
10x 10
8