Modeling Biosystems

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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

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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.

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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.

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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.

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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 ltlt the speed of light
• The time interval gt the time interval of quantum
  mechanics
• No nuclear reactions

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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

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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)

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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.

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Modeling Biosystems

• The rate form of conservation
• The time period is infinitesimally small

• Mathematical expression differential equations

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Modeling Biosystems
Example How to derive Nernst equation? Backgroun
d 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
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Modeling Biosystems
Example How to derive Nernst equation? Diffusion
Ficks law

• J flow due to diffusion
• D diffusive constant (m2/S)
• I the ion concentration
• the concentration gradient

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Modeling Biosystems
Example How to derive Nernst equation? Drift
Ohms law

• J flow due to drift
• ? mobility (m2/SV)
• I the ion concentration
• Z ionic valence
• v the voltage across the membrane

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Modeling Biosystems
Example How to derive Nernst equation? Einstein
relationship the relationship between
diffusivity and mobility

• K Boltzmanns constant (1.38x10-23J/K)
• T the absolute temperature in degrees Kelvin
• q the magnitude of the electric charge
  (1.60186x10-19C)

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Modeling Biosystems
Example How to derive Nernst equation?

K
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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 ugtgt1, the relative error is a better
  reflection of the difference between u and v.

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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!

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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

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Concepts of Numerical Analysis

• h10.(-200.50)
• dif_fsin(0.5h)-sin(0.5)./h numerical
  derivative for sin(0.5)
• deltaabs(dif_f-cos(0.5)) absolute errors
• loglog(h,delta,'-')

• hgt10-8, truncation errors dominate roundoff
  errors
• hlt10-8, roundoff errors dominate truncation
  errors

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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 e0)

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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) 0.52(1-t)

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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 xy has a large absolute error
• If yltlt1, then x/y has large relative and
  absolute errors. The same is true for xy if
  ygtgt1
• If x ?y, then x-y has a large relative error
  (cancellation error)

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Concepts of Numerical Analysis
The ill-posed problem The problem is sensitive
to small error
Example Consider evaluating the integrals
n0,1,2,25
n1,2,3,25
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Concepts of Numerical Analysis
The ill-posed problem The problem is sensitive
to small error
yzeros(1,26) allocate memory for
y y(1)log(11)-log(10) y0 for
n226,y(n)1/(n-1)-10y(n-1)end plot(025,y)