ECE602%20BME%20I

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• ECE602 BME I
• Linear Models of Biological System

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• Examples of linear models of biological system
• Gauss elimination with partial row pivoting
• Jacobi iterative method

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Linear Models of Biosystems

• Algebraic reconstruction of a biomedical image
• A square grid is imposed on the image f(x,y)
• fj the constant value in the jth cell
• N the total number of cells (n2)
• A ray is a fat line running through the (x, y)
  plane
• pi ray-sum measured with the ith ray, i1,2,M

• wij weighting factor representing the
  contribution of the jth cell to pi

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Linear Models of Biosystems

• Application of spectroscopy for biological
  solutions
• Spectroscopy study of the interaction between
  radiation (electromagnetic radiation, or light,
  as well as particle radiation) and matter

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Linear Models of Biosystems

• Application of spectroscopy for biological
  solutions
• Quantitative determination of the concentration
  of biological molecules in solution

Ai the absorbance of a solution at wavelength
i,(log10(I0/I) where I0 the intensity of
radiation before passing through the sample I
the intensity after passing through the sample
the extinction coefficient of all components k
(from 1 to N) obtained at wavelength i. If the
radiation can pass through easily, then the
coefficient is low.
C(k) the concentration of the component k
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Linear Models of Biosystems

• References
• Image reconstruction
• A. Rosenfeld and A.C. Kak, Digital picture
  Processing, 2nd ed. New York Academic Press,
  1982.
• Spectroscopy
• I. Tinoco, K. Sauer, J. C. Wang, and J. D.
  Puglini, Physical Chemistry Principles and
  Applications in Biological Sciences. Upper Saddle
  River, NJ Prentice Hall, 2002.

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Gaussian Elimination with Row Partial Pivoting
Three Steps to solve A0xb (A0n x n) Step 1
Initialization Step 2 Partial row pivoting and
forward elimination Step 3 Back substitution
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Gaussian Elimination with Row Partial Pivoting
Step 1 Initialization AA0 b (A n x (n1))
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Gaussian Elimination with Row Partial Pivoting
Step 2 Partial row pivoting and forward
elimination For k1 n-1 pivota(k,k) (pivot
element) pk
(pivot row) For ik1 n If a(i,k)gtpivot
pivota(i,k) (updating the pivot
element)
pi
(updating the pivot row)

End End If pgtk Interchange rows k and
p End For ik1n
(Loop on rows) For jn1-1k (Loop on
columns for a given row) a(i,j)a(i,j)-a(k,j)
(a(i,k) / a(k,k)) End End End
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Gaussian Elimination with Row Partial Pivoting
Step 3 Back substitution x(n)a(n,n1)/a(n,n)
For in-1-11, (Loop of rows, working
backwards)
End
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The Jacobi Iterative method
Step 1 Choose
Step 2 Do
Until
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The Jacobi Iterative method
Example
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The Jacobi Iterative method
Example
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The Jacobi Iterative method
The Jacobi method is only applied to diagonally
dominant matrix