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Nonlinear Equations in Biomedical Engineering

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Presentation on theme: "Nonlinear Equations in Biomedical Engineering"— Presentation transcript:

1 Nonlinear Equations in Biomedical Engineering
ECE602 BME I Nonlinear Equations in Biomedical Engineering

2 Examples of nonlinear equations in BME
The Method of Successive Substitution The Method of False Position (Linear Interpolation) The Newton-Raphson Method Newton’s method for Simultaneous Nonlinear Equations Implementation issues

3 Nonlinear Equations in BME
Colebrook Equation Laminar flow of blood when catheter is used b: the friction factor, dimensionless, used to determine the change in the velocity Re: Reynolds number, used to characterize flows (Re>2000, turbulent)

4 Nonlinear Equations in BME
Receptor occupancy during receptor-ligand dynamics Cell membrane receptors: bind to specific ligands (hormones, growth factors), and trigger intracellular signaling in mammalian cells. Multimetric receptors: possess multiple binding units –making the binding chemistry nonlinear

5 Nonlinear Equations in BME
Receptor occupancy during receptor-ligand dynamics Example: A: trimetrc receptor B: dimetric receptor L: the ligand D: receptor—bound ligand K: equilibrium constant C: concentration

6 Nonlinear Equations in BME
Receptor occupancy during receptor-ligand dynamics Example: x1: fractional occupancy of A; x2: fractional occupancy of B CA,0, CB,0, CL,0: the initial concentrations

7 Successive Substitution
x0: initial guess Iterative formula: Terminating condition: Convergence condition: for all x in the search interval

8 Successive Substitution
Advantages It can be started with only a single point No need for calculating the derivative of the function

9 False Position (Linear Interpolation)
Initial guess: x0, x1 ( ) Iteration: Determine x+, x- Terminating condition: x+: the point at which x-: the point at which

10 False Position (Linear Interpolation)
Advantage Doe not require evaluating derivatives. Disadvantage The accuracy and speed of convergence depend on the choice of initial guesses.


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