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ECE602 BME I Nonlinear Equations in Biomedical Engineering

Explore nonlinear equations in biomedical engineering, from the Colebrook Equation to receptor-ligand dynamics. Learn methods like Newton-Raphson and examples of receptor occupancy calculations.

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ECE602 BME I Nonlinear Equations in Biomedical Engineering

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

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

  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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