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.