Maxwell’s Equations in Vasculature Modeling

1. Maxwell’s Equations in Vasculature Modeling Sukhi Basati 6/11/08 Laboratory for Product and Process Design Advisor: Andreas Linninger H. Xue, J. Hajnal, D. Rueckert, Extraction and Registration of Neonatal Cerebral Vasculature, Imperial College London.

2. Overview of Presentation • Maxwell’s equations background • Electrostatics • Relationship between diffusion equation and electrostatics • Electrical analogy of fluid flow through a tube • Single rigid tube • Single elastic tube • Tubes in series • Tubes Converging and Diverging • 2d electric analogy of fluid through a single tube example • Using Maxwell’s eqns.

3. Motivation • What are Maxwell’s Equations? • What are the applications? • Molecular biology, Magnetohydrodynamics, Bioengineering • Can we predict the effect of the fields on particles, molecules, electrodes, etc.?

4. Maxwell’s Equations Comes from Gauss’s Law Faraday’s Law of induction • Changing Magnetic Field Also derived from Gaussian Theorem Ampere’s Law • Electric Current induces Magnetic Field • Divergence and Curl of a vector are needed for examination. Griffiths, D., Introduction to Electrodynamics, Prentice-Hall, 1981

5. Diffusion Equation • PDE of the form: Where is the diffusion coefficient, and is a scalar value • Quick derivation: • Fick’s first law tells us: • The continuity equation tells us: • By substituting: www.en.wikipedia.org

6. Electrostatics Equation General Divergence of a vector field Integration of the divergence of a vector field over a certain volume is the vector field itself integrated over the surface that contains the volume. Coulumbs law gives us Maxwell’s 1st eqn. Since Charge free domain, Laplace’s Eqn.

7. Continuity Equation • General form of Conservation of mass, energy, momentum, charge: • One of maxwell’s eqns tells us • If you take the divergence of both sides • The divergence of a curl is zero, so • Maxwell’s 1st eqn tells us • By Substitution • But if no current leaves the volume • Diffusion Equation: or or

8. Applications • Poisson-Boltzmann equation describes the electrostatic interactions between molecules in an ionic solution. • Important in molecular dynamics and biophysics. Boltzmann constant Temperature Factor for accessibility for position of ions Position dependent dielectric Electrostatic potential Charge density of solute Concentration of ion at inf. From solute Charge of ion Charge of proton Isosurfaces of calculated electrostatic potentials (solid) and analytic potentials (mesh) for the aspirin molecule at 0.5A grid spacing.1 P. Labute, M. Santavy, “Numerical Solution of the Non-Linear Poisson-Boltzmann Equation”, Chemical Computing Group, Montreal, Canada

9. Applications • Sensor for determining volume changes in a cavity. • Principle: • As volume changes, the electric field distribution changes. • Measurable by voltage sensing electrodes. Equation is Laplace’s equation (considering no initial charge in the domain)

10. Fluid-Electrical Analogy Table Electric current/ fluid flow analogy Volume Flow Current (Amps) Pressure (Pa) Voltage (Volts) Fluid Resistance Resistance (Ohms) Tube Compliance Capacitance (Farad) Volume Charge (Coulomb)

11. A Single Rigid Tube P0 P1 F0 F1 Continuity: Momentum: R0 V1 V0 I0 I1 Note that there is no effect of the change in area on output current. Kirchoff Current Law: Ohms Law:

12. A Single Elastic Tube P* A0 P0 P1 F0 F1 Continuity Momentum Distensibility Kirchoff Current Law: R0 V0 V1 I1 I0 Ohms Law: Q0 V* Capacitance Charge

13. Tube 0 Tube 1 Tube 2 F2in F2out F0in F1in A2 A0 P0 A1 P2 P1 P1 P3 Tubes in Series R R R 2 1 0 V V V V 0 1 2 3 Q Q Q Q 1 2 0 0 * * V * V V Continuity Momentum X3 X3 Distensibility +3 BC +3 BC = 12 equations = 12 equations

14. Tube 1 Tube 0 Tube 3 Tube 2 R1 Q1 R0 R3 V0 V1 V2 V* Q3 Q0 R2 V* V* Q2 V* Tubes in Parallel (Divergence and Convergence) Tube 1 Tube 2 Tube 3 Tube 4 V3

15. Return to Maxwell’s eqns for fluid flow • Electrostatic assumptions are made. Electrostatics eqn solved in ADINA for single rigid tube Electrostatics equations solved in ADINA for a single compliant tube N - S eqns solved in ADINA for single rigid tube N – S equations solved in ADINA for a single compliant tube

16. Summary • Introduction to Maxwell’s Equations • Primarily focusing on Electrostatic Assumptions. • Relationship between Diffusion and Electrostatic Equation. • Derived from continuity equation. • 1D fluid flow through tubes as an electrical analogy. • Rigid, compliant tubes. • Computer simulated tubes using Electrostatic assumptions may be an appropriate model instead of 1D circuit models.