Dennis Sullivan, Ph.D. Professor of Electrical Engineering University of Idaho Moscow, ID USA

1. The Finite-Difference Time-Domain (FDTD) Method as the Most Practical Tool for Hyperthermia Planning? Dennis Sullivan, Ph.D. Professor of Electrical Engineering University of Idaho Moscow, ID USA 83844-1023

2. Advantages of FDTD • It is a direct implementation of Maxwell’s • equations (or the bio-heat equation). There is no complicated additional mathematics, e.g., matrix inversion, field singularities, etc. 2. There is no complicated mesh to generate. FDTD uses cubes. • The needed resources increase only linearly • with the size of the problem space, e.g., a problem with 10,000 cells only takes twice as long as a problem with 5,000 cells.

3. Advantages of FDTD (continued) • It is a time-domain method. The actual radiation or temperature rise can be observed. • Also, signal processing techniques can be brought to bear on simulation problems

4. Possible Disadvantages of FDTD • The cubic structure can lead to “stair-casing,” 2. The problem space must be truncated properly or reflections will give erroneous fields.

5. Direct physics-based implementation Electromagnetic radiation is governed by the Maxwell’s equations

6. Direct physics-based implementation In one dimension in free space they become

7. Direct physics-based implementation To put these equations in a computer, take the finite-difference approximations of the partial derivatives in time and space ~ = ~ =

8. Direct physics-based implementation This is a time-domain method. Each new value of the electric field E or the magnetic field H is determined by the previous values

9. Direct physics-based implementation The k represents the location in an array in a computer while n represents time

10. Direct physics-based implementation This results in the following two equations of code in the C program language ex[k] = ex[k] + 0.5*( hy[k-1] - hy[k]) hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1])

11. Direct physics-based implementation n=n+1 Calculate En+1/2 Calculate Hn+1 Each time step represents an increment in the total time T = n Dt.

12. Direct physics-based implementation The following is a one-dimensional simulation of an EM pulse propagation in free space. {T represents the number of time steps.}