XIX Sympozjum Srodowiskowe PTZE - materialy.pdf

XIX Sympozjum PTZE, Worliny 2009
human
body
92
excitation
coil
Figure 1. Schematic view of human body surrounded by wire with excitation current.
Dimensions are given in meters.
The human body is considered as homogeneous medium with averaging material parameters.
The current in exciting wire is flowing in counter clock-wise direction. The radius of this coil
has value r1. The exciting current in the coil generates sinusoidal electromagnetic field, which
next induces eddy currents in human body. These currents are sources of heat and after some
transient time, a temperature distribution in body is established. In order to calculate
temperature, first distribution of electromagnetic field generated by circular coil with exciting
current has to be calculated. Following partial differential equation should be solved:
⎛ 1 ˆ ⎞ 2
∇× ( ωε jωσ)
ˆ ˆ
⎜ ∇× A⎟− + A = J i
(1)
⎝µ ⎠
In two dimensions we assume symmetry with respect to coordinate φ, so magnetic vector
potential can be written as ˆ A = Aϕ (, rz)
e ϕ and Jˆ i = Jie φ thus equation (1) can be written in simpler
form
( rAφ) ( rAφ)
⎡
∂ ⎛ 1 ∂ ⎞ ∂ ⎛ 1 ∂ ⎞⎤
2
− ⎢ ⎜ ⎟+ ⎜ ⎟⎥−
( ωε+ jωσ) Aφ= J
⎢∂r⎜µ r ∂r ⎟ ∂z⎜µ r ∂z
⎟
⎝ ⎠ ⎝ ⎠
⎥
⎣ ⎦
As boundary condition we assume zero potential on symmetry axis z. The above equation is
solved by finite element method over given computational domain. If σ >> ωε then
displacement current can be neglected. Unlike the computation of the electric currents in
body, for which agreement exists accordingly a derived physical model, no clear consensus
exists for an appropriate mathematical model for the evaluation of temperature field
distribution in biological tissues. An extremely important work in the modeling of heat
transfer in biological tissues was done over half a century ago by Pennes [2, 3]. The equation,
which he derived, is named bioheat equation, and it can be derived from the classical Fourier
law of heat conduction. This model is based on the simple assumption of the energy exchange
between the blood flowing in vessels and the surrounding the tumor tissues. Pennes model
may provide suitable information on temperature distributions in whole body, organ under
consideration, and tumor analysis under study.
Pennes model states, that the total heat exchange between tissue surrounding a vessel and
blood flowing in it, is proportional to the volumetric heat flow and the temperature difference
between the blood and the tissue. The expression of Pennes bioheat equation in a body with
uniform material properties in steady state is given by [4]
i
(2)