Page 325 - Biomedical Engineering and Design Handbook Volume 2, Applications

THE PRINCIPLES OF X-RAY COMPUTED TOMOGRAPHY 303

element of the surface, the initial electron beam current dI will be redistributed to a current density
dJ at x(r , z ). The corresponding volume density of power dissipated, dW, will be given by
1 1
dV
dW =− dJ (10.117)
dx
For uniform diffusion from D, the current density dJ is given by
dI
dJ = 2 (10.118)
−
4π( xz )
d
where it is assumed that z > z . Substituting Eq. (10.118) into Eq. (10.117) gives
1 d
kdI
dW = (10.119)
2
2
−
8π( xz ) ( V − kx) I 2/
d 0
The total power density dissipation at x(r , z ) is determined by integrating over the beam elements
1 1
dI. To do this, we consider a gaussian beam current density profile written as
I ⎛ r 2 ⎞
Jr () = exp − (10.120)
2
. 144π a 2 ⎜ ⎝ . 144 a ⎠ ⎟
where I = total beam current
r = radial distance at the surface
a = beam radius at which the current density is J /2
(r=0)
Hence, the current element dI is given by J dS. The region of beam cross section that is within the
2 1/2
2
range of x(r , z ) is bounded by a circle on the surface with radius [(x − z ) − (z − z ) ] and cen-
1 1 0 d 1 d
ter at x(r , 0). If z < z then x(r , z ) lies between the surface and the depth of diffusion. In this case
1 1 d 1 1
there is an additional contribution to dW from the incoming beam at r = r . This is found by replacing
1
dJ in Eq. (10.117) by J at r = r from Eq. (10.120), to give
1
I ⎛ r 2 ⎞ k
dW = exp ⎜ − 1 2 ⎟ (10.121)
0
144π a 2 ⎝ 144 a ⎠ 2( V − ) 1/2 2
2
.
.
kx
0
The steady-state temperature u, in the presence of a volume distributed heat input W(r, z), is deter-
mined by the Poisson equation, which for axisymmetric geometry is written as
2
2
(,
∂ u + 1 ∂u + ∂ u =− Wr z)
2 2 (10.122)
∂r r ∂r ∂z κ
where k is the thermal conductivity. With the appropriate boundary conditions, solutions to Eq. (10.122)
can be computed numerically. In this process it is convenient to normalize the temperature u by the disk-
heating model temperature v at the beam center, so that
0
.
u 0 099 W 0
u = and v = (10.123)
0
12 /
v 0 πκ a
If the power that is absorbed in the target is W and the backscattered power in the shaded area (Fig. 10.37)
a
is W , the total beam power W = W + W = V I . The power retention factor p = W /W =
b 0 a b 0 0 a 0
1 − W /W , which is a function of atomic number Z and is independent of voltage V (Fig. 10.39).
b 0
According to Eqs. (10.115) and (10.116), scaling the electron beam voltage by f will increase the
2
2
range and the depth of diffusion by f , thus altering the power distribution by 1/f , which in effect is