Appendix: Determination of Amplitude and Duration of Rectangular EMR Pulses

The appendix summarizes the EMR pulse parameters for the treatment of the basic targets exhibiting a high degree of symmetry, that is, the planar, cylindrical, and spherical ones. The notations and the basic parameters of the problem are explained in Table 3.A1 [59].

Table 3.A1

Variable	Dimensionality	Name	Assumptions and Relations
k	2 -1 cm[1] [2] s 1	Thermal diffusivity	Assumed to be the same all over the target
P	-3 g cm [3]	Density	Assumed to be the same all over the target
c	J/(g K)	Specific heat	Assumed to be the same all over the target
Pa	cm-1	Tissue absorption coefficient	Assumed to be zero outside the heater
q	a. u.	The ratio of radiance to the input power density
d1	cm	Thickness or diameter of the heater
d2	cm	Thickness or diameter of the target	d2 > d1
d3	cm	Mean spacing between the targets	d3 > d2
T>	°C	Initial temperature of both the target and the surrounding tissue	T0 = 37°C
T 1max	°C	Temperature of heater absorption loss	T1max = 100-250°C
T1	°C	Maximum temperature of the heater(absorber)	T2 < T1 – T1max
T T2	°C	Temperature of irreversible damage of the tissue	T2 = 70°C
A	a. u.	Temperature factor, temperature ratio	t—H V і i III <1
x	a. u.	Geometrical factor, diameter ratio	x = d2/d1 > 1

Our analysis was based on the heat conduction equation. We have found approximate analytic solutions of TDT and input power density P.

The final expressions for the important variables in question are outlined in simplified form in Table 3.A2 [59]. It is implied that the thermal constants, that is, the density, the thermal diffusivity, and the thermal conductivity do not vary significantly within the target and the surrounding tissues.

The present discussion is restricted to rectangular EMR pulses only. Our goal herewith is to determine the pulsewidth and the pulse amplitude. This may be performed in the following order:

T2, initial tissue temperature T0, the temperature factor A = (T2 – TO /(T1 – T0) can be determined.

3. Using these parameters and formulas, we can use the formulas in Table 3.A2, rows 1-3 to determine tr, TRT and finally TDT. As explained earlier, the TDT is approximately equal to the optimum duration of the EMR pulse t0.

4. Based upon an estimate of the EMR attenuation factor at the depth of target q and the absorption coefficient of pigmented area and the heater temperature T1, the power density on the skin P using can be determined by using the formulas from Table 3.A2 row 4. The power density should be limited in order not to bleach the pigmented area, but it should be significantly high to rich the temperature of target damage T2.

5. Treatment fluence is given by F= P • TDT or F= P • t0.

In contrast to both the planar and cylindrical targets, the TDT for the spherical target may be evaluated to infinity (see Table 3.A2, row 3, rightmost column). This means that one cannot ensure the safety of the heater in an attempt to damage the whole target. After a part of the target becomes damaged, the heater temperature reaches the crucial value T1. This gives rise to phase transitions, bleaching, bubble formation, and other nonlinear processes lying outside the scope of this chapter. Therefore, our simple theory provides the means to describe thermal damage of sufficiently small spherical targets only. More precisely, for a given value of A the diameter ratio x must not exceed the value obtained from the equation D = 0, where variable D is a function of x determined by the last expression of Table 3.A2, row 3, rightmost column.