Treatment Parameters and Applications

3.6.2.1 Treatment Parameters for Planar, Cylindrical, and Spherical Targets

Heat diffusion is strongly dependent on the heater and target geometry. In this section, we discuss three basic geometries: planar, cylindrical, and spherical (Fig. 3.19). In all cases, we

Figure 3.19 Three types of target with different geometry [59]: (a) planar, (b) cylindrical, (c) spherical. 1 is the heater (absorber), 2 is the target, d1 is the size of the heater, d2 is the size of the target.

assume a heater with size d1 located in the center of a target with size d2. We define the ratio x = d2/d1 to be the geometrical factor of the target. As before, T1 is the maximum temperature of the pigmented area, and T2 is the target damage temperature (T1 > T2). As we show in the Appendix, the target thermal damage time can be expressed by the following formula:

TDT = TRT x r(x, A),

where r(x, A) is a function of the geometrical factor x and temperature factor A defined as A = (T2 – T0)/(T1 – T0), T0 is the target and heater temperature before irradiation. Normally T0 is the body temperature and is equal to 373C. TDT is proportional to the TRT. In the Appendix, we present formulas for the TDT of planar, cylindrical, and spherical targets.

Figure 3.20 shows the ratio r(x, A) = TDT:TRT as a function of geometrical factor x for two heating modes: rectangular EMR pulse and flattop temperature pulse. The calculation parameters were T1 = 100°C, T2 = 65°C, T0 = 37°C (A = 0.52).

We emphasize that in the framework of our analytic theory, the ratio r(x, A) does not depend on the size of the entire target and the tissue thermal properties. Several important conclusions follow from Fig. 3.20.

First, the ratio TDT/TRT is an increasing function of geometrical factor x.

Second, the actual value of this ratio is very different for plane, cylindrical, and spherical targets. For a plane target, the TDT is several times higher than the TRT. The TDT exhibits appreciable growth with increasing the target dimensionality. It is implied herein that the planar, cylindrical, and spherical targets are one-, two-, and three-dimensional, respectively.

Third, for the same TDT/TRT, the relative size of the damaged zone x is smallest for the spherical target. Next in this order is the cylindrical target. The plane target exhibits the largest damage area. The relative size x of the damaged zone around the heater decreases when increasing the target dimensionality. The latter two conclusions are intuitively appar­ent. Actually, conductive heating of a weakly absorbing tissue should proceed more effec­tively for a low-dimensional target. This “dimensionality” concept is a useful target parameter. It is also applicable to nonsymmetrical targets. The temperature profile is sharper and better localized for the spherical heater compared to the cylindrical one, and it is better for the cylindrical than the planar heater. For the classical case of selective photothermoly­sis, the target geometry is not important because thermal damage is confined to the same area as the EMR absorption and direct heating. In our case, thermal damage due to heat diffusion is confined to an area that is distinct from the heater. The dependence of heat dif­fusion on heater geometry is very strong.

Fourth, the ratio TDT/TRT depends strongly on heating mode. The rectangular EMR pulse mode (Fig. 3.18a) represents the gentlest heating mode because the heater tempera­ture reaches maximum T1 at the end of the pulse (Fig. 3.18b). The ratio TDT/TRT is maxi­mum for this mode. The flattop temperature pulse mode (Fig. 3.18d) represents the most aggressive heating mode because the heater temperature reaches a maximum just after the beginning of the pulse and the maximum heater temperature takes place during the EMR pulse. The ratio TDT/TRT is a minimum for the flattop temperature pulse mode. As men­tioned earlier, the flattop temperature pulse mode can be realized by using an EMR pulse with a special temporal profile. The initial power density should be very high to raise the heater temperature abruptly (for a time interval of the order of or shorter than the TRT). After the maximum temperature is reached, the power density should undergo a steep fall

to prevent overheating. Then, to maintain the heater temperature at the prescribed level (Fig. 3.18d), the power density should fade gradually to compensate the heat flow out of the heater. The pulse power should be precisely adjusted to keep the heater temperature below the temperature of heater absorption loss. The power depends on heater absorption and size and the EMR attenuation in tissue (see Appendix). In reality, it is probably difficult to exactly create these two modes. Thus, the real value of TDT can be between these two extreme cases.

The ratio TDT/TRT depends on the temperature factor A = (T2 – ^/(Tj – T0). Table 3.2 shows the influence of initial target temperature T0 and maximum heater temperature T1. All calculations were done for a cylindrical target with the same size as Fig. 3.18 for the rectangular EMR pulse. The TDT increases by a factor of 2.6 by precooling from 37°C to 27°C and decreases by a factor of 2.3 by preheating to 45°C. The fluence should be changed at the same time. If the heater temperature can reach a high value without losing absorp­tion, the TDT can be significantly reduced. The ratio TDT/TRT is about 1.5-2 (TDT = (1.5-2)-TRT) for the case when the heater temperature is 200-250°C. In biological tissue, such a high temperature can be expected in melanin in the hair shaft or in an exogenous chromophore such as carbon. But we must remember that the thermal diffusivity can drop in the tissue surrounding the heater due to water vaporization. So this case is very difficult to predict.

As we have shown here, heat diffusion from the heater is very different for different target geometries. The heater temperature should depend on the heater geometry. Figure 3.21 shows the heater center temperature as a function of pulsewidth for a rectangu­lar EMR pulse with the same power. Spherical, cylindrical, and planar heaters have similar sizes d1, thermal properties, and EMR absorption coefficients. The thermal relaxation time of the heater tr depends on geometry, and the ratio is 1:2:3 for spherical, cylindrical, and planar heaters, respectively. If the pulsewidth t is significantly shorter than the thermal relaxation time of the heater tr (t << tr) the temperature rise of all the heaters exhibit the same elevation of temperature. However, as shown in Fig. 3.21, the temperature behavior of heaters with different geometries is very different for pulsewidth equal to or longer than tr. A steady-state heater temperature for a rectangular EMR pulse is possible only for a spheri­cal heater. For cylindrical and especially for planar heaters, the temperature is continuously rising when the pulsewidth is increasing (the power density should be constant, the energy density should be proportional to the pulsewidth). This is because 3D heat diffusion from the spherical heater (in contrast with 2D and 1D heat diffusion from the cylindrical and

planar heaters, respectively) can compensate constant heating from the rectangular EMR pulse. This phenomenon is very important for uniform targets when the heater and target are the same. For example, the temperature of epidermis (planar target) for a rectangular optical pulse at a wavelength strongly absorbed by melanin will continuously rise during a long pulse (t >> тг). However, for a spherical target such as the hair bulb matrix, the tem­perature will stabilize at a steady-state level. To produce a constant heater temperature (flattop temperature pulse as shown in Fig. 3.18d, the EMR pulse shape must be special (Fig. 3.18c), with a strong peak in the beginning and decaying amplitude.