Blood perfusion and thermal conduction effects in Gaussian beam, minimum time single-pulse thermal therapies

A previous analytical study has shown that the minimum obtainable treatment time for a single pulse that delivers a given thermal dose to a specified point at a specified time occurs when the temperature at that point is rapidly raised to its maximum allowable value. The present study extends that result by investigating the spatial distribution of thermal effects of a single Gaussian shaped focal zone pulse that reaches that maximum allowable temperature at the center point of the focal zone. Analytical solutions are obtained that separately include the effects of perfusion and conduction. This situation is analyzed for a conservative treatment strategy in which the desired thermal dose is delivered when the tumor cools down to basal conditions. The results show that for a specified thermal dose delivered by a spherical Gaussian beam with focal widths below approximately 4 mm, the maximum allowable temperature, the minimum obtainable treatment time, and the size of the treatment zone (as a percentage of the size of the Gaussian beam) are all independent of the tissue blood perfusion, and are only functions of the focal zone size. Conversely, for focal widths above approximately 20 cm, these results are independent of the focal width and are only functions of blood perfusion. Between these two sizes (where most practical treatments will occur, since single pulses with widths of <4 mm and >20 cm will be uncommon in practice) a transition zone exists in which both perfusion and conduction effects are important. Thus while it is possible to implement a truly perfusion-independent, single pulse thermal treatment by using focal widths of <4 mm, in practice many such pulses will be needed to treat most tumors. This is especially true since the nonlinear temperature/thermal dose relationship causes the width of the delivered dose distribution to be only approximately 25%-30% of the width of the focal zone. However, shorter overall treatment times can be obtained when multiple pulses are linked together by using larger focal zone sizes, but this gain in treatment time is accompanied by increased effects of perfusion, illustrating the conflict between attaining both perfusion-independence and minimal treatment time for multiple-pulse thermal treatments.