Closed-form solution for the thermal dose delivered during single pulse thermal therapies

This study provides a closed form, analytical expression for the thermal dose delivered by a single heating pulse. The solution is derived using the effective cooling method and the non-linear Sapareto-Dewey equation to determine the thermal dose delivered by the time-temperature history of a treatment. The analytical solutions are used to determine the optimal treatment conditions, i.e. those that exactly deliver the desired thermal dose at a specified time. For purposes of illustration, this study focuses on a 'conservative' clinical approach in which the desired thermal dose is delivered at the end of the 'cool down' period. The analytical results show that, after a clinical strategy has been chosen ( e. g. conservative, aggressive or intermediate), the user can only specify two free variables for such an optimal treatment. Results are presented which suggest that a practical approach would be to specify both ( 1) the desired thermal dose to be delivered to the target ( the clinically relevant outcome) and ( 2) the peak temperature to be reached ( a measurable, clinically useful, patient dependent response variable that can be employed in feedback control systems); and then determine the associated, optimal heating magnitude and duration that need to be used to reach that dose and temperature. The results also reveal that, with a given patient condition and power deposition distribution ( together specifying an effective cooling time constant for the treatment) and a specified thermal dose, there is a maximum allowable peak temperature that, if exceeded, will result in 'over-dosing' the heated tissue. The results also show that avoiding such non-optimal 'over-dosing' will be difficult in most high temperature therapies since, when high temperatures are produced in tissues, the temperature decay must be very fast in order to avoid over-dosing during the cooling period. Such rapid cooling can only occur if short effective cooling time constants are present - either as a result of large tissue blood flows in the patient or due to large conduction effects induced by the use of highly localized power deposition sources.