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.