A Spatial Model of Time-Dose-Area Relationships in Radiation Therapy

Abstract

Parameters affected by more than two simultaneously changing variables are not easily expressed in nonmathematical terms. This problem was noted in teaching some of the fundamentals of clinical radiation therapy to medical students and resident physicians. In particular, it was difficult to explain the relationships of the cumulative radiation dose, the time of administration of this dose, and the volume or area of exposed normal and malignant tissue to the subsequent biological reactions. In order to give a visual appreciation of these relationships, a three-dimensional model was constructed which has proved to be of benefit in teaching and investigation. Materials and Methods A graph depicting isoeffect curves for skin damage and skin cancer curability has been previously published (1). This graph was modified for demonstration in a three-dimensional form (Fig. 1). From this modification a model was constructed (Fig. 2) with two planes indicating normal tissue and malignant tumor responses respectively. The three axes of the model are dose, time, and area on logarithmic scales. The more steeply inclined plane is the normal-tissue-damage isoeffect plane. All points on this plane have co-ordinates of dose, time, and area that are anticipated to result in a 3 per cent probability of late skin necrosis. The other plane is the tumor-cure isoeffect plane. All points on it have co-ordinates giving a 99 per cent probability of permanent tumor regression. The triangular space between the planes, but always above the tumor-cure plane and below the skin-necrosis plane, represents the area of favorable therapeutic ratio. The dose-time-area co-ordinates of any point in this area will give a probability of tumor cure of more than 99 per cent and a probability of late skin necrosis of less than 3 per cent. Table I shows the equations of the slopes and planes. The isoeffect planes correspond to the response surfaces described by Box (2). Although this model is primarily made for teaching purposes, it also serves as a convenient reference for the selection of treatment factors for superficial skin cancers. The slopes and intercepts were originally derived from data on the results of clinical radiation therapy of skin and lip cancer. It is likely that the slopes and intercepts of radiation isoeffect planes vary with tumor type and anatomical location, as well as with numerous other variables. The principles shown may, however, have a general application in the field of radiation therapy of cancer. Summary A three-dimensional model depicting time-dose-area relationships has been described. The model demonstrates the relationships of overall time, cumulative radiation dose, and tissue area to normal and malignant tissue responses.