A Mathematical Model for Gamma Ray Transport in the Cardiac Region

We study gamma ray transport in a medium composed of three time-dependent regions. The inner region V1is bounded by the time-dependent surface S1. The second region V2is bounded by S1and by the surface S2 which is homothetic to S1. This is used to model radionuclide imaging of the cardiac region: V1 is, roughly, the blood within the heart of a patient, V2represents the heart walls, and V3is the rest of the body. We prove that the quasi-stationary solution of the corresponding photon transport problem, which satisfies a stationary-like equation, is a good approximation to the exact solution because the speed of the heart walls is much smaller than the speed of light. Furthermore, we show that the quasi-stationary solution n t. and its properties may be used to convert the gamma ray densities n tqd., n tq2d., . . . , detected at later times, to the first detection n t..