In tomographic imaging it is commonly assumed that the properties of the object do not vary in time during the measurement of the projections. This ensures that all the projections view the same object and they differ only in the perspective of the view. Even in this time-independent case the problem is ill-posed and the reconstruction of the activity distribution from the measured projections requires some sort of regularization.
In SPECT the requirement of time-independence is often violated due to relatively long acquisition times and short half life of the radioactive tracers. There are also studies, where the time-dependence of the activity is just the information that we are interested in. The traditional approach to imaging of time-varying object has been shortening of the acquisition time.
Time-varying reconstruction in SPECT
By combining the linear modeling of the measurement system, that is, the observation model or the space equation, and the evolution model of the time-varying activity distribution, called the state equation, we may construct a state-space model for emission tomography. Using the state-space model the filtering problem is to estimate the activity distribution at discrete times 0,...,t using the projection data up to time t. This can be done recursively with, for example, the well-known Kalman filter .
Since the reconstruction of the images is usually done off-line, we may use the full data set of the study in the estimation of activity distribution at any time, that is, we may use also the future measurements in estimation of the activity distribution at given time. This is then a so called smoothing problem that can also be solved recursively using fixed interval Kalman smoother.
More detailed discussion of the reconstruction methods is given
in the Section Mathematics of
We have simulated two different transportation models, compartmental and diffuse transportation. In both cases the measurement system is chosen to be single headed gamma camera. The reconstructions are done in 2D, that is, only one slice of the activity distribution is considered. Collimator blur, attenuation or scatter are not modeled. In this simple simulation the image plane is discretized with 42 X 42 square pixels and the projections are vectors of 42 elements. The projections are acquired with full 360 degree rotation with 10 degree increments. The time scale is discretized in a way that we estimate the activity distribution corresponding to each measured projection.
In the case of compartmental transport the evolution model is assumed to be totally unknown and therefore we use so called random-walk model for time-evolution. However, we may include prior information about the compartments in the reconstruction by assuming that the activity is constant within each compartment at given time t. This may be used to decrease the number of unknowns in the reconstruction and thereby regularize the problem (Fig. 1).
In the case of diffuse transport there are no constant areas in the image plane but we may approximate the evolution model with simply matrix-vector multiplication. In the reconstructions we use evolution model that is formed with greatly underestimated (20% of the correct value) diffusion coefficient (Fig. 2).
|Figure 1. Simulation of compartmental transport. At left, reconstructions without prior information of the compartments, at right, reconstructions with three known compartments. At the top are shown the projections and the time-activity curves of the compartments (ROI's) with solid line for correct activity, dashed for Kalman smoother and dotted for Kalman filter. At the bottom are shown the Kalman filter and smoother estimates.|
|Figure 2. Simulation of diffuse transport. In the figure at the top are shown the projections and vertical and horizontal profiles of the activity distribution, true profile with solid line, Kalman smoother estimate with dashed and Kalman filter estimate with dotted line. The profiles are taken from the center of the image plane. At the bottom are shown the true distribution, Kalman filter estimate and Kalman smoother estimate.|