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Mathematics of dynamic SPECT

Stationary observation model

If the requirement of time-independence of the object is fulfilled, we may combine all the measurements in matrix form:

(1)

or

(2)

where v is the additive noise. The exact form of matrix H depends on the complete imaging system and, for example, attenuation, scatter and the collimator blurring can be taken into account in construction of H. The estimation problem in stationary SPECT is then to solve the activity distribution, given Eq. (2) and the noisy observations p . The observation model in SPECT is usually underdetermined and some regularization method is necessary to obtain reasonable solution of Eq. (2).

Time-dependent observation model

Let us now assume that the activity of the slice depends on the time. For the simplicity we assume that the discretization interval of the activity equals the acquisition time of one projection. That is, we assume that the activity remains stationary during the measurement of one projection. The equation describing the measurements is then

(3)

where t refers now to both time and direction of measurement. If we denote the total number of projection measurements and the steps in time discretization with T, the system of equations describing all the acquisition directions contains TN2 unknowns compared with N2 unknowns of the time-invariant situation, while the number of observations remains the same. Therefore the time-varying model is highly underdetermined and further information about the parameters is required. Natural approach is then to approximate the time-evolution of the activity distribution with some simple model.

Evolution model

One of the simplest models for the time-variation of the activity distribution is the Markov model. In the first order Markov model we assume that the time-variation of the parameters can be written in the form

(4)

with some initial parameters and Gaussian, zero mean disturbance sequence w. The evolution matrix F relates the state at time t to the previous state at time t+1. This system of N2 equations is called the state equation and the term w is the so-called state noise process.

It can be shown, that the first order Markov model can be used to approximate two common tracer evolution models in SPECT, namely the diffuse and compartmental transportation models.

State estimation

The filtering problem is now to estimate the state at time t based on a subset of the observations where the last available measurement is from time t. The most common recursive estimator for is the Kalman filter algorithm. With the linear state-space model (3-4). Kalman filter can be written in a form of simple recursion:

(5)
(6)
(7)
(8)
(9)

with some initialization for the Kalman predictor and prediction covariance.  In SPECT the reconstruction of the images is done usually off-line. This means that generally the full set of projections from a dynamic study can be used to estimate the time-varying activities. The recursive algorithm that calculates the estimate based on the full data set is called the Kalman Smoother. The equations that are needed in addition to the Kalman filter Eq. (5)-(9) are:

(10)
(11)
(12)

The calculation of the smoother estimates thus requires the calculation of the prediction and the filter estimates with the Kalman filter algorithm. The so-called backward gain matrices A has also been stored during the filtering. The smoother estimates are then obtained by applying the Eq. (10)-(12) in time-reversed (backward) order.

Non-negative estimates

The activity distribution of the object to be imaged is non-negative by the definition. This constraint is not included in the Kalman filter and smoother algorithms and therefore the estimates can have negative values. The non-negative estimate can be obtained by finding the non-negative maximizers of the Gaussian probability densities

and ,

associated with the Kalman filtering and smoothing, respectively. The non-negative filter estimate at time t, is defined as:

(13)

This is equivalent to

(14)
  (15)

where . This inequality constrained minimization problem can be solved using iterative methods, for example with NNLS-algorithm. Non-negative Kalman smoother estimate is obtained by using corresponding parameters in in the Eq. (14).

Variables

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