^{1}

^{2}

^{1}

^{2}

Multipinhole SPECT system design is largely a trial-and-error process. General principles can give system designers a general idea of how a system with certain characteristics will perform. However, the specific performance of any particular system is unknown before the system is tested. The development of an objective evaluation method that is not based on experimentation would facilitate the optimization of multipinhole systems. We derive a figure of merit for prediction of SPECT system performance based on the entire singular value spectrum of the system. This figure of merit contains significantly more information than the condition number of the system, and is therefore more revealing of system performance. This figure is then compared with simulated results of several SPECT systems and is shown to correlate well to the results of the simulations. The proposed figure of merit is useful for predicting system performance, but additional steps could be taken to improve its accuracy and applicability. The limits of the proposed method are discussed, and possible improvements to it are proposed.

Small-animal SPECT imaging provides the opportunity for advanced monitoring and analysis of cancer drug tests in laboratory animals. In order to be effective, a small-animal SPECT system must have high spatial resolution and high sensitivity. The design of multipinhole systems involves many subtle factors which affect both resolution and sensitivity in ways that are difficult to model. Currently, systems are designed based only on general principles; optimization is not a part of the design procedure. Once an aperture is designed, it is tested and analyzed. A method whereby system performance could be predicted, and therefore optimized, in the design phase would allow system designers to experiment with a wider range of design possibilities and to achieve better design results overall.

The main problem in deriving a system performance predictor is the definition of system performance. An optimal system obtains a balance between high spatial resolution and low system noise. Therefore, an objective error predictor must favor both system characteristics equally; an optimal system, as defined by the error predictor, must give low noise and allow for detection of small lesions. We present an error predictor which is shown to account for both spatial resolution and noise, and therefore correlates to image quality in terms of usefulness to the clinician.

In addition, the error predictor provides an objective measure of system performance. Current evaluations of SPECT systems include simulation and actual physical imaging. Of those performing physical experiments, some use laboratory animals [

In order to obtain this error prediction, the singular value spectrum must be calculated. The matrix-based representation of clinical systems is far too large to store the entire system matrix in the computer memory. We show that the application of the power method to the analysis of SPECT imaging systems is valuable because of the ability to use simulation to find the singular value spectrum of a system. This allows for a frequency-based analysis of systems involving attenuation, photon scattering, and other complex and random phenomena, for which the creation of a system matrix would be complicated.

The theoretical
background of the proposed error estimate is presented in Section

In Section

Section

For any given
system, a matrix

Several
algorithms exist for solving this type of problems [

In this form, the
problem is always consistent, and a unique, least-squares solution can be
found, which is also the least-squares solution of

The singular value decomposition (SVD) of the projection-backprojection matrix

Because

This can be used to solve the original problem:

From the singular
value decomposition,

For any imaging
system, there is therefore a threshold for acceptable values of

It follows that for

The regularized solution

The reconstructed image

The
projection-backprojection and reconstruction process can be visualized as in
Figure

Block diagram of projection-backprojection operation. Noise is introduced at the projection operation,

In SPECT, the
projection data noise is Poisson distributed, that is, its variance equals its
mean. We can assume that in the
backprojected image, the noise

A less accurate
but less computationally expensive estimate of the noise amplification involves
the condition number

Although not as
precise as the noise amplification factor, the condition number does relate to
how well-posed the problem

For real-world
systems, it is not feasible to obtain singular values from

Of course, for
large systems, access to

Preliminary simulations were run for a
two-dimensional phantom with a one-dimensional detector. The phantom was a

The MR algorithm
is used in place of the more popular ML-EM algorithm because of its natural
applicability to the singular value decomposition. Although the ML-EM algorithm models Poisson
noise properly [

Because of the
relatively small system size, the errors in these images were compared to two
error predictors: one based on the condition number of the system, and the
other based on the

Final simulations were run for a three-dimensional phantom with a
two-dimensional detector. The phantom
used (Figure

Custom phantom used in simulation.

Apertures used in final simulation.

The noisy
projections were backprojected to create the image vector

The acceptance
angle of all pinholes in both experiments is

Illustration of acceptance angle.

The error plots of the preliminary simulations
are shown in Figure

Normalized error predictions and actual error of two-dimensional systems.

The reconstructed
two-dimensional images are shown after (a) 1, (b) 9, (c) 17, and (d) 25
iterations in Figure

Comparison of reconstructed images in two-dimensional
simulation after (a) 1, (b) 9, (c) 17, and (d) 25 iterations. (e) The image reconstructed as

The reconstructed images from the final (three-dimensional) simulations are shown in Figure

Comparison of reconstructed images from three-dimensional simulations.

Predicted and measured errors for three-dimensional simulations.

Illustration of line used for profile comparison.

Comparison of system profiles for (a) original phantom, (b) system A, (c) system C, and (d) system E.

The two-dimensional simulations show
the condition number and singular values to be useful in determining relative
uncertainty in reconstructed images.
Both error estimates (

The composite error for the
three-dimensional system, as previously defined, was created in order to
measure both system sensitivity and spatial resolution. For example, systems A and B are able to
resolve the three lesions in the bottom half of the phantom, but contain
substantial amounts of noise, as evidenced by the noisy reconstruction of the
large bright circle at the top half of the phantom (Figures

Note that
although the condition-number-based error predictor described in (

The objective of
this research was to create an error estimate that could predict the relative
performance of pinhole-based SPECT systems with a reasonable degree of
accuracy. To achieve this, the singular
value decomposition of the system’s projection-backprojection matrix was
analyzed. The singular value
decomposition allows for a frequency-based analysis, similar to a Fourier
analysis. It was based on a function termed the

These error predictors were shown to be useful in the prediction of system performance. Six systems with varying numbers and arrangements of pinholes were used to compare predicted and actual errors. The predictions were shown to be useful in determining a preferred system configuration.

The design of a pinhole-based SPECT system is a problem of system design with many variables. The number of pinholes, arrangement of pinholes, detector size and distance from the aperture, acceptance angle, and many other variables all affects the efficacy of an SPECT system in ways that are interrelated. Thus, system optimization cannot be reduced to a combination of single-variable optimizations. Simulation of each possible system configuration is also unfeasible, because of the nearly infinite number of configurations available, and because results will vary depending on the phantom used. For this reason, an unbiased error predictor, based only on the system configuration and not on any empirical data, will provide great benefits to system designers.

A drawback of the SVD-based analysis is the case of an overspecified system. In such a case, the condition number is infinity because the singular values corresponding to high frequencies are zero. In this case, the system resolution must be decreased to a point that all systems under consideration can be analyzed.

The most obvious use for an unbiased error predictor, such as the one described in this paper, is in system optimization. It is therefore the most important of the extensions of this research. However, in order to move to the goal of system optimization, research in this preliminary stage of performance prediction must be expanded.

In the mathematical derivations presented in this paper, Poisson noise was added at the detector. In the frequency-based analysis of the system, this noise was modeled as having equal power at all frequencies. An analysis of the Poisson noise in terms of the singular-value-based frequency spectrum, and incorporation of this knowledge into the error estimate, would add another degree of accuracy to the present error predictions.

When using the MR
algorithm for image reconstruction, iteration of the algorithm is terminated
after a certain number of iterations.
Because of this, high-frequency information is attenuated in the
reconstructed image. In order to reflect
this in the error predictor, the singular value spectrum must be truncated, as
shown in (

Because calculation of the entire set of singular values for a real-world system is computationally expensive, a function of the condition number was used in this paper to predict system performance. However, it is very unlikely that this is the optimal predictor, even if only using the condition number of the system. The present system could be vastly improved and a detailed system analysis could be made much simpler if a method could be devised to create a rough estimate of the singular value spectrum, or if a better estimate of the noise amplification factor could be derived. If not, a more refined estimation of the noise amplification factor, based on the condition number, would still improve the error estimate somewhat.

This work was supported in part by the National Institutes of Health under Grants no. R33 EB001489 and R21 CA100181.