A collimator optimization method for quantitative imaging: Application to Y-90 bremsstrahlung SPECT

Abstract

Purpose: Post-therapy quantitative ⁹⁰Y bremsstrahlung single photon emission computed tomography (SPECT) has shown great potential to provide reliable activity estimates, which are essential for dose verification. Typically ⁹⁰Y imaging is performed with high- or medium-energy collimators. However, the energy spectrum of ⁹⁰Y bremsstrahlung photons is substantially different than typical for these collimators. In addition, dosimetry requires quantitative images, and collimators are not typically optimized for such tasks. Optimizing a collimator for ⁹⁰Y imaging is both novel and potentially important. Conventional optimization methods are not appropriate for ⁹⁰Y bremsstrahlung photons, which have a continuous and broad energy distribution. In this work, the authors developed a parallel-hole collimator optimization method for quantitative tasks that is particularly applicable to radionuclides with complex emission energy spectra. The authors applied the proposed method to develop an optimal collimator for quantitative ⁹⁰Y bremsstrahlung SPECT in the context of microsphere radioembolization.

Methods: To account for the effects of the collimator on both the bias and the variance of the activity estimates, the authors used the root mean squared error (RMSE) of the volume of interest activity estimates as the figure of merit (FOM). In the FOM, the bias due to the null space of the image formation process was taken in account. The RMSE was weighted by the inverse mass to reflect the application to dosimetry; for a different application, more relevant weighting could easily be adopted. The authors proposed a parameterization for the collimator that facilitates the incorporation of the important factors (geometric sensitivity, geometric resolution, and septal penetration fraction) determining collimator performance, while keeping the number of free parameters describing the collimator small (i.e., two parameters). To make the optimization results for quantitative ⁹⁰Y bremsstrahlung SPECT more general, the authors simulated multiple tumors of various sizes in the liver. The authors realistically simulated human anatomy using a digital phantom and the image formation process using a previously validated and computationally efficient method for modeling the image-degrading effects including object scatter, attenuation, and the full collimator-detector response (CDR). The scatter kernels and CDR function tables used in the modeling method were generated using a previously validated Monte Carlo simulation code.

Results: The hole length, hole diameter, and septal thickness of the obtained optimal collimator were 84, 3.5, and 1.4 mm, respectively. Compared to a commercial high-energy general-purpose collimator, the optimal collimator improved the resolution and FOM by 27% and 18%, respectively.

Conclusions: The proposed collimator optimization method may be useful for improving quantitative SPECT imaging for radionuclides with complex energy spectra. The obtained optimal collimator provided a substantial improvement in quantitative performance for the microsphere radioembolization task considered.

INTRODUCTION

In targeted radionuclide therapy (TRT),^{1, 2} yttrium-90 (⁹⁰Y) is used clinically for treating hepatic primary and metastatic tumors (via microsphere radioembolization) and non-Hodgkin's lymphoma (via the Zevalin^® radioimmunotherapy regimen).^{3, 4, 5, 6} Reliable (acceptably accurate and precise) post-therapy estimates of the ⁹⁰Y activity distribution are essential to achieve reliable verification of the delivered absorbed dose.^{7, 8} Post-therapy quantitative ⁹⁰Y bremsstrahlung single photon emission computed tomography (SPECT) has the potential to provide reliable estimates of the activity distribution. Recently, quantitative SPECT (QSPECT) reconstruction methods have been proposed that improve image quality and quantitative accuracy.^{9, 10} However, to the best of our knowledge, the collimator, one of the most crucial elements in SPECT imaging, has never been optimized for quantitative ⁹⁰Y bremsstrahlung SPECT. As a result, clinically, high- or medium-energy collimators designed for radionuclides, such as gallium-67 or iodine-131, are currently used for ⁹⁰Y imaging.

For quantitative imaging, the collimator has a major impact on the reliability of the activity estimates. The variance of the activity estimates is influenced by the geometric sensitivity, geometric resolution, and the amount of septal penetration and scatter; the bias, largely due to the null space¹¹ of the image formation process when state-of-the-art reconstruction methods are used, is affected by the geometric resolution and the amount of septal penetration and scatter. The null space of the system matrix, $\mathit{H}\in {\mathbb{R}}^{M\times N}$, is composed of $\forall \mathit{x}{\in }^{N\times 1}$ that satisfies Hx = 0. Thus the projection image does not contain any information about any components of an input activity distribution that lie in the null space. For quantitative ⁹⁰Y bremsstrahlung SPECT, in particular, although the administered activity is relatively high in TRT, high sensitivity is still essential to achieve small variance of the activity estimates due to the low abundance (approximately 2% for photons with energies above 50 keV in soft tissues) of ⁹⁰Y bremsstrahlung photons. High resolution is crucial to achieving both low variance and low bias of the activity estimates for tumors, especially those with small volumes. Different from conventional radionuclides used for imaging, ⁹⁰Y bremsstrahlung photons have a continuous and broad energy distribution, and the highest energy is more than 2 MeV. High-energy photons result in septal scatter and penetration, which cause serious image-degrading effects in both high- and low-energy windows since photons that have penetrated the collimator septa can be detected in low-energy windows due to partial deposition in the crystal or backscatter from structures behind the crystal. These effects increase both the bias and the variance of the activity estimates. Thus, collimators with a small amount of septal scatter and penetration are preferable for reliable quantitative imaging.

Unfortunately, high resolution, high sensitivity, and small amount of septal penetration and scatter are usually mutually exclusive factors in parallel-hole collimator design.¹² For a given amount of septal penetration and scatter, geometric resolution is improved only at the expense of decreased geometric sensitivity, and vice versa.¹³ Similarly, decreasing septal penetration and scatter comes at the expense of degrading resolution or decreasing sensitivity.

A conventional approach to optimizing parallel-hole collimators is to find collimator parameters (hole length, hole diameter, and septal thickness) that maximize the geometric sensitivity for a predetermined geometric resolution and septal penetration fraction.^{14, 15} However, this requires selecting the desired resolution and septal penetration fraction a priori. This a priori selection ignores the fact that task performance, i.e., quantifying activity in tumors or other objects, depends on the combination of factors in a complex way.

A better approach is to optimize the collimator explicitly for the relevant imaging task.¹⁶ For quantitative imaging,¹⁷ in particular, a better approach would account for the effects of the collimator on both the bias and the variance of the activity estimates. Since the reliability of the activity estimates is affected by the geometric resolution, geometric sensitivity and the amount of septal penetration and scatter, a better approach would incorporate all these three factors into the optimization procedure rather than predetermine the resolution and septal penetration fraction. In addition, due to high computational complexity of collimator optimization, it is crucial for practical reasons to keep the number of free parameters describing the collimator as small as feasible.¹⁸ Similarly, the parameters used should characterize the collimator performance in a way that provides a straight-forward mechanism to select the collimator designs that will be considered in the investigation.

Another limitation of the conventional optimization approaches is that they are formulated for radionuclides with a single emission energy.¹⁴ For example, the penetration criterion is based on the attenuation coefficient for a specific energy (i.e., photopeak).¹⁹ For radionuclides with complex emission energy spectra, such as multiple energy peaks (e.g., ⁶⁷Ga and ¹³¹I) or continuous energy distribution (e.g., ⁹⁰Y bremsstrahlung), photons with energies above the acquisition energy window can significantly contribute to images due to septal scatter and penetration, partial deposition in the crystal, and backscatter from structures behind the crystal. In addition, in the case of ⁹⁰Y bremsstrahlung imaging, using a single attenuation coefficient for different collimator designs (e.g., to calculate the effective hole length) is highly inaccurate. This is because of the continuous nature of the emission energy spectrum and the resulting dependence of the effective attenuation coefficient on the thickness of material the beam traverses due to beam-hardening. Thus, the collimator design should not be based on a single emission energy: the whole energy spectrum must be considered in the analysis.

The aim of this work was twofold. First, we developed a parallel-hole collimator optimization method for quantitative tasks that is especially applicable to radionuclides with complex emission energy spectra (e.g., ⁹⁰Y bremsstrahlung, ⁶⁷Ga, and ¹³¹I). In this method, we used the inverse-mass-weighted root mean squared error (RMSE) of the volume of interest (VOI) activity estimates as the figure of merit (FOM) to account for the effects of the collimator on both the bias due to the null space of the image formation process and the variance of the activity estimates; the inverse mass weighting reflects the present application to dosimetry but could be replaced by another weighting for another application. We have also proposed a parameterization for the collimator that facilitates the incorporation of the important factors (geometric sensitivity, geometric resolution, and septal penetration fraction) determining collimator performance into the optimization procedure while keeping the number of free parameters describing the collimator small. Second, we applied the proposed method to optimize the collimator for quantitative ⁹⁰Y bremsstrahlung SPECT in the context of microsphere brachytherapy (also referred to as radioembolization). To make the optimization results more general, we simulated multiple tumors of various sizes in the liver. We realistically simulated human anatomy using a digital phantom and the image formation process using a previously validated and computationally efficient nonstochastic method for modeling the effects of object scatter, attenuation, and the full CDR, including septal penetration and scatter, partial deposition, and backscatter. To demonstrate the achieved improvement in quantitative performance, we compared the obtained optimal collimator to a commercial collimator currently used for clinical ⁹⁰Y imaging.

MATERIALS AND METHODS

FOM

We used the inverse-mass-weighted RMSE of VOI activity estimates as the FOM for the optimization:²⁰

$$\begin{array}{ccc}\hfill \mathrm{FOM}& =& \frac{1}{N}\sum _{i=1}^N\frac{\mathrm{RMSE}\left(A_i\right)}{m_i}\hfill \\ & =& \frac{1}{N}\sum _{i=1}^N\frac{\sqrt{\mathrm{Variance}\left(A_i\right)+{\left(\mathrm{Bias}\left(A_i\right)\right)}^2}}{m_i},\hfill \end{array}$$ (1)

where A_i and m_i denote the estimated activity and the mass of the ith VOI, respectively, and N is the number of VOIs. Bias(A_i) denotes the bias of the ith VOI activity estimate, which is the difference between the ensemble mean of the ith VOI activity estimate and the true activity in the ith VOI, and Variance(A_i) denotes the variance of the activity estimate of the ith VOI. Since absorbed dose is the deposited energy per unit mass of tissue, this FOM represents the reliability of the activity estimates related to absorbed dose calculation. The collimator with the smallest value of the FOM is deemed optimal.

It should be noted that both Bias(A_i) and Variance(A_i) depend not only on the collimator but also on the reconstruction algorithm. In particular, the bias of the activity estimate is affected by both the null space of the system matrix (which is mainly determined by the collimator) and the amount of model-mismatch (i.e., the difference between the actual image formation process and the model of it used in reconstruction). In practice, it is more difficult and expensive to design and manufacture a new collimator for a SPECT system than to improve the reconstruction algorithm (e.g., improve the model of image formation process). Thus, it is more practical to design and manufacture an optimal collimator assuming a perfect model of the image formation process and then continue to refine and improve the model of image formation process in reconstruction than to design using an imperfect image formation model and manufacture a new optimal collimator every time when the image formation model in the reconstruction is improved. Considering this, in this work we assumed a perfect model of image formation process in reconstruction and computed the bias resulting from the null space of the system matrix. The ensemble mean of the activity estimate for each VOI was obtained by estimating voxel activities using an iterative reconstruction method from simulated noise-free projection data and summing the obtained voxel activities inside the VOI. We used a very large number of iterations to guarantee convergence. The generation of the noise-free projection data and the reconstruction method will be discussed in detail below.

Since it was computationally too expensive to calculate the variance of the activity estimates using ensemble techniques for all collimators investigated, the variance of the activity estimate of the ith VOI was estimated approximately as follows:^{11, 20}

$$\mathrm{Variance}\left(A_i\right)=\frac{1}{t^2}·{\left[{\left({\mathit{P}}^{\mathrm{T}}\mathit{B}\mathit{P}\right)}^{-1}\right]}_{ii},$$ (2)

where t is the imaging time per projection view, [·]_ii denotes the ith diagonal element of a matrix, $\mathit{P}\in {\mathbb{R}}^{M\times N}$ is the system matrix that defines the mapping from image space to projection space (M is the number of projection data bins), and the diagonal matrix $\mathit{B}=\mathrm{diag}\left\{1/\mathrm{E}\left[y_j\right]\right\}$ is the inverse of the covariance matrix of the projection data, which are independent Poisson random variables. In this, E[y_j] is the ensemble mean of the photon counts, y_j, in the jth projection bin and can be obtained directly from the simulated noise-free projection data. The generation of the system matrix will be discussed in detail below.

Collimator parameterization

Since calculating FOM values required imaging simulation and iterative reconstruction for each collimator investigated, it is computationally expensive. Thus, it is practically necessary to limit the number of collimators investigated while making sure that a near-optimal collimator is included in the investigation. Thus, it is crucial to use as few parameters as necessary to parameterize the collimator. Also important, the parameters used should characterize the collimator performance in a way that provides a reasonable strategy to select collimators that will be investigated. To this end, we propose a parameterization for hexagonal-hole collimators. Instead of using three geometric design parameters (hole length, hole diameter, and septal thickness) that do not directly characterize the collimator performance, we used two parameters that give direct indications of the collimator performance.

Let T, D, and H denote the hole length, hole diameter (flat-to-flat distance for hexagonal holes), and septal thickness, respectively. Then the geometric resolution (full width at half maximum), R, and geometric sensitivity, S, of a hexagonal-hole collimator can be expressed as

$$R=\sqrt{2\ln 2}\frac{D}{T}(F+T)$$ (3)

and

$$S=\frac{\sqrt{3}}{8\pi }\frac{D^4}{T^2{(D+H)}^2},$$ (4)

respectively,¹⁴ where F is the sum of the source-to-collimator distance and the distance between the collimator back and the image plane. The shortest path length, w, that can be traversed by a photon passing through a single septum is¹³

$$w\approx \frac{TH}{2D+H}.$$ (5)

Here, we introduce a parameter used in the proposed parameterization:

$$a\genfrac{}{}{0pt}{}{\Delta }{=}\sqrt{\frac{16\pi \ln 2}{\sqrt{3}}}\frac{\sqrt{S}}{R}.$$ (6)

The meaning and utility of a will be described in more detail below. From Eqs. 3, 4, 5, 6, we have

$$w=-T+\frac{2}{a}·\left(1-\frac{\frac{1}{a}+F}{T+\frac{1}{a}+F}\right).$$ (7)

For a given R and S, we assume that the collimator with the smallest amount of penetration through a single septum is optimal. This collimator is the one with the largest value of w. Thus, the hole length, T_opt, of this collimator satisfies

$${\left.\frac{dw}{dT}\right|}_{T=T_{\mathrm{opt}}}=0.$$ (8)

Substituting Eq. 7 into Eq. 8 and after some algebraic manipulations, we find that

$$T_{\mathrm{opt}}=\sqrt{\frac{2}{a}\left(F+\frac{1}{a}\right)}-\left(F+\frac{1}{a}\right).$$ (9)

From Eqs. 3, 4, we have

$$D_{\mathrm{opt}}=\frac{R·T_{\mathrm{opt}}}{\sqrt{2\ln 2}·(T_{\mathrm{opt}}+F)},$$ (10)

$$H_{\mathrm{opt}}=D_{\mathrm{opt}}·\left[\frac{1}{a(T_{\mathrm{opt}}+F)}-1\right].$$ (11)

Equations 9, 10, 11 represent the parameterization of the collimator using the pair of parameters R and a.

The proposed parameter, a, is closely related to the collimator performance. For a fixed septal penetration fraction, a characterizes the well-known trade-off between collimator resolution and sensitivity:¹⁴ for a fixed value of a, the geometric sensitivity is proportional to the square of the geometric resolution. Among the collimators parameterized by the proposed method, for a fixed geometric resolution, the collimator with a larger value of a has a higher geometric sensitivity and a larger image-degrading effect due to septal penetration and scatter, as will be demonstrated later (see Fig. 1). Thus, the proposed parameterization method facilitates the incorporation of the important factors (geometric resolution, geometric sensitivity, and septal penetration) determining the collimator performance into the optimization procedure in a low dimensional (2D) parameter space.

Figure 1.

a (cm⁻¹) as a function of EGF for R = 1.4 cm. A smaller value of EGF means a larger image-degrading effect due to the CDR.

Parameter value selection strategy

Since calculating FOM values is computationally expensive, the values of R and a should be chosen carefully to limit both the number of collimators investigated and guarantee the inclusion of a near-optimal collimator. In this optimization study for quantitative ⁹⁰Y bremsstrahlung SPECT, we investigated five values of R: 1.0, 1.2, 1.4, 1.6, and 1.8 cm for a source-to-collimator distance of 15 cm, a distance often used in collimator optimization, and a distance between the back of the collimator and the image plane of 1.1 cm, a representative value for commercial SPECT systems. We believed that the collimators with R > 1.8 cm would not be optimal due to the large bias of the activity estimates due to poor resolution and resulting large null space. Similarly, we believed that the collimators with R < 1.0 cm would not be optimal due to the large variance of the activity estimates (for a small R, the sensitivity for a reasonable amount of septal penetration is very low). This will be confirmed later by the optimization results.

As mentioned earlier, for a given R, a collimator with a larger value of a has a higher geometric sensitivity and a larger image-degrading effect due to septal penetration and scatter. Thus, for a given R, we limited the search for the optimal collimator to within a range [a_mina_max]. Collimators with a < a_min would not be optimal because the geometric sensitivity would be too low; collimators with a > a_max would not be optimal because the image-degrading effects due to septal penetration and scatter would be too large. For each of the five values of R, we adopted a strategy, described below, to determine a reasonable range [a_mina_max], and used six values uniformly distributed in this range in the investigation. Thus a total of 30 collimators were investigated.

For a given R, the range [a_mina_max] was chosen based on a measure of the image degrading effects due to CDR. For emissions with multiple energy peaks or a continuous energy distribution, high-energy photons can contribute to images in low-energy windows due to septal scatter and penetration, partial deposition in the crystal and backscatter from structures behind the crystal. The single-septal penetration fraction used in conventional optimization method is not suitable for characterizing all of these effects. In addition, some photons that have penetrated the septa can still be detected inside the support of the geometric response (i.e., the region in the projection image where photons emitted from a point that travel through the collimator holes without traversing septa are detected). Examples of these nongeometrically collimated photons that contribute good information about the location of the source include photons backscattered from structures behind the crystal and detected in geometrically collimated regions or photons that penetrate a small distance through the bottom edge of one septum (i.e., edge penetration). To account for these effects, we defined a parameter termed the “effective geometric fraction” (EGF). An increase in the EGF corresponds to an increase in the fraction of photons carrying good information about the source position compared to photons carrying bad information. As will be demonstrated later (see Fig. 1), for a given R, a is a monotonically decreasing function of EGF: a = f_R(EGF). Thus by choosing a reasonable range for the EGF that applies to all resolutions, we can determine a unique range [a_mina_max] for each resolution.

For a point source, EGF is defined as

$$\mathrm{EGF}\genfrac{}{}{0pt}{}{\Delta }{=}\frac{g+m}{n}\times 100\%,$$ (12)

where g is the number of geometrically collimated photons detected, n is the number of all photons detected, and m is the weighted number of nongeometrically collimated photons detected, i.e.,

$$m=\sum _{\vec{r}}W\left(\vec{r}\right)·I_{\mathrm{ng}}\left(\vec{r}\right),$$ (13)

where

$$W\left(\vec{r}\right)=\frac{I_{\mathrm{g}}(0.5\times \vec{r})}{I_{\mathrm{g}}\left(0\right)},$$ (14)

and $I_{\mathrm{g}}\left(\vec{r}\right)$ and $I_{\mathrm{ng}}\left(\vec{r}\right)$ are the projection images of the point source for geometrically collimated and nongeometrically collimated photons, respectively, and $\vec{r}$ denotes the 2D spatial coordinate in the image plane. The point $\vec{r}=0$ corresponds to the position of the perpendicular projection of the point source onto the image plane. The function $W\left(\vec{r}\right)$ weights nongeometrically collimated photons as a function of the distance to the projected position of the source in order to take into account the fact that some nongeometrically collimated photons provide information about the source position: the larger the distance from the projected position ($\vec{r}=0$) the less information these photons are assumed to contain. Nongeometrically collimated photons detected at the center of the geometrically collimated region are scaled by the maximum weighting factor, 1, and are thus equivalent to geometrically collimated photons. Nongeometrically collimated photons detected further from the projected position have a smaller weight. The weighting function used is proportional to $I_{\mathrm{g}}(0.5\times \vec{r})$ and thus has wider support than $I_{\mathrm{g}}\left(\vec{r}\right)$ so that the edge penetration of the septa is included in the EGF. We chose the value of 0.5 in function $I_{\mathrm{g}}(0.5\times \vec{r})$ empirically to give a reasonable but not excessive weight to photons outside geometric regions.

For each of the five values of R, to estimate the function a = f_R(EGF), we used a previously validated version of SIMIND Monte Carlo (MC) simulation code^{10, 21} to simulate a ⁹⁰Y point source at a distance of 15 cm from collimators for a series of values of a. Other parameters simulated such as the crystal thickness and acquisition energy window were the same as that in the simulations performed for generating system matrices and noise-free projection data, which will be described in detail below. All image degrading effects including septal scatter and penetration, partial deposition in the crystal and backscatter from structures behind the crystal were simulated. For each value of a, we calculated the EGF from the simulated projection images (128 × 128 pixels for each image). To eliminate edge effects, only 80 × 80 pixels in the image center were used to calculate the EGF. Linear interpolation was then used to approximate the function a = f_R(EGF) at intermediate values. Figure 1 shows the obtained function for R = 1.4 cm. We can see clearly that a decreased monotonically as EGF increased. Thus larger values of a correspond to smaller values of EGF, and thus a relatively larger fraction of “bad” position information.

In this optimization study, we chose [15% 50%] as the range of EGFs to investigate. This range was chosen empirically to guarantee the inclusion of the optimal collimator while excluding as many collimators as possible that a priori would be expected to have suboptimal behavior. We assumed that the collimators with an EGF < 15% would not be optimal because the image degrading effects due to CDR would be too large. We assumed that collimators having an EGF > 50% would not be optimal because the geometric sensitivity would be too low. As will be demonstrated by the optimization results later, this was indeed a reasonable choice.

For each of the five values of R, six values of a uniformly distributed in the range [a_mina_max] = [f_R(50%)f_R(15%)] were investigated in the optimization study. Table 1 shows these values for each of the five values of R investigated.

Table 1.

Range [a_mina_max] = [f_R(50%) f_R(15%)] for each of the five values of R.

R (cm)	amin (cm−1)	amax (cm−1)
1.0	0.0240	0.0327
1.2	0.0253	0.0349
1.4	0.0264	0.0370
1.6	0.0276	0.0387
1.8	0.0288	0.0400

Object model

The XCAT phantom code²² was used to simulate realistic human anatomy and generate both the activity distribution and the attenuation map. To make our optimization results more general, we simulated three tumors with diameters of 1.1, 1.3, and 1.6 cm, respectively, in the liver. The sizes of tumors were chosen to represent challenging but relevant sizes: the quantification for larger tumors is less sensitive to the choice of the collimator due to smaller partial volume effects. Since the dose to normal liver tissues affects the normal-tissue complication probability, we also included normal liver tissues in the object model: normal liver tissues were treated as one VOI. Thus a total of four VOIs were simulated. The four VOIs did not overlap in 3D space. The inverse mass weighting in Eq. 1 implies that the quantification of the tumors had a larger effect on the FOM than the quantification for the normal liver tissues. This is reasonable considering the fact that the quantification for the VOI representing the normal liver tissues is less sensitive to the choice of the collimator than the quantification for the tumors: the quantification for the normal liver tissues is less affected by the partial volume effects due to its larger volume. We simulated a total activity of 1500 MBq in the liver, a typical activity used in ⁹⁰Y microsphere radioembolization. In the simulation, no activity was distributed outside the liver, as would also be typical in this application. The activity was uniformly distributed in each VOI. The three tumors had the same activity concentration. The tumor-to-normal activity concentration ratio was 5:1. Figure 2 illustrates the positions of the three tumors.

Figure 2.

(From left to right) Right lateral, anterior and inferior maximum intensity projections of three hepatic tumors’ and normal tissue activity distributions; and a sample coronal slice of the attenuation map, generated using the XCAT phantom.

Generation of system matrices and noise-free projection data

In previous work, we have developed a method, termed the multiple energy range (MER) method, for modeling the image formation process for quantitative ⁹⁰Y bremsstrahlung SPECT.¹⁰ This method models all the image degrading effects including object scatter, attenuation, and the full CDR. To model the energy dependence of these effects, we separated the modeling for bremsstrahlung photons in multiple categories and energy ranges. We have validated the MER method using both a physical phantom experiment and an XCAT phantom simulation. The method gave excellent agreement between measured and simulated projections with errors in the total counts of 1%.

In this work, we used the MER method to simulate the image formation process for each of the 30 collimators investigated. We used an acquisition energy window of 100–300 keV. In the MER method, for both the primary and scattered photons, we divided the energy range into three subranges: 0–300, 300–700, and 700–2000 keV. The scatter kernels²³ and CDR function (CDRF) tables²⁴ used in the MER method were generated using the SIMIND MC simulation code. A very large number of photons were simulated with various variance reduction techniques to generate essentially noise-free scatter kernels and CDRF tables. We simulated a collimator material of Pb-Sb alloy (2% Sb) with a density of 11.247 g/cm³, typical of what is used in commercial collimators, and a crystal thickness of 9.525 mm. The compartment behind the crystal was modeled as a slab made of SiO₂ with density of 2.6 g/cm³ and thickness of 6 cm.²⁵ We modeled a Gaussian energy resolution with a FWHM of 9.5% at 140 keV and an intrinsic spatial resolution of 3.4 mm, both with an energy dependence of 1/$\sqrt{E}$, where E represents the deposited energy.

We generated four VOI maps, each of which corresponded to the image where the voxel values were the same inside the corresponding VOI and zero outside the VOI and the sum of the voxel values inside the VOI was 1. For each of the 30 collimators, a forward projection code^{10, 23, 24, 26, 27, 28} incorporating the MER method was used to generate projections of each of the four VOI maps. Data were simulated at 128 projection views over 360° and the matrix size for each projection view was 128 × 128 with a pixel size of 4.664 mm. We simulated a body-contouring noncircular orbit by adjusting the radius of rotation for each view based on the attenuation map so that the SPECT detector face was close to the body surface. Since the scatter kernels and CDRF tables used in the MER method were essentially noise-free, the generated projection data for each VOI were also essentially noise-free. Each of the four columns of the system matrix was generated by arranging the corresponding projection data to form a vector. In this optimization study, we assumed that the imaging was performed using a dual-head SPECT system with a total imaging time of 30 min. The projection data for individual VOIs were scaled according to the activities in the corresponding VOIs and the imaging time and then summed to form the essentially noise-free projection data for the object.

Calculation of FOM values

For each of the 30 collimators, the variances of the VOI activity estimates were calculated based on Eq. 2 using the generated system matrix and noise-free projection data. To estimate the bias of the activity estimates due to the null space, the ordered subsets–expectation maximization (OS-EM) algorithm^{29, 30, 31} incorporating the MER method was used to reconstruct the noise-free projection data. The reconstructed image had 128 × 128 × 128 voxels and the voxel size was 4.664 mm. We used 1000 iterations with 32 subsets per iteration to guarantee the convergence of the bias. The ensemble mean of the activity estimate for each VOI was obtained by summing up the voxel activities in the VOI. The bias of the VOI activity estimate was then obtained by calculating the difference between the ensemble mean of the VOI activity estimate and the true VOI activity.

After obtaining the bias and the variance of the VOI activity estimates, the FOM values were calculated using Eq. 1. We assumed that the mass density was constant in the liver and equal to the density of water.

RESULTS AND DISCUSSION

Figure 3 shows the obtained FOM values for the 30 collimators investigated. The collimator with R = 1.2 cm and a = 0.0291 cm⁻¹ had the smallest FOM value: 651 kBq/g. Interestingly, the values of a for the collimators having the smallest FOM for each value of R were similar (between 0.028 and 0.03 cm⁻¹). This confirms that the parameter a was indeed an important and meaningful indicator of the collimator performance.

Figure 3.

FOM values (kBq/g) as a function of a (cm⁻¹) for each of the five values of R. The marks correspond to each of the 30 collimators investigated. The collimator having the smallest FOM value was deemed optimal: R = 1.2 cm and a = 0.0291 cm⁻¹.

Figure 4 shows the standard deviation and bias of the activity estimate for the tumor with a diameter of 1.1 cm as a function of a for each of the five values of R. We can see that for a given R, as a was increased from a relatively small value the variance decreased, likely due to increased geometric sensitivity; however, as a was further increased, the variance started increasing, likely due to increased image-degrading effects caused by septal penetration and scatter (decreased EGF) and resulting increase in noise amplification in the reconstruction. For a given R, bias always increased with increasing a due to decreased EGF (and resulting larger null space). For a given a, with improved geometric resolution, bias decreased; however, since resolution could only be improved at the expense of decreased sensitivity for a given a, variance increased. This further illustrates the important role the parameter a plays in the collimator performance, and confirms that a collimator can achieve the optimal quantitative performance only when its resolution, sensitivity and image-degrading effects due to collimator penetration and scatter are appropriately balanced.

Figure 4.

Standard deviation (kBq) and bias (kBq) of the activity estimate for the tumor with a diameter of 1.1 cm as a function of a (cm⁻¹) for each of the five values of R.

To demonstrate the improvement in quantitative performance achieved with the optimal collimator compared to a commercial collimator, we also calculated the FOM value for a high-energy general-purpose (HEGP) collimator of a Philips Precedence SPECT/CT system using the same method described above. The FOM value for this collimator was 791 kBq/g, which was substantially larger than that obtained with the optimal collimator (651 kBq/g). Table 2 shows the parameter values of the optimal collimator and the HEGP collimator, respectively. Figure 5 shows sample axial slices of OS-EM reconstructed images for the optimal and the Precedence HEGP collimators.

Table 2.

Parameter values of the obtained optimal collimator and the HEGP collimator of a Philips Precedence SPECT/CT system.

	FOM (kBq/g)	R (cm)	a (cm−1)	T (mm)	D (mm)	H (mm)
Optimal	651	1.2	0.0291	84	3.50	1.40
HEGP	791	1.65	0.0311	60	3.81	1.727

Note: T: hole length; D: hole diameter; H: septal thickness.

Figure 5.

Sample axial slices of OS-EM (10 iterations, 32 subsets) reconstructed images for (left to right) the optimal collimator with noise-free data, optimal collimator with noisy data, Precedence HEGP collimator with noise-free data and Precedence HEGP collimator with noisy data corresponding to 1500 MBq total activity and 30 min imaging time. The second and forth images were filtered using a 3D Butterworth postreconstruction filter with order 8 and cutoff 0.2 pixel⁻¹.

CONCLUSION

In this work, we have developed a parallel-hole collimator optimization method for quantitative tasks that is applicable to radionuclides with complex emission energy spectra. This method accounts for the effects of the collimator on both the bias due to the null space of the image formation process and the variance of the activity estimates. Further, it incorporates the effects of all the important factors (geometric resolution, geometric sensitivity, and image-degrading effects due to septal penetration and scatter) determining the collimator performance into the optimization procedure in a low dimensional parameter space. We applied this method to optimize the collimator for quantitative ⁹⁰Y bremsstrahlung SPECT in microsphere radioembolization. The obtained optimal collimator had better resolution than a commercial HEGP collimator (27% decrease in FWHM) and reduced the proposed FOM, the inverse-mass-weighted root mean squared error of the activity estimates for normal liver and tumor VOIs, by 18%.