A Design Strategy for Suppression of Hole-Pattern ... - IEEE Xplore

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 53, NO. 3, JUNE 2006

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A Design Strategy for Suppression of Hole-Pattern Artifacts in High-Energy Collimators Andreas Robert Formiconi, Donald Lee Gunter, and Eleonora Vanzi

Abstract—Multihole collimators are the most important piece of instrumentation in determining the tradeoff between spatial resolution and noise in nuclear medicine images. Recently, we found that the collimator-to-detector gap greatly influences the hole-pattern artifacts. In addition, we also showed that specific values of the gap exist which minimize the artifacts: here we call this phenomenon “penumbral masking” (PM). In this paper, we study the matter in greater detail and we take PM into account in the problematic context of designing high-energy collimators by, basically, substituting the conventional hole-array pattern constraint with a mathematical expression of the PM effect. With this approach we found that a solution to the problem of maximizing sensitivity for a given spatial resolution always exists while, without the PM concept, in the case of lead (Pb), a solution exists below 300 KeV, a suboptimal solution can be found within the 300–320 KeV range and no solution at all exists beyond 320 KeV. Thus, one is able to minimize the hole-pattern artifacts while designing high -energy collimators even if a somewhat lower sensitivity has to be accepted: for example, with lead, at 511 KeV, sensitivity is reduced by 20% compared to what it would be without the introduction of PM. When high energies are involved, even with our approach the weight constraint is a major problem, but this does not affect the capability of reducing the hole-pattern artifacts. However, even when the theoretical solution may not be feasible, this approach clarifies the design of high-energy collimators. Index Terms—Hole-array artifacts, multihole collimators.

I. INTRODUCTION

I

N nuclear medicine, the hole-pattern of the collimator can generate imaging artifacts that may limit the diagnostic value of the images. Research over the past two decades has produced a strategy for the design of low-energy collimators [1]–[3]. This strategy ignores the collimator hole-pattern, because, for low-energy radiation ( 300 KeV), collimators which, for example, allow less than 5% single-septal penetration do not cause hole-pattern artifacts. At such low energies, in fact, the collimator holes are smaller than the intrinsic resolution of gamma camera (3–4 mm), hence the structure of individual holes is not seen. However, for high-energy radiation ( 300 KeV) the problem of septal penetration requires the collimators to be thicker, with thicker septa and larger holes. Therefore, the hole-pattern of the collimator becomes visible in the images. Evidence of both Manuscript received November 15, 2004; revised February 20, 2006. A. R. Formiconi is with the Department of Clinical Pathophysiology, University of Florence, 50134 Firenze, Italy, and also with the Istituto Nazionale per la Fisica della Materia (INFM), 16146 Genova, Italy (e-mail: [email protected]). D. L. Gunter is with the Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550 USA. E. Vanzi is with the Department of Clinical Pathophysiology, University of Florence, 50134 Firenze, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TNS.2006.874952

Fig. 1. Example of a vial filled with

I.

the penetration and hole-pattern artifacts are exhibited in recent publications [4]–[8]. As these references indicate, collimators for both positron annihilation radiation (511 KeV) as well I (364 KeV) are currently of significant clinical interest, as mostly in cardiac and in tumor imaging. When speaking with nuclear medicine doctors, one discovers that the hole-pattern visibility is a strange problem: there are cameras that show heavy hole-pattern artifacts, when equipped with high-energy collimators, while others produce acceptable images. Apparently, there are no reasons for these differences, since similar cameras with high-energy collimators of the same type should produce similar artifacts. In the case of high-energy imaging, it may even happen that an old camera is preferred to a newer one. To give an idea of the impact the hole-pattern artifacts may I-filled calihave, in Fig. 1 we show the projection of an bration vial which was acquired using a high-energy all-purpose collimator. The distance of the vial was 10 cm from the I MIBG whole-body collimator surface. Fig. 2 presents an scan performed with the same camera and collimator. Here, the hole-pattern was spread longitudinally by the camera movement. The problem of visible hole-pattern has been tackled in several ways. One solution that has already been implemented is “colli-wobbling” or collimator rotation ([9], [10], and, more recently, [11]), whereby the hole-pattern is averaged out by collimator motion. The drawback of this method is the additional equipment required to move the collimator. Another possible solution is the development of new, denser alloys or amalgams of high-Z materials, that would permit shorter, smaller holes and reduce the hole-pattern artifacts. Finally, the effects of hole-pattern artifacts might be minimized by more clever collimator designs, such as the tapering of the septa near the front and back of the collimator.

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Fig. 2. Example of a total body with


I.

Fig. 3. Collimator and geometrical parameters.

A new idea, called “penumbral masking” (PM) was recently shown [12] to reduce the hole-pattern artifacts by the simple introduction of a precisely-spaced gap between both the collimator and imaging crystal in the gamma camera. With regard I and positron annihilation, PM may have to imaging with significant impact on collimator design. In this paper, we study the inclusion of PM in the overall strategy of collimator design. With the previous approach, based on work of Tsui et al. [1] and Gunter [3], once the collimator material, shape of the holes, and lattice are chosen, the collimator design is determined in a three-dimensional (3-D) space, the dimensions of which are , hole diameter ( , “equiva(Fig. 3) collimator thickness lent diameter”), and hole separation ( , defined by the lattice structure of the hole-pattern as the shorter lattice basis vector). The “equivalent diameter” is defined by

hole area

(1)

The problem is posed by the fact that what is required is a for a source at a given distance given spatial resolution from the collimator, as well as by seeking the collimator with maximum sensitivity. The solution is based on two equations, which express the spatial resolution and the system sensitivity in terms of collimator parameters, as well as on two inequalities which describe the need to reduce two effects: (i) the penetration of radiation through the septa and (ii) the hole-pattern artifacts. In [3] it was shown that: • below a certain energy level, a design can be found which satisfies both the penetration and the hole-pattern constraints; • within an intermediate energy range just beyond this threshold, it is not possible to obtain such a solution, although a suboptimal one which satisfies both constraints can be found; • beyond this range, it is not possible to simultaneously satisfy both constraints. These energy limits depend on the collimator material. For lead, a solution exists below 300 KeV, a suboptimal one can be found between 300–320 KeV, whereas no solution at all exists beyond 320 KeV. Therefore, while the results of this study provide definite criteria for designing low energy collimators, they also lead one to conclude that the design of high-energy collimators is a virtually impossible task. Recently, Formiconi et al. [12] found that the distance between the collimator and the imaging crystal in the camera greatly influences the amplitude of the hole-patterns artifacts. By expanding the gap between the imaging plane and the back of the collimator, one can arrange for the radiation from adjacent holes to overlap in the imaging plane so as to produce a penumbral region behind the septa, rather than a completely dark umbral region. As a result, the hole-pattern becomes less visible. In the new design approach we include the PM idea by substituting the inequality related to the hole-pattern artifacts with an expression of the PM effect. Basically, this means including the collimator–crystal gap amongst the parameters that have to be determined in the problem, rather than considering it as a fixed parameter. The paper is organized as follows: Section II outlines the basic imaging equations; Section III recaps the design method for a fixed value; and Section IV introduces the “penumbral

FORMICONI et al.: A DESIGN STRATEGY FOR SUPPRESSION OF HOLE-PATTERN ARTIFACTS

masking” concept into the method. A derivation of the solution in simple algebraic equations is given in the Appendix. A discussion can be found in Section V.

Thus, the indices harmonics. Now, the image expressed as

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and

are referred to as the hole-pattern

of a source distribution

can be

II. BASIC IMAGING EQUATIONS In order to include the PM in the method, it is necessary to review the basic equations for imaging with a collimated camera. For the description of the imaging process, the point spread function (PSF) of the system is needed. It has been shown [13], [14] that the PSF of a parallel collimator, expressed as a function of , position vector in the image plane, for a point source located in position of such plane at a distance from it, is given by

and, if we express the PSF in terms of the hole-pattern harmonics by means of (5):

(2)

(9)

The first factor is the geometric flux, whereas the second one is the collimator response function. The latter is a shift variant response, since it depends on the source position . is an approximation related to the fact The factor vanishes quite rapidly with respect to usual that the distances . is given by The collimator response function

This is the basic imaging equation in the space domain. However, it is not easy to manipulate, because the harmonics are expressed in terms of the aperture function ((6) and (3)) which may be quite awkward. It turns out that it is much easier to work with their Fourier transform:

PSF

PSF

(8)

(10) Thus, by means of the convolution theorem we can write the basic imaging equation in the Fourier domain:

(3) where (11) (4) are the lattice basis vectors and is the aperture function of the hole, that is assumed to be the same at both sides of the collimator for parallel holes. Since the collimator response function is periodic with respect to the source position, it can be broken down into a Fourier series:

(5) where

(6) and defined by the relations

. The dual basis lattice vectors are

and are the two-dimensional (2-D) Fourier where transforms of and , respectively. Since, for the analysis of the hole-pattern artifacts, the intrinsic resolution of the gamma camera plays an important role, we have to include it in our imaging equations, as follows:

(12) is the Gaussian width of the intrinsic camera resoluHere , by tion that is related to the full-width at half-maximum, . In (12), the term gives the image without any hole-pattern and the other terms give the effects of the hole-pattern. Thus, one can eliminate hole-pattern artifacts in images by suppressing all terms other than the one. We will call, for convenience, the terms of the series, so that

(13) (7)

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In order to make the analysis possible, in [12] it was assumed that the source object was a uniform planar distribution, parallel from it. Thus, in the to the image plane and at the distance 2-D frequency domain, this object is given by (14) Therefore, the terms in the series (13) become:

(15) This equation allows one to see in detail how the intrinsic resolution may suppress the hole-pattern artifacts. Basically, the intrinsic resolution Gaussian factor kills off for large and . However, the smaller terms like ( ) or ) may contribute significantly. The amount of ( this contribution depends on the balance between the intrinsic resolution, , and the periodicity of the hole-pattern, described . by

Fig. 4. Characteristic collimator thickness as a function of the gamma-ray energy for lead and tungsten collimator material. The jumps in at low energies are caused by the K-shell thresholds in the attenuation coefficient.

III. COLLIMATOR DESIGN FOR FIXED B In this section we briefly recall the derivation given in previous work [1], [3]. In this approach, the collimator–crystal gap 0.75 cm. B is fixed as usual. A typical value may be The method is based on four basic relationships. The first two are the well-known expressions of resolution and sensitivity [14]. The others are two inequalities related to the penetration and hole-pattern effects. 1) Collimator Resolution: In the absence of penetration, the collimator resolution is given by:

(16) where is a dimensionless constant that depends on the shape of the hole. 2) Collimator Sensitivity: In the absence of penetration, the collimator sensitivity is given by:

(17) where is a dimensionless constant that depends on the lattice structure of the hole-pattern. 3) Collimator Penetration: Here, an empirical relation is used that predicts whether or not collimator penetration will significantly affect image quality. The relation was proposed by Beck about 40 years ago [15], [16]. This criterion, that has successively been studied by using ray tracing techniques [17], [18], predicts that the images will exhibit acceptable levels of septal penetration provided that

(18)

where is a dimensionless constant that depends on the holeshape and array-pattern and is the attenuation-coefficient of the absorbing material in the collimator at the energy of the incident radiation. Therefore, is a characteristic parameter, dependent on the geometrical features of the collimator as well as on the material. Larger values of correspond to a reduced probability of stopping photons travelling in the wrong direction. The dependence of on energy is shown in Fig. 4 for lead and tungsten. Equation (18) means that the collimator thickness, weighted by the ratio of the volume of the absorbing material in the collimator to the total volume of the collimator, must be greater or at least equal to the photons mean free path length in the material . weighted by the empirical constant If a collimator satisfies (18) as an equality, it is considered to be on the “penetration threshold.” This is a desirable property for a collimator, because any further reduction in septal thickness would produce unacceptable penetration; otherwise, the septa are too thick and absorb more photons than necessary. 4) Hole-Pattern Invisibility Criterion: Hole-pattern invisibility criterion arises from the observation that in the Fourier transform of the image of an object, the harmonics of order greater than zero represent the hole-pattern artifacts. In point of fact, the criterion derives from the requirement that the exponent of the Gaussian factor in (15) should be large:

(19) will Whatever the explicit aperture function, the vectors , so that be inversely related to the hole separation , where are dimensionless vectors dependent only on the hole shape. Therefore, recalling that the intrinsic


resolution of the gamma camera can be expressed as , we obtain :

(20) where is the minimum allowed value of the hole-pattern parameter (generally between 0.5 and 1). The harmonics can means that each of be strongly reduced if is near to 1: the higher order harmonics contributes 5% of the order zero term. In general, we can estimate the effect of the hole-pattern by noting that the harmonics that have the greatest effect (first : implies invisiorder) are suppressed by the term bility of the hole-pattern; a small value of generally implies that hole-pattern will be a problem. By looking at the previous equations and rearranging (18) and (20) with (16), we can immediately observe that the ratio is bounded from above and below:

(21) We have already observed that sensitivity is maximized if assumes the upper bound value. Since is determined by (16)

(22)

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hole-pattern criterion is violated; in this case, a less desirable solution that satisfies both constraints is obtained by taking instead of . It is important to observe, at this point, that the lower bound is the same for all radiation energies, while the upper one depends on energy through the parameter (Fig. 4). As increases, the upper bound, representing the penetration constraint, becomes lower and, consequently, the two constraints overlap in a narincreases and decreases). For rower interval of ( 15 cm, 1.25 cm, example, for lead collimator material, 0.7 cm, and 1, , which also increases with energy (24), becomes greater than when the energy is above 302 and converge to the same value when enKeV and ergy equals 325 KeV. This means that there is no overlap of the two constraints for energies greater than 325 KeV, and therefore, no collimator can satisfy both constraints. Therefore, for lead, , and , a solution can be with the previous values of found for radiation energies below 302 KeV, a suboptimal one between 302 and 325 KeV, while no solution at all can be found that simultaneously satisfies the penetration and the hole-pattern constraints over 325 KeV. IV. PENUMBRAL MASKING As reported in Section II, the harmonic terms are determined by the aperture function of the collimator holes and by the . In the previous approach, , four parameters are adjusted in order to produce maximum sensitivity. and The idea of the penumbral masking is to adjust to satisfy

the separation between holes is given by: (25) (23) Now we can substitute (22) and (23) in (17) and, by maximizing with respect to , we obtain the collimator thickness that gives maximum sensitivity. For example, in the specific case of hexagonal holes arranged in a 120-degrees rhombic array:

In [12] it was shown that, in the absence of collimator penetration, the harmonic terms for a uniform planar spatial distribution are

(26) (24) By substituting this value in (22) and (23), we compute the values and . provides a design with maxThe collimator thickness imum sensitivity, but ignores the visibility of the collimator hole-pattern in images. In fact, the result was obtained simply equal to its upper bound and then varying by setting until sensitivity was maximized. However, the lower bound in (21), which represents the hole-pattern constraint, must also be is less than the upper bound, considered. If its value with then the hole-pattern is not a problem. On the other hand, if the is greater than the upper one, the design lower bound in must be rejected, because the hole-pattern criterion is violated. In other words, collimators are acceptable within the limited for which the upper bound exceeds the range of , the lower bound. When the designed thickness

where is the Fourier transform of the hole aperture function and is the area of the lattice cell . The solution follows from the observation that the function has zeroes; that is, spaexist for which . Consequently, tial frequencies one may hope to set

(27) such that

vanishes (one might also use the condition rather than (27), but these solutions generally require a larger gap ; consequently, the solution given by (27) is preferable). Regardless of the explicit aperture function, the zeroes must be inversely related to the hole size, that is where are dimensionless vectors, dependent only on the holeare inversely related to the shape. Similarly, the vectors

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Fig. 5. Collimator thickness versus for the two design methods. The fixed value of B is 0.75 cm. Fig. 7. Collimator hole side (equal to the two design methods.

pD=2 for square holes) versus for

Fig. 6. Collimator gap B versus for the two design methods. Values of B < 0.75 cm are not practical. For 511 KeV radiation, a gap of 5–6 cm is needed.

hole separation, so that comes

. Therefore, (27) be-

(28) and one finds a relation

Fig. 8. Collimator septal thickness (D methods.

0 D) versus for the two design

and from (30) and (23) we obtain

(32) (29)

where is a dimensionless constant determined by a detailed examination of the aperture function and the lattice structure. Different values have been derived in [12] for various holeshapes and arrangements. The value of that minimizes hole-pattern artifacts is, then

(30) Condition (30) can be used in the design strategy of Section III, instead of (20). From (16) and (17) it arises that

(31)

can be substituted in (31) in order This expression for to calculate the value that maximizes sensitivity. Then, , and can also be evaluated. A derivation of the solution, that permits writing the algorithm with simple algebraic equations, is given in the Appendix. In Figs. 5–9 we compare the collimators found with the two methods (fixed and variable ) when square holes in a square 0.472 cm for array are considered. For lead collimators, 140 KeV, 3.36 cm for 364 KeV, 7.58 cm for 511 method, 0.75 cm was assumed: KeV. With the fixed it is a standard value and cannot be reduced much below this limit, because of the depth of interaction of the radiation in the detector crystal and the need to protect the crystal from physical impacts. Therefore, designs incorporating PM that require 0.75 cm should be considered unphysical. When using a variable gap , its value increases with energy and it is relatively large for high energy radiation (Fig. 6). This


Fig. 9. Collimator sensitivity versus for the two design methods. For large values of the loss of S using penumbral masking is about 20%.

means that one must pay a price in sensitivity, if artifact-free images are desired, since depends on as described by (31). As shown in Fig. 9, the loss in sensitivity, when PM is considered in the design, is about 10% at 364 KeV and 20% at 511 KeV. V. DISCUSSION The basic result of the new design approach is that a solution to the problem always exists. As far as the practical feasibility is concerned, this does not mean that a feasible implementation can be found. In the worst case of 511 KeV radiation, for instance, very thick collimators (16–17 cm) are required, as shown in Fig. 5, and these collimators are much heavier than current gantries can support. However, the collimator weight is determined essentially by the necessity to avoid penetration, and is not affected by the introduction of penumbral masking. Thus, the weight is the same for both design strategies. The key to reducing the hole-pattern artifacts is that penumbral masking can be implemented at a cost in system sensitivity for equal resolution: about 10% at 364 keV (energy of rays I) and 20% at 511 keV. This point is mitigated emitted by by the fact that, usually, commercial collimators are already in some measure suboptimal and the possibility of reducing the hole-pattern artifacts with high-energy-emitting radionuclides may be worth the price. Alternatively, one can also consider reducing the cost in sensitivity by accepting a modest level of hole-pattern artifacts; however, such a tradeoff depends on the specific clinical task. A detailed specification of the quantitation task or of the human detection task, is needed to analyze the tradeoff in a useful way and this could be the object of further investigations. Certainly, the treatment we have adopted so far should be sufficient to mitigate situations such as those shown in Figs. 1 and 2. In the derivation [12] of the penumbral masking condition that determines the gap , it has been assumed that the radiation source was an infinite uniform sheet, and this assumption was crucial in suppressing the hole-pattern harmonics. Nonuniform sources contribute to these harmonics over a range of spatial frequencies and, therefore, their complete suppression is not possible. However, it is worth mentioning that the harmonics which describe the hole-pattern artifacts (26) do not depend on the distance of the sheet. Therefore, the same penumbral masking

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condition is also valid for an infinite uniform slab source of arbitrary thickness. This is a somewhat less strange case than the sheet source, but it is still far from reality, in which all the sources are limited also along planes parallel to the detector. On the basis of this consideration, it follows that we can expect PM to be more effective in regions where sources are flat, and less effective next to source boundaries. The reason for this behavior is simple. The infinite uniform extension of the source parallel to the detector permits the representation of the source image in frequency space as the series in (26) whose terms are eval. The PM is uated at regularly spaced frequency values only possible thanks to such regular frequency sampling. If one drops the assumption of infinite uniform source extension, the series becomes a generic function of frequency. As the source tends to be extended and vary slowly, this function tends to have while becoming 0 elsewhere. It is difficult to maxima in predict the degree to which PM will fail for a given real source distribution unless appropriate experiments or simulations are performed. Another important assumption underlying (26) is the absence of septal penetration, even if the major application of PM is for high-energy radiation, that, by its nature, penetrates septa. Probably, this penetration would blur the sharp edges of the holes, and, consequently, reduce the need for PM. It is difficult to evaluate the phenomenon simply by analyzing these equations and, hence, appropriate simulations are needed. However, let us make the common assumption that the single-septum penetration effect can be accounted for by simply considering a shorter instead of , where is total linear attenthickness: uation coefficient of the collimator material for the energy of the incident radiation. In reality, we have a distribution of penetrating photons, and we could think of the smearing effect as the superposition of images obtained with collimators having a distribution of thicknesses less than or equal to . Of course, a does not produce any smearing but single thickness it averages in some way the effect of single-septum penetration. Therefore, we can use this thickness to have some hint of the effect of single-septum penetration on hole-pattern artifacts. For instance, we can estimate how the total amplitude of the hole-pattern artifacts varies changing the thickness from to . This can be done by considering all the harmonics in . It turns out that, for a (26), that is the terms for which 4 cm, 3.2 high-energy collimator used with I ( cm for the energy 364 keV, standard gap 7.5 mm), the total amplitude of the hole-pattern artifacts diminishes by 4.7% . In the case when the thickness is decreased from to of an ultra-high-energy collimator [7] used with 511 keV pho7.7 cm, tons emitted by positrons annihilations ( 6.4 cm) the total amplitude of the artifacts diminishes by 1.7%. Therefore, we do not expect single-septum penetration to have a great impact on hole-pattern artifacts. Finally, the analysis in this paper concerns planar imaging, but it may be worthwhile extending it to SPECT or whole-body scans, that involve camera motion. APPENDIX I In this Appendix, we show how it is possible to analytically evaluate the solution of the design problem with variable .

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If we express, for convenience, , that

, and

in terms of , so

must fall

one concludes that the three real roots within the intervals

(41)

(33)

Only the last root is permitted by the physical constraints. We introduce a [3/2] Padé approximation for of the form

by using (32) and (23), (31) becomes:

(34) If now we call

, then (34) becomes

(42) which satisfies the known asymptotic conditions

(35) The solution is given by which yields

,

(43) and, furthermore

(44) (36) We should immediately observe that only solutions implies satisfy the penetration criterion (since Therefore, the extremum condition requires that

can ).

so that is proportional to . A detailed calculation of , and yields two approximations and the coefficients

(37) (45) is not of physical interest, so only The case will be considered. The fact that and imis always negative; plies that the second term in consequently, the first one must be positive and, therefore, must . For , the expression be is positive, so that the roots of the polynomial

(38) that are greater than 2, must be solutions of convenience, the parameter can be replaced by so that

. For ,

(39)

(46) with less than that bound the actual root over the range 1.5% error . With this result, the algorithm can be explicitly written in simple algebraic equations. The method associated with penumbral masking assumes that the collimator designer specifies the desired resolution as for a source at distance in front of the collimator (where is a characteristic distance for the application). The designer must also determine the parameter from the attenuation coefficient of collimator material at the energy of the incident radiation. Using these parameters, the collimator is designed in a six step procedure: 1) Calculate :

A careful examination of this fifth-order polynomial reveals that it has three real roots. From the specific values (47) 2) Evaluate

by the approximations (45) and (46): (48)

(40)

3) Find the collimator thickness:

.


4) Determine the gap between the collimator and the imaging plane using (32):

(49) 5) Determine the hole diameter from (22)

(50) 6) Determine the hole separation from (23)

(51) The constant parameters and are determined by the collimator hole shape and lattice structure. ACKNOWLEDGMENT The authors wish to thank Dr. Carlo Chiesa, at the Istituto Nazionale Tumori, Milan, Italy, for kindly providing the images in Fig. 1 and in Fig. 2. The authors are also grateful to Dr. Elizabeth Guerin for revising the manuscript. REFERENCES [1] B. M. W. Tsui, D. L. Gunter, D. L. Beck, and J. Patton, “Physics of collimator design,” in Diagnostic Nuclear Medicine, M. Sandler, Ed. Baltimore, MD: Williams & Wilkins, 1996, pp. 67–79. [2] S. C. Moore, K. Kouris, and I. Cullom, “Collimator design for single photon emission tomography,” Eur. J. Nucl. Med., vol. 19, pp. 138–150, 1992. [3] D. L. Gunter, “Collimator characteristics and design,” in Nuclear Medicine, R. E. Henkin, Ed. St. Louis, MO: Mosby, 1996, pp. 96–124.

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[4] A. Van Lingen, C. P. Huijgens, C. F. Visser, J. G. Ossenkoppele, S. O. Hoekstra, J. M. H. Martens, H. Huitink, D. M. K. Herscheid, V. M. Green, and J. J. G Teule, “Performance characteristics of a 511-KeV collimator for imaging positron emitters with a standard gamma camera,” Eur. J. Nucl. Med., vol. 19, pp. 315–321, 1992. [5] H. J. Ostertag, G. Sroka-Perez, and W. K. Kubler, “Results for a gamma camera with a new 511 KeV collimator,” Eur. J. Nucl. Med., vol. 19, pp. 917–918, 1992. [6] W. H. Martin, D. Delebeke, J. A. Patton, B. Hendrix, Z. Weinfeld, I. Ohnana, R. M. Kessler, and M. P. Sandler, “FDG-SPECT: Correlation with FDG-PET,” J. Nucl. Med., vol. 36, pp. 988–995, 1995. [7] P. K. Leichner, H. T. Morgan, K. P. Holdeman, K. A. Harrison, F. Valentino, R. Lexa, R. F. Kelly, W. G. Hawkins, and G. V. Dalrymple, “SPECT imaging of Fluorine-18,” J. Nucl. Med., vol. 36, pp. 1472–1475, 1995. [8] Y. K. Dewaraja, M. Ljumberg, and K. F. Koral, “Accuracy of I-131 tumor quantification in radioimmunotherapy using SPECT imaging with an ultra-high-energy collimator: Monte carlo study,” J. Nucl. Med., vol. 41, pp. 1760–1767, 2000. [9] B. Brunsden, P. V. Harper, and R. N. Beck, “Elimination of collimator hole-pattern by double displacement of a hexagonal array,” J. Nucl. Med., vol. 16, p. 517, 1975. [10] R. J. Wilks, J. R. Mallard, and C. G. Taylor, “Instrumental and technical note: The collywobbler—A moving collimator image-processing device for stationary deterctors in radioisotope scanning,” Br. J. Radiol., vol. 42, pp. 705–709, 1969. [11] M. F. Smith and R. J. Jaszczak, “A rotating parallel hole collimator for high resolution imaging of medium energy radionuclides,” IEEE Trans. Nucl. Sci., vol. 45, no. 4, pp. 2102–2112, Aug. 1998. [12] A. R. Formiconi, F. Di Martino, D. Volterrani, and A. Passeri, “Study of high-energy multihole collimators,” IEEE Trans. Nucl. Sci., vol. 49, no. 1, pp. 25–30, Feb. 2002. [13] B. M. W. Tsui and G. T. Gullberg, “The geometrical transfer function for cone and fan beam collimators,” Phys. Med. Biol., vol. 35, pp. 81–93, 1990. [14] A. R. Formiconi, “Geometrical response of multihole collimators,” Phys. Med. Biol., vol. 43, pp. 3359–3379, 1998. [15] R. N. Beck, “A theoretical evaluation of brain scanning systems,” J. Nucl. Med., vol. 2, pp. 314–324, 1961. [16] R. N. Beck, “A theory of radiological scanning systems,” in Medical Radioisotope Scanning. Wien, Austria: International Atomic Energy Agency, 1964, vol. 1, pp. 35–56. [17] R. N. Beck and D. L. Gunter, “Collimator design using ray-tracing techniques,” IEEE Trans. Nucl. Sci., vol. NS-32, pp. 865–869, 1985. [18] D. L. Gunter and R. N. Beck, “Generalized collimator transfer function,” Radiology, vol. 169, p. 324, 1988.