International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 915958 | 18 pages | https://doi.org/10.1155/2010/915958

Geometric Sensitivity of a Pinhole Collimator

Academic Editor: Harvinder S. Sidhu
Received25 Aug 2009
Revised07 Dec 2009
Accepted19 Feb 2010
Published01 Apr 2010

Abstract

Geometric sensitivity for single photon emission computerized tomography (SPECT) is given by a double integral over the detection plane. It would be useful to be able to explicitly evaluate this quantity. This paper shows that the inner integral can be evaluated in the situation where there is no gamma ray penetration of the material surrounding the pinhole aperture. This is done by converting the integral to an integral in the complex plane and using Cauchy's theorem to replace it by one which can be evaluated in terms of elliptic functions.

1. Introduction

Nuclear-medicine imaging provides images that assess how the body is functioning [1, 2], as opposed to anatomical modalities (e.g., X-ray computed tomography, commonly known as a CT or “CAT” scans) that provide little or no information about function, but great detail of the body's structure. Nuclear-medicine images the biodistribution of radiolabeled molecules that are typically intravenously injected into the patient in tracer (i.e., nonpharmacological) quantities. The compounds may have different biochemical properties that affect the biodistribution and, hence, the choice of pharmaceuticals used to assess the disease state of a patient.

Two common nuclear-medicine techniques are single photon emission computed tomography (SPECT) and positron emission tomography (PET). Molecules labeled with a SPECT tracer emit a single photon. PET tracers emit a positron, which annihilates with a nearby electron to produce two nearly back-to-back photons at 511 keV, the rest energy of an electron. The line of these two photons contains the emission point of the positron. Many photon pairs are detected in coincidence and reconstructed into a three-dimensional (3D) image of the tracer's distribution.

Since SPECT tracers emit only one photon, the detection of the photon itself gives little information about the location of the emission since neither the direction nor origin of the emission is known. Consequently, SPECT uses collimation that allows only photons along certain lines to be detected. The origin is still unknown, but the 3D image may be reconstructed when photons from numerous angles are detected [4, 5]. Pinhole collimation is often used when a small organ (e.g., the thyroid) is imaged by a large detector with a SPECT tracer. Figure 1 is a diagram of the apparatus. It is not hard to see that the pinhole collimation's geometric efficiency, namely, the fraction of emitted photons that pass through the circular aperture of the collimator, is approximately given by

𝑑𝑔=2sin3𝜃16ℎ2,(1.1) where 𝑑 is the diameter of the pinhole, ℎ is the perpendicular distance, and 𝜃 is the incidence angle of the photon on the aperture plane at the center of aperture. This formula becomes exact in the limit that 𝑑 goes to zero. Also, an exact formula is easily obtained when 𝜃=𝜋/2

𝑔𝜋𝜃=2=12−12𝑑1+2ℎ2−1/2≈12−1211−2𝑑2ℎ2=𝑑216ℎ2.(1.2) We derive stronger versions of these two results in the next section. But the main goal of this paper is to simplify the integral representation for the geometric sensitivity without taking 𝑑 or 𝜃−𝜋/2 to be small. We are concerned with the case where there is no penetration in the sense that not all gamma rays are stopped by the attenuating material surrounding the pinhole aperture. We take 𝜃≠𝜋/2. The geometric sensitivity, as defined above, is given by the integral over the aperture of the flux times the sine of the incidence angle at that point on the aperture:

1𝑔=4ğœ‹â„Ž20𝑑/2𝜌𝑑𝜌02𝜋𝑑𝛽sin3ğœƒğ‘Ž,(1.3) where 𝜌 and 𝛽 are polar coordinates on the aperture plane and ğœƒğ‘Ž is the incidence angle of the photon at a particular value of 𝜌 and 𝛽, and is given by [3]

cot2ğœƒğ‘Ž=cot2𝜌𝜃−2â„ŽğœŒcot𝜃cos(𝛽−𝜙)+2ℎ2,(1.4) where 𝜙 is the third coordinate of the point source. For integrals over 2𝜋 the difference 𝛽−𝜙 can be replaced with only 𝛽 without loss of generality. This yields

sin2ğœƒğ‘Ž=csc2𝜌𝜃−2â„ŽğœŒcot𝜃cos𝛽+2ℎ2−1.(1.5)

Applying this to (1.3),

1𝑔=4ğœ‹â„Ž20𝑑/2𝜌𝑑𝜌02𝜋𝑑𝛽csc2𝜌𝜃−2â„ŽğœŒcot𝜃cos𝛽+2ℎ2−3/2.(1.6)

When there is penetration the above would be modified to

1𝑔=4ğœ‹â„Ž2∞0𝜌𝑑𝜌02𝜋𝑑𝛽csc2𝜌𝜃−2â„ŽğœŒcot𝜃cos𝛽+2ℎ2−3/2𝑒−𝜇Δ𝐿,(1.7) where the path length through attenuating material, Δ𝐿, is zero for 𝜌<𝑑/2 and is otherwise given by (11) in [3].

Thus, (1.7) can be rewritten as

𝑔=𝑔NP+14ğœ‹â„Ž2î€œâˆžğ‘‘/2𝜌𝑑𝜌02𝜋𝑑𝛽csc2𝜌𝜃−2â„ŽğœŒcot𝜃cos𝛽+2ℎ2−3/2𝑒−𝜇Δ𝐿=𝑔NP+𝑔PEN,(1.8) where 𝑔PEN is the efficiency due to penetration and 𝑔NP, the efficiency under the assumption of no penetration, is given by (1.6).

In this paper we use complex variable methods to calculate the 𝛽 integral in (1.6).

2. Two Approximations for 𝑔𝑁𝑃

The special form of 𝑔NP makes it easy to derive approximations for 𝑑/ℎ small and for 𝜃−𝜋/2 small which generalize (1.1) and (1.2). Before doing this, we observe that only even powers of 𝑑 and of 𝜃−𝜋/2 can occur in the expansion of 𝑔NP. We see this by writing the integral as the sum of integrals from 0 to 𝜋 and from 𝜋 to 2𝜋.

2.1. The Expansion for 𝑑 Small

Let us, in this section, denote 𝑔NP by 𝑔. Upon substituting 𝑟=𝑑/2ℎ and 𝜂=𝜌/ℎ into (1.6) we obtain

1𝑔(𝑟,𝜃)=4𝜋02𝜋𝑟0csc2𝜃−2𝜂cot𝜃cos𝛽+𝜂2−3/2=1𝜂𝑑𝜂𝑑𝛽,4𝜋02𝜋𝑟0𝜂𝜙(𝛽,𝜂,𝜃)𝑑𝜂𝑑𝛽.(2.1) Clearly, 𝑔(0,𝜃)=0. Since

𝑔𝑟1(𝑟,𝜃)=4𝜋02𝜋𝑟𝜙(𝛽,𝑟,𝜃)𝑑𝛽,(2.2) we have 𝑔𝑟(0,𝜃)=0. Differentiating (2.12) three more times with respect to 𝑟 we obtain

𝑔𝑟𝑟1(𝑟,𝜃)=4𝜋02𝜋𝑟𝜙𝑟𝑔(𝛽,0,𝜃)+𝜙(𝛽,𝑟,𝜃)𝑑𝛽,𝑟𝑟𝑟1(𝑟,𝜃)=4𝜋02𝜋𝑟𝜙𝑟𝑟(𝛽,0,𝜃)+2𝜙𝑟𝑔(𝛽,𝑟,𝜃)𝑑𝛽,𝑟𝑟𝑟𝑟1(𝑟,𝜃)=4𝜋02𝜋𝑟𝜙𝑟𝑟𝑟(𝛽,0,𝜃)+3𝜙𝑟𝑟(𝛽,𝑟,𝜃)𝑑𝛽,(2.3) and then setting 𝑟=0 in each of the resulting equations we obtain

𝑔𝑟𝑟1(0,𝜃)=4𝜋02𝜋𝑔𝜙(𝛽,0,𝜃)𝑑𝛽,𝑟𝑟𝑟1(0,𝜃)=2𝜋02𝜋𝜙𝑟𝑔(𝛽,0,𝜃)𝑑𝛽,𝑟𝑟𝑟𝑟3(0,𝜃)=4𝜋02𝜋𝜙𝑟𝑟(𝛽,0,𝜃)𝑑𝛽.(2.4) For convenience let us write

𝑢=csc2𝜃,𝑣=cot𝜃,(2.5) so 𝜙 becomes

𝜙(𝛽,𝑟,𝜃)=𝑢−2𝑣𝑟cos𝛽+𝑟2−3/2.(2.6) Then

𝜙(𝛽,0,𝜃)=𝑢−3/2,𝜙𝑟(𝛽,0,𝜃)=3𝑢−5/2𝜙𝑣cos𝛽,𝑟𝑟(𝛽,0,𝜃)=15𝑢−7/2𝑣2cos2𝛽−3𝑢−5/2.(2.7) We substitute into the Taylor expansion

𝑔(𝑟,𝜃)=𝑔(0,𝜃)+𝑔𝑟1(0,𝜃)𝑟+𝑔2!𝑟𝑟(0,𝜃)𝑟2+1𝑔3!𝑟𝑟𝑟(0,𝜃)𝑟3+1𝑔4!𝑟𝑟𝑟𝑟(0,𝜃)𝑟4+⋯(2.8) and obtain

1𝑔(𝑟,𝜃)=214𝜋02𝜋𝑢−3/2𝑟𝑑𝛽2+133!2𝜋02𝜋6𝑢−5/2𝑟𝑣cos𝛽𝑑𝛽3+114!4𝜋02𝜋315𝑢−7/2𝑣2cos2𝛽−3𝑢−5/2𝑟𝑑𝛽4+⋯.(2.9) Since

02𝜋cos𝛽𝑑𝛽=0,02𝜋cos2𝛽𝑑𝛽=𝜋(2.10) and 𝑢−1=sin2𝜃 we obtain

𝑟𝑔(𝑟,𝜃)=24sin33𝜃+325sin5𝜃cos2𝜃−2sin5𝜃𝑟4+⋯.(2.11) Finally we replace cos2𝜃 by 1−sin2𝜃 and 𝑟 by 𝑑/2ℎ and obtain

𝑔NP=116sin3ğœƒî‚€ğ‘‘â„Žî‚2+3512sin5𝜃3−5sin2ğœƒî€¸î‚€ğ‘‘â„Žî‚4𝑑+𝒪6.(2.12)

2.2. The Expansion for 𝜃−𝜋/𝟐 Small

To start we let

𝑓(𝜃,𝜂,𝛽)=csc2𝜃−2𝜂cot𝜃cos𝛽+𝜂2−3/2(2.13) and compute its Taylor expansion at 𝜃=𝜋/2,

𝑓(𝜃,𝜂,𝛽)=1+𝜂2−3/2−31+𝜂2−5/2𝜋𝜂cos𝛽𝜃−2+12!151+𝜂2−7/2𝜂2cos2𝛽−31+𝜂2−5/2𝜋𝜃−22𝜋+𝒪𝜃−23.(2.14)

Since

𝑔NP=14𝜋02𝜋0𝑑/2â„Žğ‘“(𝜃,𝜂,𝛽)𝜂𝑑𝜂𝑑𝛽(2.15) and since we know that only even powers of 𝜃−𝜋/2 occur, we obtain

𝑔NP=1211−√1+(𝑑/2ℎ)2−38(𝑑/2ℎ)21+(𝑑/2ℎ)25/2𝜋𝜃−22𝜋+𝒪𝜃−24.(2.16)

3. The Contour Integral

We have

𝐼=02𝜋𝑑𝛽csc2𝜌𝜃−2â„ŽğœŒcot𝜃cos𝛽+2ℎ2−3/2,𝑔(3.1)NP=14ğœ‹â„Ž20𝑑/2𝐼𝜌𝑑𝜌.(3.2)𝐼 may be recast as a contour integral by setting 𝑧=𝑒𝑖𝛽: 𝐼=Γ𝑑𝑧𝑖𝑧−𝑏𝑧2𝑧2ğ‘Žâˆ’2𝑏𝑧+1−3/2=2𝑏3/21𝑖Γ(𝑧−𝑥0)(𝑥1−𝑧)𝑧−3/2𝑑𝑧𝑧,(3.3) where

Γistheunitcircle,ğ‘Ž=csc2𝜌𝜃+2ℎ2,𝜌𝑏=2â„Žğ‘¥cot𝜃,0=ğ‘Žğ‘îƒ¬î‚™1−𝑏1−2ğ‘Ž2,𝑥1=ğ‘Žğ‘îƒ¬î‚™1+𝑏1−2ğ‘Ž2.(3.4) We note that ğ‘Ž>𝑏 and so 0<𝑥0<1<𝑥1.

In this section we take 𝜌,𝜃, and ℎ to be constant. We write

𝑧𝐹(𝑧)=𝑧−𝑥0𝑥1𝑧−𝑧((𝑧−𝑥0)(𝑥1−𝑧))1/21𝑧=1𝑧−𝑥0𝑥1𝑧−𝑧(𝑧−𝑥0)(𝑥1−𝑧)1/2,2𝐼=𝑏3/21𝑖Γ𝐹(𝑧)𝑑𝑧.(3.5) We need to be careful because of the square roots. We use the usual convention that

lim𝑦→0±√−1+𝑖𝑦=±𝑖.(3.6) We wish to show that 𝐹(𝑧) is holomorphic (and single valued) in

{𝑧∶|𝑧|<1}−𝑥∶0<𝑥<𝑥0.(3.7) To isolate the square root factor in 𝐹 we set

𝑧𝑔(𝑧)=𝑧−𝑥0𝑥1.−𝑧(3.8) We take |𝑧|≤1. If in addition Re𝑧<0 and |Im𝑧|≪|𝑧−𝑥0| then Re(𝑧−𝑥0)(𝑥1−𝑧)<0. This implies that 𝑔(𝑧) never takes on negative real values and so has a single valued square root in Re𝑧<0. Similarly, if |𝑧|≤1, and Re𝑧>𝑥0 (no condition on Im𝑧 now) then Re(𝑧−𝑥0)(𝑥1−𝑧)>0 and 𝑔(𝑧) has a single valued square root in Re𝑧>𝑥0.

It follows from Cauchy's theorem that the integral of 𝐹(𝑧) over the unit circle Γ is equal to the integral over the “ barbell” 𝛾 around 0 and 𝑥0. More explicitly, 𝛾 is the boundary of

𝐵=𝐵𝜖(0)∪𝐵𝜖𝑥0∪𝑅,(3.9) where 𝐵𝜖(𝑥) is the ball of radius 𝜖 and centered at 𝑥 and 𝑅 is the rectangle

𝑅=𝑧=𝑥+𝑖𝑦∶0≤𝑥≤𝑥0,−𝛿≤𝑦≤𝛿(3.10) with 𝛿≪𝜖, see Figure 2.

The curve 𝛾 decomposes into smooth pieces which we describe using the angle 𝛼=arcsin(𝛿/𝜖)

𝛾𝑟=||𝑧∶𝑧−𝑥0||=𝜖,−𝜋+𝛼≤arg𝑧−𝑥0,𝛾≤𝜋−𝛼𝑙𝛾={𝑧∶|𝑧|=𝜖,𝛼≤arg(𝑧)≤2𝜋−𝛼},±=𝑧∶𝑧=𝑥±𝑖𝛿,𝜖cos𝛼≤𝑥≤𝑥0.−𝜖cos𝛼(3.11)

To bound the integral over 𝛾𝑙 we set 𝑧=𝜖𝑒𝑖𝜃

||||𝛾𝑙𝐹||||=||||(𝑧)𝑑𝑧𝛼2𝜋−𝛼𝐹𝜖𝑒𝑖𝜃𝜖𝑒𝑖𝜃||||𝜖𝑑𝜃=𝒪3/2.(3.12) To evaluate ∫𝛾𝑟𝐹(𝑧)𝑑𝑧 we set 𝑧=𝑥0+𝜖𝑒𝑖𝜃. Then

𝐹𝑥0+𝜖𝑒𝑖𝜃=𝐶𝜖−3/2𝑒−(3/2)𝑖𝜃𝜖+𝒪−1/2(3.13) with

𝑥𝐶=01/2𝑥1−𝑥03/2,(3.14) and so

𝛾𝑟𝐹(𝑧)𝑑𝑧=𝜋−𝛼−𝜋+𝛼𝜖−3/2𝐶𝑒−(3/2)𝑖𝜃𝜖+𝒪−1/2𝜖𝑖𝑒𝑖𝜃=(1+𝒪(𝜖))𝑑𝜃𝜋−𝛼−𝜋+𝛼𝐶𝑖𝜖−1/2𝑒−(1/2)𝑖𝜃𝜖+𝒪1/2𝑑𝜃.(3.15) Since 𝛼=𝑂(𝜖), we may set 𝛼=0 and absorb the error in the 𝒪(𝜖1/2) term. We then do the integration and obtain

𝛾𝑟𝐹(𝑧)𝑑𝑧=4𝑖𝐶√𝜖𝜖+𝒪1/2.(3.16) We want to compute the integral of 𝐹 over 𝛾+. We need to be careful about the square root. Recall that we are assuming

𝜖cos𝛼<𝑥<𝑥0−𝜖cos𝛼,𝛿≪𝜖.(3.17)

Lemma 3.1. On 𝛾+𝑥𝐹(𝑧)=𝑖0𝑥−𝑥1−𝑥−3/2𝑥1/2+𝒪(𝛿)(3.18) and on 𝛾−𝑥𝐹(𝑧)=−𝑖0𝑥−𝑥1−𝑥−3/2𝑥1/2+𝒪(𝛿).(3.19)

Proof. This is a consequence of the branching of the square root function. Let 𝑧𝐻(𝑧)=𝑧−𝑥0𝑥1=𝑧−𝑧𝑧−𝑥0𝑥1−𝑧||𝑧−𝑥0||2||𝑥1||−𝑧2,ℎ(𝑧)=𝑧𝑧−𝑥0𝑥1−𝑧.(3.20) We see that ℎ(𝑥+𝑖𝛿)=𝑥𝑥−𝑥0𝑥1𝑥−𝑥+𝑖𝛿2𝛿−1+𝒪2.(3.21) Recall that 𝛿 is replaced by −𝛿 on 𝛾− and that 0<𝑥<𝑥0<𝑥1. So Re𝐻(𝑧)<0 on both 𝛾+ and 𝛾− while Im𝐻(𝑧)<0 on 𝛾+ and Im𝐻(𝑧)>0 on 𝛾−. Thus on 𝛾+𝐻(𝑧)1/2||||𝑥=−𝑖(𝑥−𝑥0)(𝑥1||||−𝑥)1/2𝑥+𝒪(𝛿)=−𝑖(𝑥0−𝑥)(𝑥1−𝑥)1/2+𝒪(𝛿),(3.22) while on 𝛾−𝐻(𝑧)1/2𝑥=𝑖(𝑥0−𝑥)(𝑥1−𝑥)1/2+𝒪(𝛿).(3.23) Since 1𝐹(𝑧)=𝐹(𝑥)+𝒪(𝛿)=𝑥−𝑥0𝑥1−𝑥𝐻(𝑧)1/21+𝒪(𝛿)=−𝑥0𝑥−𝑥1𝐻−𝑥(𝑧)1/21+𝒪(𝛿)=−𝑥0𝑥−𝑥1𝑥−𝑥−𝑖(𝑥0−𝑥)(𝑥1−𝑥)1/2+𝒪(𝛿),(3.24) the lemma follows.

We also need to be careful about the orientation. The curve 𝛾 is oriented counter-clockwise. This means that the integration over 𝛾+ goes in the direction of 𝑥0 to 0. We need to reverse its sign in order to write it in the conventional way

𝛾+𝐹(𝑧)𝑑𝑧=−𝑥0−𝜖cos𝛼𝜖cos𝛼𝐹(𝑥+𝑖𝛿)𝑑𝑥=−𝑥0−𝜖cos𝛼𝜖cos𝛼𝑖𝑥1/2𝑑𝑥𝑥0−𝑥3/2𝑥1−𝑥3/2+𝒪(𝜖).(3.25) We let 𝑥=𝑥0𝑡 and Δ=𝜖𝑥0−1cos𝛼 and use 𝛿≪𝜖 to obtain

𝛾+𝐹(𝑧)𝑑𝑧=−Δ1−Δ𝑖𝑥03/2𝑡1/2𝑑𝑥𝑥0−𝑥0𝑡3/2𝑥1−𝑥0𝑡3/2+𝒪(𝜖).(3.26) Now let

𝑥𝜏=1𝑥0>1.(3.27) We have

𝛾+𝑖𝐹(𝑧)𝑑𝑧=−𝑥03/2Δ1−Δ𝑡1/2𝑑𝑡(𝜏−𝑡)3/2(1−𝑡)3/2+𝒪(𝜖).(3.28) We may let Δ→0 in the lower limit, but not in the upper limit. So we now have

𝛾+𝑖𝐹(𝑧)𝑑𝑧=−𝑥03/201−Δ𝑡1/2𝑑𝑡(𝜏−𝑡)3/2(1−𝑡)3/2+𝒪(𝜖).(3.29) Here Δ=𝜖𝑥0−1cos𝛼. We may let 𝛿→0 and redefine Δ to be 𝜖𝑥0−1.

We get the same value for the integral over 𝛾−. This is because the integrand is the negative of that in (3.29) but the orientation is reversed and so the two negatives cancel.

For any function 𝑔(𝑡), continuously differentiable on the interval [0,1], we have

01−Δ𝑔(𝑡)(1−𝑡)3/2𝑑𝑡=2𝑔(1)Δ1/2−2𝑔(0)−210(1−𝑡)−1/2ğ‘”î…žî€·Î”(𝑡)𝑑𝑡+𝑂1/2(3.30) as can easily be seen by an integration by parts.

We apply this to 𝑔(𝑡)=𝑡1/2/(𝜏−𝑡)3/2 as in (3.29). This yields

𝛾+𝐹(𝑧)𝑑𝑧=−2𝑖(𝜏−1)3/2𝑥03/2Δ1/2+𝑖𝑥03/210𝑡−1/2(𝜏−𝑡)−3/2+3𝑡1/2(𝜏−𝑡)−5/2𝜖𝑑𝑡+𝒪1/2.(3.31) We now take twice this quantity and add to it the results of (3.14) and (3.16),

|𝑧|=1𝐹(𝑧)𝑑𝑧=𝛾=𝐹(𝑧)𝑑𝑧4𝑖𝐶√𝜖√+2−2𝑖𝑥0𝑥1−𝑥0−3/2√𝜖+𝑖𝑥03/210(1−𝑡)−1/2(𝜏−𝑡)−3/2𝑡−1/2+3𝑡1/2(𝜏−𝑡)−5/2𝜖𝑑𝑡+𝒪1/2.(3.32) We substitute the value for 𝐶 from (3.14) and let 𝜖→0

|𝑧|=1𝐹(𝑧)𝑑𝑧=2𝑖𝑥03/210(1−𝑡)−1/2(𝜏−𝑡)−3/2𝑡−1/2+3𝑡1/2(𝜏−𝑡)−5/2𝑑𝑡.(3.33) Finally, we evaluate this integral using Mathematica and obtain

|𝑧|=1𝐹(𝑧)𝑑𝑧=2𝑖𝑥03/2[][])2(2𝜏𝐸1/𝜏−(−1+𝜏)𝐾1/𝜏(𝜏−1)2√𝜏,(3.34) where 𝐾 and 𝐸 are the complete elliptic functions of the first and second kinds

𝐾[𝑚]=101−𝑚𝑡2−1/21−𝑡2−1/2𝐸[𝑚]=𝑑𝑡,101−𝑚𝑡21/21−𝑡2−1/2𝑑𝑡.(3.35)

We now have

2𝐼=𝑏3/21𝑖Γ=2𝐹(𝑧)𝑑𝑧𝑏3/22𝑥03/22(2𝜏𝐸(1/𝜏)−(𝜏−1)𝐾(1/𝜏))(𝜏−1)2√𝜏=2𝑏3/24𝜏1/4(𝜏−1)2𝑔(2𝜏𝐸(1/𝜏)−(𝜏−1)𝐾(1/𝜏)),(3.36)NP=14ğœ‹â„Ž20𝑑/2𝜌𝐼𝑑𝜌(3.37) with

𝜌𝑏=2ℎcot𝜃,ğ‘Ž=csc2𝜌𝜃+2ℎ2,𝑥(3.38)𝜏=1𝑥0=1𝑥02,𝑥0=ğ‘Žğ‘îƒ¬î‚™1−𝑏1−2ğ‘Ž2,𝑥1=ğ‘Žğ‘îƒ¬î‚™1+𝑏1−2ğ‘Ž2.(3.39)

4. Limiting Cases

We validate (3.36) by considering two limiting cases.

4.1. Expansion for Small 𝑑

We look for an expansion of 𝑔NP under the assumption that 𝑑/ℎ is small and rederive (2.12). The dependence of 𝑥0 on 𝜌 is given in (3.39). The two-term Taylor expansion of 𝑥0 as a function of 𝜌 is

𝑥0=𝐴𝜌+𝐵𝜌3𝜌+𝒪5(4.1) with

𝐴=cos𝜃sinğœƒâ„Ž,𝐵=𝐴3âˆ’ğ´â„Ž2csc2𝜃.(4.2) Since 1𝜏=𝑥02,(4.3) we have 1𝜏=𝐴2𝜌2𝜌+𝒪4,(4.4)1(𝜏−1)2=1𝜏2+2𝜏3𝜌+𝒪8.(4.5) It is convenient to work with the quantity 𝜌2𝜏, so we note, using (4.1),

𝜌21𝜏=𝐴2−2𝐵𝐴3𝜌2𝜌+𝒪4.(4.6) We start with

2𝐼=𝑏3/24𝜏1/4(𝜏−1)212𝜏𝐸𝜏1−(𝜏−1)𝐾𝜏.(4.7) Substituting the expression for 𝑏 from (3.38) and for (𝜏−1)2 from (4.5) we obtain, after some manipulation

îƒ©î‚€â„Žğ¼=cot𝜃3/24𝜌2𝜏3/41𝜏𝜌+𝒪412𝜏𝐸𝜏1−(𝜏−1)𝐾𝜏.(4.8)

We now use the expansions (as given by Mathematica)

𝜋𝐸(𝑡)=2−𝜋𝑡8𝑡+𝒪2,𝐾𝜋(𝑡)=2+𝜋𝑡8𝑡+𝒪2.(4.9) So

î‚€â„Žğ¼=cot𝜃3/24𝜌2𝜏3/4𝜋2+𝜋+𝜋8𝜏𝜏𝜌+𝒪4.(4.10) From the expression for 𝜌2𝜏 in (4.6), we derive

1𝜌2𝜏3/4=𝐴3/231+2𝐵𝐴𝜌2𝜌+𝒪4,(4.11) and this leads to

î‚€â„Žğ¼=4cot𝜃3/2𝐴3/2𝜋2+𝜋+𝜋8𝜏𝜏+3𝐵𝜌2𝜋𝜌4𝐴+𝒪4.(4.12) We now substitute for 𝐴 and 𝐵 using (4.2) and also for 𝜏−1 using (4.4)

𝐼=sin3𝜃2𝜋+3𝜋23sin2𝜃−5sin4𝜃𝜌2ℎ2𝜌+𝒪4.(4.13) Substituting this into (3.37) and integrating, we again derive (2.12).

4.2. An Expansion for 𝜃 Near 𝜋/𝟐

We show how (4.7) also gives the expansion of Section 2.2. From (3.38) we see that

ğœŒğ‘Ž=1+2ℎ2+𝜋𝜃−22𝜋+𝒪𝜃−24,(4.14)𝑏=−2ğœŒâ„Žî‚€ğœ‹ğœƒâˆ’2𝜋+𝒪𝜃−23.(4.15) So from (3.39) we have

𝑥0=𝑏𝑏2ğ‘Ž1+24ğ‘Ž2𝜋+𝒪𝜃−23.(4.16) Thus

𝜏=4ğ‘Ž2𝑏2𝜋−2+𝒪𝜃−22,1(4.17)(𝜏−1)2=1𝜏221+𝜏𝜋+𝒪𝜃−24,1𝜏12𝜏𝐸𝜏1−(𝜏−1)𝐾𝜏=𝜋2+9𝜋𝜋8𝜏+𝒪𝜃−24.(4.18) These expansions let us write (4.7) as

2𝐼=𝑏3/24𝜏3/421+𝜏𝜋2+9𝜋𝜋8𝜏+𝒪𝜃−24=27/2𝑏2𝜏3/421+𝜏𝜋2+9𝜋𝜋8𝜏+𝒪𝜃−24.(4.19) Using (4.15) and (4.17) we obtain a substitute for (4.1)

1𝑏2𝜏3/4=4−3/4ğ‘Žâˆ’3/21+3𝑏28ğ‘Ž2𝜋+𝒪𝜃−24.(4.20) This gives, in light of (4.14),

𝐼=2ğœ‹ğ‘Žâˆ’3/2+158𝜋𝑏2ğ‘Žâˆ’7/2𝜋+𝒪𝜃−24𝜌=2𝜋1+2ℎ2−3/2𝜌−3𝜋1+2ℎ2−5/2𝜋𝜃−22+152𝜋𝜌2ℎ2𝜌1+2ℎ2−7/2𝜋𝜃−22𝜋+𝒪𝜃−24.(4.21) Finally, we integrate term by term.

1𝑔=4ğœ‹â„Ž20𝑑/2=1𝜌𝐼(𝜌)𝑑𝜌4ğœ‹â„Ž22ğœ‹â„Ž2𝑑1−1+2ℎ2−1/2−3ğœ‹â„Ž213𝑑1−1+2ℎ2−3/2𝜋𝜃−22+152ğœ‹â„Ž22−152+5(𝑑/2ℎ)2𝑑151+2ℎ2−5/2𝜋𝜃−22𝜋+𝒪𝜃−24=12𝑑1−1+2ℎ2−1/2−38𝑑1+2ℎ2−5/2𝑑2ℎ2𝜋𝜃−22𝜋+𝒪𝜃−24.(4.22)

5. Numerical Confirmation

The result in (3.36) is further validated in this section with numerical integration of (3.1). In this original formulation, the integral depends on the ratio of 𝜌/ℎ and on the value of 𝜃. We analytically considered one limiting case for each of these quantities in the previous section. We consider these and other values for these quantities in this section using numerical methods. For fixed values of 𝜌/ℎ, the numerical integration (solid lines) is shown with the results of the residue calculation (open circles) in Figure 3 as a function of 𝜃. Figure 4 shows the numerical integration and residue results for fixed values of 𝜃 as a function of 𝜌/ℎ. All corresponding values of the residue calculation and the numerical integration agree.

6. Conclusions

The accuracy of collimator efficiency is important for design and for quantitative reconstruction programs. Analytic forms obviate the need for numerical models, which do not readily offer insight into the interplay of collimator parameters. This paper approaches the problem by recasting one of the integrals in terms of contour integration. This integration was completed, but the authors were unable to execute the second integral to yield a complete analytic solution. The result was studied for special cases with known outcomes for validation. Future efforts may be able to build on this validated result to find a complete analytic solution.

Acknowledgment

Scott D. Metzler was supported by the National Institute for Biomedical Imaging and Bioengineering of the National Institutes of Health under Grant R01-EB-6558.

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Copyright © 2010 Howard Jacobowitz and Scott D. Metzler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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