# The Influence of Photon Attenuation on Tumor-to-Background and Signal-to-Noise Ratios for SPECT Imaging.

**ABSTRACT** Expanding on the work of Nuyts et. al [1], Bai et. al. [2], and Bai and Shao [3], who all studied the effects of attenuation and attenuation correction on tumor-to-background ratios and signal detection, we have derived a general expression for the tumor-to-background ratio (TBR) for SPECT attenuated data that have been reconstructed with a linear, non-iterative reconstruction operator O. A special case of this is when O represents discrete filtered back-projection (FBP). The TBR of the reconstructed, uncorrected attenuated data (TBR(no-AC)) can be written as a weighted sum of the TBR of the FBP-reconstructed unattenuated data (TBR(FBP)) and the TBR of the FBP-reconstructed "difference" projection data (TBR(diff)). We evaluated the expression for TBR(no-AC) for a variety of objects and attenuation conditions. The ideal observer signal-to-noise ratio (SNR(ideal)) was also computed in projection space, in order to obtain an upper bound on signal detectability for a signal-known-exactly/background-known-exactly (SKE/BKE) detection task. The results generally show that SNR(ideal) is lower for tumors located deeper within the attenuating medium and increases for tumors nearer the edge of the object. In addition, larger values for the uniform attenuation coefficient μ lead to lower values for SNR(ideal). The TBR for FBP-reconstructed, uncorrected attenuated data can both under- and over-estimate the true TBR, depending on several properties of the attenuating medium, including the shape of the attenuator, the uniformity of the attenuator, and the degree to which the data are attenuated.

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The Influence of Photon Attenuation on Tumor-to-Background

and Signal-to-Noise Ratios for SPECT Imaging

Edward J. Soares, Ph.D.*,†, Michael A. King, Ph.D.†, Charles L. Byrne, Ph.D.‡, Howard C.

Gifford†, and Andre Lehovich, Ph.D.†

* Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610

† Department of Radiology, University of Massachusetts Medical School, Worcester, MA 01655

‡ Department of Mathematical Sciences, University of Massachusetts at Lowell, Lowell, MA 01854

Abstract

Expanding on the work of Nuyts et. al [1], Bai et. al. [2], and Bai and Shao [3], who all studied the

effects of attenuation and attenuation correction on tumor-to-background ratios and signal detection,

we have derived a general expression for the tumor-to-background ratio (TBR) for SPECT attenuated

data that have been reconstructed with a linear, non-iterative reconstruction operator O. A special

case of this is when O represents discrete filtered back-projection (FBP). The TBR of the

reconstructed, uncorrected attenuated data (TBRno-AC) can be written as a weighted sum of the TBR

of the FBP-reconstructed unattenuated data (TBRFBP) and the TBR of the FBP-reconstructed

“difference” projection data (TBRdiff). We evaluated the expression for TBRno-AC for a variety of

objects and attenuation conditions. The ideal observer signal-to-noise ratio (SNRideal) was also

computed in projection space, in order to obtain an upper bound on signal detectability for a signal-

known-exactly/background-known-exactly (SKE/BKE) detection task. The results generally show

that SNRideal is lower for tumors located deeper within the attenuating medium and increases for

tumors nearer the edge of the object. In addition, larger values for the uniform attenuation coefficient

μ lead to lower values for SNRideal. The TBR for FBP-reconstructed, uncorrected attenuated data

can both under- and over-estimate the true TBR, depending on several properties of the attenuating

medium, including the shape of the attenuator, the uniformity of the attenuator, and the degree to

which the data are attenuated.

I. Introduction

Over the last decade, several authors have investigated the usefulness of attenuation correction

on lesion detection and localization in single-photon emission computed tomography (SPECT)

and positron emission tomography (PET). In 2000, Wells et. al. [4] compared ordered-subset

expectation-maximization (OS-EM) reconstruction with filtered back-projection (FBP) to

determine if attenuation correction helped for detection and localization of thoracic gallium-

labeled lesions in SPECT. Using psychophysical studies, they found no difference in observer

performance using images derived from FBP reconstruction, with and without attenuation

compensation (AC). While OS-EM with AC did improve observer performance relative to the

other methods, OS-EM without AC yielded comparable detection to images reconstructed with

E.J. Soares is with the Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610 USA

(telephone: 508-793-3368, e-mail: esoares@holycross.edu).

M.A. King, H.C. Gifford, and A. Lehovich are with the Department of Radiology, Division of Nuclear Medicine, University of

Massachusetts Medical School, Worcester, MA 01655 USA (telephone: 508-856-4255, e-mail: Michael.King@umassmed.edu).

C.L. Byrne is with the Department of Mathematical Sciences, University of Massachusetts at Lowell, Lowell, MA 01854 (telephone:

978-934-2447, e-mail: Charles Byrne@uml.edu).

NIH Public Access

Author Manuscript

IEEE Nucl Sci Symp Conf Rec (1997). Author manuscript; available in PMC 2009 January 23.

Published in final edited form as:

IEEE Nucl Sci Symp Conf Rec (1997). 2007 ; 5: 3609–3615. doi:10.1109/NSSMIC.2007.4436905.

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FBP and no AC. Narayanan et. al. [5] studied compensation strategies for 99mTc myocardial

perfusion imaging. They compared FBP with no AC, with OS-EM reconstruction with AC,

with AC and scatter correction, and with AC, scatter correction, and resolution compensation.

The results of their psychophysical studies suggested that accurate modeling and compensation

for these degrading effects improved detection of coronary artery disease. FBP exhibited the

lowest observer performance, while performance increased as image-degrading factors were

included in the OS-EM reconstruction process.

Nuyts et. al [1] examined the effect of the attenuation artifact on the tumor-to-background ratio

(TBR) and tumor detection for PET. They showed the attenuation artifact had scaling and

shifting components. The scaling component was the result of loss of counts due to attenuation,

while the shifting component is a smooth negative contribution that causes an artificially high

TBR in some circumstances. Bai et. al. [2] also investigated the effects of attenuation in PET.

They noted that images without AC could have locally enhanced contrast in regions of uniform

attenuation. In regions of nonuniform attenuation such as the thorax, high count foci can

disappear in images without AC, or even appear photopenic. This effect is dependent on the

size, location, and density of the foci. Bai and Shao [3] extended their analysis to SPECT

oncological imaging. They found similar behavior in tumor detectability as compared with

their analysis of PET imaging. Chang’s attenuation correction method was shown to not

improve estimation of TBR, relative to no-AC.

Motivated by this previous work, we have attempted to analytically quantify the effect of

photon attenuation in the projection data and the lack of compensation for the degradation in

SPECT reconstructed images. We have derived an analytic expression that relates the TBR of

FBP-reconstructed, uncorrected attenuated data to that of FBP-reconstructed, unattenuated

data and FBP-reconstructed, “difference” projection data. Using simulation studies, we

demonstrate the effect of object size and shape, non-uniformity of attenuator, and degree of

attenuation on the TBR and ideal observer signal-to-noise ratio (SNR).

II. Theory

A. Imaging Model

In our analysis, we consider a simple signal-known-exactly/background-known-exactly (SKE/

BKE) detection task. Let f0 denote the discretized signal-absent object and f1 denote the

discretized signal-present object. Thus, the background and tumor (signal) are given,

respectively, by

(1)

The discrete Radon transform and the discrete attenuated Radon transform can be expressed

as matrices, H = (hij) and Ha = (aij hij), respectively, where hij is the weight for determining

the contribution of object pixel j to measurement i and aij is the associated attenuation factor.

Next, we define the difference transform ΔH, where ΔH = H − Ha = ((1 −aij)hij). The

importance of this operator becomes clear in our subsequent derivations.

Both signal-absent and signal-present objects are then projected incorporating the effects of

photon attenuation. Thus, we may write the mean attenuated projections of the signal-absent

and signal-present objects as

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(2)

The object-dependent noise n is defined as the deviation between a measurement and its mean

(3)

which, when incorporated with (2), yields the imaging model

(4)

Next, the data are reconstructed with a linear, non-iterative operator O. An example of O would

be discrete filtered back-projection (FBP). Thus, we may write the reconstructed signal-absent

and signal-present images as

(5)

with mean reconstructed signal-absent and signal-present images given by

(6)

since the noise in (3) is defined tacitly as having zero mean. Finally, using equations (1) and

(6), we see that the mean reconstructed background and tumor images may be written as

(7)

and

(8)

respectively.

B. Signal-to-Noise Ratio (SNR)

In order to objectively assess the effect of photon attenuation on tumor detectability, we

computed the ideal observer signal-to-noise ratio (SNRideal) using the projection data. This is

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slightly unconventional, as it is the reconstructed images that are typically used for diagnostic

purposes and thus image quality tends to be assessed in reconstruction space. But evaluating

SNRideal in projection space serves two purposes. First, it sets an upper bound on detectability,

as SNRideal is invariant to invertible reconstruction operators O and may only decrease after

image reconstruction. One explanation for such a decrease would be if the reconstruction

operator O contains null components. Second, it will allow us to determine which signal

locations within the background produce data with higher detectability measures. In some

sense, it’s a measure of favorability of tumor location under our SKE/BKE paradigm.

The ideal observer uses the full joint probability distributions of both signal-present and signal-

absent data to compute a test statistic based on maximizing the likelihood ratio. For simplicity,

we used the normal approximation to the Poisson-distributed data

(9)

where the signal-absent and signal-present class covariance matrices are diagonal and given

by

(10)

Assuming the probabilities of occurrence of each class are P0 and P1, respectively, then

SNRideal can be calculated as

(11)

where ḡ ḡT are the mean attenuated projections of the tumor. The reader will note that the inverse

of the average class covariance matrix (P0Σg0 + P1Σg1)is easily computed, as this matrix is

diagonal.

C. Tumor-to-Background Ratio (TBR)

The tumor-to-background ratio (TBR) is a local measure of contrast between the tumor activity

and that of the surrounding background. Intuitively, we might hypothesize that a higher TBR

would correlate with higher detectability, as the tumor “stands out” more against the local

background. However, tumor detection also depends on the noise properties of image,

particularly the noise correlations. It is well known that FBP reconstruction introduces noise

correlations that inhibit human observer performance in SKE/BKE detection tasks. Thus, we

wished to compare this contrast measure with our detectability figure-of-merit SNRideal to

better understand any relationships that may exist.

In the following, we use the mean reconstructed background and tumor (eqns. (7) and (8))

rather than those containing the noise terms. This is due to the fact that the inclusion of the

noise terms would introduce an additional level of complexity in our derivation. Also, the

average activity in both the reconstructed background and tumor is computed within a region-

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of-interest (ROI), and this averaging process would most likely minimize the influence of the

noise terms on the overall TBR.

To estimate the average tumor activity and average background activity, we must first define

a ROI-observer template w that is non-zero for pixels within the ROI and zero otherwise. Often,

the template values are designed to sum to unity. We can now compute the estimated average

tumor activity (T̂) in the ROI as

(12)

as well as the estimated average background activity (B̂) in the ROI as

(13)

Finally, we define the TBR of the reconstructed, uncorrected attenuated data as

(14)

The quotient above can be simplified using component-operation long division as discussed

in [6] to yield

(15)

We note that

(16)

Dividing both sides by wtOHafB (assuming it is non-zero), we yield the identity

(17)

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