Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 9830386, 9 pages

https://doi.org/10.1155/2017/9830386

## Uncertainty Analysis in the Calibration of an Emission Tomography System for Quantitative Imaging

^{1}ENEA, National Institute of Ionizing Radiation Metrology, Via Anguillarese 301, 00123 Rome, Italy^{2}National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK

Correspondence should be addressed to Marco D’Arienzo

Received 12 June 2017; Accepted 15 August 2017; Published 12 October 2017

Academic Editor: David A. Winkler

Copyright © 2017 Marco D’Arienzo and Maurice Cox. 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.

#### Abstract

It is generally acknowledged that calibration of the imaging system (be it a SPECT or a PET scanner) is one of the critical components associated with in vivo activity quantification in nuclear medicine. The system calibration is generally performed through the acquisition of a source with a known amount of radioactivity. The decay-corrected calibration factor is the “output” quantity in a measurement model for the process. This quantity is a function of a number of “input” variables, including total counts in the volume of interest (VOI), radionuclide activity concentration, source volume, acquisition duration, radionuclide half-life, and calibration time of the radionuclide. Uncertainties in the input variables propagate through the calculation to the “combined” uncertainty in the output quantity. In the present study, using the general formula given in the GUM (Guide to the Expression of Uncertainty in Measurement) for aggregating uncertainty components, we derive a practical relation to assess the combined standard uncertainty for the calibration factor of an emission tomography system. At a time of increasing need for accuracy in quantification studies, the proposed approach has the potential to be easily implemented in clinical practice.

#### 1. Introduction

Positron emission tomography (PET) and single-photon emission computed tomography (SPECT) provide a means of evaluating the biological function of cells and organs, producing three-dimensional images of the distribution of radiopharmaceuticals introduced into the patient’s body. These molecular imaging techniques rely on radiolabelled molecules (generally consisting of a radionuclide and a molecule that determines the localization) that build up in areas of disease allowing for the collection of metabolic and functional information in vivo.

Over the last decade radionuclide imaging has gained popularity as a quantitative technique. In particular, recent advances in image processing software and the advent of hybrid SPECT/CT and PET/CT scanners have paved the way for accurate quantitative analysis, that is, the determination of activity concentration within a given tissue of interest in absolute units, for example, becquerel per millilitre or becquerel per cubic centimetre.

PET was developed as a quantitative tool and the standardized uptake value (SUV) is probably the most widely used indicator for the quantification of ^{18}F-FDG PET studies. SUV is a measure of how much cellular activity occurs in the region of uptake and is mathematically defined as the concentration of the radionuclide in the volume of interest (VOI) divided by the injected activity normalized for the patient’s body weight. High SUV values are likely to represent pathological conditions, from inflammation to infection to cancer, with higher numbers being most suggestive of cancer.

On the other hand, SPECT has traditionally been seen as nonquantitative. This is because quantification using SPECT images is a time-consuming process, requiring accurate methods that correct for a number of degrading factors, among which are attenuation, scatter, dead time, and partial volume effects. However, the advent of hybrid SPECT/CT scanners has made quantitative SPECT viable in a manner similar to quantitative PET. An extensive review of potential uses for quantitative SPECT is given in [1].

Calibration of the imaging system is an essential prerequisite to convert reconstructed voxel values to absolute activity or activity concentration, both in SPECT and in PET. As a consequence, it is generally acknowledged that calibration of the imaging system in emission tomography is a critical requirement for producing accurate quantitative data both in diagnosis and in therapy [2].

Gamma camera calibration can be performed either in air or in water. Calibration in air consists of determining the gamma camera sensitivity through the acquisition of a small volume of activity, for example, point-like source, Petri dish, line source, or spherical source. On the other hand, gamma camera calibration in water involves the use of an extended volume source mimicking the clinical conditions encountered in patient studies [3]. The general formalism for the evaluation of the system calibration factor is given in NEMA publication NU 1-2012 [4]. Absolute calibration of the PET system (often referred to as “well counter calibration”) is generally performed by scanning a large water-filled phantom that contains a known amount of activity. This procedure allows counts per second to be transformed to activity concentration. Following the absolute activity calibration, the voxel intensity in any PET image is divided by the calibration factor to obtain calibrated images in terms of Bqcm^{−3}. Further details on the procedure for evaluating the performance of positron emission tomographs are reported in NEMA Standards Publication NU 2-2007 [5].

Without loss of generality, the decay-corrected scanner calibration factor in emission tomography, the output quantity, can be written in terms of the input quantities , the summed counts over a given VOI in the image , the volume of interest , the radionuclide activity concentration , the acquisition start time , the time of activity calibration , the radionuclide physical half-life , the acquisition durationas follows [4, 5]:In this formalism all input quantities have associated standard uncertainties, which propagate through the calculation to the “combined” standard uncertainty in the output quantity. In the present study, using the general approach proposed in the GUM (Guide to the Expression of Uncertainty in Measurement) [7] for combining uncertainty components (the “law of propagation of uncertainty”), we derive a relation to assess the combined standard uncertainty for the calibration factor.

#### 2. Evaluation of Uncertainty

Let denote a set of “input” quantities and an “output” quantity or measurand. The GUM [7] considers the generic measurement modelthat is, a known functional relationship between the input and the output quantities. Given estimates of the input quantities, the GUM usesas the corresponding estimate of . Further, given standard uncertainties associated with , the GUM applies the law of propagation of uncertainty (LPU) to evaluate the combined standard uncertainty associated with . For independent input quantities, LPU is described by the following expression:in which denotes evaluated at .

Therefore, with reference to expression (1), an estimate of the calibration factor is given by evaluating this expression for estimates of the input quantities , , , , , , and , which are assumed independent. Further, the standard uncertainty associated with this estimate of is given by the following (for notational simplicity we do not distinguish between a quantity and an estimate of the quantity):

Although the use of expression (5), after evaluating the necessary partial derivatives, will deliver the required standard uncertainty , this uncertainty can be obtained more conveniently by rewriting expression (1). Using the substitutions, (1) can be expressed as Since each quantity on the right sides of equations (6) to (9) does not appear in any others of these equations, the independence of these quantities implies the independence of , , , and . Hence, in place of expression (5) we haveDefiningeach partial derivative appearing in expression (11) can simply be expressed in terms of , , , , , , and : Expression (11) can then be recast asor, in terms of relative standard uncertainties, where, for instance, denotes ,It is noted that for most radionuclides the acquisition time is generally much smaller than the radionuclide half-life (i.e., and, with reference to expression (13), ). This is especially true for therapeutic radionuclides, whose half-life is typically a few days, while acquisition times are generally in the range of 10 min to 30 min. Recalling that provided is reasonably small, (16) can be written in simplified form:

Computing the combined standard uncertainty on the calibration factor according to (17) requires minimal effort and has the potential to be easily implemented in clinical practice. The following paragraphs describe how each source of uncertainty appearing in expression (17) can be practically and effectively estimated.

##### 2.1. Evaluation of

Using a variant of the law of propagation of uncertainty that applies to a model in product/quotient form [7], the squared relative standard uncertainty on the term (see (6)) can be expressed as the sum of the squares of the relative standard uncertainties in each of the quantities appearing in (6):In turn the radionuclide activity concentration is given by the absolute activity of the radionuclide divided by the volume of the liquid solution in which it is dispersed; namely, Thus, (18) can be rewritten asConsiderations on how the above uncertainties can be evaluated are given below.(i)The relative standard uncertainty of the counts in a given VOI depends on both physical factors affecting activity quantification and the noise level generated during the process of reconstructing the image. Physical factors include photon interactions in the patient, loss of spatial accuracy due to limited system resolution, partial volume effects, and noise resulting from the random nature of radioactive decay and absorption. Counting statistics and acquisition time play an additional role. Both in SPECT [8] and in PET [9] the coefficient of variation, that is, the ratio of the standard deviation to the average signal measured in the VOI, is a viable approach to assessing the noise level.(ii)The relative standard uncertainty is generated by the voxelization of the VOI, that is, the process of converting the continuous geometric representation of the VOI into a set of voxels that approximates the continuous object. By simply selecting all voxels that are intersected by the continuous VOI (that may, e.g., be approximately spherical or cylindrical), the generated digital object may as a result be too coarse, including more or fewer voxels than are necessary. For simplicity, we consider a spherical VOI represented in terms of cubical voxels of side . We consider an indicative standard uncertainty of the diameter to be size of a voxel; namely, . It follows that the standard uncertainty associated with the radius of the sphere will be half the side of the voxel; that is, . The standard uncertainty translates into a standard uncertainty associated with the volume delineated by the VOI. For some constant , . Hence , and, accordingly, Thus, a relative standard uncertainty associated with a radius induces a relative standard uncertainty associated with a volume that is three times as large. For instance, for a plan view of a sphere of some 150 mm in radius with a voxel side equal to 3 mm, . Thus, and . In practice the VOI will differ from a sphere, but a mean radius of the VOI can instead be considered. The standard uncertainty translates into a standard uncertainty associated with the volume delineated by the VOI, and the relationship still holds.(iii) is the relative standard uncertainty associated with activity measurement. At a clinical level, activity measurements are generally made using commercially available radionuclide calibrators traceable to a national standards laboratory for the geometry being measured. The typical instrument for assaying radiopharmaceuticals is the pressurized, well-type ionization chamber. These instruments are capable of providing radioactivity measurements with varying degrees of accuracy, depending on the radionuclide and the sample configurations (e.g., glass vials and/or plastic syringes). From a regulatory standpoint, in most countries the standard of good practice is that the administered activity should be within 10% of the prescribed activity [10]. As a consequence, given the other sources of error involved in the administration of the radiopharmaceutical, radionuclide calibrators should provide an expanded uncertainty below 10%, perhaps 5% (for a coverage factor of , giving approximately 95% coverage). The achievable uncertainty in clinical practice is reported in the AAPM guidelines [11]. For radionuclide calibrator field instruments an expanded uncertainty no greater than 5% is recommended for photon emitters > 100 keV. An expanded uncertainty no greater than 10% for photon emitters < 100 keV is recommended [11]. For medium and high-energy beta emitters, a radionuclide calibrator expanded uncertainty no greater than 5% is suggested, while for low-energy beta emitters an expanded uncertainty no greater than 10% is recommended [11]. Secondary standard radionuclide calibrators and reference radionuclide calibrators should be calibrated to be within an expanded uncertainty no greater than 2% for photon emitters > 100 keV and medium and high-energy beta emitters. For the same instruments an expanded uncertainty no greater than 5% is recommended for photon emitters < 100 keV and low-energy beta emitters [11].(iv)The relative standard uncertainty is associated with volume measurement, which typically translates into weighing of masses. As a general rule, the significant factors that contribute to measurement uncertainty across the weighing range are repeatability, eccentricity (the error associated with not placing the weight in the centre of the weighing pan), nonlinearity (the error due to the nonlinear behaviour of the balance upon increasing the load on the weighing pan), and sensitivity (i.e., systematic deviation). If analytic balances are used for the measurements of small masses, uncertainties below 0.001% can be achieved [12]. If large masses need to be weighed (e.g., large phantoms filled with water mixed with radionuclide) laboratory balances with weighing capacity of up to 100 kg to 150 kg are commercially available, yielding typical relative standard uncertainties below 0.05%.

##### 2.2. Evaluation of

The relative standard uncertainty is associated with a possible time offset between the clocks used to assess the reference calibration time and the acquisition start time . With reference to (7),where is the standard uncertainty associated with the time difference between the acquisition start time and the reference calibration time . The absolute time offset between the two clocks used to determine and can be considered representative of .

It is worth noting that the overall impact of the time offset on the calibration factor uncertainty does not depend on the absolute time difference . In fact, with reference to (16) (also see (12)), the absolute time difference cancels leaving a dependence solely on the terms and :

Figure 1 illustrates the impact of time offset on the final relative standard uncertainty, , as a function of radionuclide half-life for short-lived radionuclides. The case for two widely used diagnostic radionuclides, ^{18}F () and (), is shown. as obtained from (16) is plotted as a function of the radionuclide half-life for different values of , and the absolute time offset between the two clocks is used to determine and . With reference to (16) and (18), the following relative standard uncertainties were considered: , , %, %, and . As a general rule, the greater the ratio between the time offset and the radionuclide half-life is, the larger the impact on the calibration factor relative standard uncertainty is. Figure 2 reports the same data for long-lived radionuclides, for example, ^{90}Y (), ^{177}Lu (), and ^{223}Ra (). The extreme case of is presented.