EANM practical guidance on uncertainty analysis for molecular radiotherapy absorbed dose calculations
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Abstract
A framework is proposed for modelling the uncertainty in the measurement processes constituting the dosimetry chain that are involved in internal absorbed dose calculations. The starting point is the basic model for absorbed dose in a site of interest as the product of the cumulated activity and a dose factor. In turn, the cumulated activity is given by the area under a time–activity curve derived from a time sequence of activity values. Each activity value is obtained in terms of a count rate, a calibration factor and a recovery coefficient (a correction for partial volume effects). The method to determine the recovery coefficient and the dose factor, both of which are dependent on the size of the volume of interest (VOI), are described. Consideration is given to propagating estimates of the quantities concerned and their associated uncertainties through the dosimetry chain to obtain an estimate of mean absorbed dose in the VOI and its associated uncertainty. This approach is demonstrated in a clinical example.
Keywords
Dosimetry Uncertainty analysisIntroduction
Internal dosimetry following the administration of radiolabelled pharmaceuticals for diagnostic and therapeutic purposes is a prerequisite for radiation protection, imaging optimization, patient-specific administrations and treatment planning. The medical internal radiation dose (MIRD) schema [1] has become the most widely accepted formalism for internal dose calculations. The general approach in medicine to determine the validity of a measurement is to compare the accuracy and precision against a “gold standard” measurement. To date, investigations of uncertainties for internal dosimetry have mainly used phantoms or simulated data [2, 3, 4] as the gold standard comparison. However, due to the diversity of dosimetry data, a subset of phantom experiments cannot necessarily validate the accuracy of measurements made for an in vitro population. It is therefore more appropriate to express the accuracy of a result by characterizing the uncertainty. This involves identification of the major processes and variables within the dose calculation and evaluation of their potential effect on the measurement. An uncertainty estimate should address all systematic and random sources of error and characterize the range of values within which the measured value can be said to lie with a specified level of confidence. The general relevance of performing and providing uncertainty information has been discussed in previous guidelines [5]. Flux et al. [6] provided a method to determine the uncertainty of whole-body absorbed doses calculated from external probe measurement data. Whilst whole-body dosimetry is used to predict toxicity in some procedures [7, 8], organ and tumour dosimetry are required for treatment planning and in cases where haematotoxicity is not the limiting factor for treatment tolerance.
Specific aspects of uncertainty within the dosimetry chain have been addressed, including the selection of measured time points, [9, 10], the chosen fit function [11, 12] and uncertainty of model parameters. A comprehensive analysis of propagation of every aspect of the dosimetry calculation chain has yet to be obtained.
Gustafsson et al. [13] adopted a Monte Carlo (MC) approach to investigate the propagation of uncertainties to obtain an uncertainty in estimated kidney absorbed dose in ^{177}Lu-DOTATATE therapy, using simulated gamma-camera images of anthropomorphic computer phantoms. In principle, this approach allows all aspects of the dosimetry process to be taken into account, but the need for multiple samplings from the assigned probability distributions for the quantities involved makes it computationally intensive, and its use for uncertainty assessment in an individual patient basis is challenging.
The Joint Committee for Guides in Metrology (JCGM) Guide to the Expression of Uncertainty in Measurement (GUM) [14] provides a generalized schema for propagating uncertainties. This EANM Dosimetry Committee guidance document provides recommendations on how to determine uncertainties for dosimetry calculations and apply the law of propagation of uncertainty (LPU) given in the GUM to the MIRD schema. This guidance document is presented in the form of an uncertainty propagation schema, and the recommendations are designed to be implemented with the resources available in all nuclear medicine departments offering radionuclide therapy, and are presented using terminology and nomenclature that adhere as far as possible to the GUM.
The uncertainty propagation schema examines each step of the absorbed dose calculation to estimate the standard uncertainty in the mean absorbed dose measured at the organ or tumour level using SPECT imaging. The examples given have been simplified and concentrate only on the mean absorbed dose to a target. However, the approach can be used in different scenarios and expanded to include more complex dose calculations, including cross-dose and multiexponential time–activity curves (TACs). Similarly, aspects of the methodologies described can be implemented in different applications of dosimetry such as those utilizing a hybrid imaging approach or those used to generate 3D dose maps.
Theory
The law of propagation of uncertainty
If the two variables X_{1} and X_{2} are mutually independent, the covariance term of Eq. 11 is zero, and therefore the standard fractional uncertainties u(x_{1})/x_{1} and u(x_{2})/x_{2} are simply added in quadrature.
Absorbed dose
It follows that the standard uncertainties \( u\left(\overset{\sim }{A}\right) \) and u(S) and the covariance \( u\left(\overset{\sim }{A},S\right) \) are needed to obtain the standard uncertainty \( u\left(\overline{D}\right) \) associated with \( \overline{D} \).
For the general form of the MIRD equation with meaningful contributions outside the target volume (cross-dose), uncertainties and covariances associated with additional quantities of the form \( \overset{\sim }{A} \) and S should also be considered.
Volume uncertainty
The volume or mass of an organ or tumour is generally obtained from a volume of interest (VOI) outlined on anatomical or functional imaging data. It is therefore possible to estimate the outlining accuracy by considering factors that affect delineation. The method used will depend largely on the information and resources available at the time of outlining and the method employed by the operator or operators to define the VOI. The following concerns an outline drawn manually by a single operator across all images that comprise the dosimetry dataset.
Operator variability
The historical datasets should be carefully chosen to match the current study, as differences could lead to an inaccurate estimate of the final standard deviation.
Analytical approach
When historical outlines are not available, it is possible to use an analytical method to determine uncertainty. This approach provides an estimate of the most significant contributions to the uncertainty in the outlining process but is not necessarily exhaustive.
With hybrid imaging it is often possible to use morphological information from CT imaging to aid functional VOI delineation. In this situation the VOI is drawn on the CT dataset and copied to the registered SPECT image. The coordinates of the original boundary will therefore be rounded to the nearest voxel coordinates of the SPECT image. Hence, the SPECT voxel size should be used in Eq. 17.
Hence, a fractional standard uncertainty associated with a volume is three times the fractional standard uncertainty associated with the mean diameter of that volume.
Count rate
The total reconstructed count rate, C, within a VOI depends on the VOI delineation, and can be described as a function of volume. Propagation of volume uncertainty into the measurement of count rate is therefore required. As no prior knowledge of the count distribution is generally available, the variation in C within the VOI must be approximated.
Recovery coefficient
Calibration factor
Activity
It follows that the covariances u(R, C_{i}) and u(C_{i}, C_{j}) associated with Q, R and the C_{i} must be derived.
Time–activity curve parameters
In addition to the systematic uncertainties associated with quantification and volume determination, uncertainties in the TAC data can arise from other sources such as image noise, patient motion, registration and other imperfect post-acquisition operations such as image reconstruction, including scatter and attenuation corrections. Due to the complexity of these operations, it is assumed that the uncertainties associated with the compensation for effects such as attenuation and scatter are negligible in comparison with the uncertainty associated with the compensation for partial volume effects [13]. It is therefore more appropriate to measure the causality of imperfects in these corrections, and to derive uncertainties in the fit parameters of the TAC.
Cumulated activity
S-factor
The fractional standard uncertainty associated with S is thus proportional to the fractional standard uncertainty associated with volume v, the proportionality constant being the magnitude |c_{2}| of the slope of the fitting function on a log-log scale.
Absorbed dose
Patient example
An example to demonstrate the implementation of the approach described in this paper is given in the following sections, with details of the methodology used to obtain the absorbed dose data and the associated uncertainty analysis. The example given is that of a 47-year-old patient who presented with weight loss, lethargy and upper abdominal cramps. Upper gastrointestinal endoscopy showed a mass in the third part of the duodenum. A subsequent contrast-enhanced CT scan and ^{68}Ga-DOTATATE PET/CT investigation showed a 6.5-cm mass arising from the pancreatic head and a 3-cm mass within segment 4 of the liver, in keeping with a neuroendocrine tumour arising from the pancreas. The patient underwent ^{90}Y-DOTATATE radiopeptide therapy in combination with ^{111}In-DOTATATE for imaging. The administered activity was 4,318 MBq of ^{90}Y with ^{111}In given at a ratio of 1:25.
Image acquisition
Volume
Volumes and associated standard uncertainties for liver and pancreatic lesions
Volume (cm^{3}) | Fractional standard uncertainty (%) | |||
---|---|---|---|---|
Due to voxelization | Due to resolution | Combined | ||
Liver lesion | 13.9 | 19.1 | 54.4 | 57.6 |
Pancreatic lesion | 142.0 | 8.8 | 25.1 | 26.6 |
Recovery coefficient
Recovery curve fit parameters and associated standard uncertainties
Parameter | Value | Standard uncertainty | Fractional standard uncertainty (%) |
---|---|---|---|
b _{1} | 21.1 ml | 1.2 ml | 5.8 |
b _{2} | 1.06 | 0.06 | 5.6 |
Recovery coefficients and associated standard uncertainties with and without the volume component for liver and pancreatic lesions
R | u(R)_{[b]}/R | u(R)_{[b,v]}/R | |
---|---|---|---|
Liver lesion | 0.39 | 4.3% | 37.4% |
Pancreatic lesion | 0.88 | 1.4% | 3.6% |
Count rate
Count rates and associated standard uncertainties for liver and pancreatic lesions
VOI | Count rate (cps) | Fractional standard uncertainty (%) | ||
---|---|---|---|---|
Scan 1 | Scan 2 | Scan 3 | ||
Liver lesion | 56.8 | 23.2 | 18.0 | 58.6 |
Pancreatic lesion | 865 | 480 | 292 | 13.6 |
Covariance values of count rate and recovery coefficient at each scan for the liver and pancreatic lesions
Scan | u(R,C) | |
---|---|---|
Liver lesion | Pancreatic lesion | |
1 | 4.84 | 3.42 |
2 | 1.98 | 1.90 |
3 | 1.54 | 1.16 |
Calibration
Activity
^{90}Y activities and associated fractional standard uncertainty for liver and pancreatic lesions
VOI | ^{90}Y activity (MBq) | Fractional standard uncertainty (%) | ||
---|---|---|---|---|
Scan 1 | Scan 2 | Scan 3 | ||
Liver lesion | 13.1 | 5.3 | 4.0 | 22.1 |
Pancreatic lesion | 88.3 | 48.2 | 29.0 | 10.9 |
TAC fitting
TAC parameters and associated standard uncertainties for liver and pancreatic lesions
Liver lesion | Pancreatic lesion | |||||
---|---|---|---|---|---|---|
Fitted value | Standard uncertainty | Fractional uncertainty (%) | Fitted value | Standard uncertainty | Fractional uncertainty (%) | |
A_{0} (fitting) (MBq) | 19.6 | 5.10 | 26.1 | 141.2 | 0.2 | 0.1 |
λ (fitting) (h^{−1}) | 0.026 | 0.008 | 32.4 | 0.0238 | 3.0 × 10^{−5} | 0.1 |
Cumulated activity
Covariance and gradient matrices used to calculate random (fitting) component of uncertainty in cumulated activity for liver and pancreatic lesions
Covariance matrix V_{p} | Gradient matrix g_{p} | Fractional standard uncertainty in \( \overset{\sim }{A} \) (%) | |
---|---|---|---|
Liver lesion | \( \left[\begin{array}{cc}26.1& 3.77\times {10}^{-2}\\ {}3.77\times {10}^{-2}& 6.91\times {10}^{-5}\end{array}\right] \) | \( \left[\begin{array}{c}39.00\\ {}-29722\end{array}\right] \) | 15.1 |
Pancreatic lesion | \( \left[\begin{array}{cc}0.0240& 4.66\times {10}^{-6}\\ {}4.66\times {10}^{-6}& 1.15\times {10}^{-9}\end{array}\right] \) | \( \left[\begin{array}{c}42.0\\ {}-2.49\times {10}^5\end{array}\right] \) | 0.1 |
Cumulated activity and associated components of standard uncertainties for liver and pancreatic lesions
Liver lesion | Pancreatic lesion | |||||
---|---|---|---|---|---|---|
Value (MBq h) | \( \mathrm{u}\left(\overset{\sim }{\mathrm{A}}\right) \) (MBq h) | Fractional uncertainty (%) | Value (MBq h) | \( \mathrm{u}\left(\overset{\sim }{\mathrm{A}}\right) \) (MBq h) | Fractional uncertainty (%) | |
\( \overset{\sim }{A} \) (fitting) | 115 | 15.1 | 4.0 | 0.0678 | ||
\( \overset{\sim }{A} \) (systematic) | 762 | 168 | 22.1 | 5,933 | 644 | 10.9 |
\( \overset{\sim }{A} \) (total) | 204 | 26.7 | 644 | 10.9 |
S-factors
Fit parameters and associated standard uncertainties for S-factor data of unit density spheres
Parameter | Fitted value | Standard uncertainty | Fractional uncertainty (%) |
---|---|---|---|
c _{1} | 0.429 | 3.7 × 10^{−3} | 0.4 |
c _{2} | −0.961 | 5.1 × 10^{−3} | 1.2 |
Summary of VOI S-factor data with standard uncertainties u(S)
S-factor (Gy/MBq h) | Standard uncertainty | Fractional uncertainty (%) | |
---|---|---|---|
Liver lesion | 3.4 × 10^{−2} | 1.9 × 10^{−2} | 55.5 |
Pancreatic lesion | 3.7 × 10^{−3} | 0.9 × 10^{−3} | 25.5 |
Absorbed dose
VOI S-factor data with standard uncertainties u(S)
\( \overset{\sim }{\mathrm{A}} \) | \( \mathrm{u}\left(\overset{\sim }{\mathrm{A}}\right) \) | S (Gy/MBq h) | u(S) (Gy/MBq h) | \( \mathrm{u}\left(\overset{\sim }{\mathrm{A}},\mathrm{S}\right) \) | \( r\left(\overset{\sim }{\mathrm{A}},\mathrm{S}\right) \) | |
---|---|---|---|---|---|---|
Liver lesion | 762 | 203 | 3.4 × 10^{−2} | 1.9 × 10^{−2} | −3.09 | −0.80 |
Pancreatic lesion | 5,932 | 1,046 | 3.7 × 10^{−3} | 0.9 × 10^{−3} | −0.57 | −0.95 |
Absorbed dose parameters and associated standard uncertainties for liver and pancreatic lesions
Absorbed dose (Gy) | Covariance matrix, \( {V}_{\left[\overset{\sim }{A},S\right]} \) | Gradient matrix, \( {g}_{\left[\overset{\sim }{A},S\right]} \) | Fraction uncertainty in \( \overline{D} \) (%) | |
---|---|---|---|---|
Liver lesion | 26.1 | \( \left[\begin{array}{cc}4.16\times {10}^4& -3.09\\ {}-3.09& 3.60\times {10}^{-4}\end{array}\right] \) | \( \left[\begin{array}{c}0.0342\\ {}762\end{array}\right] \) | 37.6 |
Pancreatic lesion | 21.7 | \( \left[\begin{array}{cc}4.15\times {10}^5& -0.572\\ {}-0.572& 8.71\times {10}^{-7}\end{array}\right] \) | \( \left[\begin{array}{c}0.00365\\ {}5932\end{array}\right] \) | 15.6 |
Absorbed doses with standard uncertainties for lesions and normal organs over treatment cycles
Cycle 1 | Cycle 2 | Cycle 3 | Cycle 4 | |||||
---|---|---|---|---|---|---|---|---|
\( \overline{D} \) (Gy) | \( u\left(\overline{D}\right) \) (Gy) | \( \overline{D} \) (Gy) | \( u\left(\overline{D}\right) \) (Gy) | \( \overline{D} \) (Gy) | \( u\left(\overline{D}\right) \) (Gy) | \( \overline{D} \) (Gy) | \( u\left(\overline{D}\right) \) (Gy) | |
Liver lesion | 26.1 | 9.8 | 17.8 | 6.2 | 13.4 | 4.6 | 11.5 | 4.1 |
Pancreatic lesion | 21.7 | 3.4 | 14.3 | 2.2 | 10.8 | 2.2 | 9.9 | 1.8 |
Kidneys | 6.68 | 0.3 | 6.9 | 0.8 | 6.1 | 0.7 | 8.0 | 0.8 |
Spleen | 15.4 | 1.9 | 15.4 | 2.5 | 12.3 | 2.2 | 13.0 | 2.6 |
Liver | 2.54 | 0.6 | 2.4 | 0.2 | 2.1 | 0.1 | 2.0 | 0.2 |
Propagation of uncertainties from a tracer to a therapy study
In the limiting case of infinite biological retention, the ratio of the uncertainties for the tracer and therapeutic agent will be the ratio of the physical half-lives. An uncertainty in the absorbed dose calculated for ^{111}In (physical half-life T_{phys} = 67.3 h) will therefore produce a similar uncertainty in the absorbed dose for a ^{90}Y calculation (T_{phys} = 64.1 h). However, a small uncertainty in, for example, an absorbed dose calculation for ^{68}Ga (T_{phys} = 1.13 h) would propagate by a factor of ~60 to give potentially significant uncertainty in a ^{90}Y calculation.
It is important to note that nonconformance, for example different administered amounts or affinities between diagnostic and therapeutic radiopharmaceuticals, will also introduce additional uncertainties into the prediction [26, 27]. For example, it is assumed that the biokinetics of ^{111}In- and ^{90}Y-DOTATATE are equal, whereas the renal uptake of the indium-labelled compound might be higher [28].
Discussion
The methodology presented allows uncertainty analysis to be incorporated into absorbed dose calculations using the MIRD schema [1], the most widely adopted approach for molecular radiotherapy (MRT) dosimetry. The methodology is based on the recommendations described within the GUM [14] and necessarily involves the formation of covariance matrices for several steps of the dosimetry process.
The main objective of this uncertainty propagation schema is to evaluate the standard uncertainty in absorbed dose to a target. The tasks that directly support that objective are the determination of cumulated activity and the S-factor. The cumulated activity, given by the area under a TAC, is obtained from a sequence of quantitative images. Each activity value is expressed in terms of an observed count rate, a calibration factor and a recovery coefficient. The recovery coefficient is based on a recovery curve derived from multiple phantom scans. The presence of a common calibration and recovery factor in all activity values, and the covariance between volume, recovery and measured count rate, can be considered as a systematic uncertainty applied across all TAC data points, and therefore may be applied directly to cumulated activity. For the effects of uncertainties associated with random components of TAC data, a statistical approach using a “goodness of fit” measure is used.
Within the described schema particular functions are used to fit the acquired data, for example for the TAC, recovery and S-factor models. The choice of these functions is not discussed, and an obvious fit function from theory may not always be known. In this case an optimal function can be found, and uncertainties reduced by using model selection criteria and model averaging [10, 29, 30].
It is suggested that the major factors affecting uncertainty in the absorbed dose originate from the uncertainty in the delineation of the VOI. Two approaches to determine this uncertainty using statistical and analytical methods are presented. In this example an assumption is made that only a single VOI is applied to all datasets. An alternative approach involves the individual delineation of VOIs for each time point, for which the described methods may need to be varied, taking care to account for any commonalities applied across time points.
Propagation of these uncertainties to derive those further along the dosimetry chain requires the covariance between parameters to be evaluated. An understanding of the variation in VOI counts with VOI uncertainty is challenging as there is no prior knowledge of the count distribution. A method for estimating the count distribution is therefore proposed. However, this approach does not model noise or background counts spilling into the VOI. A more rigorous approach would be to determine a function for change in counts versus volume for the dataset being analysed. However, it is considered that the approach suggested is sufficient since it does not overly complicate the methodology or require additional image processing or analysis, which is not available to the wider nuclear medicine community.
An important feature of the schema is that it can be easily implemented using standard nuclear medicine image processing techniques. This feature is demonstrated in the clinical example in which absorbed dose calculations were performed using a standard image processing workstation and a commercial spreadsheet with curve-fitting software. Clinical implementation of this approach clearly demonstrates how different aspects of the dosimetry calculation can influence uncertainty. Uncertainty pertaining to a smaller lesion is clearly affected by the ability to define precisely the lesion volume and can be significant. For larger organs (such as the liver) volume delineation is less significant and the fit to the TAC begins to dominate. The ability to determine the source of larger uncertainties facilitates optimization of dosimetry protocols.
The clinical example given in the appendix demonstrates the importance of uncertainty in reviewing the significance of results. Figure 12 shows the variation in absorbed doses measured in different treatment cycles. With the presence of uncertainties indicated by error bars, it is possible to determine where a significant difference in delivered absorbed dose occurs. If absorbed dose measurements are to be used to aid future treatment (the goal of MRT dosimetry) it is possible that different treatment strategies could be adopted if the absorbed doses delivered are seen to be constant or decrease with sequential cycles. The uncertainty given in the example demonstrates the utility of the guidance to help identify aspects of the calculations that can be addressed to improve accuracy. It is important to note that the scale of uncertainties should be considered in relation to the range of absorbed doses that are delivered from standard administrations.
Whilst the clinical example demonstrates the use of the schema for SPECT-based dosimetry, the methodology can easily be adapted to suit alternative dosimetry protocols (that is, for multiexponential TAC models, external probe counting or 3D dosimetry). However, variations to the proposed schema should always follow the uncertainty guidelines set out by the GUM.
Uncertainty analysis is important for any measured or calculated parameter, whether physical or biological. Such calculations for MRT are rare [5] and leave room for systematic improvement. With the rapid expansion of MRT and an increase in the number of centres performing dosimetry, it is important for adequate interpretation of the data in clinical practice that measurement uncertainties are quoted alongside absorbed dose values. The application of uncertainty analysis may increase the validity of dosimetry results and may become the basis for quality assurance and quality control. Uncertainty analysis may help identify and reduce errors, aiming at an increased likelihood of observing actual dose–response relationships, which in turn would lead to improved treatment regimens.
Notes
Acknowledgments
This guidance document summarizes the views of the Dosimetry Committee of the EANM and reflects recommendations for which the EANM cannot be held responsible. The recommendations should be taken into context of good practice of nuclear medicine and do not substitute for national and international legal or regulatory provisions.
The guidance document was brought to the attention of all other EANM Committees and of national societies of nuclear medicine. The comments and suggestions from the Radiopharmacy Committee and the Dutch and the Italian national societies are highly appreciated and were considered in the development of this guidance document.
Compliance with ethical standards
Conflicts of interest
Jonathan Gear, Katarina Sjögreen Gleisner, and Mark Konijnenberg are members of the EANM Dosimetry Committee, Gerhard Glatting and Glenn Flux are members of the EANM Radiation Protection Committee. All authors declare that they have no conflicts of interest.
Ethical approval
All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the principles of the 1964 Declaration of Helsinki and its later amendments or comparable ethical standards. This article does not describe any studies with animals performed by any of the authors.
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