MIRD Pamphlet No. 30: MIRDfit—A Tool for Fitting of Biodistribution Time–Activity Data for Internal Dosimetry

Study Setup

The study setup module specifies the radionuclide of interest and allows the selection of a source region list. Preset source region lists for the International Commission on Radiological Protection (ICRP) reference phantom are available (13–15). A utility to define tumor source regions is also provided, which allows tumor volume and composition to be specified easily. Alternatively, the user may manually define a source region list.

Biodistribution Input

On selection of a source region, the user may enter the respective biokinetic data; the source region uptake values may be expressed as percentage of administered activity decay-corrected to time of administration (i.e., reflecting biologic uptake and clearance [biologic percentage injected activity]) or as non–decay-corrected values (i.e., reflecting effective uptake and clearance [effective percentage injected activity]). The times after administration of the uptake values must be entered in units of hours. Notably, the uptake values can be obtained from various measurements, including regions or volumes of interest defined on images, γ-counting of blood samples or biopsy samples, dissective biodistribution measurements (e.g., for small-animal dosimetry), in vitro cellular assays (e.g., for cell-level dosimetry), or other methods tailored to the user’s specific needs.

Optionally, for each datum the user may supply uncertainties that define the weighting factors used in weighted least-squares regression. The uncertainties may be input either as the SD of each observation or as fractional SD (i.e., coefficient of variation); in these cases, the weighting factors are calculated as the inverse of the variance. The weights for each datum may also be entered directly. If the user opts to ignore these inputs, a default fractional SD of 0.1 will be assigned for the purposes of data weighting, as the mathematics of curve fitting requires some assignment of weight.

Trapezoidal Integration

The trapezoidal method generally involves 3 steps: the first is to approximate the area under the curve between successive measurements as trapezoidal areas, the second is to estimate the area under the curve between the time of administration of the radiopharmaceutical and the time of the first measurement, and the third is to estimate the area under the curve between the final time of measurement and the end of the integration period (commonly taken as infinity). For steps 2 and 3, various built-in assumptions may be applied. For example, elimination of activity beyond the last measurement time point is often assumed to occur by radioactive decay only; however, sometimes a linear or an exponential function is fit to the last 2 time points. Numerous methods for estimating these contributions to the total area under the curve are user-selectable in MIRDfit. Because of the subjective nature of these choices, MIRDfit provides no estimate of the uncertainty of the TIAC result for the trapezoidal method.

The trapezoidal method has been implemented in MIRDfit for several reasons. First, it continues to be widely used in the field (Fig. 1). Second, it provides a robust and quickly computable initial estimate for the TIAC that is useful for visualization and QC of regression results. Finally, it may be useful in the event that the time–activity curve cannot be reliably fit by any mathematic model.

Regression-Based Integration

Radiopharmaceutical biokinetics are often adequately described by simple sums of exponential functions. The nuclear medicine–appropriate exponential models included in MIRDfit are given in Table 1. Models become available to the user once the appropriate number of observations (time–activity data values) have been entered into the biodistribution input module. The user may select any or all available models to carry forward for regression analysis.

MIRDfit implements the SOLVER and SOLVERSTAT Excel add-ins for weighted least-squares regression and statistics analysis; these add-ins are described in detail elsewhere (16). SOLVER and SOLVERSTAT have been in use for over a decade and have been extensively tested and validated. Briefly, SOLVER is an optimizer that, in the present case, finds values of model parameters that minimize the sum of squares of weighted residuals. SOLVER’s generalized reduced gradient nonlinear optimization method is used in MIRDfit. SOLVERSTAT provides statistical analysis of the result, including the estimates of parameter precision, covariances, correlation, and discrimination between different models. Minor modifications to the SOLVERSTAT Visual Basic for Applications code were made to enable automation, but no changes were made to its order of operations or formulae. Using the SOLVER-optimized parameters, MIRDfit computes TIACs using the analytic expressions for the integrals of the fit functions. MIRDfit computes the SE of the TIAC via error propagation using the SEs of the parameters and covariances output by SOLVERSTAT.

For each candidate model, MIRDfit reports the TIAC, the TIAC uncertainty, and metrics for evaluation of goodness of the fit and model discrimination. Visual aids for identifying potential issues are also provided. If there are at least 2 more observations than model parameters (i.e., ≥2 degrees of freedom), model discrimination can be performed using the corrected Akaike information criterion (AICc). MIRDfit adopts the approach of the NUKFIT software (4) for model discrimination. Briefly, the AICc is a reproducible and objective metric for model discrimination (17). It balances the goodness of the fit with the simplicity of the model; that is, both overfitting (excess parameters in a model) and underfitting (discrepancy between the optimized model and data) are penalized. MIRDfit computes the Akaike weight (4,18), , for each candidate model from the respective AICc: Eq. 1where is the corrected AIC for the ith model, is the minimum AICc of all candidate models, and n is the number of candidate models being compared. The Akaike weight indicates the relative probability that a particular model is the best model; the Akaike weights are used for model averaging if multiple models are supported by the data.

MIRDfit Shelf (Saved Fit Results)

It is the responsibility of the user to perform reasonable QC and select the best fit based on the corrected AIC and other fit metrics, knowledge of realistic values for model parameters, distribution of weighted residuals, and other relevant factors. The user may elect to use a single fit result (e.g., if one model is clearly superior) or model averaging (e.g., if multiple models are supported by the data). Alternatively, the trapezoidal method result may be used (e.g., if no model adequately fits the data). For each source region, the user’s choices are saved in a table designated the shelf, so that the complete set of source region TIACs can be reviewed and exported once the time–activity data for all desired source regions have been fit. The exported results are formatted for immediate import into the MIRDcalc software for dosimetry calculations.

Voiding Bladder Module

The voiding bladder model of Cloutier et al. (19–21) is implemented as a module of MIRDfit. This model allows the TIAC for the urinary bladder content source region to be estimated by a whole-body retention curve fit and an assumed regular voiding schedule.

Source Region Visualization

A source region display widget has been implemented as an educational and QC feature. This widget displays a graphic of selected source regions of ICRP anthropomorphic phantoms; it may assist users in, for example, confirming that the appropriate source region is selected or defining volumes of interest in tomographic images such that they match the source regions of the phantoms appropriately.