[18F]fluoro-2-deoxy-D-glucose (FDG) is one of the most utilized tracers for positron emission tomography (PET) applications in oncology. FDG-PET relies on higher glycolytic activity in tumors compared to normal structures as the basis of image contrast. As a glucose analog, FDG is transported into malignant cells which typically exhibit an increased radioactivity. However, different from glucose, FDG is not reabsorbed by the renal system and is excreted to the bladder. The present paper describes a novel computational method for the quantitative assessment of this excretion process. The method is based on a compartmental analysis of FDG-PET data in which the excretion process is explicitly accounted for by the bladder compartment and on the application of an ant colony optimization (ACO) algorithm for the determination of the tracer coefficients describing the FDG transport effectiveness. The validation of this approach is performed by means of both synthetic data and real measurements acquired by a PET device for small animals (micro-PET). Possible oncological applications of the results are discussed in the final section.
Positron emission tomography (PET) [
A typical way to assess the FDG excretion in the urine is to compute the average clearance defined as the ratio between the (asymptotic) activity in the bladder and the time integral of the tracer concentration in blood [
The first novelty of our approach is in the kind of compartmental model adopted. Unlike the typical schemes for the study of renal physiology [ As in [ A third compartment, the urine, is localized in the bladder and is characterized by one single input and no corresponding output. The time activity curve (TAC) describes the input of tracer in the system and is determined from regions of interest (ROIs) of the left ventricle drawn on FDG-PET maps at different time steps. Estimates of the tracer concentration for the bladder and for the two-compartment system made of parenchyma and preurine are obtained by means of ROIs including bladder and kidneys, respectively. Six exchange coefficients describe the efficiency of tracer transmission between the different compartments (further exchange coefficients, e.g., the ones describing the circulation for the bladder back to the kidneys, are set to zero for well-established physiological reasons).
The compartmental model adopted in this paper.
From a mathematical viewpoint, the time-dependent concentrations of tracer in each compartment constitute the state variables; the time evolution of the state variables (the kinetics of the system) is modeled by a linear system of ordinary differential equations for the concentrations, expressing the conservation of tracer during flow between compartments; the (constant) coefficients describing the input/output rates of tracer for each compartment, called
The paper content is organized as follows. In Section
The state variables of the three-compartment model adopted in this paper are the tracer concentrations in the tissue (
Conservation of tracer exchanged between compartments leads to the following system of linear ordinary differential equations with constant coefficients:
Let us first assume that
We now suppose that
We then consider the case
Finally, we consider the diagonal case
The model equations obtained in the previous section describe the time behavior of the tracer concentration in the three compartments of the renal system, given the TAC for tracer concentration in blood and the transmission coefficients. Given such equations, compartmental analysis requires the determination of the tracer coefficients by utilizing measurements of the tracer concentrations provided by nuclear imaging; applying an optimization scheme for the solution of the inverse problem.
In nuclear imaging experiments, the reconstructed images can provide information on the tracer concentration in the kidneys and in the bladder as well as in the input arterial blood as measured in the left ventricle. Specifically, an acquisition sequence is set up providing count data sets collected at subsequent time intervals. For each data set, an image reconstruction algorithm is applied, ROIs are drawn within the left ventricle, the kidneys, and the bladder, and the corresponding tracer concentrations are computed. Obviously, the tracer concentration in the kidneys is an estimate of
The minimization of the functional
The implementation of ACO for the optimization of the exchange coefficients in The four ACO parameters are fixed as follows. where The values of the tracer coefficients are initialized to six random numbers picked up in the interval, respectively, The ACO procedure is then run using
In the following section, we show how this statistics-based compartmental analysis works in the case of synthetic data and real measurements recorded by a micro-PET system. In this specific application, the advantages of ACO with respect to deterministic optimization are that it does not suffer local minima and singularities in the functional gradient.
Compartmental analysis is a valid approach to physiological studies of animal models by means of PET data. An “Albira” micro-PET system produced by Carestream Health is currently operational at the IRCCS San Martino IST, Genova, Italy, and experiments with mice are currently performed by using different tracers, mainly for applications to oncology. In this section, we describe the performance of our approach to compartmental analysis in the case of synthetic data simulated by mimicking “Albira” acquisition for FDG-PET experiments. Then, we will describe the results of data analysis for five real experiments performed by using FDG.
In order to produce the synthetic data, we started from six initial values for the tracer coefficients. These selected values generate a matrix
Simulated values of tracer coefficients providing different cases for the matrix
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g. t. | 1 | 0.02 | 0.02 | 0.08 | 0.3 | 0.3 |
ACO |
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LM |
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g. t. | 0.8 | 0 | 0.02 | 0.1 | 0.4 | 0.2 |
ACO |
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LM |
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g. t. | 0.6 | 0.03 | 0 | 0.1 | 0.35 | 0.35 |
ACO |
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g. t. | 0.7 | 0 | 0 | 0.2 | 0.2 | 0.4 |
ACO |
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LM |
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In the first columns: g. t. stands for ground truth, u. t. for upper triangular, l. t. for lower triangular, and diag. for diagonal.
Simulated experiment with a full matrix
The main advantage of using ACO for the reduction of this compartmental model is in the fact that this statistical approach, as most evolutionary methods, is particularly effective in exploring the solution space. This property becomes evident by performing the same analysis of synthetic data by means of a standard least-squares method. Therefore, we have utilized the Levenberg-Marquardt (LM) approach [
We agree that in these tests, the procedure for generating the synthetic concentrations and the one for reconstructing the tracer coefficients from them are based on the same equations (in a sort of “inverse crime” procedure). However, the synthetic data are affected by Poisson noise, and in any case, the aim of these numerical applications was simply to validate the reliability and stability of ACO when applied, for the first time, to a compartmental analysis problem.
We considered five healthy murine models injected with FDG and acquired the corresponding activity by means of a dynamic acquisition paradigm over 27 experimental time points. The images have been reconstructed by applying an expectation-maximization iterative algorithm [
Analysis of real data from one of the murine models. Results obtained with the following ACO parameters:
The results of this analysis for all models are given in Table
Results of the data analysis in the case of 5 murine models. Reconstructed average values and standard deviations over both ACO (30 runs over the same random initialization) and LM (30 different random initializations).
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1ACO |
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1LM |
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2ACO |
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2LM |
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3ACO |
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3LM |
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4ACO |
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4LM |
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5ACO |
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5LM |
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In the first columns, 1ACO indicates the results concerning the first murine model provided by ACO and so on.
Correlation between the average clearances
This paper deals with the renal flow of a radioactive tracer, [18F]-FDG, injected into a mouse. The time evolution of tracer concentrations inside kidneys and from kidneys to bladder has been modeled by a linear system of ordinary differential equations with constant coefficients. The time variation of the total concentration of activity inside kidneys and bladder (essentially, the sum of the solutions) has been estimated through an analysis of micro-PET data. The six constant exchange coefficients, which provide information on FDG metabolism, have been regarded as unknowns. The related inverse problem has been solved by applying an algorithm based on ACO. Resulting applications to real and synthetic data have been shown and discussed also in comparison with the results provided by a Levenberg-Marquardt algorithm.
The mathematical approach described in this paper provides estimates of the six unknown coefficients. Unlike techniques based on graphical analysis, it does not require any distinction between irreversible or reversible uptake of tracer nor identification of a time value after which suitable expressions evaluated from the data become linear in time [
The physiological basis for this study relies on the broad utilization of FDG in the diagnosing and staging of cancer. In fasting patients, this tracer accurately maps the insulin independent glucose metabolism as an index of aggressiveness and growth rate of neoplastic lesions. We agree that MRglc determined by means of, for example, Patlak analysis would provide a reliable quantitative index of glucose consumption. However, these measurements imply the use of dynamic imaging whose long acquisition time (50–60 minutes) would hardly fit with the operational procedures of a PET lab. Accordingly, clinical PET imaging almost always implies the acquisition of only one image at the late (equilibrium) time. Under this condition, only tracer uptake can be measured. SUV is largely used to define cancer glucose consumption [
The financial support of the “Fondazione Cassa di Risparmio di Genova” is kindly acknowledged.