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Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions ieee. Abstract The estimation and analysis of kinetic parameters in dynamic PET is frequently confounded by tissue heterogeneity and partial volume effects. We propose a new constrained model of dynamic PET to address these limitations. The proposed formulation incorporates an explicit mixture model in which each image voxel is represented as a mixture of different pure tissue types with distinct temporal dynamics. As a result, we propose a constrained formulation of the estimation problem that we solve using a two-stage algorithm. In the first stage, a sparse signal processing method is applied to estimate the rate parameters for the different tissue compartments from the noisy PET time series. In the second stage, tissue fractions and the linear parameters of different time activity curves are estimated using a combination of spatial-regularity and fractional mixture constraints. A block coordinate descent algorithm is combined with a manifold search to robustly estimate these parameters. The method is evaluated with both simulated and experimental dynamic PET data. Compared to conventional static PET, which reconstructs a single image representing the time-averaged spatial tracer distribution, the additional temporal information in dynamic PET provides new capabilities for identifying and differentiating tissues with different metabolic characteristics. However, while this additional dimension of information can be extremely useful, the signal-to-noise ratio SNR per time point is very low compared to static PET, and novel analysis techniques are necessary to extract practically-useful information from the time series of images. Parametric pharmacokinetic models [ 1 ], [ 2 ] are commonly used to analyze PET data, and provide physiologically-meaningful kinetic parameter estimates that can play an important role in disease diagnosis and treament evaluation [ 3 ], [ 4 ]. To compensate for the low SNR in the reconstructed images of dynamic PET, conventional methods used for kinetic parameter estimation often define a region of interest ROI and then perform the estimation based on the ROI average [ 5 ], [ 6 ], [ 7 ], [ 8 ].

Principles and practices.

A project for monitoring trends in burn severity. Fire Ecology, 3 1 , 3— Fire and bark beetle interactions. Key, C. Ecological and sampling constraints on defining landscape fire severity.

Fire Ecology, 2 2 , 34— Landscape assessment LA : Sampling and analysis methods. Lentile, L. Influence of topography and forest structure on patterns of mixed severity fire in ponderosa pine forests of the South Dakota Black Hills, USA. International Journal of Wildland Fire, 15 October , — The method is evaluated with both simulated and experimental dynamic PET data.

Compared to conventional static PET, which reconstructs a single image representing the time-averaged spatial tracer distribution, the additional temporal information in dynamic PET provides new capabilities for identifying and differentiating tissues with different metabolic characteristics. However, while this additional dimension of information can be extremely useful, the signal-to-noise ratio SNR per time point is very low compared to static PET, and novel analysis techniques are necessary to extract practically-useful information from the time series of images.

Parametric pharmacokinetic models [ 1 ], [ 2 ] are commonly used to analyze PET data, and provide physiologically-meaningful kinetic parameter estimates that can play an important role in disease diagnosis and treament evaluation [ 3 ], [ 4 ]. To compensate for the low SNR in the reconstructed images of dynamic PET, conventional methods used for kinetic parameter estimation often define a region of interest ROI and then perform the estimation based on the ROI average [ 5 ], [ 6 ], [ 7 ], [ 8 ].

These approaches make the assumption that the ensemble-average signal from the ROI provides a sufficient summary, and that spatial variations in tracer kinetics within the ROI due to, e.

However, there are many situations where neglecting spatial heterogeneity might be undesirable. For example, biological heterogeneity is a common feature of malignant tumors. This aspect of malignancy has long been known and classically described with histological features and physiological characteristics.

Highly heterogeneous tumors are considered more aggressive with a higher propensity for metastasis or invasion [ 9 ]. Therefore quantification of tumor heterogeneity could prove to be a useful metric for treatment assessment or a predictive indicator for treatment failure [ 10 ]. As a result, there is a need for pharmacokinetic modeling that preserves information about the spatial heterogeneity of the dynamic PET signal.

Some studies have tried to avoid the homogeneous ROI assumption by estimating kinetic parameters voxel by voxel, resulting in a set of parametric images. Since the PET images have low SNR, existing approaches have used additional constraints to stabilize parameter estimation. Examples include the use of spatial smoothness regularization on the estimated parameter images [ 11 ], [ 12 ], the use of Tikhonov regularization to ensure that the estimated parameters are close to physiologically reasonable values [ 13 ], and the use of regularization to encourage the reconstructed parametric images to have high mutual information with reference anatomical MRI images [ 14 ].

The methods described above assume that there is a single kinetic process per voxel, and can work reasonably well when this assumption is valid. However, biological heterogeneity, low spatial resolution, and partial volume effects frequently mean that the signal from each voxel represents a mixture of multiple kinetic processes with distinct dynamic characteristics. As a result, the use of a single kinetic process per voxel can limit performance, particularly in the context of small or highly heterogeneous tumors.

To better model tissue heterogeneity, mixture models have been proposed to model the TAC from each voxel as a linear combination of several basis. Additional simulations not shown demonstrate that the sparse tissue penalty term R TS B becomes more helpful as the true fraction distribution becomes closer to a binary distribution. In addition to 29 , we also apply:. This constraint removes the bilinear scale ambiguity between F and A.

Quadratic smoothness regularization of the tissue fractions A qj , to impose the prior information that tissue contributions will often vary smoothly in standard PET images. Outside this range high bias can occur depending on the amount of zero activity tissue in the FOV. To find the optimal solution to equation 45 , we alternate between estimating F and A using a block coordinate descent BCD approach i. This is because the alternating BCD method does not allow for simultaneous update of A and F , leading to slow convergence if F and A need to be concurrently rescaled.

To overcome this problem, we introduce an additional manifold search into our BCD algorithm to resolve problems of scale ambiguity in this region of the parameter space.

A sketch of the full algorithm is given below:.

The details of our ADMM implementation are given in the appendix. Thus, we calculate. This a small scale quadratic programming problem and is solved using a standard interior point method. We simulated a 2-D scanner with the same geometry as the Siemens mCT scanner with 4 mm detectors. A total of 30 frames with varying frame duration was simulated for a min dynamic scan 8 frames of 15 seconds, 4 frames of 30 seconds, 11 frames of 1 minute, 5 frames of 5 minutes and 2 frames of 10 minutes.

There were a total of million counts across all 30 frames.

Each frame was reconstructed using MAP [ 40 ] reconstruction with 50 iterations of the algorithm. The spatial smoothness regularization parameter of MAP reconstruction was set to 0. The dynamic images were prewhitened for unit variance based on the variance estimated using the method in [ 37 ]. Spatial correlation between voxels was ignored. We simulated two dynamic image sequences, each representing tumors at different stages.

Images of the true fractions are shown in the top halves of Figs. In both simulations, the tumors are embedded in a background of normal tissue, with a large simulated blood vessel at the center of the ROI. The regularization parameters were chosen as follows. Representative estimated tissue fraction images are shown in the bottom halves of Figs. For comparison, we also applied a single tissue model on hand-drawn tumor and normal tissue ROIs marked in red and green respectively in Figs.

Metastatic tumor simulation. The top row corresponds to the true fractions. The bottom row corresponds to the estimated fractions using our proposed method. From left to right, each column corresponds to tumor, normal tissue, blood and necrotic tissue. The red disk shown in a marks the boundary of the tumor ROI used for the single tissue model comparison. The green ellipse in b marks the boundary of the normal tissue ROI. Necrotic tumor simulation. The red circle shown in a marks the boundary of the tumor ROI used for the single tissue model comparison.

Tissue TACs estimated from the metastasic tumor simulation. The error bars show the standard deviation of the estimated TACs. Note the estimated normal tissue curves overlap with each other. Tissue TACs estimated from the necrotic tumor simulation. Our results show that in both cases, by explicitly modelling the tissue heterogeneity, our proposed method provides accurate estimates of the tissue TACs with smaller bias than obtained with the single tissue model, which suffers from low scanner resolution and partial volume effects.

The tumor TAC is estimated more accurately for the necrotic tumor simulation than it was in the metastatic tumor simulation. This is because there are many more tumor-containing voxels in the necrotic tumor simulation. For further quantitative evaluation of performance, we estimate tumor volume by summing the estimated tumor volume based on the fractions obtained from the mixture model.

For comparison, we also estimate tumor volume based on the estimated tissue parameters from the single-tissue model using these values as inputs to the second stage of our proposed algorithm. While the variance of both methods is comparable, the low bias of the mixture model means that it has a much smaller mean-squared error.

Dynamic list mode data were binned into 30 inhomogeneous time frames with increasing frame durations the frame durations were the same as in the simulation , and reconstructed images were generated using MAP reconstruction with a 2 mm voxel size. The dynamic images were prewhitened for unit variance as in the simulations. Spatial correlation between voxels were ignored. The blood input function was acquired by manually segmenting blood signal in the left ventricle. The parameters were chosen as follows.

We chose our hyperparameters in the similar fashion as we did in Section V-A. Results are shown in Fig. Note we only show a 2D transaxial slice the same slice to which the first stage algorithm was applied through the 3D volume.

Arrows show selected lesions in the CT image. Left Estimated tissue fractions from patient data.

Right Fusion images of the tissue fractions and the CT image. From top to bottom, the fraction components are tumor, normal tissue, blood and zero activity tissue. Arrow shows the descending aorta. The CT image in Fig. From the coregistered static PET images in Fig. From the tumor and normal tissue fraction image, we can see enhanced tumor contrast compared to static PET images.

To quantify the difference in contrast, we compute mean values in tumor and normal tissue ROIs Fig. Contrast is defined as the ratio of the mean values in tumor and normal tissue ROIs. The statistics are summarized in Table III , which shows that the proposed method substantially improves the contrast between tumor and normal contrast. Overall the tissue fractions appear consistent with the anatomy and lesions seen in the CT. Note for example the high blood fraction in the abdominal aorta, and the relatively uniform normal tissue regions in the liver where lesions are not visible.

The tumor and zero-activity fractions reveal tumors with active boundaries and necrotic zero-activity cores. The computation time of the two stage algorithm is mainly limited by the first stage where the nonlinear parameters need to be estimated using combinatorial algorithms. The detailed computation time based on a by 2D slice is shown in Table IV.

Because of the high computational cost for first stage, for 3D data we typically compute the nonlinear tissue parameters based on a 2D slice and then apply the second stage of the algorithm to the whole 3D volume.

In this paper we did not study the effects of an inaccurate blood input function. We assume a parametric model 23 is fitted to sample values derived from the dynamic images. This will lead to some errors in the input function a combination of bias and variance. We have not yet explored the relative impact of these errors on our approach vs. Since we jointly estimate two or more sets of kinetic parameters the estimation problem is more illposed as explored in Section III so we can expect somewhat increased sensitivity to errors in the input function.

Fully exploring the impact of this error is beyond the scope of this paper, but is an interesting topic for future work. Here in the mixture model, the problem is more difficult since a more complex model is used. Investigation of the degree of variability and development of methods to produce approximate invariance, analogous to those in [ 37 ] and [ 38 ], is an interesting topic for future work.

In this paper, we proposed and investigated a new approach to estimating PET kinetic parameters with a mixture model.

Theoretical analysis using the CRLB and empirical studies demonstrated that the estimation of mixture model parameters is greatly enhanced by incorporating prior knowledge about sparsity. Based on these findings, we proposed a two stage algorithm that successfully estimates tissue kinetic parameters and tissue fractions for simulated and real data with relatively high noise.

We anticipate that the estimated tissue fractions could be used to measure tumor heterogeneity, which could in turn be used for tumor staging, treatment assessment and optimization.

This appendix describes how we solve equation 47 using the ADMM algorithm as in [ 39 ]. First, by integrating the constraints into the cost function, we note that constrained optimization problem in equation 47 is equivalent to the unconstrained optimization problem in equation In particular, the update procedure is:.

Equation 58 is a quadratic optimization problem with equality constraints for which we used a projected conjugate gradient method. Equation 59 also has a closed form solution, given by:. As a result, each step of the algorithm can be computed quickly. A discussion of the theoretical issues underlying selection of this parameter can be found in [ 42 ] and [ 43 ]. Practically we found that the best parameter for the proposed method was between 10 —6 and 10 —4 for both the simulated and in vivo data.

Justin P. Peter S. Richard M. National Center for Biotechnology Information , U. Author manuscript; available in PMC May 7. Conti , and Richard M. Author information Copyright and License information Disclaimer.

Copyright notice. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions ieee. See other articles in PMC that cite the published article. Abstract The estimation and analysis of kinetic parameters in dynamic PET is frequently confounded by tissue heterogeneity and partial volume effects.

Modeling Assumptions A.

Pharmacokinetic Model This work assumes that tracer pharmacokinetics follow a compartment model [ 27 ] that describes the PET tracer as it interacts with biological tissues and transitions between different states. Kinetics within a homogeneous ROI are governed by rate parameters k 1 , k 2 , k 3 and k 4 and the following ordinary differential equations: Open in a separate window. Mixture Tissue Model The previous section described the kinetic model for a single homogeneous tisue component.

Model stability While the mixture model 7 has been used before e.

Effect of the tissue fraction distribution In this section, we study how the distribution of the mixing coefficients A qj can influence the stability of parameter estimation. In addition, we assume the blood input function satisfies the following parametric model [ 32 ]: Effect of prior knowledge of the tissue fraction distribution The previous section demonstrated the effects of the tissue fraction distribution for a case in which prior information about the PDF of the tissue fraction distribution was unavailable to the estimator.

Given fractions, the likelihood function is: Mixture model parameter Estimation The previous section demonstrated that the use of constraints is important for the stable estimation of mixture model parameters.

Stage 1: Stage 2: