Blind identification of the kinetic parameters in three-compartment models

Optimization-derived blood input function using a kernel method and its evaluation with total-body PET for brain parametric imaging

Dynamic PET allows quantification of physiological parameters through tracer kinetic modeling. For dynamic imaging of brain or head and neck cancer on conventional PET scanners with a short axial field of view, the image-derived input function (ID-IF) from intracranial blood vessels such as the carotid artery (CA) suffers from severe partial volume effects. Alternatively, optimization-derived input function (OD-IF) by the simultaneous estimation (SIME) method does not rely on an ID-IF but derives the input function directly from the data. However, the optimization problem is often highly ill-posed. We proposed a new method that combines the ideas of OD-IF and ID-IF together through a kernel framework. While evaluation of such a method is challenging in human subjects, we used the uEXPLORER total-body PET system that covers major blood pools to provide a reference for validation.

METHODS

The conventional SIME approach estimates an input function using a joint estimation together with kinetic parameters by fitting time activity curves from multiple regions of interests (ROIs). The input function is commonly parameterized with a highly nonlinear model which is difficult to estimate. The proposed kernel SIME method exploits the CA ID-IF as a priori information via a kernel representation to stabilize the SIME approach. The unknown parameters are linear and thus easier to estimate. The proposed method was evaluated using 18F-fluorodeoxyglucose studies with both computer simulations and 20 human-subject scans acquired on the uEXPLORER scanner. The effect of the number of ROIs on kernel SIME was also explored.

RESULTS

The estimated OD-IF by kernel SIME showed a good match with the reference input function and provided more accurate estimation of kinetic parameters for both simulation and human-subject data. The kernel SIME led to the highest correlation coefficient (R = 0.97) and the lowest mean absolute error (MAE = 10.5 %) compared to using the CA ID-IF (R = 0.86, MAE = 108.2 %) and conventional SIME (R = 0.57, MAE = 78.7 %) in the human-subject evaluation. Adding more ROIs improved the overall performance of the kernel SIME method.

CONCLUSION

The proposed kernel SIME method shows promise to provide an accurate estimation of the blood input function and kinetic parameters for brain PET parametric imaging.

An automatic framework for evaluating the vascular permeability of bone metastases from prostate cancer

Objectives.Vascular permeability can reflect tumorigenesis and metastasis. Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) can assess microvascular permeability by pharmacokinetic parameter estimation. Most estimation methods require manually selected arterial input function (AIF) or reference regions. However, the result will be unstable due to the annotation, which relies on personal experience. In this study, we propose an automatic framework for evaluating vascular permeability of bone metastases from prostate cancer without selecting AIF.Materials and methods.This retrospective study comprised of 15 prostate cancer patients with bone metastases. Based on clinical consensus for three typical DCE-MRI curve patterns, three characteristic curves as regularization constraints were introduced to the extended Tofts model (ETM) using clustering strategy, and the clustering-based blind identification of multichannel (CBM) framework was then proposed for pharmacokinetic parameter estimation. With automatic segmentation of the whole bone area, we obtained the estimation of the pharmacokinetic parameters in the bone area and quantified for bone metastases. Two experienced radiologists compared the CBM estimations with the diagnostic results and we compared the estimations with those of the ETM in bone metastasis regions to evaluate the feasibility of the CBM framework.Results.The higher signal regions ofK^transandK_epindicated the metastasis of prostate cancer, which is consistent with the cancer area marked by the radiologists. In addition, theK^transandK_epin bone metastasis regions were significantly higher than in normal bone regions (P < 0.001,P < 0.001). The consistency of estimation by using the CBM framework and conventional ETM method was confirmed by Bland-Altman analysis.Conclusion.The proposed CBM framework can provide a fully automatic and reliable quantitative estimation of vascular permeability for bone metastases in prostate cancer patients.

Robust and efficient pharmacokinetic parameter non-linear least squares estimation for dynamic contrast enhanced MRI of the prostate

PURPOSE

To describe an efficient numerical optimization technique using non-linear least squares to estimate perfusion parameters for the Tofts and extended Tofts models from dynamic contrast enhanced (DCE) MRI data and apply the technique to prostate cancer.

METHODS

Parameters were estimated by fitting the two Tofts-based perfusion models to the acquired data via non-linear least squares. We apply Variable Projection (VP) to convert the fitting problem from a multi-dimensional to a one-dimensional line search to improve computational efficiency and robustness. Using simulation and DCE-MRI studies in twenty patients with suspected prostate cancer, the VP-based solver was compared against the traditional Levenberg-Marquardt (LM) strategy for accuracy, noise amplification, robustness to converge, and computation time.

RESULTS

The simulation demonstrated that VP and LM were both accurate in that the medians closely matched assumed values across typical signal to noise ratio (SNR) levels for both Tofts models. VP and LM showed similar noise sensitivity. Studies using the patient data showed that the VP method reliably converged and matched results from LM with approximate 3× and 2× reductions in computation time for the standard (two-parameter) and extended (three-parameter) Tofts models. While LM failed to converge in 14% of the patient data, VP converged in the ideal 100%.

CONCLUSION

The VP-based method for non-linear least squares estimation of perfusion parameters for prostate MRI is equivalent in accuracy and robustness to noise, while being more reliably (100%) convergent and computationally about 3× (TM) and 2× (ETM) faster than the LM-based method.

Semi-parametric arterial input functions for quantitative dynamic contrast enhanced magnetic resonance imaging in mice

OBJECTIVE

An extension of single- and multi-channel blind deconvolution is presented to improve the estimation of the arterial input function (AIF) in quantitative dynamic contrast enhanced magnetic resonance imaging (DCE-MRI).

METHODS

The Lucy-Richardson expectation-maximization algorithm is used to obtain estimates of the AIF and the tissue residue function (TRF). In the first part of the algorithm, nonparametric estimates of the AIF and TRF are obtained. In the second part, the decaying part of the AIF is approximated by three decaying exponential functions with the same delay, giving an almost noise free semi-parametric AIF. Simultaneously, the TRF is approximated using the adiabatic approximation of the Johnson-Wilson (aaJW) pharmacokinetic model.

RESULTS

In simulations and tests on real data, use of this AIF gave perfusion values close to those obtained with the corresponding previously published nonparametric AIF, and are more noise robust.

CONCLUSION

When used subsequently in voxelwise perfusion analysis, these semi-parametric AIFs should give more correct perfusion analysis maps less affected by recording noise than the corresponding nonparametric AIFs, and AIFs obtained from arteries.

SIGNIFICANCE

This paper presents a method to increase the noise robustness in the estimation of the perfusion parameter values in DCE-MRI.

Fundamentals of tracer kinetics for dynamic contrast-enhanced MRI

Tracer kinetic methods employed for quantitative analysis of dynamic contrast-enhanced (DCE) magnetic resonance imaging (MRI) share common roots with earlier tracer studies involving arterial-venous sampling and other dynamic imaging modalities. This article reviews the essential foundation concepts and principles in tracer kinetics that are relevant to DCE MRI, including the notions of impulse response and convolution, which are central to the analysis of DCE MRI data. We further examine the formulation and solutions of various compartmental models frequently used in the literature. Topics of recent interest in the processing of DCE MRI data, such as the account of water exchange and the use of reference tissue methods to obviate the measurement of an arterial input, are also discussed. Although the primary focus of this review is on the tracer models and methods for T(1) -weighted DCE MRI, some of these concepts and methods are also applicable for analysis of dynamic susceptibility contrast-enhanced MRI data.

Sensitivity analysis for biomedical models

This article discusses the application of sensitivity analysis (SA) in biomedical models. Sensitivity analysis is widely applied in physics, chemistry, economics, social sciences and other areas where models are developed. By assigning a prior probability distribution to each model variable, the SA framework appeals to the posterior probabilities of the model to evaluate the relative importance of these variables on the output distribution based on the principle of general variance decomposition. Within this framework, the SA paradigm serves as an objective platform to quantify the contributions of each model factor relative to their empirical range. We present statistical derivations of variance-based SA in this context and discuss its detailed properties through some practical examples. Our emphasis is on the application of SA in the biomedical field. As we show, it may provide a useful tool for model quality assessment, model reduction and factor prioritization, and improve our understanding of the model structure and underlying mechanisms. When usual approaches for calculating sensitivity index involve the employment of Monte Carlo analysis, which is computationally expensive in the large-sampling paradigm, we develop two effective numerical approximate methods for quick SA evaluations based on the unscented transformation (UT) that utilize a deterministic sampling approach in place of random sampling to calculate posterior statistics. We show that these methods achieve an excellent compromise between computational burden and calculation precision. In addition, a clear guideline is absent to evaluate the importance of variable for model reduction, we also present an objective statistical criterion to quantitatively decide whether or not a descriptive parameter is nominal and may be discarded in ensuing model-based analysis without significant loss of information on model behavior.

Dynamic single photon emission computed tomography--basic principles and cardiac applications

The very nature of nuclear medicine, the visual representation of injected radiopharmaceuticals, implies imaging of dynamic processes such as the uptake and wash-out of radiotracers from body organs. For years, nuclear medicine has been touted as the modality of choice for evaluating function in health and disease. This evaluation is greatly enhanced using single photon emission computed tomography (SPECT), which permits three-dimensional (3D) visualization of tracer distributions in the body. However, to fully realize the potential of the technique requires the imaging of in vivo dynamic processes of flow and metabolism. Tissue motion and deformation must also be addressed. Absolute quantification of these dynamic processes in the body has the potential to improve diagnosis. This paper presents a review of advancements toward the realization of the potential of dynamic SPECT imaging and a brief history of the development of the instrumentation. A major portion of the paper is devoted to the review of special data processing methods that have been developed for extracting kinetics from dynamic cardiac SPECT data acquired using rotating detector heads that move as radiopharmaceuticals exchange between biological compartments. Recent developments in multi-resolution spatiotemporal methods enable one to estimate kinetic parameters of compartment models of dynamic processes using data acquired from a single camera head with slow gantry rotation. The estimation of kinetic parameters directly from projection measurements improves bias and variance over the conventional method of first reconstructing 3D dynamic images, generating time-activity curves from selected regions of interest and then estimating the kinetic parameters from the generated time-activity curves. Although the potential applications of SPECT for imaging dynamic processes have not been fully realized in the clinic, it is hoped that this review illuminates the potential of SPECT for dynamic imaging, especially in light of new developments that enable measurement of dynamic processes directly from projection measurements.

A model-constrained Monte Carlo method for blind arterial input function estimation in dynamic contrast-enhanced MRI: II. In vivo results

Accurate quantification of pharmacokinetic model parameters in tracer kinetic imaging experiments requires correspondingly accurate determination of the arterial input function (AIF). Despite significant effort expended on methods of directly measuring patient-specific AIFs in modalities as diverse as dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI), dynamic positron emission tomography (PET), and perfusion computed tomography (CT), fundamental and technical difficulties have made consistent and reliable achievement of that goal elusive. Here, we validate a new algorithm for AIF determination, the Monte Carlo blind estimation (MCBE) method (which is described in detail and characterized by extensive simulations in a companion paper), by comparing AIFs measured in DCE-MRI studies of eight brain tumor patients with results of blind estimation. Blind AIFs calculated with the MCBE method using a pool of concentration-time curves from a region of normal brain tissue were found to be quite similar to the measured AIFs, with statistically significant decreases in fit residuals observed in six of eight patients. Biases between the blind and measured pharmacokinetic parameters were the dominant source of error. Averaged over all eight patients, the mean biases were +7% in K(trans), 0% in k(ep), -11% in v(p) and +10% in v(e). Corresponding uncertainties (median absolute deviation from the best fit line) were 0.0043 min(-1) in K(trans), 0.0491 min(-1) in k(ep), 0.29% in v(p) and 0.45% in v(e). The use of a published population-averaged AIF resulted in larger mean biases in three of the four parameters (-23% in K(trans), -22% in k(ep), -63% in v(p)), with the bias in v(e) unchanged, and led to larger uncertainties in all four parameters (0.0083 min(-1) in K(trans), 0.1038 min(-1) in k(ep), 0.31% in v(p) and 0.95% in v(e)). When blind AIFs were calculated from a region of tumor tissue, statistically significant decreases in fit residuals were observed in all eight patients despite larger deviations of these blind AIFs from the measured AIFs. The observed decrease in root-mean-square fit residuals between the normal brain and tumor tissue blind AIFs suggests that the local blood supply in tumors is measurably different from that in normal brain tissue and that the proposed method is able to discriminate between the two. We have shown the feasibility of applying the MCBE algorithm to DCE-MRI data acquired in brain, finding generally good agreement with measured AIFs and decreased biases and uncertainties relative to the use of a population-averaged AIF. These results demonstrate that the MCBE algorithm is a useful alternative to direct AIF measurement in cases where acquisition of high-quality arterial input function data is difficult or impossible.

A model-constrained Monte Carlo method for blind arterial input function estimation in dynamic contrast-enhanced MRI: I. Simulations

Widespread adoption of quantitative pharmacokinetic modeling methods in conjunction with dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) has led to increased recognition of the importance of obtaining accurate patient-specific arterial input function (AIF) measurements. Ideally, DCE-MRI studies use an AIF directly measured in an artery local to the tissue of interest, along with measured tissue concentration curves, to quantitatively determine pharmacokinetic parameters. However, the numerous technical and practical difficulties associated with AIF measurement have made the use of population-averaged AIF data a popular, if sub-optimal, alternative to AIF measurement. In this work, we present and characterize a new algorithm for determining the AIF solely from the measured tissue concentration curves. This Monte Carlo blind estimation (MCBE) algorithm estimates the AIF from the subsets of D concentration-time curves drawn from a larger pool of M candidate curves via nonlinear optimization, doing so for multiple (Q) subsets and statistically averaging these repeated estimates. The MCBE algorithm can be viewed as a generalization of previously published methods that employ clustering of concentration-time curves and only estimate the AIF once. Extensive computer simulations were performed over physiologically and experimentally realistic ranges of imaging and tissue parameters, and the impact of choosing different values of D and Q was investigated. We found the algorithm to be robust, computationally efficient and capable of accurately estimating the AIF even for relatively high noise levels, long sampling intervals and low diversity of tissue curves. With the incorporation of bootstrapping initialization, we further demonstrated the ability to blindly estimate AIFs that deviate substantially in shape from the population-averaged initial guess. Pharmacokinetic parameter estimates for K(trans), k(ep), v(p) and v(e) all showed relative biases and uncertainties of less than 10% for measurements having a temporal sampling rate of 4 s and a concentration measurement noise level of sigma = 0.04 mM. A companion paper discusses the application of the MCBE algorithm to DCE-MRI data acquired in eight patients with malignant brain tumors.

Reproducibility assessment of a multiple reference tissue method for quantitative dynamic contrast enhanced-MRI analysis

Bone metastases of 16 prostate cancer patients were scanned twice 1 week apart by dynamic contrast enhanced (DCE)-MRI at 2-s resolution using a two-dimensional gradient-echo pulse sequence. With a multiple reference tissue method (MRTM), the local tissue arterial input function (AIF) was estimated using the contrast agent enhancement data from tumor subregions and muscle. The 32 individual AIFs estimated by the MRTM, which had considerable intra-patient and inter-patient variability, were similar to directly measured AIFs in the literature and using the MRTM AIFs in a pharmacokinetic model to derive estimated individual cardiac outputs provided physiologically reasonable results. The MRTM individual AIFs gave better fits with smaller sum of squared errors and equally reproducible estimate of kinetic parameters compared with a previous reported population AIF measured from remote arteries. The individual MRTM AIFs were also used to obtain a mean local tissue AIF for the unique population of this study, which further improved the reproducibility of the estimated kinetic parameters. The MRTM can be applied to DCE-MRI studies of bone metastases from prostate cancers to provide an AIF from which reproducible quantitative DCE-MRI parameters can be derived, thus help standardize DCE-MRI studies in cancer patients.

An activity-subspace approach for estimating the integrated input function and relative distribution volume in PET parametric imaging

Dynamic positron emission tomography (PET) imaging technique enables the measurement of neuroreceptor distributions corresponding to anatomic structures, and thus, allows image-wide quantification of physiological and biochemical parameters. Accurate quantification of the concentration of neuroreceptor has been the objective of many research efforts. Compartment modeling is the most widely used approach for receptor binding studies. However, current compartment-model-based methods often either require intrusive collection of accurate arterial blood measurements as the input function, or assume the existence of a reference region. To obviate the need for the input function or a reference region, in this paper, we propose to estimate the input function. We propose a novel concept of activity subspace, and estimate the input function by the analysis of the intersection of the activity subspaces. Then, the input function and the distribution volume (DV) parameter are refined and estimated iteratively. Thus, the underlying parametric image of the total DV is obtained. The proposed method is compared with a blind estimation method, iterative quadratic maximum-likelihood (IQML) via simulation, and the proposed method outperforms IQML. The proposed method is also evaluated in a brain PET dataset.

Multiple reference tissue method for contrast agent arterial input function estimation

A precise contrast agent (CA) arterial input function (AIF) is important for accurate quantitative analysis of dynamic contrast-enhanced (DCE)-MRI. This paper proposes a method to estimate the AIF using the dynamic data from multiple reference tissues, assuming that their AIFs have the same shape, with a possible difference in bolus arrival time. By minimizing a cost function, one can simultaneously estimate the parameters and underlying AIF of the reference tissues. The method is computationally efficient and the estimated AIF is smooth and can have higher temporal resolution than the original data. Simulations suggest that this method can provide a reliable estimate of the AIF for DCE-MRI data with a moderate signal-to-noise ratio (SNR) and temporal resolution, and its performance increases significantly as the SNR and temporal resolution increase. As demonstrated by its clinical application, sufficient reference tissues can be easily obtained from normal tissues and subregions segmented from a tumor region of interest (ROI), which suggests this method can be generally applied to cancer-based DCE-MRI studies to estimate the AIF. This method is applicable to general kinetic models in DCE-MRI, as well as other CE imaging modalities.

Bayesian methods for pharmacokinetic models in dynamic contrast-enhanced magnetic resonance imaging

This paper proposes a new method for estimating kinetic parameters of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) based on adaptive Gaussian Markov random fields. Kinetic parameter estimates using neighboring voxels reduce the observed variability in local tumor regions while preserving sharp transitions between heterogeneous tissue boundaries. Asymptotic results for standard errors from likelihood-based nonlinear regression are compared with those derived from the posterior distribution using Bayesian estimation with and without neighborhood information. Application of the method to the analysis of breast tumors based on kinetic parameters has shown that the use of Bayesian analysis combined with adaptive Gaussian Markov random fields provides improved convergence behavior and more consistent morphological and functional statistics.