Precision analysis of kinetic modelling estimates in dynamic contrast enhanced MRI

The Potential of Phase Constraints for Non-Fourier Radiofrequency-Encoded MRI

In modern magnetic resonance imaging, it is common to use phase constraints to reduce sampling requirements along Fourier-encoded spatial dimensions. In this work, we investigate whether phase constraints might also be beneficial to reduce sampling requirements along spatial dimensions that are measured using non-Fourier encoding techniques, with direct relevance to approaches that use tailored spatially-selective radiofrequency (RF) pulses to perform spatial encoding along the slice dimension in a 3D imaging experiment. In the first part of the paper, we use the Cramér-Rao lower bound to examine the potential estimation theoretic benefits of using phase constraints. The results suggest that phase constraints can be used to improve experimental efficiency and enable acceleration, but only if the RF encoding matrix is complex-valued and appropriately designed. In the second part of the paper, we use simulations of RF-encoded data to test the benefits of phase constraints combined with optimized RF-encodings, and find that the theoretical benefits are indeed borne out empirically. These results provide new insights into the potential benefits of phase constraints for RF-encoded data, and provide a solid theoretical foundation for future practical explorations.

On Ambiguity in Linear Inverse Problems: Entrywise Bounds on Nearly Data-Consistent Solutions and Entrywise Condition Numbers

Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree of ambiguity that may exist about its solution. Traditional measures of ill-posedness, such as the condition number of a matrix, provide characterizations that are global in nature. While such characterizations can be powerful, they can also fail to provide full insight into situations where certain entries of the solution vector are more or less ambiguous than others. In this work, we derive novel theoretical lower- and upper-bounds that apply to individual entries of the solution vector, and are valid for all potential solution vectors that are nearly data-consistent. These bounds are agnostic to the noise statistics and the specific method used to solve the inverse problem, and are also shown to be tight. In addition, our results also lead us to introduce an entrywise version of the traditional condition number, which provides a substantially more nuanced characterization of scenarios where certain elements of the solution vector are less sensitive to perturbations than others. Our results are illustrated in an application to magnetic resonance imaging reconstruction, and we include discussions of practical computation methods for large-scale inverse problems, connections between our new theory and the traditional Cramér-Rao bound under statistical modeling assumptions, and potential extensions to cases involving constraints beyond just data-consistency.

A multi-modality medical imaging head and neck phantom: Part 2. Medical imaging

The head and neck phantom discussed in an accompanying paper (part 1), is imaged with MRI, X-ray CT, PET and ultrasound. MRI scans show a distinct image contrast between the brain compartment and other anatomical regions of the head. The silicone matrix that was used to create a porous brain compartment has a relatively high proton density and a spin-spin relaxation time (T₂) that is long enough to provide an MRI signal. While the longitudinal magnetization was found to recover according to a mono-exponential, a bi-exponential decay was observed for the transverse relaxation with a slow T₂ relaxation component corresponding to the perfusate and a fast T₂ relaxation component corresponding to the silicone. The fraction of the slow T₂ relaxation component increases upon perfusion. A dynamic contrast enhanced (DCE) MRI experiment is conducted in which the injection rate of the contrast agent is varied. Parametric DCE maps are created and reveal regional differences in contrast agent kinetics as a result of differences in porosity. The skull, vertebra and the brain compartment are clearly visible on X-ray CT. Dynamic PET scanning has been performed while the carotic arterial input function is monitored by use of a Geiger-Müller counter. Similar regions of perfusion are found in the PET study as in the DCE MRI study. By doping the perfusate with a lipid micelle emulsion, the phantom is applicable for carotic Doppler ultrasound demonstration and validation.

Efficient DCE-MRI Parameter and Uncertainty Estimation Using a Neural Network

Quantitative DCE-MRI provides voxel-wise estimates of tracer-kinetic parameters that are valuable in the assessment of health and disease. These maps suffer from many known sources of variability. This variability is expensive to compute using current methods, and is typically not reported. Here, we demonstrate a novel approach for simultaneous estimation of tracer-kinetic parameters and their uncertainty due to intrinsic characteristics of the tracer-kinetic model, with very low computation time. We train and use a neural network to estimate the approximate joint posterior distribution of tracer-kinetic parameters. Uncertainties are estimated for each voxel and are specific to the patient, exam, and lesion. We demonstrate the methods' ability to produce accurate tracer-kinetic maps. We compare predicted parameter ranges with uncertainties introduced by noise and by differences in post-processing in a digital reference object. The predicted parameter ranges correlate well with tracer-kinetic parameter ranges observed across different noise realizations and regression algorithms. We also demonstrate the value of this approach to differentiate significant from insignificant changes in brain tumor pharmacokinetics over time. This is achieved by enforcing consistency in resolving model singularities in the applied tracer-kinetic model.

Impact of (k,t) sampling on DCE MRI tracer kinetic parameter estimation in digital reference objects

Purpose

To evaluate the impact of (k,t) data sampling on the variance of tracer‐kinetic parameter (TK) estimation in high‐resolution whole‐brain dynamic contrast enhanced magnetic resonance imaging (DCE‐MRI) using digital reference objects. We study this in the context of TK model constraints, and in the absence of other constraints.

Methods

Three anatomically and physiologically realistic brain‐tumor digital reference objects were generated. Data sampling strategies included uniform and variable density; zone‐based, lattice, pseudo‐random, and pseudo‐radial; with 50‐time frames and 4‐fold to 25‐fold undersampling. In all cases, we assume a fully sampled first time frame, and prior knowledge of the arterial input function. TK parameters were estimated by indirect estimation (i.e., image‐time‐series reconstruction followed by model fitting), and direct estimation from the under‐sampled data. We evaluated methods based on the Cramér‐Rao bound and Monte‐Carlo simulations, over the range of signal‐to‐noise ratio (SNR) seen in clinical brain DCE‐MRI.

Results

Lattice‐based sampling provided the lowest SDs, followed by pseudo‐random, pseudo‐radial, and zone‐based. This ranking was consistent for the Patlak and extended Tofts model. Pseudo‐random sampling resulted in 19% higher averaged SD compared to lattice‐based sampling. Zone‐based sampling resulted in substantially higher SD at undersampling factors above 10. CRB analysis showed only a small difference between uniform and variable density for both lattice‐based and pseudo‐random sampling up to undersampling factors of 25.

Conclusion

Lattice sampling provided the lowest SDs, although the differences between sampling schemes were not substantial at low undersampling factors. The differences between lattice‐based and pseudo‐random sampling strategies with both uniform and variable density were within the range of error induced by other sources, at up to 25‐fold undersampling.

Spatially regularized estimation of the tissue homogeneity model parameters in DCE-MRI using proximal minimization

PURPOSE

The Tofts and the extended Tofts models are the pharmacokinetic models commonly used in dynamic contrast-enhanced MRI (DCE-MRI) perfusion analysis, although they do not provide two important biological markers, namely, the plasma flow and the permeability-surface area product. Estimates of such markers are possible using advanced pharmacokinetic models describing the vascular distribution phase, such as the tissue homogeneity model. However, the disadvantage of the advanced models lies in biased and uncertain estimates, especially when the estimates are computed voxelwise. The goal of this work is to improve the reliability of the estimates by including information from neighboring voxels.

THEORY AND METHODS

Information from the neighboring voxels is incorporated in the estimation process through spatial regularization in the form of total variation. The spatial regularization is applied on five maps of perfusion parameters estimated using the tissue homogeneity model. Since the total variation is not differentiable, two proximal techniques of convex optimization are used to solve the problem numerically.

RESULTS

The proposed algorithm helps to reduce noise in the estimated perfusion-parameter maps together with improving accuracy of the estimates. These conclusions are proved using a numerical phantom. In addition, experiments on real data show improved spatial consistency and readability of perfusion maps without considerable lowering of the quality of fit.

CONCLUSION

The reliability of the DCE-MRI perfusion analysis using the tissue homogeneity model can be improved by employing spatial regularization. The proposed utilization of modern optimization techniques implies only slightly higher computational costs compared to the standard approach without spatial regularization.

OEDIPUS: An Experiment Design Framework for Sparsity-Constrained MRI

This paper introduces a new estimation-theoretic framework for experiment design in the context of MR image reconstruction under sparsity constraints. The new framework is called OEDIPUS (Oracle-based Experiment Design for Imaging Parsimoniously Under Sparsity constraints) and is based on combining the constrained Cramér-Rao bound with classical experiment design techniques. Compared to popular random sampling approaches, OEDIPUS is fully deterministic and automatically tailors the sampling pattern to the specific imaging context of interest (i.e., accounting for coil geometry, anatomy, image contrast, etc.). OEDIPUS-based experiment designs are evaluated using retrospectively subsampled in vivo MRI data in several different contexts. The results demonstrate that OEDIPUS-based experiment designs have some desirable characteristics relative to conventional MRI sampling approaches.

Fisher information and Cramér-Rao lower bound for experimental design in parallel imaging

PURPOSE

The Cramér-Rao lower bound (CRLB) is widely used in the design of magnetic resonance (MR) experiments for parameter estimation. Previous work has considered only Gaussian or Rician noise distributions in this calculation. However, the noise distribution for multi-coil acquisitions, such as in parallel imaging, obeys the noncentral χ-distribution under many circumstances. The purpose of this paper is to present the CRLB calculation for parameter estimation from multi-coil acquisitions.

THEORY AND METHODS

We perform explicit calculations of Fisher matrix elements and the associated CRLB for noise distributions following the noncentral χ-distribution. The special case of diffusion kurtosis is examined as an important example. For comparison with analytic results, Monte Carlo (MC) simulations were conducted to evaluate experimental minimum standard deviations (SDs) in the estimation of diffusion kurtosis model parameters. Results were obtained for a range of signal-to-noise ratios (SNRs), and for both the conventional case of Gaussian noise distribution and noncentral χ-distribution with different numbers of coils, m.

RESULTS

At low-to-moderate SNR, the noncentral χ-distribution deviates substantially from the Gaussian distribution. Our results indicate that this departure is more pronounced for larger values of m. As expected, the minimum SDs (i.e., CRLB) in derived diffusion kurtosis model parameters assuming a noncentral χ-distribution provided a closer match to the MC simulations as compared to the Gaussian results.

CONCLUSION

Estimates of minimum variance for parameter estimation and experimental design provided by the CRLB must account for the noncentral χ-distribution of noise in multi-coil acquisitions, especially in the low-to-moderate SNR regime. Magn Reson Med 79:3249-3255, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Direct estimation of tracer-kinetic parameter maps from highly undersampled brain dynamic contrast enhanced MRI

PURPOSE

The purpose of this work was to develop and evaluate a T₁ -weighted dynamic contrast enhanced (DCE) MRI methodology where tracer-kinetic (TK) parameter maps are directly estimated from undersampled (k,t)-space data.

THEORY AND METHODS

The proposed reconstruction involves solving a nonlinear least squares optimization problem that includes explicit use of a full forward model to convert parameter maps to (k,t)-space, utilizing the Patlak TK model. The proposed scheme is compared against an indirect method that creates intermediate images by parallel imaging and compressed sensing before to TK modeling. Thirteen fully sampled brain tumor DCE-MRI scans with 5-second temporal resolution are retrospectively undersampled at rates R = 20, 40, 60, 80, and 100 for each dynamic frame. TK maps are quantitatively compared based on root mean-squared-error (rMSE) and Bland-Altman analysis. The approach is also applied to four prospectively R = 30 undersampled whole-brain DCE-MRI data sets.

RESULTS

In the retrospective study, the proposed method performed statistically better than indirect method at R ≥ 80 for all 13 cases. This approach provided restoration of TK parameter values with less errors in tumor regions of interest, an improvement compared to a state-of-the-art indirect method. Applied prospectively, the proposed method provided whole-brain, high-resolution TK maps with good image quality.

CONCLUSION

Model-based direct estimation of TK maps from k,t-space DCE-MRI data is feasible and is compatible up to 100-fold undersampling. Magn Reson Med 78:1566-1578, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Dynamic contrast-enhanced MRI in mice: an investigation of model parameter uncertainties

PURPOSE

To establish the experimental factors that dominate the uncertainty of hemodynamic parameters in commonly used pharmacokinetic models.

METHODS

By fitting simulation results from a multiregion tissue exchange model (Multiple path, Multiple tracer, Indicator Dilution, 4 region), the precision and accuracy of hemodynamic parameters in dynamic contrast-enhanced MRI with four tracer kinetic models is investigated. The impact of various injection rates as well as imprecise knowledge of the arterial input functions is examined.

RESULTS

Fast injections are beneficial for K(trans) precision within the extended Tofts model and within the two-compartment exchange model but do not affect the other models under investigation. Biases from errors in the arterial input functions are mostly consistent in size and direction for the simple and the extended Tofts model, while they are hardly predictable for the other models. Errors in the hematocrit introduce the greatest loss in parameter accuracy, amounting to an average K(trans) bias of 40% for a 30% overestimation throughout all models.

CONCLUSION

This simulation study allows the detailed inspection of the isolated impact from various experimental conditions on parameter uncertainty. Because parameter uncertainty comparable to human studies was found, this study represents a validation of preclinical dynamic contrast-enhanced MRI for modeling human tumor physiology.

Analyzing the blood-brain barrier: the benefits of medical imaging in research and clinical practice

A dysfunctional BBB is a common feature in a variety of brain disorders, a fact stressing the need for diagnostic tools designed to assess brain vessels' permeability in space and time. Biological research has benefited over the years various means to analyze BBB integrity. The use of biomarkers for improper BBB functionality is abundant. Systemic administration of BBB impermeable tracers can both visualize brain regions characterized by BBB impairment, as well as lead to its quantification. Additionally, locating molecular, physiological content in regions from which it is restricted under normal BBB functionality undoubtedly indicates brain pathology-related BBB disruption. However, in-depth research into the BBB's phenotype demands higher analytical complexity than functional vs. pathological BBB; criteria which biomarker based BBB permeability analyses do not meet. The involvement of accurate and engineering sciences in recent brain research, has led to improvements in the field, in the form of more accurate, sensitive imaging-based methods. Improvements in the spatiotemporal resolution of many imaging modalities and in image processing techniques, make up for the inadequacies of biomarker based analyses. In pre-clinical research, imaging approaches involving invasive procedures, enable microscopic evaluation of BBB integrity, and benefit high levels of sensitivity and accuracy. However, invasive techniques may alter normal physiological function, thus generating a modality-based impact on vessel's permeability, which needs to be corrected for. Non-invasive approaches do not affect proper functionality of the inspected system, but lack in spatiotemporal resolution. Nevertheless, the benefit of medical imaging, even in pre-clinical phases, outweighs its disadvantages. The innovations in pre-clinical imaging and the development of novel processing techniques, have led to their implementation in clinical use as well. Specialized analyses of vessels' permeability add valuable information to standard anatomical inspections which do not take the latter into consideration.

Simulating the effect of input errors on the accuracy of Tofts' pharmacokinetic model parameters

Pharmacokinetic modeling in Dynamic Contrast Enhanced (DCE)-MRI is an elegant and useful method that provides valuable insight into angiogenesis in cancer and inflammatory diseases. Despite its widespread use, the reliability of the model results is still questioned, as many factors hamper the calculation of the model's parameters, resulting in the poor reproducibility and accuracy of the method. Pharmacokinetic modeling relies on the knowledge of inputs such as the Arterial Input Function (AIF) and of the tissue contrast agent concentration, both of which are difficult to accurately measure. Any errors in the measurement of either of the inputs propagate into the calculated pharmacokinetic model parameters (PMPs), and the significance of the effect depends on the source of the measurement error. In this work we systematically investigate the effect of the incorrect estimation of the parameters describing the inputs of the model on the calculated PMPs when using Tofts' model. Furthermore, we analyze the dependence of these errors on the native values of the PMPs. We show that errors on the measurement of the native T1 as well as errors on the parameters describing the initial peak of the AIF have the largest impact on the calculated PMPs. The parameter whose error has the least effect is the one describing the slow decay of the AIF. The effect of input parameter (IP) errors on the calculated PMPs is found to be dependent on the native set of PMPs: this is particularly true for the errors in the flip angle, and for the errors in parameters describing the initial AIF peak. Conversely the effect of T1 and AIF scaling errors on the calculated PMPs is only slightly dependent on the native PMPs.

Assessment of neovascular permeability in a pancreatic tumor model using dynamic contrast-enhanced (DCE) MRI with contrast agents of different molecular weights

OBJECT

We evaluated the relationship of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI)-derived pharmacokinetic parameters and contrast agents with different molecular weights (MW) in a pancreatic tumor mouse model.

MATERIALS AND METHODS

Panc02 tumors were induced in mice at the hind leg. DCE-MRI was performed using Gadolinium (Gd)-based contrast agents with different MW: Gd-DOTA (0.5 kDa), P846 (3.5 kDa), and P792 (6.47 kDa). Quantitative vascular parameters (AUC, K(trans), V(e), and V(p)) were calculated according to a modified Tofts two-compartment model. Values for all contrast groups were compared for tumor and control (muscle) tissues.

RESULTS

Values for K(trans) and V(e) were significantly higher in tumor tissue than in muscle tissue. When comparing contrast agents, lowest absolute K(trans) values were observed using P792. The relative increase in K(trans) in tumor tissue compared with normal tissue was highest after the use of P792. In both tumor and normal tissues, K(trans) decreased with increasing molecular weight of the contrast agent used.

CONCLUSION

It was demonstrated that values for the different DCE-MRI vascular (permeability) parameters are highly dependent on the contrast agent used. Due to their potential to better differentiate tumor from muscle tissue, higher molecular weight contrast agents show promise when evaluating tumors using DCE-MRI.