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Vastly accelerated linear least‐squares fitting with numerical optimization for dual‐input delay‐compensated quantitative liver perfusion mapping
Magnetic Resonance in Medicine
Abstract
Purpose:
To propose an efficient algorithm to perform dual input compartment modeling for generating perfusion maps in the liver.
Methods:
We implemented whole field-of-view linear least squares (LLS) to fit a delay-compensated dual-input single-compartment model to very high temporal resolution (four frames per second) contrast-enhanced 3D liver data, to calculate kinetic parameter maps. Using simulated data and experimental data in healthy subjects and patients, whole-field LLS was compared with the conventional voxel-wise nonlinear least-squares (NLLS) approach in terms of accuracy, performance, and computation time.
Results:
Simulations showed good agreement between LLS and NLLS for a range of kinetic parameters. The whole-field LLS method allowed generating liver perfusion maps approximately 160-fold faster than voxel-wise NLLS, while obtaining similar perfusion parameters.
Conclusions:
Delay-compensated dual-input liver perfusion analysis using whole-field LLS allows generating perfusion maps with a considerable speedup compared with conventional voxel-wise NLLS fitting. Magn Reson Med, 2017. © 2017 International Society for Magnetic Resonance in Medicine.
... N t − 1} the time index with N t as the number of time frames, ξ = (ξ x , ξ y , ξ z ) is voxel index in a volume of (N x, , N y , N z ) voxels along (x, y, z) axis, ∂ t is the time derivative, c a (t) is the tracer concentration of feeding artery (global AIF), LBF is liver blood flow, V(ξ) is the volume fraction of vascular space, and c(ξ, t) is the tracer concentration scalar field. Eq. (1) is a linear equation system for LBF and LBF V , and LBF can be solved using linear least squared method [25]. All the reconstruction is performed using MATLAB R2018a (Natick, Massachusetts: The MathWorks Inc.). ...
... (2) and (3) was solved using linear least squared method [25]. ...
Objective:
We quantify liver perfusion using quantitative transport mapping (QTM) method that is free of arterial input function (AIF). QTM method is validated in a vasculature computational fluid dynamics (CFD) simulation and is applied for processing dynamic contrast enhanced (DCE) MRI images in differentiating liver with nonalcoholic fatty liver disease (NAFLD) from healthy controls using pathology reference in a preclinical rabbit model.
Methods:
QTM method was validated on a liver perfusion simulation based on fluid dynamics using a rat liver vasculature model and the mass transport equation. In the NAFLD grading task, DCE MRI images of 7 adult rabbits with methionine choline-deficient diet-induced nonalcoholic steatohepatitis (NASH), 8 adult rabbits with simple steatosis (SS) were acquired and processed using QTM method and dual-input two compartment Kety's method respectively. Statistical analysis was performed on six perfusion parameters: velocity magnitude [Formula: see text] derived from QTM, liver arterial blood flow [Formula: see text], liver venous blood flow [Formula: see text], permeability [Formula: see text], blood volume [Formula: see text] and extravascular space volume [Formula: see text] averaged in liver ROI.
Results:
In the simulation, QTM method successfully reconstructed blood flow, reduced error by 48% compared to Kety's method. In the preclinical study, only QTM |u| showed significant difference between high grade NAFLD group and low grade NAFLD group.
Conclusion:
QTM postprocesses DCE-MRI automatically through deconvolution in space and time to solve the inverse problem of the transport equation. Comparing with Kety's method, QTM method showed higher accuracy and better differentiation in NAFLD classification task.
Significance:
We propose to apply QTM method in liver DCE MRI perfusion quantification.
... Equation (3) is a linear equation system for K trans and k ep = K trans V e , but is nonlinear to τ. A voxel-wise non-linear least squares method is used to solve for kinetic parameters and traveling delay τ of AIF with the regularization parameters λ = µ = 10 −3 chosen according to the L-curve method [27][28][29]: ...
There is no noninvasive method to estimate lung shunting fraction (LSF) in patients with liver tumors undergoing Yttrium-90 (Y90) therapy. We propose to predict LSF from noninvasive dynamic contrast enhanced (DCE) MRI using perfusion quantification. Two perfusion quantification methods were used to process DCE MRI in 25 liver tumor patients: Kety’s tracer kinetic modeling with a delay-fitted global arterial input function (AIF) and quantitative transport mapping (QTM) based on the inversion of transport equation using spatial deconvolution without AIF. LSF was measured on SPECT following Tc-99m macroaggregated albumin (MAA) administration via hepatic arterial catheter. The patient cohort was partitioned into a low-risk group (LSF ≤ 10%) and a high-risk group (LSF > 10%). Results: In this patient cohort, LSF was positively correlated with QTM velocity |u| (r = 0.61, F = 14.0363, p = 0.0021), and no significant correlation was observed with Kety’s parameters, tumor volume, patient age and gender. Between the low LSF and high LSF groups, there was a significant difference for QTM |u| (0.0760 ± 0.0440 vs. 0.1822 ± 0.1225 mm/s, p = 0.0011), and Kety’s Ktrans (0.0401 ± 0.0360 vs 0.1198 ± 0.3048, p = 0.0471) and Ve (0.0900 ± 0.0307 vs. 0.1495 ± 0.0485, p = 0.0114). The area under the curve (AUC) for distinguishing between low LSF and high LSF was 0.87 for |u|, 0.80 for Ve and 0.74 for Ktrans. Noninvasive prediction of LSF is feasible from DCE MRI with QTM velocity postprocessing.
... Correcting for input function delay is an important parameter and ignoring it can result in large errors in estimated perfusion parameters. 33,34 The initial parameters describing this functional AIF form were set to multiple variants of the population-average values to allow for the option for alternative starting points, and were then allowed to vary within a certain range in the fitting process. During each iteration, the algorithm alternates between estimation of the PK parameters based on the PK model (AIF function shape and delay held fixed), estimation of the AIF delay (PK parameters fixed, AIF function form fixed), and AIF (PK parameters fixed, AIF delay fixed), as shown in Figure 2. ...
The aim of this work is to develop a data-driven quantitative dynamic contrast enhanced (DCE) MRI technique using golden-angle radial sparse parallel MRI (GRASP) with high spatial resolution and high flexible temporal resolution and pharmacokinetic analysis with an arterial input function (AIF) estimated directly from the data obtained from each patient. DCE-MRI was performed on thirteen patients with gynecological malignancy using a 3T MRI scanner with a single continuous golden-angle stack-of-stars acquisition and image reconstruction with two temporal resolutions, by exploiting a unique feature in GRASP that reconstructs acquired data with user-defined temporal resolution. Joint estimation of the AIF (both AIF shape and delay) and pharmacokinetic (PK) parameters was performed with an iterative algorithm that alternates between AIF and PK estimation. Computer simulations were performed to determine the accuracy (expressed as percentage error (PE)) and precision of the estimated parameters. PK parameters (Ktrans , ve , and vp ) and normalized root-mean-square error (nRMSE) (%) of the fitting errors for the tumor contrast kinetic data were measured both with population-averaged AIF and data-driven AIFs. On patient data, the Wilcoxon signed rank test was performed to compare nRMSE. Simulations demonstrated that GRASP image reconstruction with temporal resolution of 1 s/frame for AIF estimation and 5 s/frame for PK analysis results in absolute PE of less than 5% in the estimation of Ktrans and ve , and less than 11% in the estimation of vp . The nRMSE (mean ± std. dev.) for the dual temporal resolution image reconstruction and data-driven AIF was 0.16±0.04 as compared to 0.27±0.10 (p<0.001) with 1 s/frame using population-averaged AIF, and 0.23±0.07 with 5 s/frame using population-averaged AIF (p<0.001). We conclude that DCE-MRI data acquired and reconstructed with the GRASP technique at dual temporal resolution can successfully be applied to jointly estimate the AIF and PK parameters from a single acquisition resulting in data-driven AIFs and voxel-wise PK parametric maps.
... Thus, potential values assigned to each pixel are quantized. A drawback of curve fitting is that an initial property guess is required to begin the algorithm, and this guess can influence the final perfusion property estimates 11 . This influence can be profound if there are local minima in the curve being analyzed in this manner. ...
Perfusion properties can be estimated from pharmacokinetic models applied to DCE-MRI data using curve fitting algorithms; however, these suffer from drawbacks including the local minimum problem and substantial computational time. Here, a dictionary matching approach is proposed as an alternative. Curve fitting and dictionary matching were applied to simulated data using the dual-input single-compartment model with known perfusion property values and 5 in vivo DCE-MRI datasets. In simulation at SNR 60 dB, the dictionary estimate had a mean percent error of 0.4–1.0% for arterial fraction, 0.5–1.4% for distribution volume, and 0.0% for mean transit time. The curve fitting estimate had a mean percent error of 1.1–2.1% for arterial fraction, 0.5–1.3% for distribution volume, and 0.2–1.8% for mean transit time. In vivo, dictionary matching and curve fitting showed no statistically significant differences in any of the perfusion property measurements in any of the 10 ROIs between the methods. In vivo, the dictionary method performed over 140-fold faster than curve fitting, obtaining whole volume perfusion maps in just over 10 s. This study establishes the feasibility of using a dictionary matching approach as a new and faster way of estimating perfusion properties from pharmacokinetic models in DCE-MRI.
Purpose
To test the feasibility of using quantitative transport mapping (QTM) method, which is based on the inversion of transport equation using spatial deconvolution without any arterial input function, for automatically postprocessing dynamic contrast enhanced MRI (DCE-MRI) to differentiate malignant and benign breast tumors.
Materials and methods
Breast DCE-MRI data with biopsy confirmed malignant (n = 13) and benign tumors (n = 13) was used to assess QTM velocity (|u|) and diffusion coefficient (D), volume transfer constant (Ktrans), volume fraction of extravascular extracellular space (Ve) from kinetics method, and traditional enhancement curve characteristics (ECC: amplitude A, wash-in rate α, wash-out rate β). A Mann-Whitney U test and receiver operating characteristic curve (ROC) analysis were performed to assess the diagnostic performance of these parameters for distinguishing between benign and malignant tumors.
Results
Between malignant and benign tumors, there was a significant difference in |u| and Ktrans, (p = 0.0066, 0.0274, respectively), but not in D, Ve, A, α and β (p = 0.1119, 0.2382, 0.4418,0.2592 and 0.9591, respectively). ROC area-under-the-curve was 0.82, 0.75 (95% confidence level 0.60–0.95, 0.51–0.90) for |u| and Ktrans, respectively.
Conclusion
QTM postprocesses DCE-MRI automatically through deconvolution in space and time to solve the inverse problem of the transport equation. Comparing with traditional kinetics method and ECC, QTM method showed better diagnostic accuracy in differentiating benign from malignant breast tumors in this study.
Delays between contrast agent (CA) arrival at the site of vascular input function (VIF) sampling and the tissue of interest affect dynamic contrast enhanced (DCE) MRI pharmacokinetic modelling. We investigate effects of altering VIF CA bolus arrival delays on liver DCE MRI perfusion parameters, propose an alternative approach to estimating delays and evaluate reproducibility.
Thirteen healthy volunteers (28.7 ± 1.9 years, seven males) underwent liver DCE MRI using dual-input single compartment modelling, with reproducibility (n = 9) measured at 7 days. Effects of VIF CA bolus arrival delays were assessed for arterial and portal venous input functions. Delays were pre-estimated using linear regression, with restricted free modelling around the pre-estimated delay. Perfusion parameters and 7 days reproducibility were compared using this method, freely modelled delays and no delays using one-way ANOVA. Reproducibility was assessed using Bland–Altman analysis of agreement.
Maximum percent change relative to parameters obtained using zero delays, were −31% for portal venous (PV) perfusion, +43% for total liver blood flow (TLBF), +3247% for hepatic arterial (HA) fraction, +150% for mean transit time and −10% for distribution volume. Differences were demonstrated between the 3 methods for PV perfusion (p = 0.0085) and HA fraction (p < 0.0001), but not other parameters. Improved mean differences and Bland–Altman 95% Limits-of-Agreement for reproducibility of PV perfusion (9.3 ml/min/100 g, ±506.1 ml/min/100 g) and TLBF (43.8 ml/min/100 g, ±586.7 ml/min/100 g) were demonstrated using pre-estimated delays with constrained free modelling.
CA bolus arrival delays cause profound differences in liver DCE MRI quantification. Pre-estimation of delays with constrained free modelling improved 7 days reproducibility of perfusion parameters in volunteers.
Purpose:
Fitting tracer kinetic models using linear methods is much faster than using their nonlinear counterparts, although this comes often at the expense of reduced accuracy and precision. The aim of this study was to derive and compare the performance of the linear compartmental tissue uptake (CTU) model with its nonlinear version with respect to their percentage error and precision.
Theory and methods:
The linear and nonlinear CTU models were initially compared using simulations with varying noise and temporal sampling. Subsequently, the clinical applicability of the linear model was demonstrated on 14 patients with locally advanced cervical cancer examined with dynamic contrast-enhanced magnetic resonance imaging.
Results:
Simulations revealed equal percentage error and precision when noise was within clinical achievable ranges (contrast-to-noise ratio >10). The linear method was significantly faster than the nonlinear method, with a minimum speedup of around 230 across all tested sampling rates. Clinical analysis revealed that parameters estimated using the linear and nonlinear CTU model were highly correlated (ρ ≥ 0.95).
Conclusion:
The linear CTU model is computationally more efficient and more stable against temporal downsampling, whereas the nonlinear method is more robust to variations in noise. The two methods may be used interchangeably within clinical achievable ranges of temporal sampling and noise. Magn Reson Med, 2016. © 2016 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Perfusion magnetic resonance imaging (MRI) studies quantify the microcirculatory status of liver parenchyma and liver lesions, and can be used for the detection of liver metastases, assessing the effectiveness of anti-angiogenic therapy, evaluating tumor viability after anti-cancer therapy or ablation, and diagnosis of liver cirrhosis and its severity. In this review, we discuss the basic concepts of perfusion MRI using tracer kinetic modeling, the common kinetic models applied for analyses, the MR scanning techniques, methods of data processing, and evidence that supports its use from published clinical and research studies. Technical standardization and further studies will help to establish and validate perfusion MRI as a clinical imaging modality.
Here we describe the parametric response map (PRM), a voxel-wise approach for image analysis and quantification of hemodynamic alterations during treatment for 44 patients with high-grade glioma. Relative cerebral blood volume (rCBV) and flow (rCBF) maps were acquired before treatment and after 1 and 3 weeks of therapy. We compared the standard approach using region-of-interest analysis for change in rCBV or rCBF to the change in perfusion parameters on the basis of PRM (PRM(rCBV) and PRM(rCBF)) for their accuracy in predicting overall survival. Neither the percentage change of rCBV or rCBF predicted survival, whereas the regional response evaluations made on the basis of PRM were highly predictive of survival. Even when accounting for baseline rCBV, which is prognostic, PRM(rCBV) proved more predictive of overall survival.
The use of curve-fitting and compartmental modelling for calculating physiological parameters from measured data has increased in popularity in recent years. Finding the 'best fit' of a model to data involves the minimization of a merit function. An example of a merit function is the sum of the squares of the differences between the data points and the model estimated points. This is facilitated by curve-fitting algorithms. Two curve-fitting methods, Levenberg-Marquardt and MINPACK-1, are investigated with respect to the search start points that they require and the accuracy of the returned fits. We have simulated one million dynamic contrast enhanced MRI curves using a range of parameters and investigated the use of single and multiple search starting points. We found that both algorithms, when used with a single starting point, return unreliable fits. When multiple start points are used, we found that both algorithms returned reliable parameters. However the MINPACK-1 method generally outperformed the Levenberg-Marquardt method. We conclude that the use of a single starting point when fitting compartmental modelling data such as this produces unsafe results and we recommend the use of multiple start points in order to find the global minima.
Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is a widely used technique for assessing tissue physiology. Spoiled gradient echo (SPGR) pulse sequences are one of the most common methods for acquisition of DCE-MRI data, providing high temporal and spatial resolution with strong T(1)-weighting. Conversion of SPGR signal to concentration is briefly reviewed, and a new closed-form expression for concentration measurement uncertainty for finite signal-to-noise ratio (SNR) and baseline scan time is derived. This result is applicable to arbitrary concentration-dependent relaxation rate and is valid over the same domain as the theoretical SPGR signal equation. Expressions for the lower and upper bounds on measurable concentration are also derived. The existence of a concentration- and tissue-dependent optimal flip angle that minimizes concentration uncertainty is demonstrated and it is shown that, for clinically relevant pulse sequence parameters, this optimal flip angle is significantly larger than the corresponding Ernst angle. Analysis of three pulse sequences from the DCE-MRI literature shows that optimization of flip angle using the methods discussed here leads to potential improvements of 10-1166% in effective SNR over the 0.5-5.0 mM concentration range with minimal or no loss of measurement accuracy down to 0.1 mM. In vivo data from three study patients provide further support for our theoretical expression for concentration measurement uncertainty, with predicted and experimental estimates agreeing to within +/- 30%. Equations for concentration bias resulting from biases in flip angle and from pre-contrast relaxation time and contrast relaxivity (both longitudinal and transverse) are also derived in closed-form. The resulting equations show the potential for significant contributions to bias in concentration measurement arising from even relatively small mis-specification of flip angle and/or pre-contrast longitudinal relaxation time, particularly at high contrast concentrations.
Due to its simplicity, computational efficiency, and reliability,
weighted linear regression (WLR) is widely used for generation of
parametric imaging in positron emission tomography (PET) studies, but
parametric images estimated by WLR usually have high image noise level.
To improve the stability and signal-to-noise ratio of the estimated
parametric images, the authors have added ridge regression, a
statistical technique that reduces estimation variability at the expense
of a small bias. To minimize the bias, spatially smoothed images
obtained with WLR are used as a constraint for ridge regression. This
new algorithm consists of two steps. First, parametric images are
generated by WLR and are spatially smoothed. Ridge regression is then
applied using the smoothed parametric images obtained in the first step
as the constraint. Since both “generalized” ridge regression
and “simple” ridge regression are used in statistical
applications, we evaluated specifically in this study the relative
advantages of the two when incorporated for generating parametric images
from dynamic O-15 water PET studies. Computer simulations of a dynamic
PET study with the spatial configuration of Hoffman's brain phantom and
a real human PET study were used as the data for the evaluation. Results
reveal ridge regressions improve image quality of parametric images for
studies with high or middle noise level. As compared to WLR, use of
generalized ridge regression offers little advantage over that of simple
ridge regression
Purpose:
The linear least squares (LLS) estimator provides a fast approach to parameter estimation in the linearized two-compartment exchange model. However, the LLS method may introduce a bias through correlated noise in the system matrix of the model. The purpose of this work is to present a new estimator for the linearized two-compartment exchange model that takes this noise into account.
Method:
To account for the noise in the system matrix, we developed an estimator based on the weighted total least squares (WTLS) method. Using simulations, the proposed WTLS estimator was compared, in terms of accuracy and precision, to an LLS estimator and a nonlinear least squares (NLLS) estimator.
Results:
The WTLS method improved the accuracy compared to the LLS method to levels comparable to the NLLS method. This improvement was at the expense of increased computational time; however, the WTLS was still faster than the NLLS method. At high signal-to-noise ratio all methods provided similar precisions while inconclusive results were observed at low signal-to-noise ratio.
Conclusion:
The proposed method provides improvements in accuracy compared to the LLS method, however, at an increased computational cost. Magn Reson Med, 2017. ? 2017 International Society for Magnetic Resonance in Medicine.
Matching the bolus arrival time (BAT) of the arterial input function (AIF) and tissue residue function (TRF) is necessary for accurate pharmacokinetic (PK) modeling of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI). We investigated the sensitivity of volume transfer constant ([Formula: see text]) and extravascular extracellular volume fraction ([Formula: see text]) to BAT and compared the results of four automatic BAT measurement methods in characterization of prostate and breast cancers. Variation in delay between AIF and TRF resulted in a monotonous change trend of [Formula: see text] and [Formula: see text] values. The results of automatic BAT estimators for clinical data were all comparable except for one BAT estimation method. Our results indicate that inaccuracies in BAT measurement can lead to variability among DCE-MRI PK model parameters, diminish the quality of model fit, and produce fewer valid voxels in a region of interest. Although the selection of the BAT method did not affect the direction of change in the treatment assessment cohort, we suggest that BAT measurement methods must be used consistently in the course of longitudinal studies to control measurement variability.
Various liver diseases lead to significant alterations of the hepatic microcirculation. Therefore, quantification of hepatic perfusion has the potential to improve the assessment and management of liver diseases. Most methods used to quantify liver perfusion are invasive or controversial. This paper describes and validates a non-invasive method for the quantification of liver perfusion using computed tomography (CT). Dynamic single-section CT of the liver was performed after intravenous bolus administration of a low-molecular-mass iodinated contrast agent. Hepatic, aortic and portal-venous time—density curves were fitted with a dual-input one-compartmental model to calculate liver perfusion. Validation studies consisted of simultaneous measurements of hepatic perfusion with CT and with radiolabelled microspheres in rabbits at rest and after adenosine infusion. The feasibility and reproducibility of the CT method in humans was assessed by three observers in 10 patients without liver disease. In rabbits, significant correlations were observed between perfusion measurements obtained with CT and with microspheres (r = 0.92 for total liver perfusion, r = 0.81 for arterial perfusion and r = 0.85 for portal perfusion). In patients, total liver plasma perfusion measured with CT was 112±28 ml·min-1·100 ml-1, arterial plasma perfusion was 18±12 ml·min-1·100 ml-1 and portal plasma perfusion was 93±31 ml·min-1·100 ml-1. The measurements obtained by the three observers were not significantly different from each other (P > 0.1). Our results indicate that dynamic CT combined with a dual-input one-compartmental model provides a valid and reliable method for the non-invasive quantification of perfusion in the normal liver.
Purpose:
Model fitting of dynamic contrast-enhanced-magnetic resonance imaging-MRI data with nonlinear least squares (NLLS) methods is slow and may be biased by the choice of initial values. The aim of this study was to develop and evaluate a linear least squares (LLS) method to fit the two-compartment exchange and -filtration models.
Methods:
A second-order linear differential equation for the measured concentrations was derived where model parameters act as coefficients. Simulations of normal and pathological data were performed to determine calculation time, accuracy and precision under different noise levels and temporal resolutions. Performance of the LLS was evaluated by comparison against the NLLS.
Results:
The LLS method is about 200 times faster, which reduces the calculation times for a 256 × 256 MR slice from 9 min to 3 s. For ideal data with low noise and high temporal resolution the LLS and NLLS were equally accurate and precise. The LLS was more accurate and precise than the NLLS at low temporal resolution, but less accurate at high noise levels.
Conclusion:
The data show that the LLS leads to a significant reduction in calculation times, and more reliable results at low noise levels. At higher noise levels the LLS becomes exceedingly inaccurate compared to the NLLS, but this may be improved using a suitable weighting strategy. Magn Reson Med, 2015. © 2015 Wiley Periodicals, Inc.
To develop an efficient method for calculating pharmacokinetic (PK) parameters in brain DCE-MRI permeability studies.
A linear least-squares fitting algorithm based on a derivative expression of the two-compartment PK model was proposed to analytically solve for the PK parameters. Noise in the expression was minimized through low-pass filtering. Simulation studies were conducted in which the proposed method was compared with two existing methods in terms of accuracy and efficiency. Five in vivo brain studies were demonstrated for potential clinical application.
In the simulation studies using chosen parameter values, the calculated percent difference of K(trans) by the proposed method was <5.0% with a temporal resolution (Δt) < 5 s, and the accuracies of all parameter results were better or comparable to existing methods. When analyzed within certain parameter intensity ranges, the proposed method was more accurate than the existing methods and improved the efficiency by a factor of up to 458 for a Δt = 1 s and up to 38 for a Δt = 5 s. In the in vivo study, the calculated parameters using the proposed method were comparable to those using the existing methods with improved efficiencies.
An efficient method was developed for the accurate and efficient calculation of parameters in brain DCE-MRI permeability studies. Magn Reson Med, 2015. © 2015 Wiley Periodicals, Inc.
© 2015 Wiley Periodicals, Inc.
High spatial-temporal four-dimensional imaging with large volume coverage is necessary to accurately capture and characterize liver lesions. Traditionally, parallel imaging and adapted sampling are used toward this goal, but they typically result in a loss of signal to noise. Furthermore, residual under-sampling artifacts can be temporally varying and complicate the quantitative analysis of contrast enhancement curves needed for pharmacokinetic modeling. We propose to overcome these problems using a novel patch-based regularization approach called Patch-based Reconstruction Of Under-sampled Data (PROUD).
PROUD produces high frame rate image reconstructions by exploiting the strong similarities in spatial patches between successive time frames to overcome the severe k-space under-sampling. To validate PROUD, a numerical liver perfusion phantom was developed to characterize contrast-to-noise ratio (CNR) performance compared with a previously proposed method, TRACER. A second numerical phantom was constructed to evaluate the temporal footprint and lag of PROUD and TRACER reconstructions. Finally, PROUD and TRACER were evaluated in a cohort of five liver donors.
In the CNR phantom, PROUD, compared with TRACER, improved peak CNR by 3.66 times while maintaining or improving temporal fidelity. In vivo, PROUD demonstrated an average increase in CNR of 60% compared with TRACER.
The results presented in this work demonstrate the feasibility of using a combination of patch based image constraints with temporal regularization to provide high SNR, high temporal frame rate and spatial resolution four dimensional imaging. Magn Reson Med, 2014. © 2014 Wiley Periodicals, Inc.
© 2014 Wiley Periodicals, Inc.
Purpose:
Detection, characterization, and monitoring the treatment of hepatocellular carcinomas (HCC) in patients with cirrhosis is challenging because of their variable and rapid arterial enhancement. Multiphase dynamic contrast-enhanced MRI is used clinically for HCC assessment; however, the method suffers from limited temporal resolution and difficulty in coordinating imaging and breath-hold timing within a narrow temporal window of interest. In this article, a volumetric, high-spatial resolution, and high-temporal resolution dynamic contrast-enhanced liver imaging method for improved detection and characterization of HCC is demonstrated.
Methods:
A time-resolved three-dimensional radial acquisition with iterative sensitivity-encoding reconstruction images the entire abdomen and thorax with high spatial and temporal resolution, using real-time three-dimensional fluoroscopy to match the breath hold to contrast arrival. The sequence was tested on 17 subjects, including eight patients with HCC or other hypervascular focal lesions.
Results:
This technique was successful in acquiring volumetric imaging of the entire liver with 2.1-mm isotropic spatial and true 4-s temporal resolution.
Conclusion:
This technique may be suitable for detecting, characterizing, and monitoring the treatment of HCC. It also holds significant potential for perfusion modeling, which may provide a noninvasive means to rapidly determine the efficacy of chemotherapeutic agents in these tumors over the entire liver volume.
To evaluate the feasibility of quantifying hepatic perfusion and function by using dynamic contrast material-enhanced (DCE) magnetic resonance (MR) imaging with the hepatobiliary contrast agent gadoxetic acid and a dual-inlet two-compartment uptake model.
The study was approved by the local institutional review board, and written informed consent was obtained from all patients. Data were acquired between October 2008 and November 2009 in 24 patients with hepatic metastases from neuroendocrine tumors (13 men, 11 women; mean age, 59.8 years). DCE MR imaging was performed at 3.0 T with a standard dose of gadoxetic acid and a three-dimensional sequence, with 48 sections of data acquired every 2.2 seconds for 5 minutes. For each patient, a plasma flow map was calculated by means of deconvolution and the model was fitted to six region-of-interest curves. Results were evaluated with goodness-of-fit analysis and, in normal-appearing liver tissue, by comparing perfusion parameters with those reported in the literature. Interobserver effects in the selection of arterial and venous input functions were assessed.
With an arterial delay parameter, the model provided a good fit to all data. Values for arterial and venous plasma flow and extracellular volume in normal-appearing liver tissue were comparable to those in the literature. The mean intracellular uptake rate is 3.4 per minute, with a standard deviation of 1.9 per minute. The model also provided a good fit in all tumor data, producing high arterial flow fraction (87%) and lower uptake (1.7 per minute). Bias due to observer-dependent differences in the selection of the input functions was negligible.
The analysis of dynamic gadoxetic acid-enhanced MR images with the dual-inlet two-compartment uptake model presents a new and practical approach for measuring arterial and venous perfusion and hepatic function in a single acquisition.
Time-resolved imaging is crucial for the accurate diagnosis of liver lesions. Current contrast enhanced liver magnetic resonance imaging acquires a few phases in sequential breath-holds. The image quality is susceptible to bolus timing errors, which could result in missing the critical arterial phase. This impairs the detection of malignant tumors that are supplied primarily by the hepatic artery. In addition, the temporal resolution may be too low to reliably separate the arterial phase from the portal venous phase. In this study, a method called temporal resolution acceleration with constrained evolution reconstruction was developed with three-dimensional volume coverage and high-temporal frame rate. Data is acquired using a stack of spirals sampling trajectory combined with a golden ratio view order using an eight-channel coil array. Temporal frames are reconstructed from vastly undersampled data sets using a nonlinear inverse algorithm assuming that the temporal changes are small at short time intervals. Numerical and phantom experimental validation is presented. Preliminary in vivo results demonstrated high spatial resolution dynamic three-dimensional images of the whole liver with high frame rates, from which numerous subarterial phases could be easily identified retrospectively. Magn Reson Med, 2012. © 2012 Wiley Periodicals, Inc.
Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.
We performed an error analysis of the quantification of liver perfusion from dynamic contrast-enhanced computed tomography (DCE-CT) data using a dual-input single-compartment model for various disease severities, based on computer simulations. In the simulations, the time-density curves (TDCs) in the liver were generated from an actually measured arterial input function using a theoretical equation describing the kinetic behavior of the contrast agent (CA) in the liver. The rate constants for the transfer of CA from the hepatic artery to the liver (K(1a)), from the portal vein to the liver (K(1p)), and from the liver to the plasma (k(2)) were estimated from simulated TDCs with various plasma volumes (V(0)s). To investigate the effect of the shapes of input functions, the original arterial and portal-venous input functions were stretched in the time direction by factors of 2, 3 and 4 (stretching factors). The above parameters were estimated with the linear least-squares (LLSQ) and nonlinear least-squares (NLSQ) methods, and the root mean square errors (RMSEs) between the true and estimated values were calculated. Sensitivity and identifiability analyses were also performed. The RMSE of V(0) was the smallest, followed by those of K(1a), k(2) and K(1p) in an increasing order. The RMSEs of K(1a), K(1p) and k(2) increased with increasing V(0), while that of V(0) tended to decrease. The stretching factor also affected parameter estimation in both methods. The LLSQ method estimated the above parameters faster and with smaller variations than the NLSQ method. Sensitivity analysis showed that the magnitude of the sensitivity function of V(0) was the greatest, followed by those of K(1a), K(1p) and k(2) in a decreasing order, while the variance of V(0) obtained from the covariance matrices was the smallest, followed by those of K(1a), K(1p) and k(2) in an increasing order. The magnitude of the sensitivity function and the variance increased and decreased, respectively, with increasing disease severity and decreased and increased, respectively, with increasing stretching factor except for V(0). Identifiability analysis showed that the identifiability between K(1)(p) and k(2) was lower than that between K(1)(a) and k(2) or between K(1a) and K(1p). In conclusion, this study will be useful for understanding the accuracy and reliability of the quantitative measurement of liver perfusion using a dual-input single-compartment model and DCE-CT data.
A linear algorithm for the rapid calculation of local rate constants is proposed. The method is applicable to the three-compartment models currently used in the analysis of positron camera measurements with [11C]methionine, [11C]deoxyglucose, and [11C]glucose. The same technique can also be used for the regional measurement of local blood flow with the aid of a freely diffusible tracer. The algorithm was applied to measurements on humans with [11C]glucose. As a comparison, the same data were also analyzed with a standard nonlinear technique. Good agreement between the two methods was obtained.
The original generalized linear least squares (GLLS) algorithm was developed for non-uniformly sampled biomedical system parameter estimation using finely sampled instantaneous measurements (D. Feng, S.C. Huang, Z. Wang, D. Ho, An unbiased parametric imaging algorithm for non-uniformly sampled biomedical system parameter estimation, IEEE Trans. Med. Imag. 15 (1996) 512-518). This algorithm is particularly useful for image-wide generation of parametric images with positron emission tomography (PET), as it is computationally efficient and statistically reliable (D. Feng, D. Ho, Chen, K., L.C. Wu, J.K. Wang, R.S. Liu, S.H. Yeh, An evaluation of the algorithms for determining local cerebral metabolic rates of glucose using positron emission tomography dynamic data, IEEE Trans. Med. Imag. 14 (1995) 697-710). However, when dynamic PET image data are sampled according to the optimal image sampling schedule (OISS) to reduce memory and storage space (X. Li, D. Feng, K. Chen, Optimal image sampling schedule: A new effective way to reduce dynamic image storage space and functional image processing time, IEEE Trans. Med. Imag. 15 (1996) 710-718), only a few temporal image frames are recorded (e.g. only four images are recorded for the four parameter fluoro-deoxy-glucose (FDG) model). These image frames are recorded in terms of accumulated radio-activity counts and as a result, the direct application of GLLS is not reliable as instantaneous measurement samples can no longer be approximated by averaging of accumulated measurements over the sampling intervals. In this paper, we extend GLLS to OISS-GLLS which deals with the fewer accumulated measurement samples obtained from OISS dynamic systems. The theory and algorithm of this new technique are formulated and studied extensively. To investigate statistical reliability and computational efficiency of OISS-GLLS, a simulation study using dynamic PET data was performed. OISS-GLLS using 4-measurement samples was compared to the non-linear least squares (NLS) method using 22-measurement samples, GLLS using 22-measurement samples and OISS-NLS using 4-measurement samples. Results demonstrated that OISS-GLLS was able to achieve parameter estimates of equivalent accuracy and reliability in comparison to NLS or GLLS using finely sampled measurements (22-measurement samples), or OISS-NLS using optimally sampled measurements (4-measurement samples). Further more, as fewer measurement samples are used in OISS-GLLS, this algorithm is computationally faster than NLS or GLLS. Therefore, OISS-GLLS is well-suited for image-wide parameter estimation when PET image data are recorded according to the optimal image sampling schedule.
Dynamic susceptibility contrast (DSC) MRI is now increasingly used for measuring perfusion in many different applications. The quantification of DSC data requires the measurement of the arterial input function (AIF) and the deconvolution of the tissue concentration time curve. One of the most accepted deconvolution methods is the use of singular value decomposition (SVD). Simulations were performed to evaluate the effects on DSC quantification of the presence of delay and dispersion in the estimated AIF. Both delay and dispersion were found to introduce significant underestimation of cerebral blood flow (CBF) and overestimation of mean transit time (MTT). While the error introduced by the delay can be corrected by using the information of the arrival time of the bolus, the correction for the dispersion is less straightforward and requires a model for the vasculature.
Various liver diseases lead to significant alterations of the hepatic microcirculation. Therefore, quantification of hepatic perfusion has the potential to improve the assessment and management of liver diseases. Most methods used to quantify liver perfusion are invasive or controversial. This paper describes and validates a non-invasive method for the quantification of liver perfusion using computed tomography (CT). Dynamic single-section CT of the liver was performed after intravenous bolus administration of a low-molecular-mass iodinated contrast agent. Hepatic, aortic and portal-venous time-density curves were fitted with a dual-input one-compartmental model to calculate liver perfusion. Validation studies consisted of simultaneous measurements of hepatic perfusion with CT and with radiolabelled microspheres in rabbits at rest and after adenosine infusion. The feasibility and reproducibility of the CT method in humans was assessed by three observers in 10 patients without liver disease. In rabbits, significant correlations were observed between perfusion measurements obtained with CT and with microspheres (r=0.92 for total liver perfusion, r=0.81 for arterial perfusion and r=0.85 for portal perfusion). In patients, total liver plasma perfusion measured with CT was 112+/-28 ml.min(-1).100 ml(-1), arterial plasma perfusion was 18+/-12 ml.min(-1).100 ml(-1) and portal plasma perfusion was 93+/-31 ml.min(-1).100 ml(-1). The measurements obtained by the three observers were not significantly different from each other (P>0.1). Our results indicate that dynamic CT combined with a dual-input one-compartmental model provides a valid and reliable method for the non-invasive quantification of perfusion in the normal liver.
It has become increasingly important to quantitatively estimate tissue physiological parameters such as perfusion, capillary permeability, and the volume of extravascular-extracellular space (EES) using T(1)-weighted dynamic contrast-enhanced MRI (DCE-MRI). A linear equation was derived by integrating the differential equation describing the kinetic behavior of contrast agent (CA) in tissue, from which K(1) (rate constant for the transfer of CA from plasma to EES), k(2) (rate constant for the transfer from EES to plasma), and V(p) (plasma volume) can be easily obtained by the linear least-squares (LLSQ) method. The usefulness of this method was investigated by means of computer simulations, in comparison with the nonlinear least-squares (NLSQ) method. The new method calculated the above parameters faster than the NLSQ method by a factor of approximately 6, and estimated them more accurately than the NLSQ method at a signal-to-noise ratio (SNR) of < approximately 10. This method will be useful for generating functional images of K(1), k(2), and V(p) from DCE-MRI data.
The purpose of this study was to quantify microcirculation and microvasculature in breast lesions by pharmacokinetic analysis of Gd-DTPA-enhanced MRI series. Strongly T1-weighted MR images were acquired in 18 patients with breast lesions using a saturation-recovery-TurboFLASH sequence. Concentration-time courses were determined for blood, pectoral muscle, and breast masses and subsequently analyzed by a two-compartment model to estimate plasma flow and the capillary transfer coefficient per unit of plasma volume (F/VP, KPS/VP) as well as fractional volumes of the plasma and interstitial space (fP, fI). Tissue parameters determined for pectoral muscle (fP = 0.04 +/- 0.01, fI = 0.09 +/- 0.01, F/VP = 2.4 +/- 1.3 min(-1), and KPS/VP = 1.2 +/- 0.5 min(-1)) and 10 histologically proven carcinomas (fP = 0.20 +/- 0.07, fI = 0.34 +/- 0.16, F/VP = 2.4 +/- 0.7 min(-1), and KPS/VP = 0.86 +/- 0.62 min(-1)) agreed reasonable well with literature data. Best separation between malignant and benign lesions was obtained by the ratio KPS/F (0.35 +/- 0.17 vs. 1.23 +/- 0.65). The functional imaging technique presented appears promising to quantitatively characterize tumor pathophysiology. Its impact on diagnosis and therapy management of breast tumors, however, has to be evaluated in larger patient studies.
The purpose of this study was to investigate the accuracy of a quantitative method for estimating arterial hepatic blood flow and portal hepatic blood flow separately using a dual-input single-compartment model compared with the maximum slope method using computer simulations and clinical data. In computer simulations, the rate constants for the transfer of contrast agent (CA) from the hepatic artery to the liver (K(1a)), from the portal vein to the liver (K(1p)) and from the liver to the blood (k(2)) were estimated from simulated time-density curves with various transit times of CA from the aorta to the liver (tau(a)) and from the portal vein to the liver (tau(p)) using the linear least-squares (LLSQ) method. In clinical studies, dynamic CT data were acquired from 27 patients, and parametric maps of K(1a), K(1p) and k(2) were generated by applying the LLSQ method pixel by pixel. In simulation studies, tau(a) and tau(p) were found to have a large and a small effect on the estimates of K(1a) and K(1p), respectively. In clinical studies, the K(1a) and K(1p) values estimated with the maximum slope method were underestimated by 60+/-29% and 37+/-12%, respectively, compared with those estimated by the LLSQ method. In conclusion, our results suggest that correction of tau(a) is necessary for accurately estimating K(1a) and K(1p). Our method is therefore promising for the evaluation of hepatic blood flow in various liver diseases because it allows us to evaluate arterial hepatic blood flow and portal hepatic blood flow separately and visually.
Measurement of the local cerebral metabolic rate of glucose
(LCMRGlc) and the individual rate constant parameters of the [<sup>18
</sup>F]2-fluoro-2-deoxy-D-glucose (FDG) model can provide a clearer
understanding and insight to the physiological processes in the human
brain, and a quicker and more accurate means of diagnosis in clinical
applications. A systematic study using simulated and clinical tissue
time activity data is presented to evaluate several existing and newly
developed major algorithms used for determining LCMRGlc and the
individual rate constants from positron emission tomography dynamic
data. The computational and statistical properties of the
autoradiographic approach, weighted and unweighted nonlinear least
squares methods, Patlak graphic approach, weighted integration method,
linear least squares and generalized linear least squares methods are
investigated and discussed in this paper
With the advent of positron emission tomography (PET), a variety
of techniques have been developed to measure local cerebral blood flow
(LCBF) noninvasively in humans. A potential class of techniques, which
includes linear least squares (LS), linear weighted least squares (WLS),
linear generalized least squares (GLS), and linear generalized weighted
least squares (GWLS), is proposed. The statistical characteristics of
these methods are examined by computer simulation. The authors present a
comparison of these four methods with two other rapid estimation
techniques developed by Huang et al. (1982) and Alpert (1984), and two
classical methods, the unweighted and weighted nonlinear least squares
regression. The results show that these methods can take full advantage
of the contribution from the fine temporal sampling data of modern
tomographs, and thus provide statistically reliable estimates that are
comparable to those obtained from nonlinear LS regression. These methods
also have high computational efficiency, and the parameters can be
estimated directly from operational equations in one single step.
Therefore, they can potentially be used in image-wide estimation of
local cerebral blood flow and distribution volume with PET
A study on statistically reliable and computationally efficient algorithms for generating local cerebral blood flow parametric images with positron emission tomography
- Feng