Spatially regularized T₁ estimation from variable flip angles MRI

Abstract

Purpose: To develop efficient algorithms for fast voxel-by-voxel quantification of tissue longitudinal relaxation time (T₁) from variable flip angles magnetic resonance images (MRI) to reduce voxel-level noise without blurring tissue edges.

Methods: T₁ estimations regularized by total variation (TV) and quadratic penalty are developed to measure T₁ from fast variable flip angles MRI and to reduce voxel-level noise without decreasing the accuracy of the estimates. First, a quadratic surrogate for a log likelihood cost function of T₁ estimation is derived based upon the majorization principle, and then the TV-regularized surrogate function is optimized by the fast iterative shrinkage thresholding algorithm. A fast optimization algorithm for the quadratically regularized T₁ estimation is also presented. The proposed methods are evaluated by the simulated and experimental MR data.

Results: The means of the T₁ values in the simulated brain data estimated by the conventional, TV-regularized, and quadratically regularized methods have less than 3% error from the true T₁ in both GM and WM tissues with image noise up to 9%. The relative standard deviations (SDs) of the T₁ values estimated by the conventional method are more than 12% and 15% when the images have 7% and 9% noise, respectively. In comparison, the TV-regularized and quadratically regularized methods are able to suppress the relative SDs of the estimated T₁ to be less than 2% and 3%, respectively, regardless of the image noise level. However, the quadratically regularized method tends to overblur the edges compared to the TV-regularized method.

Conclusions: The spatially regularized methods improve quality of T₁ estimation from multiflip angles MRI. Quantification of dynamic contrast-enhanced MRI can benefit from the high quality measurement of native T₁.

INTRODUCTION

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) has shown its value for diagnosis of neurological disorders,¹ detection of tumors, and evaluation of tissue response to therapies.^{2, 3, 4} Quantification of tissue longitudinal relaxation time (T₁)-weighted DCE-MRI using pharmacokinetic models requires measurement of T₁ prior to contrast injection.⁵ Since T₁ values vary in tissue and tumor and can change during and after therapy, an accurate T₁ measurement is vital for characterization of perfusion parameters from DCE-MRI. Furthermore, quantitative tissue T₁ could be a distinctive metric for tissue discrimination, disease detection, and therapy monitoring.⁶

The conventional T₁ estimation is based upon either an inversion-recovery (IR) or a saturation-recovery (SR) pulse sequence. Although the methods generate accurate results, the prolonged acquisition time of these methods makes them less practical to be a part of a DCE-MRI protocol in clinical setting.^{7, 8, 9, 10} The approach that is widely used in a DCE-MRI study is to acquire gradient-echo images with variable flip angles (VFA) and with one or more short TRs.^{11, 12, 13, 14, 15, 16} The scanning time of VFA imaging can be further decreased by undersampling acquisition.¹⁷ Estimation of T₁ values is usually done by nonlinear least-squares fitting (NLS) of the VFA MRI,¹⁸ but also can be done by linear least-squares fitting after transforming intensities of MRI into a linear form with T₁.^{9, 13, 19} Several authors have shown that the VFA method can achieve accuracy of the T₁ estimation similar to those by the IR and SR techniques for the image data having a high signal-noise ratio (SNR), which are usually accomplished by performing computation in a region of interest.^{14, 19} However, voxel-by-voxel estimated T₁ values show a large amount of fluctuation due to the noise in the original images and the limited number of flip angles. This poor repeatability of the T₁ estimation affects utilization of the T₁ map in voxel-based DCE quantification. In order to reduce the variation in the T₁ estimation, a different approach is needed. T₁, as a characteristic property of the tissue, should exhibit locally spatial continuity, except at the boundary of tissue compartments. The locally spatial continuity has been successfully incorporated into the PET reconstruction,²⁰ the B1 magnetic field correction,²¹ and the DCE-MRI kinetic parameter quantification.²² In addition, the pharmacokinetic modeling of DCE-MRI with spatial regularization reduces both bias and variance of derived kinetic parameters.²²

Tikhonov quadratic regularization, the most commonly used spatial regularization in image applications,^{20, 23} has been proposed for the T₁ estimation from IR MR signals,²⁴ and improves the SNR in the resultant T₁ map. However, it is well-known that the quadratic penalty tends to oversmooth image at the boundaries of tissue compartments.²⁵ Total variation (TV), a nonquadratic regularization, can preserve edges at tissue boundary. In this study, we proposed an efficient method by incorporating the TV regularization in the T₁ NLS cost function in order to reduce voxel-level noise without blurring edges or decreasing the accuracy in the estimates. First, we develop a quadratic surrogate function from a log likelihood cost function according to the majorization principle, and then extend the fast iterative shrinkage thresholding algorithm (FISTA) for the TV-regularized (TVR) least-squares fitting of T₁. To the best of our knowledge, this has not been done before. We also present an efficient quadratically regularized (QR) method for T₁ estimation from the VFA MR images. The proposed methods were evaluated by using synthesized, phantom, and clinical MR data.

MATERIALS AND METHODS

Nonlinear least-squares T₁ estimation

Based upon the Bloch equation,²⁶ a steady-state MR signal intensity (s_k) acquired by a T₁-weighted spoiled gradient-echo sequence with a flip angle (FA) of α_k (k = 1, 2, …, N_FA; N_FA is the number of FAs) and a repetition time TR is given by

$$\begin{array}{c}\hfill s_k=\frac{s_0\sin a_k(1-E)}{1-E\cos a_k},\end{array}$$ (1)

where s₀ is the equilibrium longitudinal magnetization, and E = exp (−TR/T₁). The T₁ and s₀ values at a pixel are conventionally estimated from measured MR signals {y_k} by a NLS fitting:

$$\begin{array}{ccc}& & \Phi ({\mathrm{T}}_1,s_0)=\sum _{k=1}^{N_{\mathrm{FA}}}{(y_k-s_k)}^2\hfill \\ & & \mathrm{subjecting}\mathrm{to}:l{\mathrm{T}}_1\le {\mathrm{T}}_1\le u{\mathrm{T}}_1,\hfill \end{array}$$ (2)

where l T₁ and uT₁ are defined by the T₁ range of the tissue and utilized as a constraint to avoid unrealistic solutions and improve robustness of the computation.

It has been shown that the SNR (${\mathrm{T}}_1/{\sigma }_{{\mathrm{T}}_1}$) of the T₁ values relates to the SNR (s₀/σ) of the images by $({\mathrm{T}}_1/{\sigma }_{{\mathrm{T}}_1})\propto (\mathrm{TR}/{\mathrm{T}}_1)(s_0/\sigma )$,²⁷ indicating the noise in the original images is amplified into the T₁ map by the ratio of T₁/TR. Therefore, the voxels with large T₁ values are prone to potential errors. In addition, the image noise and the limited number of flip angles can cause a solution of the NLS fitting [Eq. 2] to be trapped into a local minimum before reaching the real solution during minimization.²² Although optimally selecting image acquisition parameters can improve the T₁ estimation,^{13, 14} the errors seem to persist. Therefore, we propose to incorporate prior knowledge of tissue T₁ spatial continuity to improve T₁ estimation without compromising its accuracy.

Spatially regularized T₁ estimation

TV-regularized T₁ estimation

TV-regularized T₁ estimation is to minimize the NLS cost function incorporated with TVR which is defined as

$$\begin{array}{ccc}& & \Psi (T_1,s_0)=L(T_1,s_0)+2\lambda TV\left(T_1\right)\hfill \\ \\ & & \mathrm{subject}\mathrm{to}:l{T}_1\le T_1\le u{T}_1,\hfill \end{array}$$ (3)

$$\begin{array}{c}\hfill \mathrm{where}L({\mathrm{T}}_1,s_0)=\sum _{i=1}^N\sum _{j=1}^M\ln \left(1+\Phi \left({\mathrm{T}}_1^{i,j},s_0^{i,j}\right)\right)\end{array}$$ (4)

$$\begin{array}{c}\hfill \mathrm{and}\mathrm{TV}\left({\mathrm{T}}_1\right)=\sum _{i=1}^N\sum _{j=1}^M\sqrt{{\left({\mathrm{T}}_1^{i+1,j}-{\mathrm{T}}_1^{i,j}\right)}^2+{\left({\mathrm{T}}_1^{i,j+1}-{\mathrm{T}}_1^{i,j}\right)}^2},\end{array}$$ (5)

where (i, j) are pixel indices in a 2D space, L is a log likelihood function, and λ is a constant that controls the relative strength of the spatial regularization.

To minimize Ψ over the two unknown parameters (T₁ and s₀), we use a block alternating approach, in which T₁ (T₁-step) and s₀ (s₀-step) are iteratively determined by minimizing one parameter at a time while holding the other at the previously obtained value. To minimize Ψ in the s₀-step, an analytic solution of s₀ is given by

$$\begin{array}{c}\hfill s_0^{i,j}=\sum _{k=1}^{N_{FA}}{f}_k\left(T_1^{i,j}\right)y_k^{i,j}/\sum _{k=1}^{N_{FA}}{\left(f_k\left(T_1^{i,j}\right)\right)}^2,\end{array}$$ (6)

where $f_k\left({\mathrm{T}}_1^{i,j}\right)=\sin a_k(1-E^{i,j})/(1-E^{i,j}\cos a_k).$ However, minimizing Ψ for T₁ in the T₁-step is a nontrivial problem, because Ψ is nonlinearly related to T₁ and the TV function [Eq. 5] is not continuously differentiable. We develop a fast iterative method to minimize Ψ with respect to T₁. First, we convert the log likelihood L to a quadratic surrogate function using the majorization principle. Then, we minimize the TV-regularized surrogate function using an efficient algorithm based on a gradient-based dual approach.

Quadratic surrogate

We develop a quadratic surrogate function of the log likelihood L using the majorization principle.²⁸ The derivation is provided in the Appendix. In brief, L(T₁) is approximated by ‖T₁ − z‖² for T₁ near the nth iteration solution T_1,n, where z = T_{1, n} − v/(2μ) is given in the Appendix. Then, using a matrix format, the TV-based cost function [Eq. 3] at the (n + 1)th iteration becomes

$$\begin{array}{c}\hfill {\Psi }_{n+1}=\mu {‖{\mathrm{T}}_1-z‖}^2+2\lambda \mathrm{TV}\left({\mathrm{T}}_1\right),\end{array}$$ (7)

where μ is the spatially variant weighting defined in the Appendix [Eq. A3]. As demonstrated by Eq. A1 in the Appendix, the quadratic function is always greater than L except at T_1,n, and so can be a surrogate of L.²⁸ Figure 1 plots an original log likelihood function L(T₁) with the true minimum at T₁* = 1000 and quadratic surrogate functions at T_1,n = 700 ms and T_1,n = 1300 ms. At each iteration, the minimum of the surrogate function moves toward the minimum of L (T₁*).

Figure 1.

Quadratic surrogates of the likelihood term of the cost function. The likelihood function has a minimum at T₁ = 1000. The quadratic surrogates equate the likelihood function at T₁ = 700 and 1300, but are above the likelihood curve at all other locations. The minima of the quadratic functions are converging toward the minimum of the likelihood function over iterations.

Optimization of the TV-regularized surrogate function

The surrogate function transforms the original cost function [Eq. 3] to Eq. 7 which is a typical TV-based denoising problem with a spatially variant weight μ and a spatially invariant weight λ. Minimization of Eq. 7 still is a challenging problem due to the noncontinuously differentiable TV term. Beck²⁹ proposes to convert the TV-based minimization to a smooth dual problem and then solves the dual problem using FISTA. According to the approach, two new spatially distributed parameters (p, q) are introduced to convert the TV-based cost function equation 7 to a smooth dual function. The minimizer of the smooth dual function (p, q) is determined by iterating

$$\begin{array}{ccc}\hfill (p_t,q_t)& =& (p_{t-1},q_{t-1})\hfill \\ & & +\frac{1}{8(\lambda /\mu )}{\Gamma }^T\left(P_c\left(z-\frac{\lambda }{\mu }\Gamma (p_{t-1},q_{t-1})\right)\right),\hfill \end{array}$$ (8)

where P_c and Γ are two projection operators,²⁹ and t is the iteration index. The iteration started with (p, q) being 0. After convergence of iteration 8, the T₁ solution at the (n + 1)th step is computed as

$$\begin{array}{c}\hfill {\mathrm{T}}_{1,n+1}=P_c\left[z-\frac{\lambda }{\mu }\Gamma (p,q)\right].\end{array}$$ (9)

The final solutions of s₀ and T₁ are sought via these interleaved iterations of s₀-step and T₁-step.

Quadratic-regularized T₁ estimation

We also develop a method to estimate T₁ with the conventional quadratic regularization, namely, QR, which is to minimize

$$\begin{array}{c}\hfill \overline{\Psi }(T_1,s_0)=L(T_1,s_0)+\beta R\left(T_1\right)\end{array}$$ (10)

with $R\left({\mathrm{T}}_1\right)=\frac{1}{2}{\sum }_{i=1}^N{\sum }_{j=1}^M{\sum }_{m,n}{({\mathrm{T}}_1^{i,j}-{\mathrm{T}}_1^{i-m,j-n})}^2$, the pair index (m, n) = (1,0), (–1,0), (0,1), (0,–1) denotes the coordinate offsets of the four nearest neighbors, and β is a weighting parameter for the quadratic regularization. The minimization of Eq. 10 is also done by interleaved optimizations of s₀ and T₁. At the T₁-step, again, the log likelihood L(T₁, s₀) is converted to the quadratic surrogate function. As both terms in Eq. 10 become quadratic, T₁ at the (n + 1)th iteration is analytically solved as

$$\begin{array}{c}\hfill {\mathrm{T}}_{1,n+1}={\mathrm{T}}_{1,n}-\frac{1}{\mu +4\beta }\left(\frac{\nu }{2}+\frac{\beta }{2}\nabla R\left({\mathrm{T}}_{1,n}\right)\right),\end{array}$$ (11)

where matrix μ and v are given in the Appendix. The final solutions of the QR method are obtained by iterating s₀-step [Eq. 6] and T₁-step [Eq. 11].

Implementation of T₁ estimation algorithms

NLS T₁ estimation

We implement the voxel-based NLS T₁ estimation (Eq. 3 with λ = 0) by using the quadratic surrogate function, and refer it as QS-NLS method. The computation is initialized with T₁ = 800 ms at each voxel and cycles through s₀-step [Eq. 6] and T₁-step. In T₁-step, the solution at the (n + 1)th iteration is analytically determined as

$$\begin{array}{c}\hfill {\mathrm{T}}_{1,n+1}={\mathrm{T}}_{1,n}-\frac{v}{2\mu }.\end{array}$$ (12)

The iteration is terminated when the T₁ tolerance meets |T_{1, n + 1} − T_{1, n}|/|T_{1, n}| ⩽ 10⁻⁶ or the number of iterations n ⩾ 500.

TV-regularized T₁ estimation

The TVR minimization begins with the T₁ estimated by the QS-NLS method. Then s₀ and T₁ are sought iteratively through s₀-step [Eq. 6] and T₁-step [Eq. 9]. This iterative process is terminated when the relative change in T₁ < 10⁻⁶ or the number of iterations n ⩾ 250. In each T₁-step, the T₁ [Eq. 9] is determined by (p, q) that result from iteration of Eq. 8. The iteration to obtain (p, q) terminates when the (p, q) tolerance is <10⁻⁶ or the iteration numbern ⩾ 100. Selection of λ will be described below.

Quadratic regularized T₁ estimation

The QR optimization is also initialized with the T₁ obtained by the QS-NLS method, and the iterative process is terminated when T₁ tolerance is less than 10⁻⁶ or the number of iterations n ⩾ 250.

Selection of weighting parameters

The “hyperparameter” λ and β weigh the spatial regularization terms in the TVR and QR cost functions, respectively. Large values of the parameters can cause overweighing of the spatial regularizations, and lead to oversmoothing in the resulting T₁ map; conversely, small values can result in insufficient noise reduction. To determine an appropriate weighting value of λ (or β), we estimated T₁ maps of a central slice in the simulated brain MR data with various noise levels using the TVR (or QR) method and varying λ (or β) values on a regular grid. We selected the λ (or β) value that minimized the averaged difference between the true and estimated T₁ in the central slice. We found that λ = 1 and β = 0.001 were appropriate for the images with noise levels less than 7%, and λ = 10 and β = 0.01 for the data with higher level noise. We applied these values to both the simulated and experimental image data for evaluation of the methods.

T₁ bound constraints

For all three methods of QS-NLS, QR, and TVR, T₁ bound conditions were set to be l T₁ = 50 ms and uT₁ = 3000 ms in the simulated and phantom experiments, and uT₁ = 6000 ms for the patient data.

Simulation studies

In order to evaluate the performance of the proposed methods, we utilized the Brainweb MRI simulator to simulate VFA MRI data.³⁰ This simulator accounts for the effects of various image acquisition parameters, including partial volume averaging, noise, and sampling in Fourier domain, and intensity inhomogeneity in the brain tissue.³¹ We simulated T₁-weighted MRI of a normal brain phantom using a spoiled gradient-echo pulse sequence (TR / TE of 18/10 ms, matrix size of 217 × 181 × 60, resolution of 1 × 1 × 3 mm³, and flip angles of 5°, 10°, 20°, 30°, and 40°). Gaussian noise was added onto both real and imaginary components of MR signals in the Fourier domain to obtain noise levels of 1%, 3%, 5%, 7%, and 9% in the images. The QS-NLS, QR, and TVR methods were applied to the simulated data to estimate T₁. Because of the high in-plane resolution, spatial regularizations in both QR and TVR methods were applied in 2D images.

In order to quantitatively evaluate the methods, relative mean and relative standard deviation (rSD) are computed as the percentages of the mean and SD of the estimated T₁ to the true T₁ value, respectively. For the rSD calculation, the original T₁ variations in WM or GM were removed. The relative mean and SD measure the accuracy and stability of the methods, respectively. For statistical analysis in GM and WM, we excluded the voxels within a 2-pixel wide band from the boundary to remove the effect of blurring edges.

We also examined whether the spatially regularized methods can estimate a T₁ map with spatial variations. We created a digital phantom that consists of the regions with (a) linear and quadratic T₁ spatial variations, (b) uniform but high T₁ value (1200 ms), and (c) uniform but low T₁ values (200 ms). MR images of the digital phantom were simulated using the Bloch equation [Eq. 1] with five FAs (10°, 20°, 30°, 50°, and 70°) and TR 13 ms. Random Gaussian noise was added to make the MR data have a noise level of 5%.

Experimental MRI studies

We applied the methods to experimental MR data of a EuroSpin TO5 phantom from the RIDER project,³ which was designed to evaluate the accuracy and repeatability of MR T₁ measurement. The phantom consisting of 18-compartments with different T₁ contrasts was imaged on a 1.5 T GE scanner.³ The acquisition protocol included a 2D IR spin-echo sequence and a 3D multiple flip angles fast-spoiled gradient-echo sequence (FSPGR). The FSPGR images were acquired with seven flip angles (2°, 5°, 10°, 15°, 20°, 25°, and 30°), TR/TE = 6.4/1.2 ms, and a resolution of 0.55 × 0.55 × 5 mm³. Our methods were applied to the MRI data of five of the seven flip angles (5°, 10°, 15°, 20°, and 25°) to estimate T₁. The correlation of the compartmental means of the T₁ estimated from the IR and the VFA data was computed to evaluate the accuracy of the T₁ quantification by the proposed methods.

The methods were also tested on the brain MRI of a patient who was enrolled in a prospective DCE-MRI study of brain radiation therapy. Multiflip angle MR data were acquired on a clinical 3 T MR scanner (Ingenia, Philips Medical Systems, Best, Netherlands) with a fast spoiled gradient-echo sequence. The image parameters were TR/TE of 30 ms/2.8 ms, matrix size of 256 × 256 × 80, resolution of 1 × 1 × 2 mm³, and flip angles of 5°, 15°, 20°, and 45°. To evaluate the proposed methods, the patient was also imaged using a SR sequence with the TR of 100, 200, 500, 1000, and 2000 ms. The SR images have the same resolution and matrix as the VFA MRI. T₁ values were estimated voxel-by-voxel from the SR MRI by nonlinear least-square fitting.

RESULTS

Simulated brain

Figure 2 shows estimated T₁ maps of a slice in the simulated brain with 5% of noise using the QS-NLS, QR, and TVR methods, as well as relative differences between the estimated T₁ and the ground truth. There are substantial amounts of noise in the regions of GM, WM, and cerebrospinal fluid (CSF) of the T₁ map and the difference map calculated by the QS-NLS method [Figs. 2a, 2d], indicating the QS-NLS method propagates or even amplifies noise from the original images onto the T₁ map. In contrast, the QR and TVR methods reduce the noise in each of tissue compartments substantially [Figs. 2b, 2c, 2e, 2f]. However, the relative differences between the T₁ estimated by the QR method and the true values are greater at edges of tissue compartments and in the small regions [see edge enhancement in Fig. 2e], indicating the effect of overblurring of the QR method. The TVR method preserves boundaries well while producing marked reduction of the intraregion noise in the T₁ map [Figs. 2c, 2f].

Figure 2.

T₁ maps (top) estimated from the simulated brain dataset with five FAs and 5% noise by the QS-NLS (a), QR (b), and TVR (c). The relative differences between the estimated T₁ and the ground truth are displayed in bottom row (d)–(f). The T₁ values range from 0 to 3000 ms in (a)–(c). The relative differences vary from –0.2 to 0.2 in (d)–(f).

Quantitative evaluations of the performance of the methods with respect to noise on the simulated data are shown in Fig. 3. The means of the T₁ estimated by all the three methods show less than 3% errors from the true T₁ values in both GM and WM with image noise up to 9% [Figs. 3a, 3b]. The relative SDs of the T₁ estimated by the QS-NLS method increase nonlinearly with noise in the images, reaching approximately 12% and 15% when there are 7% and 9% noise [Figs. 3c, 3d], respectively. In comparison, the TVR method is able to suppress the relative standard deviations of the T₁ values to less than 2% in both WM and GM regardless of the noise level in the image. The QR method also reduces the noise-related variations in the T₁ values to be less than 3%, which is slightly greater than the one obtained by the TVR method and possibly due to the edge-blurring effect of the quadratic regularization [Figs. 3c, 3d].

Figure 3.

The relative means and standard deviations of the T₁ values in WM (a) and GM (b) estimated by the QS-NLS, QR, and TVR methods vs noise levels in the original images. (c) and (d) are the relative standard deviations for WM and GM with the inner-structure T₁ variation (rSD at noise level 0%) removed. The three methods have a similar accuracy, but the QR and TVR methods substantially decrease the standard deviations of the estimated T₁ values for noise >3%.

Simulated digital phantom with spatial variations

Evaluation of the performance of the spatially regularized methods on the simulated phantom with linear and quadratic spatial variations in T₁ is shown in Fig. 4. As expected, the T₁ map estimated by the QS-NLS method is very noisy, and the central region, where the T₁ value is homogeneous and high, cannot be recognized [Figs. 4c, 4e]. The noise in the T₁ map estimated by the TVR method is reduced substantially, and the T₁ estimates clearly differentiate the homogeneous region from the surrounding [Fig. 4d]. Figure 4e plots profiles of the T₁ values along a horizontal line of the phantom estimated by the QS-NLS and TVR methods. Compared to the true T₁ values along the line, the TVR method reduces the variance in the T₁ values to be 3% from 7% by the QS-NLS method.

Figure 4.

T₁ maps of a simulated digital phantom with spatial variations of T₁: homogeneous low T₁ values at the left and right peripheries, homogeneous high T₁ values at the central region, and linearly and quadratically changed T₁ values from the low to high T₁ values in left and right, respectively. (a) The simulated T₁-weighted MRI with 5% of noise and a flip angle of 20°; (b) the T₁ map without noise; (c) the T₁ map estimated by the QS-NLS methods; (d) the T₁ map obtained by the TVR method; and (e) central-line profiles of the T₁ maps obtained by the QS-NLS and TVR methods compared to the true values.

Phantom experiments

Figure 5 shows T₁ maps of the 18-compartment RIDER phantom estimated by the QS-NLS, QR, and TVR methods from the VFA MR data. Using the T₁ values in each of the compartments estimated from the IR acquisition as a reference, the pixelwise relative differences between the T₁ values estimated by the three VFA-based methods and the reference are shown in Figs. 5d, 5e, 5f. Again, the TVR method produces the T₁ map with better noise reduction and better edge preservation compared to the QS-NLS and QR methods. It is interesting to see that the QR method causes underestimation of T₁ values [enhancement in Fig. 5e] around the tube edges due to overblurring of the quadratic regularization.

Figure 5.

The T₁ maps (top) of a phantom estimated from the VFA MR data using the QS-NLS (a), QR (b), and TVR methods (c), and the relative differences (bottom) of the estimated T₁ between the three VFA-based methods and the IR method (d)–(f). Gray bars denote the ranges of the T₁ values (top) and the relative differences (bottom).

T₁ measurements of the RIDER phantom are shown in Fig. 6. There are strong linear correlations (R² > 0.99) between the compartmental averaged T₁ values estimated from the VFA MR data using the QS-NLS, QR, and TVR methods and the T₁ measured from the IR MRI (a reference measure). The SDs obtained by the QS-NLS method increase with the T₁ values but not by the TVR method. Overall, the TVR method reduces the SDs of the T₁ values by a factor of 2–4 in most of the compartments compared to the QS-NLS method. The SDs obtained by the QR method are similar to the ones obtained by the QS-NLS method due to the edge-blurring effect of the QR method.

Figure 6.

Plots of the compartmental means of the estimated T₁ from the phantom MRI using the QS-NLS (left), QR(middle), and TVR methods (right) vs T₁ values from the IR MRI data. The error bars are the standard deviations of the T₁ values estimated from the VFA MR data in each compartment.

Human brain study

Figure 7 shows T₁ maps of the human brain of a patient by using the QS-NLS, QR, and TVR methods. Compared with the T₁ estimated by the SR method [Figs. 7d, 7h], the T₁ maps estimated by the QS-NLS method [Figs. 7a, 7e] are presented with noise, which could compromise physiological parameters derived from DCE-MRI if used in DCE quantification. The T₁ maps obtained by the QR method [Figs. 7b, 7f] show noise reduction but also edge blurring, as evidenced by enhanced edges on the relative difference maps between by the QR and SR methods. However, the TVR method [Figs. 7c, 7g] not only reduces noise but also preserves structure boundaries in the T₁ maps, as shown in the relative difference maps between the TVR and SR methods.

Figure 7.

T₁ maps of two brain slices (first and third rows) from a patient data by the QS-NLS (a) and (e); QR (b) and (f); TVR (c) and (g); and SR (d) and (h) methods. The relative differences of the three VFA-based T₁ estimates to the SR results are shown, respectively, in the second and forth rows for the two slices.

The ROIs related to five different tissue structures were manually delineated on the brain images. Compared with the T₁ values obtained by the SR method, the mean T₁ in the ROIs estimated by QS-NLS, QR, and TVR methods has errors less than 5% (Fig. 8). But the QS-NLS results have greater T₁ variations in the ROIs compared with the spatially regularized methods. Due to edge blurring, the QR method increases the standard deviation of T₁ estimates in gray matter, which is also shown in the difference maps in Fig. 7.

Figure 8.

Mean T₁ in the five brain ROIs on the patient images by the QS-NLS, QR, TVR, and SR methods.

DISCUSSION

In this paper, we propose spatially regularized methods to improve T₁ estimation based upon prior knowledge of T₁ local continuity. We develop a theoretical framework for fast iterative optimization of the spatially regularized T₁ quantification, including majorizing the log likelihood function to a quadratic surrogate function, converting the TV-based optimization to a smooth dual problem, and providing an iterative solution for T₁ calculation. The TVR method substantially reduces the noise in the T₁ map without image blurring or decreasing the accuracy of the estimated T₁ values. Given that the quadratic surrogate only uses the T₁ gradient of the MR signals, our methods can be generalized to estimate T₁ values from MR data acquired with variable flip angles, variable TRs, or a combination of the two. Incorporating the T₁ produced by our methods into quantification of DCE-MRI will improve the robustness of the pharmacokinetic analysis, reduce potential errors in local perfusion estimates, and possibly improve the spatial discrimination of local changes in perfusion parameters.

The basic assumption of spatial regularization is the T₁ local continuity in the tissue. To reduce the noise influence on the T₁ estimation, we enforce local continuity of the T₁ values while permitting rapid changes of T₁ at tissue boundaries. Our results show that the quadratic regularization, although reducing noise in the T₁ map compared to the QS-NLS method, overblurs the boundaries at the tissue compartments. In contrast, the TVR method is able to overcome the overblurring problem present in the quadratic regularization method as well as reduce the noise in the T₁ map. A gradual T₁ change in a region can also be preserved by the TVR method. These two spatial regularization methods can be selected in the clinical application based upon the organ or anatomy of interest. The organs with fine tissue compartments, e.g., brain and kidney, could benefit from the TVR method. One of the potential benefits of the spatially regularized methods for the T₁ estimation is voxel-by-voxel quantification of DCE-MRI using pharmacokinetic models, e.g., Toft model.³² The noise in the T₁ map can propagate into the DCE quantification and reduce reproducibility of derived kinetic parameters,³³ which could subsequently reduce the ability of these quantitative imaging parameters as a biomarker for assessment of tumor and normal tissue response to therapy. The spatially regularized methods have the potential to overcome this challenge.

The total variation, as an edge-preserving regularization, has been long considered for image restoration. However, the total variation has not been widely used in the medical image field due to the complexity of the method, which involves optimizing a noncontinuously differentiable TV function. The optimization becomes even more computation demanding when the likelihood function is a complicate nonlinear function, i.e., the T₁ problem. A quadratic surrogate can be used to replace the nonlinear least-squares likelihood function, and therefore to simplify the optimization process. However, direct majorization of the original nonlinear least-squares cost function [Eq. 2] to a quadratic surrogate results in a slow converging process in the T₁ minimization due to the large Lipschitz constant.³⁴ Instead, we majorize a logarithm likelihood function to a quadratic surrogate that converges much more rapidly. Furthermore, we apply the gradient-based dual approach that has been theoretically proven to have a converging rate in the order magnitude of 1/t² (t is the iteration number).²⁹ Our experiments show that the T₁ solution can converge within 100 iterations with a tolerance of 10⁻⁶.

The computation of the QS-NLS method is much faster than the conventional NLS T₁ estimation,¹⁸ due to that the QS-NLS method updates T₁ of all pixels in an image slice simultaneously in each of the iterations, while the conventional NLS method minimizes the cost function voxel-by-voxel. For the computation of the simulated data on a Xeon 2.668 GHz machine and using MATLAB 2010b, it takes ∼6 s to compute T₁ of a 256 × 256 slice by the QS-NLS method, but 8 min by the conventional NLS method (using “fminsearch” in MATLAB). Also, the differences between the T₁ computed by the conventional NLS and the QS-NLS method are less than 1%, indicating there is no compromise in the accuracy of the estimated T₁ using the QS-NLS method. With the spatial regularizations, both the QR and TVR minimizations are converged within 100 iterations (Fig. 9). Initialized with the T₁ estimated by the QS-NLS method, the QR and TVR methods take additional 1.2 s and 15 s to compute the final T₁ values of a 256 × 256 slice, respectively.

Figure 9.

Convergence of the QR and TVR methods from the simulated brain data with five FAs and 5% noise. The cost functions are normalized to the maximal value at the initial iteration. The two methods reach a tolerance of 1 × 10^–6 within 100 iterations.

The TVR method preserves edges at tissue boundaries in a T₁ map. However, the minimum size of an object or a lesion, which can be detected by the TVR method, has not been tested and compared to the conventional methods. Like other VFA-based methods, B1 inhomogeneity can bias the T₁ estimated by the proposed methods because B1 inhomogeneity causes variation of flip angles. Approaches have been proposed to map B1 inhomogeneity.^{16, 35} Therefore, correcting the flip angles can be performed prior to using the proposed methods.

CONCLUSION

T₁ estimation based upon variable flip angles gradient-echo MRI has been improved by applying the prior knowledge of spatial continuity. Spatial regularization, either QR or TVR, can reduce random fluctuation in the T₁ estimates compared to the conventional NLS method, and thereby improve the repeatability of the pixelwise estimates. The TV regularization can preserve the sharp transitions between the tissue compartments better than the QR method.

APPENDIX: QUANDRATIC SURROGATE

The log likelihood L near the nth iteration solution of T_1,n at pixel (i, j) can be approximated by

$$\begin{array}{ccc}& & L\left(T_1^{i,j}\right)=\ln [1+\sum _{k=1}^{N_{FA}}{\left(y_k^{i,j}-s_0^{i,j}{f}_k\left(T_1^{i,j}\right)\right)}^2]\le L\left(T_{1,n}^{i,j}\right)\hfill \\ & & +\frac{{\sum }_{k=1}^{N_{FA}}\left[{\left(y_k^{i,j}-s_0^{i,j}{f}_k\left(T_1^{i,j}\right)\right)}^2-{\left(y_k^{i,j}-s_0^{i,j}{f}_k\left(T_{1,n}^{i,j}\right)\right)}^2\right]}{1+{\sum }_{k=1}^{N_{FA}}{\left(y_k^{i,j}-s_0^{i,j}{f}_k\left(T_{1,n}^{i,j}\right)\right)}^2}.\hfill \end{array}$$ (A1)

The right-hand side of the inequality is greater than or equal to the original $L\left({\mathrm{T}}_1^{i,j}\right)$, and therefore, can be a surrogate for L.²⁸ By defining $d_k\left({\mathrm{T}}_1^{i,j}\right)=1/(1-E^{i,j}\cos \left({\alpha }_k\right))$, a_k = tan α_k, and b_k = tan α_k(cos α_k − 1), we have f_k(T₁) = a_k + b_kd_k(T₁). Substituting f_k(T₁) into the right side of Eq. A1 and applying the first order Taylor expansion to d_k(T₁), Eq. 3 in matrix format at the (n + 1)th iteration of T₁-step becomes

$$\begin{array}{c}\hfill {\Psi }_{n+1}=\mu {‖{\mathrm{T}}_1-z‖}^2+2\lambda \mathrm{TV}\left({\mathrm{T}}_1\right),\end{array}$$ (A2)

where z = T_{1, n} − v/(2μ), μ = {μ^{i, j}}, and ν = {ν^{i, j}} are matrix formats of μ^{i, j} and ν^{i, j} over the images computed as

$$\begin{array}{ccc}\hfill {\mu }^{i,j}& =& \frac{{\left(s_0^{i,j}\right)}^2{\sum }_{k=1}^{N_{\mathrm{FA}}}{\left(b_k\nabla d_k\left({\mathrm{T}}_{1,n}^{i,j}\right)\right)}^2}{1+{\sum }_{k=1}^{N_{\mathrm{FA}}}{\left(y_k^{i,j}-s_0^{i,j}{f}_k\left({\mathrm{T}}_{1,n}^{i,j}\right)\right)}^2}\hfill \\ \hfill v^{i,j}& =& \frac{2{\left(s_0^{i,j}\right)}^2{\sum }_{k=1}^{N_{\mathrm{FA}}}{b}_k^2\left(d_k\left({\mathrm{T}}_{1,n}^{i,j}\right)+\frac{1}{b_k}\left(a_k-\frac{y_k^{i,j}}{s_0^{i,j}}\right)\nabla d_k\left({\mathrm{T}}_{1,n}^{i,j}\right)\right)}{1+{\sum }_{k=1}^{N_{\mathrm{FA}}}{\left(y_k^{i,j}-s_0^{i,j}{f}_k\left({\mathrm{T}}_{1,n}^{i,j}\right)\right)}^2},\hfill \end{array}$$ (A3)

where $\nabla d_k\left({\mathrm{T}}_{1,n}^{i,j}\right)$ is a gradient of d_k with respect to T₁ at T_{1, n}. The quadratic surrogate function is always above L except at the point T_1,n, and thus iteratively converges to the minimum of L. Now, minimization of Eq. 3 becomes a conventional TV-based denoise problem.³⁶