T1-weighted dynamic contrast enhanced (DCE) MRI followed by pharmacokinetic (PK) modeling provides quantitative kinetic parameter maps (e.g. K^trans, K_ep, v_p) that aid in the diagnosis and assessment of treatment response. Recently, efforts have been made to directly reconstruct PK maps from under-sampled k,t-space data [1-3] and provide acceleration and/or reduced PK parameter variability compared to conventional indirect methods that perform PK modeling after reconstruction of a time series of anatomic images. Our prior work [3] utilized a Patlak model which assumes no backflux from interstitium to plasma space, and is applicable to studies of slow blood-brain-barrier (BBB) leakage [4] and short duration scans [5]. In this work, we have developed an extension of this direct reconstruction to the extended-Tofts model (eTofts) [6], which is known to be more appropriate for assessing brain tumor margin and areas with high BBB leakage [7]. We demonstrate convergence behavior, computational efficiency, and application to retrospectively under-sampled brain DCE-MRI.

Figure 1 illustrates the DCE-MRI forward model of mapping PK parameter maps to k,t-space data. The eTofts model is defined as $$$C_t(t)=K^{trans}\int_{0}^{t}C_p(u)e^{-K_{ep}(t-u)}du+v_pC_p(t) $$$, where C_t(t) is the contrast concentration in the tissue, C_p(t) is the arterial input function (AIF). A population-averaged AIF [8] was used in this study. We formulate the estimation of K^trans, K_ep and v_p as the following least-square optimization problem: $$(K^{trans},K_{ep},v_p)=\underset{K^{trans},K_{ep},v_p}{argmin}||k_u-y(K^{trans},K_{ep},v_p)||_2^2+\lambda_1||\Psi K^{trans}||_1+\lambda_2||\Psi K_{ep}||_1+\lambda_3||\Psi v_p||_1$$

,where K^trans, K_ep, and v_p maps are consistent to under-sampled k-space k_u by a general function y that incorporates all steps including eTofts modeling, T1-weighted signal equation, coil sensitivity, and under-sampling matrix, as illustrated in Figure 1. Sparsity is enforced by minimizing l₁ norm of the wavelet transform domain (Ψ) of the parameter maps. A closed form gradient of the cost function with respect to each PK parameter is evaluated, and a gradient-based l-BFGS algorithm is used to efficiently solve the optimization problem [9]. Five fully-sampled DCE data sets from brain tumor patients were acquired in a 3T GE scanner (FOV: 22×22cm, spatial resolution: 0.9×1.3×7.0mm³, 5 sec temporal resolution, 50 time frames, fast spoiled gradient echo sequence). Patient data were retrospectively under-sampled in the k_x-k_y plane, simulating the k_y-k_z plane in a 3D whole-brain acquisition [10], using a randomized golden-angle sampling pattern [11]. Reconstruction results at undersampling factor of 20 (R=20) were compared to PK parameter maps computed from fully-sampled images using eTofts and Patlak modeling.

Figure 2 shows K^trans maps from two representative cases. Consistent with literature [7], Patlak modeling underestimated K^trans in the tumor margin, compared to eTofts modeling. K^trans maps computed using direct reconstruction of fully sampled k-space data are consistent with the conventional approach, the averaged root Mean-Square-Error (rMSE) in the tumor regions of interest is 0.0043. The direct reconstruction provide faithful restoration of K^trans values at R=20, providing accurate depiction of tumor boundaries and tumor values (rMSE=0.0195). Estimation of K_ep in eTofts modeling has high variance even from fully-sampled data (not shown). However, accounting for K_ep improves the accuracy of K^trans maps. Figure 3 shows the objective function as a function of iteration number for different initial PK map guesses. The reconstruction converged to the same solution irrespective of all the initialization, suggesting the robustness of the optimization to local minima. The reconstruction for a single slice dynamic images (k-space data matrix size 256x256x50x8) took approximately 6 minutes on a laptop computer using Matlab. This is significantly faster than nonlinear least-square fitting algorithms commonly used in DCE-MRI (eg. around 50 minutes using ROCKETSHIP, a recent DCE-MRI toolbox [12]) to solve PK parameters from fully-sampled data sets. The gains in computation times are achieved because an analytic gradient is used instead of a numerical difference as gradient approximation. We demonstrate the feasibility of direct reconstruction of DCE-MRI kinetic parameter maps from highly under-sampled k,t-space using the extended-Tofts model. The direct reconstruction allows for better utilization of what is known about contrast agent kinetics, and allows for efficient parameter estimation. A gradient-based algorithm was found to be robust to local minima, providing fast reconstruction times. Extensions to incorporate more sophisticated model such as the 2-compartment exchange model may be feasible. Increasing model complexity does have the effect of improving data fit, but increasing the variance of estimated PK parameter maps. More temporal frames and higher temporal resolution may be required; this is a well-known a trade-off in PK model selection for DCE-MRI [13]. A population-averaged AIF was used in this study [8], but it is possible to extract patient-specific AIFs from the under-sampled data [14], or jointly estimate the AIF with the PK maps.