Brain dynamic contrast enhanced MRI (DCE-MRI) is a powerful technique to non-invasively assess blood brain barrier (BBB) permeability, and other neurovascular parameters. These include plasma volume (v_p), the forward (K^trans) and backward (K^ep) transfer constants between plasma and extravascular extracellular spaces. Sparse sampling with constrained reconstruction has been applied to improve imaging tradeoffs in DCE-MRI. Pre-determined transform constraints have been used (eg. finite difference, wavelet) [1,2]. Recent work demonstrates kinetic model based constraints to provide superior fidelity in parameter mapping [3,4]. Here, we propose to improve a previous model [4] in several aspects. We generalize it to include constraints derived from the extended-Tofts model. We perform noise sensitivity analysis to determine the accuracy and precision of parameter mapping with the proposed e-Tofts derived temporal bases. We demonstrate its utility in retrospectively accelerating brain tumor DCE datasets with different tumor characteristics.

Construction of temporal dictionary: As depicted in Fig.1, we simulate concentration vs. time profiles for a broad range of physiological kinetic parameters (K^trans=0-0.4 min^-1 in steps of 0.01 min^-1, Kep = 0-0.6 min^-1 in steps of 0.01 min^-1, vp=0-40% in steps of 1%), using a population based arterial input function [5], and the e-Tofts model [6]. We utilize k-SVD [7] to construct a dictionary of temporal basis functions (denoted V_rxN) that sparsely represent the simulated concentration profiles; r denotes the number of bases in the dictionary, and N denotes the total number of time instances. For over-completeness, the dictionary size r=100 was chosen to be larger than N=50. The sparsity parameter k was chosen based on noise sensitivity analysis described below, where the objective was to determine the smallest k for which the resulting k-sparse projected dictionary modeled profiles yield no parameter bias, and comparable variance, after e-Tofts modeling.

Noise sensitivity analysis: We analyzed the statistics of kinetic parameter estimation from 100 realizations of concentration profiles corrupted by white Gaussian noise (zero mean, standard deviation=0.001). From Fig.2, note that as k is increased the error statistics in estimating the kinetic parameters from noisy data converge towards error statistics of kinetic parameter mapping with the e-Tofts model. For low values of k ≤ 2, we observed bias, and reduced variance, and for high values of k > 10 (not shown), we observe increased uncertainty. k=3 or 4 provided the best compromise, closely mimicking e-Tofts modeling.

Reconstruction: We estimate concentration time profiles X_MxN (M-number of pixels; N- number of time frames) from the under-sampled k-t space data (b) as:$$ \min_{\mathbf X, \mathbf U}\|\mathbf X-\mathbf U\mathbf V\|_2^{2}; \mbox{such that } \|u_i\|_0=4; \|A(\mathbf X)-\mathbf b\|_2^{2}<\epsilon$$ A models Fourier under-sampling, coil-sensitivity encoding, and transformation between concentration and signal using knowledge of T1, M0, and flip angle maps obtained from calibration data; ε denotes the noise level in k-t space. U_MxrV_rxN denotes the 4-sparse projection of X in the dictionary V. The above is solved by iterating between (a) updating U using orthogonal matching pursuit sparse projection [7], and (b) enforcing consistency with acquired data X. We iterate until a stopping criterion of ||X_i-X_i-1||₂< 10^-6 is achieved. After the concentration-time profiles are obtained, we estimate kinetic parameters by fitting the profiles to the e-Tofts model.

Analysis: We perform retrospective under-sampling experiments on fully-sampled DCE-MRI data sets (3T, Cartesian T1 weighted spoiled gradient echo, FOV: 22x22x4.2cm³ resolution: 0.9x1.3x7 mm³; 5 sec temporal resolution) from three glioblastoma brain tumor patients with different tumor characteristics (shape, size, and heterogeneity). k-t undersampling was performed using a randomized golden angle trajectory [8].

Fig. 3 shows comparisons on a large glioblastoma tumor (~6.5cm). The kinetic maps derived from fully sampled data after 4-sparse projection on V (second column) are in excellent agreement with parameter maps derived directly from fully sampled data (R=1) (first column). This is attributed to the equivalence of 4-sparse projection based dictionary modeling with e-Tofts model (also verified in Fig. 2). With 20-fold undersampling, the proposed approach depict good fidelity in the parameter maps, depicting the various characteristics of the heterogonous tumor well including thin rim margins, tumor core. Fig. 4 shows comparisons on two glioblastoma tumors with different shapes and characteristics (only K^trans maps shown), where consistent performance with the proposed approach was observed. We proposed a tracer-kinetic model based reconstruction to enable highly accelerated DCE-MRI. The approach efficiently leverages information of contrast agent kinetic modeling into the reconstruction, and provides a novel alternative to current constraints that are blind to tracer kinetic modeling. The approach is flexible, and could leverage complementary spatial-sparsity constraints. Although demonstrated on brain DCE-MRI, it can be applied to other body parts.