Direct Reconstruction of Kinetic Parameter Maps in Accelerated Brain DCE-MRI using the Extended-Tofts Model

Yi Guo^{1}, Sajan Goud Lingala^{1}, Yinghua Zhu^{1}, R. Marc Lebel^{2}, and Krishna S Nayak^{1}

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 l_{1} 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: 0.9×1.3×7.0mm^{3},
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 1. Flowchart of the mapping from PK parameter maps (K^{trans}, K_{ep}, and v_{p} via the
eTofts model) to multi-coil under-sampled
k,t-space data.

Figure 2. K^{trans}
maps from two glioblastoma patients. Direct reconstruction from fully-sampled
(k,t)-space (third column) almost exactly matched conventional eTofts modeling from
fully-sampled images (second column), with rMSE=0.0043. The eTofts based direct
reconstruction provides faithful restoration of K^{trans} values at R=20 (fourth column), with rMSE=0.0195,
overcoming the underestimation using Patlak model (first column).

Figure 3. Objective
function versus iteration number for three initial PK map estimates (left), and
cropped portions of these initial and final Ktrans maps at an
undersampling factor of 20x (right). All initial guesses converged to the same
final solution, indicating that this method is robust to local minima. This
observation was made for both datasets.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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