Sajan Goud Lingala^{1}, Yi Guo^{1}, Naren Nallapareddy^{2}, Yannick Bliesener^{1}, R Marc Lebel^{3}, and Krishna S Nayak^{1}

We propose a novel nested tracer-kinetic (TK) model based constrained reconstruction method for DCE-MRI reconstruction from under-sampled data. This approach models the concentration time profiles as a sparse linear combination of temporal bases constructed from TK models of varying complexity. Subspaces from the models of plasma volume, Patlak, and the extended-Tofts are constructed. A spatial mask determining the TK model complexity at every pixel location is derived. Reconstruction involves iteration between data consistency and pixel wise projection of the concentration profiles on one of the three subspaces. We demonstrate its utility in retrospective under-sampled reconstruction of brain tumor DCE-MRI datasets.

Construction of TK model subspaces: The pipeline of
constructing temporal sub-spaces from different TK models is shown in Figure 1.
For the v_{p} model, we use a known arterial input function (AIF) as
the subspace. For the Patlak, and the e-Tofts models, libraries of
concentration-time profiles are simulated in the range of K^{trans}=0-0.8 min^{-1}, v_{p}=0-60%, and v_{e}=0-80%. A broad range of parameters is chosen to ensure coverage of the physiological kinetic parameter space. k-SVD ^{9} is used to derive subspaces
(denoted by $$$V_{r \times N}$$$ ), where any time profile in the
library is modeled as a “k-sparse” combination of bases from
the sub-space. Here $$$r$$$ represents the number of temporal bases and $$$N$$$ the number
of time instances. Previous studies showed that k=2 and 3 respectively
was adequate for modeling time profiles from the Patlak and e-Tofts
models (also see Figure 2).

Nested TK model-based reconstruction: We pose the estimation of the concentration-profiles ( $$$\mathbf \Gamma_{M \times N}$$$ $$$M$$$- number of pixels; $$$N$$$ - number of time frames), and the sparse coefficient matrix $$$\mathbf U_{M \times r}$$$ from under-sampled k-t data $$$(\mathbf b)$$$ as:$$\begin{align} min_{\mathbf \Gamma, \mathbf U}\|A(\mathbf \Gamma)-\mathbf b\|_2^{2}; \\ s.t., \mathbf \Gamma=\mathbf U\mathbf V_{v_p}; \|u_p\|_0\leq 1; p \epsilon \{\psi_{\rm v_p}\} \\ s.t., \mathbf \Gamma=\mathbf U\mathbf V_{\rm Patlak}; \|u_p\|_0\leq 2; p \epsilon \{\psi_{\rm Patlak}\} \\ s.t., \mathbf \Gamma=\mathbf U\mathbf V_{\rm e-Tofts}; \|u_p\|_0\leq 3; p \epsilon \{\psi_{\rm e-Tofts}\} \end{align}$$

$$$p$$$ is indexed over pixels. $$$\{\psi_{\rm v_p}, \psi_{\rm Patlak}, \psi_{\rm e-Tofts}\}$$$ are
sets that denote the pixel locations respectively where the plasma volume,
Patlak, and the e-Tofts models are appropriate; this constitutes the
TK model order mask. This is
obtained from a temporal finite difference (tFD)
reconstruction, where a F-test statistic at p=10^{-5} is used to
determine if a higher model order is warranted (see Figure 3)^{10-11}. $$$\gamma(\mathbf x,t)$$$ is constrained to be either a 1, 2, or a 3-sparse combination of bases respectively from the
subspaces $$$\mathbf V_{\rm vp}, \mathbf V_{\rm Patlak}, \mathbf V_{\rm e-Tofts}$$$. $$$A$$$ encompasses Fourier under-sampling, coil-sensitivity encoding, and mapping from concentration to signal intensity. The above is solved by iterating
between (a) updating $$$\mathbf U$$$ on a pixel-by-pixel basis using orthogonal matching
pursuit projection^{9} onto the subspace as determined by the model order mask, and (b) enforcing data consistency. After
the concentration-time profiles are obtained, TK parameters are estimated
by fitting the profiles to the appropriate TK model.

Analysis:
We perform
retrospective under-sampling experiments on fully-sampled DCE-MRI data sets
(3T, Cartesian T1-weighted spoiled gradient echo, FOV: 22x22x4.2cm^{3} resolution: 0.9x1.3x7 mm^{3};
5 sec temporal resolution) from two brain tumor patients. AIFs were identified
from a major vessel in the fully-sampled data and were used during construction
of the subspaces. Under-sampling was
performed using a randomized golden angle trajectory at 20 fold reduction factor. We compare against the tFD reconstruction.

Figure 3 represents the model order mask. Both the cases demonstrate spatially similar patterns from the tFD and reference reconstructions. Figure 4 shows comparisons on the meningioma case. In comparison to the proposed approach, tFD approach under-estimated K^{trans} and v_{e}. The region of interest (ROI) histograms on tumor pixels reveal these differences between the two reconstructions. Figure 5 shows comparisons on the glioblastoma case. v_{e} and v_{p} maps are comparable between the two approaches. K^{trans} from the proposed approach depict subtle noisy oscillations in the spatial maps, however the overall distribution in the ROI histogram is preserved.

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Figure 1: Construction of temporal subspaces from (a) the v_{p} model; (b) Patlak
model; (c) e-Tofts model. The 1-parameter model in (a) has the AIF as the basis
function in the subspace V_{vp}. For the higher parameter models in
(b),(c), a library of concentration time profiles is generated based on a
physiological range of TK parameters and the AIF. Using k-SVD, the library is
then reduced to a smaller set of temporal basis functions to derive the subspaces
(V_{Patlak}, V_{e-Tofts}).

Figure 2: TK model generated concentration v.s time
profiles and their representation using k-SVD derived temporal bases. (a) and
(b) respectively show representative profiles depicting different tumor
enhancement dynamics from the Patlak, and e-Tofts models. The maximum and
average approximation errors (Err_{max}/Err_{mean}) are
evaluated over arange of TK parameters. A model-sparsity choice of k=2 was
determined to be adequate for the Patlak model (Err_{max}/Err_{mean}=10^{-28}%/10^{-30}%). Similarly, k=3 was adequate for the e-Tofts model (Err_{max}/Err_{mean}=2%/0.008%).

Figure 3: TK model order
masks derived from (a) reference (R=1), and (b) temporal FD
reconstruction (R=20). Two cases with different tumor characteristics are shown. Both (a) and
(b) depict qualitatively similar spatial patterns. In the case of meningioma,
the e-Tofts model was assigned to majority of the tumor pixels. In the glioblastoma case, pixels in the tumor rim were assigned the e-Tofts
model, the pixels in the tumor core were assigned either the vp model or the
Patlak model. This pattern was consistent with brain tumor DCE-MRI literature ^{10}.

Figure 4: Comparison of TK parameters
on a Meningioma case: (a) shows the TK parameter maps; (b) shows the histograms
of the pixels as marked by the ROI in (a). The Ktrans, and ve maps from the
proposed reconstruction depict closer trends to the reference, while the maps
from tFD reconstruction depict underestimation. This is also evident in the ROI
histograms.

Figure 5: Comparison
of TK parameters on a glioblastoma case: (a) shows the TK parameter maps; (b)
shows the histograms of the pixels as marked by the ROI in (a). The ve, and vp
maps from the proposed reconstruction and tfD reconstruction depict similar
trends as the reference. In contrast to tFD approach, the Ktrans map from the
proposed approach depicts small noisy oscillations,
however the ROI histograms reveal that the overall distribution is preserved.