Accelerated brain DCE-MRI using Contrast Agent Kinetic Models as Temporal Constraints
Sajan Goud Lingala1, Yi Guo1, Yinghua Zhu1, Naren Nallapareddy1, R. Marc Lebel2, Meng Law3, and Krishna Nayak1

1Electrical Engineering, University of Southern California, Los Angeles, CA, United States, 2GE Health care, Calgary, Canada, 3Radiology, University of Southern California, Los Angeles, CA, United States

### Synopsis

We propose a novel tracer-kinetic model based constrained reconstruction scheme to enable highly accelerated DCE-MRI. The proposed approach efficiently leverages information of the contrast agent kinetic modeling into the reconstruction, and provides a novel alternative to current constraints that are blind to tracer kinetic modeling. We develop the frame-work to include constraints derived from the extended-Tofts (e-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.

### INTRODUCTION

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 (vp), the forward (Ktrans) and backward (Kep) 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.

### METHODS

Construction of temporal dictionary: As depicted in Fig.1, we simulate concentration vs. time profiles for a broad range of physiological kinetic parameters (Ktrans=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 VrxN) 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 XMxN (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. UMxrVrxN 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 ||Xi-Xi-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].

### RESULTS

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 Ktrans maps shown), where consistent performance with the proposed approach was observed.

### CONCLUSION

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.

### Acknowledgements

No acknowledgement found.

### References

[1] RM Lebel, J Jones, JC Ferre, M Law, KS Nayak. Highly accelerated dynamic contrast enhanced imaging. Magnetic Resonance in Medicine. 71(2):635-644. February 2014.

[2] H. Wang, Y. Miao, K. Zhou, Y. Yu, S. Bao, Q. He, Y. Dai, Feasibility of high temporal resolution breast DCE-MRI using compressed sensing theory, Med. Phys. 37(9), p. 4971, 2010.

[3] Y Guo, Y Zhu, SG Lingala, RM Lebel, KS Nayak. "Highly Accelerated Brain DCE MRI with Direct Estimation of Pharmacokinetic Parameter Maps." Proc. ISMRM 23rd Scientific Sessions, Toronto, June 2015, p573.

[4] SG Lingala, Y Guo, Y Zhu, S Barnes, RM Lebel, KS Nayak. "Accelerated DCE MRI Using Constrained Reconstruction Based On Pharmaco-kinetic Model Dictionaries." Proc. ISMRM 23rd Scientific Sessions, Toronto, June 2015, p196

[5] Parker et al, " Experimentally-Derived Functional Form for a Population-Averaged High-Temporal-Resolution Arterial Input Function for Dynamic Contrast-Enhanced MRI ". Magnetic Resonance in Medicine; 56: 993-1000.

[6] S.P. Sourbron, DL Buckley. "On the scope and interpretation of the Tofts models for DCE-MRI" Magnetic Resonance in Medicine; 66, 735-45, 2011

[7] M. Aharon, et al, "k-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation" IEEE Trans. Sign Processing; 54 (11): 4311-4322, 2006,

[8] Y Zhu, Y Guo, RM Lebel, M Law, KS Nayak. "Randomized Golden Ratio Sampling for Highly Accelerated Dynamic Imaging." Proc. ISMRM 22nd Scientific Sessions, Milan, May 2014, p4365.

### Figures

Figure 1: Construction of temporal basis functions (in a dictionary) from a specified tracer kinetic model. Basis functions derived from the e-Tofts model are considered, and are used as temporal constraints to enable accelerated DCE-MRI.

Figure 2: Error statistics in estimating kinetic parameters using the e-Tofts model from the (a) noisy, (b-e) 1 to 4 sparse projected concentration time profiles. Estimating (top row) Ktrans with vp =0.04, Kep=0.2 min-1, (middle row) Kep with vp=0.2, Ktrans=0.2 min-1, (bottom row) vp with Ktrans=0.2 min-1, Kep=0.4 min-1 .

Fig.3: Comparison of parameter maps from (left) fully-sampled reference, (middle) fully sampled with dictionary modeling, (right) 20x undersampled with dictionary modeling. Note the extremely low nMSE from the (x1) dictionary approach. At x 20, the nMSE is increased, however the thin tumor margins are depicted with good fidelity (see arrows).

Comparison of Ktrans maps from two glioblastoma tumors with different shape and heterogeneity. Similar to the previous figure, note the low nMSE with the (x1) dictionary approach, and good fidelity of mapping the tumor shape and heterogeneity at (x20); see arrows.

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