Yannick Bliesener^{1}, Sajan G. Lingala^{1}, Justin P. Haldar^{1}, and Krishna S. Nayak^{1}

We demonstrate an approach to evaluate and compare (k,t) sampling patterns for DCE-MRI. We compute Cramér-Rao lower bounds on the variance of pharmacokinetic (PK) parameter estimates, using pathologically- and anatomically-realistic digital reference objects. The framework allows for the optimization of sampling patterns independent of any specific estimator. We apply this framework to a 2D reference object for four sampling patterns: keyhole, TRICKS, lattice, and golden angle sampling. It is shown that TRICKS, lattice, and golden angle sampling enable low variance estimation for low undersampling factors. Out of these, lattice sampling keeps variances lowest with increasing undersampling factors.

**Framework.**
The Cramér-Rao bound (CRB) gives a lower bound on variances of any unbiased
estimator, and has been widely used to optimize MRI experiment design.^{2,3} Evaluation of the CRB requires
the derivatives of the DCE-MRI forward model with respect to the parameters
being estimated. The forward model was simulated by PK
modeling based on the extended Tofts model, an SPGR sequence, sensitivity
encoding, and Fourier undersampling. A population based arterial input function
was chosen for this simulation.^{4} Coil sensitivities for an eight channel head array and noise
covariance matrix were taken from measurements. The derivative of the forward
model needs to be evaluated at the parameter being estimated. Hence, a pathologically- and
anatomically-realistic digital reference object (DRO) ^{5} was taken to be the ground-truth
(Figure 1). CRBs were computed for SNR=20, flip angle of 24°, and 50 time frames at 5s temporal resolution.

**Sampling scheme comparison.** Four sampling
schemes were compared (Figure 2): keyhole,^{6} TRICKS,^{7} lattice,^{8} and golden angle (GA) sampling.^{9} CRBs for each sampling pattern with different undersampling factors,
i.e., R=2-5,10,15, were examined to determine whether they allow for precise
and reproducible parameter estimation (Figure 3,4).

To verify the approach and correctness of the computed CRBs, the results of the framework were confirmed by Monte-Carlo simulations (not shown).$$$$$$

Figure 3 illustrates the spatial pattern of achievable variances for PK parameters $$$K^\text{trans}$$$ and $$$v_\text{p}$$$ across the DRO for R=3. Results for $$$v_\text{e}$$$ are not shown, but are similar. While keyhole sampling leads to high variances across the patch, TRICKS, lattice, and GA sampling show highest variances in the tumor region.

A comparison of best precision achievable with unbiased estimators for varying undersampling factors is shown in Figure 4. Keyhole sampling leads to high variances irrespective the undersampling factor. TRICKS, lattice, and golden angle sampling have reasonably low standard deviation (SD) at low undersampling rates with SDs of less than two percent for $$$v_\text{p}$$$, and one magnitude lower than the $$$K^\text{trans}$$$ to be estimated for R=2,3. As undersampling factors increase, TRICKS sampling shows the strongest rise of bounds, whereas variances for lattice sampling increase most slowly. Nonetheless, SDs are in the same order of magnitude as the parameter to be estimated for R=10,15.

The CRB-based approach is computationally efficient, as it does not require expensive Monte-Carlo simulations. Thus, it provides an easier approach to evaluate sampling patterns for DCE-MRI with non-linear PK models. It directly compares the acquired information and allows one to optimize sampling schemes separately from the estimator. Hence, evaluation and optimization of this bound can be seen as a step of providing the most possible information to the estimator. As reconstruction is commonly closely tied to the sampling scheme, the challenge remains to find such unbiased estimator for an optimized pattern.

Despite being faster than Monte-Carlo simulations involving non-linear model fitting, the major challenge of this framework is the rapid increase in computational complexity with matrix size. Let $$$N$$$ be the number of voxels, the Fisher information matrix to be inverted in order to obtain the CRB grows according to $$$(3N)^2$$$. This limits the size of the DRO that can be used for optimization and analysis, but nonetheless allows one to gain insight into properties of sampling patterns.

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