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

We investigate the influence of 2D $$$(k_y,k_z,t)$$$ sampling strategies on the minimum achievable variance without bias for pharmaco-kinetic parameter estimation in 3D whole-brain DCE-MRI (equivalent to the best possible precision without bias). Cramér-Rao analysis is combined with a pathologically- and anatomically-realistic digital reference object to objectively compare measurement procedures independent of any estimator. This study did not identify any significant difference between lattice and random undersampling, or between their uniform and variable density variants.

**Introduction**

**Methods**

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 ^{3-5}. 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 Patlak model, an SPGR
sequence, sensitivity encoding, and Fourier undersampling. A population based
arterial input function (AIF) was chosen for this simulation ^{6}. Coil sensitivities and noise
covariance matrix for an eight-channel head array 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) was taken to be the ground-truth (Figure 1) ^{7}. CRBs were computed for pre-contrast
white matter SNR=10, flip angle of 24°, and 50 time frames at 5s temporal
resolution.
As
a linear system model allows the existence of the minimum variance unbiased
estimator in the Gaussian noise case, pharmaco-kinetic model and flip angle were
chosen to operate the signal equation in the linear regime ^{5}.

Sampling schemes

Cartesian sampling schemes can broadly be
classified into k-space region sampling (e.g., Keyhole, TRICKS),
lattice-based, and random sampling with uniform or
variable density variants (e.g., kt-SENSE, DISCO, TWIST, kt-SPARSE). Early
in this study, we found that k-space region sampling schemes often lead to very ill-conditioned
or even singular Fisher Information matrices, which indicate that certain PK parameters
cannot be estimated with finite variance ^{8}. Hence, zone-based sampling is omitted a-priori from
the comparison of candidate sampling patterns, shown in Figure 2. The
undersampling factors for each sampling schemes shown in Figure 2 were
R=2,5,10,15.

**Results**

**Discussion**

In this work, we used Cramér-Rao analysis
to objectively compare lattice-based sampling with random sampling both in
their uniform and variable density variants. Extending on the results in a
previous study ^{3} which analyzed common 1D undersampling strategies, the
sampling patterns in this study had no discernible impact on the minimum achievable variance for PK
parameter estimation. Note that the actual performance of a DCE measurement also depends on the
availability of a numerically stable post processing procedure to estimate PK
parameters with the predicted variance.

An important limitation of this study is that we assumed perfect knowledge of the AIF, bolus arrival time, and coil sensitivities, and no motion. During a clinical scan these assumptions do not hold. Sampling pattern may indirectly influence PK measurement precision through the ability to more precisely estimate AIF, bolus arrival time, and coil sensitivities, and to correct for motion artifacts.

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Cramér-Rao lower bounds for pharmaco-kinetic parameters $$$\boldsymbol{v}_p$$$ (top row) and $$$\boldsymbol{K}_t$$$ (bottom row) for undersampling rates R=2,5,10,15. Bounds
are computed for pre-contrast white matter SNR=10, flip angle of 24°, and 50
time frames at 5s temporal resolution. First column plots true PK parameters,
all other columns illustrate the bounds in the tumor ROI as specified in Figure
1. $$$\boldsymbol{v}_p$$$ is measured in percent, $$$\boldsymbol{K}_t$$$ in $$$\left[\text{min}^{-1}\right]$$$. Horizontal axes show pixels in
image array. Vertical axes show standard deviations of PK parameters. Confirming the results of Figure 3, there is no difference in the minimum achievable variance for pharmaco-kinetic parameter estimation between the sampling schemes.