Josiah John Simeth^{1,2} and Yue Cao^{1,2}

Though methods exist for quantifying regional liver function from dynamic gadoxetic-acid enhanced (DGE) MRI, errors are introduced when using the clinically typical temporally sparse acquisition scheme (6 volumes over 20 minutes) relative to a temporally dense dynamic acquisition (volumes every 5-10 sec over a similar period). This motivates a data driven approach. An artificial neural network (ANN) was trained to reproduce the results of the fully characterized analysis using only the restricted dataset. Across the patients evaluated the ANN solution resulted in lower mean and median WMAPE, as well as a reduction in MSE in most cases.

**Introduction**

**Methods**

Thirty-two high-temporal resolution (HTR) (figure 1) DGE MRI
scans of the liver were acquired up to 20 min in 25 patients with hepatic
cancers on a 3T scanner. To estimate k_{1}, the HTR DGE signals were
fitted to the LSITCM:

$$\overbrace{\frac{(1-Hct)C_{t}(t)}{C_{a}(t)}}^{\textrm{y}} \approx \overbrace{v_{dis} k_1}^{\textrm{slope}} \overbrace{\frac{\int_0^t C_{a}(\tau) d\tau }{C_{a}(t)}}^{\textrm{x}} + \overbrace{v_{dis}}^{\textrm{intercept}} [1]$$

where C_{t} and C_{a} are contrast concentrations of artery and
tissue, respectively, v_{dis} is a distribution volume, and Hct is hematocrit.
To train an ANN, a population of voxels was randomly selected using100,000 voxels from each liver of 32 exams.
To create the LTR DGE data, a time sampling scheme was used to select 6
time points. The initial 4 points started at zero and were uniformly spaced 5 to 50 seconds apart. Each of the initial 4 points was further perturbed by a fraction of their spacing. An
intermediate point and terminating point were randomly selected from the central 50% and final 10% of the acquisition duration. The time variations of the time points prevents an identical x vector across all voxels in each exam. Intensities
of the input curves at the 6 selected time points were defined through mean
values over a 14 second period centered on the selected times using a linear interpolation.
Then, x and y vectors were created using equation [1]. A small amount of noise was added to the x and y vectors for each
voxel, to prevent overfitting.
After removing the zero initial points, the x and y vectors were
combined to form the input to the ANN. In
addition to input and output layers, the ANN had 4 hidden layers, 2 with 10
neurons, followed by 2 with 5 neurons. The feed forward NN was trained with targets
of k1 values from fitting the HTR dataset to equation [1]. The network was trained in
Matlab using Levenberg-Marquardt backpropagation. A leave-one-out cross-validation
was performed to evaluate k_{1} values of each patient’s exams using the ANN
trained by only the other patient’s data. During ANN training, 75% of 2 million voxels
of data were used for training and 25% for validation.
Finally, to compare with the k_{1} values estimated from the
ANN, the input LTR curves were directly fitted to the LSITCM. To test the
model, both LSITCM and ANN solutions were applied to with the time interval of
early points varied from 15 s to 35 s, maintaining the final point at the end
of the acquisition, and choosing the intermediate point exactly equidistant.

**Results**

The leave-one-out cross validation showed that k_{1} values of
32 exams estimated by the ANN had a lower median and mean relative MSE (from
-80.6% to -14.2%) than those from direct fitting of equation [1] (see figure 2). WMAPEs
were also lower for the ANN as compared to direct fitting. Qualitatively, the ANN solutions reduced
runaway noise and edge effects (see figure 3).

Both relative MSE and WMAPE of k1 by both methods decreased with an increase in the intervals between the first four time points, particularly the time of the 4th point. As seen in figure 4, the ANN approach was less sensitive to closely spaced early time points, but as the spacing increased both methods became comparable (mean WMAPE differed by only 0.64%).

**Conclusion and Discussion**

This figure illustrates the relative
characteristics of densely sampled high temporal resolution (HTR) and more
sparsely sampled low temporal resolution (LTR) datasets. HTR data is regularly
sampled at 5-10 s intervals for the duration of the 5 to 20 min scan. LTR data
involves the acquisition of three post contrast samples uniformly spaced at
intervals of 15 to 35 seconds, followed by a point each at roughly 10 and 20
min post contrast. LTR data is the clinical norm.

This table shows
the relative difference in mean squared error (MSE) and weighted mean absolute
percent error (WMAPE) for the results of the LSITCM and an ANN as the spacing
of the early time points is varied from 15 to 35 seconds.

Comparison
of uptake rates derived from densely sampled LSITC (left), sparsely sampled
LSITC (middle), and sparsely sampled with an artificial neural network derived
solution (right). Note that some non-ideal behavior exists even in the
LSITC-HTR data, such as the edge effects apparent on the top patient.

The
mean WMAPE trended downward for both methods as the interval between early
points was increased. For very short times, the LSITCM fit had particular
difficulty, with error drastically increasing near 15 s.