Marco Bertleff^{1}, Sebastian Domsch^{1}, and Lothar Schad^{1}

^{1}Computer Assisted Clinical Medicine, Heidelberg University, Mannheim, Germany

### Synopsis

In
this work we present an evaluation approach based on artificial neural networks
(ANN) for fitting the IVIM-Kurtosis model parameters on the basis of simulated
DWI data. The ANN approach is compared to an ordinary bounded least squares
regression (LSR) in terms of correlation between estimates and ground truth, systematic,
statistical and total estimation error. While for *D* and *K* high correlations
and low errors were found for both LSR and ANN, a significant improvement was
observed for *f* and *D** regarding correlation coefficients,
precision and the total estimation error when using ANN.

### Purpose

Both the intravoxel incoherent motion (IVIM)^{1} and
the kurtosis^{2} model are extensions to the monoexponential apparent diffusion coefficient (ADC) model
for the evaluation of diffusion weighted imaging (DWI). The biexponential IVIM
model considers flow effects in perfused tissue while the Kurtosis model takes
non-Gaussian diffusion into account. Both models provide valuable information for
the assessment of various pathologies such as cancer or stroke^{3,4}.
Hence, the simultaneous access to the parameters of the combined IVIM-Kurtosis model is highly desirable. Different
approaches have been used to overcome the poor fit stability of a conventional
least squares regression (LSR) when applying the combined model,
which is due to the
number of unknown fit parameters. The purpose of this simulation study is to
present an artificial neural network (ANN) approach for fitting the
IVIM-Kurtosis model parameters and to compare it to an ordinary bounded LSR.### Methods

Four
feed forward ANNs with 10 hidden layers were implemented differing in the signal-to-noise-ratio
(SNR) of the datasets they were trained with (Neural Network Toolbox provided by Matlab R2014a). The input layer is fed with
16 signal intensities corresponding to the following b-values: b = {0,
10, 20, 40, 70, 100, 150, 200, 300, 500, 700, 1000, 1500, 2000, 2400, 2800}
s mm^{-2} . The output layer returns the parameter estimates
for the IVIM-Kurtosis model: $$$S(b)=S_0\cdot(f\cdot\exp{(-b(D^*+D))}+(1-f)\cdot\exp{(-bD+Kb^2D^2/6)})$$$.
The ANNs were trained with 20^{5} simulated DWI signals within the parameter
ranges typical for brain tissue of *S*_{0} = {0, …, 1500}, *f* = {0, …, 0.14}, *D* = {0, …, 1.8} ∙ 10^{-3} mm^{2} s^{-1},
*D** = {0.9, …, 17} ∙ 10^{-3} mm^{2} s^{-1}
and *K* = {0, …, 2}
with additional zero mean Gaussian noise. Thereby the training SNR levels of
25, 50, 100 and 150 yielded the corresponding ANNs further referred to as
ANN25, ANN50, ANN100 and ANN150. In a Monte-Carlo simulation noisy DWI signals
(SNR = 200) were generated sweeping the range of each parameter while
keeping the remaining parameters constant. These signals were evaluated with a
conventional bounded LSR and the different ANNs. From the resulting correlation
curves the systematic error was determined by the root mean squared error (RMSE)
between a first order polynomial fit through the data and the ground
truth. The statistical error was determined by the standard deviation with
respect to the fit and the total error was calculated by the RMSE of the data
with respect to the ground truth. All errors were calculated relative to the
respective parameter means.### Results

Figure
1 shows the correlation curves for all model parameters for LSR and ANN (i.e. ANN25 & ANN100). Both
LSR and ANN show high correlations for *D*
and *K*.
For *f* and *D** many non-converged values can be observed particularly for LSR
while ANN shows higher systematical deviations. For
*D* and *K* the correlation coefficient was between *R*=98% and *R*=100%
for both LSR and ANN. By using ANN the correlation was significantly improved
for *f* from *R*=77.0% (LSR) to *R*=86.1% (ANN100) and *R*=96.4% (ANN25). For *D** the correlation was improved from *R*=70.3% (LSR) to *R*=80.5% (ANN100) and *R*=90.2% (ANN25). Figure 2 shows the systematic
(first row), statistical (second row) and total errors (third row) for all parameters using LSR and each ANN. For *D*
and *K* both LSR and ANN reveal
comparable errors clearly below 10% and significantly smaller than for the
parameters *f* and *D**. For *f* and *D** the lowest systematic error was achieved
with LSR. Increasing the ANN training SNR led to a decreased systematic error
and for high training SNR the systematic errors were comparable to LSR. On the
contrary, the statistical errors of the ANNs were clearly decreased with
decreasing training SNR and were significantly smaller compared to LSR.
Overall, the total error was decreased by all ANNs compared to LSR with no
apparent trend regarding the training SNR.### Discussion

It was successfully demonstrated that the
proposed ANN approach performed superior over a conventional LSR in estimating
diffusion parameters of the combined IVIM-Kurtosis model. While the choice of
the evaluation method had a rather low impact on the performance of the *D* and *K* estimation, significant differences were found for *f* and *D**. Here the correlation coefficients between estimates and ground
truth could significantly be improved by using the proposed ANN method. The
increased systematic error observed for ANN was traded for a simultaneously
decreased statistical error resulting in an overall improved estimation. As a
consequence the ANN approach appears preferable to a conventional LSR in
providing valuable information for the assessment of brain pathologies such
as cancer or stroke.### Acknowledgements

The first author is funded by the
Carl-Zeiss-Stiftung in the form of a PhD scholarship.### References

1. Le
Bihan, D., et al., MR imaging of
intravoxel incoherent motions: application to diffusion and perfusion in
neurologic disorders. Radiology, 1986. 161(2):
p. 401-407.

2. Jensen,
J.H., et al., Diffusional kurtosis
imaging: The quantification of non-gaussian water diffusion by means of
magnetic resonance imaging. Magnetic Resonance in Medicine, 2005. 53(6): p. 1432-1440.

3. Bisdas,
S., et al., Intravoxel incoherent motion
diffusion-weighted MR imaging of gliomas: feasibility of the method and initial
results. Neuroradiology, 2013. 55(10):
p. 1189-1196.

4. Hui,
E.S., et al., Stroke assessment with
diffusional kurtosis imaging. Stroke, 2012. 43(11): p. 2968-2973.