Michael Germuska^{1}, Hannah L Chandler^{1}, Rachael C Stickland^{1}, Catherine Foster^{2}, Jessica Steventon^{1}, Valentina Tomassini^{1}, Kevin Murphy^{1}, and Richard G Wise^{1}

A frequency-domain machine learning method is presented that significantly reduces the bias and variance in dual-calibrated estimation of oxygen extraction fraction, as demonstrated with simulation and in-vivo imaging. In addition, the method substantially reduces the processing time compared to previous robust analysis methods.

Simulated data were used to train ANNs to
estimate resting cerebral blood flow (CBF_{0}) and CMRO_{2},
from which OEF_{0} was calculated. Two ANNs were cascaded such that the
results of the CBF network were fed into the CMRO_{2} network. The data
were simulated using standard physiological models ^{2} and constrained
to be physiologically plausible. Constraints on the cerebral physiology were
applied according to a simple model of oxygen transport ^{5}.
Physiological limits were chosen to encompass both healthy and pathological
brain tissue and are listed in figure 1. Simulated BOLD and ASL time-courses
were generated according to an 18-minute gas paradigm ^{6}, with added
noise.

Time
series data was high-pass filtered (BOLD data only, 320 seconds) and then
Fourier transformed. The input feature vector for the CBF ANN consisted of the
first 15 points of the magnitude and phase data for both the ASL and BOLD
timeseries, [Hb], hyperoxic ΔPaO_{2}, SaO_{2,0}, CaO_{2,0},
and the post-label delay (65 data points in total). The feature vector for the
CMRO_{2} ANN also included the CBF_{0} estimate (66 data points
in total). Simulated datasets were constructed with 1x10^{6} simulations
(10% used for early stopping). Networks had 2 hidden layers (50 nodes in each)
and a relu activation function. An additional data set (OEF_{0} range
0.1 to 0.6) was simulated to compare the performance of the FML networks with
previously published regularised non-linear least squares (R-NLS) methods ^{6,7}.
Each method (FML and R-NLS) was also
applied to data acquired (3T Siemens Prisma) in healthy volunteers (n=16, 10 male,
mean age 34.5 years) (TR 4.4
seconds, 15 slices, in-plane resolution 3.4 x 3.4 mm and slice thickness 7 mm).^{7 }
In-vivo data was processed in the same manner as the
simulated data, or as previously described ^{7} (regularisation was only applied to
oxygen diffusivity for the in-vivo analysis as this had the best performance in
simulations). No spatial smoothing was applied to the data prior
to analysis.

1. Bulte D. et al. Quantitative measurement of cerebral physiology using respiratory-calibrated MRI. NeuroImage. 2012; 60: 582-591

2. Gauthier, C.J. et al Absolute quantification of resting oxygen metabolism and metabolic reactivity during functional activation using QUO2 MRI. Neuroimage. 2012. 63, 1353-1363.

3. Lee Y and Oh S-H. Input Noise Immunity of Multilayer Perceptrons. ETRI Journal. 1994; 16(1): 35-43.

4. Hertel L, Phan H, and Mertins A. Comparing Time and Frequency Domain for Audio Event Recognition Using Deep Learning. International Joint Conference on Neural Networks. 2016: 3407-3411

5. Gjedde A. The pathway for oxygen in brain. APMIS Suppl. 2003; 109: 146-53

6. Germuska M. et al. A forward modelling approach for the estimation of oxygen extraction fraction by calibrated fMRI. NeuroImage. 2016; 139: 313-323

7. Germuska M. et al. Dual-calibrated fMRI measurement of absolute cerebral metabolic rate of oxygen consumption and effective oxygen diffusivity. NeuroImage. 2018. doi:10.1016/j.neuroimage.2018.09.035

8. Ibaraki M. et al. Interindividual variations of cerebral blood flow, oxygen delivery, and metabolism in relation to hemoglobin concentration measured by positron emission tomography in humans. JCBFM 2010; 30(7): 1296-1305

Range of physiological parameters used in simulation of dual-calibrated fMRI time series

Error and bias in OEF_{0} estimates for FML and R-NLS fitting methods based on simulated data. R-NLS fitting simulated with a range of regularisation weights to explore the variance vs bias trade-off

Example OEF_{0} maps and histogram of grey matter values from an individual subject. The FML method produces OEF_{0} estimates with reduced variance, consistent with lower RMSE predicted by the simulations.

Scatter plot and best-fit regression lines comparing
expected physiological variation in OEF_{0}, given CBF_{0} and [Hb], against grey
matter OEF_{0} estimates calculated with the FML and R-NLS methods.

Scatter plot and best-fit regression lines comparing [Hb] against grey matter OEF_{0} estimates calculated with the FML and R-NLS methods.