Matthew T Cherukara^{1,2}, Michael A Chappell^{1,2}, and Nicholas P Blockley^{2,3}

The effect of diffusion is believed to cause underestimation of oxygen extraction fraction (OEF) in streamlined quantitative BOLD (qBOLD). We propose physically motivated modifications to the qBOLD model in order to account for this effect, and demonstrate, using Monte Carlo simulations, and analysis of healthy in vivo data, that simple scalar corrections to the inferred reversible relaxation rate $$$R_2'$$$, can substantially reduce the error in OEF and deoxygenated blood volume (DBV) estimation. These corrections have the potential to be improved further by taking other physiological and acquisition-related parameters into account.

The streamlined qBOLD method uses asymmetric spin echo (ASE) to quantify $$$R_2'$$$ and DBV, from which OEF can be computed[8,9]. In the regime of large spin-echo offsets $$$\tau$$$, the ASE signal is: $$$\operatorname{ln}S(\tau)=-R_2'\cdot{}\tau$$$, where

$$R_2'=k\cdot{}Hct\cdot{}OEF\cdot{}DBV\tag{1}$$

k is a constant that subsumes vessel geometry,
susceptibility differences, and magnetic field strength, and Hct is the
fractional haematocrit (assumed constant). When applied to real data, and
compared to Monte Carlo simulations, this method underestimates $$$R_2'$$$ (hence
OEF), as illustrated in Figure 1. This may be the result of diffusive
dephasing^{[3]}. To rectify this, we propose defining $$$R_2'$$$:

$$R_2'=\kappa\cdot{}[dHb]^\beta\cdot{}DBV\tag{2}$$

where [dHb] is the deoxyhaemoglobin concentration ($$$[dHb]=Hct\cdot{}OEF/k'$$$) which is related to the BOLD signal by exponent
$$$\beta$$$^{[10,11]}; and $$$\kappa$$$ is a scaling constant that subsumes k
and k'. We expect that $$$\kappa$$$ will depend on OEF, and define it
parametrically:$$\kappa=A+B\cdot{}OEF\tag{3}$$where A and B will be defined by optimization.
We do not expect such variance in $$$\beta$$$.

**Simulations:** In
MATLAB (MathWorks, Natick, MA), Monte Carlo methods were used to generate
simulated ASE signals for a 100x100 matrix of OEF and DBV in physiological
ranges^{[3]}. This involved filling a space with randomly oriented cylinders
following a known distribution of vessel radii^{[12]} and calculating the phase
accrued by protons that diffuse around them. Three optimizations were
performed; (i) minimization of the error in OEF estimate using $$$\kappa$$$
alone, (ii) minimization of error using $$$\kappa$$$ and $$$\beta$$$ and (iii) minimization
of error where $$$\kappa$$$ is a function of OEF and $$$\beta$$$ is scalar.

**Data Acquisition:**
Five healthy participants (24-35, 2F) were scanned using a 3T Verio system
(Siemens Healthineers, Germany). GESEPI-ASE scans^{[8]} were acquired (FOV=240mm2,
96x96 matrix, eight 5mm slices, TR/TE=3s,82ms, TIFLAIR=1210ms,
$$$\tau$$$=-16:64ms in steps of 8ms) along with a high-resolution anatomical
for segmentation.

**In Vivo Data
Analysis:** Variational Bayesian inference was performed using FABBER^{[13]} to
estimate $$$R_2'$$$ and DBV^{[9]} using Equations 1-3 with optimized values. OEF was
calculated voxel-wise from $$$R_2'$$$ and DBV, and average grey-matter values were
obtained using anatomical data segmented with FSL-FAST^{[14]}.

**Simulations:** Figure
2 shows the results of parameter estimates obtained using simulated data.
Although the error in deoxyhaemoglobin content estimation is not uniform across
OEF-DBV space, a value of $$$\kappa=14.72$$$ was found which minimized the
average error, reducing it from 45.0% to 19.8%. Simultaneously optimizing $$$\kappa$$$
and $$$\beta$$$ ($$$\kappa=21.51$$$, $$$\beta=1.2$$$) reduced this error even
further, to 6.9%. Applying scalar $$$\kappa$$$ correction to OEF and DBV
estimation also reduced their relative errors, as shown in Table 1. The effects
of including $$$\beta$$$ correction and of parametrically defining $$$\kappa$$$
as a function of OEF, are summarized in Table 1.

**In Vivo Data:** Figure
3 shows example OEF maps obtained from the several correction methods. Mean
grey matter estimates of OEF and DBV were obtained using each, the group
averages are shown in Table 2. Subject-wise ANOVA showed statistically
significant differences between OEF distributions of all methods.

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