Junghun Cho^{1}, Shun Zhang^{2}, Youngwook Kee^{3}, Pascal Spincemaille^{3}, Thanh Nguyen^{3}, Simon Hubertus^{4}, Ajay Gupta^{3}, and Yi Wang^{1,3}

We propose the use of machine-learning to improve the accuracy of a QSM+qBOLD model based Cerebral metabolic rate of oxygen (CMRO2) and oxygen extraction fraction (OEF) mapping. The proposed method, data-driven regularized inversion or DRI, significantly outperformed, in simulation, the current method at all SNR levels. In n=11 healthy subjects, uniform OEF maps were obtained as expected. In n=18 ischemic stroke patients, low OEF regions were clearly located within the lesion region as defined by DWI.

CMRO2 (μmol/100g/min) and OEF (%)
can be expressed as $$CMRO2=CBF\cdot OEF\cdot [H]_a$$ $$OEF=1-\frac{Y}{Y_a}$$ where CBF is the cerebral blood flow, $$$[H]_a$$$ is the oxygenated heme molar concentration in
the arteriole (7.377 μmol/ml)^{7}, $$$Y$$$ and $$$Y_a$$$ are venous and arterial (0.98) oxygenation. The QSM+qBOLD model consists of solving $$Y^*,v^*,\chi^*_{nb},S^{0*},R^*_2=argmin_{Y,v,\chi_{nb},S^{0},R_2}|||S|-F_{qBOLD}(Y,v,\chi_{nb},S^{0},R_2)||_2^2$$ $$Y^*,v^*,\chi^*_{nb}=argmin_{Y,v,\chi_{nb}}||F_{QSM}(Y,v,\chi_{nb})-QSM||_2^2$$ where $$$S^0$$$ the GRE signal at TE=0 and $$$R_2$$$ the cellular contribution to signal decay. $$F_{qBOLD}(Y,v,\chi_{nb},S^{0},R_2)=S^0 \cdot e^{-R_2\cdot TE}\cdot F_{BOLD}(v,\delta \omega(Y,\chi_{nb}),TE)\cdot G(TE)$$ where $$$F_{BOLD}$$$ and $$$G$$$ are extravascular and macroscopic contribution
to magnitude decay^{6,8}. $$$\delta \omega(Y,\chi_{nb})$$$ is the frequency difference between
deoxygenated blood and the surrounding tissue: $$\delta \omega(Y,\chi_{nb})=\frac{1}{3}\cdot \gamma \cdot B_0 \cdot [\Delta \chi\cdot (1-Y)+\chi_{ba}-\chi_{nb}]$$ $$$\gamma$$$ is the gyromagnetic ratio (267.513 MHz/T), $$$B_0$$$=3T, $$$\Delta \chi$$$ is the susceptibility difference between fully
oxygenated and fully deoxygenated blood ($$$0.357\times 4\pi \times 0.27$$$ ppm)^{9}, $$$\chi_{ba}$$$ is purely oxygenated blood susceptibility (
-108.3 ppb). $$F_{QSM}(Y,v,\chi_{nb})=\left[\frac{\chi_{ba}}{\alpha}+\psi_{Hb}\cdot \Delta \chi_{Hb}\cdot (-Y+\frac{1-(1-\alpha)\cdot Y_a}{\alpha})\right]\cdot v + \left(1-\frac{v}{\alpha}\right)\cdot \chi_{nb}$$ where $$$\alpha$$$ is the ratio between vein and total blood
volume assumed (0.77), $$$\psi_{Hb}$$$ the hemoglobin volume fraction (0.0909 for
tissue and 0.1197 for vein), $$$\Delta \chi_{Hb}$$$ the susceptibility difference between deoxy-
and oxy-hemoglobin (12522 ppb)^{7,10,11}.

Data-driven Regularized Inversion

The idea behind DRI is that voxels with similar GRE signal curves should have similar parameters ($$$Y,v,R_2$$$). To identify clusters of similar signal, k-means clustering was applied. Once clusters are obtained, the parameters ($$$Y,v,R_2$$$) are assumed to be uniform within each cluster, thereby effectively increasing SNR for inversion significantly.

Optimization

An
initial guess for $$$Y_0$$$ was obtained from the sagittal sinus, $$$\chi_{nb,0}$$$ set to $$$\chi_{ba}$$$, and $$$R_{2,0}$$$ obtained from a mono-exponential fit against
Eq. 5 with $$$Y_0,v_0,\chi_{nb,0}$$$.Three initial values
for $$$v$$$ were selected (1, 2, and 3 %). The $$$Y,v,R_2$$$ were scaled by their initial guess: $$$x\mapsto \frac{x}{avg(x_0)+4\cdot std(x_0)}$$$. The L-BFGS-B algorithm^{12,13} was used for
constrained optimization. At each iteration, $$$\chi_{nb}$$$ was updated from Eq. 4 and the other unknowns
were updated from Eq. 3. The solution with the smallest residual across the
three $$$v_0$$$ values was selected. After performing the
optimization with ($$$Y,v,R_2$$$) constant within each
cluster, the $$$Y,v,R_2$$$ values were updated voxel-to-voxel using the cluster-based
result as initial guess.

Validation

The proposed DRI-based QSM+qBOLD was
compared with voxel-wise QSM+qBOLD in a numerical simulation (Fig. 1) and with the previous QSM+qBOLD with the
constant OEF initial guess^{4} in n=11 healthy
subjects at 3T: 3D ASL (20 cm FOV, 1.56x1.56x3.5 mm^{3} voxel size,
1500 ms labeling period, 1525 ms post-label delay) and 3D spoiled Gradient Echo
(SPGR, 0.78x0.78x1.2 mm^{3} voxel size, 7 echoes, TE1=2.3ms, 3.9 ms, 30.5 ms), and n=18 ischemic stroke patients at
3T: 3D ASL (24 cm FOV, 1.9x1.9x2.0 mm^{3} voxel size, 1500 ms labeling
period, 1525 ms post-label delay) and 3D spoiled Gradient Echo (SPGR,
0.47x0.47x2 mm^{3} voxel size, 8 echoes, TE1=4.5ms, 4.9 ms, 42.8 ms).

1. Derdeyn CP, Videen TO, Yundt KD, Fritsch SM, Carpenter DA, Grubb RL, Powers WJ. Variability of cerebral blood volume and oxygen extraction: stages of cerebral haemodynamic impairment revisited. Brain : a journal of neurology 2002;125(Pt 3):595-607.

2. Gupta A, Chazen JL, Hartman M, Delgado D, Anumula N, Shao H, Mazumdar M, Segal AZ, Kamel H, Leifer D, Sanelli PC. Cerebrovascular reserve and stroke risk in patients with carotid stenosis or occlusion: a systematic review and meta-analysis. Stroke 2012;43(11):2884-2891.

3. Gupta A, Baradaran H, Schweitzer AD, Kamel H, Pandya A, Delgado D, Wright D, Hurtado-Rua S, Wang Y, Sanelli PC. Oxygen Extraction Fraction and Stroke Risk in Patients with Carotid Stenosis or Occlusion: A Systematic Review and Meta-Analysis. American Journal of Neuroradiology 2014;35(2):250-255.

4. Cho J, Kee Y, Spincemaille P, Nguyen TD, Zhang J, Gupta A, Zhang S, Wang Y. Cerebral metabolic rate of oxygen (CMRO2) mapping by combining quantitative susceptibility mapping (QSM) and quantitative blood oxygenation level-dependent imaging (qBOLD). Magnetic resonance in medicine 2018;80(4):1595-1604.

5. Lee H, Englund EK, Wehrli FW. Interleaved quantitative BOLD: Combining extravascular R2' - and intravascular R2-measurements for estimation of deoxygenated blood volume and hemoglobin oxygen saturation. Neuroimage 2018;174:420-431.

6. Yablonskiy DA, Sukstanskii AL, He X. BOLD-based Techniques for Quantifying Brain Hemodynamic and Metabolic Properties – Theoretical Models and Experimental Approaches. NMR in biomedicine 2013;26(8):963-986.

7. Zhang J, Cho J, Zhou D, Nguyen TD, Spincemaille P, Gupta A, Wang Y. Quantitative susceptibility mapping-based cerebral metabolic rate of oxygen mapping with minimum local variance. Magn Reson Med 2017.

8. X. U, DA. Y. Enhancing image contrast in human brain by voxel spread function method. In Proceedings of the 22nd Annual Meeting of ISMRM;Milan, Italy, Abstract 3197.

9. Ulrich X, Yablonskiy DA. Separation of cellular and BOLD contributions to T2* signal relaxation. Magnetic resonance in medicine 2016;75(2):606-615.

10. Zhang J, Liu T, Gupta A, Spincemaille P, Nguyen TD, Wang Y. Quantitative mapping of cerebral metabolic rate of oxygen (CMRO2) using quantitative susceptibility mapping (QSM). Magnetic Resonance in Medicine 2015;74(4):945-952.

11. Zhang J, Zhou D, Nguyen TD, Spincemaille P, Gupta A, Wang Y. Cerebral metabolic rate of oxygen (CMRO2) mapping with hyperventilation challenge using quantitative susceptibility mapping (QSM). Magnetic resonance in medicine 2017;77(5):1762-1773.

12. Liu DC, Nocedal J. On the limited memory BFGS method for large scale optimization. Mathematical programming 1989;45(1):503-528.

13. Byrd RH, Lu P, Nocedal J, Zhu C. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing 1995;16(5):1190-1208.

14. Raichle ME, MacLeod AM, Snyder AZ, Powers WJ, Gusnard DA, Shulman GL. A default mode of brain function. Proceedings of the National Academy of Sciences 2001;98(2):676-682.

15. Grotta JC, Lo EH. Stroke: Pathophysiology, Diagnosis, and Management: Elsevier; 2015.

16. Guadagno JV, Warburton EA, Jones PS, Fryer TD, Day DJ, Gillard JH, Carpenter TA, Aigbirhio FI, Price CJ, Baron JC. The Diffusion-Weighted Lesion in Acute Stroke: Heterogeneous Patterns of Flow/Metabolism Uncoupling as Assessed by Quantitative Positron Emission Tomography. Cerebrovascular Diseases 2005;19(4):239-246.

17. Bolar DS, Rosen BR, Sorensen A, Adalsteinsson E. QUantitative Imaging of eXtraction of oxygen and TIssue consumption (QUIXOTIC) using venular‐targeted velocity‐selective spin labeling. Magnetic resonance in medicine 2011;66(6):1550-1562.

18. Stone AJ, Blockley NP. A streamlined acquisition for mapping baseline brain oxygenation using quantitative BOLD. Neuroimage 2017;147:79-88.

19. Sun X, He G, Qing H, Zhou W, Dobie F, Cai F, Staufenbiel M, Huang LE, Song W. Hypoxia facilitates Alzheimer's disease pathogenesis by up-regulating BACE1 gene expression. Proceedings of the National Academy of Sciences of the United States of America 2006;103(49):18727-18732.

20. Acosta-Cabronero J, Williams GB, Cardenas-Blanco A, Arnold RJ, Lupson V, Nestor PJ. In vivo quantitative susceptibility mapping (QSM) in Alzheimer's disease. PloS one 2013;8(11):e81093.

21. Trapp BD, Stys PK. Virtual hypoxia and chronic necrosis of demyelinated axons in multiple sclerosis. The Lancet Neurology 2009;8(3):280-291.

22. Stadlbauer A, Zimmermann M, Kitzwogerer M, Oberndorfer S, Rossler K, Dorfler A, Buchfelder M, Heinz G. MR Imaging-derived Oxygen Metabolism and Neovascularization Characterization for Grading and IDH Gene Mutation Detection of Gliomas. Radiology 2017;283(3):799-809.

23. Kudo K, Liu T, Murakami T, Goodwin J, Uwano I, Yamashita F, Higuchi S, Wang Y, Ogasawara K, Ogawa A, Sasaki M. Oxygen extraction fraction measurement using quantitative susceptibility mapping: Comparison with positron emission tomography. Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism 2016;36(8):1424-1433.

Numerical simulation of the accuracy comparison between voxel-wise and DRI-based QSM+qBOLD at different SNRs. The brain mask of the 6 days onset stroke
patient was divided into the two: lesion (OEF 10%) and normal (OEF 35%). The GRE signals
and the QSM values were simulated using $$$F_{qBOLD}$$$ and $$$F_{QSM}$$$. Then, gaussian noise
was added to simulated data: no noise, SNR 1000, 100, and 50. As SNR decreases,
the DRI-based QSM+qBOLD provides highly more accurate OEF values than voxel-wise
QSM+qBOLD, e.g. 39.2 % and 13.4% vs. 31.0% 26.8% for ground truth 35% and 10%
at SNR 50.

The comparison of CMRO2,
OEF, $$$v$$$, $$$R_2$$$, and $$$\chi_{nb}$$$ map between previous QSM+qBOLD and DRI-based
QSM+qBOLD in a healthy subject. DRI-based QSM+qBOLD shows a good
CMRO2 contrast between CGM and WM without extremely high values (>400
μmol/100g/min), uniform OEF, higher $$$R_2$$$vales, and less noisy $$$\chi_{nb}$$$ than previous QSM+qBOLD.

The ROI analysis of CMRO2, OEF, $$$v$$$, $$$R_2$$$, and $$$\chi_{nb}$$$ map between previous QSM+qBOLD and DRI-based QSM+qBOLD in cortical gray matter of the healthy subjects (N=11). CMRO2 was 184.2 ± 17.6 and 127.5 ± 23.5
μmol/100g/min, OEF was 40.8 ± 2.3 and 28.0 ± 3.9 %, $$$v$$$ was 4.48 ± 0.41 and 2.93
± 0.0%, $$$R_2$$$ was 12.9 ± 0.5 and 15.6 ± 0.8 Hz, and $$$\chi_{nb}$$$ was -19.8 ± 3.5 and -13.3 ± 3.8 ppb for previous QSM+qBOLD and DRI-based QSM+qBOLD, respectively.

The OEF comparison between previous QSM+qBOLD and DRI-based QSM+qBOLD
in the 5 stroke patients (18hour, 4, 6, 7, and 12day onset). DRI-based QSM+qBOLD shows a
good agreement between low OEF area and the lesion in DWI. On the other hand, previous QSM+qBOLD does not show a clear low OEF lesion which coincides with the lesion
in DWI.

The ratio of the average
OEF between the lesion and its mirror side in 18 stroke patients: previous QSM+qBOLD (black) vs. DRI-based QSM+qBOLD (red). $$$\frac{\overline{OEF_{low,lesion}}}{\overline{OEF_{mirror}}}$$$ and $$$\frac{\overline{OEF_{high,lesion}}}{\overline{OEF_{mirror}}}$$$ is the ratio between the average of the ‘low’
and ‘high’ OEF group in the lesion (2-means) and the average of the mirror side,
respectively. Each dot on the left indicates each patient. DRI-based QSM+qBOLD (red)
show smaller $$$\frac{\overline{OEF_{low,lesion}}}{\overline{OEF_{mirror}}}$$$. For $$$\frac{\overline{OEF_{high,lesion}}}{\overline{OEF_{mirror}}}$$$, both previous QSM+qBOLD and DRI-based QSM+qBOLD show almost 100%. More distinctive difference between acute and
sub-acute phase with DRI-based QSM+qBOLD is shown in the low OEF lesion group.