Table 1.

Simple theoretical framework for estimating CMRO₂ from multimodal measurements.

Theoretical Framework
VARIABLES
f = CBF normalized to baseline
r = CMRO2 normalized to baseline
s = BOLD signal normalized to baseline
E = O2 extraction fraction
pC = mean blood pO2 (mmHg)
pT = mean tissue pO2 (mmHg)
REQUIRED BASELINE PARAMETERS (TYPICAL VALUES)
E0 = 0.4
pT0 = 25 mmHg
MODEL PARAMETERS
α = 0.4 (blood volume effects)
β = 1.5 (intravascular signal changes, diffusion)
M = local scaling factor, determined by calibration
p50 = 26 mmHg (O2-hemoglobin 50% saturation)
h = 2.8 (Hill exponent)
MODEL EQUATIONS
(1) Mass balance:
$r=\frac{E}{E_0}f$
(2) BOLD signal (Davis et al., 1998):
$s=M\left[1-f^{\text{α}}{\left(\frac{E}{E_0}\right)}^{\text{β}}\right]$
(3) Blood/tissue O2 gradient:
$r=\frac{p_C-p_T}{p_{C0}-p_{T0}}$
(4) Mean blood pO2 (Gjedde, 2005a,b):
$p_C=p_{50}{\left[\frac{2}{E}-1\right]}^{1/h}$

The equations describe: (1) Mass balance for O₂, expressed in terms of variables normalized to their baseline values; (2) Davis model for the BOLD signal, with the parameter α describing effects of blood volume changes, and β approximating differential diffusion effects around large and small vessels estimated from Monte Carlo simulations; (3) Assumption that the tissue/blood pO₂ gradient increases to match a change in CMRO₂, and that this gradient is determined solely by the difference of the mean O₂ concentrations in blood and tissue (no capillary recruitment); and (4) Expression for mean blood pO₂ assuming the Hill equation for the O₂-hemoglobin saturation curve, with exponent h and half-saturation at a pO₂ of p₅₀, and assuming that the pO₂ corresponding to half of the total extracted O₂ is the mean value for blood.