A General Model to Calculate the Spin–Lattice Relaxation Rate (R1) of Blood, Accounting for Hematocrit, Oxygen Saturation, Oxygen Partial Pressure, and Magnetic Field Strength Under Hyperoxic Conditions

Abstract

Background: Under normal physiological conditions, the spin-lattice relaxation rate (R1) in blood is influenced by many factors, including hematocrit, field strength, and the paramagnetic effects of deoxyhemoglobin and dissolved oxygen. In addition, techniques such as oxygen-enhanced magnetic resonance imaging (MRI) require high fractions of inspired oxygen to induce hyperoxia, which complicates the R1 signal further. A quantitative model relating total blood oxygen content to R1 could help explain these effects. Purpose: To propose and assess a general model to estimate the R1 of blood, accounting for hematocrit, SO 2 , PO 2 , and B0 under both normal physiological and hyperoxic conditions. Study Type: Mathematical modeling. Population: One hundred and twenty-six published values of R1 from phantoms and animal models. Field Strength/Sequence: 5-8.45 T. Assessment: We propose a two-compartment nonlinear model to calculate R1 as a function of hematocrit, PO 2 , and B0. The Akaike Information Criterion (AIC) was used to select the best-performing model with the fewest parameters. A previous model of R1 as a function of hematocrit, SO 2 , and B0 has been proposed by Hales et al, and our work builds upon this work to make the model applicable under hyperoxic conditions (SO 2 > 0.99). Models were assessed using the AIC, mean squared error (MSE), coefficient of determination (R 2 ), and Bland-Altman analysis. The effect of volume fraction constants W RBC and W plasma was assessed by the SD of resulting R1. The range of the model was determined by the maximum and minimum B0, hematocrit, SO 2 , and PO 2 of the literature data points. Statistical Tests: Bland-Altman, AIC, MSE, coefficient of determination (R 2 ), SD. Results: The model estimates agreed well with the literature values of R1 of blood (R 2 = 0.93, MSE = 0.0013 s À2 ), and its performance was consistent across the range of parameters: B0 = 1.5-8.45 T, SO 2 = 0.40-1, PO 2 = 30-700 mmHg. Data Conclusion: Using the results from this model, we have quantified and explained the contradictory decrease in R1 reported in oxygen-enhanced MRI and oxygen-delivery experiments. Level of Evidence: 3 Technical Efficacy: Stage 1

A simplified empirical model to estimate oxygen relaxivity at different magnetic fields

“…Finally, this model does not represent the r 1Ox in blood and tissues, where the addition of proteins, lipids, and deoxyhemoglobin will affect the R 1 -pO 2 relationship; however, we have created a separate general model to calculate the R 1 of blood, accounting for hematocrit, oxygen saturation, oxygen partial pressure, and magnetic field strength under hyperoxic conditions. 46 For convenience, however, we have listed the reported literature values found in tissue and blood in Supplementary Table S4-nine values from blood and three from tissues. For the purpose of this model, only the 28 r 1Ox values in water, saline, vitreous fluid, and plasma were used.…”

Modeling the Effect of Hyperoxia on the Spin–Lattice Relaxation Rate R1 of Tissues

“…It is known that the values estimated by the r1 Ox model and R1 Blood models by Bluemke et al 15,19 both agree well with empirical measurements (R 2 = 0.93 and 0.93); however, since the tissue compartment of this model contains variables that were not measured at the time of OE-MRI data collection (i.e., hematocrit, changes in arterial PO 2 , tumor blood volume), it is not possible to quantitatively compare the model ΔR1 predictions to the measured ΔR1 with metrics such as R 2 and MSE. Instead, we used a variety of reported ΔR1 from OE-MRI literature to gain a rough estimation of the accuracy of this model: qualitative ΔR1 responses, categorized by different tumor tissue types by the authors of the OE-MRI literature, are listed in Supporting Information Table S1.…”

“…The ΔR1 induced from supplemental oxygen in the arterial (ΔR1 B,A ) and venous (ΔR1 B,V ) blood compartments can both be calculated from the following equation $$$$ \Delta R{1}_B=R{1}_b\left(P{O}_{2, ox}\right)-R{1}_b\left(P{O}_{2, air}\right) $$$$ where R1 b (PO 2 ) is the general equation for calculating the R1 of blood by Bluemke et al 19 (shown in Figure 3): $$$$ {\displaystyle \begin{array}{cc}R{1}_b\left(P{O}_2\right)& ={f}_e\left(R{1}_{eox}+r{1}_{dHb}\left[ Hb\right]\left(1-\frac{P{O_2}^n}{P{O_2}^n+P{50}^n}\right)\right)\\ {}& \kern1em +\left(1-{f}_e\right)\left(R{1}_p+r{1}_{pOx}P{O}_2\right)\end{array}} $$$$ where R1 b is the relaxation rate of whole blood, R1 eox is the relaxation rate of erythrocytes when SO 2 = 100%, [Hb] is the mean corpuscular hemoglobin concentration (5.15 mmol Hb tetramer/L plasma), r1 dHb is the molar relaxivity of deoxyhemoglobin (in s −1 L plasma in erythrocyte/mmol Hb tetramer), n is the Hill exponent for hemoglobin (typically 2.7), 30 R1 p is the longitudinal relaxation rate of plasma (s −1 ), and r1 pOx is the relaxivity of dissolved oxygen in the plasma in s −1 mmHg −1 oxygen. The variable f e is the fraction of water in whole blood that resides in erythrocytes (0–1), which is described by the equation: …”

“…The variable f e is the fraction of water in whole blood that resides in erythrocytes (0–1), which is described by the equation: $$$$ {f}_e(Hct)=\frac{W_{RBC} Hct}{W_{RBC} Hct+{W}_{plasma}\left(1- Hct\right)} $$$$ where Hct is the hematocrit (0–1), W RBC is the volume fraction of water within the erythrocyte (typically valued at 0.70 due to hemoglobin occupying approximately 30% of the erythrocyte volume), and W plasma is the volume fraction of water within the plasma (typically valued at 0.95, where the other 5% volume is occupied by plasma proteins such as albumin) 32 . In addition, Bluemke et al 19 modeled R1 eox , R1 p , r1 dHb , and r1 pOx to have a linear dependence on B0 (where β 0 and β 1 are the y‐intercept and slope for the linear fit): $$$$ R{1}_{eox}(B0)={\beta}_{0,R{1}_{eox}}+{\beta}_{1,R{1}_{eox}}B0 $$$$ $$$$ R{1}_p(B0)={\beta}_{0,R{1}_p}+{\beta}_{1,R{1}_p}B0 $$$$ $$$$ r{1}_{dHb}(B0)={\beta}_{0,r{1}_{dHb}}+{\beta}_{1,r{1}_{dHb}}B0 $$$$ <...…”

“…In addition, Bluemke et al 19 modeled R1 eox , R1 p , r1 dHb , and r1 pOx to have a linear dependence on B0 (where β 0 and β 1 are the y‐intercept and slope for the linear fit): $$$$ R{1}_{eox}(B0)={\beta}_{0,R{1}_{eox}}+{\beta}_{1,R{1}_{eox}}B0 $$$$ $$$$ R{1}_p(B0)={\beta}_{0,R{1}_p}+{\beta}_{1,R{1}_p}B0 $$$$ $$$$ r{1}_{dHb}(B0)={\beta}_{0,r{1}_{dHb}}+{\beta}_{1,r{1}_{dHb}}B0 $$$$ $$$$ r{1}_{pOx}(B0)={\beta}_{0,r{1}_{pOx}}+{\beta}_{1,r{1}_{pOx B0}}\ B0. $$$$ The values used for all empirically derived parameters and constants in the blood model are listed in the original publication 19 . The behavior of this model is shown in Figure 3.…”