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

The change in longitudinal relaxation rate (R1) produced by oxygen has been used as a means of inferring oxygenation levels in magnetic resonance imaging in numerous applications. The relationship between oxygen partial pressure (pO2) and R1 is linear and reproducible, and the slope represents the relaxivity of oxygen (r1Ox) in that material. However, there is considerable variability in the values of r1Ox reported, and they have been shown to vary by field strength and temperature. Therefore, we have compiled 28 reported empirical values of the relaxivity of oxygen as a resource for researchers. Furthermore, we provide an empirical model for estimating the relaxivity of oxygen in water, saline, plasma, and vitreous fluids, accounting for magnetic field strength and temperature. The model agrees well (R2 = 0.93) with the data gathered from the literature for fields ranging from 0.011 to 8.45 T and temperatures of 21‐40 °C. This provides a useful resource for researchers seeking to quantify pO2 in simple fluids in their studies, such as water and saline phantoms, or bodily fluids such as vitreous fluids, cerebrospinal fluids, and amniotic fluids.


| INTRODUCTION
Many researchers have investigated using the paramagnetic relaxivity effect of oxygen on longitudinal relaxation as a means of inferring oxygenation levels for applications ranging from cancer therapy to seawater analysis. [1][2][3][4][5] For example, measurements of the longitudinal relaxation rate R 1 (1/T 1 ) have been used to infer oxygen levels in vitreous fluid as a noninvasive alternative to the highly invasive oxygen electrodes used to measure retinal hypoxia, 6-8 bladder urine 9 and urine in the renal pelvis to create a noninvasive detection of renal dysfunction, 5 and cerebrospinal fluid, 9,10 and this relationship between pO 2 and R 1 is also the basis for oxygen-enhanced MRI techniques. [11][12][13] In the linear relationship between pO 2 and R 1 , the slope represents the relaxivity of oxygen, or r 1Ox , in that material. Unfortunately, however, there is considerable variability in the values reported for r 1Ox from empirical measurements, and consequently reliable quantification of pO 2 from R 1 measurements presents a challenge.
This paper provides a summary of the empirical measurements reported in the literature, investigating the relationship between R 1 and the partial pressure of oxygen in phantoms, saline and water solutions, and vitreous fluid. These experiments have been performed using different equipment and field strengths and reported with a variety of units. For consistency, therefore, all T 1 values will be reported in ms, R 1 in s À1 , and relaxivity in s À1 /mmHg oxygen, with the corresponding field strength, temperature and material specified where these data are available. We then Abbreviations: AIC, Akaike information criterion; B 0 , main static magnetic field in MRI scanner; MSE, mean squared error; pO 2 , partial pressure of oxygen; R 1 , longitudinal relaxation rate; r 1Ox , relaxivity of oxygen.
propose a simplified empirical model for estimating r 1Ox in water, saline, plasma, and vitreous fluids based on these reported literature values and report the resulting model parameters. Our aim is to provide a useful review and tool for researchers seeking to quantify pO 2 in simple fluids, such as water and saline phantoms, or bodily fluids such as vitreous fluids, cerebrospinal fluids, and amniotic fluids. This model does not represent the r 1Ox in blood or tissue, due to the addition of proteins, structure, cells, lipids, and deoxyhemoglobin, which will affect the R 1 -pO 2 relationship, to be addressed in a separate manuscript.
2 | METHODS 2.1 | Model theory T 1 (measured in ms), and its inverse, R 1 (typically reported in s À1 ), have both been used in the literature when reporting the relaxivity effect of oxygen. R 1 is linearly dependent on the concentration of paramagnetic particles, 14,15 in this case dissolved molecular oxygen in the solution, with the following equation: where R 1Ox is the relaxation rate in the solution with oxygen added, R 1,0 is the relaxation rate in the solution without oxygen, C is the concentration or partial pressure of oxygen, and r 1Ox is the relaxivity of oxygen in that solution (whose units depend on the oxygen measurement used in the constant C) (shown in Figure 1). Since the partial pressure of oxygen (pO 2 ) is a common measurement in biomedicine and clinical applications, in this manuscript, we report C as pO 2 in mmHg and r 1Ox in s À1 /mmHg. Conversion factors to other common units such as kPa, Torr, mmol/L, mg/L, and mL/L can be found in Supplementary Table S1.
Changes in both T 1 and R 1 have been used to report changes in pO 2 in the past. 16 However, although an increase in oxygen could still qualitatively roughly be inferred from a shortening of T 1 , it is important to note that the linear relationship exists with 1/T 1 (R 1 )-not T 1 -and The relationship between T 1 and pO 2 (A) and R 1 and pO 2 (B) in a solution, with the initial T 1 of 3000 ms. The values are calculated using a range of r 1Ox (relaxivity) reported in the literature at 1.5 T, in units s À1 /mmHg oxygen The relationship between ΔT 1 and initial T 1 (A) and ΔR 1 and initial R 1 (B), for a ΔpO 2 of 200 mmHg. The values are calculated using a range of r 1Ox (relaxivity) reported in the literature at 1.5 T, in units s À1 /mmHg oxygen therefore the change in T 1 caused by oxygen will be dependent on the original T 1 (shown in Figure 2). Therefore, for a quantitative inference of pO 2 change it is necessary to discuss changes with respect to R 1 .
The r 1Ox , or relaxivity of oxygen, is affected by various experimental factors, including field strength. Equations already exist to determine the relationship between field strength and R 1 ; in 2001, Teng et al. 17 measured the proton spin-lattice relaxation rate in water as a function of magnetic field strength at 1 atm of oxygen (approximately 760 mmHg) and found that the magnetic relaxation dispersion due to the paramagnetic contribution from molecular oxygen is well approximated by a Lorentzian shape, for which they proposed the following equation: where A and B are constants, ω s is the electron Larmor frequency, and τ is the correlation time for the electron-nuclear coupling (empirically measured to be 6.8 ± 0.5 ps in water). 17 One variable of particular interest for medical imaging research is field strength, which can be related to the electron Larmor frequency above (ω s ) by the electron gyromagnetic ratio (γ e = 1.76 Â 10 11 rad s À1 T À1 ) and Larmor equation ω s = γ e B 0 . By substituting ω s = γ e B 0 into Equation 2, we can see that From Equation 1 we know that 1/T 1Ox is proportional to the relaxivity (r 1Ox ), and from Reference 17 we know that the Lorentzian magnetic relaxation dispersion is due to the paramagnetic contribution from molecular oxygen. Therefore, we propose that the relationship between r 1Ox and field strength will be well approximated by a similar Lorentzian equation with new constants: where C 1 , C 2 , and C 3 are new constants, and B 0 is the magnetic field strength (T). It is worth noting that in Equation 4 C 1 will not be equal to Aτ as it is in Equation 3, since Equation 3 is calculating R 1 and Equation 4 is calculating r 1Ox -while R 1 and r 1Ox should be proportional, there are additional multiplying or dividing factors that will be encompassed by C 1 .
Finally, the relaxivity of oxygen is also reported to be affected by temperature, as seen in the varied relaxivity measurements reported by Muir et al. 8 To account for this, we add a fourth constant (C Temp ), which represents the linear slope of relaxivity change due to temperature, resulting in the final equation:

| Data collection and analysis
The 28 reported values for oxygen relaxivity were collected from the literature, and all units were converted to s À1 /mmHg oxygen, shown in Table 1 alongside the field strength and material used in each experiment. If data extraction from graphs was necessary, a digital plot analyzer was used to reliably extract values. 25 The SciPy optimize function for nonlinear least-squares fitting was used. 26 Equation 5 was fitted using a randomized subset of 90% of the 28 literature data points in Table 1, fitting the B 0 and temperature values simultaneously. The dataset was split into randomized subsets for fitting using the sklearn train_test_fit function with shuffling. 27 This process was iterated 1000 times, and the median and 95% confidence interval of the distribution of fitted values for each parameter were used as the final parameters (listed in Table 2

| RESULTS
As shown in Table 1, we found 28 total measurements of r 1Ox : 7 measured in saline solutions, 7,9,16,18,19 18 in water, 3,8,9,[20][21][22][23] 1 in vitreous fluid, 7 and 2 in plasma (ex vivo). 16,24 The measurements were collected at field strengths ranging from 0.011 to 8.45 T and temperatures ranging from 21 to 40 C. 12 additional values of r 1Ox , measured in blood (ex vivo 16,24,28 and in vivo 20,29 ) and tissues (lung 30 and brain 31 ), were also found T A B L E 1 A collection of 28 reported values for oxygen relaxivity from the literature with all units as s À1 /mmHg oxygen alongside the field strength and material used in each experiment. Temperature is indicated where it was reported. The MRI acquisition details from each experiment are listed in Supplementary Table S4 Reference (provided in Supplementary Table S2); however, these were not included in the model fitting because blood and tissues will contain factors not accounted for in this model (see Section 4).
The final values for the four parameters C 1 , C 2 , C 3 , and C Temp (with lower and upper 95% confidence intervals) are listed in Table 2 To understand the behavior of the resulting model, the effect of varying field strength on r 1Ox is illustrated using synthetic data under varying temperatures (Figure 4), and the linear effect of varying temperature on r 1Ox is illustrated using synthetic data under varying field strengths (Supplementary Figure S4). The resulting model prediction is also shown over a scatterplot of the r 1Ox of the 28 literature points in Figure 5.
Since Equation 5 contains four parameters, the fitting process was repeated for all combinations of fewer parameters (eg removing C Temp ) and the Akaike information criterion (AIC) was calculated for each version of the model-the best-fit model according to the AIC is the model that explains the greatest amount of variation using the fewest possible independent variables. 32 The AIC score, R 2 , and MSE results of the different models tested are listed in Supplementary Table S3 temperature, and that this variation was well approximated by a Lorentzian function and linear relationship with temperature. Therefore, while the table of reported r 1Ox empirical measurements can be referred to for future oxygen-MRI experiments, the r 1Ox can also be estimated using the proposed simplified model for estimating the r 1Ox in water, saline, plasma, and vitreous fluids that agrees well with the empirical measurements.
The relationship between longitudinal relaxation and the paramagnetism of oxygen has a long history in NMR, and there is a large body of both theoretical and empirical work. 1,14,15,[33][34][35][36][37] For materials containing paramagnetic contrast agents, there can be complex relationships between R 1 and field strength, and these relationships are affected by various factors inherent to the contrast agent. 38 This relationship between contrast agent relaxivity and the magnetic field is important, as it can obscure the reproducibility of MRI-oxygen experiments if performed at different field strengths and temperatures.
The physical mechanisms that explain the relationship between field strength and relaxivity are complex, and although there are sophisticated explanations for specific contrast agents, 39-42 much of the modelling of this relaxivity relies on empirical measurements. 43 While there has been previous evidence for modelling relaxivity-B 0 relationships as linear 44 or logarithmic, 45 one major limitation is that the majority of relaxivity measurements of contrast agents are performed at only two field strengths (1.5 and 3 T), which makes it difficult to properly describe the true relationship from empirical results. Interestingly, to address this issue, an experiment by Chou et al measured the relaxivity of a gadolinium-based contrast agent at a large range of field strengths, 0-12 T, revealing a curve that peaks around 1-2 T and drops off (in a Lorentzian shape) as field strength increases. 39 While these data are from a gadolinium-based contrast agent rather than oxygen, they suggest that the relationship of oxygen relaxivity and magnetic field strength may also follow a more complex curve than simply linear, or logarithmic. Therefore, for any modelling of relaxivity, it is important to state the range of field strengths over which the model is valid. The lowest field strength used to fit this model was 0.011 T, and below this field strength it is likely that the r 1Ox curves back down to zero as field strength approaches zero, in a similar manner to the pattern seen in Figure 4 of Chou et al, 39 which we have reproduced in Supplementary Figure S6.

| Limitations
The relaxivity values collected were from experiments performed over a timespan of five decades, with a huge variation in experimental equipment and temperature measurement techniques. Experimental measurement of r 1Ox can be difficult even within a relatively simple system  Supplementary   Table S4. MRI technology has advanced significantly since the measurements made in the 1980s, which account for the measurements made below 0.5 T, and low-field systems usually also suffer from poor signal to noise ratio. In addition, the values have often been extracted from original plots, some hand drawn, which is a source of error, and converted from the various original units to s À1 /mmHg, which can be another source of error. Finally, one experiment did not report the temperature of the solution during the experiment, and it was assumed to be 37 C.
These limitations inevitably represent a large source of potential error in the accuracy of this model. It is hoped, however, that this can be addressed as the NMR community produces new measurements of r 1Ox at a range of field strengths and temperatures. As a future direction of this work, we have hosted the open-source model code and current r 1Ox measurements on GitHub (github.com/BulteGroup/ OxygenRelaxivityModel) and invite the NMR community to share new r 1Ox measurement results and refit the model to improve the accuracy and enhance the utility of the model.
Furthermore, for the fitting of this model, we have combined values from water, saline, plasma, and vitreous fluids. In reality, there are factors that would cause the relaxivity in these solutions to differ, even amongst saline solutions with different compositions and concentrations. The decision to combine them was due to a lack of sufficient data points at a range of field strengths and temperatures in each solution; however, this is a considerable limitation, as it seems to fit more accurately to the water samples (R 2 = 0.94) than saline samples (R 2 = 0.73) alone (Supplementary Figure S3). Most importantly, values from the vitreous fluid and plasma would ideally be fit with separate models, as there are extra proteins in the plasma and vitreous fluid that have been shown to decrease the relaxivity slightly. 7,16 However, this simply was not possible due to a lack of available data points in the literature. As above, we very much hope that this issue will be addressed as new data are acquired by future researchers.
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.

| CONCLUSION
In conclusion, we have provided an overview of the literature reporting a relationship between longitudinal relaxation and oxygen in phantoms, saline and water solutions, and vitreous fluid ranging from 0.011 to 8.45 T and 21 to 40 C. In addition, we have provided a simplified model for estimating the r 1Ox in water, saline, plasma, and vitreous fluids that agrees well (R 2 = 0.93) with the empirical measurements. We hope that this will provide a useful resource for researchers seeking to quantify pO 2 in simple fluids, such as water and saline phantoms, or bodily fluids such as vitreous fluids, cerebrospinal fluids, and amniotic fluids.