Avery JL Berman^{1,2}, Jonathan R Polimeni^{1,3}, and G Bruce Pike^{2,4}

The weak
field approximation (WFA) is a theory that relates T_{2} relaxation from tissue to
the underlying tissue properties and is commonly applied to the analysis of relaxation
from red blood cells (RBCs) in blood. This study examines the hematocrit-dependence
of the different parameters of the WFA using simulated populations of RBCs and published
experimental relaxometry results from two studies. Both the simulations and the
experimental results show an unexpected result that the characteristic
perturber size estimate is not constant with hematocrit but is negatively
correlated with it. This has important implications for the implementation and
interpretation of the WFA theory on blood relaxometry data.

The WFA was derived by assuming that the field offsets generated by the system of perturbers have a radial correlation, *G*(*r*), that follows the relationship$$G(r)=G_0\text{exp}[-(r/r_c)^2],$$ where *G*_{0} is the mean square field inhomogeneity of the system and *r _{c}* is a characteristic length of the field perturbations.

In a Carr-Purcell-Meiboom-Gill (CPMG) experiment, the measured relaxation rates' dependence on the WFA parameters is$$ R_2=R_{2,0}+\gamma^2\,G_0\,\frac{r_c^2}{2D}\,F\left(\tau_{180}\frac{2D}{r_c^2}\right)$$ where τ_{180} is the refocusing interval, *R*_{2,0} is the relaxation rate in the limit τ_{180}$$$\rightarrow0$$$, *D* is the diffusion coefficient, and *F*(*x*) is defined in ref 2. *R*_{2,0}’s relationship to hematocrit and SO_{2} is well documented and not explored here.^{8}

The impact of varying hematocrit on the WFA parameters was examined experimentally using two published relaxometry datasets with wide ranges of hematocrit and SO_{2}. One consisted of *R*_{2} measurements at 1.5T in 88 human umbilical cord blood specimens from 6 caesarian deliveries.^{10} The second study consisted of *R*_{2} measurements at 3T in 20 bovine blood specimens.^{11} The *R*_{2} values from each specimen were fit to Eq. (3), resulting in estimates for *G*_{0}, *r _{c}*, and

Example radial correlation functions from three different sphere distributions, along with their fits to Eq. (1), are shown in Figs. 2a–c. The fitted *G*_{0} and *r _{c}* for all the distributions are shown in Figs. 2d–e.

Similar trends were observed in the experimental data of both studies (Fig. 5). Although not all tests were statistically significant, a quadratic fit of *G*_{0} vs. hematocrit outperformed the linear fit, and *r _{c}* did tend to negatively correlate with hematocrit.

This study shows for the first time how *r _{c}* decreased in proportion to hematocrit when RBCs cannot overlap. The rate of decrease also depended on the spatial distribution of the RBCs. This is unexplained by current theory, where no hematocrit dependence of

The experimental results corroborated these findings, although experimental and biological variability likely decreased the statistical significance of some of the relationships. This suggests that fits to blood relaxometry data may require a correction to account for the hematocrit dependence of *r _{c}*.

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- Grgac, K., et al., Transverse water relaxation in whole blood and erythrocytes at 3T, 7T, 9.4T, 11.7T and 16.4T; determination of intracellular hemoglobin and extracellular albumin relaxivities. Magn Reson Imaging, 2017. 38: p. 234-249.

Fig 1: Example cross-sections through the three three-dimensional sphere distributions, each populated to 40% hematocrit. (a) An overlapping distribution where the sphere positions were uncorrelated and allowed to overlap. (b) A random non-overlapping distribution where sphere positions were randomly assigned without overlap until the desired hematocrit was reached. (c) A hexagonal close packed distribution where the spheres were randomly assigned to the lattice elements of a hexagonal close packed distribution until reaching the desired hematocrit. The sphere boundaries are emphasized in black to highlight the differences between the distributions.

Fig. 2: (a–c) Radial correlation functions from the three sphere distributions at hematocrit = 40% (circle markers) and their fits to Eq. (1) (dashed line). These have been normalized such that their values at r = 0 should equal Hct, or Hct(1–Hct) for the non-overlapping networks. (d) *G*_{0} values from the sphere distributions vs. hematocrit. The dashed and dash-dotted lines show the cases where *G*_{0} is proportional to Hct or Hct(1–Hct), respectively. (e) The characteristic length, *r*_{c}, obtained from the fits to the radial correlation functions as a function of hematocrit. The dashed black line is the predicted *r*_{c}.

Fig. 3: The mean simulated signals vs. the CFS for Hct = 40%, τ_{180} = 40 ms, and the three different sphere distributions: (a) overlapping, (b) random non-overlapping, and (c) hexagonal close packed. The shaded bands represent the mean ± standard deviation of the simulated signals and the dashed lines represent the CFS for each SO_{2} as predicted using the fitted *G*_{0} and *r*_{c} values from the radial correlation functions. The plotted colours correspond to the SO_{2} values in the bar on the right. Simulation parameters were B_{0} = 3T, D = 2.7 μm^{2}/ms, and intrinsic T_{2} was ignored.

Fig. 4: Accuracy of the CFS compared to the simulations across all volume fractions for the three sphere distributions as assessed using Δ*R*_{2}*.* Mean simulated Δ*R*_{2} are represented by the circle symbols and the CFS-predicted Δ*R*_{2} are represented by the x’s. The sphere distributions are overlapping (a), non-overlapping (b), and hexagonal close packed (c). The symbol colours correspond to the SO_{2} values in the colour bar on the right. Note that the Δ*R*_{2} scale in (a) is four times those of (b) and (c).

Fig. 5: Fitted *G*_{0} and *r*_{c} estimates from the experimental relaxometry data of Portnoy et al.^{11} (top row) and Grgac et al.^{12} (bottom row). Dotted lines show the linear regression of *G*_{0} and *r*_{c} vs. Hct. Dashed lines show the quadratic fit of *G*_{0} vs. Hct. To account for *G*_{0}’s simultaneous SO_{2}-dependence, the *G*_{0} estimates were normalized by (1–SO_{2})^{2}. The coefficients of determination (R^{2}) and P-values for each regression are listed within each figure.