A method to remove the influence of fixative concentration on postmortem T2 maps using a kinetic tensor model

Abstract Formalin fixation has been shown to substantially reduce T2 estimates, primarily driven by the presence of fixative in tissue. Prior to scanning, post‐mortem samples are often placed into a fluid that has more favourable imaging properties. This study investigates whether there is evidence for a change in T2 in regions close to the tissue surface due to fixative outflux into this surrounding fluid. Furthermore, we investigate whether a simulated spatial map of fixative concentration can be used as a confound regressor to reduce T2 inhomogeneity. To achieve this, T2 maps and diffusion tensor estimates were obtained in 14 whole, formalin‐fixed post‐mortem brains placed in Fluorinert approximately 48 hr prior to scanning. Seven brains were fixed with 10% formalin and seven brains were fixed with 10% neutral buffered formalin (NBF). Fixative outflux was modelled using a proposed kinetic tensor (KT) model, which incorporates voxelwise diffusion tensor estimates to account for diffusion anisotropy and tissue‐specific diffusion coefficients. Brains fixed with 10% NBF revealed a spatial T2 pattern consistent with modelled fixative outflux. Confound regression of fixative concentration reduced T2 inhomogeneity across both white and grey matter, with the greatest reduction attributed to the KT model versus simpler models of fixative outflux. No such effect was observed in brains fixed with 10% formalin. Correlations between the transverse relaxation rate R 2 and ferritin/myelin proteolipid protein (PLP) histology lead to an increased similarity for the relationship between R 2 and PLP for the two fixative types after KT correction.

: Acquisition parameters for the TSE scans used in this study. Details of individual brains provided in Table S1.

Estimating T2 using an Extended Phase Graph (EPG) model
The . At 7T, B1 inhomogeneity produces a spatially varying flip angle across the brain ( Fig. S1b), leading to refocusing pulses that vary from 180 o . Under these conditions, the signal evolution can deviate substantially from a mono-exponential signal model ( Fig. S1a -purple, blue and green lines). Fitting a mono-exponential signal model to these data will lead to estimates of T2 that strongly depend on the B1 profile (Fig. S1c).
Extended Phase Graphs (EPG) provide a more accurate description of signal evolution under a variety of MRI sequences and conditions (1)(2)(3). In this work, we use an EPG framework to describe the signal evolution of the TSE sequence under a refocusing pulse of flip angle a. Our approach fits the measured TSE signal to estimate the voxelwise T2 and a (via estimation of B1). Fitting with an EPG model is shown to reduce the bias on T2 estimates in areas of low B1, leading to more homogeneous T2 maps across the brain (Fig.   S1d).
To achieve this, we performed a two-step fitting approach, where we first fit the TSE signal to an EPG model to estimate T2 and B1 across the brain. It was found that in regions of very low B1, sharp discontinuities were observed in the B1 maps. Given that B1 is expected to smoothly vary across the brain, we subsequently smoothed the B1 map and repeated the fitting for T2, keeping B1 as a fixed parameter.
Step one: For each postmortem brain we simulated the TSE sequence (parameters provided in Table   S2) under an EPG framework, obtaining voxelwise estimates of B1 and T2 by minimising: where TSE $%& is the experimental TSE data over all six echoes (TE ':) ) and TSE *+, is the simulated TSE signal using the EPG framework given a value of ' , -, ' , and the echo times. To avoid fitting for the signal amplitude ( . ), the experimental data and simulated signal were normalised (e.g. division by 3∑ TSE / ). In a series of evaluations, T1 was found to have very little effect on our T2 estimates, after which it was set equal to a fixed constant (450 ms -the approximate value of measured T1 in our postmortem datasets).
Fitting was performed in MATLAB (version 2019b, The MathWorks, Inc., Natick, MA) based on code from (3). Eq. [S1] was minimised using lsqnonlin. Figures S2a and b show the resulting B1 and T2 maps for a single postmortem brain fit with this approach. The spatial inhomogeneity across the T2 map is substantially reduced in comparison to the mono-exponential fit (Fig. S1c). However, B1 maps are expected to vary smoothly across the brain and Fig. S2a reveals changes in B1 that depend on the tissue contrast (e.g. blue arrow), in addition to underestimation of B1 close to the brain boundary Figure S1: Motivation for the EPG framework. When the refocusing angle = 180 o (a -red line), the TSE signal evolves via a mono-exponential decay. However, at different flip angles the signal evolution can deviate substantially from a mono-exponential signal model (a -purple, blue and green lines). At 7T, brain samples experience B1 inhomogeneity, leading to a spatially varying flip angle across the brain (b). Fitting these data with a mono-exponential model leads to a T2 map that depends strongly on the B1 profile (c). An EPG framework is able to account for the spatially varying flip angle across the brain, leading to T2 maps with more homogeneous contrast (d). (a) simulated using an EPG framework (TE = 10 -60 ms with a 10 ms echo spacing, T2 = 30 ms), with resulting curves normalised to the signal at TE = 10 ms to aid visualisation of the deviation from a mono-exponential decay. Note that there is a degeneracy when a > 180 o , where flip angle a produces the same signal evolution as 360 oa. Step two: To ensure spatial smoothness of B1, the B1 maps from step 1 were subsequently filtered using a local 3D polynomial filter (order = 2, kernel volume = 10x10x10 mm 3 ), with voxels weighted by the inverse of the standard error on the B1 estimates. Figure S2c displays an example B1 map after filtering, revealing a smoothly varying B1 profile across the entire brain. Figure S2: T2 estimates with EPG and smoothing B1.
Step 1 -Our EPG model estimates B1 maps a with decreasing B1 as the brain boundary is approached (a), in addition to T2 maps that do not have a strong dependence on the B1 profile (b). The B1 profile is expected to be smoothly varying across the brain.
However, anatomical contrast is visible within the B1 map (a -blue arrow), in addition to sharp discontinuities in areas close to the brain boundary (a -green arrows), regions associated with low SNR. This leads to artefacts and subtle contrast changes in the resulting T2 map (b -blue and green arrows).
Step 2 -By smoothing the B1 maps with a polynomial filter (c) and fixing B1 to the resulting map in a second stage estimate of T2, we obtain consistent T2 estimates across the brain (d). Here the B1 maps (a and c) are  T2 estimates were subsequently regenerated across the brain using the smoothed B1 map as follows: where B1 was fixed to the value of the smoothed B1 map. Figure S2d displays the resulting T2 map, where the most notable change is in regions of the brain where the B1 was previously underestimated (Fig. 2d green arrows) -in these regions the T2 estimates now match those of the surrounding tissue.
In regions very close to the brain boundary (characterised by very low B1 and low SNR), in some brains the T2 estimates were found to be highly sensitive to noise, leading to spurious values (Fig. S3a). To correct for this, regularisation was added to Eq.
[S2] as follows: where is a regularisation constant ( = 2), -,,$4 is the median value of T2 across the entire brain from the first step fitting and W is a scalar weight 3∑ TSE $%&,/

Correcting for spuriously high diffusion coefficients
It is not possible to determine diffusion coefficients in voxels with unreliable signal estimates. In this study, unreliable signal estimates arose (1) in voxels in very close proximity to remaining air bubbles, (2) in regions of extremely low SNR, and (3) due to imperfect masking (resulting in a small number of non-brain tissue voxels remaining in the brain mask, e.g. small quantities of signal-producing formalin remaining in brain sulci and ventricles). In these voxels, diffusion coefficients manifested as spuriously high values on our resulting diffusion tensor estimates.
The number of unreliable voxels was small, with an average of 1.9 ± 2.1% of voxels / brain requiring correction, below 0.6% in 7 of the 14 brains. However, it is critical to correct for these voxels for two reasons. First, an accurate brain shape is important for modelling fixative dynamics -removal of these voxels would lead to a change the brain shape, or holes in the brain. Second, spuriously high diffusion coefficients can lead to artefacts in the resulting fixative concentration maps (Fig. S4).     formalin. The corrections presented in this work assume that presence of fixative in tissue is the source of variation in T2 across the brains. However, the predicted patterns of fixative are similar to the spatial pattern of B1. Here, we consider how much of the variation in T2 may be explained by the B1 spatial profile.
A dependency is observed vs B1 in our post-mortem cohort (a), with a stronger effect in brains fixed with 10% NBF (left) vs 10% formalin (right), consistent with the other three models investigated in this study (Main Text Figs. 7 and 8). However, regressing out the influence of B1 using Eq. [6] (b) leads to a higher remaining inhomogeneity as shown in Table S3 vs the D2S, KI and KT models for the brains fixed with 10% NBF (Main Text Table 1), and similar performance (no change) for brains fixed with 10% formalin. Results displayed as the mean ± standard deviation across brains.    Regressing out the influence of the KI model incorporating these mean diffusivity maps using Eq.