The distortions of the free water model for diffusion MRI data when assuming single compartment relaxometry and proton density

Abstract

Objective:

To document the bias of the simplified free water model of diffusion MRI (dMRI) signal vis-à-vis a specific model which, in addition to diffusion, incorporates compartment-specific proton density (PD), T1 recovery during repetition time (TR), and T2 decay during echo time (TE).

Approach:

Both models assume that volume fraction f of the total signal in any voxel arises from the free water compartment (fw) such as cerebrospinal fluid (CSF) or edema, and the remainder (1-f) from hindered water (hw) which is constrained by cellular structures such as white matter (WM). The specific and simplified models are compared on a synthetic dataset, using a range of PD, T1 and T2 values. We then fit the models to an in vivo healthy brain dMRI dataset. For both synthetic and in vivo data we use experimentally feasible TR, TE, signal-to-noise ratio (SNR) and physiologically plausible diffusion profiles.

Main Results:

From the simulations we see that the difference between the estimated simplified f and specific f is largest for mid-range ground-truth f, and it increases as SNR increases. The estimation of volume fraction f is sensitive to the choice of model, simplified or specific, but the estimated diffusion parameters are robust. Specific f is more accurate and precise than simplified f. In the white matter (WM) regions of the in vivo images, specific f is lower than simplified f.

Significance:

In dMRI models for free water, accounting for compartment specific PD, T1 and T2, in addition to diffusion, improves the estimation of model parameters. This extra model specification attenuates the estimation bias of compartmental volume fraction without affecting the estimation of other diffusion parameters.

Introduction

Diffusion MRI (dMRI) probes the microstructure of biological tissue through water diffusion. The standard sequence for collecting the dMRI signal is single-shot echo-planar imaging (ss-EPI) ¹, with gradients following a pulsed-gradient spin-echo (PGSE) scheme ², which is defined by repetition time (TR), echo time (TE), and the diffusion settings (diffusion time, pulse time, and gradient strength). The acquired signal is modeled as a product of four terms arising from four properties or effects: proton density (PD), T1 recovery during time TR, T2 decay during time TE, and diffusion (b>0 ms/μm²) ³. The modeling accuracy of each term is dependent on factors such as tissue type, field strength, temperature and, especially, the homogeneity of the tissue producing such signal.

When assuming a single water compartment, before fitting the model to the data, the total acquired signal described above is stripped of the T1, T2 and PD weights. The most common of these methods, the “normalization” of the signal, divides the diffusion weighted (b>0) signal by the non-diffusion-weighted (b=0) signal ^4,5. Such procedure has been shown to keep the estimated diffusion parameters broadly unaffected: Celik ⁶ observed that the estimated apparent diffusion coefficient (ADC) from a phantom remains stable for TR higher than 3 s at room temperature (22°C). One can infer that such normalization would also not distort the characterization of diffusion in 3D, commonly described via a diffusion tensor (DT) ⁷.

When the voxel signal contains partial volumes of fluid or tissue, a model consisting of multiple compartments can describe the underlying tissue more specifically ^8,9. The (no-exchange) multi-compartment models describe the total signal in each voxel’s microstructure as a sum of signals from each constituent tissue compartment (where, further, each compartment signal is described as a product of PD, T1, T2 and diffusion terms, as described above). An example of a multi-compartment model is the “free water” model, which seeks to separate tissue from surrounding fluid, e.g., to separate white matter (WM) from cerebrospinal fluid (CSF) ^10,11,12,13.

Although occasionally an adjustment for T1 and/or T2 is made ^12,14,15,16, in the majority of dMRI studies that use multicompartment models, the correct accounting for all three (PD, T1, T2) terms, as recently done by Abbas et al. ¹⁷ or Grussu et al. ¹⁸, is the exception rather than the rule. Specifically for the free water model, the PD, T1, T2 terms are usually assumed to be the same across the compartments; although the studies that use such models often acknowledge potential differences in compartment specific T1, T2 and PD, and that experiment choices for TR and TE would alter the relative weight of these terms ¹⁹. So, although it is well known that T1, T2 and PD affect the dMRI signal, the size of this collective effect is not.

With this study we seek to evaluate this volume fraction bias through analytical simulations and a clinically feasible in vivo brain dataset, examining the effect that PD, T1 and T2 relaxation terms have on the compartmentalization of the signal and the estimated parameters of the free water model.

Methods

We will refer to the free water model that combines a tensor for tissue and an isotropic tensor for water as FWDT. We start by defining the specific FWDT model, where the tissue and free water compartments have their own PD, T1, T2, and diffusion terms. The simplified model specifies only diffusion terms. Then, we describe the synthetic data generated and the in vivo data acquired for comparing these two models.

Analytical Expression of Signal

The specific signal equation for the FWDT model is the linear sum of the signal from the free water (fw) compartment and the signal from the tissue compartment, which we refer to as hindered water (hw), as indicated by the bracketed terms below ^8,10.

$$S=\left[S_0^{hw}·\left(1-f\right)·S_{DW}^{hw}\right]+\left[S_0^{fw}·f·S_{DW}^{fw}\right]$$ [1]

The weighting term f is the volume fraction of fw relative to the total volume. Each hw and fw term has its non-diffusion-weighted signal S₀ and its diffusion-weighted signal S_DW.

The diffusion-weighted signals are:

$$S_{DW}^{fw}=\exp \left(-b{g}^t{D}^{hw}g\right)$$ [2]

$$S_{DW}^{fw}=\exp \left(-b·d^{fw}\right)$$ [3]

where b is the diffusion weighting and g is the diffusion-encoding direction (both set a priori in the experiment). The unknown D^hw is a 3X3 second order tensor (i.e., a square-symmetric matrix) with six unique entries, whereas the free water diffusivity is fixed at d^fw = 3.0 μm²/ms.

For a spin-echo MRI sequence, the non-diffusion-weighted signal contains the PD, T1 and T2 relaxation for each hw and fw compartment signal ^3,20:

$$S_0^{hw}=\kappa ·{\rho }_0^{hw}·\exp \left(-\frac{TE}{T_2^{hw}}\right)·\left[1-\exp \left(-\frac{TR}{T_1^{hw}}\right)\right]$$ [4]

$$S_0^{fw}=\kappa ·{\rho }_0^{fw}·\exp \left(-\frac{TE}{T_2^{fw}}\right)·\left[1-\exp \left(-\frac{TR}{T_1^{fw}}\right)\right]$$ [5]

where κ is an instrumental scaling constant (here assumed to be 1, without loss of generality, as it is independent of the choice of compartment). The effective PDs are set at ${\rho }_0^{hw}=0.7$ and ${\rho }_0^{fw}=1.0$ ^21,22,23,17. The relaxations T1 and T2 take values as specified in Table 1; these are approximate tissue parameters for a 3T field strength ^{24,25,21,26,27,28,29,30,17,31}.

Table 1:

The experimental assumptions for the synthetic and in vivo data at 3T

EXPERIMENT SPECIFICS	VALUE
— — — — — —	— — — — — —
b-values (ms/μm2)	0.5; 0.75; 1.0; 1.25; 1.5
G (mT/m)	55
nr directions/shell	18, optimally spaced in 3D
TR (s)	1; 5; 10; 15
TE in simulation (ms)	30; 70; 110; 150
TE in vivo (ms)	65.8
— — — — — —	— — — — — —
TISSUE SPECIFICS	VALUE
— — — — — —	— — — — — —
fw (CSF) T1 (ms)	4000
fw (CSF) T2 (ms)	2000
fw (CSF) d (μm2/ms)	3
hw (WM) T1 (ms)	850
hw (WM) T2 (ms)	75
— — — — — —	— — — — — —

Assuming that the non-diffusion-weighted signals S₀ from hw and S₀ from fw (Eqns. 4 and 5) are the same, Eq.1 reduces to the simplified signal expression:

$$S=S_0·\left[\left(1-f\right)·S_{DW}^{hw}+f·S_{DW}^{fw}\right]$$ [6]

Parameter Differences from Simulated Data

To illustrate the bias of the estimated free water parameter f-fit from ground-truth f-set, we use Eq.1 (for the specific FWDT model) and Eq.6 (for the simplified model). After setting T1 and T2 to the values shown in Table 1, we generate a dataset for each f-set:

• first, for three noise profiles, with SNR at 20, 40 and 80; and three sets of b-value shells;

• second, for four TR profiles at 1, 5, 10, 15 s; and four TE profiles at 30, 70, 110, 150 ms;

• third, for three fractional anisotropy (FA) profiles of hw, at 0.21, 0.49, 0.76, in all cases choosing axial and radial diffusivities so that the mean diffusivity (MD) is kept at 0.8 μm²/ms.

The choice of b-value range (0.50 to 1.50 μm²/ms) follows from theoretical considerations ⁸ and a previous optimization study ¹³ which uses three points in that range: 0.50, 1.00, 1.50 μm²/ms ¹³. We add two extra b-values to that set (giving b-values: 0.50, 0.75 1.00, 1.25, 1.50 μm²/ms), and use these simulations to see what difference such b-sampling makes. However, unless otherwise specified, the diffusion weighting scheme will contain the five b-value set.

Although the models have 8 unknowns (1 for S₀, 6 for D^hw, 1 for f), thus necessitating at least 8 data points, we acquire many more measurements to improve the SNR. The upper bound on this number is determined by an experimentally acceptable scan time, in our case 10 mins. Therefore, each of the five b-values in that range are sampled in 18 directions, optimally spaced in 3D ³².

The simulations assume that the signal magnitude has Gaussian noise in both the real and the imaginary part of the complex signal model ^33,34,35:

$$S_{noisy}=\sqrt{{\left(S_{noisefree}+{\text{n}}_r\right)}^2+{\left({\text{n}}_i\right)}^2}$$ [7]

where each of n_r and n_i are sampled from a normal distribution with mean 0 and standard deviation $\sigma =\frac{S_0}{SNR}$ (here S₀ = 1). The first squared term under the square-root is the real part of the complex signal and the second term is the imaginary part. One thousand random perturbations of noise at levels SNR=20, 40 and 80 are added to the synthesized signal at each f-set point of each FA profile. In every perturbation, the two (Euler) angles of a cylindrically symmetric tensor are sampled from the sphere parameterized by θ = arccos (2v − 1) and ϕ = 2πu, where u and v are two randomly distributed variables between 0 and 1 on the real line.

Each of the two models is fitted to the dataset via the dMRI processing platform DIPY³⁶, implemented in Python. To obtain the specific model fitting, we made minor modifications to the in-built function fwdti which implements the simplified model ³⁷. From the resulting 1,000 sets of parameter estimates, due to 1000 noisy perturbations, we report the mean and standard deviation.

We examine the stability of the specific parameters to slight changes in T1 and T2 relaxation, after fixing TE=70 ms, TR=10 s, the hw compartment FA=0.76, and MD=0.8 μm²/ms. The T1 and T2 values for hw are perturbed either way by 15%. Due to its relative size and variation, the T1 and T2 values for fw are perturbed either way by 25%.

Parameter Differences from In Vivo Data

After providing informed written consent, a healthy 53-year-old volunteer was scanned using a GE Discovery MR750 self-shielded 3.0 T system with 55 mT/m gradient and a slew-rate of 200 T/m/s (see Table 1). We fix TE and sample four experimentally feasible TR values. We make this choice because, compared to TE, the effect of TR is less studied, as it tends to have the lesser effect on the signal, especially for long TR of, say, 15 or 20 s.

On the mid-axial slice, we select two regions of interest (ROI), each consisting of two non-adjacent areas. The fw ROI lies in the CSF of both atria of the lateral ventricles, whereas the hw ROI lies in the WM of the genu and splenium.

To test the stability of parameter statistics to variations in ROI, we resample the estimated indices through bootstrapped ROIs ³⁸. For this classical bootstrap, given one dataset of n measurements (such as f, FA or MD) pertaining to n voxels within an ROI, a new bootstrap dataset is constructed by sampling with replacement the same number of elements, that is n, from the original dataset. This procedure is repeated one-thousand times, giving a distribution of 1000 ROI-specific metrics, such as mean and standard deviation (again, of f, FA or MD). When reporting statistics on ROIs, we take the average across these 1000 bootstrapped metrics.

Results

We first show the non-diffusion-weighted signal profiles for each compartment. Then we demonstrate the differences that the current (specific) and the previous (simplified) modeling approaches have on the fitted parameters to the synthetic and in vivo datasets.

Analytical Expression of Signal

Fig.1 shows the non-diffusion-weighted (b=0) component of each compartment hw and fw as defined by Eqns. 4 and 5, i.e., $S_0^{hw}$ and $S_0^{fw}$. The signals for both compartments increase with increasing TR and decreasing TE. The two (compartments’) manifolds cross where $S_0^{hw}=S_0^{fw}$, providing a region of TR and TE points for which one would expect little or no difference in the estimation of the (specific vs. simplified) model parameters: e.g., for TE=60 ms the corresponding TR is around 1 s.

Figure 1:

Given an experimental TR and TE, this plot shows the resulting S₀ of the fw compartment (upper sheet) and the S₀ of the hw compartment (lower sheet). Normalization of signal by b=0 signal assumes that the two profiles are the same. Only for the TE and TR values on the line of intersection between the two surfaces does this assumption hold. E.g. at TR=1 s one would expect both (simplified and specific) methods to yield similar S₀ coefficients, and so similar volume fractions f; but to see clear differences between these associated f values at TR=15 s.

Analytical Simulations

Fig.2 shows the difference between the specific and simplified models in recovering the ground-truth f (the “f-set”), for various experimental setups of noise and diffusion weights. In each of the top three subplots, SNR is the only changing variable, and the rest are kept fixed. For this hw-fw mixture the hw signal assumes a typically dense tissue structure with FA=0.76 and TR=10 s. Each subplot represents a specific SNR value: 20, 40 and 80. The f-bias, that is the difference of each model’s f-fit estimate from the ground-truth f-set, as shown in the plot with a green line, is higher for the simplified than the specific model. For example, for SNR=40 (top-middle subplot), when f-set=0.1 (i.e., 90% hw mixed with 10% fw), the simplified f-fit bias is 18% and the specific f-fit bias is 1%, whereas at f-set=0.4, the simplified f-fit bias is 31% and the specific f-fit bias is 3%. The measurements’ variance decreases as SNR increases, with standard deviations of both specific and simplified measurement ranging from around 4% (SNR=20, f-set=0.4) to 1% (SNR=80, f-set=0.4).

Figure 2:

The predicted f-fit using both methods, simplified (in red) and specific (in black) for noise and diffusion weighting variations in experimental setup. For SNR, in the top panel, the method difference is highest for mid-range f-fit and, as SNR increases from left to right of subplots, the f-fit precision increases. For the different b-value sets, the bottom subplots compare f-bias for the specific and the simplified methods at SNR=40 as the number of b-values decreases from five (top subplots) to two and three. The parameter estimation variance increases as the sampling of the b-value range decreases.

The bottom two subplots of Fig.2 use fixed SNR=40, but different b-value sets. In addition to the previous set of five b-values (top-middle subplot), here we acquire signal for scanning protocols with only two and three b-values. The results are comparable between these three sets of b-values, with variance increasing as the number of b-values decreases: for f-set=0.4, std=0.04 for two b-values, but std=0.02 for the five b-value set.

Fig.3 plots the specific and simplified model parameters as TE and TR are varied. Each subplot in the top row was generated by fixing TE=80 ms and using four TR values: 1, 5, 10 or 15 s. Each subplot in the bottom row was generated by fixing TR=10 s and using four TE values: 30, 70, 110 or 150 ms. Here FA=0.76, and noise is added at SNR=40 level. Though at the extremes of no or all water (where f-set approaches 0 or 1) the two models converge, for the in-between partial volume scenarios the f-bias increases more for the simplified than the specific model. For example, when f-set =0.4, at TR=1 s, the simplified f-fit=0.43±0.03 and the specific f-fit=0.41±0.03, but at TR=15 s the simplified f-fit=0.71±0.02 and the specific f-fit=0.43±0.02. Similarly, when f-set =0.4, at TE=30 ms, the simplified f-fit=0.59±0.02 and the specific f-fit=0.42±0.02, but at TE=150 ms the simplified f-fit=0.89±0.02 and the simplified f-fit=0.47±0.04.

Figure 3:

As TR or TE increases, the original simplified fitting routine (shown in red) overestimates the fw volume in all but the cases of all-water or no-water. Top panel: in each TR profile, the fitted fw volume (y-axis) is plotted against the ground-truth fw volume f-set (x-axis), with fixed TE=80ms, SNR=40, FA=0.76. Bottom panel: TR=10s, SNR=40, FA=0.76.

Fig.4 is an extension of Fig.3, after fixing SNR=40, TE=70 ms and TR=10 s at three levels of hw FA. The top three subplots show that the bias in f observed for the FA=0.76 case is replicated across the other two profiles, FA=0.21 and FA=0.49. A general observation for FA and MD (second and third row) subplots is that the estimation of these diffusion indices remains unchanged between the specific and simplified methods. The bias and variance for FA-fit and MD-fit rise rapidly, however, as f-set increases, particularly for f-set > 0.3. For example, for the FA=0.76 profile, at f-set=0.3 the model estimates FA=0.80±0.03 and MD=0.71±0.04 μm²/ms, but for f-set=0.6 the model estimates FA=0.90±0.06 and MD=0.52±0.09 μm²/ms.

Figure 4:

This is an extension to Fig.3 for SNR=40, TE=70 ms and TR=10 s, showing more FA profiles (columns) and parameters (estimated fw volume f-fit, FA and MD in each row), all plotted against ground-truth fw volume f-set. A reminder: the FA and MD here refer to just the hw compartment (as those for fw are fixed by the FWDT model to 0 and 3.0 μm²/ms, respectively). From first row subplots it is clear that the difference between the specific and simplified f is largest for mid-range f-set. The second and third row subplots show that the diffusion parameter estimation (FA and MD) is independent of the model approach; the bias and the variance in both methods is almost identical. The bias and variance of FA and MD increase with f-set because, as fw volume f-set increases, the proportion of hw signal inside the voxel decreases, thus reducing the effective SNR and making the tensor indices FA and MD less accurate and precise.

In Fig.5 we observe that the tensor parameter estimates, FA and MD, are stable for slight perturbations in T1 and T2 relaxation. We see that FA and MD are little sensitive to such changes, and most of the variation is absorbed by water volume f. As can be seen from the first subplot, the perturbation with the highest impact is the T2 of hw compartment. Like the earlier results for Fig.4, as the fw volume f-set increases, and the remaining signal of hw decreases, the FA and MD becomes less accurate.

Figure 5:

The sensitivity of the parameter estimates to perturbations in the assumed T1 and T2 relaxations for each compartment. Here, TR=10 s, and FA=0.76 for the hw compartment. In the top panel subplots the data has SNR=4000. Each hw and fw parameter is perturbed either way by 15% and 25%, respectively (e.g., in the legend, T1hw+ refers to T1 of hw taking a value 15% more than that specified in the main simulation). FA and MD are little sensitive to such changes (note the scale on the y-axis; MD is in μm²/ms units). Water volume is within 5% of estimated f and the largest variation is a result of T2hw+. In the bottom panel subplots the T2hw+ case has SNR=40. At each f-set point (on the x-axis) we show the mean and std of the deviation from the noise-free estimated indices across 1000 random initializations (of noise and tensor angles). The (f, FA and MD) parameter estimation becomes less accurate as water f increases. Units on y-axes: for the first subplot, estimated f (for each scenario as in the legend) minus the baseline (zero-perturbation scenario); similarly for the second and third subplots of the FA and MD indices.

In Vivo Human Brain Data

Fig.6 illustrates qualitatively the differences between the specific and the simplified method of fitting for FWDT. Drawn with red contours, in this mid-axial brain slice, are the ROIs for WM in the genu and splenium of the corpus callosum and the ROI for CSF in the lateral atria. The ROIs of WM contain 194+121 voxels and those of CSF contain 35+31 voxels.

Figure 6:

The water volume fraction estimated by the simplified method (left), and the specific method (right). The ROIs are shown in red. The statistics show mean±std for each bootstrapped ROI. f-CSF is broadly stable for both methods, at around 1.00. At TR=1s, both methods show similar f-WM, as predicted by Figs.1 and 3. However, as TR increases, simplified f-WM reaches just over 0.09, but specific f-WM decreases to just over 0.03.

Across all TR values, the estimated fw volume in the ventricles, i.e. f-CSF, remains unchanged at around 1 (i.e., 100%). But there are obvious differences in the free water volume estimation of the corpus callosum, i.e., f-WM. For TR=1s, the simplified vs. specific f are approximately the same, as predicted by Fig.1. However, beyond TR=1s, differences between the simplified vs. specific f emerge. As TR increases, while the simplified f-WM rises relatively modestly, the specific f-WM decreases. To illustrate this, in the selected ROI of the corpus callosum, f-WM of the specific model drops from 0.071±0.036 at TR=1 s to 0.035±0.021 at TR=10 s, while f-WM of the simplified model goes up from 0.076±0.038 at TR=1 to 0.091±0.046 at TR=10 s.

Discussion

This work examined the stability of the estimated parameters of multicompartment dMRI models to compartmental T1-, T2- and PD-weighted terms. Specifically, for a model that separates water diffusion into two compartments, this evaluation demonstrated the sensitivity of the compartment volume fraction f to experimental settings TR and TE. Documenting the bias in the estimation of compartmental volume fraction is crucial for clinical studies that employ multicompartment dMRI models. In particular, establishing the accurate presence of water in a contaminated voxel is important not just for the accuracy of the estimated volume fraction but also of the DT indices.

In the simulations (Fig.3) we observe that, while the specific model shows little bias from the set ground-truth across the TR values, the simplified model overestimates the water content, especially over the mid-range f. Such f corresponds roughly to the partial volume areas observed in WM regions of the brain. This observed overestimation in simulations is consistent with in vivo results (Fig.6): as TR increases, the simplified water content values in the WM (f-WM) are higher than the respective specific values; for TR >1 s, almost double.

We saw (in Fig.5) that slight perturbations in relaxation values in turn slightly affect water fraction f. Beyond this, it is difficult to apportion the quantitative changes in f to specific factors. Although other variables were fixed for TR changes, the scanning, the processing, and the modeling can make the quantification of the changes in f difficult. In particular, during the in vivo image acquisition, each TR-specific dataset has its own scan optimization for the TR and other experimental tunables, as well as the inevitable subject movement. During image preprocessing, the choice of ROI within the selected slice and the choice of slice across the brain is TR dataset-specific too. Not least, when fitting the model to the data, f will have a varying extent of (however slight) decoupling from other DT parameters, estimated on the assumption of fixed T1 and T2 across the compartments.

We also saw (Fig.5) that variations in TR affect f, but affect very little the DT parameters of the WM. However, the bias of DT parameter estimates is a function of the amount of water in the contaminated voxel (Fig.4). The reason for this is that, while the noise in any given voxel signal remains unchanged, the fw volume increases at the expense of the remaining hw. This reduces the effective SNR for the DT estimation, and therefore increases the observed bias and variance in FA and MD.

In this work, we calculate directly the T1-, T2- and PD-weighted terms, later fitting other parameters (of DT and f), so as to provide a less biased estimate of fw volume. Two other studies are relevant. The first, by Abbas et al.¹⁷, estimates PD, T1 and T2* (which is the “observed” T2) on a cohort of 25 healthy subjects. Differently from our work, the model fitted does not include dMRI signal or modeling, and it is intended for homogeneous areas of the brain which are free of partial volume contamination.

Another relevant study is by Frigo et al. ³⁹, which has similar aim to ours, eliminating volume fraction bias in multicompartment models. They stress that estimating volume fraction without taking account of other non-diffusion compartment properties like T1 and T2 produces “signal” rather than “volume” fraction. In particular, normalizing the compartmental signal produces biased parameter estimates. Using multi-shell dMRI data, their optimization routine first finds the signal fractions, later teasing out the volume fraction, while assuming that the compartments’ dMRI signals (of a single TE and TR) are linearly independent. However, the technique has so far been used with only an essentially white matter model (the NODDI model ⁴⁰), fitted to multiple gradient-dense b-value shells with very good SNR (extracted from the HPC database), and needs further theoretical and experimental exploration.

Limitations

The results from this study are subject to some caveats and limitations. The first is the assumption of WM for the non-water (hw) compartment, and water for the CSF (fw) compartment. This is common in FWDT models, but it needs to be borne in mind when interpreting the parameter map over the whole brain, say in grey matter regions.

Water properties depend on the physiological and the physical environment it is found. Water in the CSF has different properties to, say, water around the parenchyma or in extracellular space. For example, previous studies have found that edematous water has different relaxation coefficients from CSF water (T1=1500 ms vs. 4000 ms, respectively) ⁴¹. There are also diffusivity variations: CSF diffusivity is approximately the same as that of pure water at 37°C, but fluid diffusivity in tumors and edema can be around half the diffusivity of pure water ⁴¹. Separately, ex vivo tissue needs a different set of relaxation and diffusivity parameters from in vivo tissue ⁴². Variations in T1 and T2 values can arise due to the environment temperature ³, as well as sex, age, and pathology ⁴³. Pathology also affects the water content (hence PD), and multiple such examples are given in the book by Paul Tofts ²⁰. For example, demyelinated WM in subacute sclerosing leukoencephalitis ⁴⁴ and WM lesions in Multiple Sclerosis show increased water content (ex vivo evidence from Tourtellotte ⁴⁵; in vivo studies by MacKay ⁴⁶ and Whittall ²¹); this has also been observed in glioblastomas ⁴⁷.

The best fix to attenuate volume fraction bias is to obtain T1, T2 and PD maps before fitting the other parameters ⁴⁸. Accounting for T2 relaxation may be sufficient ^15,49,50 if not explicitly modeling fluid (which would have a much higher T1 than that of white/grey matter) and the TR is high, or indeed if assuming that the volume fraction captures PD and T1-weighted effects of the modeled compartments ^51,52,53. For a more advanced approach, there is a 2D MRI relaxometry and diffusometry framework proposed by Benjamini and Basser ⁵⁴, the recent MESMERISED framework ⁵⁵ and other more specific applications that employ diffusion and relaxometry, as discussed in Pizzolato et al. ⁵⁶ or the review by Slator et al. ⁴⁹.

The last three limitations to the results concern the in vivo acquisition. The first is that we do not employ cardiac gating. However, although used for mitigating the effects of pulsation, cardiac gating needs to be used with caution, as too low a TR (say 1 s, which is in the order of a heartbeat) can increase the variability in the effective TR, and thus affect the signal in unpredictable ways.

Second, our preprocessing ignored the inevitable presence of Gibbs ringing ^57,58. These arise as a result of truncation of the image’s Fourier representation. Although we do not use them here, various correction methods have been proposed that reduce the bias in the estimated parameters ^59,60,61.

Finally, we primarily compared the two modelling approaches with each-other, without investigating the bias introduced to the parameter estimation as a result of the selected orientation of the gradient directions, or the uncertainty due to the number of measurements (hence SNR). Other studies have explored these ^62,63, especially for the FWDT model ¹³. Here, we also scan only one subject, primarily to illustrate the results from the simulations. Future studies with more subjects, including those with pathology, will determine the wider applicability of our findings.

Conclusion

In dMRI studies that aim to capture free water via two or more diffusion compartments, accounting for the PD and relaxation of each compartment improves the parameter estimation. Specifically, it attenuates the bias and variance of the estimated volume fractions, which can potentially confound or produce false contrast in the diffusion parameter maps. This will improve the ability of dMRI multicompartment models to characterize neuronal tissue integrity or degradation in clinical populations.