Implications of fitting a two-compartment model in single-shell diffusion MRI

Abstract

It is becoming increasingly common for studies to fit single-shell diffusion MRI data to a two-compartment model, which comprises a hindered cellular compartment and a freely diffusing isotropic compartment. These studies consistently find that the fraction of the isotropic compartment $(f)$ is sensitive to white matter (WM) conditions and pathologies, although the actual biological source of changes in $f$ has not been validated. In this work we put aside the biological interpretation of $f$ and study the sensitivity implications of fitting single-shell data to a two-compartment model. We identify a nonlinear transformation between the one-compartment model (diffusion tensor imaging, DTI) and a two-compartment model in which the mean diffusivities of both compartments are effectively fixed. While the analytic relationship implies that fitting this two-compartment model does not offer any more information than DTI, it explains why metrics derived from a two-compartment model can exhibit enhanced sensitivity over DTI to certain types of WM processes, such as age-related WM differences. The sensitivity enhancement should not be viewed as a substitute for acquiring multi-shell data. Rather, the results of this study provide insight into the consequences of choosing a two-compartment model when only single-shell data is available.

1. Introduction

Recent advances in multi-shell diffusion MRI have made it possible to probe complex heterogeneities in white matter (WM) tissue microstructure in vivo (for reviews, see e.g. Novikov et al 2019, Jelescu et al 2020). And yet, the majority of diffusion MRI studies continue to employ conventional single-shell data, as single-shell data are less time-consuming to acquire and more widely available in legacy data (Davis et al 2022). Many single-shell studies fit the one-compartment diffusion tensor imaging (DTI) model to the data, and report the mean diffusivity (MD) to be sensitive to various conditions and pathologies (e.g. Estevez-Fraga et al 2023, Gazes et al 2023, Lopez-Soley et al 2023, Mayer et al 2023, Rashid et al 2023, Rasmussen et al 2023, Schwarz et al 2023, Tian et al 2023, Yilmaz et al 2023, Li et al 2023b, Zhang et al 2023b). Other single-shell studies fit a two-compartment model, which includes a hindered cellular compartment and a freely diffusing isotropic compartment, and report the fraction of the isotropic compartment $(f)$ to be sensitive to various conditions and pathologies (e.g. Andica et al 2023, Gustavson et al 2023, Jing et al 2023, Keijzer et al 2023, Ottoy et al 2023, Read et al 2023, Schumacher et al 2023, Wang et al 2023, Yu et al 2023, Li et al 2023a, Zhang et al 2023a).

It is currently unknown how MD and $f$ compare as single-shell markers of WM microstructure, in part because the interpretation of $f$ is unclear (Golub et al 2020). MD is a measure of the average of all diffusivities across biological compartments that contribute to the diffusion MRI signal. $f$, on the other hand, is designed to quantify the fraction of the diffusion MRI signal attributed to a compartment of free isotropic diffusion, which can only exist in large enough isotropic spaces on the order of at least 20 μm in diameter. The cellular and extracellular spaces described in microscopy studies of normal WM tissue are typically smaller than this (Meier-Ruge et al 1992), so the biological source of a nonzero $f$ in WM is not clear, and $f$ cannot necessarily be interpreted as a literal measure of physically-free water. One source of the ambiguous interpretation of $f$ is that the single-shell fit of the two-compartment model is ill-posed when both $f$ and the mean diffusivity of the cellular compartment are fitted as free parameters, thus requiring model simplifications or regularization (Pasternak et al 2009). Despite this ambiguity, the two-compartment model remains commonly used in practice due to its sensitivity to WM conditions. In this work, then, we put aside the biophysical interpretation of $f$ and set out to identify whether there are sensitivity differences between fitting the two-compartment model and conventional DTI.

To do so, we compare the sensitivity of metrics derived from the one and two-compartment model to WM differences associated with normal aging—a relevant test case for the application of these models (Chad et al 2018, 2021). Rather than solely comparing the sensitivities empirically, we study a nonlinear relationship between the two models which allows us to devise theoretical predictions about their respective sensitivities. To explain the sensitivity of $f$ in single-shell studies specifically, we study a simplified version of the two-compartment model that effectively fixes the mean diffusivity of both compartments. As this simplified two-compartment model does not offer more biologically-interpretable information over DTI, our aim is to determine whether the nonlinear transformation from the one to two-compartment model can enhance sensitivity in terms of signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR) of the corresponding metrics, regardless of the underlying biology.

2. Theory

In this section we devise theoretical predictions about the sensitivities of metrics derived from the one and two-compartment models, and we use simulations to illustrate these predictions. We then test the predictions empirically in the subsequent sections.

2.1. Background

Diffusion MRI data consists of a signal-intensity set $\mathbf{\text{s}}=\left\{{\text{s}}_{\text{k}}\right\}$ where the kth volume is weighted by diffusion along direction ${\widehat{\mathbf{\text{q}}}}_{\text{k}}$ at diffusion weighting $b_{\text{k}}$. Whereas multi-shell diffusion MRI datasets consist of data acquired at multiple diffusion weightings, here we consider single-shell datasets acquired with only one nonzero $b_{\text{k}}\equiv b$ across multiple directions, in addition to non-diffusion weighted signal ${\text{s}}_0$ acquired with $b=0$. Many studies fit the single-shell data to a one-compartment model (Basser et al 1994),

$${\text{s}}_k={\text{s}}_0{\text{e}}^{-b{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}\mathbf{\text{D}}{\widehat{\mathbf{\text{q}}}}_k}$$ (1)

which models Gaussian anisotropic diffusion with a single 3 × 3 symmetric positive-definite matrix $\mathbf{D}$. Other studies fit the data to a two-compartment model (Pasternak et al 2009),

$${\text{s}}_k={\text{s}}_0\left[(1-f){\text{e}}^{-\text{b}{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{\mathbf{\text{D}}}_{\text{c}}{\widehat{\mathbf{q}}}_k}+f{\text{e}}^{-{\mathit{\text{bd}}}_{\text{iso}}}\right]$$ (2)

where ${\mathbf{D}}_{\text{c}}$ is the diffusion tensor of the cellular compartment, $f$ is the signal fraction of the isotropic compartment, and $d_{\text{iso}}$ is the diffusion coefficient of the isotropic compartment. In practice, $d_{\text{iso}}$ is fixed to the diffusion coefficient of free water at body temperature—$3\times {10}^{-3}$ mm² s⁻¹.

2.2. Analytic relationship between the one and two-compartment models

To obtain an analytic relationship between the two models, we first consider one-dimensional versions of the models in which the diffusion tensor of the cellular compartment is replaced by a diffusion coefficient. The one-compartment apparent diffusion coefficient is denoted by $d$ and the cellular compartment diffusion coefficient is denoted by $d_c$ such that $d_{\text{c}}<d_{\text{iso}}$. To investigate how variations in $d$ (one-compartment model) correspond to variations in the fraction $f$ (two-compartments model), we equate the two models

$${\text{e}}^{-bd}=(1-f){\text{e}}^{-b{d}_{\text{c}}}+f{\text{e}}^{-b{d}_{\text{iso}}}$$ (3)

Solving for $d$ (appendix A), we arrive at a nonlinear relationship between the one-compartment parameter $d$ and the two-compartment parameter $f$:

$$d=d_{\text{c}}+\beta \sum _{n=1}^{\infty }\frac{{(\alpha f)}^n}{n}$$ (4)

where $\alpha =1-{\text{e}}^{-b\left(d_{\text{iso}}-d_{\text{c}}\right)}$ and $\beta =\frac{1}{b}$. This relationship forms the basis of the current study.

If we hold $d_{\text{c}}$ constant, two important properties about the correspondence between changes in $f$ and changes in $d$ can be seen (figure 1(A)):

Figure 1.

Nonlinear relationship between fraction of the isotropic compartment ($f$) and one-compartment diffusion coefficient ($d$) (A) In 1D, a linear increase in $f$ corresponds to an accelerated increase in $d$, especially for lower values of $d_{\text{c}}$ (with $d_{\text{c}}$ held constant). (B) In 3D, a linear increase in $f$ corresponds to an accelerated increase in one-compartment diffusivity, greater radially than axially (with the cellular diffusion tensor held constant). The simulated cellular tensor is diagonal with eigenvalues 1.4, 0.6 and 0.2 μm² ms⁻¹. $d_{\text{iso}}=3\mu {\text{m}}^2$ ms⁻¹ and b = 1000 s mm⁻² in all plots.

1. For a linear increase in $f$, $d$ increases nonlinearly;

2. For a linear increase in $f$, $d$ increases faster at lower values of $d_{\text{c}}$.

Property 1 implies that a linear increase in $f$ corresponds to an accelerated increase in one-compartment diffusivity. Property 2 implies that, in voxels containing a single fiber bundle (such that $d_{\text{c}}$ is lower perpendicular to the fiber than parallel to the fiber), the accelerated increase in one-compartment diffusivity is greater perpendicular to the fiber than parallel to the fiber. As shown in appendix C, the effective three-dimensional extension of equation (4) that analytically relates $f$ and the mean diffusivity (MD) in single-shell studies is

$$\text{MD}=\text{MDc}+\beta \sum _{n=1}^{\infty }\frac{{\left({\alpha }^{\prime }f\right)}^n}{n}$$ (5)

where ${\alpha }^{\prime }=1-{\text{e}}^{-b\left(d_{\text{iso}}-\text{MDc}\right)}$ and with $d_{\text{iso}}$ and MDc assigned to fixed values; MDc is the mean diffusivity of the cellular compartment. With the cellular tensor held constant, figure 1(B) illustrates the three-dimensional extensions of properties 1 and 2 above.

The one-compartment phenomenon corresponding to a linear increase in $f$—a faster increase in one-compartment diffusivity radially than axially, and at an accelerated rate—is known to occur in normal adult aging (Sexton et al 2014, Vinke et al 2018), suggesting that a linear increase in $f$ can be used as a mathematical construct to help model normal age-related change. See appendix B and supplementary figure 1 for other important properties of equation (4), such as the effect of b-value selection.

2.3. Sensitivity analysis

We examine two key sensitivity considerations that determine the utility of a biological marker, and consider the case of age-related changes as an example. The first consideration is its robustness to noise, which can be gleaned by its signal-to-noise ratio (SNR): $\text{SNR}[y]☰\text{E}[y]/\text{std}[y]$, where $\text{E}[y]$ is the expected value of the marker $y$ across voxels of interest and $\text{std}[y]$ is the standard deviation. Higher SNR implies more robust estimation of benchmarks (such as reference values for a given age) as well as more robust estimation of deviations from established baselines, for example in pathologies that deviate from normal aging. The second consideration is its sensitivity to biological variability, which can be gleaned by its contrast-to-noise ratio (CNR): $\text{CNR}[y]=\text{C}[y]\cdot \text{SNR}[y]$, where $\text{C}[y]=\text{Δy}/\text{E}[\text{y}]$ is the fractional effect size and $\text{Δy}$ represents some variability of interest (e.g. change in marker $y$ per year). Higher CNR implies higher sensitivity to biological differences and/or changes over time, such as age-related changes (Kingsley and Monahan 2005).

Markers derived from the one and two-compartment models exhibit distinct SNRs and CNRs. The specific shape of the nonlinear relationship between the two models determines their different SNR and CNR properties. Appendix D provides a detailed analysis. In brief, if the plot of $y$ versus $x$ is concave up, then $\text{SNR}[y]$ is necessarily lower than $\text{SNR}[x]$ (figure 2(A)). However, increased SNR does not necessarily imply increased CNR. Typically, studies focus on CNR within the framework of a linear model that assumes a normal distribution of model fit residuals around a line. In cases where $\text{SNR}[y]<\text{SNR}[x]$ and the residuals in $x$ adhere more closely to a normal distribution than those in $y$, it follows that $\text{CNR}[y]<\text{CNR}[x]$.

Figure 2.

The effect of one-to-one nonlinear transformations $x\leftrightarrow y$ on SNR (assuming monotonicity and $y=0$ when $x=0$). (A) If $y$ versus $x$ is concave up such that the values of $y$ are lower than if $y$ were a linear transformation of $x$ (dashed line), then SNR[y] < SNR[x]. (B) RA versus FAis concave up, so SNR[RA] < SNR[FA]. (C) MD−MDc versus $f$ is concave up, so SNR[MD−MDc] < SNR[f].

A famous example of a nonlinear relationship between a pair of diffusion MRI markers that implicates SNR and CNR is the relationship between two scale-invariant measures of anisotropy: relative anisotropy (RA) and fractional anisotropy (FA). The plot of RA as a function of FA follows a concave-up trend (figure 2(B)), which explains the established finding that $\text{SNR}[\text{RA}]<\text{SNR}[\text{FA}]$ (Papadakis et al 1999, Hasan et al 2004). FA is also known to exhibit higher CNR than RA (Alexander et al 2000), as the distribution of RA values is skewed towards higher values whereas the distribution of FA is more normal (Kingsley and Monahan 2005).

Analogous to the pair of RA and FA, in the next two sections we will explore the SNR and CNR relationship between two pairs of metrics derived from the one- and two-compartment models, based on a similar type of nonlinear transformation:

1. MD (one-compartment model) and $f$ (two-compartment model), which both capture variability in isotropic diffusion;

2. Norm of anisotropy (NA; one-compartment model) and FA of the cellular compartment (FAc; two-compartment model), which both capture variability in anisotropic diffusion that is disentangled from variability in isotropic diffusion.

2.3.1. Analysis of the relationship between MD and $f$

For the SNR analysis, we compare $f$ with $\text{MD}-\text{MDc}$ (that is, the deviation of the total mean diffusivity from that of the cellular compartment), since $\text{MD}-\text{MDc}=0$ where $f=0$. Like that of RA and FA, the plot of $\text{MD}-\text{MDc}$ as a function of $f$ is concave up (figure 2(C)), which implies that $\text{SNR}[\text{MD}-\text{MDc}]<\text{SNR}[f]$. As derived in appendix D, the ratio of their respective SNRs is

$$\frac{\text{SNR}[f]}{\text{SNR}[\text{MD}-\text{MDc}]}\approx 1+\sum _{n=1}^{\infty }\frac{1}{(n+1)!}{b}^n{(\text{MD}-\text{MDc})}^n>1$$ (6)

$f$ thus provides a more robust estimate of the deviation in diffusivity from $\text{MDc}$ than quantifying $\text{MD}-\text{MDc}$ directly. With a fixed MDc, the SNR advantage of $f$ over $\text{MD}-\text{MDc}$ increases as diffusivity is elevated. In the example of aging, the SNR advantage of $f$ over $\text{MD}-\text{MDc}$ is enhanced among older adults, because diffusivity is elevated with advancing age. Accordingly, while the SNR of $\text{MD}-\text{MDc}$ is expected to decrease with advancing age (due to increased variability in MD at older ages), the SNR of $f$ should decrease at a slower rate, meaning that benchmarks (i.e. estimated standardized values) of $f$ are expected to be more stable than of MD at older ages.

In addition to the SNR advantage of $f$ over MD, $f$ can also have greater CNR than MD if the underlying diffusivity is expected to change at an accelerated rate (as is the case in normal adult aging). As derived in appendix D, the ratio of their respective CNRs is

$$\frac{\text{CNR}[f]}{\text{CNR}[\text{MD}]}\approx 1+b\left(\langle \text{MD}\rangle -\frac{{\text{MD}}_1+{\text{MD}}_2}{2}\right)$$ (7)

where ${\text{MD}}_1$ and ${\text{MD}}_2$ are respectively the lowest and highest MD values in the linear regression. $\langle \text{MD}\rangle$ is the average of MD values across the sample. If MD is accelerating relative to the linear regression, $\langle \text{MD}\rangle$ can be greater than $\left({\text{MD}}_1+{\text{MD}}_2\right)/2$ and hence $\text{CNR}[f]/\text{CNR}[\text{MD}]>1$. An example of $\text{CNR}[f]/\text{CNR}[\text{MD}]>1$ when MD is accelerating is displayed in figures 3(A)–(B).

Figure 3.

Simulated relationship between one- and two-compartment models. Three scenarios in which $f$ increases at a linear rate and the corresponding one-compartment diffusivities increase at an accelerated rate. (A) Scenario 1: $f$ is increased while FAc is not changing (FAc = 0.85). (B) In this scenario, CNR[f] > CNR[MD]. The panel displays a continuous spectrum of CNR values, where CNR is measured via a linear regression from $f_{\text{i}}=\{0.0,0.1,0.2\}$ to each subsequent value (Δf). (C) Scenario 2: $f$ is increased while FAc decreases (from 0.85 to 0; compare the plots in panels C and A near the green arrows). (D) In this scenario, in addition to CNR [f] > CNR[MD], |CNR[FAc]| < |CNR[NA]|. (E) Scenario 3: $f$ is increased while FAc increases (from 0.85 to 1; compare the plots in panels E and A near the green arrows). (F) In this scenario, in addition to CNR[f] > CNR[MD], |CNR[FAc]| > |CNR[NA]|. Note that the x-axis of the plot in panel (F) is cut off at 0.4, because for greater increases in $f$, NA begins to decrease, so FAc and NA are no longer both increasing. See appendix E for simulation parameters.

2.3.2. Analysis of the relationship between NA and FAc

The second pair of metrics derived from the one- and two-compartment models are measures of anisotropy. In the two-compartment model, the fractional anisotropy of the cellular compartment (FAc) is intended to probe anisotropy that is disentangled from variation in the isotropic compartment. An equivalent one-compartment metric is the norm of anisotropy (NA), which quantifies anisotropy that is disentangled from an equal variation in diffusivity across directions (Ennis and Kindlmann 2006, Chad et al 2021) (defined in appendix D). We compare FAc with NA rather than FA because FA entangles isotropic and anisotropic variation. Note that the fixed value of MDc biases the values of both $f$ and FAc: fixing a lower value of MDc allots more diffusivity to the isotropic compartment, rendering a more anisotropic cellular compartment regardless of the underlying biology. As with $f$, then, we put aside the biological interpretation of FAc values and merely focus on sensitivity properties.

Similar to the analysis of the relationship between MD and $f$, we can derive relationships between NA and FAc for SNR and CNR. The relationship between NA and FAc is, however, more complicated and depends on the covariability of FAc and $f$ (see property 2 above—variation in $f$ affects one-compartment diffusivity differently along different directions). Nevertheless, for small values of $f$, the SNR relationship between FAc and NA (as derived in appendix D) can be approximated as:

$$\frac{\text{SNR}[\text{FAc}]}{\text{SNR}[\text{NA}]}\approx 1+\frac{{\text{NA}}^2}{3{\text{MDc}}^2}>1$$ (8)

The SNR advantage of FAc over NA is enhanced at higher values of NA. For example, if NA increases with advancing age, the SNR advantage of FAc over NA will increase with advancing age.

In the previous section, we demonstrated a CNR advantage of $f$ over MD in the case of an accelerated increase in diffusivity. While the CNR relationship between NA and FAc is more complex, we consider two scenarios in which there is an accelerated increase in diffusivity (that renders NA versus age to be concave down; see appendix D). In the first scenario, anisotropy decreases, in which case the CNR of NA is higher than that of FAc (figures 3(C)–(D)):

$$\frac{\text{CNR}[\text{FAc}]}{\text{CNR}[\text{NA}]}<1,\text{if}\text{Δ}f>0\text{and}\text{ΔFAc}<0$$ (9)

In the second scenario, anisotropy increases, in which case the CNR of FAc is higher of that of NA (figures 3(E)–(F)):

$$\frac{\text{CNR}[\text{FAc}]}{\text{CNR}[\text{NA}]}>1,\text{if}\text{Δ}f>0\text{and}\text{ΔFAc}>0$$ (10)

As mentioned above, an accelerated increase in MD is expected in aging, so for the aging example, the CNR of FAc is expected to be less than that of NA in most degenerating fiber bundles, where anisotropy decreases (equation (9)). Conversely, the CNR of FAc is expected to be greater than that of NA where anisotropy increases (equation (10)), which in the normal aging case can arise due to, for example, selective degeneration of secondary crossing fibers (Han et al 2023).

3. Methods

To garner empirical support for the principles developed in the Theory section, we adduce diffusion MRI data from a sample of UK Biobank participants. We use this data to assess SNR and CNR properties of metrics derived from the one- and two-compartment metrics in the case of studying normal age associations.

3.1. Diffusion MRI data

All subjects of the present study are participants of the UK Biobank initiative. The UK Biobank (https://ukbiobank.ac.uk) recruited 500 000 participants aged 40–69 across the United Kingdom between 2006 and 2010, and has since embarked on a plan to image 100 000 of these participants. The UK Biobank received ethical approval under Research Ethics Committee (REC) reference number 16/NW/0274 and all participants provided written consent.

The present study is conducted as a component of UK Biobank Application 40922. At the time of application in 2018, 22 427 participants aged 44–80 years had been imaged. Subjects were excluded from our study if they reported a neurological, psychological or psychiatric disorder, neurological injury, or history of stroke. Subjects were then divided up into seven age categories of [46–50, 51–55, 56–60, 61–65, 66–70, 71–75, 76–80] years, and 50 male and 50 female subjects were randomly selected from each of the seven age categories. The final sample thus consists of 700 subjects aged 46–80 (50% male, 50% female). This sample was also used in our previous work (Chad et al 2021).

Detailed information about MRI data and processing from the UK Biobank can be found in (Alfaro-Almagro et al 2018). In brief, diffusion MRI data were acquired on a Siemens Skyra 3 T system with 5 b = 0 and 50 b = 1000 s mm⁻² at TR = 3.6 s, TE = 92 ms, matrix size = 104 × 104 × 72 with (2 mm)³ resolution, 6/8 partial Fourier, 3x multislice acceleration and no in-plane acceleration. An additional 3 b = 0 volumes were acquired with a reversed phase encoding. The data were corrected for eddy-current and susceptibility-related distortions via EDDY (Andersson and Sotiropoulos 2016) and TOPUP (Andersson et al 2003), respectively. The UK Biobank also includes diffusion MRI data acquired with b = 2000 s mm⁻², but we have omitted these data in the current study for the purpose of studying single-shell approaches.

3.2. Processing

The one-compartment DTI model was fit by the UK Biobank initiative using FSL’s dtifit. NA is not automatically outputted from dtifit, so NA was computed using in-house code. The two-compartment model was fit by computing $f$ from a rearrangement of equation (5).

$$f=\frac{{\text{e}}^{-b\text{MD}}-{\text{e}}^{-b\text{MDc}}}{{\text{e}}^{-b{d}_{\text{iso}}}-{\text{e}}^{-b\text{MDc}}}$$ (11)

with b = 1000 s mm⁻², and fixing $\text{MDc}=0.6\mu {\text{m}}^2$ ms⁻¹ (the minimum one-compartment mean diffusivity value expected in human WM (Tofts and Collins 2011)) and $d_{\text{iso}}=3\mu {\text{m}}^2$ ms⁻¹ (the maximum possible diffusivity, that is, the diffusivity of free water at body temperature). These values of MDc and $d_{\text{iso}}$ maximize their difference within the realm of biophysically plausibility and thus maximize the nonlinear relationship between MD and $f$. Note that if $\text{MD}<\text{MDc}$, $f$ is negative, so the low value of MDc is needed to be able to transform all values of MD into a non-negative $f$. In voxels where $\text{MDc}=0.6\mu {\text{m}}^2$ ms⁻¹ (which was the case for an average of 3% of WM skeleton voxels per participant), $f$ was defined as 0 to enforce non-negative $f$. $f$ was then plugged into equation (2) and the corresponding ${\mathbf{\text{D}}}_{\text{c}}$ was computed using linear least squares. FAc was then computed as the fractional anisotropy of the eigenvalues of ${\mathbf{\text{D}}}_{\text{c}}$.

3.3. Statistical analysis

FSL’s TBSS (Smith et al 2006) was used with default parameters to obtain a WM skeleton based on the 700-subject sample registered to MNI152 space. For voxelwise analysis, a significant correlation of metric with age was defined by p < 0.05 with correction for multiple comparisons (cluster-wise controlling family-wise error rate with 500 permutations) as per FSL’s randomise (Winkler et al 2014). The effect size of the correlation (difference in metric per year), defined as the slope of the regression line of metric versus age, was calculated in each voxel using a general linear model (mri_glmfit) as implemented in FreeSurfer (Fischl 2012). To control for potential sex-differences in aging trajectories (Kodiweera et al 2016), the linear model was fit separately for males and females and the two regression line slopes were averaged (i.e. mri_glmfit controlled for sex). Contrast-to-noise ratio (CNR) was defined by dividing the effect size by the standard deviation of the metric across the 700-subject sample. Signal-to-noise ratio (SNR) was defined as the ratio of the average value in the WM skeleton divided by the standard deviation of the metric across the WM skeleton.

4. Results

Figure 4 shows that, as predicted by the theoretical analysis, the SNR of two-compartment metrics $f$ and FAc averaged across the WM skeleton are always higher than the analogous one-compartment metrics MD−MDc and NA. While the SNR of these metrics are all negatively associated with age, the SNR advantage of $f$ over MD-MDc increases with age, while the SNR advantage of FAc over NA decreases with age.

Figure 4.

SNR comparison of one- and two-compartment metrics averaged across the WM skeleton. (A) SNR [f] is consistently higher than SNR[MD−MDc], especially at older ages, so the negative age associations of SNR[f] are not as steep as those of SNR[MD−MDc] (slopes of −0.0041 versus −0.0067). (B) SNR[FAc] is consistently higher than SNR[NA], but less so at older ages, so the negative age associations of SNR[FAc] are steeper than those of SNR[NA] (slopes of −0.0056 versus −0.0033). Note that, as shown in the Theory section, the reason that SNR[FAc] versus age is steeper than SNR[NA] versus age is that the WM skeleton-averaged values of NA and FAc are negatively associated with age; in regions where FAc and NA are positively associated with age, the negative age associations of SNR[FAc] are less steep than those of SNR[NA].

A CNR comparison of MD and $f$ is displayed in figure 5. The CNR of these metrics are always positive, indicating an increase in diffusivity with advancing age. Due to their one-to-one relationship, regions with the highest CNR for MD correspond to regions with the highest CNR for $f$ and vice versa. CNR values for $f$ are higher than the CNR values for MD through most of the WM skeleton (bottom row of figure 5, $\text{CNR}\left[f\right]/\text{CNR}\left[\text{MD}\right]>1$), as predicted in the Theory section for regions where MD exhibits a monotonic accelerated increase with age. A paired t-test confirms that MD exhibits significantly (p < 0.05) higher CNR than $f$ on average across the WM skeleton; the 95% confidence interval for the average $\text{CNR[MD]}-\text{CNR}\left[f\right]$ is [0.00056, 0.00057]. The average CNR value for $f$ is 4.4% greater than the average CNR value for MD. However, some posterior and inferior WM regions display lower CNR values for $f$ than for MD (bottom row of figure 5, $\text{CNR}\left[f\right]/\text{CNR}\left[\text{MD}\right]<1$), including parts of the posterior thalamic radiations and parts of the corpus callosum, where MD is known to be less likely to exhibit a monotonic accelerated increase with age (Sexton et al 2014, Vinke et al 2018). Supplementary figure 4 illustrates the accelerated and non-accelerated increase with age in sample voxels that exhibit $\text{CNR}\left[f\right]/\text{CNR}\left[\text{MD}\right]>1$ and $\text{CNR}\left[f\right]/\text{CNR}\left[\text{MD}\right]<1$, respectively.

Figure 5.

CNR comparison of MD and $f$ aging. MD and $f$ both exhibit positive CNR across the WM skeleton, but their CNR is not exactly equal. CNR[f] is greater than CNR[MD] through most of the WM, especially in anterior regions, as can be seen by the red and yellow in the bottom row. Note that CNR is defined as the difference in metric per year, derived from a linear regression, divided by the standard deviation of the metric across participants, and displayed only in voxels with significant correlations with age. Voxels of the WM skeleton without significant correlations with age are displayed in white.

A CNR comparison of NA and FAc is displayed in figure 6. As expected in aging, NA and FAc exhibit negative CNR in much of the WM, indicating a decrease in anisotropy with advancing age. However, both anisotropy measures exhibit positive CNR throughout the superior and posterior corona radiata, posterior part of the internal capsule, and splenium of the corpus callosum—crossing fibers regions known to exhibit selective degeneration with aging (Chad et al 2021). The magnitude of CNR[FAc] is generally greater than that of CNR [NA] in regions where anisotropy is positively associated with age (bottom row of figure 6, $\text{CNR}\left[\text{FAc}\right]/\text{CNR}\left[\text{NA}\right]>1$), whereas it is generally smaller than that of CNR[NA] in regions where anisotropy is negatively associated with age (bottom row of figure 6, $\text{CNR}\left[\text{FAc}\right]/\text{CNR}\left[\text{NA}\right]>1$). The average positive CNR value for FAc is 2.9% greater than the average corresponding CNR value for NA, and the average negative CNR value for FAc is 1.6% lower than the average corresponding CNR value for NA. A paired t-test confirms that FAc exhibits significantly (p < 0.05) higher CNR than NA on average across the WM skeleton; the 95% confidence interval for the average CNR[FAc]−CNR[NA] is [0.00023, 0.00028]. An illustration of the actual values of metrics derived from the one- and two-compartment models are displayed in supplementary figure 5.

Figure 6.

CNR[NA] and CNR[FAc] in aging. NA and FAc exhibit positive CNR throughout the medial WM, including regions where FA is unassociated with age (see figure 4). CNR[FAc] is generally greater than CNR[NA] in these medial regions where they are positively associated with age, whereas the absolute value of CNR[FAc] is generally lower than that of CNR[NA] in lateral regions where they are negatively associated with age, as can be seen by the blue and cyan in the bottom row. Note that CNR is defined as the difference in metric per year, derived from a linear regression, divided by the standard deviation of the metric across participants, and displayed only in voxels with significant correlations with age. Voxels of the WM skeleton without significant correlations with age are displayed in white.

5. Discussion

In this paper we investigated the relationship between analogous metrics derived from the one- and two-compartment models when fit to single-shell diffusion MRI data. We found a nonlinear relationship that renders the respective metrics with different SNR and CNR values, and thus different sensitivities to WM aging.

With MDc set to a constant, variability in isotropic diffusivity is fully reflected in both the metrics MD (one-compartment model) and $f$ (two-compartment model), while variability in anisotropy that is disentangled from isotropic diffusivity is reflected in both NA (one-compartment model) and FAc (two-compartment model). We found that $f$ and FAc always exhibit higher SNR than MD−MDc and NA, respectively (MDc must be subtracted from MD for a proper SNR comparison, as described in the Theory). We also found that the SNR of these measures are all negatively associated with age. However, the finding that the negative age associations of $\text{SNR}\left[f\right]$ are not as steep as those of SNR[MD−MDc] implies that the SNR improvement of $f$ over MD−MDc is enhanced at older ages (figure 4(A)). The greater SNR improvement at older ages is a consequence of higher diffusivity at older ages, as described in the Theory section. Conversely, the finding that the negative age associations of SNR [FAc] are steeper than those of SNR[NA] imply that the SNR improvement of FAc over NA is subdued at older ages (figure 4(B)). This is a consequence of lower anisotropy at older ages, as described in the Theory section—although in some regions, anisotropy is higher at older ages, and in such regions the negative age associations of SNR[FAc] are less steep than those of SNR[NA], implying a greater SNR improvement of FAc over NA with advancing age.

Our empirical analysis of age-effects supported the theoretical prediction that $f$ always has an SNR advantage over MD, as well as the prediction that $f$ will typically have a CNR advantage over MD, where MD exhibits a monotonic accelerated increase with age. An exception to $f$ having a CNR advantage, in the case of aging, is in posterior and inferior WM regions (see figure 5). These are regions known to exhibit very small age-effects on MD across the age range used in our study (46–80 years) (Sexton et al 2014, Vinke et al 2018)). Therefore, CNR [f]/CNR[MD] can itself serve as a potential biophysical marker of aging, such that positive values of the ratio CNR[f]/CNR[MD] reflect the monotonic accelerated diffusional elevations that are typical of WM degeneration, whereas non-positive values of the CNR ratio may reflect other types of trajectories (see supplementary figure 4). The association of this CNR ratio with other clinical markers of aging, and the regional distribution of such associations, should be explored in future research.

In terms of the anisotropy measures, CNR[FAc] is generally lower than CNR[NA] in regions where FAc and NA are negatively associated with age, and CNR[FAc] is generally higher than CNR[NA] in regions where FAc and NA are positively associated with age. This means that in WM regions consisting of degenerating single-fiber bundles, NA has a CNR advantage over FAc. Conversely, the case in which FAc and NA are positively associated with age is an important case, since a simultaneous increase in both isotropic diffusivity and cellular anisotropy effectively cancel each other out and become invisible from the perspective of scale-invariant measures of anisotropy (i.e. RA or FA). As demonstrated in our previous work, an increase in anisotropy in aging may occur in areas of crossing fiber bundles that exhibit selective degeneration of one of the bundles, and NA or FAc are needed to detect such an increase in anisotropy (Chad et al 2018, 2021). In this scenario, the two-compartment metrics {f, FAc} exhibit both higher SNR and higher CNR than the analogous one-compartment metrics {MD, NA}. The nonlinear transformation from the one- to two-compartment model can thus be a useful procedure for enhancing biological sensitivity in linear model analyses.

A corollary of the theoretical analysis is that an elevation in $f$ predicts a greater elevation in diffusivity radially than axially, which is aligned with observations in age-related single-fiber degeneration. A change in the one-compartment anisotropy can thus be explained by the two-compartment parameter $f$. A linear elevation in $f$ can also explain the accelerated diffusivity elevations in normal aging. Both these properties derive from the nonlinear transformation of $f$ to one-compartment diffusivity. Therefore, similar to how the parameter RA was historically refined to FA (Kingsley and Monahan 2005), $f$ may be viewed as a refinement of the parameter MD to be more linearly aligned with age. The biological specificity of $f$ is unclear, but it can be presumed that age-related increases in $f$ capture increased water movement associated with cellular breakdown during age-related degeneration.

While single-shell studies do not explicitly fix MDc, we showed in appendix C that when MDc is initialized as a constant across all voxels (Parker et al 2020), MDc is likely to remain spatially uniform even when regularization is applied. This means that estimating MDc from single-shell data is generally not informative about cellular microstructure. Instead, fixing MDc ensures that all diffusivity variations are allotted to variations in $f$, rendering FAc completely unaffected by an elevated fraction of the isotropic compartment and thus analogous to NA. The assumption of a fixed mean cellular diffusivity is clearly violated in certain pathologies such as tumors or stroke, where MD values drop substantially below 0.6 μm² ms⁻¹ (Parker et al 2020). In such cases, the fitted values of $f$ and FAc cannot be assumed to be accurate measures of physical tissue properties. Obtaining accurate measures of physical tissue properties is a major challenge when fitting other diffusion MRI models as well, as most diffusion measures are biased by modeling assumptions (Jelescu et al 2016). Nonetheless, a fixed MDc assumption may not always be entirely biophysically implausible, as changes in parallel and perpendicular diffusivity during fiber degeneration do not necessarily always affect the mean (Pierpaoli et al 2001). As previously demonstrated in this dataset (Chad et al 2021), regions with elevated MD in aging exhibit elevations in both axial and radial diffusivity, whereas regions without elevations in MD exhibit opposite diffusivity patterns axially and radially (reductions in axial diffusivity and elevations in radial diffusivity), which cancel each other out such that no age association of MD is observed. An interpretation of this finding is that elevated MD mainly occurs in the extracellular domain (represented by elevated $f$), whereas cellular myelin degradation along the two radial dimensions produces myelin fragments that slow axial diffusivity twice as much compared to each radial dimension such that MDc remains constant. It is therefore possible that the biophysical mean diffusivity of WM fibers is approximately fixed over the course of WM fiber degeneration in normal adult aging, and that elevated MD in normal aging mainly reflects elevated extracellular isotropic diffusivity. Multi-shell diffusion MRI is required to test this hypothesis. Nevertheless, we emphasize that metrics derived from the single-shell fitted two-compartment model should not be interpreted literally, as they likely will not reflect the same biological processes conveyed by their multi-shell counterparts. We therefore cannot pinpoint specific neurobiological processes involved in degeneration based on the results of the current study.

By adding a second shell at a lower b-value with as few as six directions, a more biologically-plausible value of MDc could be estimated in the fitting process for the two-compartment model (Tristán-Vega et al 2022). This would allow for accounting for two types of elevated isotropic diffusion: that within a cellular compartment (MDc), and that arising from partial voluming with a separate free water compartment ($f$). However, since this approach will naturally split variation in MD into the two compartments (i.e. increased MD would be split between increased $f$ and increased MDc), an increase in MD will yield a smaller increase in $f$ and thus $f$ will likely not be as sensitive to degeneration. Comparing the sensitivity of parameters derived from the two-compartment model with and without MDc fixed could not be done in the current study, since the second shell of the UK Biobank data (b = 2000 s mm⁻²) is too high for modeling the cellular compartment with a single tensor (Liu et al 2005), but such a comparison is an important topic for future research. Another relevant topic is to consider alternative types of diffusion models, such as those with multiple cellular compartments (Barazany et al 2009, Zhang et al 2012), as well as those whose isotropic compartment diffusivity is not fixed to that of free water (Roine et al 2014, Tax et al 2020), and those that account for more abstract shapes of diffusion without necessarily representing specific cellular compartments (Alexander et al 2001).

Overall, despite limitations in interpreting the single-shell fitted isotropic compartment beyond a mathematical construct, this paper suggests that the concept of an ‘effective’ isotropic compartment may prove useful for characterizing and monitoring WM degeneration regardless of its biophysical interpretation.

6. Conclusion

In this work, we obtained relationships between metrics derived from fitting the one- and two-compartment models to single-shell data. We found that $f$ exhibits greater sensitivity to age-related variability than MD, analogous to how FA has been found to exhibit greater sensitivity to biological variability than RA. The SNR and CNR advantages of $f$ over MD suggest that $f$ is a useful metric for quantifying age-related WM degeneration regardless of biological interpretation. Linear elevations in $f$ further help to explain the well-documented observations of (i) accelerated diffusivity elevations and (ii) greater radial than axial diffusivity elevations in aging. We also demonstrated a benefit of fitting the two-compartment model for observing a simultaneous increase in isotropic diffusivity and cellular anisotropy (e.g. resulting from selective degeneration of secondary crossing fibers): the two processes can be individually identified by employing either {MD, NA} (one-compartment model) or {f, FAc} (two-compartment model), but we found that the latter pair exhibits both higher SNR and higher CNR than the former. Thus, while multi-shell data are necessary for improved biophysical interpretation, this study provides sensitivity considerations for studies that are limited to using single-shell data.

Appendix A. Analytic relationship between $f$ and $\text{d}$

The diffusion coefficient d in the one-compartment model is related to $d_{\text{c}}$ and $f$ in the two-compartment model by

$$\begin{gathered} d=-\frac{1}{b}\ln \left[(1-f){\text{e}}^{-b{d}_{\text{c}}}+f{\text{e}}^{-b{d}_{\text{iso}}}\right] \\ =-\frac{1}{b}\ln \left[{\text{e}}^{-b{d}_c}\left(1-f+f{\text{e}}^{-b\left(d_{\text{iso}}-d_{\text{c}}\right)}\right)\right] \\ =d_{\text{c}}-\frac{1}{b}\ln \left[1-\left(1-{\text{e}}^{-b\left(d_{\text{iso}}-d_{\text{c}}\right)}\right)f\right] \\ =d_c-\beta \ln [1-\alpha f] \\ =d_c+\beta \sum _{n=1}^{\infty }\frac{{(\alpha f)}^n}{n} \end{gathered}$$ (A1)

where

$$\alpha \equiv 1-{\text{e}}^{-b\left(d_{\text{sso}}-d_{\text{c}}\right)}$$ (A2)

$$\beta \equiv \frac{1}{b}$$ (A3)

Appendix B. Properties of the analytic relationship between $f$ and $d$

Properties A and B in the Theory section depend on the $b$-value (supplementary figure 1). The variation in $d$ that corresponds to variation in $f$ is larger for lower $b$-values: the maximum possible variation in $d$ corresponding to a given elevation in $f$ is when $\underset{b\to 0}{\text{lim}}d=d_c+\left(d_{\text{iso}}-d_{\text{c}}\right)f$, whereas variation in $f$ has virtually no effect on $d$ (as long as $f$ doesn’t reach 1) when $\underset{b\to \infty }{\text{lim}}d=d_{\text{c}}$. At high $b$-values, the nonlinearity is also more apparent.

By extension, if there are $N$ compartments each with signal fraction $f_{\text{i}}$, then

$$d=d_{\text{c}}+\beta \sum _{n=1}^{\infty }\frac{{\left({\sum }_{n=1}^N{\alpha }_i{f}_i\right)}^n}{n}$$ (B1)

where

$${\alpha }_i\equiv 1-{\text{e}}^{-b\left(d_{\text{i}}-d_{\text{c}}\right)}$$ (B2)

In a more generalized form, if we choose not to designate any diffusion coefficient as ‘cellular’ (i.e. $d_{\text{c}}=0$), the diffusion coefficient can be written as

$$d=\beta \sum _{n=1}^{\infty }\frac{{\left({\sum }_{n=1}^N{\alpha }_i{f}_i\right)}^n}{n}$$ (B3)

At extremely low b-values, ${\alpha }_i=1-{\text{e}}^{-b{d}_{\text{i}}}\approx b{d}_{\text{i}}$ and $n>1$ terms are negligible, with

$$\underset{b\to 0}{\text{lim}}d=\sum _{n=1}^N{f}_i{d}_i$$ (B4)

At larger $b$-values, the nonlinearities in equation (B3) are apparent. It is seen that the relative effect of compartments with large $d_{\text{i}}\left({\alpha }_{\text{i}}\approx 1\right)$ is smaller at higher $b$-values (due to $d$ being scaled by $\beta =1/b$), whereas the relative effect of compartments with lower $d_{\text{i}}\left({\alpha }_{\text{i}}\approx b{d}_{\text{i}}+1/2b^2{d}_{\text{i}}^2+\dots \right)$ are larger at higher $b$-values, with

$$\underset{b\to \infty }{\text{lim}}d=\min \left(d_{\text{i}}\right)$$ (B5)

Appendix C. Analytic relationship between $f$ and MD

Equation (4) defines an analytic relationship between $f$ and a one-dimensional $d$. There is no natural extension of equation (4) into three dimensions, because the one- and two-compartment models are not analytically equivalent across all directions in 3D space (i.e. the one-compartment model represents diffusion as a single ellipsoid whereas the two-compartment model does not). In this work the 3D extension is defined as

$$\text{MD}=\text{MDc}+\beta \sum _{n=1}^{\infty }\frac{{\left({\alpha }^{\prime }f\right)}^n}{n}$$ (C1)

where ${\alpha }^{\prime }=1-{\text{e}}^{-b\left(d_{\text{iso}}-{\text{MD}}_{\text{c}}\right)}$ and with MDc and $d_{\text{iso}}$ assigned to fixed values. This definition enforces a one-to-one relationship between MD and $f$, which is necessary for any change in isotropic diffusivity to be fully captured by both MD and $f$, and hence for both NA and FAc to be unbiased by elevated isotropic diffusivity (see section 2.3.2). This definition is also consistent with what has been used in existing single-shell studies that fit the two-compartment model. Equation (C1) was originally defined to compute an initial value of $f$ for the two-compartment model fit after obtaining MD by fitting the one-compartment model, with MDc only fixed during initialization (Parker et al 2020). However, subsequently fitting for both $f$ and MDc as free parameters requires multi-shell data, because deviations between the one- and two-compartment models within a single shell generally fall well below the noise floor (Bergmann et al 2016). Single-shell studies that have fit the two-compartment model to date nevertheless left both $f$ and MDc freely varying by implementing regularization to aid with denoising after initializing the fit with equation (C1), but the final results are known to maintain a relatively fixed and spatially uniform MDc (Golub et al 2020, Parker et al 2020). In what follows, we show that this fixed and spatially uniform MDc is a consequence of the initialization, where a fixed and spatially uniform MDc is a local minimum of the cost function used for the regularization.

Fitting the two-compartment model requires minimization of

$$\text{L}=\frac{1}{2}\sum _k{\delta }_k^2$$ (C2)

where

$${\delta }_k={\text{s}}_0\left[(1-f){\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{\mathbf{\text{D}}}_c{\widehat{\mathbf{\text{q}}}}_k}+f{\text{e}}^{-b_k{d}_{\text{iso}}}\right]-{\widehat{\text{s}}}_k$$ (C3)

and ${\widehat{\text{s}}}_k$ is the measured data. In previous studies, this functional has left both $f$ and ${\mathbf{D}}_{\text{c}}$ as free parameters. However, with single-shell data, every possible $f$ has a corresponding ${\mathbf{D}}_{\text{c}}$ that yields an (approximately) equal L (Bergmann et al 2016), that is, ${\sum }_k{\delta }_k\approx 0$, where ${\delta }_{\text{k}}$ is a function of the pair $\left\{f,{\mathbf{\text{D}}}_{\text{c}}\right\}$. Thus, assuming the initialized ${\mathbf{D}}_{\text{c}}$ properly corresponds to the initialized $f$, the single-shell fit is initialized at a local minimum of L (or more precisely, initialized such that $|L-\text{min}(L)|<\delta$ with $\delta$ small enough such that the fit cannot escape the initialization at a typical step size), so minimization of L has no effect, and regularization is needed to move the fit past initialization.

Here we show that if the fit is initialized with a spatially uniform distribution of MDc values, then the Laplace-regularized single-shell fit maintains this uniform MDc distribution. In other words, we show that if MDc is uniform at iteration $t-1$, then MDc remains uniform at iteration $t$.

The minimization is performed iteratively, where ${\mathbf{D}}_{\text{c}}^{\text{t}}$ is the cellular tensor at the tth iteration. The effect of Laplace regularization per iteration is

$${{\mathbf{D}}_{\text{c}}}^{\text{t}}={{\mathbf{D}}_{\text{c}}}^{\text{t}-1}+\eta {\nabla }^2{{\mathbf{D}}_{\text{c}}}^{\text{t}-1}$$ (C4)

where $\eta$ is the step size. Since the Laplace operator ${\nabla }^2$ is linear, $\text{Tr}\left[{\mathbf{D}}_{\text{c}}\right]=3\text{MDc}$ is unaltered by Laplace regularization

$$\text{Tr}\left[{{\mathbf{\text{D}}}_{\text{c}}}^{\text{t}}\right]=\text{Tr}\left[{{\text{D}}_{\text{c}}}^{\text{t}-1}+\eta {\nabla }^2{{\mathbf{\text{D}}}_{\text{c}}}^{\text{t}-1}\right]=\text{Tr}\left[{{\mathbf{\text{D}}}_{\text{c}}}^{\text{t}-1}\right]+\eta {\nabla }^2\text{Tr}\left[{{\mathbf{\text{D}}}_{\text{c}}}^{\text{t}-1}\right]=\text{Tr}\left[{{\mathbf{\text{D}}}_{\text{c}}}^{\text{t}-1}\right]$$ (C5)

because we assume that MDc is spatially uniform at iteration $t-1$ such that ${\nabla }^2\text{Tr}\left[{{\text{D}}_{\text{c}}}^{\text{t}-1}\right]=3{\nabla }^2$ $\text{MDc}=0$.

We now assess if minimization of L alters MDc following the regularization step. For convenience, we define ${\delta }_{\text{k}}$ as the residual before the regularization step

$${\delta }_k={\text{s}}_0[(1-f){\text{e}}^{-b_k{\widehat{\mathbf{\text{q}}}}_k{{\mathbf{\text{D}}}_c}^{\text{t}-1}{\widehat{\mathbf{\text{q}}}}_k}+f{\text{e}}^{-b_k{d}_{\text{iso}}]}-{\widehat{\text{s}}}_k$$ (C6)

After the regularization step, the functional at the tth iteration of the fit is given by

$$\begin{gathered} \text{L}=\frac{1}{2}\sum _k{\left({\text{s}}_0\left[(1-f)\left({\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}+f{\text{e}}^{-b_k{d}_{\text{iso}}}\right)\right]-{\widehat{\text{s}}}_{\text{k}}\right)}^2 \\ =\frac{1}{2}\sum _k{\left({\text{s}}_0(1-f)\left({\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}-{\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}-1}{\widehat{\mathbf{\text{q}}}}_k}\right)+{\delta }_k\right)}^2\text{(plugging in equation (C6))} \\ =\frac{{{\text{s}}_0}^2(1-f)}{2}\sum _k{\left({\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}-{\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}-1}{\widehat{\mathbf{\text{q}}}}_k}\right)}^2 \\ +{\text{s}}_0(1-f)\sum _k{\delta }_k\left({\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}-{\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}-1}{\widehat{\mathbf{\text{q}}}}_k}\right)+\frac{1}{2}\sum _k{{\delta }_k}^2 \end{gathered}$$ (C7)

Taking the derivative with respect to MDc,

$$\begin{gathered} \frac{\partial \text{L}}{\partial \text{MDc}}=-{{\text{s}}_0}^2{(1-f)}^2\sum _k{b}_k{\text{e}}^{-2b_k\text{MDc}}\left({\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}-{\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}-1}{\widehat{\mathbf{\text{q}}}}_k}\right) \\ \left({\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\mathbf{\text{D}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}\right)-{\text{s}}_0(1-f)\sum _k{\delta }_k{b}_k{\text{e}}^{-b_k\text{MDc}}{\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k} \end{gathered}$$ (C8)

where ${\dot{\mathbf{D}}}_{\text{c}}$ is the demeaned tensor. When the tensor is demeaned, it is valid to approximate ${\text{e}}^{-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k}\approx 1-b_k{{\widehat{\mathbf{\text{q}}}}_k}^{\text{T}}{{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}{\widehat{\mathbf{\text{q}}}}_k$, so, for single-shell data with only one nonzero ${\text{b}}_{\text{k}}\equiv \text{b}$,

$$\begin{gathered} \frac{\partial \text{L}}{\partial \text{MDc}}\approx -b{e}^{-2b\text{MDc}}{{\text{s}}_0}^2\left(1-f\right)\sum _k\left({{\widehat{\mathbf{q}}}_k}^T\text{Δ}{{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}{\widehat{\mathbf{q}}}_k\right)\left(1-b{{\widehat{\mathbf{q}}}_k}^T{{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}{\widehat{\mathbf{q}}}_k\right) \\ -b{e}^{-b\text{MDc}}{\text{s}}_0\left(1-f\right)\sum _k{\delta }_k\left(1-b{{\widehat{\mathbf{q}}}_k}^T{{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}{\widehat{\mathbf{q}}}_k\right) \end{gathered}$$ (C9)

where $\text{Δ}{\dot{\mathbf{D}}}_c={{\dot{\mathbf{\text{D}}}}_c}^{\text{t}}-{{\dot{\mathbf{D}}}_c}^{\text{t}-1}$. From $\sum _k\left({{\widehat{\mathbf{q}}}_k}^T\text{Δ}{\dot{\mathbf{D}}}_c{\widehat{\mathbf{q}}}_k\right)=0$ for equally spaced ${\widehat{\mathbf{q}}}_{\text{k}}$ within each b-shell (since the trace of $\text{Δ}{\dot{\mathbf{D}}}_{\text{c}}$is 0, i.e. the regularization does not affect the trace) and approximating all second order small terms as 0, we get

$$\frac{\partial \text{L}}{\partial \text{MDc}}\approx -b{e}^{-\text{bMDc}}{s}_0\left(1-f\right)\sum _k{\delta }_k$$ (C10)

Since ${\sum }_k{\text{δ}}_k\approx 0$ for single-shell data,

$$\frac{\partial \text{L}}{\partial \text{MDc}}\approx 0$$ (C11)

That is, whereas the regularizer pulls the solution towards a point that varies spatially smoothly (with a constant gradient = zero Laplacian), and does so along a path that holds MDc constant, the minimization of L pulls the solution back towards the space of solutions which corresponds to the data, and does so approximately along a path that holds MDc constant.

Therefore, fixing MDc and implementing the analytic relationship between MD and $f$ as per equation (C1) is the approach that has been effectively employed in existing single-shell studies. It is in turn the approach we take in the current study, rendering our analysis consistent with the existing studies.

Appendix D. SNR and CNR analysis

SNR

According to standard propagation of errors theory, if $y$ is a function of $x$, we can approximate the standard deviation (std) of $y$ as (Benaroya et al 2005)

$$\text{std}[y]\approx \frac{dy}{dx}\text{std}[x]$$ (D1)

Defining $\text{SNR}[y]\equiv \text{E}[y]/\text{std}[y]$, and assuming $x$ and $y$ are unbiased, we have

$$\frac{\text{SNR}[x]}{\text{SNR}[y]}=\frac{dy/dx}{y/x}$$ (D2)

such that, if $y=0$ when $x=0$, $\text{SNR}[x]>\text{SNR}[y]\Rightarrow \text{d}y/\text{d}x>y/x\Rightarrow {\text{d}}^{2}y/\text{d}{x}^2>0$, i.e. the function of $y$ versus $x$ is concave up (figure 2(A)).

CNR

Defining $\text{CNR}[y]\equiv \text{Δ}y/\text{std}[y]$, equation (D1) leads to

$$\frac{\text{CNR}[x]}{\text{CNR}[y]}=\frac{dy/{dx|}_{y=<y>}}{\text{Δ}y/\text{Δ}x}$$ (D3)

($<>$represents average) so $\text{CNR}\left[x\right]>\text{CNR}\left[y\right]$ if $\text{Δ}y$ predicted in the linear regression is less than the true contrast in $y$. An example would be the case of a change in y with age accelerating such that the extreme values are outliers, rendering the average value of $y$ in the regression line to be less than $<y>$.

Analysis of the relationship between MD and $f$

Applying equation (C1) to $\text{f}$ and $\text{MD}-\text{MDc}$,

$$\text{std}[f]=\frac{b{\text{e}}^{-\text{bMD}}}{{\text{e}}^{-b\text{MDc}}-{\text{e}}^{-b{d}_{iso}}}\text{std}[\text{MD}-\text{MDc}]$$ (D4)

such that

$$\frac{\text{SNR}[f]}{\text{SNR}[\text{MD}-\text{MDc}]}=\frac{{\text{e}}^{b(\text{MD}-\text{MDc})}-1}{b(\text{MD}-\text{MDc})}=1+\sum _{n=1}^{\infty }\frac{1}{(n+1)!}{b}^n{(\text{MD}-\text{MDc})}^n$$ (D5)

and

$$\frac{\text{CNR}[f]}{\text{CNR}[\text{MD}]}=\frac{{\text{e}}^{-b\left({\text{MD}}_1-<\text{MD}>\right)}-{\text{e}}^{-b\left({\text{MD}}_2-<\text{MD}>\right)}}{b\left({\text{MD}}_2-{\text{MD}}_1\right)}\approx 1+b\left(<\text{MD}>-\frac{{\text{MD}}_1+{\text{MD}}_2}{2}\right)$$ (D6)

(these relationships assume that the estimated values of $f$ and $\text{MD}-\text{MDc}$ are unbiased; see Supplementary figure 2 for an evaluation of the unbiased assumption). Note that if ${\text{MD}}_1=\text{MDc}$ and ${\text{MD}}_2=\langle \text{MD}\rangle$, equation (D6) is equivalent to equation (D5).

Analysis of the relationship between NA and FAc

Here we analyze metrics of anisotropy for a diffusion tensor of eigenvalues $\lambda$. Whereas the fractional anisotropy (FA) is defined as

$$FA=\frac{3\text{std}(\mathbf{\lambda })}{\sqrt{2}\Vert \mathbf{\lambda }\Vert }$$ (D7)

the norm of anisotropy (NA) is defined as (Ennis and Kindlmann 2006)

$$\text{NA}=\sqrt{3}\text{std}(\mathbf{\lambda })$$ (D8)

We first provide a derivation of the SNR and CNR relationships between NA and NAc, under the approximation that $f$ and NAc have a one-to-one relationship (since $f$ and NAc are both expected to progress monotonically during the course of degeneration). Writing $\text{NA}=\text{NAc}-(\text{NAc}-\text{NA})$,

$$\text{std}[\text{NAc}]=\sqrt{\text{Var}[\text{NAc}]+\text{Var}[\text{NAc}-\text{NA}]-2\mathrm{cov}[\text{NAc},\text{NAc}-\text{NA}]}$$ (D9)

The ratio of SNR[NAc] to SNR[NA] is

$$\frac{\text{SNR}[\text{NAc}]}{\text{SNR}[\text{NA}]}=\frac{\text{NAc}}{\text{NA}}\sqrt{1+\frac{\text{Var}[\text{NAc}-\text{NA}]-2\mathrm{cov}[\text{NAc},\text{NAc}-\text{NA}]}{\text{Var}[\text{NAc}]}}$$ (D10)

In the presence of random variability, SNR[NAc] is slightly less than SNR[NA], because the difference between NAc and NA is greater for higher values of NAc such that cov[NAc,NAc-NA] is slightly greater than $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\text{Var}\left[\text{NAc}-\text{NA}\right]$ and the term under the square root is less than one (i.e. the function NA versus NAc is slightly concave down). However, in the presence of biological variability such that higher NAc is associated with lower $f$ (as would be expected for single WM fiber tracts), SNR[NAc] can be slightly greater than SNR[NA] because lower $f$ is associated with a lower difference between NAc and NA such that cov[NAc,NAc-NA] is negative and the term under the square root is greater than 1; in other words, the function NA versus NAc is concave up when NAc is negatively associated with $f$ (see supplementary figure 3(A)).

In any case, the SNR of NA and NAc is very similar. Conversely, FAc has significantly higher SNR than NA and NAc. We can derive a relationship between the SNR of NAc and FAc using the approximation in equation (D1)

$$\text{std}[\text{FAc}]=\frac{\text{FAc}}{\text{NAc}\left(1+\frac{{\text{NAc}}^2}{3{\text{MDc}}^2}\right)}\text{std}[\text{NAc}]$$ (D11)

to get

$$\frac{\text{SNR}[\text{FAc}]}{\text{SNR}[\text{NA}]}=\left(1+\frac{{\text{NAc}}^2}{3{\text{MDc}}^2}\right)\frac{\text{SNR}[\text{NAc}]}{\text{SNR}[\text{NA}]}$$ (D12)

In the presence of random variability, FAc versus NA is concave up (see Supplementary figure 3(B)) and thus SNR[FAc] > SNR[NA]. The only scenario in which SNR[FAc < SNR[NA] is if NA and $f$ are positively correlated (as could be the case in complex fiber architecture), such that $\mathrm{cov}[\text{NAc},\text{NAc}-\text{NA}]>\frac{1}{2}\text{Var}[\text{NAc}-\text{NA}]+\frac{1}{2}\text{Var}[\text{NAc}]\left(1-\frac{{\text{NA}}^2}{{\text{NAc}}^2{\left(1+\frac{{\text{NAc}}^2}{3{\text{MDc}}^2}\right)}^2}\right)$.

Next, we can assess the CNR in a linear regression. Assuming that $f$ changes with age approximately linearly, the function NA versus $f$ has negative second and third derivatives (since a linear increase in $f$ causes the discrepancy between eigenvalues to decrease at an accelerated rate). There are two main situations to assess:

(i). NA and FAc decrease with advancing age:

Since NA versus $f$ has negative second and third derivatives, the decrease in NA with advancing age accelerates at older ages, rendering the contrast in NA to be underestimated in a linear regression. The decrease in FAc with advancing age accelerates even more at older ages, since FAc versus NA have negative second and third derivatives (supplementary figure 3), so the contrast in FAc is even more underestimated than that of NA. Thus,

$$\frac{\text{CNR}[\text{FAc}]}{\text{CNR}[\text{NA}]}<,\text{if}\text{Δ}f>0\text{and}\text{ΔFAc}<0$$ (D13)

(ii). NA and FAc increase with advancing age:

Since NA versus $f$ has negative second and third derivatives, the increase in NA with advancing age decelerates at older ages, rendering the contrast in NA to be overestimated in a linear regression. The increase in FAc with advancing age decelerates even more at older ages, since FAc versus NA have positive second and third derivatives (supplementary figure 3), so the contrast in FAc is even more overestimated than that of NA. Thus,

$$\frac{\text{CNR}[\text{FAc}]}{\text{CNR}[\text{NA}]}>1,\text{if}\text{Δ}f>0\text{and}\text{ΔFAc}>0$$ (D14)

Appendix E. Simulation parameters

Figure 3:

In panels A and B, FAc is fixed as 0.85.

In panels Cand D, FAc varies according to $\text{NAc}={(1-f)}^{1/4}$ [chosen to render a linear change in FA, FA = 0.85*(1−f), to be consistent with (Cox et al 2016)].

In panels E and F, $\text{NAc}={(1+f)}^{1/4}$.

Supplementary figures 2 and 3: Signal s was simulated by synthesizing equation (1) with parameters corresponding to the UK Biobank data, that is, 50 directions at b = 1000 s mm⁻² and 5 b = 0. A diagonal tensor D was simulated. For the case of an isotropic tensor, all eigenvalues equal MD. For the case of an anisotropic tensor, an axially-symmetric tensor with NA = 1 μm² ms⁻¹ was simulated. Rician noise was then added to s with SNR = 20, defined by $\text{SNR}={\text{s}}_0/\sigma (\eta )$ where $\eta$ is randomly taken from a normal distribution independently for a real and imaginary channel and $\sigma$ is its standard deviation. One-compartment DTI parameters (MD, NA) were estimated via weighted linear least squares (using the two-step approach described by (Veraart et al 2013)). MD was converted to $f$ via equation (11), and cellular-compartment metrics (NAc, FAc) were computed for the given $f$ via WLLS with the same weight matrix used for fitting conventional DTI parameters. To evaluate CNR in linear model analyses, a linear regression was applied from the parameter value corresponding to MD = MDc to each subsequent parameter value using the Matlab polyfit function.

Data availability statement

The data that support the findings of this study are openly available at the following URL/DOI: https://www.ukbiobank.ac.uk/enable-your-research/apply-for-access.