Personalized dMRI Harmonization on Cortical Surface

Abstract

The inter-site variability of diffusion magnetic resonance imaging (dMRI) hinders the aggregation of dMRI data from multiple centers. This necessitates dMRI harmonization for removing non-biological site-effects. Recently, the emergence of high-resolution dMRI data across various connectome imaging studies allows the large-scale analysis of cortical micro-structure. Existing harmonization methods, however, perform poorly in the harmonization of dMRI data in cortical areas because they rely on image registration methods to factor out anatomical variations, which have known difficulty in aligning cortical folding patterns. To overcome this fundamental challenge in dMRI harmonization, we propose a framework of personalized dMRI harmonization on the cortical surface to improve the dMRI harmonization of gray matter by adaptively estimating the inter-site harmonization mappings. In our experiments, we demonstrate the effectiveness of the proposed method by applying it to harmonize dMRI across the Human Connectome Project (HCP) and the Lifespan Human Connectome Projects in Development (HCPD) studies and achieved much better performance in comparison with conventional methods based on image registration.

1. Introduction

Diffusion MRI (dMRI) [1] is widely used for the in vivo and non-invasive investigation of brain connectivity. For large-scale imaging studies, it is often necessary to pool data across multiple sites for increasing statistical power [2]. However, the incompatibility of dMRI data across sites, owing to inter-site variations in the magnetic field strength, scanner vendor, and acquisition protocol, poses significant challenges for multi-site data aggregation [3,4]. The harmonization of dMRI data thus plays an important role in modern connectome imaging research.

According to observations in [5,6], the inter-site variability in dMRI is tissue- and region-specific, which implies that accurate alignment of brain anatomy is a key prerequisite in dMRI harmonization. For the harmonization of data from two different sites, in essence this requires the construction of two reference sets with comparable anatomy for each voxel to be harmonized. After that, various transformation models can be applied to remove scanner related differences. In [6], the ComBat method was used to harmonize the diffusion tensor imaging (DTI) by regressing out the site-specific factors voxel-wisely in a co-registered space. For the harmonization of diffusion weighted imaging (DWI), Mirzaalian et al. [5] utilized rotation invariant spherical harmonics (RISH) features as anchor features and the harmonization is conducted by linearly scaling spherical harmonic (SH) coefficients of dMRI signal. Following a similar framework, Huynh et al. presented a method of moments based harmonization approach [16]. Although, these harmonization frameworks demonstrate overall effectiveness in the reduction of site-wise variation, they all rely on conventional image registration to establish anatomical correspondences across subjects, which inevitably will suffer from significant misalignment of brain anatomy in cortical areas with high variability across population and hence result in degraded harmonization.

Recent emergence of high resolution dMRI data from connectome imaging studies allows the analysis of cortical microstructure. The dMRI harmonization of gray matter (GM) in high resolution is thus an increasingly urgent problem. While, due to the geometric complexity of the cortex and the inter-subject variability, the dMRI harmonization of gray matter is hard to achieve for previous methods described above. As illustrated in the Fig. 1, the co-registration cannot properly build the anatomical correspondence across subjects, especially in the GM regions. This will no doubt lead to a highly undesirable scenario of harmonization based on reference voxels from drastically different anatomy and hence intensity distributions, which will confound the estimation of harmonization transformations. Alternatively, surface-based registration can alleviate some of this anatomy misalignment problem for dMRI harmonization, but it is still insufficient to resolve this challenge [14]. This is especially evident in association cortices where topographically different folding patterns are commonly present across subjects.

Fig. 1:

Comparison between the anatomy of the individual subjects and the sample mean template: (a) the sample mean template of multiple FA images of HCP subjects and co-registered FA images of HCP subjects in (b)

To robustly conduct dMRI harmonization of cortical gray matter in high resolution, we present a framework of personalized dMRI harmonization on the cortical surface by adaptively estimating the inter-site harmonization mappings. We integrated our personalized mechanism with the RISH based dMRI harmonization framework [15] for the dMRI harmonization across the Human Connectome Project (HCP) [8] and the Lifespan Human Connectome Projects in Development (HCPD) [9] studies. Our experiments demonstrate that the proposed method achieved better performance comparing with the conventional approach. Our main contributions include: (1) we proposed a surface based dMRI harmonization to improve the robustness of gray matter harmonization; (2) without relying on image or surface registration, we proposed a general local correspondence detection mechanism to enhance harmonization performance.

2. Method

The proposed dMRI harmonization framework is depicted in Fig. 2. The core of the personalized dMRI harmonization is the construction of local reference sets, which mitigates the anatomical mismatching problem by adaptively selecting corresponding locations according to geometric and biomedical features on the cortical surface. Then, site-specific templates could be reliably estimated in a personalized manner and used for the estimation of site-wise harmonization mapping.

Fig. 2:

Overview of personalized dMRI harmonization on the cortical surface. (a) illustrates the detection of inter-subject local correspondence and (b) shows the overall framework of personalized dMRI harmonization on the cortical surface

2.1. Diffusion MRI harmonization and Linear RISH framework

Let’s denote the dMRI datasets acquired from a source site $\left(\mathit{\text{src}}\right)$ and a target site $\left(\mathit{\text{tar}}\right)$ as ${𝒮}^{\mathit{\text{src}}}={\left\{S_i^{\mathit{\text{src}}}\right\}}_{i=1}^{N^{\mathit{\text{src}}}}$ and ${𝒮}^{\mathit{\text{tar}}}={\left\{S_i^{\mathit{\text{tar}}}\right\}}_{i=1}^{N^{\mathit{\text{tar}}}}$, where $S$ represents diffusion MRI signal. The harmonization of dMRI is to mitigate the site-wise differences by mapping the dMRI data from the source site to the target one: ${\hat{S}}^{\mathit{\text{src}}}=\Psi \left(S^{\mathit{\text{src}}}\right)$. Without explicitly paired scans from two sites for mapping estimation, the harmonization mapping can be determinated by comparing the site-specific templates of diffusion representations such as the rotation invariant spherical harmonic (RISH) features [15]. With spherical harmonics, we can represent diffusion signal as $S\simeq {\sum }_{\mathit{\text{lm}}} C_{\mathit{\text{lm}}}{Y}_{\mathit{\text{lm}}}$, where $Y_{\mathit{\text{lm}}}$ and $C_{\mathit{\text{lm}}}$ are the spherical harmonics basis and corresponding coefficients of order $l$ and degree $m$. The linear RISH based harmonization is conducted by scaling the spherical harmonic coefficients of the source site dMRI: ${\hat{C}}_{\mathit{\text{lm}}}^{src}={\Phi }_l{C}_{\mathit{\text{lm}}}^{src}$, where ${\Phi }_l$ is the linear scale map. The harmonized data can be generated according to: ${\hat{S}}^{\mathit{\text{src}}}={\sum }_{lm} {\hat{C}}_{lm}^{\mathit{\text{src}}}{Y}_{lm}$. The estimation of the mapping ${\Phi }_l$ relies on the estimation of the site-specific RISH templates: ${\Phi }_l=\sqrt{\frac{E_l^{tar}}{E_l^{\mathit{\text{src}}}+ϵ}}$, where $E_l^{\mathit{\text{tar}}}$ and $E_l^{\mathit{\text{src}}}$ are the $l$-order’s RISH templates for the target and source site, respectively.

2.2. Personalized dMRI Harmonization on Cortical Surface

The estimation of a site-specific template $E$ is conducted by pooling the representative references of anatomical correspondence from multiple subjects: $E=G\left(\mathbf{F}\right)$, where $\mathbf{F}={\left[F_1,\dots ,F_N\right]}^T$ is a stack of dMRI representations. The coregistration is conventionally used to build the anatomical correspondence across subjects. However, being aware of the inter-subject variability in cortical folding, we aim to improve the performance of harmonization in gray matter regions by adaptively generating the inter-site dMRI mapping function for each subject from the source site (namely source subject) on the cortical surface. Locally, at a certain vertex $v$ on the cortical surface of a source subject, we construct personalized reference sets $ℛ(v{)}^{\mathit{\text{src}}}$ and $ℛ(v{)}^{\mathit{\text{tar}}}$ for the source site and target site. Then the personalized dMRI representations, e.g. RISH features, can be estimated accordingly:

$$E(v)=\frac{1}{\left|ℛ(v)\right|}\sum _{v^{\prime }\in ℛ(v)}F\left(v^{\prime }\right),$$ (1)

where $F\left(v^{\text{'}}\right)$ is the local dMRI representation at vertex $v^{\text{'}}$ on the cortical surface of a reference subject and $\left|ℛ\right(v\left)\right|$ is the size of reference set.

Inter-subject local correspondence detection on the cortical surface

For a real-valued function $f$ on a Riemannian manifold $M$, the Laplace-Beltrami (LB) operator ${\Delta }_M$ is defined as: ${\Delta }_{M}f:=div(\mathit{\text{grad}} f)$ with $\mathit{\text{grad}} f$ the gradient of $f$ and $\mathit{div}$ the divergence. At a vertex $v$ on the manifold $M$, the LB embedding defined in [17] is a infinite-dimensional vector: ${ℰ}_M\left(v\right)=\left[\frac{{\varphi }_1\left(v\right)}{\sqrt{{\lambda }_1}},\frac{{\varphi }_2\left(v\right)}{\sqrt{{\lambda }_2}},\frac{{\varphi }_3\left(v\right)}{\sqrt{{\lambda }_3}},\dots \right]$, where ${\lambda }_i$ and ${\varphi }_i$ represent eigenfunction and eigenvalue of the LB operator. Given two surfaces $M_1$ and $M_2$, we can use the spectral distance: $d_g(v,w)=\parallel {ℰ}_{M_1}\left(v\right)-{ℰ}_{M_2}\left(w\right){\parallel }_2$ for the corresponding vertex detection, where $v$ and $w$ are on surfaces $M_1$ and $M_2$ respectively. With the spectral distance defined in the context of the whole surface, we can only coarsely screen the corresponding vertices and narrow down the searching space. To refine the searching of corresponding vertices we examine the similarity of the local LB embedding. Within the LB framework, we construct the local LB embedding: ${\mathscr{H}}_{𝒩\left(k\right)}\left(v\right)=\left[\frac{{\varphi }_1^{𝒩\left(k\right)}\left(v\right)}{\sqrt{{\lambda }_1^{𝒩\left(k\right)}}},{\sum }_{i=2} \frac{{\varphi }_i^{𝒩\left(k\right)}\left(v\right)}{\sqrt{{\lambda }_i^{𝒩\left(k\right)}}}\right]$, where $𝒩\left(k\right)$ is the k-ring patch with the center point at the vertex $v$. ${\varphi }_i^{𝒩\left(k\right)}$ and ${\lambda }_i^{𝒩\left(k\right)}$ are the i-th eigenfunction and eigenvalue of the patch $𝒩\left(k\right)$. In this paper, we calculate the local LB embedding for $k=\{1,3,5,7,9\}$ and rearrange them into three group $𝒦=\left\{\right\{1,3,5\},\{3,5,7\},\{5,7,9\left\}\right\}$ to represent the geometric characteristics of different localities. Then, the local embedding distance is defined as follow:

$$d_l(v,w)=\underset{K\in 𝒦}{\text{min}} \underset{k\in K}{\text{max}} {∥{\mathscr{H}}_{𝒩\left(k\right)}\left(v\right)-{\mathscr{H}}_{𝒩\left(k\right)}\left(w\right)∥}_1.$$ (2)

Besides the geometry of the cortical surface, the biomedical measurement such as cortical thickness could be employed to further regularize the construction of reference. The thickness difference between two locations can be defined as follow:

$$d_t(v,w)=\parallel T\left(v\right)-T\left(w\right){\parallel }_1,$$ (3)

where $T\left(v\right)$ and $T\left(w\right)$ are the local cortical thickness.

Construction of personalized reference sets

Equipped with the vertex-wise similarity measure, we can retrieve reliable references for any location $v$ on the cortex surface of a source subject from a reference subject:

$$R\left(v\right)=\left\{w:w\in M^{ref};d_l(v,w)\le {\theta }_l\wedge d_t(v,w)\le {\theta }_t\right\}$$ (4)

where $M^{ref}$ is the cortical surface of the reference subject, ${\theta }_l$ is the local LB embedding distance threshold, and ${\theta }_t$ is the thickness difference threshold. The collection of the reference set for multiple subjects from a site src or tar forms the personalized reference set for that site:

$$ℛ\left(v\right)=\bigcup _i R_i\left(v\right).$$ (5)

3. Experiments and Results

3.1. Implementation Details

Our experiments focus on the dMRI harmonization between the HCP and HCPD studies. The MRI scans for HCP and HCPD subjects are acquired on a customized Siemens “Connectome Skyra” $(100\mathit{\text{mT}}/m)$ and 3T Siemens Prisma scanners $(80\mathit{\text{mT}}/m)$, respectively. We performed the harmonization on the single shell $\mathrm{b}\text{-}\text{value}=3000s/{mm}^2$, which is overlapped across two studies. There are 90 and 46 directions in the selected shell for the HCP and HCPD studies, respectively. We followed the minimal preprocessing pipelines [10] for the distortion correction of dMRI [11] and generating the pial and white matter surface mesh [12]. From each dataset we selected 119 healthy subjects with well-matched age and genders for the cross-study dMRI harmonization, with 62 females and 57 males in HCP and 66 females and 53 males in HCPD. All selected subjects from each study were recruited as references for the site-specific template construction and harmonization mapping estimation. For the personalized dMRI harmonization on the cortical surface, we generated (left hemisphere and right hemisphere) gray matter surfaces which are the middle surfaces of the corresponding pial and white surfaces to reliably extract the dMRI signal of gray matter. The dMRI data were linearly interpolated onto these mesh surfaces to represent the gray matter dMRI. To construct the personalized reference sets at each query location, we empirically set the local LB embedding distance threshold ${\theta }_l=0.2$ and the thickness difference threshold ${\theta }_t=0.15$.

For comparison, we implement volume-based dMRI harmonization by following Linear RISH framework in [15]. The resolution of HCP data was downsampled from $1.25\mathit{\text{mm}}$ to $1.5\mathit{\text{mm}}$ isotropic to match HCPD’s. We calculated five RISH templates of order $l=\{0,2,4,6,8\}$ in all harmonization tasks.

3.2. Results

Inter-subject local correspondence

The results of local correspondence detection on the cortical surface are illustrated in Fig. 3. One can see that for each query location (left in each subfigure) multiple locations with both geometric and anatomical similarity are detected (see mid and right in each subfigure). The corresponding parcels of cortex in Desikan-Killiany’ cortical atlas [7] are used as references of anatomy (in yellow).

Fig. 3:

Examples of inter-subject local correspondence detection at two typical locations. The query vertex (a red dot) on a source cortical surface (left) and corresponding reference sets (dots in red) of two reference subjects (middle and right) are shown in each subfigure.

Distribution of DTI features before and after harmonization

Fig. 4 shows the density curve of fractional anisotropy (FA) and mean diffusivity (MD) on the cortical surface of all subjects before and after harmonization. Note that as we focus on the single shell dMRI harmonization, all DTI features are calculated from b=3000 shell. It is observable that variability across HCP (red solid line) and HCPD (red dash line) datasets exists for both FA and MD features. Comparing with the volume-based harmonization (green solid and green dash lines), the surface-based harmonization (blue solid and blue dash lines) has supervisor performance as corresponding histograms have higher similarity to which of target site.

Fig. 4:

FA and MD distributions before and after harmonization. The density curves for FA on left hemisphere, FA on right hemisphere, MD on left hemisphere, and MD on right hemisphere are shown in (a), (b), (c), and (d).

Jensen-Shannon divergence of DTI distributions

To quantitatively demonstrate the performance of harmonization, we measure the distribution differences in DTI features across sites by using the Jensen–Shannon divergence (JSD). Due to the lack of subject-wise correspondence across HCP and HCPD studies, we extend the Jensen-Shannon divergence to measure the subject-set divergence:

$$\mathit{\text{JSD}}(P\parallel 𝒬)=\underset{Q\in 𝒬}{\text{inf}} \left(\mathit{\text{JSD}}\right(P\parallel Q\left)\right)$$ (6)

where $\mathit{\text{JSD}}(P\parallel Q)$ is the JSD of distributions $P$ and $Q,𝒬={\left\{Q_i\right\}}_{i=1}^{N_{𝒬}}$ is the distribution set. We computed the subject-set divergence between the DTI distribution of each harmonized subject from the $src$ site and corresponding distributions of subjects from the $\mathit{\text{tar}}$ set to reflect the performance of harmonization. A smaller JSD, indicating less inter-site variability, is preferred. The Jensen-Shannon divergences (magnified 1000 times) of DTI distributions across HCP and HCPD datasets for both left and right hemispheres (lh and rh) are summarized in Table 1. The results show that the volume-based harmonization can only mitigate the differences between datasets when harmonizing subjects from the HCPD study to HCP dataset $(\mathit{\text{HCP}}+\mathit{\text{HCPD}} v\mathit{\text{RISH}})$. In contrast, the proposed method can properly cope with both HCP harmonization task $(\mathit{\text{HCP}} \mathit{\text{cRISH}}+\mathit{\text{HCPD}})$ and HCPD harmonization task $(\mathit{\text{HCP}}+\mathit{\text{HCPD}} \mathit{\text{cRISH}})$. Further performance comparisons are conducted via paired Student’s t-test, and the improvements achieved by our method over volume-based method are statistically significant $\left(\mathrm{p}<{10}^{-6}\right)$ in all cases.

Table 1:

Jensen-Shannon divergence of DTI distributions across HCP and HCPD studies.

Harmonization task	FA		MD
	lh	rh	lh	rh
HCP + HCPD	3.69 ± 3.39	4.09 ± 3.60	5.98 ± 3.39	6.53 ± 4.17
HCP vRISH + HCPD	3.79 ± 2.03	4.98 ± 3.17	8.16 ± 3.88	18.80 ± 10.34
HCP cRISH + HCPD	2.12 ± 0.82	2.19 ± 1.19	3.20 ± 1.55	2.86 ± 1.03
HCP + HCPD vRISH	3.28 ± 2.84	3.15 ± 2.45	3.55 ± 1.95	4.16 ± 1.62
HCP + HCPD cRISH	2.05 ± 0.90	1.80 ± 0.73	2.84 ± 1.30	2.62 ± 0.89

Regional coefficient of variation of DTI features

We investigated the effectiveness of harmonization on reducing the inter-site variation of regional DTI features. Fig. 5 shows the FA and MD inter-site coefficient of variations (CoVs) for major cerebral cortex lobes. From the results we can find that the volume-based harmonization framework cannot reliably mitigate the inter-site CoVs of DTI features for lobes, such as left parietal and right occipital, in either harmonization task of $\mathit{\text{HCP}} \mathit{\text{vRISH}}+\mathit{\text{HCPD}}$ or $\mathit{\text{HCPD}} \mathit{\text{vRISH}}+\mathit{\text{HCP}}$. By using the proposed method, the CoVs of both FA and MD reduce for all regions in both tasks.

Fig. 5:

FA and MD inter-site coefficient of variation.

4. Discussion and Conclusions

In the experiments, we focused on the harmonization of dMRI on the cortical surface, where the conventional registration methods cannot establish comparable anatomical correspondences across subjects. This anatomical misalignment dramatically increases the likelihood of failure in harmonization, as demonstrated in our experiments, which reaffirms the central role of anatomical correspondences in the site-wise harmonization. In this paper, we proposed a novel framework for dMRI harmonization on the cortical surface. By adaptively estimating the inter-site harmonization mappings according to underlying anatomical and biomedical information, our method outperformed the baseline framework. The proposed adaptive reference detection is a general mechanism and can be integrated into other harmonization framework to regularize the harmonization mapping estimation.