Robust Cortical Thickness Estimation in the Presence of Partial Volumes using Adaptive Diffusion Equation

Abstract

Background:

Automated estimation of cortical thickness in brain MRI is a critical step when investigating neuroanatomical population differences and changes associated with normal development and aging, as well as in neurodegenerative diseases such as Alzheimer’s and Parkinson’s. The limited spatial resolution of the scanner leads to partial volume effects, where each voxel in the scanned image may represent a mixture of more than one type of tissue. Due to the highly convoluted structure of the cortex, this can have a significant impact on the accuracy of thickness estimates, particularly if a hard intensity threshold is used to delineate cortical boundaries.

New method:

In this paper, we describe a novel method based on an adaptive diffusion equation (ADE) that explicitly accounts for the presence of partial tissue volumes to estimate cortical thickness more accurately. The diffusivity term uses gray matter fractions to incorporate partial tissue volumes into the thickness calculation.

Results:

We show that the proposed method is robust to the effects of finite voxel resolution and blurring. The method was validated through simulations, comparisons with histological measurements reported in the literature, and single- and multi-scanner test-retest studies.

Comparison with existing methods:

The proposed method was compared with methods based on the Laplace equation, a linked distance metric, and the FreeSurfer software package.

Conclusion:

We introduced a novel method (ADE) for estimating cortical thickness that is robust to variations in image resolution and scanner field strength. ADE yields accurate, histologically consistent thickness estimates and demonstrates superior consistency in multi-scanner test-retest studies.

Graphical Abstract

1. Introduction

Cortical thickness can typically vary between 2 to 5 mm across a population of healthy adult subjects as well as across different brain regions within an individual [1]. The thickness of the human cerebral cortex is an important phenotypical feature that serves as a biomarker for a range of neurological diseases and disorders, as well as for measuring brain development. An automated approach for measuring thickness from T1-weighted MRI scans is essential to perform cortical thickness studies of large populations.

Several approaches for cortical thickness estimation are based on first estimating the pial and inner gray/white surfaces and then defining the cortical thickness measures based on the distance between the two [2, 3, 4]. The Linked Distance method (LD) uses the distance between corresponding nodes in the two surfaces as a thickness measure. FreeSurfer (FS) [5] defines cortical thickness as the average of the shortest distance between the two surfaces computed in both directions. Cortical Pattern Matching (CPM) [6] finds the shortest distance between the inner and pial surfaces using the Eikonal equation. In contrast to surface-based approaches, voxel-based methods compute thickness based on line integrals [7, 8], the Laplace Equation (LE) [9, 1, 10], or using image registration [11]. The accuracies of both surface- and voxel-based methods are impacted when individual voxels are composed of a mixture of multiple tissue types leading to partial volume effects. The convoluted geometry of the cortex, together with the point spread function associated with the finite resolution makes partial volume effects inevitable. Despite this, most methods use crisp definitions of cortical boundaries for surface-based calculation. In some cases, partial volume effects have been accounted for by modifying the cortical surface boundaries using Eulerian or Lagrangian partial differential equations (PDEs) [10, 12, 13], registering inner and pial cortical surfaces [11], computing closest point distances between inner and pial cortical surfaces measured in both directions [14], or using an electric field model together with a topology-preserving level set approach [15]. While these methods account for partial volume effects in defining inner and pial cortical surfaces, they do not explicitly account for the actual partial volume fractions that lie between the two boundaries once they are defined. Partial volume effects are primarily caused by blurring between tissue types due to limited resolution and finite voxel size. When a tissue boundary, such as the boundaries of the cortical ribbon, intersects a group of voxels, those voxels will contain different concentrations of gray matter than regions that are completely gray matter. Additionally, some regions of gray matter may have lower concentrations of cells due to anatomical variations, and these differences may appear as partial volume voxels in the segmented image. The density of gray matter in a particular region of the brain is known to correlate with various abilities and skills [16, 17, 18], as well as neurological, psychological conditions and disorders [19, 20, 21], as well as age and gender-associated differences [22, 23]. Therefore, it is important to model the gray matter fraction in the computation of cortical thickness.

The method of Jones et al. [9] introduced the Laplace Equation (LE) method for the computation of cortical thickness, solving the PDE with boundary constraints on inner and pial cortical surfaces. Here, we use a similar model but extend the approach to incorporate tissue fractions into the formulation using a PDE framework. To account for partial volumes, we use the adaptive form of the diffusion equation [24, 25], in which we model the spatially-varying diffusion coefficient as a function of local gray matter tissue fraction. We then solve for the steady-state solution. We note that this reduces to the LE method [9] when all tissue fractions are set to ‘1’ or ‘0’, i.e., for hard thresholding. We also describe a closed-form expression for computing thickness after solving the PDE, avoiding the need for costly computation of streamlines as needed for the LE method. We show in Sec. 2.2, using a 1D analogy, that this closed-form expression is equivalent to the result obtained using the streamline method and that the thickness measurement is robust to blurring of the image with a unit-integral kernel. Finally, we present results that compare accuracy and robustness with alternative methods for cortical thickness calculation. A preliminary version of the proposed method was presented in Joshi et al. [26]; here, we provide more details of the approach with comprehensive comparisons and evaluations.

2. Materials and Methods

2.1. The (partial-volume) Adaptive Diffusion Equation (ADE) Method

We assume that the brain image has been segmented using a partial volume classification scheme such that each voxel is assigned a fraction of gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF), with the constraint that the sum of the fractions is unity [27]. We model the cortex as a thin sheet constrained by inner and outer cortical surfaces that are set to temperatures 0 and 1, respectively. We then model the propagation of heat between the two boundary layers of the cortex using the diffusion equation in a formulation inspired by the seminal paper by Perona and Malik on anisotropic diffusion for image smoothing [25]. In our partial-volume adaptive diffusion equation model, the diffusion coefficient is set to be inversely proportional to the fraction of gray matter in each voxel, such that pure white matter and CSF are modeled as perfect heat conductors. The temperature $\varphi (v,t)$ as a function of spatial location $v$ and time $t$ is given by

$$\frac{\partial \varphi (v,t)}{\partial t}=\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{1}{f(v)}\nabla \varphi (v,t)\right),\text{subject}\text{to}\varphi (v,t)=\left\{\begin{array}{ll}0& v\in (\partial \mathrm{\Omega }{)}_{\mathit{inner}}\\ 1& v\in (\partial \mathrm{\Omega }{)}_{\mathit{pial}}\end{array}\right.$$ (1)

where $\mathrm{\Omega }$ is the domain of the computation bounded by the closed inner surface $(\partial \mathrm{\Omega }{)}_{\mathit{inner}}$ and the closed pial (outer) surface $(\partial \mathrm{\Omega }{)}_{\mathit{pial}}$, and $f(v)$ represents the gray matter fraction at location $v$. In this formulation, it is important that all partial volume voxels that contain cortical gray matter are included within the surfaces that bound $\mathrm{\Omega }$. Although similar image-weighted approaches are historically referred to as ‘anisotropic diffusion’ [25], the locally adaptive diffusion coefficient in (Eq. 1) is not orientation dependent and therefore isotropic [28]. Therefore, here we adopt the terminology ‘Adaptive Diffusion Equation’ (ADE).

We solve for the equilibrium condition, ${\varphi }_{\mathrm{\infty }}={\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to \mathrm{\infty }}{\varphi }_t$:

$$\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{1}{f(v)}\nabla {\varphi }_{\mathrm{\infty }}(v)\right)=0,$$ (2)

subject to the earlier boundary conditions. Using the calculus of variations, this equation can be reduced to the harmonic energy minimization problem:

$${\varphi }_{\mathrm{\infty }}(v)=\underset{\psi }{\arg \mathrm{m}\mathrm{i}\mathrm{n}}{\int }_{\mathrm{\Omega }}\frac{1}{f\left(v\right)}\Vert \nabla \psi \left(v\right){\Vert }^{2}dv.$$ (3)

subject to the boundary conditions mentioned above. We use the variational formulation since it reduces the order of the PDE and allows the use of a smaller, 2-point stencil for the discretization of the derivatives. This becomes particularly important in the case of complex and heterogeneous domains such as the cortical sheet, which in some areas, such as the cerebellum and occipital pole, is only 1–4 voxels thick von Economo and Koskinas [29]. We note that in the case where partial fractions are not included and $f(v)=1$, Eq. 2 reduces to the Laplace equation and our approach is equivalent to the LE method of Jones et al. [9].

Fig. 1 illustrates the temperature distribution solution to the ADE in a 2D section of cortex in comparison to the solution of the LE and the LD method within $\mathrm{\Omega }$. We also show corresponding streamlines for the ADE using partial tissue fractions relative to those for the LE method. Note that the green streamlines in Fig. 1d extend into the white matter due to the non-zero gray matter fraction in this region. Integrals of ${\varphi }_{\mathrm{\infty }}(v)$ over these streamlines from inner to pial surface can be used to compute thickness [30].

Figure 1:

Illustration of different cortical thickness computation approaches. (a) Inner and pial surface boundaries overlaid on a T1-weighted image; (b) cortical thickness computation based on linked distance (LD) overlaid on the gray matter fraction image. The solutions of (c) the isotropic Laplace Equation (LE) and (d) the (partial-volume) Adaptive Diffusion Equation (ADE) method are shown as a color-coded temperature distribution with green lines depicting corresponding streamlines. The yellow arrow depicts an example region where the three methods differ significantly.

Rather than compute streamlines, we propose an simple alternative analytic expression for cortical thickness $T(v)$, computed at each point on the mid-cortical surface as:

$$T(v)=f(v)\frac{1}{‖\nabla {\varphi }_{\mathrm{\infty }}(v)‖},\mathrm{where}v\in (\partial \mathrm{\Omega }{)}_{mid}.$$ (4)

Here the mid-cortical surface $(\partial \mathrm{\Omega }{)}_{\mathit{mid}}$ is defined as the level set:

$$(\partial \mathrm{\Omega }{)}_{mid}=\left\{v\in \mathrm{\Omega }\left|{\varphi }_{\mathrm{\infty }}(v)=\frac{1}{2}\right.\right\}.$$ (5)

The analytic expression in Eq. 4 is shown in Sec. 2.2 to be equivalent to the path integral for the 1-dimensional solution to the ADE, not only at $(\partial \mathrm{\Omega }{)}_{\mathit{mid}}$ but at all points with non-zero gray matter fractions. An intuitive explanation of why this approximation works is as follows. We impose boundary conditions of temperatures 0 and 1 on the inner and pial surfaces, respectively. Thus, for homogeneous gray matter the temperature gradient between the two surfaces, $‖\nabla {\varphi }_{\mathrm{\infty }}(v)‖$, will be inversely proportional to thickness. Consequently, the calculation of the reciprocal of the gradient at the midcortical surface should produce a good estimate of thickness. For the ADE, we account for the increased flux in partial volume voxels that may lie on the mid-cortical surface by scaling by the gray matter fraction $f(v)$. Note that the above explanation is only for an intuitive understanding of how the method works, and the argument does not necessarily extend to all configurations of cortical folding patterns in 3D. However, we assume that for the brain, the model suggested largely holds for most of the cortex.

The analytic expression in Eq. 4 was also empirically verified to be consistent with the thickness computed using line-integrals along the streamlines for human brain images in the supplemental material (Supp. Sec. 1). Examples of the steady-state temperature distribution for human brain data using LE and ADE are shown in Fig. 2. The figure depicts the temperature profiles from a point on the inner cortical surface to the corresponding point on the pial surface as defined by the streamline from the inner surface point. While the temperature gradient is unaffected by the gray matter tissue fraction for LE, it is modulated by the gray matter tissue fraction for ADE.

Figure 2:

Illustration of steady-state (SS) temperature distribution for (left) LE and (right) ADE as a function of gray matter tissue fraction from a common point on the inner cortical surface, along their respective streamlines, to the pial surface. Note that the streamlines for LE and ADE are not the same, and therefore, the gray matter fraction measured along the streamline is different.

2.2. Analysis using a 1D Model

Here, for simplicity and for developing intuitive understanding, we use a 1D model to justify the use of Eq. 4 to compute thickness, and show that the ADE method is robust to unit kernel blurring. In this model, we assume the region from $x=-\mathrm{\infty }$ to $-L$ is pure white matter, the region from $x=-L$ to $L$ is the cortex that may contain non-uniform gray matter, and the region from $x=L$ to $\mathrm{\infty }$ is pure CSF. We assume an arbitrary gray matter fraction distribution $f(x)$ on $(-L,L)$. In this case, we expect the gold standard thickness to be $T={\int }_{-L}^{L}f(x)dx$. When this model is observed with limited spatial resolution, the distribution $f(x)$ is blurred a kernel $g(x)$ with property ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(x)=1$. Following Eq. 4, we obtain the thickness

$$T=f(x){\left(\frac{d{\varphi }_{\mathrm{\infty }}\left(x\right)}{dx}\right)}^{-1}\mathrm{at}x:{\varphi }_{\mathrm{\infty }}(x)=0.5.$$ (6)

Note that the absolute value in Eq. 4 is not needed in the 1D case since ${\varphi }_{\mathrm{\infty }}$ is monotonic due to the mean value property [30]. The 1D adaptive diffusion equation is

$$\frac{\partial \varphi (x,t)}{\partial t}=\frac{\partial }{\partial x}\left(\frac{1}{f(x)}\frac{\partial \varphi (x,t)}{\partial x}\right),$$ (7)

with boundary conditions $\varphi (-L,t)=0$ and $\varphi (L,\mathrm{t})=1$. Solving this equation gives

$${\varphi }_{\mathrm{\infty }}\left(x\right)=\frac{{\int }_{-L}^{x}f\left(y\right)dy}{{\int }_{-L}^{L}f\left(y\right)dy}.$$ (8)

Substituting this into Eq. 6 gives the correct value $T={\int }_{-L}^{L}f(y)dy$. It is interesting that the thickness value computed in this manner does not depend on the point $x$ at which it is computed in Eq. 6, although for consistency of definition, we always compute it at the midpoint between inner and outer boundary.

When resolution is limited through blurring kernel $g(x)$, we replace $f(x)$ with its blurred version and again solve the ADE. Substituting in Eq. 6 still yields the correct thickness $T={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)g(y-x)dxdy={\int }_{-L}^{L}f(x)dx$. In other words, the thickness calculation using ADE is unaffected by blurring with a sufficiently narrow kernel with unit integral, provided the blurred gray matter fraction of the cortex lies within the bounds defined by the inner and pial surfaces. It should be noted that in the 3D case, blurring of the image will lead to some smoothing of the computed thickness values along the surface tangents, which may slightly reduce the resolution of surface thickness measures but should not lead to consistent biases.

2.3. Numerical Implementation

We used BrainSuite (Version 18a), available for download from (http://brainsuite.org) [31, 32], to estimate the partial volume tissue fractions from T1-weighted brain MRI. BrainSuite is a collection of open-source software tools that enable largely automated processing of magnetic resonance images (MRI) of the human brain. The extracted tissue fraction volumes were then up-sampled to a resolution of 0.5×0.5×0.5mm³ using linear interpolation to produce an appropriate grid resolution for solving the ADE. The discrete differential operator $L$ for the gradient was computed for the volume in matrix form using the forward difference method [33]. Using this discretization, the cost function $C(\overrightarrow{\varphi })$ in Eq. 3 can be expressed as:

$$C(\overrightarrow{\varphi })=\sum _{i,j,k}\frac{1}{f_{i,j,k}}{\left(L_U\overrightarrow{\varphi }+L_I\overrightarrow{0}+L_P\overrightarrow{1}\right)}_{ijk}^2,$$ (9)

where $f_{i,j,k}$ is a discretized version of the tissue fraction $f(x,y,z)$, $\overrightarrow{\varphi }$ represents the equilibrium temperatures in cerebral cortex, $L_U$, $L_I$, and $L_P$ are differential operator explained later, and $\overrightarrow{0}$ and $\overrightarrow{1}$ represent constant vectors of 0 s and 1 s, respectively. The matrix $L$ has size $N_v\times N_v$ where $N_v$ denotes the number of voxels in $\mathrm{\Omega }$. Let $I$ and $P$ denote the set of indices of voxels corresponding to the inner cortical surface $(\partial \mathrm{\Omega }{)}_{\mathit{inner}}$ and pial cortical surfaces $(\partial \mathrm{\Omega }{)}_{\mathit{pial}}$, respectively. The constraint of 0 and 1 temperatures on inner and pial cortical surfaces is imposed by selecting columns of $L$ corresponding to these indices and composing new matrices $L_I$ and $L_P$, where $L_I$ represents a matrix composed of columns of $L$ with indices $I$, and $L_P$ represents a matrix composed of columns of $L$ with indices $P$. The matrix $L_U$ is similarly composed and represents the columns of the $L$ matrix corresponding to the remaining region.

Multiplying $L_I$ by $\overrightarrow{0}$ and $L_P$ by $\overrightarrow{1}$ in Eq. 9 enforces the temperature boundary condition on inner and outer surfaces, respectively. The resulting unconstrained quadratic cost function minimization problem is solved for the equilibrium temperature values $\varphi$ for each voxel using the conjugate gradient method with a Jacobi preconditioner [34]. The tessellated pial surface $(\partial \mathrm{\Omega }{)}_{\mathit{pial}}$ generated by BrainSuite is an expanded version of the inner surface $(\partial \mathrm{\Omega }{)}_{\mathit{inner}}$, so it has an identical triangulation. We find a midcortical level-set surface with identical topology to these two surfaces using the coordinates along the lines joining corresponding nodes on inner and pial surface at which the temperature $\varphi =0.5$. The thickness is then calculated for every vertex on this mid-cortical surface using the gradient of the temperature distribution as a discretized form of Eq. 4 given by:

$$T(i,j,k)=f(i,j,k)\frac{1}{‖L\varphi \left(i,j,k\right)‖},\mathrm{where}(i,j,k)\in (\partial \mathrm{\Omega }{)}_{mid}.$$ (10)

An example showing partial volume fractions of gray matter, the estimated temperature distribution, and the resulting cortical thickness maps with ADE is shown in Fig. 3.

Figure 3:

ADE thickness estimation. (a) Gray-matter fraction estimated using partial volume model. (b) Temperature map obtained using the (partial-volume) adaptive diffusion equation (ADE). (c) Thickness estimate using the ADE method shown on the estimated mid-cortical surface.

3. Experiments and Results

3.1. Average Cortical Thickness Study

We evaluated the differences in estimates of cortical thickness as obtained by different approaches and how they compare with prior histological observations. For this purpose, we analyzed 3D structural brain MRI scans of 198 normal right-handed subjects (76 male, 122 female, age range: 18–26 years) obtained from the Beijing-Zang subset [35] of the FCON-1000 Project data collection (http://fcon_1000.projects.nitrc.org/fcpClassic/FcpTable.html). MPRAGE images were acquired on a Siemens TRIO 3T scanner: TR =2530 ms, TE = 3.39 ms, slice thickness = 1.33 mm, flip angle = 7°, inversion time = 1100 ms, FOV = 256 mm × 256 mm, in-plane resolution = 256 × 192, 128 slices. We used BrainSuite to estimate the partial volume fractions and to extract the inner and pial cortical surfaces for each subject. The cortical surfaces were registered to a common atlas space, the USCBrain atlas [36]. We used the SVReg module in BrainSuite [37] to generate a mapping from each subject’s cortical surface to the atlas cortical surface.

We generated cortical thickness estimates for each of the 198 subjects using four approaches: Linked Distance (LD), Laplace Equation (LE), (partial-volume) Adaptive Diffusion Equation (ADE), and FreeSurfer (FS). The cortical thickness estimates were smoothed in the original subject surface space using Laplace-Beltrami isotropic smoothing [38] with a ~10 mm FWHM kernel. This smoothing was applied to compensate for discretization and small misregistration errors. The smoothed thickness estimates were then mapped to the atlas surface to compute vertex-wise average cortical thickness measures on the surface for each of the three methods (LD, LE, and ADE). We also processed the same set of subjects using FreeSurfer (FS) (Version 5.3.0) and used FreeSurfer’s average atlas surface for population averaging [39]. The FreeSurfer pipeline uses its own implementations and parameters for cortical surface mesh, tissue fractions, levels of smoothing, and method for mapping of the estimated cortical thickness to a common atlas, but is included here because of its widespread use in cortical thickness studies. To compute average cortical thickness across the population, we used a robust mean estimate in which outliers (the 5% most extreme values) were first removed for each vertex on the surface.

3.1.1. Qualitative comparison

Maps of estimated average cortical thickness are shown in Fig. 4(b)–(e). For comparison to histological measurements, we include in Fig. 4(a) a pseudo-colored version of von Economo’s map of cortical thickness from [29] (see supplemental material for more details). MRI surfaces were reoriented to match the orientation of von Economo’s (recolored) drawings. It should be noted that the results using FreeSurfer were generated using its own cortical surface generation pipeline, while the remaining results were generated using BrainSuite-generated cortical surfaces. In supplemental material (Supp. Fig. 9 and 10), we present more comparisons that use the same cortical surfaces with our implementation of FreeSurfer’s thickness computation to remove any confounding effects of other parts of the processing chain.

Figure 4:

Comparison of cortical thickness estimates using different approaches. (a) Histology-based thickness map from von Economo [29, 40]; (b)-(e) Average cortical thickness maps of left and right hemispheres. Lateral (left) and medial (right) views from N = 198 adult subjects computed using: (b) (partial-volume) Adaptive Diffusion Equation (ADE), (c) FreeSurfer method (FS), (d) Isotropic Laplace Equation (LE), and (e) Linked Distance (LD).

Fig. 4 indicates that the patterns of thickness variation across the cortex were similar for all methods, however, the range of values was quite different. The range of cortical thickness estimates found using ADE and FS were more consistent with the von Economo estimates than other methods. Overall, the FS maps are consistently smoother than those from ADE and showed a lower range of thickness estimates across the brain. An earlier evaluation of FreeSurfer-based cortical thickness computation showed similar findings [41, 42]. LE shows a thicker cortex throughout the brain than von Economo, with LD showing even thicker values.

3.1.2. Quantitative comparison

For further quantitative comparison, we used the MRI Von Economo - Koskinas atlas made available by the Dutch Connectome lab (http://www.dutchconnectomelab.nl/economo/) [40], which was mapped to our common USCBrain space for thickness analysis (see supplemental material, Supp. Sec. 5 for details). This allowed us to quantitatively compare our thickness estimates to the measurements made by von Economo using histology on 43 different brain areas [43, 29].

Since von Economo’s measurements are not hemisphere-specific, they do not account for the asymmetry of the cortex. For comparison, the data from left and right hemispheres was averaged. Table 1 shows the average cortical thickness for representative ROIs. These ROIs were chosen, as suggested by Kabani et al. [44], as representative cases for evaluation of cortical thickness algorithms as they range from thickest (precentral) to thinnest (occipital), and include hidden (insula) and medial (cingulate) surfaces. Von Economo provided both cortical layer-wise thickness, which can be summed to obtain cortical thickness, and photomicrographic plates for a range of cortical thicknesses. The layer-wise cortical thicknesses are available from the digitized von Economo atlas [40], and ranges of thicknesses for regions are provided in Kabani et al. [44]. We show both in Table 1. As in Kabani et al. [44], no statistical tests were performed as the data from the two hemispheres was merged, and similar regions in the two hemispheres may not have the same thickness.

Table 1:

Average cortical thicknesses (in mm) estimated using different approaches compared to those reported from von Economo’s histology measurements.

Atlas Region	Economo Region	Economo (layers)	Economo (plates)	LD	LE	ADE	FS
Anterior Cingulate	LA		2.50	4.12	3.08	2.80	2.81
Cuneus	OA	2.17	1.3–2.6	3.46	2.40	2.17	2.05
Insula	IA&IB	2.92	2.8–3.5	5.11	3.73	3.65	3.67
Posterior Cingulate	LC2		2.5–3	4.01	3.04	2.72	2.96
Precentral (lateral surface)	FAy	3.71	3.2–4.5	3.95	3.02	2.79	2.70
Postcentral	PC	3.20	3–3.3	3.73	3.78	3.18	2.17
Superior Frontal	FB	3.59	3.2–4	4.46	4.04	3.56	2.73
	FC	2.30	2.9–3.5	4.73	4.14	2.53	2.70
Supramarginal	PF	3.42	3–3.5	4.36	3.18	3.14	2.52
Superior Temporal	TA	2.85	3.00	4.36	3.36	3.10	2.25

3.2. Effect of Resolution Differences: BrainWeb Simulated MRI

In this section, we analyzed the effect of resolution on the cortical thickness computation for different methods. We used BrainWeb (https://brainweb.bic.mni.mcgill.ca/cgi/brainweb1) MRI simulator [45] to generate MRIs of two different slice thicknesses: 1 mm and 3 mm, with in-plane pixel size 1 mm × 1 mm. We simulated T1 MRI with 3% noise and 20% RF inhomogeneity. We calculated cortical thickness using the ADE, LE, LD and FS methods. In this experiment, we computed cortical thickness of 1 mm and 3 mm simulated BrainWeb phantoms using the BrainSuite generated tissue fraction maps. The method generates cortical thickness at each gray matter voxel. We mapped this volumetric thickness measurement to BrainSuite generated mid-cortical surface of the phantom using linear interpolation. The cortical thickness maps were coregistered to the USCBrain atlas. At each point on the atlas cortex, we computed the absolute difference between 1 mm and 3 mm phantom’s cortical thicknesses. The cortical maps (Fig. 5) and the histograms (Fig. 6) of the absolute differences are shown. It can be seen from the results that the ADE method shows the most consistency of cortical thicknesses for the two phantoms. All methods were most inconsistent in the frontal and temporal poles. LE, LD, and FS tended to be additionally inconsistent in the medial and ventral surfaces of the brain. ADE, followed by LE, had lower FWHM and showed higher consistencies compared to FS and LD. FS and LD had similar consistency error distributions.

Figure 5:

Effect of resolution differences of BrainWeb phantom on cortical thickness estimates for different methods: pointwise cortical maps of absolute thickness difference between estimates from the 1mm and 3mm versions of the phantom.

Figure 6:

Effect of resolution differences of BrainWeb phantom on cortical thickness estimates for different methods is shown using histograms of absolute thickness difference between estimates from the 1mm and 3mm versions of the phantom.

3.3. Effect of Resolution Differences: Real MRI at Original vs 2 mm Resolution

We investigated the effect of a change in scan resolution on thickness estimates in a real MRI study. For this, we started with T1 MRI images from 50 subjects selected randomly from the Human Connectome Project (HCP) database [46]. These images are originally acquired at an image resolution of 0.7 mm isotropic voxels. For comparison, we also downsampled these images to a resolution of 2 mm isotropic voxels. Both the original and downsampled images were preprocessed using BrainSuite, and the cortical thickness computation was performed using the ADE, LE, and LD methods. We also compute the cortical thickness using FreeSurfer to compare these approaches with the FS method. The thickness maps were coregistered to an atlas as part of the processing sequence. The pointwise absolute difference between thickness measurements for original and downsampled images of each subject was computed. Smaller differences indicate higher robustness to resolution changes.

The maps and histograms of absolute difference in the cortical thickness estimates, averaged across the 50 subjects, between high and low-resolution scans for the four methods are shown in Fig. 7 and Fig. 8, respectively. ADE shows the lowest difference, indicating greater robustness to resolution changes than the alternative methods. FS shows the most consistency across the brain, although the difference is higher overall compared to ADE.

Figure 7:

Effect of resolution on cortical thickness estimates for different methods: pointwise cortical maps of absolute differences between cortical thickness estimates, averaged over the 50 HCP subjects, for high- (0.7 mm isotropic) and low-resolution (2 mm isotropic) images. The mean (over subjects) absolute difference for the four methods were: ADE (0.08 ± 0.1 mm); LE (0.11 ± 0.12 mm); LD (0.3 ± 0.32 mm); FreeSurfer (0.21 ± 0.03 mm).

Figure 8:

Effect of resolution on cortical thickness estimates for different methods is illustrated by the histogram of the mean absolute differences between thickness estimates, averaged over the 50 HCP subjects, for high- (0.7 mm isotropic) and low-resolution (2 mm isotropic) images.

3.4. Short-term Test-Retest Reliability Study for Same Scanner

To test the repeatability of the measurements we obtain from the thickness estimation methods, we applied the methods to a test-retest dataset collected at New York University. The New York University Child Study Center Test-Retest (NYU CSC Test-Retest) dataset [47] includes anatomical images of 25 participants. The MRI images were collected on several occasions: (a) The first scan in a scan session; (b) 5–11 months after the first resting-state scan; (c) about 30 (< 45) minutes after acquisition in (b). Here, we show the results of the short-term test-retest study, i.e., the comparison between the thickness estimates obtained from scanning sessions (b) and (c). We estimated the cortical thickness using the four methods. For this purpose, the images were preprocessed using BrainSuite and FreeSurfer, and the cortical thickness computation was performed using the ADE, LE, LD and FS methods. The thickness maps were coregistered to an atlas as part of the processing sequence. The pointwise absolute difference between thickness measurements for the test and retest images of each subject was computed. Smaller differences indicate higher test-retest reliability. The maps and histograms of absolute difference in the cortical thickness estimates, averaged across the subjects, between the test and retest images for the four methods are shown in Fig. 9 and Fig. 10, respectively. ADE shows the lowest difference, indicating higher test-retest reliability than alternative methods. FS shows most consistency across the brain, although the difference is higher overall compared to ADE.

Figure 9:

Difference in cortical thickness estimates computed for short-term test-retest scans using the four methods: absolute thickness difference between estimates from the two scans, averaged over subjects. The mean absolute differences for the four methods were: ADE (0.09 ± 0.21 mm); LE (0.21 ± 0.23 mm); LD (0.35 ± 0.45 mm); FreeSurfer (0.21 ± 0.03 mm).

Figure 10:

Histogram of the average (across subjects) absolute difference between thickness measures from the test and retest scans for ADE, LE, LD, and FS.

3.5. Test-Retest Reliability Study for Multiple Scanners

We studied test-retest reliability of different methods using data from five normal subjects that were scanned using two different scanners over a three-day period at the University of Iowa. The first MPRAGE scan was acquired using a Siemens Trio 3T scanner (slice thickness 1 mm, TR 2530 ms, TE 3.99 ms, inversion time 1100 ms, in-plane resolution 1 mm², and flip angle 10°). The second MPRAGE scan was acquired using a Siemens Avanto 1.5T scanner (slice thickness 1.5 mm, TE 7 ms, in-plane resolution 1.066 mm², and flip angle 30°). The data were processed using the procedure described in Sec. 2.2 to obtain thickness estimates using all methods. The absolute values of thickness differences corresponding to 3T and 1.5T Siemens scanners, averaged over the five subjects, were computed at each vertex for all four methods. Fig. 11. and Fig. 12 show the maps and histograms of these differences, respectively. Similar to previous observations in Sec. 3.2, the ADE method showed the smallest absolute difference between the 3T and 1.5T scanners relative to LE, LD, and FreeSurfer. Note that this comparison tests reliability across different magnetic field strengths (1.5T vs 3T) as well as different resolutions (1.7 mm³ vs 1.0 mm³). A test-retest reliability study for same scanner was also performed for short, medium and long-term (Sec. 3.4, and in the Supplemental Material, Supp. Sec. 2, Supp. Sec. 4) which again showed similar performance for ADE, LE and FS, while LD showed the largest difference across sessions.

Figure 11:

Effect of scanner differences on cortical thickness estimates for four methods: absolute thickness difference between estimates from the Siemens Trio 3T and Siemens Avanto 1.5T scanners, averaged over 5 subjects. The mean (over subjects) absolute difference for different methods were: ADE (0.1575 ± 0.1034 mm); LE (0.1786 ± 0.1163 mm); LD (0.2480 ± 0.1520 mm); FreeSurfer (0.1679 ± 0.0821 mm).

Figure 12:

Histogram of the average (across five subjects) absolute differences between thickness measures for ADE, LE, LD and FS for 1.5T Siemens Avanto and 3T Siemens Trio scanners.

3.6. Comparison of Thickness Methods for Group Analysis Study

In this section, we compare the thickness computation methods when used for a statistical analysis study, specifically for group analysis. As a demonstrative study, we used OASIS-3 dataset (http://www.oasis-brains.org) [48] for MMSE (Mini Mental State Exam [49]) regression. We chose 235 subjects from the dataset. The demographic information of this study population is included in the supplemental material (Supp. Sec. 7). We processed the subjects using BrainSuite to generate their cortical surface representations and tissue fraction maps. Then we computed the cortical thicknesses using LE, LD, ADE, and FS methods and coregistered the thickness maps to USCBrain atlas. At each point on the atlas cortical surface, we computed the Pearson correlation of subjects’ MMSE scores to cortical thicknesses estimated using different methods. The Pearson correlations were converted to p-values using Fisher Z-transform [50] at each point on the atlas cortex. We corrected for the multiple comparisons using False Discovery Rate (FDR) with Benjamini-Hochberg correction [51]. The significance maps at α = 0.05 are shown in Fig. 13. The proposed ADE method shows the highest significance, followed by FS, LD, and LE methods. It should be noted that the ground truth of the MMSE-thickness association is unknown, but higher significance possibly indicates more statistical power for the proposed ADE method compared to the alternatives.

Figure 13:

p-value map of Pearson correlation between point-wise cortical thickness and MMSE scores in the study population. The p-values were adjusted for multiple comparisons using FDR and thresholded at α = 0.05.

4. Discussion

Previous studies of average cortical thickness have reported a wide range of thickness values depending on the estimation method and software used to process the data [52, 44, 53, 54]. Martinez et al. [52] observed that cortical thickness measurements vary widely across surface extraction pipelines and thickness computation methods. In that study for the CIVET protocol [55], the highest values were found in the insular cortex, the medial temporal pole/entorhinal cortex, and the posterior portion of the medial orbitofrontal gyrus. Using the LD methods, as implemented in earlier versions of BrainSuite, the highest values were found mainly in the anterior cingulate, the medial superior frontal gyrus, the anterior portion of the lateral superior temporal gyrus, the medial temporal pole/entorhinal/anterior parahippocampal gyrus, and the insular cortex. In both the LD and CIVET methods, the thinnest areas were observed in the lateral and medial occipital, and in the lateral superior and medial portions of the precentral and postcentral gyri. Finally, the distribution of cortical thickness calculated by the CPM pipeline [56], as reported in Martinez et al. [52], differed from both CIVET and LD, except in some occipital regions, insular cortex, and the medial temporal pole, where all methods converged. Another detailed comparison of various methods [53] showed, surprisingly, given the results above, that the linked distance method was the most precise, showing good single-subject reproducibility among the methods evaluated. The linked distance method depends on reliable surface extraction and therefore is particularly sensitive to differences in resolution, blurring, SNR, and other scan parameters. Surface-based and voxel-based methods were compared in Velázquez et al. [57] for clinical applications where they showed a similar trend but different measurement biases.

We compared our ADE method to the LD, LE, and FS methods. As noted above, the FS method is implemented using a completely different MRI processing pipeline than the BrainSuite pipeline used by ADE, LE, and LD. Therefore the results of FS reflect not just differences in the cortical thickness computation methods, but differences in the entire processing pipeline, including independent estimates of tissue fraction and cortical surfaces. In supplemental material (Supp. Sec. 6), we also explored the differences in behavior between ADE and FS thickness estimation methods with the same pre-processing, i.e., when input cortical surfaces and tissue fractions were identical. To do this, we developed our own implementation of the FS method and used it with surfaces and tissue fraction estimates that were used with other methods. These results showed that among the other three methods (LD, LE, ADE), FreeSurfer thickness estimates were most similar to LE, which is consistent with the fact that partial volumes are not explicitly taken into account by these methods. A detailed comparison of FreeSurfer’s thickness estimates with von Economo’s maps was presented by Scholtens et al. [41], where it was shown that the patterns of variation over the cortex were similar, which is consistent with our finding in Sec. 3.1. FreeSurfer’s segmentation in regions such as the insula, claustrum, and putamen is generally more suitable for cortical thickness estimation pipelines. In this study, we used the existing BrainSuite segmentation without modification; however, incorporating targeted refinements to the insular region segmentation could further enhance the accuracy of cortical thickness measurements in and around the insula.

Compared to LD and LE, the thickness estimates found using the ADE method were more consistent with the literature, with the ADE thickness map showing regions similar to those presented by von Economo [58], Scholtens et al. [41], and Kim et al. [13]. The literature suggests that the thickness values of normal subjects peak at 2–3 mm, which is consistent with the estimates from the ADE method. Among the four compared methods, the LD approach was observed to be the fastest method, but tended to overestimate cortical thickness. Computation time for the LD method is ~1 sec, while LE and ADE take ~10 min and ~15 min, respectively, using a MATLAB^® implementation on a typical desktop computer. Note that the introduction of adaptive diffusivity in our formulation leads to an increase in cost relative to the Laplacian-based (LE) method. This is compensated to some extent by computing Eq. 4 instead of using streamline computation, although our closed-form expression could also be used with the LE method by setting the diffusivity term to a constant.

One issue that impacts the accuracy of thickness computation is the bias field in the MR images. The inhomogeneities in B1 transmit and receive fields result in a low-frequency bias in the image intensity that will impact tissue classification. Because the proposed thickness computation method depends on accurate tissue classification, this bias field can introduce errors in the result. This problem is addressed by including bias correction as a pre-processing step in the MR processing sequence [59, 60, 31]. In the dataset used in Sec. 3.5 and Sec. 3.1, we found that the optimal processing steps using BrainSuite software include four repetitions of the bias field correction (BFC) step for 3T scans.

The comparison of the cortical thickness measured from MRI and von Economo’s histological measurements shown in Sec. 3.1.1 and Sec. 3.1.2 should be interpreted with caution. The demographics of the subjects used in the histological study were different from those of the imaged population. von Economo and Koskinas used brains from Caucasian subjects, 30–40 years of age [43], whereas the population for the data we used had an age range of 18–26 and were scanned in Beijing. Apart from the difference in demographics, another possible source of error is shrinkage caused by fixation [61, 62, 63]. Another factor is that the cortical thickness measurements derived from histological sections can be estimated with relatively high accuracy only where the plane of sectioning is parallel to cell columns. Instead of the commonly used method of sectioning the whole brain serially, perpendicular to its fronto-occipital axis, von Economo and Koskinas obtained tissue sections always perpendicular to the axis of each gyrus or sulcus and in directions corresponding to their convoluted pattern [43]. Despite this, local differences in orientation relative to the cutting plane can lead to overestimation of cortical thickness. In summary, von Economo’s histological measurements suffer from two possible biases, both underestimation (due to shrinkage) and overestimation (due to cutting plane) [61, 64]. Nevertheless, von Economo’s histological measurements are one of the most widely accepted and commonly used measurements, and they represent a useful baseline against which to compare imaging-based measures.

Studies of cortical thickness usually compare differences in thickness in homologous areas between two groups, or changes in thickness over time during maturation, aging, or disease progression, often in a multi-site study. For this reason, the consistency and robustness of thickness estimates are possibly as important as absolute accuracy. Therefore, we examined not only the average cortical thickness over a relatively large population but also the consistency of thickness estimates among subjects scanned using two different scanners, as well as test-retest using the same scanner. The consistency study results in Fig. 11 and Fig. 12 show a consistent bias in all methods, i.e., the peak of these histograms is not at zero. This could be an effect of slightly different bias correction results at two different resolutions resulting in small change in tissue fraction estimates. Nonetheless, the histograms of absolute differences in Fig. 8 and Fig. 12 confirm that ADE has lower sensitivity to differences in resolution and scanner relative to FS, LE, and LD. Additonal comparisons presented in the Supplemental Material also demonstrate the superior performance of the ADE method.

Although our results demonstrate that ADE can reduce the effect of inter-scanner differences, they also indicate that it is important that scanner-dependent effects are factored into any subsequent analysis. Further evaluation is required to understand the behavior of different methods with different imaging hardware, such as from different equipment manufacturers, different imaging protocols, and different reconstruction approaches, all of which may lead to subtle but important contrast differences.

5. Conclusion

In summary, we have presented a novel method to estimate cortical thickness that is robust to differences in image resolution and scanner field strength. Our method uses the (partial-volume) Adaptive Diffusion Equation (ADE), which makes use of partial volume tissue fractions to obtain accurate thickness estimates. Results presented here indicate that ADE is capable of producing population average cortical thickness estimates that are largely consistent with those reported from histological measurements. ADE also demonstrated the most consistent estimation in multi-scanner test-retest study with resolution differences. The proposed ADE method is implemented in BrainSuite as the thicknessPVC module and is available as open-source software through the BrainSuite website (http://brainsuite.org).

• We propose an Adaptive Diffusion Equation (ADE) method for cortical thickness measurement.

• ADE uses partial tissue volumes for improved accuracy of thickness estimates.

• ADE estimates are more consistent with the histological measurements.

• ADE also showed improved same-scanner as well as cross-scanner test-retest reliability.

• ADE is implemented in BrainSuite software (thicknessPVC program).

Data and Code Availability

The software implementation (binary executables for Mac, Windows, and Linux) of the method proposed in this paper is available in our BrainSuite software. The source code is also available on the BrainSuite website http://brainsuite.org under GPL (v2) along with the source code for BrainSuite.

This study used several publicly available datasets. The Beijing-Zang dataset, part of the 1000 Functional Connectomes Project, can be downloaded from http://fcon\_1000.projects.nitrc.org/fcpClassic/FcpTable.html. The OASIS-3 dataset was obtained from https://www.oasis-brains.org/. Human Connectome Project (HCP) data were accessed via https://www.humanconnectome.org/. The NYU CSC Test-Retest dataset was provided through the 1000 Functional Connectomes Project and is available at https://www.nitrc.org/projects/nyu_trt. Simulated MRI data were generated using the BrainWeb simulator, available at https://brainweb.bic.mni.mcgill.ca/. Additionally, the MRI version of the Von Economo-Koskinas atlas was obtained from the Dutch Connectome Lab at http://www.dutchconnectomelab.nl/economo/. All datasets were used in accordance with their respective data usage agreements and licenses.