Registration, atlas generation, and statistical analysis of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions 5

5 Geodesic Regression of Orientation Distribution Functions with its Application to Aging Study After the HARDI dataset across a population has been warped into the common atlas, the non-linear nature of HARDI data presents a challenge to HARDI-based statistical analysis (see Figure 5.1) Regression analysis is a fundamental statistical tool to determine how a measured variable is related to one or more independent variables The most widely used regression model is linear regression, because of its simplicity, ease of interpretation, and ability to model many phenomena However, if the response variable takes values in a nonlinear manifold, a linear model is not applicable Such manifold-valued measurements arise in many applications, including those that involve directional data, transformations, tensors, and shapes Several works have studied the regression problem on manifolds e.g., [69, 70] In this chapter, we adapt the framework of geodesic regression, proposed in [70], to the HARDI data and apply it to the aging 85 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY study First we derive the algorithm for the geodesic regression on Riemannian manifold of ODFs and conduct the simulation experiment to evaluate its performance Finally, we examine the effects of aging via geodesic regression of ODFs in a large group of healthy men and women, spanning the adult age range HARDI data ODF Reconstruction Data Acquisition Subjects ODF images Atlas Generation ODF atlas Registration serve as common space in registration ODF images in common space Statistical Analysis Biomarkers/ Inference Figure 5.1: The role of Chapter in the ODF-based analysis framework 5.1 Geodesic Regression on the ODF manifold In statistics, the simple linear regression is an approach to modeling the relationship between a scalar dependent variable Y and a non-random scalar variable denoted as X 86 5.1 Geodesic Regression on the ODF manifold A linear regression model of this relationship can be given as Y = β0 + β1 X + , (5.1) where β0 is an unknown intercept parameter, β1 is an unknown slope parameter, and is an unknown random variable representing the error drawn from distributions with zero mean and finite variance Given n observations, i.e., (xi , yi ), for i = 1, 2, · · · , n, ˆ ˆ the least square estimates, β0 and β1 , for the intercept and slope can be computed by minimizing the square errors ˆ ˆ (β0 , β1 ) = arg (β0 ,β1 ) n yi − β0 − β1 xi (5.2) i=1 This minimization problem can be analytically solved The observations, yi can be approximated as ˆ ˆ yi = β0 + β1 xi , i = 1, 2, · · · , n ˆ We will now extend the above simple linear regression to the one modeling the relationship of the ODF and one non-random scalar variable, X, by adopting the general framework of geodesic regression in [70, 81] We now formulate the geodesic regression that models the relationship of a Ψ-valued random variable Ψ and a non-random variable X ∈ R Such a geodesic relationship cannot be simply written as the summation of the intercept β0 and the scalar variable X weighted by β1 , where β0 and β1 are scalar measures as shown in Eq (5.1) Rather, in this case (see Figure 5.2), it needs to be adopted to the manifold setting in which the exponential map in Eq (2.4) will be used to replace the addition in the Euclidean space Hence, we introduce the parameters ψ and ξ, where ψ is an element on Ψ and 87 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY ξ ∈ Tψ Ψ is a tangent vector at ψ on Ψ This pair of parameters (ψ, ξ) provides an intercept ψ and a slope ξ, analogous to the β0 and β1 parameters in the simple linear regression in Eq (5.1) The geodesic regression model can thus be written as Ψ = exp exp(ψ, Xξ), , (5.3) where is a random variable taking values in the tangent space at exp(ψ, Xξ) Notice that in a Euclidean space, the exponential map is simply addition, i.e., exp(ψ, Xξ) = ψ+Xξ Thus, when the manifold is a Euclidean space, the geodesic model is equivalent to the simple linear regression in Eq (5.1) Hence, the geodesic regression is the generalization of the simple linear regression to the manifold setting Figure 5.2: Geodesic regression on manifold Ψ 88 5.1 Geodesic Regression on the ODF manifold 5.1.1 Least-Squares Estimation and Algorithm Given n observations, (xi , ψi ) ∈ R × Ψ, i = 1, 2, · · · , n, we estimate (ψ, ξ) via ˆ ˆ solving a least-squares problem by computing the least-squares estimates ψ and ξ, which minimizes n E(ψ, ξ) = i=1 dist ψi , exp(ψ, xi ξ) 2 , (5.4) where dist(·, ·), given in Eq (2.3), is the geodesic metric distance between ψi and its ˆ approximation ψi = exp(ψ, xi ξ) Unlike the simple linear regression, this minimization problem cannot be analytically solved Therefore, we seek the estimates of (ψ, ξ) using a gradient descent algorithm In order to so, we have to derive the gradient of E with respect to ψ as well as with respect to ξ We refer readers to §5.1.1.1 for the detailed derivation of these two gradient terms Briefly, the gradient of E with respect to ψ (see Eq 5.7), denoted as ∇ψ E, can be written as n ∇ψ E = − cos(xi ξ i=1 ˆ ˆ ψi ) logψi ψi , (5.5) Furthermore, the gradient of E with respect to ξ (see Eq 5.8), denoted as ∇ξ E, can be written as n xi ∇ξ E = − i=1 logψi ψi ˆ ˆ ψi sin(xi ξ ξ + ξ ψ ξ ψ 89 ψ) logψi ψi ˆ ⊥ , (5.6) GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY where logψi ψi ˆ and logψi ψi ˆ logψi ψi ˆ ⊥ = are given as logψi ψi , ˆ ξi ξi ψ i ˆ ˆ ψi ξi , ξi ψ i ˆ and logψi ψi ˆ ⊥ = logψi ψi − logψi ψi ˆ ˆ , ˆ where ξi is the parallel translation of ξ from ψ to ψi , given as ξi = − sin xi ξ ψ ξ ψψ + cos xi ξ ψ ξ, and (·) and (·)⊥ denote components of (·) parallel and orthogonal to ξi respectively We employ the gradient descent algorithm to find the minimizer to the least-squares problem found in Eq (5.4) At the beginning of the optimization, we initialize ξ = and ψ to be the Karcher mean of the observed ODF, ψi , i = 1, 2, · · · , n, where the Karcher mean is computed using the Karcher mean algorithm given in [1] During each iteration of the optimization, the estimates of (ψ, ξ) are respectively updated using Eq (5.5) and Eq (5.6) The above computation is repeated until the change of E is sufficiently small 5.1.1.1 Derivation of the Least-Squares Estimation We now show how to compute the gradient of E in Eq (5.4) with respect to the intercept ψ and the slope ξ via the calculus of variation method The reader can skip this subsection without losing the flow of the exposition by assuming that the gradient 90 5.1 Geodesic Regression on the ODF manifold ˆ of E in Eq (5.4) holds ture For the simplicity, we denote ψi = exp(ψ, xi ξ) in the following derivation We first compute the gradient of E with respect to ψ, denoted as ∇ψ E Let ψ ε = exp(ψ, εμ) where ε is a scalar and μ ∈ Tψ Ψ is a tangent vector at ψ We now take the derivative of E at ε = We have ∂E ∂ε ε=0 = ∂ ∂ε n = (a) i=1 n ≈ i=1 n = i=1 n = i=1 n i=1 logexp(ψε ,xi ξ) ψi logψi ψi , ˆ exp(ψ ε ,xi ξ) ε=0 ∂ logexp(ψε ,xi ξ) ψi ∂ε ˆ ε=0 ψi ∂ exp(ψ ε , xi ξ) ∂ε ˆ ε=0 ψi − logψi ψi , ˆ − logψi ψi , Dψ exp(ψ, xi ξ)μ ˆ ˆ ψi † − Dψ exp(ψ, xi ξ) logψi ψi , μ ˆ ψ , where (a) is obtained from the first order approximation of logexp(ψε ,xi ξ) ψi based on the Taylor expansion of the logarithm map Dψ exp(ψ, xi ξ) is the operator that maps ˆ μ from the tangent space of ψ to that of ψi When we assume that xi ξ ψε = xi ξ ψ, where ε is sufficiently small, this operator can be directly computed according to the analytical form of the exponential map given in Eq (2.4) and the first order approximation of exp(ψ, εμ) based on the Taylor expansion of the exponential map This yields Dψ exp(ψ, xi ξ) = cos(xi ξ ψ) † It is self-adjoint Its adjoint operator, Dψ exp(ψ, xi ξ) , maps logψi ψi from the ˆ 91 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY ˆ tangent space of ψi to that of ψ Therefore, n ∇ψ E = − cos(xi ξ i=1 ˆ ˆ ψi ) logψi ψi , (5.7) Next, we compute the derivative of E with respect to ξ denoted as ∇ξ E Let ξ ε = ξ + εμ, where ε is a scalar and μ ∈ Tψ Ψ is a tangent vector at ψ We now take the derivative of E at ε = We have ∂E ∂ε ε=0 = ∂ ∂ε n = (a) i=1 n ≈ i=1 n i=1 logexp(ψ,xi ξε ) ψi logψi ψi , ˆ exp(ψ,xi ξε ) ε=0 ∂ logexp(ψ,xi ξε ) ψi ∂ε ˆ ε=0 ψi ∂ exp(ψ, xi ξ ε ) ∂ε ˆ ε=0 ψi − logψi ψi , ˆ , where (a) is obtained from the first order approximation of logexp(ψ,xi ξε ) ψi based on Taylor expansion of the logarithm map According to the analytical form of the exponential map given in Eq (2.4), we have ∂ exp(ψ, xi ξ ε ) ∂ε ε=0 = − sin(xi ξ + sin(xi ξ ξ ψ ψ )ψ ψ) + cos(xi ξ μ− ψ) ξ ,μ ξ ψ ψ ξ xi ξ ψ ξ ξ ψ ξ ,μ ξ ψ ψ For short, we denote ξi = − sin(xi ξ ξi ψi ˆ ψ )ψ + cos(xi ξ ψ) ξ , ξ ψ which is the unit tangent direction of the geodesic regression line at exp(ψ, xi ξ) based 92 5.1 Geodesic Regression on the ODF manifold on ξ at ψ by parallel transport Therefore, it yields ∂ exp(ψ, xi ξ ε ) ∂ε ε=0 = xi ξ ,μ ξ ψ ξi ξi ψ i ˆ Substituting the above equations to ∂E ∂ε ε=0 = i=1 n i=1 where Dξ exp(ψ, xi ξ) ψ) ξ , μ ψ ) ξ ψ (μ − , we have † † ψ , is the adjoint of Dξ exp(ψ, xi ξ) Therefore, we have logψi ψi ˆ and logψi ψi ˆ i=1 ˆ ψi − Dξ exp(ψ, xi ξ) logψi ψi , μ ˆ xi logψi ψi ˆ sin(xi ξ ξ ψ − logψi ψi , Dξ exp(ψ, xi ξ) μ ˆ n where ε=0 + n = ∇ξ E = − ∂E ∂ε ψ logψi ψi ˆ ˆ ψi = ⊥ sin(xi ξ ξ + ξ ψ ξ ψ ψ) logψi ψi ˆ ⊥ , (5.8) are given as logψi ψi , ˆ ξi ξi ψ i ˆ ˆ ψi ξi , ξi ψ i ˆ and logψi ψi ˆ ⊥ = logψi ψi − logψi ψi ˆ ˆ , where (·) and (·)⊥ denote components of (·) parallel and orthogonal to ξi respectively 93 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY 5.1.2 Statistical Testing We describe how to perform statistical testing on the relationship between the ODF and the independent variable X by examining the amount of the ODF variance explained by X To this end, we first introduce a reduced model of the geodesic regression in Eq (5.3) as Ψ = exp ψ, (5.9) The solution to this reduced model is the minimizer of n E(ψ) = i=1 dist ψi , ψ 2 , which is the Karcher mean of the ODFs, ψi , i = 1, 2, · · · , n, as shown in [1] We denote ¯ ψ as the least-squares solution to Eq (5.9) Now, to measure the amount of explained variance of the ODF by the variable X, we use a generalization of the coefficient of determination, i.e., R2 statistic, R =1− n ˆ i=1 dist(ψi , ψi ) n ¯ i=1 dist(ψ, ψi ) , (5.10) ˆ where ψi is the estimate of ψi from the full geodesic regression model in Eq (5.3) From this definition, the R2 statistic is always non-negative and less or equal to R2 is equal to if and only if the model in Eq (5.3) perfectly fits the data Finally, we empirically compute the distribution of the R2 statistic via a permutation test by calculating all possible values of the R2 statistic under the permutation of the labels on the observed data points In each randomized trial, X is randomly assigned 94 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY this experiment Figure 5.3: Illustration of synthetic ODFs for single (a) and crossing fibers (b) We constructed a series of simulation data by randomly generating the error term, , of the geodesic regression in Eq (5.3) We first drawed xi , i = 1, , n from a uniform distribution on [0, 1] The error term i was then generated from an isotropic Gaussian distribution in the tangent space of exp(ψ, xi ξ), with the standard deviation σ = M ξ ψ, where M is a constant determining the level of noise The resulting data (xi , ψi ) was considered as observations in the geodesic regression to estimate ˆ ˆ the parameters (ψ, ξ) according to the algorithm described in §5.1 An illustration of synthetic data at different levels of noise M is shown in Figure 5.4 This experiment was repeated for 000 times for each sample size (n = 2k , k = 3, , 8) and each level of noise (M = 0.1, 0.5, 1.0, 2.0), respectively We calculate the mean squared error ˆ ˆ (MSE) between the estimated parameters (ψ, ξ) and the ground truth (ψ, ξ) as ˆ MSE(ψ) = T T ˆ dist(ψt , ψ)2 , t=1 and ˆ MSE(ξ) = T T t=1 96 ˜ ξt − ξ ψ , 5.2 Experiments √ Figure 5.4: Illustration for synthetic ODF data, regression result and ground truth under four levels of noise (M = 0.1, 0.5, 1.0, 2.0): In each panel, each column shows the ODFs at xi = 0, 0.2, 0.4, 0.6, 0.8, 1.The first five rows illustrate the synthetic ODFs, while the next row shows the regression result The bottom row shows the ground truth for the geodesic regression for each experiment with a certain sample size and a noise level T = 1000 is the ˆ ˆ number of repeated trials, and (ψt , ξt ) is the estimate from the t-th trial It is important ˜ ˆ ˆ to note that ξt ∈ Tψ Ψ is the transformed ξt from the tangent space of ψt to the tangent space of ψ through parallel transportation Figure 5.5 shows the plots of the resulting 97 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY ˆ ˆ MSE of (ψ, ξ) As expected, the MSE approaches zero as the sample size increases for all noise levels When the noise level is high, a larger sample size is required to achieve an accurate estimation To check for consistency of the results with metrics other than the geodesic distance used in the regression framework, we compared the MSE of the geodesic distance with the MSE of the L2 norm of spherical harmonics coefficients, and the MSE of symmetric Kullback-Leibler divergence between ODFs [51], in Figure 5.6 From Figure 5.6, we see that the different metrics exhibit the same behavior √ Figure 5.5: Evaluation of the geodesic regression accuracy using synthetic ODF Panels ˆ ˆ (a) and (b) show the plots of the mean square error of ψ and ξ for estimated geodesic regression at four noise levels (M = 0.1, 0.5, 1, 2) against the number of observations n respectively ¯ We calculated mean variance of the synthetic data M SE(ψi , ψ), mean squared ˆ residuals M SE(ψi , ψi ), and R2 for the estimated geodesic regression at all noise levels under three distance measures of ODFs including the geodesic distance, the L2 norm of spherical harmonics coefficients, and the symmetric Kullback-Leibler divergence between ODFs, in Figure 5.7 As designed in this experiment, the higher level of noise, the larger the mean variance of the synthetic data (see the first column) From the results, 98 5.2 Experiments Figure 5.6: Consistency of results under different metrics including the geodesic distance, the L2 norm of spherical harmonics coefficients, and the symmetric Kullback-Leibler divergence between ODFs, we see that the R2 value decreases dramatically as the noise increases Low values of R2 statistically mean that the regression geodesic explains a small portion of the variance of the synthetic data This finding is expected due to the noise being large (for M ≥ 0.5) and high-dimensionality of the underlying space However, it is important to note here that the low R2 value does not imply that the parameters estimated by regression can be found by a chance Even when the noise level is high, the regression still provides an accurate estimation, as demonstrated in Figure 5.5 In addition, when comparing the mean variance, mean squared residuals, and R2 across each row, it demostrates the consistency of the regression results under different distance measures To investigate the effects of the order of spherical harmonics on geodesic regression √ results, we generated a simulated ODF data at the noise level M = 0.5 with the number of observations n = 64 with the coefficients of spherical harmonics from the 2nd-order and up to the 8th-order The results of geodesic regression under different spherical harmonics orders are shown in Figure 5.8 We observed that the results of geodesic regression on ODFs of 2nd-order of spherical harmonics produce “smoother” ODFs Despite a slight loss of information when due to the truncation resulting from lower order spherical harmonics, the results of geodesic regression on ODFs of 4th-order 99 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY √ Figure 5.7: Results of geodesic regression for simulated ODF data at four noise levels (M = 0.1, 0.5, 1.0, 2.0) against the number of observations n Three types of metric between ODFs: geodesic distance; L2 norm of spherical harmonics coefficients and symmetric Kullback-Leibler divergence are calculated, one for each row Under one type of ODF ¯ metrics of that row, the mean variance of synthetic data, M SE(ψi , ψ); the mean squared of the geodesic regression are ˆ residuals of the geodesic regression, M SE(ψi , ψi ); and R shown in each column respectively and above are relatively stable to the order choice 100 5.2 Experiments Figure 5.8: Results of geodesic regression under different spherical harmonics orders 5.2.2 Experiments on Real Human Brain Data: Aging Study In this section, we examined the effects of aging on brain white matter via geodesic regression of ODFs in a large group of normal Chinese subjects, spanning the entire adult age range 5.2.2.1 Image Acquisition and Preprocessing We first briefly describe the demographic information of human subjects and HARDI data processing used in this study The dataset was acquired from 185 Chinese participants (79 males and 106 females) ranging from 22 to 79 years old (mean ± standard deviation (SD): 47.7 ± 15.9 years) All participants had minimental state examination (MMSE) greater than 26 and have no history of major illnesses and mental disorders Each participant underwent HARDI using a 3T Siemens Magnetom Trio Tim scanner with a 32-channel head coil at Clinical Imaging Research Center of the National Uni- 101 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY versity of Singapore The image protocols were as follows: (i) isotropic high angular resolution diffusion imaging (single-shot echo-planar sequence; 48 slices of mm thickness; with no inter-slice gaps; matrix: 96 × 96; field of view: 256 × 256 mm; repetition time: 6800 ms; echo time: 85 ms; flip angle: 90◦ ; 91 diffusion weighted images (DWIs) with b = 1150 s/mm2 , 11 baseline images without diffusion weighting); (ii) isotropic T2-weighted imaging protocol (spin echo sequence; 48 slices with mm slice thickness; no inter-slice gaps; matrix: 96 × 96; field of view: 256 × 256 mm; repetition time: 2600 ms; echo time: 99 ms; flip angle: 150◦ ) The DWIs of each subject were first corrected for motion and eddy current distortions using affine transformation We followed the procedure detailed in [103] to correct geometric distortion of the DWIs due to b0-susceptibility differences over the brain in a single subject Briefly summarizing, the T2-weighted image was considered as the anatomical reference The deformation that relates the baseline b0 image without diffusion weighting to the T2-weighted image characterized the geometric distortion of the DWI For this, intra-subject registration was first performed using FSL’s Linear Image Registration Tool [104] to remove linear transformations (rotations and translations) between the diffusion weighted images and the T2-weighted image Then, we used the brain warping method, large deformation diffeomorphic metric mapping (LDDMM) [93], to find the optimal nonlinear transformation that deformed the baseline image without the diffusion weighting to the T2-weighted image This diffeomorphic transformation was then applied to every DWI in order to correct the nonlinear geometric distortion Finally, we estimated the ODFs represented by 4th-order real spherical harmonics using the approach considering the solid angle constraint based on DWI images described in [48] The ODFs of each subject were then warped to an ODF atlas using an ODF- 102 5.2 Experiments based registration method proposed in Chapter This registration algorithm seeks an optimal diffeomorphism of large deformation between the ODFs of two subjects in a spatial volume domain and at the same time, locally reorients an ODF in a manner such that it remains consistent with the surrounding anatomical structures The ODF atlas, as a representative of the studied dataset, was constructed with a probabilistic approach from a known hyperatlas through a flow of diffeomorphisms proposed in Chapter It is important to note that after atlas generation and registration, estimated deformations were applied to align each subject to the atlas space, using the rotation based reorientation scheme The rotation reorientation only aligned ODFs between the atlas and individual subjects but did not modify the shape of ODFs For comparison purposes, the DTI images of each subject were also aligned to the atlas space using the same estimated deformations via the scheme which preserved the principal direction of diffusion tensors [96] Such a registration algorithm does not change the anisotropy of the ODFs (a major observation of aging), and thus would allow us to optimally align the datasets spatially while minimizing its potential influence on regression results 5.2.2.2 Geodesic Regression of ODFs and Aging Effect We employed the proposed geodesic regression of ODFs to this aging dataset after the aforementioned data processes to examine aging effects on brain white matter In addition, to investigate information gain that resulted from the proposed method, we compared the proposed method against traditional linear regression based on GFA To interpret the regression results, we illustrated the age-associated changes of ODFs captured by the method, especially in the regions of fiber crossing Since the ODFs of all the subjects had been warped into the atlas space after the aforementioned processes, we were able to perform geodesic regressions in a common 103 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY atlas space across all the subjects We chose to focus our study only on the white matter region Therefore, we applied a white matter mask in the atlas space The initial mask was generated by first keeping the voxels in the atlas whose GFA greater than a threshold of 0.2 Next, we applied morphological operation to the initial mask to remove its small isolated components Finally, the voxel-wise geodesic regression of ODFs was performed with age for all the voxels within the white matter mask as described in §5.1 For the p-values for statistical significance of estimated regression, we ran the permutation tests for 10 000 times for each voxel to estimate the underlying distribution of R2 , as mentioned in §5.1.2 To study information gain resulting from the proposed method, we first compared the proposed method, referred to as ODF regression, against three other methods: (1) FA-based simple linear regression, referred to as FA regression; (2) multivariate linear regression based on diffusion tensor under Log-Euclidean metric [4], referred to as DTI regression (the DTI elliposid is normalized to unit volume in order to have a fair comparison with the shape information provided by ODF); (3) GFA-based simple linear regression, referred to as GFA regression We performed a voxel-wise FA/DTI/GFA regression for each voxel in the same manner as described above for ODF regression The corresponding p-values for FA/DTI/GFA regression was also calculated from the same permutation tests based on R2 Figure 5.9 shows the maps of uncorrected voxelwise p-values for FA, DTI, GFA, and ODF regression In addition, we applied a false discovery rate (FDR) of 0.05 for the correction for multiple comparisons [106] As shown in Table 5.1, the ODF regression captured more regions with aging effects in white matter, particularly towards to the crossing fiber region, than the other regressions Furthermore, from Table 5.1, we see that 87% of the significant voxels found in FA regression, 92% in DTI regression and 84% in GFA regression are also in ODF 104 5.2 Experiments regression Table 5.1: Numbers of voxels with age-related significance in each regressions out of 14 881 voxels in the white matter mask after the correction for multiple comparisons # of voxels # of voxels that intersects with ODF regression FA regression 037 648 (87%) 986 572 (92%) DTI regression 506 977 (84%) GFA regression 778 778 (100%) ODF regression Corrected p-value p < 0.0010 p < 0.0016 p < 0.0011 p < 0.0019 Second, we attempt to illustrate the aging effect captured by the ODF regression In Figure 5.10, the corresponding ODFs from the evolution of geodesic regression are shown in four regions of interest (ROI) in the age range from twenty to eighty years with an interval of two decades These four regions are the genu and splenium of corpus callosum (CC) (panels a,f), the left and the right regions of the fiber crossing between CC and corticospinal tracts (CST) (panels k,p) As we can see in the ROIs in Figure 5.10, the ODFs become more spherical, especially at the boundary of the white matter fiber (e.g., panels (b) and (e)), as age increases Additionally, as age increases, the anisotropy at the primary direction of CC is reduced at crossing regions between CC and CST (e.g., arrows on panels (l-o) and panels (q-t)) This provides the intuitive illustration of possible age-related demyelination that may lead to age-related decline in diffusion anisotropy This finding agrees with the general consensus found in DWIbased studies [e.g 105, 108], that is, diffusion anisotropy in white matter characterized by FA declines with advancing age The decrease in diffusion anisotropy in white matter may be indicative of mild demyelination and loss of myelinated axons observed in postmortem MRI as well as in histological studies of normal aging [109] It is well-known that ODFs in the regions of fiber crossings provide more detailed 105 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY profiles of relative diffusivity than traditional DTI To fully exploit this advantage offered by ODFs, we inspect the values of ODFs in CC and CST directions To so, the ODFs at individual ages were computed from the estimated ODF regression At most two prominent peaks of these ODFs for each voxel were identified using a peak extraction algorithm based on the analytical solutions of 4th-order spherical harmonics [110] An example is shown in panel (b) of Figure 5.11, where blue and red lines indicate the diffusion directions of CC and CST based on their ODF values After determining the directions, Figure 5.11 shows the ODF values of observations and regression lines in the corresponding directions for each voxel As could be seen in panels (c)-(h), the values of ODFs in the direction of the CC decline over age in most cases Panels (c) and (f) suggest that the values of ODFs in CC direction decreases more than in the values of ODFs in CST direction Panels (d) and (g) suggest that the values of ODFs in CC direction decreases while an increase is observed in the values of ODFs in CST direction The disproportionate changes of ODF values in the directions of CC and CST are most likely a reflection of the underlying microstructural changes Hence, our regression approach provides one of the first glimpse of in vivo patterns for age-related microstructural deterioration in the region where fibers cross, thanks to the rich information offered by ODFs 5.3 Summary In this chapter, we developed a theoretical framework for the geodesic regression on the Riemannian manifold of ODFs and evaluated its performance on synthetic data We further examined the effects of aging via the proposed geodesic regression in a large group of healthy men and women, spanning across the adult age range The results 106 5.3 Summary show that the proposed method is able to capture more regions with aging effects on white matter as compared to the conventional GFA-based regression The evolution of ODFs along the geodesic regression line depicts in great detail the changes of white matter with aging, and this finding agrees with the current consensus such as the decrease in diffusion anisotropy and the anterior-posterior gradient of corpus callosum [e.g 105, 111, 112, 113] Results also suggest that the diffusivity in corpus callosum declines more than in corticospinal tracts in the selected region, thus, providing new insights into the description and prediction of the diffusion behavior in the region of complex fiber structure 107 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY Figure 5.9: The age effects captured by linear regression based on FA, multivariate linear regression based on full tensor under Log-Euclidean metric, linear regression based on GFA extracted from ODF and geodesic regression directly on ODF For the ease of visualiztion, p−value is only illustrated for the voxels with p < 0.05 108 5.3 Summary √ Figure 5.10: Illustration of evolution of geodesic regression of ODF: exp(ψ, xi ξ) at ages xi = 20, 40, 60, 80: Panels (a)-(e) show the genu of corpus callosum, while panels (f)-(j) show the splenium The crossing regions between corpus callosum and corticospinal tracts are shown in panels (k)-(o) for the left hemisphere and panels (p)-(t) for the right hemisphere For each voxel, the underlying intensity value indicates FA of the altas Arrows on panels (l)-(o) and panels (q)-(t) point out the primary direction of the ODFs 109 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY Figure 5.11: Interpretation for the regions between corpus callosum and corticospinal tracts: Panel (a) shows the T1 image of the selected slice and ROIs Panel (b) illustrates the ODFs in the selected ROI and the peaks of each ODF, where blue and red lines indicate the diffusion directions of CC and CST based on their ODF values Panels (c)-(h) show the ODF values against the age for corresponding voxels labeled in Panel (b), where blue denotes the ODF values in the diffusion direction of CC and red denotes the ODF values in the diffusion direction of CST For both diffusion directions, the crosses (or dots) and lines represent the ODF values for observation data in the chosen direction and the projections of estimated regression geodesics on the chosen direction respectively 110 ... description and prediction of the diffusion behavior in the region of complex fiber structure 107 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY Figure 5. 9:... space of ψt to the tangent space of ψ through parallel transportation Figure 5. 5 shows the plots of the resulting 97 GEODESIC REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION... REGRESSION OF ORIENTATION DISTRIBUTION FUNCTIONS WITH ITS APPLICATION TO AGING STUDY 5. 1.2 Statistical Testing We describe how to perform statistical testing on the relationship between the ODF and