Anisotropy of Anomalous Diffusion Improves the Accuracy of Differentiating and Grading Alzheimer's Disease Using Novel Fractional Motion Model

Background and Purpose: Recent evidence shows that the fractional motion (FM) model may be a more appropriate model for describing the complex diffusion process of water in brain tissue and has shown to be beneficial in clinical applications of Alzheimer's disease (AD). However, the FM model averaged the anomalous diffusion parameter values, which omitted the impacts of anisotropy. This study aimed to investigate the potential feasibility of anisotropy of anomalous diffusion using the FM model for distinguishing and grading AD patients. Methods: Twenty-four patients with AD and 11 matched healthy controls were recruited, diffusion MRI was obtained from all participants and analyzed using the FM model. Generalized fractional anisotropy (gFA), an anisotropy metric, was introduced and the gFA values of FM-related parameters, Noah exponent (α) and the Hurst exponent (H), were calculated and compared between the healthy group and AD group and between the mild AD group and moderate AD group. The receiver-operating characteristic (ROC) analysis and the multivariate logistic regression analysis were used to assess the diagnostic performances of the anisotropy values and the directionally averaged values. Results: The gFA(α) and gFA(H) values of the moderate AD group were higher than those of the mild AD group in left hippocampus. The gFA(α) value of the moderate AD group was significantly higher than that of the healthy control group in both the left and right hippocampus. The gFA(ADC) values of the moderate AD group were significantly lower than those of the mild AD group and healthy control group in the right hippocampus. Compared with the gFA(α), gFA(H), α, and H, the ROC analysis showed larger areas under the curves for combination of α + gFA(α) and the combination of H + gFA(H) in differentiating the mild AD and moderate AD groups, and larger area under the curves for combination of α + gFA(α) in differentiating the healthy controls and AD groups. Conclusion: The anisotropy of anomalous diffusion could significantly differentiate and grade patients with AD, and the diagnostic performance was improved when the anisotropy metric was combined with commonly used directionally averaged values. The utility of anisotropic anomalous diffusion may provide novel insights to profoundly understand the neuropathology of AD.


INTRODUCTION
Alzheimer's disease (AD), manifested as progressive cognitive decline and memory loss, is the most common neurodegenerative disease (Reddy and Oliver, 2019). Approximately accounting for 60-70% of dementia patients, AD has been the most prevalent type of dementia (Wortmann, 2012;Alzheimer's, 2016;Khan et al., 2017). The underlying neuropathological mechanisms of AD include the hyperphosphorylation of tau protein and the deposition of β-amyloid (Aβ), which lead to the formation of intracellular neurofibrillary tangles (NFTs) and Aβ plaques separately (Kidd, 1963;Hyman et al., 1984;Braak and Braak, 1991;Wegmann et al., 2010;Mattsson et al., 2019) and ultimately result in the apoptosis of neurons. Neuropathological changes can occur and persist for decades before the appearance of cognitive degeneration. Recently, a variety of magnetic resonance imaging (MRI) techniques have been widely investigated for the diagnosis of AD; however, these methods are insufficient to make a specific diagnosis of AD (Cummings, 2017;Mattsson et al., 2019).
Diffusion MRI (dMRI) can non-invasively describe the random motion of water molecules in and around brain structures such as cell bodies or brain white matter fibers, which provides rich information of microscopic properties than other traditional MRI sequences (Le Bihan, 1995;Le Bihan and Johansen-Berg, 2012;Harrison et al., 2020) and has become a widely used imaging practice in clinical practice and relevant researches (La Rocca et al., 2018;Anckaerts et al., 2019;Bergamino et al., 2020;Finsterwalder et al., 2020). Directional dependence (i.e., anisotropy) is one of the most important microscopic properties obtained from the nervous system by dMRI. Anisotropy results from the dense accumulation of axons and inherent axonal membranes, which prevent the diffusion of water perpendicular to the long axis of fibers (Beaulieu, 2002). One of the most commonly used diffusion MRI technologies, the apparent diffusion coefficient (ADC), was found useful in differentiating AD patients (Takahashi et al., 2017;Xue et al., 2019) and AD transgenic mice (Thiessen et al., 2010). Moreover, the ADC value of white matter in the frontal lobe was correlated with mini-mental state examination (MMSE) scale (Xue et al., 2019). Diffusion tensor imaging (DTI) is another commonly used diffusion MRI technology to measure the anisotropy in the research (Basser et al., 1994a,b). DTI has been increasingly applied to the diagnosis of AD in both basic and clinical studies. The degree of diffusion anisotropy is mostly quantified by two DTI-derived metrics, the fractional anisotropy (FA) and mean diffusivity (MD), in patients with AD (Mayo et al., 2017;Brueggen et al., 2019;Marcos Dolado et al., 2019). Several studies found that the FA values reduced and MD values increased in the hippocampus of AD patients and amnestic mild cognitive impairment (aMCI) patients when compared with healthy control. And the FA and MD might be used to differentiate healthy controls, aMCI patients, and AD patients (Hong et al., 2013;Tang et al., 2016;Schouten et al., 2017). Moreover, the FA value or MD value of hippocampus could be used to predict the progression of AD or aMCI, which is evaluated by MMSE scale (Hong et al., 2013;Lee et al., 2017), indicating the possibility of diffusivity as a biomarker for disease progression. In addition to DTI combined with functional MRI, structural MRI can improve the diagnostic accuracy of AD (Dyrba et al., 2015;Tang et al., 2016;Bouts et al., 2018).
DTI presumes a normal diffusion process in brain tissues and is consequently quantified using a mono-exponential model, S/S 0 = exp (-b · ADC). The b-value represents the applied magnetic field gradient sequence. However, it has been recognized that the observed dMRI signal decay curve deviates from the mono-exponential form in brain tissues, especially at high b-values (De Santis et al., 2011). To solve this problem, several models have been developed based on different theories of anomalous diffusion processes to find the optimal consistency between the observed signal decay curve and the fitted curves. Representative models include the stretched exponential model (Bennett et al., 2003), the bi-exponential model (Mulkern et al., 1999), the kurtosis model (Jensen et al., 2005), and the statistical model (Yablonskiy et al., 2003). Additionally, several physics-motivated dMRI models have also been proposed (Magin et al., 2008;Ingo et al., 2014).
The fractional motion (FM) model has been proposed as a more appropriate approach to describe the complex diffusion process of biological systems (Magdziarz et al., 2009;Burnecki and Weron, 2010;Weiss, 2013). Theoretically, the FM model presumes that the diffusion process of water molecules is α-stable and H-self-similar and has stationary increments. The symbol α represents the Noah exponent, which describes the fluctuations of the random process. The symbol H represents the Hurst exponent, which depicts the self-similarity property of molecular trajectories. The FM model possesses a relevantly more excellent consistency between experimental data and fitting curves. Many studies have demonstrated the clinical feasibility of anomalous diffusion using the FM model (Kwee et al., 2010a,b;Sui et al., 2015;Karaman et al., 2016;Xu et al., 2017bXu et al., , 2018Du et al., 2020). In the aforementioned studies, researchers averaged the anomalous diffusion parameter values that were acquired in different gradient directions, which ignored the impacts of anisotropy. However, existing literature elucidated that the anisotropy of anomalous diffusion should not be neglected as it revealed a different image contrast and provided unique information (Hall and Barrick, 2012;Xu et al., 2017a). At present, the availability regarding the clinical application of the anisotropy of anomalous diffusion in AD patients remains unclear. The purpose of this study was to investigate the potential feasibility of anisotropy of anomalous diffusion for distinguishing AD patients from healthy controls and grading AD patients.  (McKhann et al., 1984;Mattsson et al., 2019). Only the mild-to-moderate AD patients (11 ≤ MMSE score ≤ 25) (Folstein et al., 1975;Perneczky et al., 2006;Tchalla et al., 2018) who met the following criteria were considered for inclusion: (a) the participants' acquired MR image had no artifacts; (b) the participants had no other brain diseases, such as cerebral ischemia or infarction; and (c) the participants had no visual and hearing impairment disorders, aphasia, and limb activity disorder. Finally, 24 patients with AD were eligible and enrolled in this study (9 males and 15 females, mean age, 69.0 years, age range, 50-79 years). Healthy controls were recruited from the local community. Inclusion criteria were as follows: (a) ages range from 50 to 79 years (including 50 and 79 years); (b) a degree of primary education or above; and (c) neurological examination showed no obvious anomalies, and the MMSE scores were between 26 and 30. Healthy controls who suffered from cardiovascular, neurologic, metabolic, and psychiatric disorders or brain abnormalities were excluded. Eventually, 11 healthy controls (2 males and 9 females, mean age 65.3 years, range 54-78 years) were enrolled in the present study. Detailed demographic and clinical characteristics of all participants are summarized in Table 1.

Image Acquisition
All participants received conventional MRI, 3D T1-weighted imaging, and dMRI. The MRI scans were performed on a 3.0-T MRI scanner (GE Healthcare, Discovery MR750, USA) equipped with an eight-channel head coil. dMR images of all participants were obtained using a special Stejskal-Tanner single-shot spinecho echo-planar-imaging sequence. To fit the FM model, we did not fix the diffusion gradient separation time ( ) during the scanning process as the conventional dMRI sequence. Specifically, was arrayed at 27.060, 39.560, and 52.060 ms. For each value, the diffusion gradient amplitude (G 0 ) was 15.67, 19.68, 24.73, 31.06, 39.01, and 49.00 mT/m in sequence, which were selected to be approximately evenly spaced on the log axis. The gradient duration constant (δ) was set to 20.676 ms. Thereafter, 18 nonzero b-values (151, 239, 377, 595, 939, and 1,481 s/mm 2 for at 27.060 ms; 245, 387, 611, 964, 1,521, and 2,399 s/mm 2 for at 39.560 ms; and 339, 535, 845, 1,333, 2,103, and 3,317 s/mm 2 for at 52.060 ms) were obtained in each gradient direction. In order to decrease the effect of diffusion anisotropy, we successively applied the diffusion gradients in three orthogonal directions (the x-axis, y-axis, and z-axis) in turn. Moreover, a total of 12 images without diffusion sensitization (b = 0) were acquired.

Image Segmentation
In the present study, the hippocampus was chosen as the region of interest (ROI) (Figure 1). At first, the hippocampus was manually drawn slice by slice using MRICRON by a radiologist (LD, 5 years' working experience) on the T1 structure images, and then the drawn ROIs were registered onto lower-resolution dMRI, to more easily define the boundary of hippocampus. The ROIs' boundary was accurately segmented, and ambiguous voxels would be eliminated in all participants. Then the average values

Image Analysis
First, the obtained images were corrected for head motion and eddy current distortions by FSL tools (Cha, 2006). In the dMRI acquisition, ADC maps were calculated using the images obtained at b-values of 0 and 954 s/mm 2 (closest to conventional 1,000 s/mm 2 b-value). We used the FM model to analyze the images. According to the FM-based dMRI theory (Sui et al., 2015), the following formula can be used to calculate diffusioninduced signal decay: where D α ,H represents the diffusion coeffcient of anomalous diffusion and γ represents the gyromagnetic ratio. G 0 represents the diffusion gradient amplitude, and ∆ represents the gradient separation time. η is a dimensionless number, which can be calculated using α, H, δ, and ∆ in the following formula (Xu et al., 2017b(Xu et al., , 2018: (2) where µ = H−1/α, and µ is the memory parameter. Along each direction, the signal attenuation at each voxel is fitted to Equation 1 separately. We used the trust-region-reflective nonlinear fitting algorithm in MATLAB (MathWorks, Natick, MA) to perform the fitting procedures.
A metric similar to FA, called gFA, is introduced to quantify anisotropy, where the sample standard deviation is divided by the root mean square [35]: where N represents the number of sampling directions, including three directions in this research, and V refers to the parameter values to be measured. V is the directionally averaged value, and V i is the value in the i-th direction. The gFA maps of α, H, and ADC were calculated.

Statistical Analysis
Among the mild AD group, moderate AD group, and healthy control group, gender was analyzed using the chi-square (χ 2 ) test, and the age, education, and MMSE score were compared using one-way ANOVA. The MoCA score was compared using a two-sample t-test between the mild AD group and moderate AD group, since the MoCA score was not assessed in healthy control. Except for gender, the data were shown in the form of mean ± SD.
The gFA values of α, H, and ADC were compared using a one-way ANOVA test and post-hoc Tukey test among the healthy group, mild AD group, and moderate AD group. Moreover, receiver-operating characteristic (ROC) curves were performed to evaluate the diagnostic capability of each gFA value in differentiating AD patients from healthy controls and distinguishing mild AD patients from moderate AD patients by the area under the curve (AUC). Additionally, multivariate logistic regression analysis was utilized to assess the diagnostic performances of the combination of the anisotropy value and the directionally averaged value. For example, the probability of the combination of H and gFA(H) can be expressed as exp(a 0 + a 1 gFA(α) + a 2 gFA(H)) 1 + exp(a 0 + a 1 gFA(α) + a 2 gFA(H)) where a 1 and a 2 are the regression coeffcients for H and gFA(H), respectively, and a 0 is a constant. The probabilities of other combinations can be expressed in a similar method.
These probability values were used as the test variables in the ROC analysis.
In AD patients, the correlations between gFA(α), gFA(H), and gFA(ADC) values and the cognitive functions evaluated by MMSE and MoCA scales were investigated using Pearson correlation analysis. P-values < 0.05 were considered statistically significant.

Characteristics of All Subjects
The demographic information and clinical cognition scores in all subjects are summarized in Table 1. Ultimately, 24 AD patients (69.0 ± 8.1 years) and 11 healthy controls (65.3 ± 6.6 years) were enrolled in this study. Then patients with AD were divided into two groups [the mild AD group (six males and six females, mean age 65.8 ± 10.1 years) and the moderate AD group (three males and nine females, mean age 72.1 ± 3.8 years)] according to their MMSE score and education level. The general division criteria are as follows: 21 ≤ MMSE score ≤ 25 was considered as mild AD, and 11 ≤ MMSE score ≤ 20 was considered as moderate AD (Perneczky et al., 2006). As demonstrated in Table 1, the age, gender, and education level of the three groups were matched (P > 0.05), while there was a significant difference in the MMSE score among the three groups (P < 0.05). MoCA score was significantly different between the mild AD group and moderate AD group, and a significant difference was found between the two groups (P < 0.05).
The locations of the bilateral hippocampus in the T1-weighted image are shown in Figure 1. Figures 2, 3 show the representative maps of a 58-year-old male patient with AD and a 60year-old male healthy patient, including the 3D T1-weighted images; directionally averaged maps of α, H, and ADC; and the gFA maps of α, H, and ADC. From Figures 2, 3, we found that there no outstanding contrasts between the hippocampus and other brain regions were observed by the naked eye.

Comparisons of gFA(α), gFA(H), and gFA(ADC) Values Among Three Groups
The gFA values of α, H, and ADC of the left and right hippocampus in all participants are summarized in Table 2. Data are presented in the form of mean ± SD. The comparisons among three groups in gFA(α), gFA(H), and gFA(ADC) are shown in Figure 4. From Figure 4, we found that the gFA(α) and gFA(H) values of the moderate AD group were higher than those of the mild AD group (P = 0.003, P = 0.008, separately) in the left ROI (Figures 4A,C). We also found that the gFA(α) values of the moderate AD group were higher than those of the healthy control group (P < 0.001, P = 0.003, separately, Figures 4A,B) in the bilateral ROI, and the gFA(ADC) values of the moderate AD group were lower than those of the healthy control group and mild AD group in the right ROI (P = 0.038, P = 0.035, separately, Figure 4F). No significant differences were found between the healthy control group and mild AD group (P > 0.05 for all, Figures 4A-F).
The performances in differentiating mild AD and moderate AD were illustrated by ROC analysis. Figure 5 depicts the ROC curves calculated from individual gFA values and directionally averaged maps of α, H, and ADC. Figure 5 shows that gFA(α) (AUC = 0.833) and gFA(H) (AUC = 0.826) of the left ROI and gFA(ADC) (AUC = 0.764) of the right ROI exhibited good capacity to differentiate the two groups. The other anisotropy measures of gFA parameters did not perform well. Similarly, ROC analysis in differentiating AD patients and healthy controls was also made.  Figure 9 shows that the gFA(α) and gFA(H) values of the left hippocampus were negatively correlated to corresponding MMSE score (P = 0.017, P = 0.037, respectively) in patients with AD. However, the correlations were not so strong, and there was no significant correlation after false discovery rate (FDR) correction and family-wise error rate (FWER) correction. Moreover, no significant correlations were found in other gFA parameters.

DISCUSSION
In the current study, we investigated the feasibility and effectiveness of the anisotropy of anomalous diffusion assessed by the FM model to differentiate and grade AD patients. We introduced a new anisotropy metric called gFA, and we explored its potencies in identifying AD patients and healthy controls and distinguishing mild AD and moderate AD patients. Our results demonstrated that the anisotropy of α and H of the left ROI exhibited good performances to grade AD patients, and the anisotropy of α of the bilateral ROI FIGURE 4 | Comparisons among three groups: healthy control group, mild AD patient group, and moderate AD patient group (A-F). Scatter plots show that gFA(α) and gFA(H) of the left hippocampus (A,C), and gFA(ADC) values of the right hippocampus (F) can readily separate the mild AD patients and moderate AD patients, and gFA(α) of the bilateral hippocampus (A,B), gFA(ADC) values of right hippocampus (F) can easily distinguish the moderate AD patients and healthy controls. n = 12 for mild AD patients, n = 12 for moderate AD patients, and n = 11 for healthy controls. P < 0.05 was considered as significant. AD, Alzheimer's disease; gFA, generalized fractional anisotropy; ADC, apparent diffusion coefficient.
possessed good potencies to differentiate AD patients and healthy controls. It was worth noting that the diagnostic accuracy was increased when combined with the anisotropy metric with the commonly used directionally averaged value, indicating that the anisotropy metric could improve the diagnostic performances of directionally averaged values in identifying and grading AD patients. An important finding in the present study was that the anisotropy of α and H showed significant superiority to distinguish mild AD and moderate AD patients and identify AD patients from healthy controls, in particular the α. In combination with the results of our previous study (Du et al., 2020), we reached a conclusion that both directionally averaged value and the anisotropy value of α exhibited excellent capacity to identify and grade AD patients, which indicated that α-related values may possess a specific advantage in the diagnosis and grading of AD. In line with the currently available data, α-related diffusion values probably already provide sufficient information regarding the differentiation and classification of AD, which is beneficial to shorten the scan time, simplify the test procedure, and improve medical efficiency. However, it should be validated by future studies with larger sample size.
Possible explanations for the diagnostic performance of gFA(α) are as follows. The symbol α describes the fluctuations  of the random process. Excessive deposition of Aβ protein and neurofibrillary tangles in brain tissue are two main pathological changes in AD patients and can lead to neuronal apoptosis . These pathological alterations that emerged along with AD progression can result in subsequent neuronal apoptosis and encephalatrophy, which eventually reduce the volume of affected brain regions (especially the hippocampus). Moreover, it is well acknowledged that the α values depend on the structural complexity of the brain regions. The non-Gaussian water molecule diffusion is more active in a more complex brain region, and accordingly, the measured α values would be higher and gFA(α) values would be lower. Consistently, the degenerative pathological alterations (neuronal apoptosis and encephalatrophy) that occurred in AD progression can markedly decrease the structural complexity of the hippocampus (Jensen et al., 2005;Grinberg et al., 2011;Yoshida et al., 2013) and  can manifest as different α values and gFA(α) values among healthy controls, mild AD, and moderate AD patients, which were observed in the current study.
Compared with other commonly used traditional techniques, such as DTI, using the FM model to calculate anisotropy of anomalous diffusion possesses several potential advantages. On the one hand, DTI quantifies the diffusion process of water molecule using a mono-exponential form. However, the observed dMRI signal decay curve in the brain deviates from the monoexponential form. In this regard, the FM model was introduced and showed a better agreement between the measured signal decay curve and the fitted curves (Magin et al., 2008). On the other hand, the detected diffusion-time dependence of the MR signal shows a non-Gaussian nature of diffusion, while the DTI model assumes the molecular displacement in brain tissues with a 3D Gaussian ellipsoid (Fieremans et al., 2016). This defect was circumvented by the FM model.
The feasibility and effectiveness of anisotropy calculated by the FM model have been verified in this research. But potential clinical applications are not limited to the FM model. As shown in Equation 1, the α is an exponent of the diffusion gradient, which is proportional to the parameters of other dMRI models, like the stretching parameter γ in the stretched-exponential model (Bennett et al., 2003;Hall and Barrick, 2008;Zhou et al., 2010). Therefore, the clinical feasibility of anisotropy may be generalized to the α-like parameters of other models because of the intrinsic consistency of their anisotropic properties (Xu et al., 2017a).
The current results indicated that the gFA(α) and gFA(H) values of the left hippocampus were negatively correlated with MMSE score in patients with AD. This finding was partially consistent with the results of our previous study using directionally averaged values (Du et al., 2020), which further provided evidence for the reliability and repeatability of our findings. However, the correlations were not so strong and even not significant after FDR and FWER correction. This may be explained by the following reasons. In addition to the hippocampus, pathological alterations of AD affected other brain regions such as the prefrontal cortex and basal ganglion region, which synergistically contributed to cognitive impairments in AD patients. So there might be no linear relationship between cognitive scores and anisotropic values in the hippocampus. Moreover, the small sample size of this study may also affect the results, and further researches with larger sample size are needed to confirm these findings.
The limitations of the present study must be acknowledged. First, this is a single-institution study with a limited number of healthy controls and AD patients, and the results should be validated by further study with larger samples. We hope the independent validation of our results can be done at separated institutions. Second, the voxel of imaging is large, and a single voxel displays an average measurement of neuronal environment, which may decrease the sensitivity for brain tissue components occupying a small part of a voxel. Third, the x-, y-, and z-axes are the only three directions being applied by the diffusion gradients, and the accuracy of this study may be affected, as increased sampling directions are conducive to measuring the anisotropy of anomalous diffusion.

CONCLUSIONS
In summary, the anisotropy of anomalous diffusion was successfully applied in differentiating and grading patients with AD. It was worth noting that the diagnostic performance was improved when the anisotropy metric was combined with commonly used directionally averaged value in identifying and grading AD patients. The anisotropy of anomalous diffusion calculated by the FM model may provide novel insights to profoundly elucidate the neuropathology process of AD.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Materials, further inquiries can be directed to the corresponding author.

ETHICS STATEMENT
The studies involving human participants were reviewed and approved by China-Japan Friendship Hospital. The participants provided their written informed consent to participate in this study. Written informed consent was obtained from the individuals for the publication of any potentially identifiable images or data included in this article.

AUTHOR CONTRIBUTIONS
LD, ZZ, and BX designed this research, analyzed the MRI data, and drafted this manuscript. SS, WG, XL, YC, YW, JL, and BL did the MRI scanning. BX and JG wrote and revised the content of the FM model theory, separately. GM revised the whole manuscript. All authors approved the final manuscript.