1. Introduction
The ability of computer methods that can predict healthy chronological age of the brain based on radiological images, such as magnetic resonance imaging (MRI), has recently led to a new research direction in computational neuroscience [
1,
2,
3,
4,
5,
6,
7]. This new type of study is important because it holds promise of being able to train computers to identify neurodegenerative disorders at an early onset, where image samples collected for brain diseases are limited for clinical inference [
8,
9]. In other words, the brain imaging data of healthy or control participants can be utilized as models for machine learning, which is then applied to estimate the brain age of a subject under study based on the brain imaging data of the subject. If the predicted brain age is older than its chronological age, then there is some evidence of its accelerated aging that indicates abnormal cognitive impairment [
10,
11], or traumatic brain injury [
12].
In fact, a normal aging process is known as the progression of gradually accumulated pathologies associated with physical decline, cognitive impairment, and brain-volume loss [
13,
14]. It is well perceived that the degenerations of the brain result in fast aging processes, thus causing brain disorders; and healthy brain imaging data can offer useful medical information for early detection and intervention of these neuropathological and cognitive changes using computational methods [
10,
15,
16]. For example, Alzheimer’s disease (AD), which is not a normal part of brain aging and known as the most common form of dementia that causes problems with memory, thinking and behavior. AD symptoms usually progress slowly and deteriorate over years to the stage at which daily tasks can be severely hindered [
17]. Because AD shares many aspects of abnormal brain aging, by applying pattern recognition methods, structural MRI-based features have been discovered as promising biomarkers to identify the early setting of mild cognitive impairment to AD based on the matching of similarity between constructed computational models of healthy and pathological brain aging patterns [
11,
16].
Most computational methods developed for the detection of early stages of neurodegenerative disease are based on the notion of brain age estimation using the MRI of the brain. The mean age prediction accuracy using hidden Markov models was within the mean absolute error of 2.41 years as reported in [
3], 4.98 years using relevance vector machine for regression (RVR) [
2], 6.3 years using quantitative brain water maps [
18], and within the root mean squared error of 6.5 years using the RVR [
19]. Regarding the use of brain age estimate for the diagnosis of traumatic brain injury, the suggestion of brain injury was reported when the brain age is estimated to be older than its chronological age with mean errors of 4.66 years and 5.97 years for gray matter (GM) and white matter (WM), respectively [
7]. While the application of computational models for estimating brain ages using structural MRI to quantify the atrophy of the brain is promising for the early diagnosis of brain diseases, different computational methods provide different results with different respective errors, making it difficult to assess the reliability of the age estimation. Furthermore, although the prediction accuracy can be achieved within an average error of about 5 years, the standard deviation can be as large as 10 years [
7], setting the physiological brain age inference with large uncertainty for medical decision making and treatment. Another issue for brain age estimation is that it is difficult to obtain sufficient MRI data of control subjects, which match the age of each individual patient and other confounding factors that correlate with both dependent and independent variables [
4]. Thus, there is a need to develop some alternative computational methodology for a more reliable prediction of the normal or control brain age using structural MRI data. This is the motivation of this paper that aims to explore the concept of brain age grouping in order to provide a more robust way of physiological inference from control chronological brain age models for more reliable diagnoses of brain disorders.
A methodologically sound procedure for brain age grouping is by way of the concept of phylogeny developed in genomics [
20]. The hypothesis of phylogeny in molecular biology is that if genomes involve by mutations, then the quantitative difference in a nucleotide sequence between a genome pair proportionally indicates how recently those two genomes shared a common ancestor. Two genomes that diverged in the recent past would be expected to have fewer differences than a pair of genomes whose common ancestor is more ancient. The implication is that by comparing three or more genomes, it is possible to infer not just only the similarities but the evolutionary relationships between them [
20]. Likewise, chronological brain ages are altered over time and can be divided into separate groups or hybridize together due the complexity of the brain structure. This is analogous to the evolutionary branching process in molecular phylogenetics, which may be depicted with a phylogenetic tree, and the place of each of the various groups of age on the tree is based on a hypothesis about the brain structure in which cerebral atrophy occurred. This hypothesis is also found in comparative linguistics, which is a branch of historical linguistics, where similar concepts are used with respect to relationships between languages [
21]. Thus, by partitioning brain ages into groups instead of estimating a single chronological age of the brain, we can better reduce the prediction uncertainty, and be able to validate the result by examining the relationships within and between the models built for the brain groups.
To enable the performance of “phylogenetic” analysis for brain age grouping, we need to extract informative features of the brain on MRI so that the dissimilarity or distance matrix between brain groups can be determined. It is known that GM is an important evidence for the assessment of cerebral atrophy revealed by MRI scans of the brain, because GM regions are found to be specifically susceptible to the cognitive decline and AD process [
22]. Furthermore, the importance of the measurement of structures and changes of the brain during its development, aging, learning, disease and evolution is recognized as an emerging field of neuro-informatics known as brain morphometry, which aims to quantify anatomical features of the brain in terms of shape, mass, and volume using typically MRI data to gain insight into the pattern of the brain and its longitudinal extent across individuals or different biological species [
23].
A feasible way for feature extraction is to transform the GM morphology from MRI scans into time series, which was found useful for a study of the complexity of brain folding, cortical surface structure in AD and aging [
16]. In fact, the human brain is relatively larger and more wrinkled than other species. Brain wrinkles increase its surface, indicating more neurons, therefore more folds of the brain means more surface area and more neurons. Declines in brain function in the elderly are sometimes due to widening of the folds (sulci), resulting in fewer neurons, in comparison with young people [
24]. The brain outer layer (cerebral cortex) is a central region in the mammalian brain that monitors complex cognitive behaviors [
25]. Recent scientific evidence has suggested that the size and extent of the folded gray matter of the brain are important factors that influence cognitive abilities and sensorimotor skills [
26]. Furthermore, the conversion of images into sequences for applications of time-series analysis tools has been utilized for solving several problems in image data mining [
27,
28,
29]. Because time-series data of the GM morphology are inherently complex, chaos and nonlinear dynamics analyses of these time-series data are therefore suitable mathematical techniques for extracting their informative statistical properties. Moreover, chaos and nonlinear dynamics have been increasingly reported as effective computational methods for analyzing complex data in medicine and biology [
30,
31,
32,
33,
34,
35,
36]. We therefore explore several methods of chaos and nonlinear dynamics to extract features from time-series data of the GM spatial structure of the brain on MRI. Such methods include approximate entropy (ApEn) [
37], sample entropy (SampEn) [
38], regularity dimension (RD) [
39], recurrence plots (RP) [
40], and the largest Lyapunov exponent (LLE) [
41,
42]. Because dynamic-time warping (DTW) is known as a popular approach for pattern matching in terms of time series [
43,
44], it is also applied in this study to establish the distance matrix for the tree reconstruction for the proposed brain age grouping, which can be compared with those obtained from the methods of chaos and nonlinear dynamics.
The rest of this paper is organized as follows.
Section 2 presents the methods of chaos, nonlinear dynamics, dynamic-time warping, and phylogenetic tree reconstruction, which are implemented in this study.
Section 3 describes the database and the preprocessing of the MRI of the brain.
Section 4 consists of experimental tests for brain age grouping obtained from several methods, discussion and comparison of the results. Finally,
Section 5 is the conclusion of the research, including the summary of new findings, limitation, and issues suggested for future work.
4. Results and Discussion
The sample entropy (SampEn), regularity dimension (RD), recurrence plots (RP), largest Lyapunov exponent (LLE), and dynamic-time warping (DTW) methods were used to obtain featured distances between the brain-age groups from the brain time-series data to construct the corresponding phylogenetic trees.
Using
m = 1, 2 and 3, and the tolerance
r = 0.2 for computing SampEn values; three corresponding trees were constructed and shown in
Figure 2.
Table 1 shows the means and variances of SampEn values for
m = 1, 2 and 3. For
m = 1, the tree topology provides a desirable result by partitioning the brain ages into two distinct clades of 20–59 and 60–86 (a clade represents a single branch on the tree to indicate a population having a common ancestor), in which 20–29 and 30–39, 40–49 and 50–59, 60–69 and 70–79 are sister groups. For
m = 2, the relationships of the brain ages are well divided into three groups: 20–29 and 30–39, 40–49 and 50–59, and 60–69 and 70–79 and 80–86, where the latter two groups are closer to each other. As for
m = 3, 20–29 and 30–39, and 40–49 are closer to 50–59, 60–69 and 70–79, while 80–86 is considered as an outgroup. In general, the three tree topologies shown in
Figure 2 display reasonable relationships of the brain ages.
Utilizing SampEn values, with
m = 1, 2 and 3,
values were obtained to establish the trees of relationships among the brain-age groups, which are shown in
Figure 3. For each fixed value of
m,
r was decreased from 0.5 to 0.05 with an interval of 0.05 [
16]. The slopes of SampEn versus
were taken at the first five points, which form a straight line, to calculate
.
Table 2 shows the means and variances of
values for
m = 1, 2 and 3. For
m = 2, the RD-based trees appropriately assign the relationships of the brain-age groups by separating the clade of 20–29 and 30–29 from the other brain-age groups, and the clade of 40–49 and 50–59 from the clade of 60–69, 70–79, and a single branch for 80–86. The RD-based tree, with
m = 1, shows the grouping of the chronological brain ages into well separated clades, except the groups of 70–79 and 40–49 belong to the same clade. For
m = 3, the RD-based tree locates the groups of 80–86 and 40–49 to the same clade, while other groups are appropriately partitioned.
Table 1.
(Mean, variance) of sample entropy (SampEn) values of brain-age groups with r = 0.2, and m = 1, 2, and 3.
Table 1.
(Mean, variance) of sample entropy (SampEn) values of brain-age groups with r = 0.2, and m = 1, 2, and 3.
Age Group | m = 1 | m = 2 | m = 3 |
---|
20–29 | (4.5079, 0.0173) | (1.5254, 0.0032) | (1.2375, 0.0031) |
30–39 | (4.5115, 0.0139) | (1.5424, 0.0028) | (1.2400, 0.0026) |
40–49 | (4.5540, 0.0145) | (1.5653, 0.0024) | (1.2504, 0.0023) |
50–59 | (4.5838, 0.0143) | (1.5897, 0.0013) | (1.2673, 0.0015) |
60–69 | (4.6531, 0.0148) | (1.6067, 0.0008) | (1.2773, 0.0015) |
70–79 | (4.7003, 0.0089) | (1.6197, 0.0006) | (1.2812, 0.0010) |
80–86 | (4.7765, 0.0018) | (1.6158, 0.0006) | (1.3029, 0.0006) |
Table 2.
(Mean, variance) of regularity dimension (RD) values of brain-age groups with m = 1, 2, and 3.
Table 2.
(Mean, variance) of regularity dimension (RD) values of brain-age groups with m = 1, 2, and 3.
Age Group | m = 1 | m = 2 | m = 3 |
---|
20–29 | (1.0482, 0.0002) | (0.7409, 0.0006) | (0.6907, 0.0020) |
30–39 | (1.0475, 0.0001) | (0.7448, 0.0005) | (0.6666, 0.0024) |
40–49 | (1.0519, 0.0001) | (0.7572, 0.0003) | (0.6441, 0.0024) |
50–59 | (1.0528, 0.0001) | (0.7664, 0.0002) | (0.6094, 0.0027) |
60–69 | (1.0524, 0.0001) | (0.7733, 0.0002) | (0.5997, 0.0024) |
70–79 | (1.0521, 0.0001) | (0.7753, 0.0002) | (0.5786, 0.0036) |
80–86 | (1.0516, 0.0001) | (0.7783, 0.0002) | (0.6282, 0.0008) |
Figure 2.
SampEn-based constructed trees of relationships of MRI-based healthy brain age groups, with m = 1 (a), m = 2 (b), and m = 3 (c); where values on the x-axis of each plot are the corresponding feature-based distances.
Figure 2.
SampEn-based constructed trees of relationships of MRI-based healthy brain age groups, with m = 1 (a), m = 2 (b), and m = 3 (c); where values on the x-axis of each plot are the corresponding feature-based distances.
The trees provided by the RP, with
ϵ = 5, using its recurrence rate (RR) feature and
m = 1, 2 and 3, are shown in
Figure 4a–c, respectively.
Table 3 shows the means and variances of RR values for
m = 1, 2 and 3. While the group of over 80 years old can be well separated from other groups in all the trees; the groups of 20–29 and 40–49 and 50–59 belong to the same clade for
m = 1; the groups of 20–29 and 60–69 are located in the same clade, also the groups of 30–39 and 50–59 belong to the same clade in the trees for
m = 2 and 3.
Table 3.
(Mean, variance) of recurrence rate (RR) values of brain-age groups with m = 1, 2, and 3.
Table 3.
(Mean, variance) of recurrence rate (RR) values of brain-age groups with m = 1, 2, and 3.
Age Group | m = 1 | m = 2 | m = 3 |
---|
20–29 | (0.0187, 0.0000) | (0.0161, 0.0000) | (0.0154, 0.0000) |
30–39 | (0.0192, 0.0000) | (0.0167, 0.0000) | (0.0159, 0.0000) |
40–49 | (0.0189, 0.0000) | (0.0165, 0.0000) | (0.0158, 0.0000) |
50–59 | (0.0189, 0.0000) | (0.0166, 0.0000) | (0.0159, 0.0000) |
60–69 | (0.0181, 0.0000) | (0.0158, 0.0000) | (0.0152, 0.0000) |
70–79 | (0.0176, 0.0000) | (0.0154, 0.0000) | (0.0147, 0.0000) |
80–86 | (0.0164, 0.0000) | (0.0142, 0.0000) | (0.0136, 0.0000) |
Figure 3.
RD-based constructed trees of relationships of MRI-based healthy brain age groups, with m = 1 (a), m = 2 (b), and m = 3 (c); where values on the x-axis of each plot are the corresponding feature-based distances.
Figure 3.
RD-based constructed trees of relationships of MRI-based healthy brain age groups, with m = 1 (a), m = 2 (b), and m = 3 (c); where values on the x-axis of each plot are the corresponding feature-based distances.
Using the recurrence entropy (RE) feature of the RP with
m = 1, and
= 2 and 6, the relationships of the brain-age groups are shown in
Figure 4d,e, respectively. The groups of 60–69 and 80–86 are assigned to the same clade in both sub-plots. Using the recurrence-entropy feature of the RP with
m = 2 and
= 2 and 6, the relationships of the brain-age groups are shown in
Figure 4f,g, respectively. For
= 2, the groups of 60–69 and 80–86 are in the same clade. A reasonable result of the tree construction is obtained using
= 6. Increasing
m to 3, the corresponding
Figure 4h,i show that the RP-based assignment of the age group of 80–86 is confused with the younger age groups (closer to 60–69 for
= 2, and closer to 40–49 for
= 6).
Table 4 and
Table 5 show the means and variances of RE values for
= 2 and 6, with
m = 1, 2 and 3; respectively.
Figure 4.
Recurrence plot (RP)-based constructed trees of relationships of MRI-based healthy brain age groups, where recurrence rate (RR) used as feature with m = 1 (a), m = 2 (b), and m = 3 (c); recurrence entropy (RE) used as feature with m = 1 and = 2 (d), m = 1 and = 6 (e), and m = 2 and = 2 (f); recurrence entropy (RE) used as feature with m = 2 and = 6 (g), m = 3 and = 2 (h), and m = 3 and = 6 (i). Values on the x-axis of each plot are the corresponding feature-based distances.
Figure 4.
Recurrence plot (RP)-based constructed trees of relationships of MRI-based healthy brain age groups, where recurrence rate (RR) used as feature with m = 1 (a), m = 2 (b), and m = 3 (c); recurrence entropy (RE) used as feature with m = 1 and = 2 (d), m = 1 and = 6 (e), and m = 2 and = 2 (f); recurrence entropy (RE) used as feature with m = 2 and = 6 (g), m = 3 and = 2 (h), and m = 3 and = 6 (i). Values on the x-axis of each plot are the corresponding feature-based distances.
Table 4.
(Mean, variance) of recurrence entropy (RE) values of brain-age groups with = 2, and m = 1, 2, and 3.
Table 4.
(Mean, variance) of recurrence entropy (RE) values of brain-age groups with = 2, and m = 1, 2, and 3.
Age Group | m = 1 | m = 2 | m = 3 |
---|
20–29 | (5.1295, 0.0134) | (4.6164, 0.0202) | (4.3049, 0.0224) |
30–39 | (5.0966, 0.0123) | (4.5690, 0.0170) | (4.2528, 0.0184) |
40–49 | (5.0435, 0.0090) | (4.5039, 0.0118) | (4.1852, 0.0125) |
50–59 | (4.9894, 0.0066) | (4.4417, 0.0087) | (4.1217, 0.0091) |
60–69 | (4.9627, 0.0056) | (4.4123, 0.0070) | (4.0917, 0.0071) |
70–79 | (4.9341, 0.0041) | (4.3826, 0.0053) | (4.0609, 0.0054) |
80–86 | (4.9502, 0.0027) | (4.4027, 0.0031) | (4.0806, 0.0033) |
Table 5.
(Mean, variance) of recurrence entropy (RE) values of brain-age groups with = 6, and m = 1, 2, and 3.
Table 5.
(Mean, variance) of recurrence entropy (RE) values of brain-age groups with = 6, and m = 1, 2, and 3.
Age Group | m = 1 | m = 2 | m = 3 |
---|
20–29 | (4.9974, 0.0243) | (4.4715, 0.0157) | (4.2885, 0.0068) |
30–39 | (4.9517, 0.0210) | (4.4329, 0.0114) | (4.2620, 0.0042) |
40–49 | (4.8813, 0.0142) | (4.3807, 0.0075) | (4.2346, 0.0027) |
50–59 | (4.8136, 0.0098) | (4.3391, 0.0043) | (4.2190, 0.0014) |
60–69 | (4.7811, 0.0081) | (4.3144, 0.0031) | (4.2106, 0.0008) |
70–79 | (4.7483, 0.0060) | (4.2935, 0.0016) | (4.2073, 0.0004) |
80–86 | (4.7649, 0.0038) | (4.2899, 0.0020) | (4.1883, 0.0002) |
On the tree reconstruction using the LLE feature,
Figure 5 shows the relationships of the chronological brain-age groups for
m = 1, 2 and 3. The time delay
L was selected to be 1, and the first five points of the divergence curve were used to calculate the slope of a straight line as the value of the LLE.
Table 6 shows the means and variances of LLE values for
m = 1, 2 and 3. It can be seen from all the three figures that the LLE-based trees tend to distinguish the clade of the age groups of 40–49, 50–59, 60–69, 70–79, and 80–86 from the clade of 20–29 and 30–39. In general, the LLE-based tree topologies show the groups of similar ages are more closely related to each other than they are to the outgroup, except for
m = 1, the group of 80–86 is closer to the clade of 40–49 and 50–59 than the clade of 60–69 and 70–79, and for
m = 3, 80–86 is slightly closer to 50–59 than the clade of 60–69 and 70–79.
On the use of the DTW for the phylogenetic tree reconstruction of the brain-age groups, the matching costs were calculated by adopting the Itakura and Sakoe-Chiba local constraints [
64].
Figure 6a–c show the DTW-based trees using the Itakura cost constraints, using the three longest (first, second, and third) time series of the GM boundaries of each age group for pattern matching, respectively.
Figure 6d–f show the DTW-based trees using the Sakoe-Chiba cost constraints, using the three longest (first, second, and third) time series of the GM boundaries of each age group for pattern matching, respectively.
Figure 6a puts the group of 80–86 closer the clade of 20–29 and 40–49, and assigns 30–39 and 50–59 to the same clade.
Figure 6b can separate the groups of 60–69, 70–79, and 80–86 from those of 20–29, 30–39, 40–49, 50–59 by assigning these two groups into two distinct clades; however, 20–29 is closer to the clade consisting of 40–49 and 50–50 than 30–39.
Figure 6c shows that 30–39 is closer to the clade consisting of 50–59 and 60–69, and 20–29 is closer to 40–49. On the adoption of the Sakoe-Chiba local constraints,
Figure 6d assigns 20–29 and 40–49 to the same clade. The topology of the tree shown in
Figure 6e represents reasonable relationships between the age groups, and distinguishes the age groups of 20 to 59 from those of 60 to 86. The age group of 20–29 is the outgroup to the others for the DTW-based tree using the Sakoe-Chiba local constraints as can be seen from
Figure 6f.
Table 6.
(Mean, variance) of largest Lyapunov exponent (LLE) values of brain-age groups with m = 1, 2, and 3.
Table 6.
(Mean, variance) of largest Lyapunov exponent (LLE) values of brain-age groups with m = 1, 2, and 3.
Age Group | m = 1 | m = 2 | m = 3 |
---|
20–29 | (0.3230, 0.0002) | (0.3546, 0.0001) | (0.3516, 0.0001) |
30–39 | (0.3267, 0.0001) | (0.3599, 0.0001) | (0.3566, 0.0001) |
40–49 | (0.3322, 0.0001) | (0.3642, 0.0001) | (0.3601, 0.0001) |
50–59 | (0.3328, 0.0001) | (0.3671, 0.0001) | (0.3624, 0.0000) |
60–69 | (0.3357, 0.0001) | (0.3677, 0.0000) | (0.3626, 0.0000) |
70–79 | (0.3361, 0.0000) | (0.3685, 0.0000) | (0.3628, 0.0000) |
80–86 | (0.3408, 0.0000) | (0.3722, 0.0000) | (0.3652, 0.0000) |
Figure 5.
LLE-based constructed trees of relationships of MRI-based healthy brain age groups, with m = 1 (a), m = 2 (b), and m = 3 (c); where values on the x-axis of each plot are the corresponding feature-based distances.
Figure 5.
LLE-based constructed trees of relationships of MRI-based healthy brain age groups, with m = 1 (a), m = 2 (b), and m = 3 (c); where values on the x-axis of each plot are the corresponding feature-based distances.
Regarding the values given in
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6, the SampEn means tend to increase with the older-age groups, and become smaller with larger values for
m. For
m = 1, the means of RD get larger with the older-age groups; but for
m = 3, the RD values are smaller for the older ages, except for the cohort of 80–86 years of age; and there is no clear trend of increasing or decreasing of the RD with the brain ages. The RD values are smaller for larger values of
m. All RR and RE values are smaller for larger
m. Except for 80–86 year-old cohort, the RE decreases with increasing ages of the brain. Except for the 20–29 year-old cohort for
m = 1, and 20–29 and 50–59 year-old cohorts for
m = 2 and 3, the RR tends to decrease with older age groups. All the means and variances presented in
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6 are truncated to four decimal digits, which make the RR variances equivalent to zero. Such small values of the RR variances can be due to both the insufficient samples of MRI data and the computational method.
In summary, with the given selection of parameters for each method, the SampEn-based trees, RD-based tree with m = 2, RP-based tree using the RE with m = 2 and = 6, and DTW-based tree using the Sakoe-Chiba constraints on the second largest GM-generated time series yield reasonable results to represent the chronological relationships of the healthy brain MRI groups. Among these mentioned reasonable tree topologies, those obtained from SampEn seem all valid for expressing the relationships of the brain age groups, and therefore the SampEn is the most preferred method with the currently available MRI data.
Figure 6.
Dynamic-time warping (DTW)-based constructed tree of relationships of MRI-based healthy brain age groups using Itakura local constraints based on: largest (a), second largest (b), and third largest (c) gray matter (GM)-generated time series of brain MRI; Sakoe-Chiba local constraints based on: largest (d), second largest (e), and third largest (f) GM-generated time series of brain MRI. Values on the x-axis of each plot are the corresponding feature-based distances.
Figure 6.
Dynamic-time warping (DTW)-based constructed tree of relationships of MRI-based healthy brain age groups using Itakura local constraints based on: largest (a), second largest (b), and third largest (c) gray matter (GM)-generated time series of brain MRI; Sakoe-Chiba local constraints based on: largest (d), second largest (e), and third largest (f) GM-generated time series of brain MRI. Values on the x-axis of each plot are the corresponding feature-based distances.
A common problem encountered with the computational methods studied here in assigning the group of 80–86 years of age to an appropriate branch of the tree is likely due to the limited data of the age group, which consists of only eight subjects. Other issues are due to data requirements and parameter selections imposed by different methods, which need to be further investigated to ensure their optimal implementations. For example, the LLE is a method for quantifying chaos (a positive LLE indicates chaos), the concept of the LLE is to keep track of two nearby orbits to determine the average logarithmic rate of separation of the orbits. If the two orbits depart too far from each other, one of them has to be adjusted back to the vicinity of the other along the line of separation, and this adjustment is known to be the most difficult and error-prone step for calculating the LLE [
65]. When underlying equations for chaos are available, the numerical calculation of the LLE is relatively simple; but for experimental data, such a calculation is very difficult [
65]. In this study, MRI-based time-series data were analyzed using the Rosenstein method [
41], which is particularly developed to compute the LLE from small recorded time series and robust to the choice of embedding dimension, reconstruction delay, and noise. In this study, the first five points of the divergence curves were used to estimate the LLEs using the least-squares fit. A practical reason for the poorer presentation of the brain-age relationships via the LLE-based tree topologies than other feature-based tree topologies is due to the inconsistency of the numbers of divergence points representing a straight line for the LLE calculation. Therefore, some method that can adaptively determine suitable numbers of points on the divergence curves for different MRI-based time series would be needed to improve the diagnostic performance of the Rosenstein method.
Finally, the idea is that if a valid tree structure that represents the chronological relationships within a group of brain ages can be constructed, the physiological age of a brain on MRI can be reliably predicted by assigning its physiological age to the age group whose feature-based distance is smallest among others. Furthermore, the constructed brain-age tree can be always updated and extended when more MRI data representing various age groups become available.