11institutetext: International Institute of Information Technology, Hyderabad
500032, India 22institutetext: TIFR Centre for Applicable Mathematics,
Bangalore 560065, India
Corresponding Author e-mail: 22email<EMAIL_ADDRESS>
# A Diffeomorphic Aging Model for Adult Human Brain from Cross-Sectional Data

Alphin J Thottupattu 11 Jayanthi Sivaswamy 11 Venkateswaran P.Krishnan 22

###### Abstract

Normative aging trends of the brain can serve as an important reference in the
assessment of neurological structural disorders. Such models are typically
developed from longitudinal brain image data – follow-up data of the same
subject over different time points. In practice, obtaining such longitudinal
data is difficult. We propose a method to develop an aging model for a given
population, in the absence of longitudinal data, by using images from
different subjects at different time points, the so-called cross sectional
data. We define an aging model as a diffeomorphic deformation on a structural
template derived from the data and propose a method that develops topology
preserving aging model close to natural aging.The proposed model is
successfully validated on two public cross sectional datasets which provide
templates constructed from different sets of subjects at different age points.

###### Keywords:

Brain Aging Model cross-sectional aging data

##  1 Introduction

Human brain morphometry varies with respect to age, gender, and population.
Since the human brain changes structurally with age, understanding the
normative aging process from structural and functional images has been of
interest both in general and within a specific population. Studies aimed at
arriving at such an understanding, either use a longitudinal or a cross
sectional design for collecting images of the study cohort. The former is
usually difficult as it is challenging to access a fixed cohort over an
extended number of years and scan them repeatedly. A more pragmatic approach
is based on a cross sectional design where a set of individuals in different
age range forms the cohort. This approach makes it easier to collect scans but
their analysis requires a disentangling of the inter-subject variations from
age-related changes which is not straight forward. A more elaborate treatment
of the differences in such approaches can be found in [1].

Regardless of the design, templates play a major role in gaining an
understanding of the aging process and derive a normative standard. Templates
are images defined using an appropriate reference coordinate space. Templates
created for the young adult Caucasian population [2, 3] are the most well
known and used, though population specific templates are also gaining
attention [4, 5]. In computational anatomy, aging is typically modelled as a
continuous deformation of a template image over time [6]. This modelling helps
to derive any age-specific template from the model, develop subject specific
growth trajectory and derive direct interpretations from the deformation field
about the aging pattern. In this paper, we propose a method to develop such an
aging model for the adult human brain from cross sectional data drawn from a
specific population. Studies using longitudinal data have modelled the aging-
based deformation in a variety of ways such as a geodesic [7, 8, 9] piece-wise
geodesic [10], as a spline [11] as well as with stationary velocity fields
parameterized path [12] and acceleration parameterized path [13, 14]. In a
cohort-based longitudinal study, the variability in inter-subject aging trends
can also be high. This was handled in [15] by considering a tubular
neighbourhood for the deformation. The spatio-temporal model suggested in [16]
also considers similar variations due to diseased data points in the dataset
and uses partial least squares regression to compute normal aging
deformations; this gives modes of aging and corresponding scores for each
subject.

A cross sectional design allows creation of larger data sets compared to that
with longitudinal data. In aging studies with a cross sectional design, the
inter-subject variability within an age group and across age groups is
disentangled to some extent by developing age-specific templates [17, 18, 19].
Several such age-specific templates are publicly accessible even though the
image-sets used for template generation are not publicly available [17, 19,
20, 21].

We are aware of only two reports that explicitly develop an aging model from
cross-sectional data. In the first [22], an image regression approach based on
weighted averaging is proposed for the aging model. In the second [23], a
global template is derived from all the age-specific templates and the mapping
between each of the age-specific template to the global template constitutes
the aging model. The consequence of the second approach is that any comparison
of a subject image with the global template space needs two transformations of
the subject image; one from the subject image to the corresponding age-
specific template and then to the global template image. Further, this is a
departure from the notion of aging as a deformation process which acts on a
template image [6].

We argue that an aging model represented by a set of image points developed by
smoothly deforming a template image is more natural than a weighted average of
image points in neighbourhoods. Hence given cross-sectional data, we explore
the use of a diffeomorphic deformation of a template image as an aging model.
In this paper, we focus on developing the aging model from age-specific
templates. The contributions of this paper are: i) a method to derive an aging
model from cross sectional data, and ii) an aging model based on a
diffeomorphic deformation process applied to the global template for all age
points, which is closer to the definition in [6]. The key features of the
proposed aging model are: it aids mapping of a subject image to the template
space with a minimum number of deformations and its ability to handle
temporally non-uniform diffeomorphic deformations.

##  2 Method

The proposed method derives an aging model from a given set of cross-sectional
data for different age groups. The aging is modelled as a diffeomorphic
deformation of a global structural template defined from all the images in the
given data. The method permits the flexibility to use only age-specific
templates instead of the entire dataset as full datasets are usually not
publicly available.
The proposed aging model has two elements derived from the supplied templates:
(i) a structural template for the brain and (ii) an aging deformation as a
function of time defined on the structural template. The diffeomorphic aging,
defined as a suitable deformation of the structural template, is explained in
Sect. 2.1. Computing the aging deformation, explained in Sect. 2.2, and
mapping it to the structural template explained in Sect. 2, are the main steps
of this method. The model derives temporally and spatially smooth deformations
to minimize the effect of cross-sectional variations in the aging deformation
computed from the cross-sectional data. These are discussed in detail below.

###  2.1 Computing the structural template

Age-specific templates are used as inputs to derive the aging model. Closely
and equally spaced age-specific templates, each of them generated from equal
number of images is preferred. Let an age-specific template $T_{i}$ at an
interval about age $i$ be a representation of the brain for the given
population and assume that we have $N$ such templates. As each of the
templates is derived from different sets of images, the template space defined
for each set need not be the same and thus the aging path will be different
for each $T_{i}$. Finding the common aging path from the unaligned
$T_{1},T_{2},\cdots,T_{N}$ is the challenge here. The final aging model is
defined using a template $G$ constructed from the set of all $T_{i}$s in the
diffeomorphic space $\mathcal{G}$. The template $G$ is the best structural
representation of all the $T_{i}$s, which is computed using a non-rigid group-
wise registration method called SyNG proposed in [24]. SyNG iteratively
computes the templates that minimizes the average distance from the template
to each of the $T_{i}$s on the diffeomorphic space $\mathcal{G}$, and the
optimal template is $G$. The distance from this template to each of the
$T_{i}$ incorporates two aspects; the cross-sectional and aging deformations.

The cross-sectional variations and aging deformations in the data affect $G$
less as it is the template developed from the whole data covering the entire
age range of interest. The template $G$ can be considered as the global
template for all $T_{i}$s and considering it as the structural template
element in the proposed model avoids biases toward any $T_{i}$. All the
$T_{i}$s are aligned to $G$ using an affine transformation before developing
the model to make it affine invariant. This simplifies the model as an affine
alignment can be done accurately from one template to another or to an image.

###  2.2 Computing the Aging Deformation

####  2.2.1 Assumptions made while computing the aging deformation

In order to compute the aging deformation from cross-sectional data, it is
useful to understand the natural aging process from a physiological/structural
perspective. It is observed that in a mature human brain (from approximately
20 years), brain tissue regions shrink and the ventricular space increases
with aging [25, 26, 27]. A key implication of this observation is that the
deformation that the brain undergoes with normal aging is spatially and
temporally smooth and topology is preserved as no new structure appears with
aging. We can therefore assume that growth-induced deformation will be the
smoothest and the most predictable among other types of spatial (cross-
sectional variation) and temporal (atrophy) deformations. The log-Euclidean
framework [28] covers such less-complex diffeomorphic deformations, and can be
used to extract aging deformations from cross-sectional data. The space of
diffeomorphisms is an infinite-dimensional manifold, and subject-images can be
generated by applying a set of diffeomorphisms $\mathcal{G}$ on a template
image. The log-Euclidean framework uses the locally Euclidean nature of the
manifold to work with diffeomorphisms in a computationally efficient manner.
This is defined by representing the diffeomorphism by Stationary Velocity
Fields (SVF). Group exponential maps are generally used to compute the
deformation $\phi$ represented by the SVF $v$, that is, $\phi=\exp(v)$.

####  2.2.2 Aging deformation modelled by two SVFs

Recall that we already have age specific templates $\\{T_{i}\\}$ and their
global representation $G$. The aging deformation is the second element in the
aging model. As mentioned in Sect. 2.1, $G$ does not carry any information
about the aging deformation. Mapping between $G$ and $T_{i}$s constitutes both
aging and cross-sectional deformations. Therefore $G$ cannot be used directly
to extract the aging deformation from the $T_{i}$s.

An age-specific template $T_{M}$ among the $T_{i}$s that needs the smallest
deformation to map $G$ to that template is used as reference template to
compute the aging deformation. The aging deformation computed with respect to
$T_{M}$ can be mapped to $G$ fairly accurately as they are close. The SVFs
$\\{v_{i}\\}$ that maps each $T_{i}$ to $G$ are computed first to find the
reference template $T_{M}$. Let $\left\|v_{i}\right\|$ be a measure of the
distance between $G$ and $T_{i}$. Then the desired $T_{M}$ is the template
corresponding to the smallest norm $\lVert v_{i}\rVert$. In other words,

$T_{M}=T_{i}\mbox{ where }i\mbox{ is such that }\lVert
v_{i}\rVert=\mbox{min}\\{\lVert v_{k}\rVert,1\leq k\leq n\\}.$ (1)

The deformation $\exp(v_{M})$ which maps $T_{M}$ to $G$ is used to map the
deformation computed with respect to $T_{M}$ to the common space.

In the proposed model, the aging deformation is considered as a temporal
relationship with a consistent trend between subsequent $T_{i}$s, with $T_{M}$
being considered as the reference template. An example of consistent trend is
the fluid-filled regions in a mature brain increasing in size with aging. This
temporally consistent aging deformation is derived from the deformations
between template pairs in the forward ($f$) and backward ($b$) directions. For
instance, the deformations $v_{j_{f}}$ between $(T_{M+(j-1)_{f}},T_{M+j_{f}})$
for $j=1,2,\cdots,(N-M)$ are the SVF parameterizations for pairs in the
forward direction. Similarly, the deformations $v_{j_{b}}$ obtained by
registering template pairs $(T_{M-(j-1)_{b}},T_{M-j_{b}})$ for
$j=1,2,\cdots,(M-1)$ are the corresponding SVF parameterizations for pairs in
the backward direction. The spatial aging trends will be locally consistent
and therefore, composing the forward/backward pairwise deformations can be
used to extract the consistent trends in the deformation with respect to
$T_{M}$ in both directions. We propose to do this by composing the
deformations sequentially using the Baker–Campbell–Hausdorff (BCH) formulation
given in Eqn. 4 below. This allows compositions of group exponentials to be
expressed as a single SVF. Let the velocity vector field obtained as a result
of repeated application of BCH formula on the forward (backward) deformations
be denoted as $\mathbf{v_{j_{f}}}$ ($\mathbf{v_{j_{b}}}$). The vector field
$\mathbf{v_{j_{f}}}$ defines the single SVF parameterization of the forward
deformation from $T_{M}$ to $T_{(M+j)_{f}}$ and $\mathbf{v_{j_{b}}}$ defines
the same for the backward deformation from $T_{M}$ to $T_{(M-j)_{b}}$. These
velocity fields are computed from Eqn. 2 and Eqn. 3 with an initialization of
$\mathbf{v_{1_{f}}}=v_{1_{f}}$ and $\mathbf{v_{1_{b}}}=v_{1_{b}}$.

$\mathbf{v_{j_{f}}}=\mbox{BCH}(\mbox{BCH}(\cdots(\mbox{BCH}(\mathbf{v_{1_{f}}},v_{2_{f}}),v_{3_{f}}),...),v_{j_{f}}),j=2...(N-M),$
(2)
$\mathbf{v_{j_{b}}}=\mbox{BCH}(\mbox{BCH}(\cdots(\mbox{BCH}(\mathbf{v_{1_{b}}},v_{2_{b}}),v_{3_{b}}),...),v_{j_{b}}),j=2...(M-1).$
(3)

The BCH formula for a pair of forward deformations is given in Eqn. 4.
Backward deformations can be computed in a similar manner.

$\mbox{BCH}(\mathbf{v_{(j-1)_{f}}},v_{j_{f}})=\log(\exp(\mathbf{v_{(j-1)_{f}}})\exp(v_{j_{f}}))\\\
=\mathbf{v_{(j-1)_{f}}}+v_{(j_{f})}+\frac{1}{2}([\mathbf{v_{(j-1)_{f}}},v_{(j_{f})}])+\\\
\frac{1}{12}([\mathbf{v_{(j-1)_{f}}},[\mathbf{v_{(j-1)_{f}}},v_{(j_{f})}]]+[{v_{j_{f}}},[{v_{(j_{f})}},\mathbf{v_{(j-1)_{f}}}]])+\cdots=\mathbf{v_{j_{f}}}.$
(4)

Here, $[\cdot,\cdot]$ denotes the Lie bracket of two vector fields.
It should be noted that since the BCH approximation is valid only for small
deformations, in practice, $v_{(j_{f})}$ is divided into $n$ smaller
deformations such that $\frac{v_{(j_{f})}}{n}<0.5\times$ voxel dimension, and
these smaller deformations are composed iteratively with
$\mathbf{v_{(j-1)_{f}}}$ to compute $\mathbf{v_{j_{f}}}$. In the proposed
method, the extracted deformation is constrained to be spatially smooth due to
the log-Euclidean framework and temporally smooth since the composing step
captures only the temporally consistent trends from the sequential data. For
simplicity, the forward aging deformation from $T_{M}$ to $T_{N}$,
$\phi_{f}=\exp(\mathbf{v_{(N-M)_{f}}}t)$ is denoted as $\exp(\mathbf{v_{f}}t)$
and the backward aging deformation from $T_{M}$ to $T_{1}$,
$\phi_{b}=\exp(\mathbf{v_{(M-1)_{b}}}t)$ is denoted as
$\exp(\mathbf{v_{b}}t)$.

The computed aging deformations $\phi_{f}$ and $\phi_{b}$ vary uniformly with
time which is not consistent with the natural aging trends whereas the aging
deformation cannot be expected to vary uniformly, for example tissue
degradation will be rapid for elderly age range [27]. Hence, a temporal
dependency is introduced in $\phi_{f}$ and $\phi_{b}$ to accommodate any non-
uniform changes in natural aging. This step is explained in the next section.

####  2.2.3 Imposing non-uniform temporal variations on aging deformation

The aging deformation need not increase linearly in time with respect to
$T_{M}$. Hence we propose a quantification for the aging deformation (denoted
as $R$) at each time point in Eqn. 5. This is defined in terms of the distance
between $T_{i}$ and $T_{M}$ as in Eqn. 5. Here $\mathbf{v^{*}}=\mathbf{v_{f}}$
in the forward direction and $\mathbf{v_{*}}=\mathbf{v_{b}}$ in the backward
direction. Further, $\mathbf{v_{i}}=\mathbf{v_{j_{f}}}$ for $i=(M+j)_{f}$ and
$\mathbf{v_{i}}=\mathbf{v_{j_{b}}}$ for $i=(M-j)_{b}$. With this, let us
define

$R(i)=\frac{d(T_{1},T_{i})}{d(T_{M},T_{i})}=\frac{\left\|\mathbf{v_{i}}\right\|}{\left\|\mathbf{v^{*}}\right\|}.$
(5)

Since $R(i)$ is a discrete sequence, whereas a continuous aging trend is of
interest, a smooth curve $\gamma(t)$ is found by fitting a curve to $R(i)$. In
our implementation a smooth spline fitting was done in the forward and
backward directions. The function $\gamma(t)$ for $t=[t_{0},t_{N}]$,
quantifies the aging deformation at a particular time point with respect to
$T_{M}$. As this deformation increases in both directions with time, the curve
will, in general, have a bilateral increasing trend about the age point
corresponding to $T_{M}$. An illustration of the proposed method to extract
the aging trends is shown in Fig. 1.

Figure 1: Illustration of the proposed aging model computation framework

###  2.3 Transferring the deformations to the global template space

The deformations captured using Eqn. 4 are mapped to the global template space
using the mapping from $T_{M}$ to $G$, i.e., $\exp(v_{M})$. The captured
deformations on the manifold $\mathcal{G}$ are parameterized by SVF. In order
to transfer the aging deformations to the global template space we use an
existing algorithm [29] for parallel transport. This is explained next.
Let the global template space images corresponding to $T_{i}$s be $G_{i}$s.
The deformations to be transported are parameterized by SVFs $\mathbf{v_{f}}$
and $\mathbf{v_{b}}$. A schematic of the deformation mapping scheme is shown
in Fig. 2. Here
$\rho_{b}=\exp\left(\frac{v_{M}}{2}\right)\circ\exp(-\mathbf{v_{b}})$ and
$\exp(\Pi(\mathbf{v_{b}}))=\exp\left(\frac{v_{M}}{2}\right)\circ\rho_{b}^{-1}$.
Therefore,

$\exp(\Pi(\mathbf{v_{b}}))=\exp\left(\frac{v_{M}}{2}\right)\circ\exp(\mathbf{v_{b}})\exp\left(\frac{-v_{M}}{2}\right),$
(6)

and similarly,

$\exp(\Pi(\mathbf{v_{f}}))=\exp\left(\frac{v_{M}}{2}\right)\circ\exp(\mathbf{v_{f}})\exp\left(\frac{-v_{M}}{2}\right).$
(7)

Figure 2: $T_{M}$ is mapped to $G$ using $\exp(v_{M})$ and the path is used to
transport $\exp(\mathbf{v_{f}})$ and $\exp(\mathbf{v_{b}})$ to the global
template space.

In Fig. 2, $G_{1}^{\prime}=G\circ\exp(-\Pi(\mathbf{v_{b}}))$ and
$G_{N+1}^{\prime}=G\circ\exp(-\Pi(\mathbf{v_{f}}))$. Thus, the inverse of the
mappings from $G$ to $G_{1}^{\prime}$ and $G_{N+1}^{\prime}$ i.e.,
$\exp(\Pi(\mathbf{v_{b}}))$ and $(\exp(\Pi(\mathbf{v_{f}}))$ gives $\phi_{b}$
and $\phi_{f}$ respectively.

Algorithm 1 Proposed Algorithm

Input: $T_{1},T_{2},\cdots,T_{N+1}$,

Result: Aging Model (${\color[rgb]{0,.5,.5}G}$,
${\color[rgb]{1,0,1}\Pi(\mathbf{v_{f}})},{\color[rgb]{0,0,1}\Pi(\mathbf{v_{b}})}$,
$\gamma(t)$)

Step 1: Compute the global template as group mean of
$T_{1},T_{2},\cdots,T_{N+1}$ $\longrightarrow{\color[rgb]{0,.5,.5}G}$

Step 2: Register $G$ to $T_{i}$ $\forall i\in[1...N]$ $\longrightarrow v_{i}$

Step 3: Compute the distance $d_{i}=||(v_{i})||$ between $G$ and each $T_{i}$

Step 4: Find $T_{i}$ which is closest to $G$ by comparing $d_{i}$
values$\longrightarrow T_{M}$

Repeat Step 5 and Step 6 for $(T_{(M+(j-1))_{f}},T_{(M+j)_{f}})$ where
$j=1,2...(N-M)$

Step 5: Register each pair $(T_{(M+(j-1))_{f}},T_{(M+j)_{f}})$ using log-
demons registration $\longrightarrow v_{j_{f}}$

Step 6: Single SVF parameterization of the composed aging deformation from
$T_{M}$ to $T_{(M+j)_{f}})$ using Eqn. 2 $\longrightarrow\mathbf{v_{j_{f}}}$

Repeat Step 7 and Step 8 for $(T_{(M-(j-1))_{b}},T_{(M-j)_{b}})$ for
$j=1,2,\cdots,(M-1)$

Step 7: Register each pair $(T_{(M-(j-1))_{b}},T_{(M-j)_{b}})$ using log-
demons registration $\longrightarrow{v_{j_{b}}}$

Step 8: Single SVF parameterization of the composed aging deformation from
$T_{M}$ to $T_{(M-j)_{b}}$ using Eqn. 3 $\longrightarrow\mathbf{v_{j_{b}}}$

Step 9: $\mathbf{v_{f}}\longleftarrow\mathbf{v_{(N-M)_{f}}}$ and
$\mathbf{v_{b}}\longleftarrow\mathbf{v_{(M-1)_{b}}}$

Step 10: Parallel Transport $\mathbf{v_{f}}$, $\mathbf{v_{b}}$ along $v_{M}$
using Eqn. 6 and 7 respectively
$\longrightarrow{\color[rgb]{1,0,1}\Pi(\mathbf{v_{f}})},{\color[rgb]{0,0,1}\Pi(\mathbf{v_{b}})}$.

Step 11: Compute a curve fitting for the discrete function R defined by Eqn. 5
using $\mathbf{v_{j_{f}}},\mathbf{v_{j_{b}}},\mathbf{v_{f}}$ and
$\mathbf{v_{b}}$ $\longrightarrow\gamma(t)$

###  2.4 The aging model

The aging model has three components, $G$, $\gamma(t)$ and the SVF
parameterization of the transported forward and backward deformations
$\Pi(\mathbf{v_{f}}),\Pi(\mathbf{v_{b}})$ respectively. An age-specific
template at any time point $t$ can be computed using the following formula:

$T(t)=\begin{cases}G\circ\exp(\Pi(\mathbf{v_{f}})\gamma(t))\mbox{ for }t\geq
M,\\\ G\circ\exp(\Pi(\mathbf{v_{b}})\gamma(t))\mbox{ for }t\leq M.\end{cases}$
(8)

##  3 Experiments

In this section, We report on valiadtion of the proposed model and
experiements with the model. All experiments, barring the one with simulated
data, were done on 3D data though only 2D central slices from the results are
shown for visual comparison. The proposed aging model is affine invariant, and
therefore results were also aligned using affine transformation prior to
comparison. We have used two databases to do experiments which are discussed
in Sect. 1 and the models derived from these datasets are discussed in Sect.
3. Experiments done to i) assess the quality of representation of the proposed
model are presented in Sect. 3.3 and ii) validate the model are presented in
Sect. 3.4.

###  3.1 Data

The proposed method to create an aging model was implemented using two cross-
sectional template datasets: (i)Brain Imaging of Normal Subjects (BRAINS) [20]
and (ii)Neurodevelopmental MRI Database [21] (NEURODEV). Only age-specific
templates are accessible in these datasets along with information on the age
interval and number of scans of subjects used to create each age-specific
template. From Table 1, it can be observed that in BRAINS, the inter-age
interval is not uniform, particularly at the upper age level, and the number
of scans used for template creation is less relative to NEURODEV. The inter-
age interval is shorter (5 years) and uniform in NEURODEV.

BRAINS[20] | NEURODEV[21]
---|---
| Age interval
---
(Template age)
| Number of scans
---
used for
template creation
| Age interval
---
(Template age)
| Number of scans
---
used for
template creation
25-34 (29.5) | 20 | 20-24 (22) | 559
35-44 (39.5) | 24 | 25-29 (27) | 525
45-54 (49.5) | 23 | 30-34 (32) | 422
55-64 (59.5) | 13 | 35-39 (37) | 73
71-74 (72.5) | 47 | 40-44 (42) | 96
75-78 (76.5) | 50 | 45-49 (47) | 82
91-93 (92) | 48 | 50-54 (52) | 72
|  | 55-59 (57) | 81
|  | 60-64 (62) | 57
|  | 65-69 (67) | 71
|  | 70-74 (72) | 65
|  | 75-79 (77) | 44
|  | 80-84 (82) | 44
|  | 85-89 (87) | 20
Table 1: Composition of the two age-specific template datasets [20, 21] used
in our study.

###  3.2 Aging Model

Recall that the proposed aging model has two elements, namely, the structural
template $G$ and the aging deformation. The aging deformation has three
components: the forward aging deformation $\phi_{f}$ parameterized by
$\mathbf{v_{f}}$, $\phi_{b}$ parameterized by $\mathbf{v_{b}}$ and the
$\gamma$ function. The proposed model developed with NEURODEV and BRAINS
datasets are shown in Fig. 3. A direct interpretation of $\gamma(t)$ plot does
not give much information about the aging trend as it represents the degree of
deformation with respect to $G$, rather than any of the end point templates.
It however does indicate the age point that corresponds to the reference
template $T_{M}$. In the case of NEURODEV this is 67 years and for BRAINS it
is 77 years where the template age ranges for these datasets are 22-87 and
30-92 years respectively.

Figure 3: The aging model computed with NEURODEV and BRAINS datasets

###  3.3 Representation Quality Analysis

Age-specific templates were generated with the proposed aging model using Eqn.
8, and were used for visual comparison to assess the quality of
representation. Comparisons are done with natural aging trends in Sect. 3.3.1,
existing spatio-temporal atlas [16] in Sect. 3.3.2 and the supplied templates
used for model creation in Sect. 3.3.3.

####  3.3.1 Compatibility with Natural Aging

Templates at increasing age points were generated with the proposed aging
model to study the structural change with aging. The BRAINS dataset [20] was
chosen to do this experiment as it covers a longer span at the elderly age end
where more changes are expected. Human brain aging literature [30, 31, 27]
indicates that a mature brain undergoes minimal cognitive and structural
changes up to the age of $\approx 50$ and more for the elderly, i.e. $\approx
60+$. This trend was verified by computing the intensity difference between
the current template and the first (at age 30) template. This difference
essentially is due to age-induced structural change.

####  3.3.2 Growth Trend across Aging Models

Huizinga et al. in [16] proposed a cross-sectional spatio-temporal reference
model for representing aging. This model does not ensure a diffeomorphic aging
deformation and the template space representation of a subject image needs a
computationally intensive group-wise registration with a training set used to
generate the model. In contrast, our model requires only one pairwise
registration from a subject to the corresponding age-specific template,
derived from the model. The aging trends observable in the templates derived
from our model was was compared with those derived using [16]; the latter
templates are available in http://www.agingbrain.nl/ for the age range of
45-92 years. Templates at the same age points were generated with the proposed
method using the NEURODEV dataset.

####  3.3.3 Age-specific Template Assessment

The generated templates with our model were visually compared with the
templates given in the 2 datasets to understand how well the model have
represented these templates. The templates for the first and last time points
in our aging model have undergone maximum deformation compared to those at
other age points. Hence, such a visual comparison is of interest.

###  3.4 Aging Model Validation

Model validation was done by analysing the ability of the model to capture
natural deformations in Sect. 3.4.1; and the similarity of model-generated
age-specific templates to a set of subject images of same age in Sect. 3.4.2.
Since our model was derived for a cross-sectional setting, we also studied its
performance in a longitudinal data setting as it is of interest. This is
presented in Sect. 3.4.3.

####  3.4.1 Topology Preservation

Since diffeomorphic deformations best fit natural deformations, we considered
aging related deformation also as a diffeomorphism. Accordingly, our model is
defined on a manifold $\mathcal{G}$ of diffeomorphisms. It is of interest to
verify if an extrapolation of the model generates deformations in
$\mathcal{G}$ itself. This was done by extrapolating the aging trend and
deriving templates in both younger and older ages. NEURODEV data which covers
that age range of 22-87 (reference template age point, M=77 years) is used for
this experiment. The templates from extrapolation in both directions were
generated for this experiment with Eqn. 8.

####  3.4.2 Validation with Segmentation

A localised assessment, i.e., of few structures, is of interest in many
situations. This requires labeling by aligning the subject image to a labeled
template and doing a label transfer. An alignment process that requires
smaller deformations indicates that the template is structurally very close to
the subject image. This will lead to better segmentation. With our age model,
this involves only a single registration step and hence potentially least
deformation. This is in contrast to the steps required when using the model in
[23] which requires two registration steps: one to transfer the labels from
the global template to the age-specific template that is closest to the given
subject age, and a second to transfer the age-specific template labels to the
subject image. Each of these registration steps can contribute to error in
labeling in addition to the structural dissimilarity of template and subject
image.

An experiment was done to quantitatively compare the accuracy of labeling
using the proposed method and with [23]. Ten subject images from MICCAI 2012
dataset [32] of ages 18,30,38,45,54,61,68,75,83 and 90 along with the ground
truth labels were used to perform the comparison. The templates corresponding
to the subject ages 18 and 90, were constructed by extrapolating the proposed
aging model. The accuracy of label transfer from a template is highly
influenced by the registration method and the global template labels being
used. Hence, efforts were taken to do a fair comparison as follows.

The Neurodev [21] templates were used to study both the aging models. A global
template $G$ was constructed with ANTs template creation algorithm [33].
Starting with a common labeled template [5], the labels were transferred to
the global template using DRAMMS non-rigid registration method [34]. The
registration algorithms and the parameter setting were fixed to be the same
for both aging models. The Dice score was used for assessing the segmentation
accuracy.

####  3.4.3 Validation with Simulated Longitudinal Data

The proposed method was aimed at handling cross-sectional data. In order to
understand how the model would handle longitudinal data, an experiment was
done using simulations as longitudinal data is unavailable. Simulated data was
generated as follows. A set ($S$) of fifty randomly deformed Shepp-Logan
phantoms were taken (to simulate a cohort) and five copies were made.
Deformations with increasing degree was applied on these five copies to
simulate aging of different subjects. The five sets thus form our longitudinal
data. For each of the five sets a template was computed separately using the
method suggested in [33]. The templates were then used as inputs for the
proposed model to generate templates at different age points. These were then
compared against the deformed versions of the Shepp-Logan phantom (proxy
ground truth).

##  4 Results

###  4.1 Estimation of the Aging Model

####  4.1.1 Compatibility with Natural Aging

Fig. 4 shows the generated sequential templates (first row) and difference
between the sequential templates and the first template (second row). The
difference images facilitate understanding the structural changes with aging.
The mean squared error or MSE (difference) is plotted in the third row. The
difference appears to be very low for the first few decades relative to the
last few decades where changes like ventricular expansion occurs. This trend
is consistent with the existing information about natural healthy aging.

Figure 4:  The proposed aging model at different time points(first row) along
with the difference image with respect to the initial time point(second row)

####  4.1.2 Growth Trend across Aging Models

Fig. 5, shows sample 2D slices of templates from [16] in odd numbered rows,
along with the ones derived with the proposed model (from the NEURODEV
dataset) in even numbered rows, for comparison. The comparison at image-level
comparison is not meaningful as the templates are generated from different
data-sets. However, one can observe growth trends. The structural similarity
across rows in a column appear to have similar trends across age indicating
growth trend to be consistent.

Figure 5: Correctness of Aging trends captured in the model: Publicly
available spatio-temporal images [16](row 1,3,5) compared with images
generated at same time points with proposed aging model using NEURODEV data at
different time points(row 2,4,6) Each highlighted row pairs compare same
slices as specified in the figure.

###  4.2 Age-specific Template Assessment

The given templates along with our generated templates are shown in Fig. 6 for
comparison. The first and last time points for the both BRAINS and NEURODEV
are shown in the top row, while the corresponding templates generated by the
proposed model are shown in the second row. As per the proposed aging model,
the template for the first and last time points are maximally deformed with
respect to the template closest to $G$, i.e., $T_{M}$. Yet, the derived
templates are visually quite similar to the templates from the two datasets.
Thus, the proposed model appears to preserve the structural details of the
given template at each time point.

Figure 6:  Comparison of templates given in the two datasets (BRAINS and
NEURODEV) (top row) with those generated by the proposed aging model (bottom
row). Only the templates for the first and last time-points are shown.

###  4.3 Aging Model Validation

####  4.3.1 Topology Preservation

Two templates, namely at age 20 and age 100, generated with the proposed model
are shown in Fig. 7 along with the Global template. These are results of
extrapolation from the data given in the NEURODEV dataset[21]. The topology
appears to be preserved even when the aging model is extrapolated in both
directions implying that the extrapolated deformations also belong to
$\mathcal{G}$. It can also been that while global similarity (in structure)
exists across age, local deformations persist. For instance, the ventricle is
much smaller at age 20 and enlarges with age, consistent with the expected
aging trend.

Figure 7: The central coronal slices of extrapolated age-templates are shown
along with the global template image

####  4.3.2 Validation with Segmentation

Table 2 lists the Dice scores for segmenting 4 pairs of sub-cortical
structures with our model and [23]. It can be observed that the proposed model
outperforms [23] as it has higher average (over 10 subjects) dice score. The
proposed model’s better performance in label transfer implies that it
generates most accurate age-point templates which is closest to each given
subject image age-point.

|  | Right
---
Hippo-
campus
| Left
---
Hippo-
campus
| Right
---
Pallidum
| Left
---
Pallidum
| Right
---
Putamen
| Left
---
Putamen
| Right
---
Thalamus
| Left
---
Thalamus
[23] | 0.6266 | 0.6229 | 0.6604 | 0.5803 | 0.6802 | 0.6444 | 0.6944 | 0.6917
| Proposed
---
model
0.6724 | 0.6900 | 0.7720 | 0.7266 | 0.7798 | 0.7880 | 0.7912 | 0.7682
Table 2: Model validation via segmentation. Average Dice scores computed over
10 subject images are listed.

####  4.3.3 Validation with Simulated Longitudinal Data

Sample images generated by applying the simulated deformations on the Shepp-
Logan phantom are shown in the first row of Fig. 8. This forms the ground
truth. The template images derived with the proposed model are shown in the
second row. The images in the 2 rows appear to be very similar to each other
at the same time points. The template images generated with the proposed model
and corresponding $\gamma$ curve is also shown in the same figure. The degree
of deformation in the simulated deformation is uniformly increasing with time
and hence it can be expected that the $\gamma$ curve will be symmetric with
respect to mid-time point. We see that, in Fig. 8, is indeed true. The
proposed model captures the applied deformation without much errors from the
simulated longitudinal data.

Figure 8:  Aging model for a simulated longitudinal dataset. A - First row:
Deformed images with known transformation and second row: images generated
with the proposed model, for the same time points (as in the first row) and
last row: the absolute difference of first and second rows; B - The $\gamma$
curve of our aging model and C - Some sample images used in $S$

##  5 Discussion

Cross-sectional images at different age points are easier to acquire than that
of the same subject. This motivated us to develop a method to generate an
aging model using cross-sectional data. The aging model is based on continuous
deformation applied to a template. Experimental results show that our aging
model can be used to generate templates at different time points in a manner
that is consistent with the natural aging trend observed by other studies; it
preserves structural details of the supplied templates and generates topology-
preserving aging deformations.

###  5.1 Limitations

The proposed aging model has a few limitations. Firstly, it is applicable only
for matured brain growth where no new brain structures are introduced. The
quality of the proposed model is completely dependent on the data.
Consequently, the number of scans in each age interval needs to be large to
generate the age-specific templates that are representative of the
cohort/population under study. What constitutes ’large’ is an open question.
Secondly, while the model reduces the effect of cross-sectional data induced
variations in the aging deformation, there is no formal proof as yet that it
completely removes the cross-sectional variation. Finally, the proposed aging
model defines a single average growth path and does not attempt to model the
cross-sectional aging variations.

###  5.2 Future Work

Since the proposed model works only to obtain a mean aging path, future work
can be a refinement in terms of defining the aging model as a distribution of
paths about the average path. The basic requirement to develop such a model
however, is the availability of scans at different age points, not the
templates alone. Our current work is directed at developing a public database
for this purpose with subject scans at different age points.

##  6 Conclusion

We have proposed a novel aging model from cross-sectional data. The spatio-
temporal smoothness and consistency are assured in the model to make it closer
to natural aging. The model has the potential to be used for clinical
purposes. Currently population specific aging trends are of interest and this
can be generated with the proposed model with less efforts. The code to
generate proposed the aging model has made publicly available in
http://dx.doi.org/10.17632/nw983x225c.1.

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