Our goal is to identify differences in spatiotemporal atrophy patterns characteristic for healthy and AD-related brain aging. Therefore, we formulate a multiphysics approach that couples mechanics-driven volume loss and the biology-driven spreading of toxic proteins (Weickenmeier et al., 2018). In our constitutive model, we pose that healthy aging is linked to a steady volume loss in gray and white matter tissues, while AD accelerates atrophy proportional to the local toxic protein level (Schäfer et al., 2019). We solve our continuum problem on an anatomically accurate finite element (FE) brain model and quantify hallmark features of cerebral atrophy including volume loss, cortical thinning, ventricular enlargement, and sulcal widening.

We implemented our continuum model in the finite element software Abaqus (Simulia, Providence RI) and solved our coupled problem as a thermo-mechanical analysis. We add the nonlinear source term of the protein spreading equation (Raz and Rodrigue, 2006) to the standard heat transfer problem using the subroutine HETVAL which requires the flux, f c = α   c   [ 1 − c ] and rate of change of heat flux per temperature, d f c / d c = α [ 1 − 2 c ] . Similarly, we incorporate our constitutive material model using the user subroutine UMAT which requires Cauchy stress and its Jaumann rate. To determine Cauchy stress at the integration point level, we calculate the atrophy factor via a finite difference scheme, ϑ ˙ = ϑ − ϑ n Δ t ,       such that       ϑ = ϑ n + [ 1 + γ ( c ) ] G h Δ t , (11) where ( ∘ ) and ( ∘ ) n denote the unknown quantity at t = t n + 1 and the converged quantity at the previous time step t = t n , respectively, and Δ t = t − t n > 0 is the current time increment. Here, we approximate the Heaviside step function ℋ in γ ( c ) (Eq. 10) as a smooth function, ℋ ( c − c crit ) = 1 1 + exp ( β ( c − c crit ) ) , (12) where β controls the transition between the two states. We store the converged atrophy factor as a state variable for post-processing, then calculate the atrophy part and the elastic part of the deformation gradient F a and F e (Eq. 4). We then calculate Cauchy stress, σ = J − 1 PF T , and its Jaumann rate, c abaqus = c + 1 2 [ σ ⊗ ¯ I + I ⊗ ¯ σ + σ ⊗¯ ¯ I + I ⊗¯ ¯ σ ] , (13) with the consistently linearized tangent stiffness matrix, c , c = 1 J e [ I ⊗ F e ] : ∂ 2 ψ ∂ F e ⊗ ∂ F e : [ I ⊗ F e T ] , (14) where we used the tensor operators { • ⊗ ¯ ∘ } i j k l = { • } i k ⊗ { ∘ } j l and { • ⊗ _ ∘ } i j k l = { • } i l ⊗ { ∘ } j k .

We created an anatomically accurate FE brain model from T1-weighted magnetic resonance images of a healthy adult male brain. We used ScanIp from Simpleware (Synopsis Inc., Mountain View CA) to semi-automatically segment the regions of interest and generate the FE mesh. Our model differentiates between gray matter (GM), white matter (WM), the hippocampus, ventricles, and cerebrospinal fluid (CSF). Figure 2A) shows representative sagittal, axial, and coronal MRI slices of the subject’s brain, as well as the volumetric reconstructions of the respective substructures. We built our model sequentially and began segmentation with reconstruction of the ventricles, followed by WM, GM, and finally CSF. We avoided reconstructing the skull by defining zero-displacement Dirichlet boundary conditions on the peripheral surface of CSF. Here, we merged the lateral ventricles, third ventricle, and fourth ventricle into a single volume in order to quantify ventricular enlargement, one of the hallmark features of brain aging. We paid close attention to the segmentation of WM tissue to accurately capture individual sulci and gyri across all lobes. To realistically simulate cortical thinning and sulcal widening, we must prevent self-contact of the cortical layer. Therefore, we inflated the WM segmentation by a constant thickness of 3 mm to obtain the GM layer. We then manually modified the GM layer to remove self-contact between lobes and folds in each slice. Ultimately, we aimed for a balance between agreement of segmentation and MRI on the one hand, and obtaining a FE mesh that may realistically predict structural shape changes of the brain on the other. Following WM and GM segmentation, we isolated the hippocampus as a separate substructure, given its relevance in AD as one of the first brain structures to markedly shrink. Finally, we inflated the GM layer by 5 mm and applied smoothing to obtain the CSF layer. This layer allows us to anchor the brain in our atrophy simulations while minimizing external forces on the GM layer.

Model Properties: Our model consists of 1,361,277 tetrahedral elements: 7,925 elements for the ventricles, 2,898 elements for the hippocampus, 121,904 elements for WM, 172,238 elements for GM, and 98,755 elements for CSF. We restricted element edge length to vary from 2.0 to 2.3 mm to minimize element distortion and obtain similarly sized elements. We imported the mesh into Abaqus for analysis. Specifically, we use linear tetrahedral elements C3D4 and define two simulation cases. We simulate healthy aging by simply solving the atrophy problem and simulate accelerated aging by running a thermo-mechanical analysis. In both cases, we only prescribe zero-displacement Dirichlet boundary conditions to the outer surface of the CSF layer to fix the model in space. In the AD case, we additionally prescribe an initial concentration of c 0 = 0.3 in the hippocampus. We used model parameters from our previous experimental and computational studies (Schaer et al., 2008; Weickenmeier et al., 2018; Weickenmeier et al., 2016) and summarize the model parameters for the atrophy and protein problem (Eq. 1, Eq. 5, Eq. 10) in Table 1. To assess long-term brain shape changes we simulate an age range of 40 years. Literature provides a myriad of large cohort studies that assess volumetric changes across this age-range (Apostolova et al., 2012; Coupé et al., 2019). Moreover, this allows us to review the impact of AD-onset time by varying the critical prion load necessary to trigger accelerated aging.

We wrote custom python codes for post-processing of our simulations in order to determine volume ratios, anterior-posterior variations of the gyrification index, sulcal widening, and cortical thinning.

To calculate relative volume ratios of WM, GM, hippocampus, and ventricles, we sum the volume of all elements belonging to one of these subregions and divide by the total brain volume; we repeat this step for each time increment to obtain longitudinal changes as shown in Figure 7.

The gyrification index (GI) is determined by slicing our 3D model into 160 coronal slices (1 mm spacing between slices) and creating a binary image showing the domain associated with brain tissues, i.e., GM, WM, hippocampus, and ventricles wherever present. The subsequent steps are based on functions in the scikit-image processing package. Specifically, we determine the convex hull that fully encapsulates the brain domain to obtain the smoothed outer circumference and extract the contour tightly lining the pial surface. We repeat this process for each slice and determine the gyrification index as the local ratio between exact pial surface length and smooth outer circumference, as shown in Figure 11.

Our cortical thickness measurement is based on the approach used in FreeSurfer (http://surfer.nmr.mgh.harvard.edu) (Han et al., 2006). We create triangulated surfaces of the outer GM surface and the outer WM surface and define cortical thickness t c as the average of two distance measures, d i j and d j k . We iterate over every node of the GM surface, n i , identify the closest node on the WM surface, n j , and save the Euclidian distance between these two nodes as d i j . We repeat this search for that particular WM node, n j , and save the Euclidian distance between n j and GM node n k as distance d j k . We ultimately obtain a cortical thickness measure at each GM surface node as t c = 0.5 [ d i j + d j k ] and plot the result as a surface plot, as shown in Figure 8. We export nodal coordinates of our surfaces in the undeformed and the deformed configuration in order to determine cortical thickness at a young and an old age.

We introduce sulcal widening as the volume increase in the fluid-filled cavity of five prominent sulci, i.e., the intra-parietal sulcus, the superior temporal sulcus, the central sulcus, the sylvian fissure, and the superior frontal sulcus, as shown in Figure 10. Similar to determining the relative volume fractions, we sum the volume of all elements of a particular sulcal fold for each time increment of our simulation.

We simulate the spreading of neurofibrillary tangles (NFT) consisting of misfolded tau protein based on the toxic protein spreading model described in §2.1. Pathological studies have shown that NFTs first appear in the entorhinal cortex and subsequently spread throughout the brain. Figure 3 shows the spatiotemporal propagation of the NFT concentration through the brain. We observe that the hippocampus is affected first, then infiltrates the temporal lobe next, followed by the parietal lobe, occipital lobe, and in the late stages reaches the frontal lobe. Our observations are in line with cadaver studies that show a similar progression pattern of NFTs (Jucker and Walker, 2018). The coronal view shows a highly symmetric protein spread in the left and right hemisphere; from the axial and coronal cross-sections, it can be seen that deep gray matter structures tend to saturate with NFTs first. Early deep gray matter involvement, such as putamen and thalamus (de Jong et al., 2008), is linked to well-known early symptoms of AD, including short-term memory loss, difficulty performing daily tasks, and mood changes. The delay between onset and cortical layer involvement is part of the long pre-symptomatic phase of AD (Hanseeuw et al., 2019) and consistent with imaging studies that observed spatially heterogeneous atrophy patterns (Anderson et al., 2012).

The atrophy model allows us to differentiate between healthy and AD aging. On top of an age-proportional atrophy factor in healthy aging, we added additional toxic protein concentration-related atrophy to simulate AD. Figure 4 shows the spatiotemporal distribution of the atrophy factor, i.e. the volume shrinking fraction, which ranges from 1 (no shrinking) to 0.8 (maximum volume loss). We differentiate between WM and GM atrophy rates due to tissue specific neurodegenerative processes. Therefore, GM and WM have the same atrophy factors in healthy aging, respectively. In AD, we see a spatially heterogeneous distribution with maximum atrophy in deep WM and GM structures and in the frontal lobe. The coronal view shows that the cortex exhibits an atrophy gradient that ranges from the temporal lobe to the frontal lobe; in WM we observe a gradient ranging from the temporal lobe to the parietal lobe. Both are consistent with imaging studies investigating regional atrophy rates in the cortex (McDonald et al., 2009).

Figure 5 shows the temporal progression of the predicted deformation field and corresponding equivalent structural image for healthy aging and AD for representative axial and coronal sections. We observe maximum displacement magnitudes of 7.17 mm for healthy aging and 8.58 mm in AD. Maximum displacements concentrate around the lateral ventricles which undergo significant enlargement, especially in the AD brain. In comparison to the atrophy factor, which affects the hippocampus first, ventricles, and the surrounding white and gray matter regions, appear to deform early, followed by cortical deformations. For late stages we observe higher displacement magnitudes for the GM layer in comparison to deep white matter structures. The structural scans reveal hallmark features of cerebral atrophy: hippocampal shrinking, early onset of deep GM shrinking, cortical thinning, and ventricular enlargement. We generally observe that these features are exacerbated in AD in comparison to healthy aging. These observations are strongly correlated with medical imaging based studies that observe hippocampal shrinking, cortical thinning, and ventricular enlargement as early predictors for AD (Apostolova et al., 2012). Previous computational studies typically prescribe a zero-boundary condition on nodes of the brainstem in order to fix the model in space (Harris et al., 2019; Schäfer et al., 2019). These boundary conditions significantly impact the simulated deformation field and limit these models’ abilities to resolve temporospatial patterns or critical features such as ventricular enlargement. Here, the cerebrum is loosely tethered to the skull via the ultrasoft CSF layer which allows for physical features to emerge naturally. Strikingly, we observe global brain involvement despite scattered atrophy features.

Figure 6 shows the gradual expansion of the lateral ventricles for healthy aging and AD. We observe significantly larger ventricles in AD, which increase by a factor 2.66, in comparison to healthy aging, where ventricles increase by a factor 1.76. The simulation predicts a predominantly uniform inflation of the entire ventricular cavity in healthy brain aging at a moderate expansion rate. In AD, we observe consistent overall ventricular dilation, but notice a significant concentration of maximum expansion in the body of the ventricles and the posterior horns. This observation is consistent with a medical imaging study that reported a temporal pattern that starts in the occipital horn, then affects the body, and ultimately reaches the frontal horns (Apostolova et al., 2012). The sagittal view of the brain shows the corresponding white and gray matter loss. As the ventricles expand, we observe a smoothing of the superior horn, temporal horn, and occipital horns with an overall decrease in curvature of the ventricular surface.

Cerebral atrophy is caused by diverse tissue damage mechanisms that culminate in brain volume loss (Oschwald et al., 2020; Blinkouskaya et al.). While healthy aging and AD share some of the gray and white matter damage mechanisms there is a distinct point during the lifespan where the atrophy trajectory in AD diverges from the healthy model due to accelerated neurodegeneration (Callaghan et al., 2014; Coupé et al., 2019). Most common damage mechanisms are neurodegeneration in GM (Farokhian et al., 2017), demyelination in WM (Vernooij et al., 2008), activation of microglia cells (Von Bernhardi et al., 2015), and cerebral small vessel disease which is associated with microbleads, lacunes, and perivascular spaces (Cuadrado-Godia et al., 2018).

In gray matter, neurons undergo morphological changes linked to a reduction in the complexity of dendrite arborization (Dickstein et al., 2007). The underlying dendritic shortening and loss of dendritic spines leads to a progressive decrease in synaptic density and synaptic transmission with major implications on cognitive decline (Dickstein et al., 2013). Unlike healthy aging, AD is accompanied by neuron death due to the ever-increasing presence of neurotoxic proteins such as amyloid beta plaques and neurofibrillary tangles (Serrano-Pozo et al., 2011). GM volume loss is therefore exacerbated in AD and manifests in accelerated atrophy rates (Anderson et al., 2012) and increased cortical thinning (Du et al., 2007). It is well established today that the very first morphological changes associated with AD appear in the entorhinal cortex and hippocampus at least 10 years before the diagnosis (Dickstein et al., 2007).

In WM the most prevalent tissue changes are characterized by partial loss of myelin, axons, and oligodendroglial cells (Xiong and Mok, 2011); mild reactive astrocytic gliosis linked to WM lesions (Rodríguez-Arellano et al., 2016); arteriolosclerosis of small vessels resulting in incomplete ischemia and cell death (Pantoni, 2002); and the emergence of perivascular spaces that interfere with the glymphatic drainage of the brain’s waste products (Rasmussen et al., 2018; Wardlaw et al., 2020).

During normal aging, amyloid beta plaques can be found in the frontal lobe, hippocampus, and entorhinal cortex of healthy elderly. In addition, neurofibrillary tangles, although much rarer than plaques, are commonly found in the medial temporal areas after 50 years of age (Dickstein et al., 2007). In AD, however, the progressive aggregation of plaques and NFTs has detrimental effects on neuronal morphology and synapses. Unlike in normal aging when neurons shrink, AD triggers sustained neuronal loss in neocortical and entorhinal regions of up to about 30% (Mattson, 2004).

Figure 7 shows brain volume fractions of GM, WM, and ventricles representative of a brain aged 40 years and older. We extracted atrophy data from Coupe et al. who identified volume changes from a cross-sectional study with 4,329 subjects (2,944 healthy subjects and 3,262 subjects with AD and mild cognitive impairment) (Coupé et al., 2019), see dashed lines. We focus on brain aging and calibrate our model parameters such that our model provides good qualitative agreement for healthy brain aging, (Figure 7A). Our model successfully reproduces GM and WM volume loss and ventricular enlargement. The offset between GM, WM, and ventricular volume fractions is due to comparison of a personalized brain model with cross-sectional data. More importantly, the numerically observed atrophy trajectories paint a representative picture that demonstrates the ability of our modeling approach to predict shape changes associated with brain aging. Our model predicts GM volume fraction to drop from 52.36% at age 40 years to 50.49% at age 80 years in healthy aging and 49.34% in AD; WM volume fraction to drop from 47.63% at age 40 years to 40.29% at age 80 years in healthy aging and 32.95% in AD; ventricular volume fraction increases from 3.22% at age 40 years to 5.66% at age 80 years in healthy aging and 8.57% in AD. AD clearly exacerbates tissue loss and exhibits an accelerating atrophy rate with increasing age, (Figure 7B). Tissue lost due to atrophy is replaced by fluid (volume fraction shown in grey) linked in one part to ventricular enlargement and in another part to sulcal widening and loss of gyrification (Scahill et al., 2003).

The cortical layer is subject to spatially heterogeneous age-related cortical thinning. The deterioration of dendritic connections and the loss of GM neurons cause volume loss that can be broken down into cortical thickness and surface area. These two properties do not necessarily follow each other chronologically (Dickerson et al., 2009). The differentiation between both measures has proven useful, however, because of increased sensitivity with respect to age-related changes (Storsve et al., 2014; Dotson et al., 2016). In our computer model, we observe a mean cortical thickness of 2.79 mm in the young brain and 2.64 mm in the aged brain. In Figure 8, we report our model’s brain thickness which ranges from 1.5 to 4.3 mm in the young brain and decreases to a range from 1.3 to 3.9 mm in the aged brain. These values compare well to results presented by Fjell et al. who observed a progressive decline in overall cortical thickness in their three subject groups aged < 40, 40–60, and > 60 (Fjell et al., 2001; Fjell and Walhovd, 2010). They report that sulci undergo more pronounced thinning than gyri and that thinning is unevenly distributed across the cortex. Based on data extracted from Fjell et al., the cortex appears to thin by roughly 0.1% per year, or 0.00745 mm, which corresponds to an overall thickness decrease of about 0.3 mm over the course of 4 decades for subjects aged > 40 (Fjell et al., 2015). The linearly decreasing relationship between cortical thickness and age across several datasets provides strong support for our modeling approach which assumes a constant atrophy rate for all ages (Fjell et al., 2001; Du et al., 2006). Despite significant efforts to identify common thinning trajectories in the human brain, cortical thinning is driven by molecular and cellular processes that are not limited to individual regions. Cross-sectional studies report that the frontal cortices are most strongly affected and that the medial-temporal cortices, i.e., parahippocampal and entorhinal cortex, are moderately affected. Lateral inferior parts of the temporal lobes show least thinning and the superior parts of the lateral temporal lobes exhibits more pronounced thinning than the inferior parts (Fjell et al., 2001; Fjell and Walhovd, 2010). In our model, we observe slightly higher thinning in the frontal and temporal region, while the occipital lobe thins less. In aging research the temporal lobes play a significant role because they are functionally related to the hippocampus and other GM structures that are associated with memory loss and cognitive decline (Dickerson et al., 2009; Dhikav et al., 2014). In the end, our model leads to fairly similar cortical thinning across the entire brain due to the prescribed constant GM atrophy rate. Coupling to the spreading of neurotoxic proteins may lead to a stronger heterogeneity in terms of thinning.

The hippocampus is one of the, if not, the earliest cortical substructures to undergo detectable atrophy in Alzheimer’s disease and related dementias (Henneman et al., 2009). Hippocampal changes can be detected as early as 10 years prior to the onset of symptoms and is therefore considered to be a strong indicator for abnormal aging processes (Ritchie et al., 2016; Kinnunen et al., 2018). Hippocampal shrinking precedes most cortical changes by up to 5 years and is reported to shrink by 5.2% per year based on data from cross-sectional brain imaging studies (Thompson et al., 2004; Henneman et al., 2009). It is primarily linked to de-arborization of subcortical GM neurons (Esiri, 2007; Dickstein et al., 2013). In comparison to healthy aging, Alzheimer’s disease accelerates neuronal degeneration due to accumulation of neurotoxic amyloid beta plaques and neurofibrillary tangles (Bobinski et al., 1999). Figure 9 shows our model’s predicted volumetric shrinking for healthy aging and AD. We observe a decrease of the hippocampal brain volume fraction by 8.87% for healthy aging and by 24.1% for AD. The direct comparison illustrates the distinct difference in the atrophy trajectory in accelerated aging in AD observed in cross-sectional studies (Coupé et al., 2019).

The brain tissue volume lost due to cerebral atrophy, is replaced by fluid. Structurally, this manifests in significant ventricular enlargement (Pagani et al., 2008; Apostolova et al., 2012) and an increase in the space between folds, i.e., sulcal widening (Liu et al., 2013; Jin et al., 2018). Ventricular enlargement is one of the most prominent features in longitudinal medical images and represents a major change in brain topology (Sengoku, 2020). Mechanically, the extent of ventricular enlargement is significant and will lead to high loads on the membrane separating ventricle and cerebrum. The ependymal cells lining the ventricular wall are likely to be fatigued with age, leading to CSF leakage into white matter and causing tissue degeneration, such as leukoaraiosis in the vicinity of ventricular horns (Milhorat et al., 1970; Todd et al., 2018). Our model predicts a uniform volumetric expansion of the entire ventricles which is reflective of findings from imaging studies (Salat et al., 2009; Coupé et al., 2019). Our simulation is able to reproduce this deformation mode due to our physically motivated boundary conditions on the FE model. Instead of constraining individual nodes in the brainstem (Harris et al., 2019; Schäfer et al., 2019), here, we suspend the brain inside the skull by mimicking CSF as an ultrasoft, highly compressible solid. The suspension of the shrinking cerebrum allows for the ventricles to expand. This leads to a fairly symmetric displacement field with respect to the left and right hemisphere. In our model, the initial ventricular volume corresponds to 2.37% of the total intracranial volume. In our simulation, we observe an increase to 4.15% of total intracranial volume, or a 75.03% volume increase in healthy aging; In AD, ventricular volume fraction increases to 6.28%, or an overall volume increase by 164.98%. Our data aligns well with data reported by Coupe et al. that observe significant acceleration of ventricular expansion at age 40 (Coupé et al., 2019). Microstructurally, ventricular expansion is accompanied by a progressive deterioration of the ventricular wall which is composed of ciliated ependymal cells that undergo significant cellular stretch during each pulsation cycle. Over the course of a lifetime, these cells accumulate significant mechanical fatigue and cause membrane failure (Milhorat et al., 1970; Jiménez et al., 2014). The subsequent leakage of CSF into white matter tissue causes leukoaraiosis and white matter deterioration.

Ventricular enlargement is accompanied by an increase in the space between folds and loss of gyrification (Hamelin et al., 2015; Aso et al., 2020). This feature is less prominent on medical images, but is another indicator for the significant topological changes of the brain (Plocharski et al., 2016; Shen et al., 2018). From a FE modeling perspective, creating an anatomically accurate mesh that properly capture sulcal widening represents a major challenge. Most folds touch each other such that the segmentation process typically does not produce a GM surface without self-contact. This leads to node sharing of elements that belong to different folds and ultimately, prevents models to allow for separation of the GM surface upon tissue atrophy. Here, we specifically address this issue and produced a FE mesh that has minimal node sharing between neighboring folds. Therefore, our model exhibits this hallmark feature of cerebral atrophy and allows us to compare model response with imaging data. Jin et al., for example, report that the mean sulcal width between primary sulci increases by ∼ 17.3% from 1.27 ± 0.17 mm in middle-aged persons to 1.49 ± 0.20 mm in older adults (71). In Figure 10 we report sulcal widening, a measure of the volume increase of the fluid between folds. We segment these volumes for five prominent sulci, the intra-parietal sulcus, the superior temporal sulcus, the central sulcus, the sylvian fissure, and the superior frontal sulcus (Kochunov et al., 2005; Liu et al., 2013). We observe that the overall volume change of all sulci follow a similar trend and increase by up to 40%. Similar to previous work, the sylvian fissure exhibits the largest increase in width and is noticeably larger in individuals with AD in comparison to cognitively normal subjects (Park et al., 2013; Cai et al., 2017). Overall, we observed that the technical challenges associated with detailed geometric interpretation of sulcal changes, such as sulcal widening and changes in sulcal depth, represent a barrier to serving as a reliable biomarker for morphological changes in the aging brain. Especially, subject-specificity will limit absolute comparisons with any healthy or diseased cohort (Shen et al., 2018).

The gyrification index (GI), defined as the ratio between actual GM surface divided by the smooth surface surrounding the cortex, is another parameter that is closely linked to the topology of brain folds (Madan, 2021). In Figure 11 we show the gyrification index for 164 coronal slices calculated for the healthy young brain, healthy aged brain, and the brain affected by Alzheimer’s disease. The GI is highest across the brain for the young brain. With aging or AD, the GI decreases due to decreased folding. We observe the highest GI in the temporal lobe with 3.28 for young, 3.27 for aged, and 3.19 for the AD brain; minimum GI is observed in the frontal lobe with 1.22 in young, 1.06 in aged, and 0.64 in the AD brain. We observe a mean GI of 2.48 ± 0.38 in the young, 2.42 ± 0.4 in the aged, and 2.32 ± 0.44 in the AD brain. The most prominent and persistent drop in GI is observed in the temporal and parietal lobes which are heavily affected by early infiltration of our neurotoxic biomarker and corresponding accelerated atrophy. Our reported values compare well with cross-sectional studies reported in literature (Jockwitz et al., 2017; Madan, 2021). In a cross-sectional study by Cao et al., the GI drops from 3.4 at age 10 to 2.6 at age 85, following the curve GI = a + b   ln ( A + c ) , with age A and parameters a = 3.4, b = −0.175, and c = −2.9991 (Cao et al., 2017). According to this formula, GI drops from 2.8 at age 40 to 2.6 at age 80, or by 4.5% between ages 40 and 80. Our model predicts a 2.7% change for the most folded coronal slice.

Our computational model is based on several assumptions and thus not without limitations. For example, when creating the FE mesh, we uniformly inflate the WM surface to create a GM layer which results in a fairly homogeneous GM thickness across the brain. In reality, the gray matter layer is characterized by thickness differences between sulci and gyri (Lin et al., 2021) and varies across the brain (Fischl and Dale, 2000). We chose this approach due to the necessity to avoid self-contact between GM folds in order to capture sulcal widening during atrophy. Furthermore, our current constitutive model differentiates between GM and WM atrophy rates, but assumes a uniform parameter across the brain. Cross-sectional studies have demonstrated significant regional variation in brain shrinking rates in healthy aging and AD (Fox and Schott, 2004; Fjell et al., 2014). The coupling of biomarker concentration and atrophy rate in our model introduces, however, a degree of heterogeneity that exacerbates spatiotemporal differences between healthy aging and AD. Our model shows good agreement with cross-sectionally observed image-based atrophy patterns. Going forward, there is a need to develop a validation approach that allows to calibrate model parameters against longitudinal imaging data of individual subjects (Rusinek et al., 2003). To that end, we will develop a non-rigid registration technique that delivers the full-field displacements of the brain between two images (Wang et al., 2021). And lastly, AD is characterized by two different protein spreading mechanisms: connectivity-based spread via intracellular diffusion of neurofibrillary tangles along the axon network and proximity-based spread of amyloid beta via extracellular aggregation of plaques (Jack and Holtzman, 2013). Here, we only consider isotropic diffusion through the bulk tissue. As a next step, we will integrate the diffusion tensor imaging-based tractome to more accurately represent intracellular spreading of tau which has shown to better correlate with neurocognitive decline (Raj et al., 2015).