Multiscale Modeling of Dementia: From Proteins to Brain Dynamics

At first, the signs were subtle; only a life of intimacy allowed his wife to notice the changes. At age 57, he had difficulty tying his necktie, could not properly handle the house finances, and seemed withdrawn. As these early lapses worsened, Mr. U, now aged 61, underwent a general neurological examination that appeared normal. However, a basic psychological test showed delayed recall and problems with episodic memory. A magnetic resonance imaging (MRI) scan revealed broad cerebral atrophy and—together with advanced psychological tests—delivered the devastating diagnosis of probable Alzheimer’s disease. From then on, Mr. U experienced a well-known pattern of decline. He struggled with abstract reasoning, memory functions, and word retrieval, and could not properly visualize the world around him. At age 65, Mr. U exhibited global deterioration in his cognitive abilities — particularly verbal memory, language, motor control, face recognition, and executive skills. Another MRI confirmed extended bilateral cerebral atrophy. Bouts of apathy, psychosis, and violent outbursts became more frequent.

At age 70, Mr. U was a pale shadow of the man he once was. His wife could not manage his condition anymore and he was admitted to a nursing home. He died three years later from pneumonia.

A Protein Story

Mr. U’s experience is unfortunately not unique, as the overall pattern of degeneration in Alzheimer’s patients is fairly homogeneous and systematic. The disease affects more than 50 million people worldwide and occurs in roughly one out of every nine adults over the age of 65. Unlike cancer and many viral and bacterial diseases, Alzheimer’s disease typically presents in codified stages that arise in most patients. This cognitive staging is associated with a systematic invasion of two key toxic proteins in the brain, which Alois Alzheimer identified in 1905: the well-known amyloid beta and the group of tau proteins that normally stabilize microtubules in axons. 

Scientists believe that a misfolded version of tau acts as a template for healthy tau proteins and promotes the formation of increasingly sized oligomers, eventually leading to large aggregates that are evident in brain tissues after death [4]. While most drug trials focus on amyloid beta, the presence of toxic tau proteins is primarily correlated with brain atrophy and cognitive symptoms. The typical pattern of tau evolution—as measured through histological staining—is called Braak and Braak staging after neuroanatomists Eva and Heiko Braak, who first proposed the mechanism in 1991 [2]. It describes the disease’s evolution in six stages, beginning in the entorhinal cortex and evolving through the hippocampal region (which is associated with memory), the temporal lobe, the occipital lobe, and all regions of the neocortex. The lesions in each region worsen with time.

Alzheimer’s disease is rather intimidating from a modeling perspective. Nevertheless, the systematic pattern of invasion through the brain—and the resulting cognitive effects—suggest that some simple underlying features of the brain may be responsible for the spatiotemporal pattern. Together with Ellen Kuhl of Stanford University, I therefore sought to obtain minimal mathematical models based on clear principles that can capture staging at the brain level as well as other characteristics of the disease, such as brain atrophy and changes in overall brain dynamics.

Fisher-KPP Strikes Again

From a phenomenological point of view, we must consider three processes: transport, expansion, and saturation. During transport, toxic proteins move throughout the brain primarily along axonal bundles, which connect different regions and act as information highways. Expansion refers to the autocatalytic nature of the aggregation process, which leads to an initial exponential increase of small toxic populations. And saturation means that each region can only support a certain level of toxic proteins. A canonical model for such a process in a continuum medium is the celebrated Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) equation. If \(c(x,t)\in[0,1]\) is the scaled concentration of the toxic protein, the equation reads

\[\frac{\partial c}{\partial t}=\nabla\cdot(\textbf{D}\nabla c)+\alpha c (1-c).\tag1\]

Here, \(\alpha>0\) is the growth rate and \(\textbf{D}\) is a transversely anisotropic diffusion tensor with a strong preferential direction along the axonal bundle. Under generic conditions, we can obtain this model as a normal form of the full aggregation-fragmentation equations that track the evolution of differently sized oligomers [7]. The parameters can hence directly relate to the aggregation and clearance rates at the microscopic level.

After using MRI scans to obtain both the full brain geometry and the direction of axonal bundles, we began with an initial seed of toxic proteins in the entorhinal cortex and simulated the evolution of the brain’s field (see Figure 1). The results were striking. Without any other ingredients, the dynamics that we obtained from this minimal model recovered the disease’s basic dynamics in great detail [8]. When we apply the same model to other neurodegenerative diseases that are associated with different toxic proteins—such as α-synuclein for Parkinson’s disease or TDP-43 for amyotrophic lateral sclerosis—it also reproduces their basic spatiotemporal patterns as well as the atrophy patterns obtained from finite-element simulations; the only difference is the region and extent of seeding. Despite the complexity and diversity of these diseases, universal features of progression emerge from the combination of an autocatalytic process and transport in an anisotropic medium.

The Role of Topology

When such strong universal patterns appear, a natural question emerges: What essential features are responsible for the invasion pattern? Since we cannot further simplify the autocatalytic dynamics, we investigated the transport term and decided to test our hypothesis that the observed patterns are a consequence of the strong transport anisotropy along the axonal bundles. These bundles serve as more than information highways; they also efficiently carry toxic proteins across the brain. We can assess this idea by coarse graining the brain and considering multiple regions and the connections between them. We then replace the full brain with a network called the connectome, wherein each node is a region and edges represent the possible connections between regions. This approach constitutes the basis for a large field of neuroscience, and various groups have successfully used the connectome to examine the effects of protein diffusion in the brain [6].

The natural discretization of the diffusion operator on a network is the weighted graph Laplacian \(\mathbf{L}\), whose weights are proportional to the number of connections and inversely proportional to the distance squared between nodes. We then define \(c_i(t)\) as the concentration of toxic proteins at node \(i\), which establishes the discrete Fisher-KPP as

\[\dot c_i = -\rho \sum_{j=1}^N L_{ij} c_j+\alpha c_i (1-c_i),\quad  i=1,\ldots,N.\tag2\]

This system of nonlinear ordinary differential equations is an excellent approximation of the dynamics that are generated by the full nonlinear partial differential equations \((1)\) with which we started [3]. Mathematically, there are two regimes of interest depending on the ratio \(\rho/\alpha\). In the diffusion-dominated regime \((\rho/\alpha\gg 1)\), the system behaves mostly homogeneously and the concentration of toxic proteins uniformly increases in all regions (as evidenced in the propagation of amyloid beta). However, our systematic study of data from the Alzheimer’s Disease Neuroimaging Initiative used hierarchical Bayesian parameter inference to demonstrate that the evolution of tau proteins occurs in the growth-dominated regime \((\rho/\alpha\ll1)\), where the primary seed invades each region. In this regime, we can implement a systematic nonlinear perturbation method to obtain an approximation of the solution (see Figure 2). We can also attach a metric to the graph based on the dynamics that provides a natural notion of propagation times between different regions [5]. Combining analytical and computational results revealed that brain topology strongly constrains the dynamics in the early stages of Alzheimer’s disease; in the latter stages, a balance of protein kinetics and geometry controls the dynamics.

Despite its simplicity, our network model serves as a basic starting point to study many different aspects of Alzheimer’s disease and test possible mechanisms. In more recent work, we considered local variations in parameters that are associated with brain inhomogeneity; identified topological signatures of the disease in graph space; and studied the coupling between amyloid beta and tau proteins, the role of clearance in the initiation and dynamics of Alzheimer’s, and the interactions between the microvasculature and toxic proteins. In addition, we analyzed the perplexing brain activity dynamics in patients who typically show periods of hyperactivity followed by hypoactivity and a shift in brain wave frequencies. When coupled to so-called neuronal mass models for brain activity, the same model allowed us to test multiple hypotheses and conclude that local damage to particular groups of neuronal cells is most likely responsible for these observations [1].

Modeling Matters

The difficulty of modeling Alzheimer’s disease stems from its combination of multiple effects at different scales—from individual proteins to the entire brain—with interactions that transpire over decades in a highly complex environment. Yet this scenario is exactly the playground of modern applied mathematics. Models help us test hypotheses and identify key mechanisms that lead to possible therapeutic targets. In particular, we can match progression models to imaging data at the macroscopic level, to the parameters that enter aggregation-fragmentation models at the microscopic level, and to clearance rates at the cellular level. Such multiscale, multiphysics interactions are key to understanding and treating diseases like Alzheimer’s. At the mathematical level, they represent a formidable and exciting intellectual challenge that synthesizes fields such as nonlinear partial differential equations, networks, dynamical systems, continuum mechanics, mathematical neuroscience, and topological data analysis.

References

[1] Alexandersen, C.G., de Haan, W., Bick, C., & Goriely, A. (2022). A mechanistic model explains oscillatory slowing and neuronal hyperactivity in Alzheimer’s disease. Preprint, bioRxiv.
[2] Braak, H., & Braak, E. (1991). Neuropathological stageing of Alzheimer-related changes. Acta Neuropathol., 82(4), 239-259.
[3] Fornari, S., Schäfer, A., Jucker, M., Goriely, A., & Kuhl, E. (2019). Prion-like spreading of Alzheimer’s disease within the brain’s connectome. Interface Roy. Soc., 16(159).
[4] Jucker, M., & Walker, L.C. (2013). Self-propagation of pathogenic protein aggregates in neurodegenerative diseases. Nature, 501(7465), 45-51.
[5] Putra, P., Oliveri, H., Thompson, T., & Goriely, A. (2022). Front propagation and arrival times in networks with application to neurodegenerative diseases. SIAM J. Appl. Math., to be published.
[6] Raj, A., Kuceyeski, A., & Weiner, M. (2012). A network diffusion model of disease progression in dementia. Neuron, 73(6), 1204-1215.
[7] Thompson, T.B., Meisl, G., Knowles, T.P.J., & Goriely, A. (2021). The role of clearance mechanisms in the kinetics of pathological protein aggregation involved in neurodegenerative diseases. J. Chem. Phys., 154(12), 125101.
[8] Weickenmeier, J., Kuhl, E., & Goriely, A. (2018). Multiphysics of prionlike diseases: Progression and atrophy. Phys. Rev. Lett., 121(15), 158101.