How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer’s Disease Research and Clinical Practices?

Abstract

The accumulation of amyloid-β (Aβ) plaques in the brain is considered a hallmark of Alzheimer’s disease (AD). Mathematical modeling, capable of predicting the motion and accumulation of Aβ, has obtained increasing interest as a potential alternative to aid the diagnosis of AD and predict disease prognosis. These mathematical models have provided insights into the pathogenesis and progression of AD that are difficult to obtain through experimental studies alone. Mathematical modeling can also simulate the effects of therapeutics on brain Aβ levels, thereby holding potential for drug efficacy simulation and the optimization of personalized treatment approaches. In this review, we provide an overview of the mathematical models that have been used to simulate brain levels of Aβ (oligomers, protofibrils, and/or plaques). We classify the models into five categories: the general ordinary differential equation models, the general partial differential equation models, the network models, the linear optimal ordinary differential equation models, and the modified partial differential equation models (i.e., Smoluchowski equation models). The assumptions, advantages and limitations of these models are discussed. Given the popularity of using the Smoluchowski equation models to simulate brain levels of Aβ, our review summarizes the history and major advancements in these models (e.g., their application to predict the onset of AD and their combined use with network models). This review is intended to bring mathematical modeling to the attention of more scientists and clinical researchers working on AD to promote cross-disciplinary research.

Keywords

Alzheimer’s disease amyloid-β disease progression mathematical modeling Smoluchowski equation

INTRODUCTION

Alzheimer’s disease (AD) is a growing public health concern, and the prevalence of AD is expected to rise as the population ages. It is estimated that the cost associated with the care of AD in the United States is predicted to increase from 6.2 million to 13.8 million by 2060 without effective treatments [1]. It not only impacts the patients and their carers, but also places a significant economic burden on the healthcare system [2]. In the early stage of AD, the accumulation of extracellular amyloid-β (Aβ) plaques is typically detected in the frontal and temporal lobes of the brain [3]. As the disease progresses, Aβ pathology can spread to other areas of the neocortex. The accumulation of extracellular Aβ plaques and hyperphosphorylation of tau impairs the normal function of neurons and lead to neuronal death [4], altogether the rate of AD progression varies between individuals [5, 6].

Mathematical modeling is a broad term that covers both descriptive and predictive models in regards to the intended use [7]. It can also be categorized as per the mathematical characteristics, such as statistical model, differential equation model, and discrete model. In recent years, the development of computational simulations and numerical solution methods for differential equation models have made mathematical modeling a powerful tool for the study and understanding of pathological processes of many diseases [8 –10], including AD [11 –15]. Mathematical models are different to statistical models. For example, differential equation models (one type of mathematical model) use equations to simulate the quantity of the target factors involved in the disease and their interactive relationships, while statistical models are often used to examine the connection between distinct factors (e.g., gender, APOE genotype) and AD, aiming to identify highly associated factors. Therefore, the mathematical modeling approach provides an insightful understanding of the underlying disease processes.

This review begins with an overview of the mathematical models based on differential equations that are used to simulate brain Aβ levels (oligomers, protofibrils, and/or plaques), which is a hallmark of AD. The major mathematical models are compared, and their advantages and limitations are assessed. Mathematical formulas are not discussed in detail as this is beyond the scope of the current review. We specifically highlight the mathematical models based on the Smoluchowski equation and network modeling, which simulate the brain levels and/or the dynamic behavior (diffusion and aggregation) of Aβ, therefore simulating a closer reflection of physiological condition. We also propose how mathematical modeling can be incorporated into future cross-disciplinary medical research.

MATHEMATICAL MODELS PREDICTING THE BRAIN LEVELS OF Aβ

Mathematical modeling is a powerful tool for studying and understanding complex systems in various fields, such as physics [16], engineering [17], and biology [18]. To simulate brain levels of Aβ, mathematical models based on differential equations are typically used. These models are constructed to predict Aβ levels at later time points based on Aβ levels measured at earlier time points [19]. Some studies have used PET imaging data (especially for simple models) [20, 21], while others have used plasma/cerebrospinal fluid (CSF) biomarkers data (usually for complex ordinary differential equation (ODE) and partial differential equation (PDE) models) [22]. In these studies, some models focus on the levels of Aβ oligomers [23] or plaques [24], while others concentrate on the total amyloid burden [25]. Additionally, one study proposes strategies to construct the model using virtual data generated by some mathematical tools (e.g., MATLAB) [26]. Pallitto and Murphy were the first that attempted to describe the kinetics of Aβ aggregation using mathematical modeling [27]. Since then, researchers have continued to develop and refine the models to mimic biological processes for better prediction of Aβ levels in the brain. The advancement and refinement of mathematical models, along with the availability of powerful computational software, have been pivotal in redefining the challenge of replicating biological processes to enhance the prediction of brain Aβ levels. Previously, the construction and numerical solution of nonlinear ODE and PDE models posed a significant computational challenge [28], this was mitigated by the development of advanced computational software such as MATLAB and MATHEMATICA [29]. The combination of powerful computational software and improved mathematical models has enabled the successful implementation of these numerical solution methods. As a result, researchers have made substantial strides in accurately predicting Aβ levels in the brain, thereby facilitating a more precise emulation of biological processes and potential application in clinical contexts.

To date, numerous mathematical models have been developed with the goal of simulating the brain levels of Aβ. Although the first mathematical model can be traced back to 2001 [27], the first breakthrough application of PDE model to predict brain levels of Aβ oligomers was reported in 2013 [30]. Therefore, we have summarized existing mathematical models used to predict Aβ levels from 2013 to 2023 in Table 1. The studies are categorized into five major models, namely the general ODE model, the general PDE model, the network model, linear optimal ODE model, and the Smoluchowski equation model (SEM). The relationship between these different types of models is shown in Fig. 1.

Fig. 1

The relationship between the mathematical models. SEM and the linear optimal ODE model are two special cases of the general PDE and ODE, respectively. The combination of PDE and network modeling is known as the PDE network, while the combination of ODE and network modeling is known as the ODE network.

Table 1

Mathematical models for Aβ accumulation from 2013 to 2023

Smoluchowski equation model
Authors	Bertsch et al. [19]; Achdou et al. [30]; Bridstrup et al. [44]; Bertsch et al. [52]; Bertsch et al. [65]; Raj et al. [66]; Franchi and Lorenzani [68]; Franchi and Lorenzani [69]; Bertsch et al. [70]; Franchi et al. [71]; Tesi [72]
Limitation	The aggregation due to collisions is simplified in these models.
	Smoluchowski equations are PDEs which are more complicated to solve than ODE models.
Advantage	SEM has performed well in describing the accumulation and diffusion behavior of Aβ, which can simulate the Aβ levels in different brain regions and at various time points.
Network model
Authors	Vosoughi et al. [41]; Presigny and Fallani [42]; Pal and Melnik [58] Proctor et al. [73]; Kashyap et al. [74]; Jafari et al. [75]
Limitation	In real-brain networks, connections between the functional components in the brain changes with the progression of AD, which is difficult to reflect in network models due to the staticity of the edges.
Advantage	It can reflect the interaction of individual components in the brain.
General ordinary differential equation model
Authors	Latulippe et al. [14]; Hadjichrysanthou et al. [34]; Yoon and Jo [76]; Whittington et al. [77]; Ranjan et al. [78]; Han et al.[79]; Hoore et al. [80]; Zhang and Wang [81]; Hu et al. [82]; Minicucci et al. [83]; Ackleh et al. [84]; Hu et al. [85]
Limitation	The components within the ODE models must be known with certainty.
	The general ODE system model can be used only to predict the levels of Aβ with time or space, but not for both.
Advantage	The computation is less complex than other models.
General partial differential equation model
Authors	Asili et al. [15]; Hao and Friedman [67]; Wilson et al. [86]; Helal et al. [87]; Chen et al. [88]; Hoore et al. [89]; Andrade et al. [90]; Pal et al. [91]
Limitation	The components within the PDE models must be known with certainty.
	In general, PDE models are more complicated to solve than ODE models, particularly in 3D space.
Advantage	It can predict the levels of Aβ temporally and spatially, which can help identify specific brain regions that are particularly vulnerable to Aβ accumulation and guide the design of interventions targeting specific regions.
Linear optimal method and ordinary differential equation model
Authors	Hao et al. [11]; Dayeh et al. [26]
Limitation	This model is not suitable for a large-scale population, and it will incur high costs if applied to many patients.
Advantage	It can identify the most appropriate ODE system model for individual patients and increases accuracy.

GENERAL ODE MODELS

The general ODE models are used to illustrate the changes over time for a single or multiple components within a system. In the context of AD, ODE models describe the change of Aβ levels over time [31], considering multiple factors such as Aβ production, diffusion, and clearance. This can provide insights into the Aβ-mediated pathogenesis in AD and help identify potential therapeutic targets [32, 33]. In addition, when considering the system of ODE models, some other factors that play an important role in the production of Aβ (e.g., tau, neurofibrillary tangles) can be incorporated in the description of brain geometry [31]. In the application, these ODE models can also be used to predict how different therapeutic interventions, such as drugs or lifestyle modifications, affect Aβ levels over time [34]. These models can be applied to predict the Aβ-lowering effect of new drugs, and therefore inform the design of clinical trials, including dosing regimen. ODE models assume that Aβ levels increase continuously over time, as they can only be applied to continuous processes [31]. The major drawback of ODE models is that the pathogenesis of AD is still not clearly understood (i.e., Aβ pathology may not be the single underlying mechanism of the disease), making it impossible to find all the components related to AD for modeling [31].

GENERAL PDE MODELS

General PDE models are equations that involve multiple independent variables and their partial derivatives. These equations describe how variables in a system change with respect to both space and time. In the context of AD, PDE models describe how the levels of Aβ vary at a given point in space and time. Unlike the ODE models, PDE models encapsulate the spatial distribution of Aβ within the brain tissue, which is important for understanding how the disease progresses in different brain regions [35]. This helps to identify specific brain regions that are particularly vulnerable to Aβ accumulation and guide the design of interventions that target specific brain regions. The PDE models assume continuous changes in the system for both space and time. Similar to the general ODE models, the interaction between all components in the PDE model system must be known [36]. The major pitfall of these models is their complexity and the difficulty in finding exact solutions, especially when dealing with complicated relationships within the model. In many cases, only numerical solutions can be found, which can lead to mathematical limitations and inaccuracies [37].

NETWORK MODEL

Network models are built upon graph theory, where the graph consists of vertices and edges. Each vertex on the graph can be associated with a PDE or ODE, and a vertex can sometimes interact with another vertex, which is therefore connected by an edge. Network models describe the behavior of a system (i.e., ODE, PDE system) in terms of interacting components, where the behavior of each component is influenced by its neighborhood [38].

The components represent different particular chemical species, such as proteins affected by AD, while the interaction between the components reflect distinct biological processes, including neuroinflammation and neuronal death or neurodegeneration [39]. One advantage of the network models is their ability to capture the complex and dynamic nature of biological interactions, allowing the prediction of how changes in one component of the system will affect behavior of the others [40]. Network models assume that the brain network is static, meaning that the connections between each functional component in the brain are fixed and time-invariant [41]. This assumption contributes to a major limitation of this model, as in real-brain networks, the connections between functional components in the brain changes with the progression of AD [42]. For example, in Haass and Selkoe’s study, large aggregates of a disease causing protein can actually be inert to neurons, although these protein aggregates are toxic to the neurons at the early stage of AD [43].

LINEAR OPTIMAL ODE MODEL

Unlike the general ODE models that aim to construct a general model for typical use, the linear optimal ODE model focuses on simulating the personalized brain levels of Aβ [11]. It employs a linear optimal method to construct a personalized model for each patient to establish the most appropriate ODE model. This approach improves the accuracy in the prediction of Aβ levels compared with the general ODE models. However, although this personalized approach can be effective, it may not be practical for implementation in a large population, which is likely to be cost prohibitive.

SEM MODEL

The SEM as a particular PDE model in statistical physics aims to describe the dynamics of aggregation of particles in fluids. It is a natural candidate for modeling the behavior of Aβ monomers [27] and suitable for simulating large numbers of Aβ monomers in a complex environment such as the brain [44]. The SEM has the potential to provide a more comprehensive understanding of the aggregation process of Aβ in AD when integrated with other models such as nucleation and growth models [45]. This may aid in the identification of critical factors that drive Aβ aggregation in AD and to guide the development of new treatments. As a specifically highlighted mathematical model, we will give more general details about SEM in the following sections.

General information about the Smoluchowski equation

Smoluchowski equation is suitable to describe aggregation phenomena [46, 47], which is similar to the aggregation of Aβ during the progression of AD. The Smoluchowski equation incorporated with the diffusion term is described in Equation 1. Further mathematical details of this equation can be found in a note written by Herrero et al. [48].

$\begin{matrix} \frac{\partial u_{i}}{\partial t} = D_{i} Δ u_{i} + \frac{1}{2} \sum_{k = 1}^{i - 1} a_{i - k, k} u_{i - k} u_{k} \\ - \sum_{k = 1}^{\infty} a_{i, k} u_{i} u_{k} (i ⩾ 1) \end{matrix}$ (1)

In Equation 1, u_i (x, t) denotes the concentration of aggregates containing i monomers at the position x in the domain of interest $D, D \subset R^{N}$ in the brain when time is t. N denotes the dimension of the position and is usually set to 2 or 3 in practice. The first term D_iΔu_i represents diffusion, and the second term $\frac{1}{2} \sum_{k = 1}^{i - 1} a_{i - k, k} u_{i - k} u_{k}$ as well as the third term $\sum_{k = 1}^{\infty} a_{i, k} u_{i} u_{k}$ represents the gain or loss of mass for the agglomerates upon collision. The diffusion coefficient D_i (D_i ⩾ 0) represents the diffusion rate, while the aggregation coefficient a_ik represents the aggregation rate of particles u_i and u_k. The index i refers to the rank of molar mass. When i = 1, the molecule is of the lowest molar mass, indicating it only loses mass after a collision. In practice, there are some advanced software such as Simbiology, Monolix that can be used to estimate the aggregation and diffusion coefficients with a relatively low error using clinical data. However, it should be noted that the determinations of these coefficients are still a challenge, which concerns the validations of all the models proposed, and the predictive character the models could eventually have.

Assumptions and limitations of the SEM

Although the SEM is a useful tool for understanding the physical processes involved in protein aggregation and diffusion, it does not incorporate the complex biology of AD. In particular, the Smoluchowski equation does not consider the various biological factors that influence Aβ aggregation, such as the presence of other proteins or cellular structures, enzyme activity, or the immune response to Aβ [49]. In addition, Aβ aggregation does not occur in a single step, but indeed takes a series of steps starting with initial seeds, soluble small oligomers, protofibrils, and insoluble polymers, which eventually form amyloid fibrils with Aβ sheet conformation [50]. In the studies of the SEM application in informing Aβ aggregation, Aβ aggregation is simplified from monomers to soluble oligomers and then to long fibrils [30 , 52]. The primary reason for this simplification is to reduce the computational challenges. Moreover, parameter estimation in SEM requires data specifying levels of Aβ from multiple time points. As a result, in SEM, the levels of Aβ is typically measured through blood or CSF biomarkers, rather than using imaging data. Despite these limitations, the capacity of SEM to simulate the accumulation and diffusion behavior gives a more detailed depiction of the dynamic mechanisms of Aβ, which helps researchers have deeper insights into comprehending the progression of AD. Furthermore, the advancements in computer science have facilitated the accessibility of substantial computing resources, this will overcome the computational challenge and make the application of SEM in simulating the several steps of accumulation of Aβ become possible.

Advantages of SEM compared with other models

Although there are many mathematical models available, the SEM offers several advantages over the other models when simulating brain levels of Aβ. Unlike general ODE models, the SEMs also provide spatial information. For example, Bertsch et al. have applied the SEM to conduct a qualitative analysis of the levels of Aβ₄₂ oligomers in the brain [51]. This model can be used to predict the levels of Aβ₄₂ oligomers in any brain region at any future time point, which has the potential to facilitate early diagnosis of AD and predict the progression of amyloid pathology. In addition, the Smoluchowski equation is a PDE model that incorporates the diffusion and aggregation terms. Both terms are essential to describe the dynamic behavior of Aβ monomers at the microscopic level. The diffusion term describes the movement of Aβ monomers in the brain, while the aggregation term expresses the formation of Aβ plaques from monomers (as illustrated in Fig. 2). The dynamic behavior of Aβ is often neglected by other PDE models that focus on AD progression at the macroscopic level (e.g., cognitive function) [12 , 53].

Fig. 2

Diffusion, aggregation, and clearance of Aβ. Aβ monomers are produced by neurons by cleavage of the amyloid precursor protein, which diffuse within the brain. The soluble Aβ monomers aggregate to form Aβ fibrillar plaques. The Aβ monomers and Aβ fibrillar plaques are cleared from the brain via the blood-brain barrier and the phagocytic activity of microglia, respectively. The figure was created in BioRender.

Modifications of SEM for modeling Aβ

Over the last decades, several research groups have modified the Smoluchowski equation to incorporate biological concepts and develop modified SEMs (summarized in Table 2). The history and development of SEM is shown in Fig. 3. The Smoluchowski equation was introduced by Smoluchowski in 1918 to describe the aggregation of particles [46], which has been applied in physics and chemistry. A few decades later, Achdou et al. and Bertch et al. pioneered the application of the Smoluchowski equation to construct an Aβ₄₀ and Aβ₄₂ aggregation and diffusion model which aimed to predict the levels of Aβ oligomers for the early diagnosis of AD [30, 51]. The brain region of interest in this model is the hippocampus and cerebral cortex, as they are the primary areas affected by AD [54]. To construct Achdou’s model, a definite spatial scale must be fixed [30]. This scale is much larger than the neurons, but it is still microscopic (e.g., a portion of the hippocampus). That is, Achdou’s model does not consider the interaction between the selected brain portion and the rest of the brain, nor the clinical manifestations of AD (i.e., macroscopic levels). Achdou’s model assumes that the Aβ monomers follow a linear diffusion. Therefore, considering the macroscopic levels and the interaction of Aβ monomers with other molecules will invalidate this model [55].

Fig. 3

Timeline of the development of SEM. The Smoluchowski equation was first introduced in 1918 and its application to Aβ aggregation and diffusion was first documented in 2013. In 2017, Betsch et al. used the SEM to predict both the Aβ brain levels and the onset of AD and validated it using magnetic resonance imaging. In 2022, Pal and Melnik employed network modeling to predict brain levels of Aβ, which was later incorporated into SEM by Bertsch et al. in 2022.

Table 2

Mathematical models for the prediction of Aβ levels in AD using the Smoluchowski equation

A qualitative model for aggregation and diffusion of A β in Alzheimer’s disease [30]
Model	The aggregation and diffusion terms were incorporated into the Smoluchowski equation (i.e., SEM) to simulate the levels of Aβ.
Application	To model the accumulation of Aβ from early disease stage, once Aβ brain levels can be detected using advanced imaging.
Limitation	The brain network was not considered.
Alzheimer’s disease: a mathematical model for onset and progression [51]
Model	Multiple time scale method and a parameter (α) that represents the extent of neuronal damage were incorporated into the Smoluchowski equation.
Application	To predict the brain levels of Aβ and the onset/progression of AD.
Limitation	The brain network was not considered.
The role of A β and tau proteins in Alzheimer’s disease: a mathematical model on graphs [59]
Model	The diffusion and aggregation terms were incorporated into the Smoluchowski equation, which was then combined with network modeling. The Aβ accumulation in this model is a multistep process rather than a single step as assumed in other models.
Application	Simulate the brain levels of Aβ and other components (e.g., tau) in AD.
Limitation	Network modeling assumes that all the components have fixed relationships. However, this is not valid as their relationships change with the progression of AD.

Achdou’s model only considered the accumulation and diffusion of Aβ without the onset of AD. In order to predict the onset of AD, in 2017, Bertsch et al. incorporated a method of treating multiple time scale into the Smoluchowski equation and constructed a SEM that concerned Aβ₄₂ deposition and predicted AD onset [51]. Bertsch’s model was developed to account for the progression of AD, that occurs over years, whereas the diffusion and aggregation of Aβ occurs in a relatively short time [51]. Two temporal scales were introduced in Bertsch’s model, i.e., a short, rapid scale (t) for the diffusion and aggregation of Aβ, and a longer, larger scale (T) for the progression of the disease. For computational efficiency, only one variable (time) is considered in a PDE system, a small constant 0 < ɛ < <1 was introduced to consider the relationship between the different time scales. Considering that neuronal damage reflects the progression of AD [56], Bertsch et al. incorporated neuronal damage (parameter α) into the Smoluchowski equation, where a value of α close to 0 indicates a healthy neuron, while a value close to 1 represents neuronal death [51]. Bertsch’s model therefore allows simulation of Aβ plaques levels in the brain and prediction of the onset of AD (i.e., brain Aβ plaques levels exceed a threshold). Additionally, the model also accounted for the clearance of Aβ plaques via the blood-brain barrier and microglia phagocytosis [57], which is a development of the diffusion-aggregation model introduced by Achdou et al. in 2013. Bertsch’s model assumed that malfunctioning neurons are detrimental to their healthy neighboring neurons, while healthy neurons do not affect the malfunctioning neurons, and that only Aβ oligomers are toxic, while monomers and plaques are not. In addition, this model assumed that the damaged neurons produce more Aβ than healthy neurons, and that Aβ is not produced by dead neurons [51]. However, whether these assumptions made for Bertsch’s model are (patho)physiologically true should be carefully evaluated. On the other hand, although the Bertsch’s model [51] addressed the significant difference between the time scale of Aβ diffusion/aggregation and the onset of AD, it neglected the effects of other factors (e.g., hyperphosphorylated tau, neurofibrillary tangles) on the production of Aβ in AD.

To overcome the limitations of Bertsch’s model, a network modeling method was developed. Pal and Melnik employed network models to simulate the Aβ oligomers levels in the brain [58]. This model considered how the pathological changes in AD affect the production of Aβ. However, Pal and Melnik’s model neglected the aggregation behavior of Aβ oligomers. Later in 2022, Bertsch et al. constructed a SEM-network model, which utilized SEM to describe the aggregation and diffusion of Aβ and employed the network model to simulate the brain network [59]. This improvement overcame the limitation of traditional SEMs that solely focus on Aβ. The aggregation-network-diffusion SEM introduced by Bertsch et al. gives a more comprehensive and detailed depiction of Aβ plaque levels within the brain network.

POTENTIAL CLINICAL APPLICATION OF THE MATHEMATICAL MODELS IN AD

The mathematical models have real potential to be used in the clinic for the prediction of brain levels of Aβ and the onset of AD (i.e., brain Aβ levels reach a predetermined threshold assessed by PET). This approach is supported by the prognostic utility of patient-specific mathematical models in cancer research, such as the prediction of patient survival through the modeling of the c-Jun N-terminal kinase pathway network dynamics in neuroblastoma [60]. Additionally, the potential prognostic and clinical utility of mathematical modeling has been demonstrated in glioblastoma research through the modeling of tumor invasiveness [61], discrimination of patient survival outcomes [62], and the optimization of radiotherapy [63]. The proposed pipeline for the implementation of mathematical modeling in AD risk prediction is shown in Fig. 4.

Fig. 4

The proposed clinical application of mathematical models.

Using mathematical models, clinicians can identify who is at risk of AD and estimate the age at which a population or an individual may develop AD. Appropriate interventions can therefore be initiated for timely treatment to improve the quality of life for those who are at risk of AD. Moreover, mathematical models are likely to be more cost-efficient. However, for most mathematical models discussed in this review, their application to PET imaging data may be limited, as data from different time points are required. The use of multiple PET scans may result in high costs [64]. Simple mathematical models, e.g., linear statistical model, can be applied to PET imaging data, where only a small number of data points are required. For instance, Villemagne et al. employed a curve fitting method to construct linear, quadratic, and cubic mathematical models for predicting the rates of Aβ decomposition [20]. Similarly, Schindler et al. [21] successfully utilized the fitting curve method to construct the model that describes brain Aβ levels over time. In their model, only a few PET scans are needed as initial measurement data for predicting the onset of AD by identifying the critical time point of the curve that reflects the rapid change in Aβ levels [21]. For complex models such as SEM, numerous parameters and longitudinal data are required for them to provide detailed information on the levels of Aβ in the brain network. In various studies, CSF or blood biomarkers data rather than PET imaging data has been used [15 , 66]. By using data collected from longitudinal dementia cohorts, scientists can develop an Aβ aggregation model from CSF or blood biomarkers. For example, in [51] the levels of Aβ₄₂ and the global brain atrophy are evaluated by numerical simulation, which demonstrates the potential practical application of SEM. Moreover, in [59], using MRI data, the total burden of Aβ and tau in the whole brain and local burden of Aβ and tau in different regions are simulated by SEM, where synergetic action of these two proteins was confirmed by the numerical simulation. Future research is required to construct a global model that can explain the interaction and behavior of various biomarkers including Aβ, phosphorylated tau and others.

In addition, mathematical models can be developed to evaluate the effectiveness and safety of potential treatments for AD before they are evaluated in clinical trials. This could help streamline the drug development process and identify the most promising treatments to be assessed in clinical trials. There have already been some advancements in this research field; for example, a mathematical model has been used to simulate the effects of aducanumab on Aβ clearance [11]. By adjusting the parameters of the model to reflect the pharmacokinetics and biology, researchers can predict how a drug would affect Aβ levels in the brain and how this would slow down cognitive decline. Using a quantitative amyloid/tau/neurodegeneration mathematical model (i.e., modified type of ODE model) with Alzheimer’s Disease Neuroimaging Initiative dataset, Roche has performed a clinical trial simulation on gantenerumab (an anti-amyloid antibody drug), which shows that amyloid PET levels decrease markedly with continued anti-amyloid treatment over 5 years [22]. Furthermore, mathematical modeling also makes it possible to evaluate the potential benefits of combined therapies. For example, combined therapy with etanercept (a tumor necrosis factor alpha inhibitor) and aducanumab (an anti-amyloid drug) could yield a significant efficacy in slowing the progression of AD [67]. By simulating the effects of different drug combinations, researchers can identify the most promising treatment strategies and prioritize them for clinical trials, which is illustrated in Fig. 5.

Fig. 5

The clinical application of mathematical models in treatments for AD.

CONCLUSIONS

This review provides an overview of the main mathematical models that have been employed to predict the brain levels of Aβ and their application in tracking the levels of Aβ during the progression of AD. It is evident that SEM has received a multitude of attention for its use in AD research and potential clinical use. Existing work has demonstrated the possibility of combining the Smoluchowski equation and network modeling to investigate the levels of Aβ in AD patients but has not yet been applied to predict the onset of AD at critical time points, which can be explored in future studies. Additionally, deriving an efficient and low error numerical method would contribute to the practical application of SEM and improve the accuracy in the prediction of Aβ levels. Moreover, constructing complex mathematical models like SEMs demands a substantial dataset. Due to the numerous parameters involved in the model, many longitudinal data points are required to accurately determine these parameters. Using blood biomarkers could be a suitable option to construct such models and evaluate their performance in future studies since longitudinal blood samples are relatively easier to be obtained than imaging data (PET).

On the other hand, close collaboration between mathematicians, biomedical scientists and clinical researchers should be encouraged to ensure that the models better reflect the known physiological and pathophysiological processes, which can help identify new targets for treatment and improve our understanding of the disease. Personalized medicine approaches could also be developed by using mathematical models to predict response to treatment and identify patients who are most likely to benefit from a particular therapy. This could help improve treatment outcomes and reduce the risk of adverse effects.

Overall, mathematical modeling has the potential to impact our understanding and treatment of AD. Given the magnitude of the global AD burden, collaboration across disciplines, including neuroscience, molecular biology, computational modeling, is critical to improving our understanding of AD and developing effective prevention, diagnosis, and treatment strategies.

Footnotes

ACKNOWLEDGMENTS

The authors have no acknowledgments to report.

FUNDING

The salary of Y.P. and APC were funded by National Health and Medical Research Council (grant number GNT2007912) and Alzheimer’s Association USA (grant number 23AARF-1020292).

CONFLICT OF INTEREST

Colin Masters is an Editorial Board Member of this journal but was not involved in the peer-review process nor had access to any information regarding its peer-review.

All other authors have no conflict of interest to report.

References

The Alzheimer’s Association (2021). 2021 Alzheimer’s disease facts and figures. Alzheimers Dement 17, 327–406.

Nichols

, Szoeke

, Vollset

, Abbasi

, Abd-Allah

, Abdela

, Aichour

MTE

, Akinyemi

, Alahdab

, Asgedom

(2019) Global, regional, and national burden of Alzheimer’s disease and other dementias, 1990–2016: A systematic analysis for the Global Burden of Disease Study 2016. Lancet Neurol 18, 88–106.

Nestor

, Fryer

, Hodges

(2006) Declarative memory impairments in Alzheimer’s disease and semantic dementia. Neuroimage 30, 1010–1020.

Masters

, Bateman

, Blennow

, Rowe

, Sperling

, Cummings

(2015) Alzheimer’s disease. Nat Rev Dis Primers 1, 15056.

Mazure

, Swendsen

(2016) Sex differences in Alzheimer’s disease and other dementias. Lancet Neurol 15, 451.

Risacher

, Anderson

, Charil

, Castelluccio

, Shcherbinin

, Saykin

, Schwarz

, Alzheimer’s Disease Neuroimaging Initiative (2017) Alzheimer disease brain atrophy subtypes are associated with cognition and rate of decline. Neurology 89, 2176–2186.

Aris

(1994) Mathematical modelling techniques, Courier Corporation.

Hormuth

, 2nd Farhat

, Christenson

, Curl

, Chad Quarles

, Chung

, Yankeelov

(2022) Opportunities for improving brain cancer treatment outcomes through imaging-based mathematical modeling of the delivery of radiotherapy and immunotherapy. Adv Drug Deliv Rev 187, 114367.

Slavkova

, Patel

, Cacini

, Kazerouni

, Gardner

, Yankeelov

, Hormuth

(2023) Mathematical modelling of the dynamics of image-informed tumor habitats in a murine model of glioma. Sci Rep 13, 2916.

10.

Sung

, Hong

, Poznansky

, Paganetti

, Grassberger

(2022) Mathematical modeling to simulate the effect of adding radiation therapy to immunotherapy and application to hepatocellular carcinoma. Int J Radiat Oncol Biol Phys 112, 1055–1062.

11.

Hao

, Lenhart

, Petrella

(2022) Optimal anti-amyloid-beta therapy for Alzheimer’s disease via a personalized mathematical model. PLoS Comput Biol 18, e1010481.

12.

Drapaca

(2022) A mathematical investigation of sex differences in Alzheimer’s disease. Fractal Fract 6, 457.

13.

Kyrtsos

, Baras

(2013) Studying the role of ApoE in Alzheimer’s disease pathogenesis using a systems biology model. J Bioinform Comput Biol 11, 1342003.

14.

Latulippe

, Lotito

, Murby

(2018) A mathematical model for the effects of amyloid beta on intracellular calcium. PLoS One 13, e0202503.

15.

Asili

, Yarahmadian

, Khani

, Sharify

(2019) A mathematical model for amyloid-β aggregation in the of metal ions: A timescale analysis for the progress of Alzheimer disease. Bull Math Biol 81, 1943–1964.

16.

Acebrón

, Bonilla

, Vicente

CJP

, Ritort

, Spigler

(2005) The Kuramoto model: A simple paradigm for synchronization phenomena. Rev Mod Phys 77, 137.

17.

Burova

, Wall

, Shipley

(2019) Mathematical and computational models for bone tissue engineering in bioreactor systems. J Tissue Eng 10, 2041731419827922.

18.

Motta

, Pappalardo

(2013) Mathematical modeling of biological systems. Brief Bioinform 14, 411–422.

19.

Bertsch

, Franchi

, Meacci

, Primicerio

, Tesi

(2021) The amyloid cascade hypothesis and Alzheimer’s disease: A mathematical model. Eur J Appl Math 32, 749–768.

20.

Villemagne

, Burnham

, Bourgeat

, Brown

, Ellis

, Salvado

, Szoeke

, Macaulay

, Martins

, Maruff

(2013) Amyloid β deposition, neurodegeneration, and cognitive decline in sporadic Alzheimer’s disease: A prospective cohort study. Lancet Neurol 12, 357–367.

21.

Schindler

, Li

, Buckles

, Gordon

, Benzinger

, Wang

, Coble

, Klunk

, Fagan

, Holtzman

(2021) Predicting symptom onset in sporadic Alzheimer disease with amyloid PET. Neurology 97, e1823–e1834.

22.

Mazer

, Hofmann

, Lott

, Gieschke

, Klein

, Boess

, Grimm

, Kerchner

, Baudler-Klein

, Smith

(2022) Development of a quantitative semi-mechanistic model of Alzheimer’s disease based on the amyloid/tau/neurodegeneration framework (the Q-ATN model). Alzheimers Dement 19, 2287–2297.

23.

Ciuperca

, Dumont

, Lakmeche

, Mazzocco

, Pujo-Menjouet

, Rezaei

, Tine

(2019) Alzheimer’s disease and prion: An mathematical model. Discrete Continuous Dyn Syst Ser B 24, 5225–5260.

24.

Lloret-Villas

, Varusai

, Juty

, Laibe

, Le Novere

, Hermjakob

, Chelliah

(2017) The impact of mathematical modeling in understanding the mechanisms underlying neurodegeneration: Evolving dimensions and future directions. CPT Pharmacometrics Syst Pharmacol 6, 73–86.

25.

Craft

, Wein

, Selkoe

(2004) The impact of novel treatments on Aβ burden in Alzheimer’s disease: Insights from a mathematical model. In Operations Research and Health Care, Brandeau ML, Sainfort F, Pierskalla WP, eds. International Series in Operations Research&Management Science, vol 70. Springer, Boston, MA.

26.

Dayeh

, Livadiotis

, Elaydi

(2018) A discrete mathematical model for the aggregation of β-amyloid. PLoS One 13, e0196402.

27.

Pallitto

, Murphy

(2001) A mathematical model of the kinetics of β-amyloid fibril growth from the denatured state. Biophys J 81, 1805–1822.

28.

De Marchi

(2005) Computational and mathematical modeling in the social sciences, Cambridge University Press.

29.

Feng

, Glowinski

, Neilan

(2013) Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev 55, 205–267.

30.

Achdou

, Franchi

, Marcello

, Tesi

(2013) A qualitative model for aggregation and diffusion of-amyloid in Alzheimer’s disease. J Math Biol 67, 1369–1392.

31.

Moravveji

, Doyon

, Duchesne

, Mashreghi

(2022) A scoping review of mathematical models covering Alzheimer’s disease progression. bioRxiv, https://doi.org/10.1101/2022.12.07.519498

32.

Han

, Park

, Mook-Jung

(2016) Amyloid β-interacting partners in Alzheimer’s disease: From accomplices to possible therapeutic targets. Prog Neurobiol 137, 17–38.

33.

, Ma

(2017) The role of amyloid beta clearance in cerebral amyloid angiopathy: More potential therapeutic targets. Transl Neurodegener 6, 22.

34.

Hadjichrysanthou

, Ower

, de Wolf

, Anderson

, Alzheimer’s Disease Neuroimaging Initiative (2018) The development of a stochastic mathematical model of Alzheimer’s disease to help improve the design of clinical trials of potential treatments. PLoS One 13, e0190615.

35.

Dai

, Sun

, Zhao

, Pu

(2020) Novel imaging and related techniques for studies of diseases of the central nervous system: A review. Cell Tissue Res 380, 415–424.

36.

Deutsch

, Dormann

(2005) Mathematical modeling of biological pattern formation. In Cellular Automaton Modeling of Biological Pattern Formation. Modeling and Simulation in Science, Engineering and Technology. Birkhauser, Boston, MA, pp. 49-61.

37.

Chen

, Chen

, Wan

, Xu

, Liu

(2021) An improved data-free surrogate model for solving partial differential equations using deep neural networks. Sci Rep 11, 19507.

38.

Porter

, Gleeson

(2016) Dynamical Systems on Networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 4. Springer, Cham.

39.

Kyrtsos

, Baras

(2012) A graph theoretic mathematical model for Alzheimer’s disease. Proceedings of 12th IEEE International Conference on Bioinformatics & Bioengineering (BIBE), pp. 472-477.

40.

Keurentjes

, Angenent

, Dicke

, Dos Santos

VAM

, Molenaar

, van der Putten

, de Ruiter

, Struik

, Thomma

(2011) Redefining plant systems biology: From cell to ecosystem. Trends Plant Sci 16, 183–190.

41.

Vosoughi

, Sadigh-Eteghad

, Ghorbani

, Shahmorad

, Farhoudi

, Rafi

, Omidi

(2020) Mathematical models to shed light on amyloid-beta and tau protein dependent pathologies in Alzheimer’s disease. Neuroscience 424, 45–57.

42.

Presigny

, Fallani

FDV

(2022) Colloquium: Multiscale modeling of brain network organization. Rev Mod Phys 94, 031002.

43.

Haass

, Selkoe

(2007) Soluble protein oligomers in neurodegeneration: Lessons from the Alzheimer’s amyloid β-peptide. Nat Rev Mol Cell Biol 8, 101–112.

44.

Bridstrup

, Yuan

J-M

, Schreck

(2022) Stochastic kinetic study of protein aggregation and molecular crowding effects of Aβ40 and Aβ42. arXiv, https://doi.org/10.48550/arXiv.2209.10630.

45.

Gillam

, MacPhee

(2013) Modelling amyloid fibril formation kinetics: Mechanisms of nucleation and growth. J Phys Condens Matter 25, 373101.

46.

Smoluchowski

(1918) Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z Phys Chem 92, 129–168.

47.

Drake

(1972) A general mathematical survey of the coagulation equation. In Topics in Current Aerosol Research, Hidy GM, Brock JR, eds., Pergamon Press, New York.

48.

Herrero

, Rodrigo

(2005) A note on Smoluchowski’s equations with diffusion. Appl Math Lett 18, 969–975.

49.

Rahman

, Lendel

(2021) Extracellular protein components of amyloid plaques and their roles in Alzheimer’s disease pathology. Mol Neurodegener 16, 1–30.

50.

Chimon

, Shaibat

, Jones

, Calero

, Aizezi

, Ishii

(2007) Evidence of fibril-like β-sheet structures in a neurotoxic amyloid intermediate of Alzheimer’s β-amyloid. Nat Struct Mol Biol 14, 1157–1164.

51.

Bertsch

, Franchi

, Marcello

, Tesi

, Tosin

(2017) Alzheimer’s disease: A mathematical model for onset and progression. Math Med Biol 34, 193–214.

52.

Bertsch

, Franchi

, Tesi

, Tosin

(2017) Microscopic and macroscopic models for the onset and progression of Alzheimer’s disease. J Phys A Math Theor 50, 414003.

53.

Puri

, Li

(2010) Mathematical modeling for the pathogenesis of Alzheimer’s disease. PLoS One 5, e15176.

54.

Braak

, Alafuzoff

, Arzberger

, Kretzschmar

, Del Tredici

(2006) Staging of Alzheimer disease-associated neurofibrillary pathology using paraffin sections and immunocytochemistry. Acta Neuropathol 112, 389–404.

55.

Bero

, Yan

, Roh

, Cirrito

, Stewart

, Raichle

, Lee

J-M

, Holtzman

(2011) Neuronal activity regulates the regional vulnerability to amyloid-β deposition. Nat Neurosci 14, 750–756.

56.

Mosconi

, Berti

, Glodzik

, Pupi

, De Santi

, de Leon

(2010) Pre-clinical detection of Alzheimer’s disease using FDG-PET, with or without amyloid imaging. J Alzheimers Dis 20, 843–854.

57.

Iliff

, Wang

, Liao

, Plogg

, Peng

, Gundersen

, Benveniste

, Vates

, Deane

, Goldman

(2012) A paravascular pathway facilitates CSF flow through the brain parenchyma and the clearance of interstitial solutes, including amyloid β. Sci Transl Med 4, 147ra11.

58.

Pal

, Melnik

(2022) Coupled Neural–glial dynamics and the role of astrocytes in Alzheimer’s disease. Math Comput Appl 27, 33.

59.

Bertsch

, Franchi

, Tesi

, Tora

(2022) The role of beta-amyloid and tau proteins in Alzheimer’s disease: A mathematical model on graph. arXiv, https://doi.org/10.48550/arXiv.2212.13868

60.

Fey

, Halasz

, Dreidax

, Kennedy

, Hastings

, Rauch

, Munoz

, Pilkington

, Fischer

, Westermann

(2015) Signaling pathway models as biomarkers: Patient-specific simulations of JNK activity predict the survival of neuroblastoma patients. Sci Signal 8, ra130–ra130.

61.

Baldock

, Ahn

, Rockne

, Johnston

, Neal

, Corwin

, Clark-Swanson

, Sterin

, Trister

, Malone

(2014) Patient-specific metrics of invasiveness reveal significant prognostic benefit of resection in a predictable subset of gliomas. PLoS One 9, e99057.

62.

Neal

, Trister

, Cloke

, Sodt

, Ahn

, Baldock

, Bridge

, Lai

, Cloughesy

, Mrugala

(2013) Discriminating survival outcomes in patients with glioblastoma using a simulation-based, patient-specific response metric. PLoS One 8, e51951.

63.

Corwin

, Holdsworth

, Rockne

, Trister

, Mrugala

, Rockhill

, Stewart

, Phillips

, Swanson

(2013) Toward patient-specific, biologically optimized radiation therapy plans for the treatment of glioblastoma. PLoS One 8, e79115.

64.

Tai

, Piccini

(2004) Applications of positron emission tomography (PET) in neurology. J Neurol Neurosurg Psychiatry 75, 669–676.

65.

Bertsch

, Franchi

, Meschini

, Tesi

, Tosin

(2021) A sensitivity analysis of a mathematical model for the synergistic interplay of amyloid beta and tau on the dynamics of Alzheimer’s disease. Brain Multiphys 2, 100020.

66.

Raj

, Tora

, Gao

, Cho

, Choi

, Ryu

, Lyoo

, Franchi

(2021) Combined model of aggregation and network diffusion recapitulates Alzheimer’s regional tau-positron emission tomography. Brain Connect 11, 624–638.

67.

Hao

, Friedman

(2016) Mathematical model on Alzheimer’s disease. BMC Syst Biol 10, 108.

68.

Franchi

, Lorenzani

(2016) From a microscopic to a macroscopic model for Alzheimer disease: Two-scale homogenization of the Smoluchowski equation in perforated domains. J Nonlinear Sci 26, 717–753.

69.

Franchi

, Lorenzani

(2017) Smoluchowski equation with variable coefficients in perforated domains: Homogenization and applications to mathematical models in medicine. Harmonic Analysis, Partial Differential Equations and Applications: In Honor of Richard L. Wheeden, pp. 49-67.

70.

Bertsch

, Franchi

, Tesi

, Tosin

(2018) Well-posedness of a mathematical model for Alzheimer’s disease. SIAM J Math Anal 50, 2362–2388.

71.

Franchi

, Heida

, Lorenzani

(2019) A mathematical model for Alzheimer’s disease: An approach via stochastic homogenization of the Smoluchowski equation. arXiv, https://doi.org/10.48550/arXiv.1904.11015.

72.

Tesi

(2021) The synergistic interplay of Amyloid beta and tau proteins in Alzheimer’s disease: A compartmental mathematical model. Bruno Pini Mathematical Analysis Seminar, pp. 15-30.

73.

Proctor

, Boche

, Gray

, Nicoll

(2013) Investigating interventions in Alzheimer’s disease with computer simulation models. PLoS One 8, e73631.

74.

Kashyap

, Bapat

, Das

, Gowaikar

, Amritkar

, Rangarajan

, Ravindranath

, Ambika

(2019) Synapse loss and progress of Alzheimer’s disease-A network model. Sci Rep 9, 6555.

75.

Jafari

, Sarbaz

, Ebrahimi-kalan

, Golabi

(2022) Novel mathematical model based on cellular automata for study of Alzheimer’s disease progress. Netw Model Anal Health Inform Bioinform 11, 26.

76.

Yoon

S-S

, Jo

(2012) Mechanisms of amyloid-β peptide clearance: Potential therapeutic targets for Alzheimer’s disease. Biomol Ther 20, 245.

77.

Whittington

, Sharp

, Gunn

(2018) Spatiotemporal distribution of β-amyloid in Alzheimer disease is the result of heterogeneous regional carrying capacities. J Nucl Med 59, 822–827.

78.

Ranjan

, Chong

, Zheng

(2018) Composite mathematical modeling of calcium signaling behind neuronal cell death in Alzheimer’s disease. BMC Syst Biol 12, 61–74.

79.

Han

, Kim

, Choi

(2020) Computational modeling of the effects of autophagy on amyloid-β peptide levels. Theor Biol Med Model 17, 2.

80.

Hoore

, Khailaie

, Montaseri

, Mitra

, Meyer-Hermann

(2020) Mathematical model shows how sleep may affect amyloid-β fibrillization. Biophys J 119, 862–872.

81.

Zhang

, Wang

(2020) Mathematical analysis for stochastic model of Alzheimer’s disease. Commun Nonlinear Sci Numer Simul 89, 105347.

82.

, Zhang

, Meyer-Baese

, Ye

(2020) Stationary distribution of a stochastic Alzheimer’s disease model. Math Methods Appl Sci 43, 9706–9718.

83.

Minicucci

, Alfond

, Demuro

, Gerberry

, Latulippe

(2021) Quantifying the dose-dependent impact of intracellular amyloid beta in a mathematical model of calcium regulation in xenopus oocyte. PLoS One 16, e0246116.

84.

Ackleh

, Elaydi

, Livadiotis

, Veprauskas

(2021) A continuous-time mathematical model and discrete approximations for the aggregation of β-amyloid. J Biol Dyn 15, 109–136.

85.

, Zhang

, Meyer-Baese

, Ye

(2022) Stability in distribution for a stochastic Alzheimer’s disease model with reaction diffusion. Nonlinear Dyn 108, 4243–4260.

86.

Wilson

, Gates

, Castellanos

(2013) Modeling the short time-scale dynamics of β-amyloid–neuron interactions. J Theor Biol 331, 28–37.

87.

Helal

, Hingant

, Pujo-Menjouet

, Webb

(2014) Alzheimer’s disease: Analysis of a mathematical model incorporating the role of prions. J Math Biol 69, 1207–1235.

88.

Chen

, Tseng

, Ward

(2019) A mathematical model demonstrating the role of interstitial fluid flow on the clearance and accumulation of amyloid β in the brain. Math Biosci 317, 108258.

89.

Hoore

, Kelling

, Sayadmanesh

, Mitra

, Schips

, Meyer-Hermann

(2020) Brain geometry matters in Alzheimer’s disease progression: A simulation study. bioRxiv, https://doi.org/10.1101/2020.07.24.220228.

90.

Andrade-Restrepo

, Ciuperca

, Lemarre

, Pujo-Menjouet

, Tine

(2021) A reaction–diffusion model of spatial propagation of Aβ oligomers in early stage Alzheimer’s disease. J Math Biol 82, 39.

91.

Pal

, Shaheen

, Melnik

(2022) The Influence of Amyloid-Beta on Calcium Dynamics in Alzheimer’s Disease: A Spatio-Temporal Study. In Computational Science and Its Applications–ICCSA 2022 Workshops: Malaga, Spain, July 4–7, 2022, Proceedings, Part I, Springer, pp. 308-322.