Mathematical Modeling for the Pathogenesis of Alzheimer's Disease

Despite extensive research, the pathogenesis of neurodegenerative Alzheimer's disease (AD) still eludes our comprehension. This is largely due to complex and dynamic cross-talks that occur among multiple cell types throughout the aging process. We present a mathematical model that helps define critical components of AD pathogenesis based on differential rate equations that represent the known cross-talks involving microglia, astroglia, neurons, and amyloid-β (Aβ). We demonstrate that the inflammatory activation of microglia serves as a key node for progressive neurodegeneration. Our analysis reveals that targeting microglia may hold potential promise in the prevention and treatment of AD.


Introduction
Alzheimer's disease (AD) is one of the most prevalent neurodegenerative disorders associated with aging, causing dementia and related severe public health concerns [1]. Despite extensive research effort and progress, the pathogenesis of AD remains incompletely understood, partly due to highly complex and intertwined intercellular cross-talks taking place throughout the aging process [2]. Consequently, despite limited treatment options to manage and slow the progression of AD, no effective cure is available.
Although the deposition of amyloid-b (Ab) peptides and formation of senile plaques in the brain is the cardinal morphological feature identifying the clinical phenotype of AD [3,4], increasing clinical and basic studies suggest that inflammatory activation of microglia may play an equally important role during the initiation and progression of the disease [5]. Microglia are resident innate immune macrophages within brain tissues, capable of expressing pro-inflammatory mediators and reactive oxygen species when activated by inflammatory signals including amyloid-b (Ab) [6]. In healthy brains, together with quiescent astroglia (Aq), resting microglia may adopt an anti-inflammatory state (M2) and in turn foster neuron survival (Ns) and prevent astroglia proliferation (Ap) [7,8]. As inflammatory signals (e.g. Ab) gradually build, microglia may adopt an activated pro-inflammatory state (M1), leading to A p proliferation and neuron death (Nd) [9,10,11]. Neuronal debris, amyloid-b (Ab), and/or proliferating astroglia (Ap) may in turn further exacerbate the inflammatory phenotype of M1 macroglia [12,13]. The multiple positive and negative feedbacks among these cells are thus crucial for neurodegeneration that eventually alters the neuronal structure and function during the pathogenesis of AD (Figure 1).
Due to its multi-cellular components and complex nature, conventional experimental approaches have failed to identify critical underlying causes for AD, contributing to the lack of an effective therapeutic treatment. Mathematical models can serve as powerful tools to understand the molecular and cellular processes that control complex diseases [14,15]. Indeed, there have been several attempts to model the process of senile plaque formation [16,17,18,19]. Specifically, these approaches focused on a nucleation step that is coupled with rates for the irreversible binding of Ab monomers to the fibril ends, the lateral aggregation of filaments into fibrils, and fibril elongation through end-to-end association. Other modeling efforts examined the signaling cascade responsible for microglia migration and activation in response to an initial inflammation-provoking stimulus involving Ab [16,20].
However, no systematic modeling approaches have been reported to examine the network cross-talks among microglia, neuron, and astroglia, and the corresponding pathological consequence. Here, we evaluate the dynamic network involving multiple cross-talks among distinct states of microglia, astroglia, and neurons through a mathematical model. Our approach has led to an intriguing insight suggesting that microglia activation in addition to a threshold for Ab may be the critical initiator for the pathogenesis of AD.

Mathematical Method
We propose a sixteen pathway AD mechanism involving seven species that is shown schematically in Fig. 1. The paths have rates a i that implicitly represent the influences of intercellular signaling along them. The mechanism is based on an assumption of constant risk of neuronal death, i.e., a single event randomly initiates cell death independently of the state of any other neuron at any instant [21]. The spatiotemporal influence of diffusion is neglected since local cell events are assumed to occur on a slower timescale than signal dispersion through chemotaxis.
The seven rate equations for the cell populations and the number of Ab molecules in an arbitrary local volume can be written through seven coupled rate equations, namely, These relate the change in each cell population or the number of Ab molecules at any instant to the values of all species at that Table 2. Sensitivities of the cell types to the initial conditions.
Value 10 4 10 2 10 5 10 3 10 3 10 5 10 3 Initial values X(0) of the initial cell populations and the number of molecules of Ab in an arbitrary local volume and orders of magnitude for the sensitivity coefficients S(N j, j = s,d ) = dN j /d(X(0)) determined after 20 years for tenfold perturbations, i.e., 106 and 10 21 6, in these initial values. doi:10.1371/journal.pone.0015176.t002 Figure 1. Schematic of the AD mechanism that incorporates feedback influences from surviving and dead neurons, N s and N d , quiescent and proliferating astroglia A q and A p , reactive and normal microglia, M 1 and M 2 , and Ab. The rates associated with the pathways are included in Table 1 The rates for each a i are specified, as shown in presented in Table 1 for each pathway. Since the literature points to the path N s R Ab being dominant, we assume that it is also the fastest. Its rate is set at 1/year, i.e., each year every N s cell stimulates the formation of a sustaining an Ab molecule. Likewise, since neuronal survival decreases significantly once disease progresses, we assume that the overall path M 2 R A q R N s is slow so that the associated rates a 1 and a 4 are also relatively the smallest. The other rates are similarly specified in terms of their relative abilities to facilitate or inhibit the formation of a cell or Ab molecule according to the particular pathway. Next, we specify the initial composition of the volume under consideration. These initial conditions for the seven species are presented in Table 2.

Results
Our objective is to be able to describe neuropathogenesis during AD in terms of the N s and N d populations. Hence, we first determine the sensitivities of these cells to changes in the rates a i , using the usual definition of the sensitivity coefficient, The sensitivity coefficients for N s , N d , M 1 and M 2 cells, presented in Table 1, are determined after 20 years for 62.5% perturbations in each a i value. A cell population is more sensitive  Table 1. The removal rate a r stabilizes the net number of Ab molecules after three years so that there is only a gradual increase in N d and corresponding decline in N s thereafter. The microglia populations are also consequently relatively stable. doi:10.1371/journal.pone.0015176.g002  Table 1 for the Ab removal rate. As a r decreases, there is an increase in neuropathogenesis so that all four populations increase. doi:10.1371/journal.pone.0015176.g003 to a change in a rate that produces a larger value of |S(N j )|. Positive values for S(N j ) imply that a rate contributes to an increase in N j while a negative value implies that its influence leads to a corresponding population decrease. The sensitivity analysis shows that the N s and N d populations are most sensitive to the path A q R N s , which increases neuronal survival and decreases neuron death. Important paths that inhibit neuropathogenesis include A q R M 2 , A q H M 1 and M 2 H M 1 , while those that enhance disease involve M 1 R N d , Ab H M 2 and Ab R M 1 .
A similar analysis that perturbs the initial cell populations and the number of Ab molecules tenfold is presented in Table 2. It shows that, in comparison to the other species, the M 1 population is most sensitive to these substantial perturbations in the initial amount of any species while N d is only sensitive to the initial amounts of M 1 and M 2 . This implies an important role for microglia during AD progression. The sensitivity coefficients S(M 1 ) and S(M 2 ), also presented in Table 1, show that, as for N s and N d , the dominant paths that inhibit neuropathogenesis by affirming M 2 and decreasing M 1 are also A q R M 2 , A q H M 1 and M 2 H M 1 . Once again, paths 7 and 14 involving Ab, i.e., Ab H M 2 and Ab R M 1 promote AD progression. The model suggests that interventions aimed at decreasing a 8 and a 13 , which involve M 1 , M 2 and Ab and contribute to AD progression, are the ones more likely to diminish neuropathogenesis. This intuitive result emphasizes that decreasing the number of reactive microglia and ensuring a sufficient population of quiescent astroglia is important in treating AD.
The temporal variation in various species for the rates in Table 1 is illustrated in Fig. 2. Figure 2(a) presents the N s , M 1 and Ab populations over 20 years, and Fig. 2(b) the corresponding values for N d and A p . Most notable is the influence of the removal rate a r , which stabilizes the number of Ab molecules after three years. Following that period, there is only a gradual increase in N d that is coupled with a corresponding decline in N s . Consequently, the microglia populations are also relatively stable. Therefore, the rates in Table 1 should be considered as being representative of a healthy population.
We examine the influence of varying a r on neuropathogeneis in Fig. 3, which presents the M 1 , A p , Ab and N d populations over 20 years for three values of a r . As a r decreases, there is an increasing neuronal death. Thus, all four populations, which are associated with AD progression, increase. While microglia play an important role in AD, Fig. 3 shows how the local Ab concentration plays a critical role in initiating and promoting AD.
We investigate this further by varying a 8 and a 13 . Figure 4 presents results for the M 1 , N d and Ab populations over 20 years for three values of a 13 . As a 13 increases, the M 1 , A p and N d populations also increase, leading to an associated decrease in neuronal survival, as illustrated through Eqs. (1) and (2) of the mathematical model. A tenfold increase in a 13 leads to a near doubling in N d after 20 years. As N s decreases so does Ab, but the smaller protein concentration is still sufficient to promote neuropathogenesis among the smaller N s population. Identical results are obtained for similar variations in the rate a 8 for Ab H M 2 , since the sensitivity coefficients for each of M 1 and N d towards paths 8 and 13 are identical.

Discussion
We present a mathematical model for neuropathogenesis during AD that involves neurons, normal and reactive glial cells, and Ab. It uses neuronal death as a surrogate for senile plaque formation. By monitoring neuronal health, we are able to identify intuitive strategies for interventions. In particular, the model suggests that the most effective intervention is one that improves the inhibition of reactive microglia and Ab by normal microglia, and ensuring a sufficient population of quiescent astroglia. Overall, neuropathogenesis proceeds through the production of reactive microglia.
Our analysis is consistent with experimental data that indicate that inflammation may be an early initiator for AD, long before the apparent senile plaque formation [22,23]. It further reinforces  Table 1 for the path Ab R M 1 . As a 13 increases, M 1 and N d also increase and, consequently, there is an associated decrease in neuronal survival. This is also illustrated through Eqs. (1) and (2)  the notion that additional studies should be directed at examining earlier inflammatory signals and alterations involving microglia as a key node so as to better define AD initiation and understand mechanisms for effective prevention and treatment of the disease.
We realize that our mathematical analysis is an initial attempt to examine AD and may not fully account for the associate intertwined cellular communication pathways. Nevertheless, it serves as a hypothesis provoking and building process that should encourage integrated analyses of AD pathogenesis. Future experimental data examining the cross-talks among microglia, astroglia, and neurons will allow us to better refine our model and implement realistic parameters in the rate equations.