Mathematical model on Alzheimer’s disease
 Wenrui Hao^{1}Email author and
 Avner Friedman^{2}
https://doi.org/10.1186/s1291801603482
© The Author(s) 2016
Received: 25 June 2016
Accepted: 25 October 2016
Published: 18 November 2016
Abstract
Background
Alzheimer disease (AD) is a progressive neurodegenerative disease that destroys memory and cognitive skills. AD is characterized by the presence of two types of neuropathological hallmarks: extracellular plaques consisting of amyloid βpeptides and intracellular neurofibrillary tangles of hyperphosphorylated tau proteins. The disease affects 5 million people in the United States and 44 million worldwide. Currently there is no drug that can cure, stop or even slow the progression of the disease. If no cure is found, by 2050 the number of alzheimer’s patients in the U.S. will reach 15 million and the cost of caring for them will exceed $ 1 trillion annually.
Results
The present paper develops a mathematical model of AD that includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. The model is represented by a system of partial differential equations. The model is used to simulate the effect of drugs that either failed in clinical trials, or are currently in clinical trials.
Conclusions
Based on these simulations it is suggested that combined therapy with TNF α inhibitor and anti amyloid β could yield significant efficacy in slowing the progression of AD.
Keywords
Background
AD is the most common form of dementia. The disease is an irreversible, progressive, brain disorder that destroys memory and cognitive skills, and eventually the ability to carry out even the simplest tasks. While the genetic inheritability of AD is in the range of 50 –80% [1, 2], the cause of the disease is mostly unknown. The disease strikes ageing people typically 65 or older, and twice more women than men. In 2015 there were more than 5 million people in the United States with AD, and 44 millions worldwide [3]. The cost of caring for AD patients in the U.S. was estimated at $226 billions for 2015 [3].
AD is characterized by the presence of two types of neuropathological hallmarks: extracellular plaques and intracellular neurofibrillary tangles (NFTs). The extracellular plaques consist primarily of amyloid βpeptide (A β) deposits. The NFTs are intraneural aggregation of hyperphosphorylated tau proteins. Reactive oxygen species (ROS) appears to be one of the early events in the progression of the disease [4]. Amyloid precursor protein (APP) on neurons membrane constitutively shed A β peptides [5]. High levels of ROS promote abnormal deposition of A β [4, 6]. Tau protein in the central nervous system (CNS) is predominantly expressed in neurons; its main role is to promote microtubles assembly and stability. Glycogen synthase kinasetype 3(GSK3) is activated by the abnormally produced A β, and it mediates the hyperphosphorylation of tau proteins [4, 6–9].
The hyperphosphorylated tau proteins cause microtuble depolymerization and destruction, as they aggregate to form neurofibrillary tangles. This results in neuronal death and release of the NFTs to the extracellular environment [4, 10].
The nonneuronal cells in the brain consist of cells that support neurons directly, mostly astrocytes, and immune cells.
Microglias are the resident macrophages in the brain. They constitute the main active immune cells in the brain. They are activated by soluble A β oligomers which build up from the A β deposits [11, 12].
Astrocytes are in close proximity to neurons. They support neuronal crosstalk, and mediate the transport of nutrients from the blood to neurons. Astrocytes are activated primarily by TNF α, but also by A β [10, 13–16]. Activated astrocytes produce A β, but at a smaller rate than neurons [16]. Activated astrocytes also produce MCP1, which attracts monocytes from the blood into the plaques [17–19]. The monocytes differentiate into proinflammatory macrophages, \(\hat {M}_{1}\), but may then change phenotype into antiinflammatory \(\hat {M}_{2}\) macrophage. Activated microglias have two phenotypes: proinflammatory M _{1} macroglia and antiinflammatory M _{2} macroglia [12, 20]. Macrophages have a major role in A β clearance [12, 20], but activated microglia are poorly phagocytic for A β compared to peripheral macrophages [21]. M _{1} and \(\hat {M}_{1}\) macrophages are neurotoxic; they produce proinflammatory cytokines TNF α, IL6, IL12 and IL1 β [20, 22, 23]. M _{2} microglias and peripheral \(\hat {M}_{2}\) macrophages produce antiinflammatory cytokines IL10, IL13, IL4 and TGF β [20]. The neuronal stress caused by the proinflammatory cytokines, is resisted by IL10, IL13 and IL4, but nevertheless it contributes to neuronal damage and death [20, 22, 23].
There are currently no drugs that can cure AD, or stop its progression. Many clinical trials of drugs aimed at preventing or clearing the A β and tau pathology have failed to demonstrate efficacy [24–27]. Currently the only treatment of AD is by medications that are used to treat the symptoms of the disease.
The role of TGF β is somewhat controversial [28]. On one hand, TGF β provides protection against neuroninflammation and neurondegeneration [29–34], but on the other hand, TGF βinduced TIAF1 interacts with amyloid fibrils to favorably support plaque formation [28], and blocking TGF βsmad2/3 in peripheral macrophages mitigates AD pathology [35].
In this paper we develop a mathematical model of AD. The model is represented by a system of partial differential equations (PDEs) based on Fig. 1. For simplicity we represent all the proinflammatory cytokines by TNF α, and all the antiinflammatory cytokines by IL10.
We shall use our model to conduct in silico trials with several drugs: TNF α inhibitor, antiA β drug, MCP1 inhibitor, and injection of TGF β. Simulations of the model show that continuous treatment with TNF α inhibitor yields a slight decrease the death of neurons, and antiA β drug yields a slight decrease in the aggregation of A β over 10 years period, while the benefits from injection of TGF β and MCP1 inhibitor drugs are negligible. This suggests that clinical trials consider combination therapy with TNF α and antiA β drugs.
We note that Fig. 1 does not display neurites: the projections of axons and dendrites from the body of neurons. It is known that the aggregations of A β mediate rapid disruption of synaptic plasticity and memory [36–39]. Thus the progression of AD in terms of reduction in dendritic complexity and synaptic dysfunction will not be considered in the present paper.
We conclude the Introduction by mentioning earlier mathematical models which deal with some aspects of AD: A β polymerization [40], A β plaque formation and the role of prions interacting with A β [41, 42], linear crosstalk among brain cells and A β [43], and the influence of SORLA on AD progression [44, 45].
Methods
Mathematical model
Model’s variables
The variables of the model; concentration and densities are in units of g/c m ^{3} for cells and g/m l for cytokines
ROS (R):  Reactive oxygen species  GSK3 (G):  Glycogen synthase kinasetype 3 
\(A_{\beta }^{i}\):  Amyloid β inside neurons  \(A_{\beta }^{o}\):  Amyloid β outside neurons 
NFT (F _{ i }):  Neuronfibrillary tangle inside neurons  NFT (F _{ o }):  Neuronfibrillary tangle outside neurons 
APP (A _{ P }):  Amyloid precursor protein  A βO (A _{ O }):  Amyloid β oligomer (soluble) 
TNF α (T _{ α }):  Tumor necrosis factor alpha  TGF β (T _{ β }):  Transforming growth factor beta 
IL10 (I _{10}):  Interleukin 10  P:  MCP1 
M _{1}:  Proinflammatory microglias  M _{2}:  Antiinflammatory microglias 
MG (M _{ G }):  Microglias  N:  Live neurons 
A :  Astrocytes  N _{ d }:  Dead neurons 
\(\hat {M}_{1}\)  Peripheral proinflammatory macrophages  \(\hat {M}_{2}\):  Peripheral antiinflammatory macrophages 
τ  hyperphosphorylated tau protein  H  High mobility group box 1 (HMGB1) 
Equations for A β
where N _{0} is the reference density of the neuron cells in the brain.
where \(\bar {K}_{A_{\beta }^{o}}\) is a MichaelisMenten coefficient. Neurons die at a rate \(\frac {\partial N}{\partial t}\), thereby releasing their \(A_{\beta }^{i}\). Hence they contribute \(A_{\beta }^{i}\left \frac {\partial N}{\partial t}\right \) to the growth rate of \(A_{\beta }^{o}\), which is the first term on the righthand side of Eq. (2). The second term on the righthand side of Eq. (2) represents A β constitutively released from APP [5], and the third term accounts for A β released by activated astrocytes [16]; A _{0} is the reference density of the astrocyte cells in the brain. \(A_{\beta }^{0}\) is cleared primarily by peripheral macrophages \(\hat {M}_{1}\) and \(\hat {M}_{2}\), but also by activated microglias M _{1} and M _{2}, so \(d_{A_{\beta }^{o}\hat {M}}>d_{A_{\beta }^{o}{M}}\) [21], and \(\hat {M}_{1}\) M _{1} are more effective in clearing \(A_{\beta }^{o}\) than \(\hat {M}_{2}\) and M _{2} [46, 47] so 0≤θ<1. APP on live neurons shed A β peptides both inside the neurons (as \(A_{\beta }^{i}\)) and outside the neurons (as \(A_{\beta }^{o}\)). We assume that most \(A_{\beta }^{o}\) are produced from dead neurons. Hence, in Eq. (2), we neglected the production of \(A_{\beta }^{o}\) by live neurons. We also assumed that ROS increases primarily the A β that are within live neurons, and thus neglected the increase of \(A_{\beta }^{o}\) by ROS.
Equation for τ
We assume that initially we already have a disease state. Thus, in particular, the tau proteins are already hyperphophorylated and ROS induces increases in the production of these proteins.
Equations for NFT
Equation for neurons
where the death rates of N caused by F _{ i } and T _{ α } are assumed to depend on their saturation levels.
Equation for astrocytes
Equation for dead neurons
where \(\bar {K}_{N_{d}}\) is a MichaelisMenten coefficient. The first two terms on the righthand side arise from the death of N cells. The last two terms account for the clearance of N _{ d } by microglias and peripheral macrophages [48].
Equation for A βO
where \(\lambda _{A_{O}}\) is the rate by which the A _{ O } are formed from the extracellular amyloid β peptides, and \(D_{A_{O}}\Delta A_{O}\) accounts for the diffusion of A _{ O }.
Equation for HMGB1
Equations for activated microglias
where \(\varepsilon _{1}=\frac {T_{\alpha }}{T_{\alpha }+K_{T_{\alpha }}}\) and \(\varepsilon _{2}=\frac {I_{10}}{I_{10}+K_{I_{10}}}\).
Microglias can travel in the brain [55]. Activated microglias are chemoattracted to dead neurons [10, 13, 15], more precisely, to the cytokines HMGB1 produced by N _{ d }, and this is represented by the second term of the lefthand side of Eqs. (11), (12). Microglias are activated by extracelluar NFTs [10, 13, 15], and by soluble oligomers A _{ O } [11, 12]. They become of M _{1} phenotype under proinflammatory signals from TNF α, and of M _{2} phenotype under antiinflammatory signals from IL10. These facts are expressed by the first term on the righthand sides of Eqs. (11), (12); \(\frac {\beta \varepsilon _{1}}{\beta \varepsilon _{1}+\varepsilon _{2}}\) is the ratio by which the activated microglias become M _{1} macrophages, and \(\frac {\varepsilon _{2}}{\beta \varepsilon _{1}+\varepsilon _{2}}\) is the ratio by which activated microglias become M _{2} macrophages. The parameter β reflects the ratio of proinflammatory/antiinflammatory environment, as determined by the relative ‘strength’ of T _{ α } v.s. I _{10}.
In addition, there is a transition M _{1}→M _{2} under the TGF β signaling [32], which is accounted by the second term on the righthand side of these equations.
Equations for macrophages
where \(\hat {M}=\hat {M}_{1}+\hat {M}_{2}\) and \(\alpha (P)=\alpha \frac {P}{P+K_{P}}\) [56].
Equations for TGF β, TNF α MCP1 and IL10
The estimates of parameters in Eqs. (1)–(18) are given in “Appendix".
Results and discussions
Parameters’ description and value
Parameter  Description  Value 

\(D_{A_{O}}\)  Diffusion coefficient of A βO  4.32×10^{−2} c m ^{2} day ^{−1} estimated 
D _{ H }  Diffusion coefficient of HMGB1  8.11×10^{−2} c m ^{2} day ^{−1} estimated 
\(D_{T_{\alpha }}\)  Diffusion coefficient for TNF α  6.55×10^{−2} c m ^{2} day ^{−1} estimated 
\(d_{T_{\beta }}\)  Diffusion coefficient of TGF β  6.55×10^{−2} c m ^{2} day ^{−1} estimated 
\(D_{I_{10}}\)  Diffusion coefficient of IL10  6.04×10^{−2} c m ^{2} day ^{−1} estimated 
D _{ P }  Diffusion coefficient of MCP1  1.2×10^{−1} c m ^{2} day ^{−1} estimated 
\(\lambda _{\beta }^{i} \)  Production rate of \(A_{\beta }^{i}\)  9.51×10^{−6} g/ml/day estimated 
λ _{ N }  Production rate of \(A_{\beta }^{o}\) by neuron  8×10^{−9} g/ml/day estimated 
λ _{ A }  Production rate of \(A_{\beta }^{o}\) by astrocytes  8×10^{−10} g/ml/day estimated 
λ _{ τ0}  Production rate of tau proteins in health  8.1×10^{−11} g/ml/day estimated 
λ _{ τ }  Production rate of tau proteins by ROS  1.35×10^{−11} g/ml estimated 
λ _{ F }  Production rate of NFT by tau  1.662×10^{−3}/day estimated 
\(\lambda _{AT_{\alpha }}\)  Production/activation rate of astrocytes by TNF α  1.54/day estimated 
\(\lambda _{AA_{\beta }^{o}}\)  Production/activation rate of astrocytes by \(A_{\beta }^{o}\)  1.793/day estimated 
\(\lambda _{A_{O}}\)  Production rate of A βO  5×10^{−2}/day estimated 
λ _{ H }  Production rate of HMGB1  3×10^{−5}/day estimated 
λ _{ MF }  Production/activation rate of microglias by NFT  2×10^{−2}/day estimated 
λ _{ MA }  Production/activation rate of microglias by astrocytes  2.3×10^{−3}/day estimated 
\(\lambda _{M1T_{\beta }}\)  Rate of M _{1}→M _{2}  6×10^{−3}/day estimated 
\(\lambda _{\hat {M}_{1}T_{\beta }}\)  Rate of \(\hat {M}_{1}\rightarrow \hat {M}_{2}\)  6×10^{−4}/day estimated 
\(\lambda _{T_{\beta } M}\)  Production rate of TGF β by M  
\(\lambda _{T_{\beta }\hat {M}}\)  Production rate of TGF β by \(\hat {M}\)  
\(\lambda _{T_{\alpha } M1}\)  Production rate of TNF α by M _{1}  3×10^{−2} day ^{−1} estimated 
\(\lambda _{T_{\alpha } \hat {M}_{1}}\)  Production rate of TNF α by \(\hat {M}_{1}\)  3×10^{−2} day ^{−1} estimated 
\(\lambda _{I_{10}M_{2}}\)  Production rate of IL10 by M _{2}  
\(\lambda _{I_{10}\hat {M}_{2}}\)  Production rate of IL10 by \(\hat {M}_{2}\)  
λ _{ PA }  Production rate of MCP1 by astrocytes  6.6×10^{−8} day ^{−1} estimated 
\(\lambda _{PM_{2}}\)  Production rate of MCP1 by M _{2}  1.32×10^{−7} day ^{−1} estimated 
θ  M _{2}/M _{1} effectivity in clearance of \(A_{\beta }^{o}\)  0.9 estimated 
α  Flux rate of macrophages  5 estimated 
β  Proinflammatory/antiinflammatory ratio  10 estimated 
γ  I _{10} inhibition ratio  1 estimated 
Parameters’ description and value
Parameter  Description  Value 

\(d_{A_{\beta }^{i}}\)  Degradation rate of \(A_{\beta }^{i}\)  9.51/day [82] 
\(d_{A_{\beta }^{o}}\)  Degradation rate of \(A_{\beta }^{o}\)  9.51/day [82] 
\(d_{A_{\beta }^{o}{M}}\)  Clearance rate of \(A_{\beta }^{o}\) by microglia  2×10^{−3}/day estimated 
\(d_{A_{\beta }^{o}\hat {M}}\)  Clearance rate of \(A_{\beta }^{o}\) by macrophages  10^{−2}/day estimated 
d _{ τ }  Degradation rate of tau proteins  0.277/day [88] 
\(d_{F_{i}}\)  Degradation rate of intracellular NFT  2.77×10^{−3}/day estimated 
\(d_{F_{o}}\)  Degradation rate of extracellular NFT  2.77×10^{−4}/day estimated 
d _{ N }  Death rate of neurons  1.9×10^{−4}/day estimated 
d _{ NF }  Death rate of neurons by NFTs  3.4×10^{−4}/day estimated 
d _{ NT }  Death rate of neurons by TNF α  1.7×10^{−4}/day estimated 
\(d_{N_{d}M}\phantom {\dot {i}\!}\)  Clearance rate of dead neurons by M  0.06/day estimated 
\(d_{N_{d}\hat {M}}\)  Clearance rate of dead neurons by \(\hat {M}\)  0.02/day estimated 
d _{ A }  Death rate of astrocytes  1.2×10^{−3} day ^{−1} estimated 
\(d_{{M}_{1}}\phantom {\dot {i}\!}\)  Death rate of M _{1} microglias  
\(d_{{M}_{2}}\phantom {\dot {i}\!}\)  Death rate of M _{2} microglias  
\(d_{\hat {M}_{1}}\phantom {\dot {i}\!}\)  Death rate of M _{1} macrophages  
\(d_{\hat {M}_{2}}\phantom {\dot {i}\!}\)  Death rate of M _{2} macrophages  
\(D_{A_{O}}\)  Degradation rate of A βO  0.951/day estimated 
d _{ H }  Degradation rate of HMGB1  58.71/day [95] 
\(D_{T_{\alpha }}\)  Degradation rate of TNF α  
\(d_{T_{\beta }}\)  Degradation rate of TGF β  
\(d_{I_{10}}\phantom {\dot {i}\!}\)  Degradation rate of IL10  16.64 day ^{−1} [47] 
d _{ P }  Degradation rate of MCP1  
R _{0}  Initial inflammation by ROS  6 estimated 
M _{0}  Monocytes concentration in blood  5×10^{−2} estimated 
N _{0}  Reference density of neuron  0.14 g/c m ^{3} estimated 
\({M_{G}^{0}}\phantom {\dot {i}\!}\)  Source of microglia  0.047 g/c m ^{3} estimated 
A _{0}  Reference density of astrocytes  0.14 g/c m ^{3} estimated 
\(\bar {K}_{A_{\beta }^{o}}\)  MichaelisMention coefficient for \(A_{\beta }^{o}\)  7×10^{−3} g/ c m ^{3} estimated 
\(\bar {K}_{N_{d}}\)  MichaelisMention coefficient for N _{ d }  10^{−3} g/ml estimated 
\(K_{I_{10}}\phantom {\dot {i}\!}\)  Halfsaturation of IL10  2.5×10^{−6} g/ c m ^{3} estimated 
\(K_{T_{\beta }}\phantom {\dot {i}\!}\)  Halfsaturation of TGF β  2.5×10^{−7} g/ml [90] 
K _{ M }  Halfsaturation of microglias  0.047 g/ml estimated 
\(K_{\hat {M}}\)  Halfsaturation of macrophages  0.047 g/ml estimated 
\(K_{M_{1}}\phantom {\dot {i}\!}\)  Halfsaturation of M _{1} microglias  0.03 g/ml estimated 
\(K_{M_{2}}\phantom {\dot {i}\!}\)  Halfsaturation of M _{2} microglias  0.017 g/ml estimated 
\(K_{\hat {M}_{1}}\)  Halfsaturation of \(\hat {M}_{1}\) macrophages  0.04 g/ml estimated 
\(K_{\hat {M}_{2}}\)  Halfsaturation of \(\hat {M}_{2}\) macrophages  0.007 g/ml estimated 
\(K_{F_{i}}\phantom {\dot {i}\!}\)  Halfsaturation of intracellular NFTs  3.36×10^{−10} g/ml [89] 
\(K_{F_{o}}\phantom {\dot {i}\!}\)  Average of extracellular NFTs  2.58×10^{−11} g/ml estimated 
\(K_{A_{O}}\phantom {\dot {i}\!}\)  Average of of A βO  1×10^{−7} g/ml estimated 
K _{ P }  Halfsaturation of MCP1  6×10^{−9} g/ml estimated 
\(K_{T_{\alpha }}\)  Halfsaturation of TNF α  4×10^{−5} g/ml estimated 
We next observe that neurons are dying at approximately the rate of 5% a year, which was one of our important assumptions that was based on clinical data. We also note that, as the disease progresses, the plaque of A β peptides, \(A_{\beta }^{o}\), and the soluble A β oligomers, A _{ O }, are increasing; \(A_{\beta }^{o}\) reaches the level of 7×10^{−6} g/ml, in agreement with clinical data [57], and the assumed average of A _{ O } concentration, \(K_{A_{O}}\), is indeed in good approximation to the average of the profile of A _{ O } in Fig. 2. The assumed average of the F _{ o } concentration, \(K_{F_{o}}\), is also in good agreement with the average of the profile of F _{ o } in Fig. 2.
We note that N _{ d } nearly stabilizes over time, at the level assumed in “Appendix," which means that, over time, macrophages and microglias clear debris of dead cells at nearly the same rate at which neurons are dying. Hence \(\Big \frac {\partial N_{d}}{\partial t}\Big \) becomes very small over time, resulting in significant decline in extracellular NFT, while intracellular NFTs (F _{ i }) maintain a comparatively high level.
We finally note that the density of activated astrocytes is slightly increasing in agreement with a mouse model [58] which reports that astrocytes become increasingly prominent with the progression of the disease. The increase in A causes P also to increase, and the average of P is approximately equal to our estimate of K _{ P } in S.I.
AntiAlzheimer drugs
Until now, all clinical trials aimed to develop drugs that can cure AD have failed. There are currently no drugs that can prevent, stop or even delay the progression of Alzheimer’s disease, and there are many ongoing clinical trials. According to the 2016 Alzheimer’s Disease Facts and Figures, and the National Institute of Aging, if no cure is found, by 2050 the number of alzheimer’s patients in the U.S. will reach 15 millions and the cost of caring for them will exceed $ 1 trillion annually.
Avenues for AD therapies include prevention of build up of plaque (antiamyloid drugs), preventing tau aggregation, and reducing inflammation. Clinical trials are concerned with both safety and efficacy. Here we shall use our mathematical model to conduct in silico trials with several drugs, addressing only the question of efficacy.
Treatment for AD causes changes in the densities of cells and concentrations of cytokines. In order to determine the efficacy of a drug, we should observe (i) to what extend it decreases the death rate of N, since slowing the death of neurons will improve cognition of patients; and (ii) to what extend it decreases \(A_{\beta }^{o}\), since A β aggregation mediates rapid dysfunction of synaptic plasticity and dendritic channels thereby causing memory loss [36–39].
TNF α inhibitor
Since TNF α is implicated in generating neurotoxicity which leads to death of neurons, TNF α inhibitor (etanercept) has been considered as a drug for Alzheimer’s patients [59]. In 2015 clinical trials phase 2 [60] the drug has shown some favorable trends but with “no statistically significant changes in cognition.” Since there were no serious adverse events, it was suggested that a larger, broader group needs to be tested before recommending etanercept for use for general Alzheimer patients.

Run the model for 300 days in order to ensure that AD has been diagnosed in patients;

Apply continuous treatment by the drug from day 300 until the end of 10 years.
TGF β injection
TGF β is an antiinflammatory cytokine which induces phenotype change from proinflammatory to antiinflammatory macrophages. It was suggested that TGF β mitigates AD pathology [29–34].
where g is proportional to the amount of injected TGF β. In steady state, T _{ β } maintains the level of \(K_{T_{\beta }}\), while its degradation rate is \(d_{T_{\beta }}\). Hence the source of T _{ β } in steady state is \(d_{T_{\beta }}T_{\beta }\). We take g to be 10 times this source, that is \(g=10d_{T_{\beta }}K_{T_{\beta }}\). We then follow the same treatment procedure for TNF α inhibitor. The lightblue profiles in Fig. 3 show the results of the treatment, compared to no treatment.
AntiA β drugs
where h is proportional to the amount of the dozing level; we take h=10.
MCP1 inhibitor
with k=10 Following the treatment procedure as in the case of of TNF α inhibitor, Fig. 4 shows no efficacy of the drug in terms of N and \(A_{\beta }^{o}\) in comparison to no treatment.
Methylthiomnium chloride (MTC) is the first identified tau aggregation inhibitor currently in Phase 3 trial [27]. In our model the drug will cause a decrease in the production of tau proteins and in their ability to turn into NFT. We model this by multiplying the production terms λ _{ τ0} and λ _{ τ } by 1/10. Following the procedure as in case of TNF α inhibitor, we found that the drug has almost negligible efficacy (not shown here).
Combination therapy
We see that the efficacy of the combined therapy is very small if f<20 or h<10, and it increases sharply with f and h in the region where {40<f<50,20<h<25}.
From Fig. 6 we see that antiA β antibody decreases the external concentration of A β (\(A_{\beta }^{o}\)) with efficacy less than 0.5 (h=20, f=0). Higher efficacy requires T _{ α } inhibitor (h=20, f=20) which will protect neuron from death and prevent astrocytes activation, and thereby reduce \(A_{\beta }^{o}\). This result can be explained by our assumptions in Eq. (2) where we neglected the production of \(A_{\beta }^{o}\) by live neurons and the increase of \(A_{\beta }^{o}\) by ROS.
The PK/PD literature employs the concept of combination index (ϕ) in order to assess the level of synergy between two drugs [67]. This concept was used in simulations of several diseases (e.g. cancer and microbial diseases) in order to determine optimal dosage regimens [67–69]. Since in our AD model it is not clear how to define ϕ, and no data are available to evaluate ϕ, we shall, instead, introduce the following concept, for example in the case of etanercept and aducanumab:
The synergy map for \(\sigma _{A_{\beta }^{o}}\) is similar to that of σ _{ N } (not shown here), and so the synergy increases when f/g is increased.
From Fig. 5 we see that although the amyloid level are controlled, cell death levels do not decrease significantly. This may suggest that other combinations of drugs may target complimentary pathways more efficiently. For example, it was suggested in [70] that Amyloid β and tau combine to induce neuron into cell cycle, which leads to cell death; accordingly, one could explore using antiA β and anti tau aggregation in combination therapy.
Sensitivity analysis
We observe that ε is negatively correlated to \(A_{\beta }^{o}\). Indeed, if ε is increased, more \(A_{\beta }^{o}\) are cleared out (by Eq. (2)). To see how this affects N we note that if \(A_{\beta }^{o}\) is decreased then A _{ O } decreases (by Eq. (9)) and correspondingly M _{1} decreases (by Eq. (11)), and then T _{ α } decreases (by Eq. (17)); so we may expect N to increase, but perhaps not much, since we have ignored other indirect interactions from the model. From Fig. 8 we see that ε is indeed positively correlated to N but the correlation is small. The correlation levels of ε with respect to N and \(A_{\beta }^{o}\) suggest that an antiA β drug, like aducanumab, will have some benefits in reducing Amyloid β, but little benefit in reducing death of neurons. This is also seen from Fig. 4.
Conclusion
AD is an irreversible progressive neuroninflammatory/neurodegenerative disease that destroys memory and cognitive skills. Currently there is no drug that can cure, stop, or even slow the progression of the disease. Life expectancy at diagnosis is 10 years, and, at death, 50% of the brain neurons have already died. AD patients show abnormal aggregation of betaamyloids (\(A_{\beta }^{o}\)) and neurofibrillary tangles (NFTs) of hyperphosphorylated tau proteins. NFTs destroy microtubles in neurons, which results in neurons death. Soluble \(A_{\beta }^{o}\) oligomers activate microglias (the resident macrophages in the brain), thereby initiating inflammatory response. Additionally, peripheral macrophages, responding to cue from MCP1 produced by astrocytes, are attracted to the brain and increase the inflammatory environment, which is harmful to neurons.
Figure 1 is a schematic network of AD: it includes neurons, astrocytes, microglias, peripheral macrophages, βamyloids, tau proteins, and several cytokines involved in the crosstalk among the cells. In the present paper, we developed a mathematical model of AD based on Fig. 1. The model can be used to explore the efficacy of drugs that may slow the progress of the disease. We conducted several in silico trials with several drugs: etanercept (TNF α inhibitor), injection of TGF β, aducanumab (AntiA β drug) and bindarit (MCP1 inhibitor). We found that at ’10fold’ level, etanercept has the largest efficacy in slowing death of neurons, while aducanumab has the largest efficacy in reducing the aggregation of \(A_{\beta }^{o}\), although these efficacies were quite small. Based on these findings we propose that clinical trials should use a combination therapy with etanercept (f) and aducanumab (h). In Fig. 6 we developed efficacy maps for any combination therapy with 0<f<50 and 0<h<25, and we used this map to derive, in Fig. 7, a synergy map for σ _{ N }=σ _{ N }(f,g). Figure 7 shows that the synergy between f and g increases if f/g increases, while f+g is kept fixed. This suggests that in an optimal regimen with fixed total amount, A, of the drugs, f should be significantly larger than h. We did not consider here, however, adverse side effects that are likely to limit the amount of drugs that can be given to a patient. When these limits become better known, one could then proceed to determine the optimal combination of etanercept and aducanumab for slowing the progression of AD.
The mathematical model developed in this paper depends on some assumptions regarding the mechanism of interactions involving amyloid, tau and neunofilaments in AD. There are currently not enough data to sort out competing assumptions. Hence the conclusion of the paper regarding combination therapy should be taken with caution.
Our mathematical model focused on the progression of AD in terms of neurons death and amyloid β aggregation. But dendritic pathologies also play an important role in the disease. Dendritic abnormalities in AD include dystrophic neuritides, reduction in dendritic complexity and loss in dendritic spines [36, 37]. In particular, A β plaques affect dendritic channels, and NFT mediates synaptic dysfunction [36–39]. Recent studies also begin to address white matter degeneracy that could help identify high risk of AD [72].
Appendix
Parameter estimation
In the sequel, in an expression of the form \(\frac {X}{X+K_{X}}\) in the context of activation, the halfsaturation parameter K _{ X } is taken to be the steady state of the species X provided X tends to a steady state. Hence in a steady state equation this factor is equal to \(\frac {1}{2}\). If X does not tend to a steady state then the parameter K _{ X } will be taken to be the estimated average of X over a period of 10 years, the average survival time of AD patients [73]. In an expression of the form \(\frac {1}{1+\gamma X/K_{X}}\) (where γ=γ(X)) in the context of inhibition, K _{ X } is again the halfsaturation of X, so that in steady state the inhibition is 1/(1+γ). If cells Y phagocytose species X, then the clearing rate is proportional to \(Y\frac {X}{X+\bar {K}_{X}}\) where the MichaelisMenten constant \(\bar {K}_{X}\) depends only on the ‘eating capacity’ of Y, so \(\bar {K}_{X}\) has no relation to the halfsaturation of X.
Diffusion coefficients
Molecular weight of A β is 24 kDa [81], so in soluble state its diffusion coefficient would be 8.64×10^{−2} c m ^{2} day ^{−1}. We assume that soluble oligomer A β O has a smaller diffusion coefficient, namely, \(D_{A_{O}}=4.32\times 10^{2}\) c m ^{2} day ^{−1}.
Eq. (1)
By [82], the halflife of \(A_{\beta }^{i}\) is 1.5–2 h in mice. Hence \(d_{A^{i}_{\beta }}=d_{A^{o}_{\beta }}=\frac {ln 2}{1.75}\times 24\)=9.51 /day. Membrane proteins APP shed amyloid β, some end up inside the cell and some outside the cell. We assume that in healthy steady state \(A^{i}_{\beta }=A^{o}_{\beta }\), however the simulation results do not change appreciably if we take \(A^{o}_{\beta }>A^{i}_{\beta }\). According to [57], the density in braingray matter of \(A_{\beta }^{o}\) is approximately 1000 ng/g in control and 7000 ng/g in AD. Hence, from the steady state of Eq. (1) in a healthy normal case, \(A^{i}_{\beta }=10^{6}\) g/ml and \(\lambda _{\beta }^{i}=d_{A_{\beta }^{i}}\times 10^{6}\)= 9.51×10^{−6} g/ml/day. From the steady state of Eq. (1) in AD and Eq. (21) we then get that R _{0}=6.
The brain has 75% water and 60% of its dry matter is fat. We assume that the average density of brain tissue is 1 g/c m ^{3}. The human brain has 100 billion neurons, and its weight is approximately 1400 g, so its volume is approximately 1400 ml. Hence its neurons number density is 7×10^{7} neurons/ c m ^{3}. The diameter of neurons is 16 μ m [83]. Accordingly, we estimate the volume of 1 neuron to be 2×10^{−9} c m ^{3}, and the neurons density is then 7×10^{7}×2×10^{−9} g/c m ^{3}, that is N _{0}=0.14 g/c m ^{3}.
Eq. (2)
The number of neurons is three times the number of microglia [55], hence \(K_{\hat {M}}=\frac {1}{3}N_{0}=0.047\) g/ml.
By [16] an astrocyte produces much less A β than a neuron, so we take \(\lambda _{A}=\frac {1}{10}\lambda _{N}\).
Microglias are the first responders to NFTs and A βO. Peripheral macrophages arrive later, and their immune response may perhaps exceed that of microglia, but this is currently not known [12, 84]. We assume that in steady state the microglias density M and the peripheral macrophages density \(\hat {M}\) are equal, so that \(\hat {M}=K_{\hat {M}}=M=K_{M}=0.047\) g/ml. Motivated by the inflammatory immune attack in AD [85], we assume that, in steady state, the proinflammatory macrophages exceed the antiinflammatory macrophages, and that proinflammatory peripheral macrophages exceed the proinflammatory microglias. Thus, in steady state, \(\hat {M}_{1}>\hat {M}_{2}\), M _{1}>M _{2} and \(\hat {M}_{1}>M_{1}\), and we take \(K_{\hat {M}_{1}}=0.04\), \(K_{\hat {M}_{2}}=0.007\), \(K_{M_{1}}=0.03\), \(K_{M_{2}}=0.017\).
We assume that \(\hat {M}_{1}\) and M _{1} are more effective than \(\hat {M}_{2}\) and M _{2} in clearing A β, and take θ=0.9.
We assume that survival time of patients with AD is 10 years, and that at the endstage 50% of their neurons have died [73]. Hence, the death rate of N is \(d_{N}=\frac {ln2}{10 ~years}=1.9\times 10^{4}\)/day.
The values of \(\left \frac {\partial N}{\partial t}\right \) for 500<t<1000 days vary very little, i.e., from 1.8×10^{−5} g/ml/day to 1.9×10^{−5} g/ml/day. We take \(\left \frac {dN}{dt}\right =1.8\times 10^{5}\) g/ml/day as the average of \(\left \frac {dN}{dt}\right \) over 10 years, but other choices do not affect significantly our simulation results. We then get that λ _{ N }=4×10^{−9} g/ml/day.
The estimate of λ _{ N } was based on the steadystate assumption in Eq. (2). However, in AD the A β peptides are continuously aggregating, so that the steady state assumption needs to be revised. We do this by increasing the value of λ _{ N }: we take λ _{ N }=2×4×10^{−9}= 8×10^{−9} g/ml/day, and then λ _{ A }=8×10^{−10} g/ml/day.
The number of astrocytes is approximately equal to the number of neurons [86, 87], hence A _{0}=N _{0}=0.14 g/ml.
Eq. (3)
Halflife of tau proteins is 60 hours [88]. Hence \(d_{\tau }=\frac {ln2}{60/24}=24ln2\)=0.277/day. Concentration of tau proteins is in healthy normal individuals is 137 pg/ml and, in AD, 490 pg/ml [89]. From the steady state of Eq. (3) in the healthy case, we have λ _{ τ0}=d _{ τ } τ, where τ=137 pg/ml. Hence λ _{ τ0}= 3.78×10^{−11} g/ml/day. Similarly, λ _{ τ0}+λ _{ τ } R=d _{ τ } τ in AD, where τ=490 pg/ml. Hence we have λ _{ τ } R=8.1×10^{−11} g/ml, or λ _{ τ }=1.35×10^{−11}/day.
Eqs. (4) and (5)
We assume that neurofibrillary tangles inside neurons are much more stable than tau proteins, taking \(d_{F_{i}}=\frac {1}{10^{2}}d_{\tau }=2.77\times 10^{3}\)/day. We also assume that extracellular NFTs do not degrade as fast as internalized NFTs, taking \(d_{F_{o}}=\frac {1}{10}d_{F_{i}}=2.77\times 10^{4}\)/day.
We also assume that 60% of the hyperphosphorglated tau proteins become neurofibrillary tangles. From the steady state of Eq. (4) we then have that \(\lambda _{F}=0.6d_{F_{o}}\phantom {\dot {i}\!}\). Hence λ _{ F }=1.662×10^{−3}/day.
Eq. (6)
It is not known whether the rate of death of neurons caused by NFT is larger or smaller than the death rate caused by T _{ α }. We take d _{ NF }=2d _{ NT }, but the simulation of the model in the case where d _{ NT }=2d _{ NF } are very similar (not shown here). Assuming that at steady state of Eq. (6) the concentrations of F _{ i }, T _{ α } and I _{10} are at halfsaturation, we get \(d_{NF}\left (\frac {1}{2}+\frac {1}{4}\frac {1}{1+\gamma }\right)=d_{N}\), so that \(d_{NF}=\frac {4+4\gamma }{3+2\gamma }\times 1.9\times 10^{4}\)/day and \(d_{NT}=\frac {2+2\gamma }{3+2\gamma }\times 1.9\times 10^{4}\)/day. In particular, if γ=1 then d _{ NF }=2.4×10^{−4}/day and d _{ NT }=1.7×10^{−4}/day. We take \(K_{I_{10}}=2\times 10^{6}\) g/ c m ^{3} (which is somewhat larger than the estimated halfsaturation of I _{10} in lung inflammation [47, 90]). We assume that in AD, 60% of hyperphosphorylated tau proteins (whose concentration in disease is 490 pg/ml [89]) are in NFT form, so that \(K_{F_{i}}=0.6\times 490\) pg/ml= 2.94×10^{−10} g/ml. In [89] the concentration of tau protein was taken uniformly in the tissue of patients. We assume, however, that the concentration of NFT is higher inside neurons than outside neurons, and take \(K_{F_{i}}=3.36\times 10^{10}\) g/ml, \(K_{F_{o}}=2.58\times 10^{11}\) g/ml. From the steady state of Eq. (17) and the estimates of \(\lambda _{T_{\alpha } M_{1}}\) and \(\lambda _{T_{\alpha }\hat {M}_{1}}\) (see under Eq. (17) below) we get T _{ α }=4×10^{−5} g/ml, so that \(K_{T_{\alpha }}=4\times 10^{5}\) g/ml.
Eq. (7)
We take the halflife of astrocytes to be the same as the halflife of ganglionic glial cells, that is, 600 days [91]. Hence d _{ A }=1.2×10^{−3}/day. We assume that the activation of astrocytes is due more to TNF α than to A β, and take \(\lambda _{A T_{\alpha }}T_{\alpha }=2\lambda _{AA_{\beta }^{o}}A_{\beta }^{o}\). By the steady state of Eq. (7) we then get \(\lambda _{AT_{\alpha }}=1.4\)/day, and \(\lambda _{AA_{\beta }^{o}}=1.63\)/day. Actually, in a mouse model of AD, the number of activated astrocytes is increasing [58]. So we compensate for this by increasing both \(\lambda _{AT_{\alpha }}\) and \(\lambda _{AA_{\beta }^{o}}\) by a factor 1.1, taking \(\lambda _{AT_{\alpha }}=1.54\)/day and \(\lambda _{AA_{\beta }^{o}}=1.793\)/day.
Eq. (8)
In mice experiments [92], macrophages phagocytosed apoptotic cells at rates that varied in the range 0.1–1.27/h. We assume that necrotic cells (and their debris) in human brain are phagocytosed by peripheral macrophages at rate \(d_{N_{d}\hat {M}}=0.2\)/day. We also assume that microglia play a greater role in clearing necrotic neurons, and take \(d_{N_{d}M}=3\times 0.2\)=0.6/day. We also take \(\bar {K}_{N_{d}}=10^{3}\) g/ml.
Eq. (9)
We assume the degradation rate of A _{ O } is much slower than that of \(A_{\beta }^{o}\), taking \(d_{A_{O}}=\frac {1}{10}d_{A_{\beta }^{o}}=0.951\)/day. The ratio of soluble A _{ O } to total \(A_{\beta }^{o}\) is approximately \(\frac {1}{25}\) [93].
From the steady state of Eq. (9) we then get \(\lambda _{A_{O}}=\frac {1}{25}d_{A_{O}}=3.8\times 10^{2}\)/day.
The estimate of \(\lambda _{A_{O}}\) was based on the steadystate assumption in Eq. (9). However, in AD the soluble A β oligomer is continuously increasing, following the increase in \(A_{\beta }^{o}\), so the steadystate assumption needs to be revised. We do this by increasing the above value of λ _{ AO }, taking the new value to be λ _{ AO }=5×10^{−2}/day.
Eq. (10)
Concentration of HMGB1 in neurons is 1.3 n g/m l [94], hence H=0.14×1.3 ng/ml= 1.8×10^{−10} g/ml. Halflife of HMGB1 is 17 minutes [95], so that d _{ H }=58.71/day. We assume that N _{ d } stabilizes somewhere below 2.5×10^{−4} g/ml. From the steady state of Eq. (10), we then get λ _{ H }=3×10^{−5}/day.
Eqs. (11) and (12)
We take \(d_{M_{1}}=d_{M_{2}}=0.015\)/day [47, 90]. Then, our assumption (under Eq. (2)) that \(K_{M_{1}}>K_{M_{2}}\) suggests that β>1. We take β=10.
We take \({M_{G}^{0}}=K_{M}=0.047\) g/ml and α=5. In the absence of data, we take the production rate λ _{ MF } of macrophages by NFT to be the same as the production rate under stimulation by M. Tuberculosis in [90], namely, λ _{ MF }=2×10^{−2}/day. We assume that production rate of macrophages by NFT is larger than the production rate by A _{ O }, and take λ _{ MA }=2.3×10^{−3}/day.
By [57] the concentration of A β in AD is 7×10^{−6} g/ml and, by [55], the ratio of A _{ O } to \(A_{\beta }^{o}\) is \(\frac {1}{25}\), so that \(K_{A_{O}}=\frac {1}{25}\times 7\times 10^{6}=2.8\times 10^{7}\) g/ml.
We assume that more NFT reside within neurons than outside them, so that \(K_{F_{o}}\) is smaller than \(K_{F_{i}}\). Recalling that \(K_{F_{i}}=3.36\times 10^{10}\) g/ml, we take \(K_{F_{o}}=2.58\times 10^{11}\) g/ml.
The coefficient \(\lambda _{{M}_{1}T_{\beta }}\phantom {\dot {i}\!}\) is the rate by which TGF β affects the change of phenotype from M _{1} to M _{2}. In the case of infection in the lung by M. tuberculosis, under inflammatory conditions caused by the pathogen, \(\lambda _{M_{1}T_{\beta }}=6\times 10^{3}\)/day [90]; we take it to be the same in the present case. We take \(K_{T_{\beta }}=2.5\times 10^{7}\) g/ml, and \(K_{I_{10}}=2.5\times 10^{6}\) g/ml.
Eqs. (13) and (14)
Peripheral macrophages immigrate into the brain of AD [96, 97]. We assume that, because of the BBB, the concentration of monocytes in the brain capillaries must be significantly higher than the concentration of peripheral macrophages already in the tissue. Recalling that in steady state \(\hat {M}=0.047\) g/ml, we take M _{0}=0.05 g/ml. The parameter α was estimated by 5, in order to make the asymptotic behavior of \(\hat {M}\) in the simulations agree with its assumed steady state of 0.047 g/ml (under Eq. (2)). When microglia cells are activated, they become either of M _{1} or M _{2} phenotype. But peripheral macrophages are initially biased toward \(\hat {M}_{1}\) phenotype rather than \(\hat {M}_{2}\) phenotype, since \(K_{T_{\alpha }}>K_{I_{10}}\). We assume, in line with this bias toward \(\hat {M}_{1}\), that the transition rate from \(\hat {M}_{1}\) into \(\hat {M}_{2}\) phenotype by TGF β is at a smaller rate than the corresponding transition rate for microglias, that is, \(\lambda _{\hat {M}_{1}T_{\beta }}<\lambda _{M_{1}T_{\beta }}\). We take \(\lambda _{\hat {M}_{1}T_{\beta }}=6\times 10^{4}\)/day.
Eq. (17)
Activated alveolar macrophages produce TNF α at rate 4.86×10^{−3}/day [47]. We assume that proinflammatory macrophages produce TNF α at a larger rate (five fold), taking \(\lambda _{T_{\alpha } M_{1}}=\lambda _{T_{\alpha } \hat {M}_{1}}=3\times 10^{2}\) g/ml.
Eq. (18)
with P=6×10^{−9} g/ml and d _{ P }=1.73/day [74], we get \(\lambda _{PM_{2}}=1.2\times 10^{7}\)/day and λ _{ PA }=6×10^{−8}/day [47].
Since A is increasing in time, also P is increasing in time. Hence the steady state assumption needs to be revised. We do it by increasing λ _{ PA } and \(\lambda _{PM_{2}}\) by a factor 1.1, taking \(\lambda _{PM_{2}}=1.32\times 10^{7}\)/day, and λ _{ PA }=6.6×10^{−8}/day.
Declarations
Acknowledgements
The authors have been supported by the Mathematical Biosciences Institute and the National Science Foundation under Grant DMS 0931642.
Availability of data and materials
The dataset supporting the conclusions of this article is included within the article.
Authors’ contributions
WH and AF developed and simulated the model, and wrote the final manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
 Gatz M, Reynolds CA, Fratiglioni L, Johansson B, Mortimer JA, Berg S, Fiske A, Pedersen NL. Role of genes and environments for explaining Alzheimer disease. Arch Gen Psychiat. 2006; 63(2):168–74.View ArticlePubMedGoogle Scholar
 Wilson RS, Barral S, Lee JH, Leurgans SE, Foroud TM, Sweet RA, GraffRadford N, Bird TD, Mayeux R, Bennett DA. Heritability of different forms of memory in the Late Onset Alzheimer’s Disease Family Study. J Alzheimers Dis. 2011; 23(2):249–55.PubMedPubMed CentralGoogle Scholar
 alzheimers, n.: 2015 Alzheimer’s Statistics. 2016. http://www.alzheimers.net/resources/alzheimersstatistics/. Accessed 1 Sept 2016.
 Liu Z, Li P, Wu J, Yi W, Ping L, Xinxin H, et al.The Cascade of Oxidative Stress and Tau Protein Autophagic Dysfunction in Alzheimer’s Disease. Alzheimer’s Dis Challenges Future. 2015;2. doi:10.5772/59980.
 Seeman P, Seeman N. Alzheimer’s disease: betaamyloid plaque formation in human brain. Synapse. 2011; 65(12):1289–97.View ArticlePubMedGoogle Scholar
 Kremer A, Louis JV, Jaworski T, Van Leuven F. GSK3 and Alzheimer’s Disease: Facts and Fiction. Front Mol Neurosci. 2011; 4:17.View ArticlePubMedPubMed CentralGoogle Scholar
 Bloom GS. Amyloidbeta and tau: the trigger and bullet in Alzheimer disease pathogenesis. JAMA Neurol. 2014; 71(4):505–8.View ArticlePubMedGoogle Scholar
 MondragonRodriguez S, Perry G, Zhu X, Boehm J. Amyloid Beta and tau proteins as therapeutic targets for Alzheimer’s disease treatment: rethinking the current strategy. Int J Alzheimers Dis. 2012; 2012:630182.PubMedPubMed CentralGoogle Scholar
 Wray S, Noble W. Linking amyloid and tau pathology in Alzheimer’s disease: the role of membrane cholesterol in Abetamediated tau toxicity. J Neurosci. 2009; 29(31):9665–7.View ArticlePubMedGoogle Scholar
 Mokhtar SH, Bakhuraysah MM, Cram DS, Petratos S. The Betaamyloid protein of Alzheimer’s disease: communication breakdown by modifying the neuronal cytoskeleton. Int J Alzheimers Dis. 2013; 2013:910502.PubMedPubMed CentralGoogle Scholar
 Joshi P, Turola E, Ruiz A, Bergami A, Libera DD, Benussi L, et. al. Microglia convert aggregated amyloidbeta into neurotoxic forms through the shedding of microvesicles. Cell Death Differ. 2014; 21(4):582–93.View ArticlePubMedGoogle Scholar
 Theriault P, ElAli A, Rivest S. The dynamics of monocytes and microglia in Alzheimer’s disease. Alzheimers Res Ther. 2015; 7(1):41.View ArticlePubMedPubMed CentralGoogle Scholar
 de Calignon A, Polydoro M, SuarezCalvet M, William C, Adamowicz DH, Kopeikina KJ, et al.Propagation of tau pathology in a model of early Alzheimer’s disease. Neuron. 2012; 73(4):685–97.View ArticlePubMedPubMed CentralGoogle Scholar
 Garwood CJ, Pooler AM, Atherton J, Hanger DP, Noble W. Astrocytes are important mediators of Abetainduced neurotoxicity and tau phosphorylation in primary culture. Cell Death Dis. 2011; 2:167.View ArticleGoogle Scholar
 Morales I, GuzmanMartinez L, CerdaTroncoso C, Farias GA, Maccioni RB. Neuroinflammation in the pathogenesis of Alzheimer’s disease. A rational framework for the search of novel therapeutic approaches. Front Cell Neurosci. 2014; 8:112.PubMedPubMed CentralGoogle Scholar
 Zhao J, O’Connor T, Vassar R. The contribution of activated astrocytes to A beta production: implications for Alzheimer’s disease pathogenesis. J Neuroinflammation. 2011; 8:150.View ArticlePubMedPubMed CentralGoogle Scholar
 Hohsfield LA, Humpel C. Migration of blood cells to betaamyloid plaques in Alzheimer’s disease. Exp Gerontol. 2015; 65:8–15.View ArticlePubMedPubMed CentralGoogle Scholar
 Li C, Zhao R, Gao K, Wei Z, Yin MY, Lau LT, Chui D, Yu AC. Astrocytes: implications for neuroinflammatory pathogenesis of Alzheimer’s disease. Curr Alzheimer Res. 2011; 8(1):67–80.View ArticlePubMedGoogle Scholar
 Porcellini E, Ianni M, Carbone I, Franceschi M, Licastro F. Monocyte chemoattractant protein1 promoter polymorphism and plasma levels in alzheimer’s disease. Immun Ageing. 2013; 10(1):6.View ArticlePubMedPubMed CentralGoogle Scholar
 Wang WY, Tan MS, Yu JT, Tan L. Role of proinflammatory cytokines released from microglia in Alzheimer’s disease. Ann Transl Med. 2015; 3(10):136.PubMedPubMed CentralGoogle Scholar
 Lai AY, McLaurin J. Clearance of amyloidbeta peptides by microglia and macrophages: the issue of what, when and where. Future Neurol. 2012; 7(2):165–76.View ArticlePubMedPubMed CentralGoogle Scholar
 Bhaskar K, Maphis N, Xu G, Varvel NH, KokikoCochran ON, Weick JP. et al. Microglial derived tumor necrosis factoralpha drives Alzheimer’s diseaserelated neuronal cell cycle events. Neurobiol Dis. 2014; 62:273–85.View ArticlePubMedGoogle Scholar
 Sharma V, Thakur V, Singh S, Guleria R. Tumor Necrosis Factor and Alzheimer’s Disease: A Cause and Consequence Relationship. Klinik Psik Bull Clin Psyc. 2012; 22:86–97.Google Scholar
 Boutajangout A, Sigurdsson EM, Krishnamurthy PK. Tau as a therapeutic target for Alzheimer’s disease. Curr Alzheimer Res. 2011; 8(6):666–77.View ArticlePubMedPubMed CentralGoogle Scholar
 HongQi Y, ZhiKun S, ShengDi C. Current advances in the treatment of Alzheimer’s disease: focused on considerations targeting Abeta and tau. Transl Neurodegener. 2012; 1(1):21.View ArticlePubMedPubMed CentralGoogle Scholar
 Lansdall C. An effective treatment for Alzheimer’s disease must consider both amyloid and tau. Biosci Horizons. 2014;7. doi:10.1093/biohorizons/hzu002.
 Wischik CM, Harrington CR, Storey JM. Tauaggregation inhibitor therapy for Alzheimer’s disease. Biochem Pharmacol. 2014; 88(4):529–39.View ArticlePubMedGoogle Scholar
 Lee MH, Lin SR, Chang JY, Schultz L, Heath J, Hsu LJ, Kuo YM, Hong Q, Chiang MF, Gong CX, Sze CI, Chang NS. TGFbeta induces TIAF1 selfaggregation via type II receptorindependent signaling that leads to generation of amyloid beta plaques in Alzheimer’s disease. Cell Death Dis. 2010; 1:110.View ArticleGoogle Scholar
 Chao CC, Hu S, Frey WH, Ala TA, Tourtellotte WW, Peterson PK. Transforming growth factor beta in Alzheimer’s disease. Clin Diagn Lab Immunol. 1994; 1(1):109–10.PubMedPubMed CentralGoogle Scholar
 Chen JH, Ke KF, Lu JH, Qiu YH, Peng YP. Protection of TGFbeta against neuroinflammation and neurodegeneration in Abeta1–42induced Alzheimer’s disease model rats. PLoS ONE. 2015; 10(2):0116549.Google Scholar
 Das P, Golde T. Dysfunction of TGFbeta signaling in Alzheimer’s disease. J Clin Invest. 2006; 116(11):2855–7.View ArticlePubMedPubMed CentralGoogle Scholar
 von Bernhardi R, Cornejo F, Parada GE, Eugenin J. Role of TGF beta signaling in the pathogenesis of Alzheimer’s disease. Front Cell Neurosci. 2015; 9:426.View ArticlePubMedPubMed CentralGoogle Scholar
 WyssCoray T. TgfBeta pathway as a potential target in neurodegeneration and Alzheimer’s. Curr Alzheimer Res. 2006; 3(3):191–5.View ArticlePubMedGoogle Scholar
 WyssCoray T, Lin C, Yan F, Yu GQ, Rohde M, McConlogue L, Masliah E, Mucke L. TGFbeta1 promotes microglial amyloidbeta clearance and reduces plaque burden in transgenic mice. Nat Med. 2001; 7(5):612–8.View ArticlePubMedGoogle Scholar
 Town T, Laouar Y, Pittenger C, Mori T, Szekely CA, Tan J, et al.Blocking TGFbetaSmad2/3 innate immune signaling mitigates Alzheimerlike pathology. Nat Med. 2008; 14(6):681–7.PubMedPubMed CentralGoogle Scholar
 Cochran JN, Hall AM, Roberson ED. The dendritic hypothesis for Alzheimer’s disease pathophysiology. Brain Res Bull. 2014; 103:18–28.View ArticlePubMedGoogle Scholar
 Dorostkar MM, Zou C, BlazquezLlorca L, Herms J. Analyzing dendritic spine pathology in Alzheimer’s disease: problems and opportunities. Acta Neuropathol. 2015; 130(1):1–19.View ArticlePubMedPubMed CentralGoogle Scholar
 Klyubin I, Cullen WK, Hu NW, Rowan MJ. Alzheimer’s disease Abeta assemblies mediating rapid disruption of synaptic plasticity and memory. Mol Brain. 2012; 5:25.View ArticlePubMedPubMed CentralGoogle Scholar
 Koffie RM, Hyman BT, SpiresJones TL. Alzheimer’s disease: synapses gone cold. Mol Neurodegener. 2011; 6(1):63.View ArticlePubMedPubMed CentralGoogle Scholar
 Craft DL, Wein LM, Selkoe DJ. A mathematical model of the impact of novel treatments on the A beta burden in the Alzheimer’s brain, CSF and plasma. Bull Math Biol. 2002; 64(5):1011–31.View ArticlePubMedGoogle Scholar
 Bertsch M, Franchi B, Marcello N, Tesi MC, Tosin A. Alzheimer’s disease: a mathematical model for onset and progression. Math Med Biol. 2016. doi:10.1093/imammb/dqw003.
 Helal M, Hingant E, PujoMenjouet L, Webb GF. Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions. J Math Biol. 2014; 69(5):1207–35.View ArticlePubMedGoogle Scholar
 Puri IK, Li L. Mathematical modeling for the pathogenesis of Alzheimer’s disease. PLoS ONE. 2010; 5(12):15176.View ArticleGoogle Scholar
 Lao A, Schmidt V, Schmitz Y, Willnow TE, Wolkenhauer O. Multicompartmental modeling of SORLA’s influence on amyloidogenic processing in Alzheimer’s disease. BMC Syst Biol. 2012; 6:74.View ArticlePubMedPubMed CentralGoogle Scholar
 Schmidt V, Baum K, Lao A, Rateitschak K, Schmitz Y, Teichmann A, et al.Quantitative modelling of amyloidogenic processing and its influence by SORLA in Alzheimer’s disease. EMBO J. 2012; 31(1):187–200.View ArticlePubMedGoogle Scholar
 Hamza T, Barnett JB, Li B. Interleukin 12 a key immunoregulatory cytokine in infection applications. Int J Mol Sci. 2010; 11(3):789–806.View ArticlePubMedPubMed CentralGoogle Scholar
 Hao W, Crouser ED, Friedman A. Mathematical model of sarcoidosis. Proc Nat Acad Sci USA. 2014; 111(45):16065–70.View ArticlePubMedPubMed CentralGoogle Scholar
 Sokolowski JD, Mandell JW. Phagocytic clearance in neurodegeneration. Am J Pathol. 2011; 178(4):1416–28.View ArticlePubMedPubMed CentralGoogle Scholar
 Haass C, Selkoe DJ. Soluble protein oligomers in neurodegeneration: lessons from the Alzheimer’s amyloid betapeptide. Nat Rev Mol Cell Biol. 2007; 8(2):101–12.View ArticlePubMedGoogle Scholar
 Waters J. The concentration of soluble extracellular amyloidbeta protein in acute brain slices from CRND8 mice. PLoS ONE. 2010; 5(12):15709.View ArticleGoogle Scholar
 Muller S, Ronfani L, Bianchi ME. Regulated expression and subcellular localization of HMGB1, a chromatin protein with a cytokine function. J Intern Med. 2004; 255(3):332–43.View ArticlePubMedGoogle Scholar
 Gao HM, Zhou H, Zhang F, Wilson BC, Kam W, Hong JS. HMGB1 acts on microglia Mac1 to mediate chronic neuroinflammation that drives progressive neurodegeneration. J Neurosci. 2011; 31(3):1081–92.View ArticlePubMedPubMed CentralGoogle Scholar
 Lotze MT, Tracey KJ. Highmobility group box 1 protein (HMGB1): nuclear weapon in the immune arsenal. Nat Rev Immunol. 2005; 5(4):331–42.View ArticlePubMedGoogle Scholar
 Zou JY, Crews FT. Release of neuronal HMGB1 by ethanol through decreased HDAC activity activates brain neuroimmune signaling. PLoS ONE. 2014; 9(2):87915.View ArticleGoogle Scholar
 Savchenko VL, McKanna JA, Nikonenko IR, Skibo GG. Microglia and astrocytes in the adult rat brain: comparative immunocytochemical analysis demonstrates the efficacy of lipocortin 1 immunoreactivity. Neuroscience. 2000; 96(1):195–203.View ArticlePubMedGoogle Scholar
 Hao W, Rovin BH, Friedman A. Mathematical model of renal interstitial fibrosis. Proc Nat Acad Sci USA. 2014; 111(39):14193–8.View ArticlePubMedPubMed CentralGoogle Scholar
 Roher AE, Esh CL, Kokjohn TA, Castano EM, Van Vickle GD, Kalback WM, et al.Amyloid beta peptides in human plasma and tissues and their significance for Alzheimer’s disease. Alzheimers Dement. 2009; 5(1):18–29.View ArticlePubMedPubMed CentralGoogle Scholar
 Furman JL, Sama DM, Gant JC, Beckett TL, Murphy MP, Bachstetter AD, Van Eldik LJ, Norris CM. Targeting astrocytes ameliorates neurologic changes in a mouse model of Alzheimer’s disease. J Neurosci. 2012; 32(46):16129–40.View ArticlePubMedPubMed CentralGoogle Scholar
 Tobinick E, Gross H, Weinberger A, Cohen H. TNFalpha modulation for treatment of Alzheimer’s disease: a 6month pilot study. MedGenMed. 2006; 8(2):25.PubMedPubMed CentralGoogle Scholar
 Butchart J, Brook L, Hopkins V, Teeling J, Puntener U, Culliford D, et al. Etanercept in Alzheimer disease: A randomized, placebocontrolled, doubleblind, phase 2 trial. Neurology. 2015; 84(21):2161–8.View ArticlePubMedPubMed CentralGoogle Scholar
 Piazza F, Winblad B. AmyloidRelated Imaging Abnormalities (ARIA) in Immunotherapy Trials for Alzheimer’s Disease: Need for Prognostic Biomarkers?J Alzheimers Dis. 2016; 52(2):417–20.View ArticlePubMedGoogle Scholar
 Karran E, Hardy J. A critique of the drug discovery and phase 3 clinical programs targeting the amyloid hypothesis for Alzheimer disease. Ann Neurol. 2014; 76(2):185–205.View ArticlePubMedPubMed CentralGoogle Scholar
 Patel KR. Biogen’s aducanumab raises hope that Alzheimer’s can be treated at its source. Manag Care. 2015; 24(6):19.Google Scholar
 Reardon S. Antibody drugs for Alzheimer’s show glimmers of promise. Nature. 2015; 523(7562):509–10.View ArticlePubMedGoogle Scholar
 Ge S, Shrestha B, Paul D, Keating C, Cone R, Guglielmotti A, Pachter JS. The CCL2 synthesis inhibitor bindarit targets cells of the neurovascular unit, and suppresses experimental autoimmune encephalomyelitis. J Neuroinflammation. 2012; 9:171.View ArticlePubMedPubMed CentralGoogle Scholar
 Severini C, Passeri PP, Ciotti M, Florenzano F, Possenti R, Zona C, et al.Bindarit, inhibitor of CCL2 synthesis, protects neurons against amyloid??induced toxicity. J Alzheimers Dis. 2014; 38(2):281–93.PubMedGoogle Scholar
 Li JY, Ren YP, Yuan Y, Ji SM, Zhou SP, Wang LJ, Mou ZZ, Li L, Lu W, Zhou TY. Preclinical PK/PD model for combined administration of erlotinib and sunitinib in the treatment of A549 human NSCLC xenograft mice. Acta Pharmacol Sin. 2016; 37(7):930–40.View ArticlePubMedPubMed CentralGoogle Scholar
 Nielsen EI, Cars O, Friberg LE. Pharmacokinetic/pharmacodynamic (PK/PD) indices of antibiotics predicted by a semimechanistic PKPD model: a step toward modelbased dose optimization. Antimicrob Agents Chemother. 2011; 55(10):4619–30.View ArticlePubMedPubMed CentralGoogle Scholar
 Yuan Y, Zhou X, Ren Y, Zhou S, Wang L, Ji S, Hua M, Li L, Lu W, Zhou T. SemiMechanismBased Pharmacokinetic/Pharmacodynamic Model for the Combination Use of Dexamethasone and Gemcitabine in Breast Cancer. J Pharm Sci. 2015; 104(12):4399–408.View ArticlePubMedGoogle Scholar
 Bloom GS. Amyloidbeta and tau: the trigger and bullet in Alzheimer disease pathogenesis. JAMA Neurol. 2014; 71(4):505–8.View ArticlePubMedGoogle Scholar
 Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008; 254(1):178–96.View ArticlePubMedPubMed CentralGoogle Scholar
 CollinsPraino LE, Francis YI, Griffith EY, Wiegman AF, Urbach J, Lawton A, Honig LS, Cortes E, Vonsattel JP, Canoll PD, Goldman JE, Brickman AM. Soluble amyloid beta levels are elevated in the white matter of Alzheimer’s patients, independent of cortical plaque severity. Acta Neuropathol Commun. 2014; 2:83.PubMedPubMed CentralGoogle Scholar
 Mohs RC, Haroutunian V. Chapter 82: Alzheimer Disease: From Earliest Symptoms to End Stage. Neuropsychopharmacology: The Fifth Generation of Progress. 1999; 8(2):1189–1197.Google Scholar
 Hao W, Friedman A. The LDLHDL profile determines the risk of atherosclerosis: a mathematical model. PLoS ONE. 2014; 9(3):90497.View ArticleGoogle Scholar
 Young ME, Carroad PA, Bell RL. Estimation of Diffusion Coefficients of Proteins. Biot Bioe. 1980; 22(5):947–55.View ArticleGoogle Scholar
 Chen D, Roda JM, Marsh CB, Eubank TD, Friedman A. Hypoxia inducible factorsmediated inhibition of cancer by GMCSF: a mathematical model. Bull Math Biol. 2012; 74(11):2752–77.PubMedPubMed CentralGoogle Scholar
 Yokochi S, Hashimoto H, Ishiwata Y, Shimokawa H, Haino M, Terashima Y, Matsushima K. An antiinflammatory drug, propagermanium, may target GPIanchored proteins associated with an MCP1 receptor, CCR2. J Interferon Cytokine Res. 2001; 21(6):389–98.View ArticlePubMedGoogle Scholar
 Dubois CM, Laprise MH, Blanchette F, Gentry LE, Leduc R. Processing of transforming growth factor beta 1 precursor by human furin convertase. J Biol Chem. 1995; 270(18):10618–24.View ArticlePubMedGoogle Scholar
 Stepanets OV, Chichasova NV, Nasonova MB, Samsonov MIU, Nasonov EL. [Soluble receptors of TNFalpha with molecular mass 55 kDa in rheumatoid arthritis: clinical role]. Klin Med (Mosk). 2003; 81(4):42–6.Google Scholar
 Bonaldi T, Talamo F, Scaffidi P, Ferrera D, Porto A, Bachi A, Rubartelli A, Agresti A, Bianchi ME. Monocytic cells hyperacetylate chromatin protein HMGB1 to redirect it towards secretion. EMBO J. 2003; 22(20):5551–60.View ArticlePubMedPubMed CentralGoogle Scholar
 Ahmed M, Davis J, Aucoin D, Sato T, Ahuja S, Aimoto S, Elliott JI, Van Nostrand WE, Smith SO. Structural conversion of neurotoxic amyloidbeta(1–42) oligomers to fibrils. Nat Struct Mol Biol. 2010; 17(5):561–7.View ArticlePubMedPubMed CentralGoogle Scholar
 Saido T, Leissring MA. Proteolytic degradation of amyloid betaprotein. Cold Spring Harb Perspect Med. 2012; 2(6):006379.View ArticleGoogle Scholar
 Cragg BG. The density of synapses and neurons in normal, mentally defective ageing human brains. Brain. 1975; 98(1):81–90.View ArticlePubMedGoogle Scholar
 Gate D, RezaiZadeh K, Jodry D, Rentsendorj A, Town T. Macrophages in Alzheimer’s disease: the bloodborne identity. J Neural Transm (Vienna). 2010; 117(8):961–70.View ArticleGoogle Scholar
 Heppner FL, Ransohoff RM, Becher B. Immune attack: the role of inflammation in Alzheimer disease. Nat Rev Neurosci. 2015; 16(6):358–72.View ArticlePubMedGoogle Scholar
 HerculanoHouzel S. The human brain in numbers: a linearly scaledup primate brain. Front Hum Neurosci. 2009; 3:31.View ArticlePubMedPubMed CentralGoogle Scholar
 HerculanoHouzel S. The glia/neuron ratio: how it varies uniformly across brain structures and species and what that means for brain physiology and evolution. Glia. 2014; 62(9):1377–91.View ArticlePubMedGoogle Scholar
 Poppek D, Keck S, Ermak G, Jung T, Stolzing A, Ullrich O, Davies KJ, Grune T. Phosphorylation inhibits turnover of the tau protein by the proteasome: influence of RCAN1 and oxidative stress. Biochem J. 2006; 400(3):511–20.View ArticlePubMedPubMed CentralGoogle Scholar
 Kapaki E, Kilidireas K, Paraskevas GP, Michalopoulou M, Patsouris E. Highly increased CSF tau protein and decreased betaamyloid (1–42) in sporadic CJD: a discrimination from Alzheimer’s disease?J Neurol Neurosurg Psychiatr. 2001; 71(3):401–3.View ArticlePubMedPubMed CentralGoogle Scholar
 Hao W, Schlesinger LS, Friedman A. Modeling Granulomas in Response to Infection in the Lung. PLoS ONE. 2016; 11(3):0148738.Google Scholar
 Elson K, Ribeiro RM, Perelson AS, Simmons A, Speck P. The life span of ganglionic glia in murine sensory ganglia estimated by uptake of bromodeoxyuridine. Exp Neurol. 2004; 186(1):99–103.View ArticlePubMedGoogle Scholar
 Maree AF, Komba M, Finegood DT, EdelsteinKeshet L. A quantitative comparison of rates of phagocytosis and digestion of apoptotic cells by macrophages from normal (BALB/c) and diabetesprone (NOD) mice. J Appl Physiol. 2008; 104(1):157–69.View ArticlePubMedGoogle Scholar
 Wang J, Dickson DW, Trojanowski JQ, Lee VM. The levels of soluble versus insoluble brain Abeta distinguish Alzheimer’s disease from normal and pathologic aging. Exp Neurol. 1999; 158(2):328–37.View ArticlePubMedGoogle Scholar
 Zhu XD, Chen JS, Zhou F, Liu QC, Chen G, Zhang JM. Relationship between plasma high mobility group box1 protein levels and clinical outcomes of aneurysmal subarachnoid hemorrhage. J Neuroinflammation. 2012; 9:194.View ArticlePubMedPubMed CentralGoogle Scholar
 Allette YM, Due MR, Wilson SM, Feldman P, Ripsch MS, Khanna R, White FA. Identification of a functional interaction of HMGB1 with Receptor for Advanced Glycation Endproducts in a model of neuropathic pain. Brain Behav Immun. 2014; 42:169–77.View ArticlePubMedPubMed CentralGoogle Scholar
 RezaiZadeh K, Gate D, Gowing G, Town T. How to get from here to there: macrophage recruitment in Alzheimer’s disease. Curr Alzheimer Res. 2011; 8(2):156–63.View ArticlePubMedPubMed CentralGoogle Scholar
 RezaiZadeh K, Gate D, Town T. CNS infiltration of peripheral immune cells: DDay for neurodegenerative disease?J Neuroimmune Pharmacol. 2009; 4(4):462–75.View ArticlePubMedPubMed CentralGoogle Scholar
 Westin K, Buchhave P, Nielsen H, Minthon L, Janciauskiene S, Hansson O. CCL2 is associated with a faster rate of cognitive decline during early stages of Alzheimer’s disease. PLoS ONE. 2012; 7(1):30525.View ArticleGoogle Scholar
 Hao W, Marsh C, Friedman A. A Mathematical Model of Idiopathic Pulmonary Fibrosis. PLoS ONE. 2015; 10(9):0135097.Google Scholar