 Research Article
 Open Access
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
 Alzheimer disease
 Mathematical modeling
 Drug treatment
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
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