Mathematical model on Alzheimer’s disease

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 β-peptides and intracellular neurofibrillary tangles of hyperphosphorylated tau proteins. The disease affects 5 million people in the United States and 44 million world-wide. 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.


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 world-wide [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 β-peptide (Aβ) deposits. The NFTs are intraneural aggregation of hyperphosphorylated tau proteins. Reactive oxygen species *Correspondence: hao.50@mbi.osu.edu 1 Department of Mathematics, The Penn State University, 16802 University Park, PA, USA Full list of author information is available at the end of the article (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 Aβ peptides [5]. High levels of ROS promote abnormal deposition of Aβ [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 kinase-type 3(GSK-3) is activated by the abnormally produced Aβ, and it mediates the hyperphosphorylation of tau proteins [4,[6][7][8][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 non-neuronal 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 Aβ oligomers which build up from the Aβ deposits [11,12].
There are currently no drugs that can cure AD, or stop its progression. Many clinical trials of drugs aimed at preventing or clearing the Aβ and tau pathology have failed to demonstrate efficacy [24][25][26][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][30][31][32][33][34], but on the other hand, TGF-β-induced TIAF1 interacts with amyloid fibrils to favorably support plaque formation [28], and blocking TGF-β-smad2/3 in peripheral macrophages mitigates AD pathology [35]. Figure 1 is a schematics of the network associated with the progression of AD. Figure 1a shows the network within a neuron which leads from ROS to NFTs and the destruction of microtubules. Figure 1b shows the network of activated cells, microglia, astrocyte and monocytederived macrophages and their effect on neurons and their microenvironment.
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 anti-inflammatory cytokines by IL-10.
We shall use our model to conduct in silico trials with several drugs: TNF-α inhibitor, anti-Aβ drug, MCP-1 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 anti-Aβ drug yields a slight decrease in the aggregation of Aβ over 10 years period, while the benefits from injection of TGF-β and MCP-1 inhibitor drugs are negligible. This suggests that clinical trials consider combination therapy with TNF-α and anti-Aβ 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 Aβ mediate rapid disruption of synaptic plasticity and memory [36][37][38][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: Aβ polymerization [40], Aβ plaque formation and the role of prions interacting with Aβ [41,42], linear crosstalk among brain cells and Aβ [43], and the influence of SORLA on AD progression [44,45].

Model's variables
The mathematical model is based on Fig. 1 and is represented by a system of partial differential equations. Table 1 lists the variables used in the model.

Equation for τ
Tau protein is constitutively produced at some rate λ τ 0 . We assume that when A i β production exceeds a threshold A i0 β , GSK-3 becomes activated and it mediates 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.

Equation for neurons
Hyberphosphorated tau proteins, forming neurofibrillary tangles, cause microtubles depolymerization and destruction, resulting in neuron death [4,[6][7][8][9]. Neuron death is also caused by stress from proinflammatory cytokines which is, however, resisted by antiinflammatory cytokines [20,22,23]. For simplicity we represent all the proinflammatory cytokines by TNF-α and all the anti-inflammatory cytokines by IL-10. Hence the equation for N takes the following form: where the death rates of N caused by F i and T α are assumed to depend on their saturation levels.

Equation for dead neurons
The equation for dead neurons, N d , is given by whereK N d is a Michaelis-Menten coefficient. The first two terms on the right-hand 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 AβO
The AβO are soluble Aβ oligomers and they can diffuse throughout the brain tissue [49,50]. Their density A O satisfies the equation: where λ A O is the rate by which the A O are formed from the extracellular amyloid β peptides, and D A O A O accounts for the diffusion of A O .

Equations for activated microglias
Activated microglias have two phenotypes: proinflammatory M 1 and anti-inflammatory M 2 . They satisfy the following equations: where ε 1 = T α T α +K Tα and ε 2 = 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 HMGB-1 produced by N d , and this is represented by the second term of the left-hand 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 anti-inflammatory signals from IL-10. These facts are expressed by the first term on the righthand sides of Eqs. (11), (12); βε 1 βε 1 +ε 2 is the ratio by which the activated microglias become M 1 macrophages, and ε 2 βε 1 +ε 2 is the ratio by which activated microglias become M 2 macrophages. The parameter β reflects the ratio of proinflammatory/anti-inflammatory 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 right-hand side of these equations.

Equations for macrophages
Peripheral macrophagesM are differentiated from monocytes which migrate through the blood vessels. They satisfy a flux condition on the boundary of the blood vessels, where n is the outward normal, M 0 is the density of the monocytes in the brain capillaries, andα(P) is a function which depends on the concentration of MCP-1. By averaging these fluxes from blood vessels, we can represent (as in [56]) the immigration ofM macrophages into the brain tissue by a termα(P)(M 0 −M). We assume that the incoming macrophages divide intoM 1 andM 2 phenotype depending on the relative concentrations of TNF-α and IL-10 [47]. MacrophagesM 1 can also change phenotype toM 2 macrophages under signaling by TGF-β. We finally note that because of the blood-brain barrier (BBB) we do not include diffusion of peripheral macrophages, but we do include chemotaxis by amyloid-β plaques or, more specifically, by the soluble A O [17][18][19]. Hence peripheral macrophages satisfy the following equations: whereM =M 1 +M 2 and α(P) = α P P+K P [56].

Results and discussions
We simulate the model (1) We take initial values, for each variable X, to be below (or above) the expected steady state for X, if X is expected to grow (or decrease) with the progression of the disease. A specific choice is given below, but the simulations do not change, after a short time, with other choices: We also prescribe the value of ROS in Eqs. (1), (3) by (21) Figure 2 shows the average density of all the 18 variables of the model over a period of 10 years. We first observe that, for all the species that tend to a steady state in Fig. 2, the steady states are approximately the same as those that we assumed in estimating some of the model parameters. Thus the steady state values of τ , F i , H, M 1 , M 2 ,M 1 ,M 2 , T β , T α and I 10 are approximately equal to the values assumed in "Appendix". We conclude that estimates of the parameters which were based on steady state assumptions on macrophages, microglias and the half-saturation parameters are consistent with the simulation results.
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 Aβ peptides, A o β , and the soluble Aβ oligomers, A O , are increasing; A o β reaches the level of 7 × 10 −6 g/ml, in agreement with clinical data [57], and the assumed average Fig. 2. The assumed average of the F o concentration, K F o , is also in  Tables 2 and 3 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 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.

Anti-Alzheimer 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 (anti-amyloid 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 o β , since Aβ aggregation mediates rapid dysfunction of synaptic plasticity and dendritic channels thereby causing memory loss [36][37][38][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.
We shall apply our model to determine how this TNFα inhibitor affects AD patients. We use the following procedure: 1. Run the model for 300 days in order to ensure that AD has been diagnosed in patients; 2. Apply continuous treatment by the drug from day 300 until the end of 10 years.
During treatment, the effect of the drug is to replace Eq. (17) for TNF-α by the equation where f is proportional to the amount of etanercept. We note that since etanercept is a soluble TNF receptor fusion protein, it stabilizes TNF-α [61] and thus TNF-α is diminished at rate fT α . The red profiles in Fig. 3 show the result of the treatment with f = 10d T α , compared to no treatment.

TGF-β injection
TGF-β is an anti-inflammatory cytokine which induces phenotype change from proinflammatory to antiinflammatory macrophages. It was suggested that TGF-β mitigates AD pathology [29][30][31][32][33][34]. We note that the effect of T β injection is to decrease M 1 andM 1 (see Eqs. (11), (13)), which results in a decrease in T α (by Eq. (17)) and hence in a decrease in neuronal death rate. To model the treatment by injection of TGF-β we replace Eq. (15) for T β by the equation where g is proportional to the amount of injected TGFβ. In steady state, T β maintains the level of K T β , while its degradation rate is d T β . Hence the source of T β in steady state is d T β T β . We take g to be 10 times this source, that is g = 10d T β K T β . We then follow the same treatment procedure for TNF-α inhibitor. The light-blue profiles in Fig. 3 show the results of the treatment, compared to no treatment.

Anti-Aβ drugs
There are several drugs in Phase 3 clinical trials that aim to reduce the effect of Aβ aggregation [62]. Among them is aducanumab, which is thought to be microglia-mediated phagocytosis and clearance of Aβ [63,64]. In our model, this drug will cause a decrease in the concentration of soluble A o β , by replacing Eq. (2) by the equation  Tables 2 and 3 where h is proportional to the amount of the dozing level; we take h = 10. Figure 4 shows the efficacy of several drugs in terms of N and A o β . The lowest curve in Fig. 4a, and the highest curve in Fig. 4b, correspond to the case where the curves of no treatment and several other drugs coincide; these drugs have negligible efficacy. Following the same treatment procedure as in the case of TNF-α inhibitor, Fig. 4 shows no efficacy of aducanumab in terms of N but significant efficacy in terms of A o β in comparison to no treatment and to treatments by TNF-α inhibitor and TGF-β injection.

MCP-1 inhibitor
Bindarit was shown to inhibit CCL2 (MCP-1) in brain tissue [65]. Hence it decreasesM 1 (by Eq. (13)), which results in a decrease in T α (by Eq. (17)) and thus also in a decrease in neuronal death rate. Bindarit was also reported to inhibit Aβ-induced neuronal death in vitro [66]. Hence it has a therapeutic potential in the treatment of neuroninflammatory/neurodegenerative diseases like AD [66]. We can conduct in silico trial with bindarit by revising Eq. (18) for P, replacing it with the equation 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 o β 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 τ 0 and λ τ by 1/10. Following the procedure as in case of TNFα inhibitor, we found that the drug has almost negligible efficacy (not shown here).

Combination therapy
The results of Fig. 4 suggest that a combination therapy with etanercept (TNF-α inhibitor) and aducanumab (anti-Aβ drug), under the same '10-fold' amount, could both slow the death rate of neurons and decrease the growth of Aβ in a significant way. Figure 5 shows the dynamics of N and A o β under such combination of drugs with different proportions of fold numbers: (etanercept,aducanumab)=(0,0) (no drugs), (10,5), (20,10), (30,15), (40,20) and (50,25). The reduction in the death of neurons, after 10 years, compared to the case of no drugs, is 3.8, 5.2, 6.4, 7.9 and 9.2%, respectively, and the respective reduction in the concentration of Aβ is 21, 32.2, 43.6, 53.9 and 64.1%.
We next consider combined therapy for any value of etanercept (f ) and aducanumab (h). We define the Nefficacy of (f, h), E N (f , h), to be where the A o β concentration is computed also at the end of 10 years. Figure 6 is an efficacy map of the combined therapy with f in range of (0,50) and h in the range of (0,25). For any pair (f,h) the color columns in Fig. 6a and 6b show the efficacy for N and anti-A o β . 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 anti-Aβ antibody decreases the external concentration of Aβ (A o β ) 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 o β . This result can be explained by our assumptions in Eq. (2) where we neglected the production of A o β by live neurons and the increase of A o β 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][68][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: We say that these two drugs at concentrations f and g have positive synergy with respect to N if E N (f , g) > E N (2f , 0) and E N (f , g) > E N (0, 2g). We accordingly define the synergy index σ N = σ N (f , g) by Thus, σ N > 1 means positive synergy and σ N < 1 means negative synergy. The above definition depends on the doses f, g. If σ N is large then the combination therapy at total amount f + g is much more effective than a single therapy, at the total same amount, in reducing the death rate of N. If σ N < 1 then a single drug is preferable. Similarly one can define the synergy index σ A o β with respect to A o β . Figure 7 shows the synergy index σ N for (f, g) in the range 0 < f < 50 and 0 < g < 25. We see that there is a positive synergy between etanercept (f ) and aducanumab (h). Furthermore, given a total amount A of the combined drugs, so that f+h=A, the synergy increases as f/g increases. This suggests that in an optimal regimen f should be significantly larger than h, provided negative side-effects are discounted.

The synergy map for σ A o
β 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 anti-Aβ and anti tau aggregation in combination therapy.

Sensitivity analysis
Sensitivity analysis on the model parameters can support the robustness of the simulation results. But it can also suggest what drugs do not work and what drugs are more likely to work. We conducted sensitivity analysis on parameters associated with production and removal rates of A o β , death rates of N, and production rates of TNF-α, TGF-β and MCP-1: where we varied λ N , λ A , d NF , d NT between 1 2 and twice their value in Tables 2 and 3, and varied γ , δ, ξ , between 1 2 and 2.
Following the sensitivity analysis method described in [71], we performed Latin hypercube sampling and generated 2000 samples to calculate the partial rank correlation coefficients (PRCC) and p-values with respect to the density of N and with respect to the concentration of A o β at  N decreases) and as a result more Aβ emerge from the increasingly dying neurons, thus raising the concentration of A o β . The fact that the correlation coefficients of d NT are significantly larger than the correlation coefficients of d NF , suggests that a drug which blocks TNF-α would be more effective than a drug which clears the F i . The other PRCC values can also be seen to be consistent with the model dynamics.
We observe that is negatively correlated to A o β . Indeed, if is increased, more A o β are cleared out (by Eq. (2)). To see how this affects N we note that if A o β is decreased then A O decreases (by Eq. (9)) and correspondingly M 1 decreases (by Eq. (11)), and then T α decreases (by   Fig. 8 The PRCC values of parameter for sensitivity analysis 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 is indeed positively correlated to N but the correlation is small. The correlation levels of with respect to N and A o β suggest that an anti-Aβ 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 beta-amyloids (A o β ) and neurofibrillary tangles (NFTs) of hyperphosphorylated tau proteins. NFTs destroy microtubles in neurons, which results in neurons death. Soluble A o β oligomers activate microglias (the resident macrophages in the brain), thereby initiating inflammatory response. Additionally, peripheral macrophages, responding to cue from MCP-1 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, β-amyloids, tau proteins, and several cytokines involved in the cross-talk 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 (TNFα inhibitor), injection of TGF-β, aducanumab (Anti-Aβ drug) and bindarit (MCP-1 inhibitor). We found that at '10-fold' level, etanercept has the largest efficacy in slowing death of neurons, while aducanumab has the largest efficacy in reducing the aggregation of A o β , 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, Aβ plaques affect dendritic channels, and NFT mediates synaptic dysfunction [36][37][38][39]. Recent studies also begin to address white matter degeneracy that could help identify high risk of AD [72].
where the Michaelis-Menten constantK X depends only on the 'eating capacity' of Y, soK X has no relation to the half-saturation of X.

Eq. (1)
By [82], the half-life of A i β is 1.  [57], the density in brain-gray matter of A o β 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 β = 10 −6 g/ml and .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/cm 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/cm 3 . The diameter of neurons is 16 μm [83]. Accordingly, we estimate the volume of 1 neuron to be 2 × 10 −9 cm 3 , and the neurons density is then 7 × 10 7 × 2 × 10 −9 g/cm 3 , that is N 0 = 0.14 g/cm 3 .
Microglias are the first responders to NFTs and AβO. 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 densityM are equal, so thatM = KM = 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 anti-inflammatory macrophages, and that proinflammatory peripheral macrophages exceed the proinflammatory microglias. Thus, in steady state,M 1 >M 2 , M 1 > M 2 andM 1 > M 1 , and we take KM 1 = 0.04, KM 2 = 0.007, Activated microglias are poorly phagocytic for Aβ compared to peripheral macrophages [6]. Accordingly we take Taking d A o βM = 10 −2 /day, we then have We assume thatM 1 and M 1 are more effective thanM 2 and M 2 in clearing Aβ, and take θ = 0.9.
We assume that survival time of patients with AD is 10 years, and that at the end-stage 50% of their neurons have died [73]. Hence, the death rate of N is d N = ln2 10 years = 1.9 × 10 −4 /day.
By [57], A o β = 7 × 10 −6 g/ml. We assume that the clearance of A o β by macrophages and microglias is nearly unlimited (i.e., it is almost linear in A o β ) by takingK A o β = 10 3 A o β = 7 × 10 −3 g/ml. To estimate λ N , we first consider the steady state of Eq. (2), To estimate the average of ∂N ∂t , we use the equation The values of ∂N ∂t 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 dN dt = 1.8×10 −5 g/ml/day as the average of dN dt 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 steady-state assumption in Eq. (2). However, in AD the Aβ 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.

Eqs. (4) and (5)
We assume that neurofibrillary tangles inside neurons are much more stable than tau proteins, taking d F i = 1 10 2 d τ = 2.77 × 10 −3 /day. We also assume that extracellular NFTs do not degrade as fast as internalized NFTs, taking d F o = 1 10 d F i = 2.77 × 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 λ F = 0.6d F o . 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 /day. We take K I 10 = 2 × 10 −6 g/cm 3 (which is somewhat larger than the estimated half-saturation 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 × 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 × 10 −10 g/ml, K F o = 2.58 × 10 −11 g/ml. From the steady state of Eq. (17) and the estimates of λ T α M 1 and λ T αM1 (see under Eq. (17) below) we get T α = 4 × 10 −5 g/ml, so that K T α = 4 × 10 −5 g/ml.

Eq. (7)
We take the half-life of astrocytes to be the same as the half-life 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 Aβ, and take λ AT α T α = 2λ AA o β A o β . By the steady state of Eq. (7) we then get λ AT α = 1.4/day, and λ AA o β = 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 λ AT α and λ AA o β by a factor 1.1, taking λ AT α = 1.54/day and λ AA o β = 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 dM = 0.2/day. We also assume that microglia play a greater role in clearing necrotic neurons, and take d N d M = 3 × 0.2=0.6/day. We also takeK N d = 10 −3 g/ml.

Eq. (9)
We assume the degradation rate of The estimate of λ A O was based on the steady-state assumption in Eq. (9). However, in AD the soluble Aβ oligomer is continuously increasing, following the increase in A o β , so the steady-state 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.
We take M 0 G = 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 Aβ in AD is 7 × 10 −6 g/ml and, by [55], the ratio of A O to A o β is 1 25 , so that K A O = 1 25 × 7 × 10 −6 = 2.8 × 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 = 3.36×10 −10 g/ml, we take K F O = 2.58×10 −11 g/ml.
The coefficient λ M 1 T β 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, λ M 1 T β = 6 × 10 −3 /day [90]; we take it to be the same in the present case. We take K T β = 2.5 × 10 −7 g/ml, and K I 10 = 2.5 × 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 stateM = 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 ofM 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 towardM 1 phenotype rather thanM 2 phenotype, since K T α > K I 10 . We assume, in line with this bias toward M 1 , that the transition rate fromM 1 intoM 2 phenotype by TGF-β is at a smaller rate than the corresponding transition rate for microglias, that is, λM 1 T β < λ M 1 T β . We take λM 1 T β = 6 × 10 −4 /day.
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 λ PM 2 by a factor 1.1, taking λ PM 2 = 1.32 × 10 −7 /day, and λ PA = 6.6 × 10 −8 /day.