Alzheimer’s Disease: PDE Model
DKWC
2021-07-28
1 Alzheimer’s disease: PDE Model
- Figure 1a shows the network within a neuron which leads from ROS to NFTs and the destruction of microtubules (sourced from Hao and Friedman (2016)).
- Figure 1b shows the network of activated cells, microglia, astrocyte and monocyte-derived macrophages and their effect on neurons and their microenvironment (sourced from Hao and Friedman (2016)).
The notaion of variables used are same as the paper of Hao and Friedman (2016).
All concentration and densities are in units of \(g/c m^3\) for cells and \(g/m l\) for cytokines
\(P\) = MCP-1
\(A\) = Astrocytes
\(τ\) = hyperphosphorylated tau protein
\(H\) = High mobility group box 1 (HMGB1)
\(M_1\) = Proinflammatory microglias
\(M_2\) = Anti-inflammatory microglias
\(N_d\) = Dead neurons
\(N\) = Live neurons
\(A^i_β\) = Amyloid β inside neurons
\(A^o_β\) = Amyloid β outside neurons
\(\hat{M}_1\) = Peripheral proinflammatory macrophages
\(\hat{M}_2\) = Peripheral anti-inflammatory macrophages
\(ROS(R)\) = Reactive oxygen species
\(GSK-3(G)\) = Glycogen synthase kinase-type 3
\(NFT(F_i)\) = Neuronfibrillary tangle inside neurons
\(NFT(F_o)\) = Neuronfibrillary tangle outside neurons
\(APP(A_P)\) = Amyloid precursor protein
\(AβO(A_O)\) = Amyloid β oligomer (soluble)
\(vTNF-α(T_α)\) = Tumor necrosis factor alpha
\(vTNF-β(T_β)\) = Transforming growth factor beta
\(MG(M_G)\) = Microglias
\(IL-10(I_{10})\) = Interleukin 10
1.1 Equation: \(A^i_β\)
1.2 Equation: \(A^o_β\)
1.3 Equation: \(\tau\)
1.4 Equation: \(NFT\)
1.5 Equation: Neurons
1.6 Equation: Astrocytes
1.7 Equation: Dead Neurons
1.8 Equation: \(A^o_β\)
1.9 Equation: \(HMGB-1\)
1.10 Equations: Activated Microglias
1.11 Equations: Macrophages \(\hat{M}\)
1.12 Other Equations
2 Simulation PDE Model
rectangular domain simulation \(\Omega ={(x,y),0≤x≤1,0≤y≤1}\)
assumption of periodic boundary conditions: \(A_{O}, H, T_{\beta}, I_{10}, T_{\alpha}\)
3 Initial Setting:
- \(N = 0.14~g/ml\)
- \(A = 0.14~g/ml\)
- \(I_{10}= 10^{-5}~g/ml\)
- \(M_{1}=M_{2}=0.02~g/ml\)
- \(T_{\beta} = 10^{-6}~g/ml\)
- \(P = 5 \times 10^{-9}~g/ml\)
- \(H=1.3\times 10^{-11}~g/ml\)
- \(A_{\beta}^{i}\,=\,10^{-6}~g/ml\)
- \(A_{\beta}^{o}\,=\,10^{-8}~g/ml\)
- \(\tau = 1.37 \times 10^{-10}~g/ml\)
- \(F^{i} =3.36 \times 10^{-10}~g/ml\)
- \(F^{o} = 3.36 \times 10^{-11}~g/ml\)
- \(T_{\alpha} = 2 \times 10^{-5}~g/ml\)
- \(\hat{M}_{1} = \hat{M}_{2} = N_{d} = 0~g/ml\)
4 Case I: TNF-\(\alpha\) Inhibitor
5 Case II: Anti-A \(\beta\) drugs
