
MBW:Extension to Mathematical Modeling of Alzheimer's DiseaseFrom MathBioThe following is a replication of the work done by Ishwar K. Puri, Liwu Li in Mathematical Modeling for the Pathogenesis of Alzheimer's Disease. For more background information go to MBW: Mathematical Modeling of Alzheimer's Disease.
ContentsExecutive SummaryThe following is a reproduction of the results of Puri et. al in Mathematical Modeling for the Pathogenesis of Alzheimer’s Disease ^{[1]}. Alzheimer's is one of the most prevalent causes of dementia and even death in people over 65 years old, however little is known about the development of the disease on a neurological level. The authors studied the mechanism of one of the proposed causes of Alzheimer's, the effect of the deposition of amyloid particles forming senile plaques in the brain. Using a system of ordinary differential equations, it is possible to model one of the mechanisms taking place in the senile plaques, the activation of active microglia that in turn send out toxins and stimulate the death of the brain cells. The results from this model show that inflammatory activation of microglia is the dominant factor to the development of Alzheimer's and that amyloid plays a principal role in the mechanism of activation of microglia, however is not itself the key player in the development of the disease. In terms of treatment, however, decreasing the presence of amyloid is shown to decrease the cell death associated with Alzheimer's.
BackgroundOne of the most common causes of dementia is Alzheimer’s, a neurodegenerative disorder that is predominant in the senile population. It is estimated that 11% of people 65 years old and older and 32% of those 85 years old and older have Alzheimer’s. In addition to its predominance in society, this disorder has a high mortality rate, at the sixth leading cause of death in the United States. Just in 2012, more than $216 billion were spent on unpaid healthcare for patients with Alzheimer's. ^{[2]} Clearly, there are a lot of societal costs. Even worse are the costs directly to patients and their families. Patients with Alzheimer's have tissue loss in the brain, resulting in brain shrinkage in a multitude of areas including the hipposcampus, which is responsible for storing longterm memories.
Mathematical ModelThe proposed mechanism accounts for the feedback from amyloid, and the distinct states of microglia, astroglia, and neurons. The intercellular crosstalks are represented as the 16 pathways in the mechanism.
Two important and justified assumptions must be addressed: 1. Constant risk of neuronal death. 2. Spatiotemporal influence of diffusion is negligible. However, within the scope of the small area of the senile plaques, both of these assumptions show external validity.
Variable Definitions1. Amount of neuron population that has survived the current time step. 2. Amount of neuron population that has died during the current time step. 3. Amount of astroglia population in a state of quiescence. 4. Amount of astroglia population in a state of proliferation. 5. Amount of normal microglia population in the resting antiinflammatory state. 6. Amount of reactive microglia population in the active proinflammatory state. 7. Number of amyloid peptide molecules. The initial conditions are give in Table 2 from Puri et al.
Parameter DefinitionsThe proposed mechanism requires 16 rates for its description, and the model and analysis call for one additional rate. The required rates for the mechanism are given implicitly as for , which represent the crosstalks. The additional rate considered is , which represents the rate of amyloid removal out of the system.
Original System of EquationsThe system is modeled with seven coupled ordinary differential equations. This system can also be represented as reaction rates or pathways between different states of cell populations: This yields the following reaction rates: (1.)
Adapted System of EquationsThese are identical to the original equations with the exception that one additional term is used to describe equation (7a.) as follows, (7b.)
Justification for the adapted model is presented in the Results section below.
Sensitivity AnalysisSensitivity analysis of the original system with respect to the parameter . (1.1)
.
ResultsCell Population ModelsFor the original model, an ordinary differential equation solver was used to produce the following plots in MatLab. The two plots in the first column are supposed to be identical to the Puri et al Figure 2 with , assuming that the results from Puri et al were indeed obtained by the model they described. The other two columns are also from the original model but with and , respectively.
The following plots are from the adapted model. The two plots in the first column are identical to the Puri et al Figure 2, where the rate . The other two columns are also the adapted model but with and , respectively.
Sensitivity ComparisonThe sensitivities for of the original and adapted models calculated in MatLab are very different, using the equations described above. It is a little unclear how Puri et al preformed the sensitivity analysis. It seems they performed a finite difference, instead, by making small perturbations between in each and determined after 20 simulation years. Even with the difference in calculation methods, the adapted model yields the same conclusions as cited in the paper by Puri et al. Below are the sensitivities calculated at simulation year 20. Sensitivity of Neuron survival with respect to As cited by Puri et al: Original model MatLab calculations: Adapted model MatLab calculations:
As cited by Puri et al: Original model MatLab calculations: Adapted model MatLab calculations:
As cited by Puri et al: Original model MatLab calculations: Adapted model MatLab calculations:
As cited by Puri et al: Original model MatLab calculations: Adapted model MatLab calculations:
The following figure plots the sensitivities with respect to time for the original model. Notice these are incompatible with the behavior of the sensitivities cited by Puri et al, in both difference of magnitudes and sign. For example, the original sensitivities indicates that promotes an increase of reactive microglia which is the opposite to that cited by Puri et al.
Variation in ParametersIt is also possible to reproduce the findings of Puri et al. with respect to changes in each of the different populations when adjusting the values of different parameters. First, changes in the reactive microglia are studied when changing the rate of amyloid removal. For higher values of amyloid removal from the system, the amount of reactive microglia decreases. Looking at the proliferating astroglia, changes with respect to amyloid removal as well, with increases in amyloid removal resulting in the decrease of Astroglia proliferation. In addition, as expected, when the rate at which amyloid is removed from the system increase, the amount of Amyloidbeta decreases.
In turn, it is clear that as the rate of creation increases, the cell death increases and the reactive microglia population initially increases, but then decreases with time.
ConclusionsThe original model as described in the paper did not yield the same results as cited by Puri et al. However, the adapted model fits the results remarkably well. The figures from the adapted model are, to the best of our knowledge, identical to the figures by Puri et al. We believe our sensitivity calculations are more accurate than what was described by Puri et al., however both ways of calculating lead to the same conclusions, that the key factors at play in this mechanism are the active microglia and the amyloid proteins. Yet again, the adapted model reproduces results closest to that of the paper.
CodeMain.m%% Main: % This program initiates the simulation and calls on Odesystem.m clear all clc format long %% Initial conditions: % Declare global variables: global y0 alpha; % Preallocating vectors: y0 = zeros(7,1); alpha = zeros(17,1); % Cell and molecule populations at time t(0) for some arbitrary volume: y0(1) = 1e+4; % N_s : Neuron survival. y0(2) = 1e+2; % N_d : Neuron death. y0(3) = 1e+5; % A_q : Astroglia quiescent. y0(4) = 1e+3; % A_p : Astroglia proliferation. y0(5) = 1e+5; % M_2 : Microglia normal. y0(6) = 1e+3; % M_1 : Microglia reactive. y0(7) = 1e+3; % A_beta : Amyloidbeta molecules. % Rates associated to the proposed mechanism. Units are [1/year]: alpha(1) = 1e5; alpha(2) = 1e3; alpha(3) = 1e2; alpha(4) = 1e4; alpha(5) = 1e2; alpha(6) = 1e2; alpha(7) = 1e4; alpha(8) = 1e2; alpha(9) = 1e2; alpha(10) = 1e2; alpha(11) = 1e2; alpha(12) = 1e4; alpha(13) = 1e2; alpha(14) = 1e4; alpha(15) = 1; alpha(16) = 1e2; alpha(17) = 1; % I/O: If Toggle(1) is active then Toggle(2) must be commented out, and vice versa. % Loop through three different Amyloidbeta removal rates for alpha_r: %% Toggle(1): Begin for j=1:3 if j==1 alpha(17) = 1; % alpha_r : Removal rate of Amyloidbeta from the system. end if j==2 alpha(17) = .1; end if j==3 alpha(17) = .01; end %% Toggle(1): End % Loop through three different alpha_13 rates for impact of A_beta > M_1: %% Toggle(2): Begin % for j=1:3 % if j==1 % alpha(13) = 1e2; % alpha_13 :. % end % % if j==2 % alpha(13) = .1; % end % % if j==3 % alpha(13) = .5; % end %% Toggle(2): End %% Time: % Move forward in time from 0 to 20. Units of time are years: t0 = 0; tf = 20; %% Solve the system of difference equations with above initial conditions: [t,y] = ode15s( @OdeSystem, [t0 tf], y0 ); %% Plots: % The plots below are to be run with Toggle(1) active above unless % stated otherwise (Plot 4 requires Toggle(2) active). zero = t; % Straight line for comparison, y=0. %% Plot 2: % (2a.1) Plot for alpha_r = 1: if j==1 %figure subplot(2,3,1); hold on plot(t,y(:,1),'g'); % N_s : Neuron survival. plot(t,y(:,6),'r'); % M_1 : Microglia reactive. plot(t,y(:,7),'b'); % A_beta : Amyloidbeta molecules. plot(t,zero,'k', 'Linewidth',1); title('\bf Populations with Amyloid{\beta} removal: {\alpha_r = 1}','FontSize',16); legend('N_s: Neuron Survival','M_1: Reactive Microglia','A{\beta}: Amyloid\beta',2); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Populations of {N_s}, {M_1}, and {A{\beta}}','FontSize',12); %axis([0 20 100 11100]); hold off end % (2a.2) Plot for alpha_r = .1: if j==2 %figure subplot(2,3,2); hold on plot(t,y(:,1),'g'); % N_s : Neuron survival. plot(t,y(:,6),'r'); % M_1 : Microglia reactive. plot(t,y(:,7),'b'); % A_beta : Amyloidbeta molecules. plot(t,zero,'k', 'Linewidth',1); title('\bf {\alpha_r = .1}','FontSize',16); %legend('N_s: Neuron Survival','M_1: Reactive Microglia','A{\beta}: Amyloid\beta',2); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Populations of {N_s}, {M_1}, and {A{\beta}}','FontSize',12); %axis([0 20 100 15100]); hold off end % (2a.3) Plot for alpha_r = .01: if j==3 %figure subplot(2,3,3); hold on plot(t,y(:,1),'g'); % N_s : Neuron survival. plot(t,y(:,6),'r'); % M_1 : Microglia reactive. plot(t,y(:,7),'b'); % A_beta : Amyloidbeta molecules. plot(t,zero,'k', 'Linewidth',1); % The line y = 0. title('\bf {\alpha_r = .01}','FontSize',16); %legend('N_s: Neuron Survival','M_1: Reactive Microglia','A{\beta}: Amyloid\beta',2); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Populations of {N_s}, {M_1}, and {A{\beta}}','FontSize',12); %axis([0 20 100 15100]); hold off end % (2b.1) Plot for alpha_r = 1: if j==1 %figure subplot(2,3,4); hold on plot(t,y(:,2),'g'); % N_d : Neuron death. plot(t,y(:,4),'b'); % A_p : Astroglia proliferation. title('\bf Neuron death and Astroglia proliferation:','FontSize',16); legend('N_d: Neuron Death','A_p: Astroglia Proliferation',2); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Populations of {N_d}, {A_p}','FontSize',12); hold off end % (2b.2) Plot for alpha_r = .1: if j==2 %figure subplot(2,3,5); hold on plot(t,y(:,2),'g'); % N_d : Neuron death. plot(t,y(:,4),'b'); % A_p : Astroglia proliferation. %title('\bf {\alpha_r = .1}','FontSize',16); %legend('N_d: Neuron Death','A_p: Astroglia Proliferation',2); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Populations of {N_d}, {A_p}','FontSize',12); hold off end % (2b.3) Plot for alpha_r = .01: if j==3 %figure subplot(2,3,6); hold on plot(t,y(:,2),'g'); % N_d : Neuron death. plot(t,y(:,4),'b'); % A_p : Astroglia proliferation. %title('\bf {\alpha_r = .01}','FontSize',16); %legend('N_d: Neuron Death','A_p: Astroglia Proliferation',2); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Populations of {N_d}, {A_p}','FontSize',12); hold off end %% Plot 3: % Must comment out plots 12 above and 45 below to plot this section: % Warning ~ Must plot 3a, 3b, 3c, and 3d ~>ONE AT A TIME<~ with All other % plots commented out. %% (3a) Reactive microglia (M_1) for three values of alpha_r: % hold on % if j==1 % % plot(t,y(:,6),'k'); % end % if j==2 % plot(t,y(:,6),'r'); % end % if j==3 % plot(t,y(:,6),'b'); % title('Reactive Microglia for different rates of Amyloidbeta removal'); % legend('{\alpha_r = 1}','{\alpha_r = .1}','{\alpha_r = .01}'); % xlabel('Time [years]'); % ylabel('Population of {M_1} for different {\alpha_r} values'); % hold off % end %% (3b) Astroglia proliferation (A_p) for three values of alpha_r: % hold on % for v=1:3 % % if j==1 % plot(t,y(:,4),'k'); % end % if j==2 % plot(t,y(:,4),'r'); % end % if j==3 % plot(t,y(:,4),'b'); % end % % end % title('Astroglia proliferation for different rates of Amyloidbeta removal'); % xlabel('Time [years]'); % ylabel('Population of {A_p} for different {\alpha_r} values'); % hold off %% (3c) Amyloidbeta (A_beta) for three values of alpha_r: % hold on % for v=1:3 % % if j==1 % plot(t,y(:,7),'k'); % end % if j==2 % plot(t,y(:,7),'r'); % end % if j==3 % plot(t,y(:,7),'b'); % end % % end % title('Amyloidbeta for different rates of Amyloidbeta removal'); % xlabel('Time [years]'); % ylabel('Population of {A\beta} for different {\alpha_r} values'); % hold off %% (3d) Neuron death (N_d) for three values of alpha_r: % hold on % for v=1:3 % % if j==1 % plot(t,y(:,2),'k'); % end % if j==2 % plot(t,y(:,2),'r'); % end % if j==3 % plot(t,y(:,2),'b'); % end % % end % title('Neuron death for different rates of Amyloidbeta removal'); % %legend('{\alpha_r = 1}','{\alpha_r = .1}','{\alpha_r = .01}'); % xlabel('Time [years]'); % ylabel('Population of {N_d} for different {\alpha_r} values'); % hold off %% Plot 4: % Must comment out plots 13 above and 5 below to plot this section: % Warning ~ Must plot 4a, 4b, and 4c one at a time with All other % plots commented out. % Must run these with Toggle(2) active above. %% (4a) Plot of M_1 population for alpha_13 = .01: % hold on % if j==1 % plot(t,y(:,6),'k','Linewidth',1.2); % end % if j==2 % plot(t,y(:,6),'r','Linewidth',1.2); % end % if j==3 % plot(t,y(:,6),'b','Linewidth',1.2); % title('Impact of Amyloidbeta on reactive Microglia for{M_1}'); % legend('{M_1 } for {\alpha_{13} = .01}','{M_1 } for {\alpha_{13} = .1}','{M_1 } for {\alpha_{13} = .5}'); % xlabel('Time [years]'); % ylabel('Population of {M_1} for different {\alpha_{13}} values'); % hold off % end %% (4b) Plot of N_d population for alpha_13 = .1: % hold on % if j==1 % plot(t,y(:,2),'k','Linewidth',1.2); % end % if j==2 % plot(t,y(:,2),'r','Linewidth',1.2); % end % if j==3 % plot(t,y(:,2),'b','Linewidth',1.2); % title('Impact of Amyloidbeta on reactive Microglia for{N_d}'); % legend('{N_d } for {\alpha_{13} = .01}','{N_d } for {\alpha_{13} = .1}','{N_d } for {\alpha_{13} = .5}'); % xlabel('Time [years]'); % ylabel('Population of {N_d} for different {\alpha_{13}} values'); % hold off % end %% (4c) Plot of A_beta population for alpha_13 = .5: % hold on % if j==1 % plot(t,y(:,7),'k','Linewidth',1.2); % end % if j==2 % plot(t,y(:,7),'r','Linewidth',1.2); % end % if j==3 % plot(t,y(:,7),'b','Linewidth',1.2); % title('Impact of Amyloidbeta on reactive Microglia for{A\beta}'); % legend('{A\beta } for {\alpha_{13} = .01}','{A\beta } for {\alpha_{13} = .1}','{A\beta } for {\alpha_{13} = .5}'); % xlabel('Time [years]'); % ylabel('Population of {A\beta} for different {\alpha_{13}} values'); % hold off % end %% Plot 5: % (1) Plot for alpha_r = 1: % if j==1 % figure % hold on % plot(t,y(:,1),'g'); % N_s : Neuron survival. % plot(t,y(:,2),'og'); % N_d : Neuron death. % plot(t,y(:,3),'b'); % A_q : Astroglia quiescent. % plot(t,y(:,4),'+b'); % A_p : Astroglia proliferation. % plot(t,y(:,5),'r'); % M_2 : Microglia normal. % plot(t,y(:,6),'+r'); % M_1 : Microglia reactive. % plot(t,y(:,7),'cy'); % A_beta : Amyloidbeta molecules. % title('Neuron survival and Neuron death with {\alpha_r = 1}'); % %legend('N_s','N_d','A_q','A_p','M_2','M_1','A{\beta}',1); % xlabel('Time [years]'); % ylabel('All Populations and Molecules'); % hold off % end % % % (2) Plot for alpha_r = .1: % if j==2 % figure % hold on % plot(t,y(:,1),'g'); % N_s : Neuron survival. % plot(t,y(:,2),'+g'); % N_d : Neuron death. % plot(t,y(:,3),'b'); % A_q : Astroglia quiescent. % plot(t,y(:,4),'+b'); % A_p : Astroglia proliferation. % plot(t,y(:,5),'r'); % M_2 : Microglia normal. % plot(t,y(:,6),'+r'); % M_1 : Microglia reactive. % plot(t,y(:,7),'cy'); % A_beta : Amyloidbeta molecules. % title('Neuron survival and Neuron death with {\alpha_r = .1}'); % %legend('N_s','N_d','A_q','A_p','M_2','M_1','A{\beta}',1); % xlabel('Time [years]'); % ylabel('All Populations and Molecules'); % hold off % end % % % (3) Plot for alpha_r = .1: % if j==3 % figure % hold on % plot(t,y(:,1),'g'); % N_s : Neuron survival. % plot(t,y(:,2),'+g'); % N_d : Neuron death. % plot(t,y(:,3),'b'); % A_q : Astroglia quiescent. % plot(t,y(:,4),'+b'); % A_p : Astroglia proliferation. % plot(t,y(:,5),'r'); % M_2 : Microglia normal. % plot(t,y(:,6),'+r'); % M_1 : Microglia reactive. % plot(t,y(:,7),'cy'); % A_beta : Amyloidbeta molecules. % title('Neuron survival and Neuron death with {\alpha_r = .01}'); % %legend('N_s','N_d','A_q','A_p','M_2','M_1','A{\beta}',1); % xlabel('Time [years]'); % ylabel('All Populations and Molecules'); % hold off % end end OdeSystem.mfunction dydt = OdeSystem(t,y0) global alpha N_s = y0(1); % Neuron survival : N_s(0) = 1.0e+4; N_d = y0(2); % Neuron death : N_d(0) = 1.0e+2; A_q = y0(3); % Astroglia quiescent : A_q(0) = 1.0e+5; A_p = y0(4); % Astroglia proliferation : A_p(0) = 1.0e+3; M_2 = y0(5); % Microglia normal : M_2(0) = 1.0e+5 M_1 = y0(6); % Microglia reactive : M_1(0) = 1.0e+3; A_beta = y0(7); % Amyloidbeta molecules : A_beta(0) = 1.0e+3; % Preallocate the solution vector: dydt = zeros(7,1); % Definitions of the seven coupled rate equations: dydt(1) = alpha(1)*A_q  alpha(2)*A_p  alpha(3)*M_1; dydt(2) = (dydt(1)); dydt(3) = alpha(4)*M_2  alpha(5)*M_1; dydt(4) = (dydt(3)); dydt(5) = (alpha(6) + alpha(11))*N_s  alpha(10)*N_d + (alpha(7) + alpha(12))*A_q ...  alpha(9)*M_1 + alpha(14)*M_2  (alpha(8) + alpha(13))*A_beta; dydt(6) = (dydt(5)); %% I/O: Can only run Main with either (7a) active and (7b) commented out, or vice versa: % (7a) Original model: %dydt(7) = alpha(15)*N_s  alpha(16)*M_2; % (7b) Adapted model: dydt(7) = alpha(15)*N_s  alpha(16)*M_2  alpha(17)*A_beta; % Note: The adapted model has one additional term required in dydt(7) for the rate at % which Amyloidbeta is removed from the system; ie: alpha(17)*A_beta. return SensitivityMain.m%% Sensitivity for alpha_1: % This program initiates the simulation and calls on SensitivitySystem.m clear all clc format long %% Initial conditions: % Declare global variables: global y0 alpha; % Preallocating vectors: y0 = zeros(7,1); alpha = zeros(17,1); % Sensitivities of cell populations at time t(0) w.r.t alpha_1: y0(1) = 1e+5; % Value of initial A_q population from derivation of new system. y0(2) = 1e+5; % This is derived from above. y0(3) = 0; y0(4) = 0; y0(5) = 0; y0(6) = 0; y0(7) = 0; % Rates associated to the proposed mechanism. Units are [1/year]: alpha(1) = 1e5; alpha(2) = 1e3; alpha(3) = 1e2; alpha(4) = 1e4; alpha(5) = 1e2; alpha(6) = 1e2; alpha(7) = 1e4; alpha(8) = 1e2; alpha(9) = 1e2; alpha(10) = 1e2; alpha(11) = 1e2; alpha(12) = 1e4; alpha(13) = 1e2; alpha(14) = 1e4; alpha(15) = 1; alpha(16) = 1e2; alpha(17) = 1; %% Time: % Move forward in time from 0 to 20. Units of time are years: t0 = 0; tf = 20; %% Solve the system of difference equations with above initial conditions: [t,y] = ode15s( @SensitivitySystem, [t0 tf], y0 ); figure hold on plot(t,y(:,1),'g'); % dNs : d(Neuron survival). plot(t,y(:,2),'og'); % dNd : d(Neuron death). plot(t,y(:,5),'r'); % dM2 : d(Microglia normal). plot(t,y(:,6),'+r'); % dM1 : d(Microglia reactive). title('\bf Sensitivity of cell populations w.r.t. {\alpha_1}','FontSize',16); legend('S(Ns)','S(Nd)','S(M2)','S(M1)',1); xlabel('\bf Time [years]','FontSize',12); ylabel('\bf Sensitivity to {\alpha_1}','FontSize',12); hold off %% Display S(Ns), S(Nd), S(M2), S(M1) with respect to alpha_1 at year 20: [n m] = size(y); disp('Sensitivity of Neuron survival with respect to alpha_1:'); SNs = y(n,1) disp(' '); disp('Sensitivity of Neuron death with respect to alpha_1:'); SNd = y(n,2) disp(' '); disp('Sensitivity of Microglia reactive with respect to alpha_1:'); SM1 = y(n,6) disp(' '); disp('Sensitivity of Microglia normal with respect to alpha_1:'); SM2 = y(n,5) disp(' '); SensitivitySystem.mfunction dydt = SensitivitySystem(t,y0) global alpha % This is for the population sensitivities with respect to % alpha_1 dNs = y0(1); % d(Neuron survival) : dNs = 1.0e+5; dNd = y0(2); % d(Neuron death) : dNd = 1.0e+5; dAq = y0(3); % d(Astroglia quiescent) : dAq = 0; dAp = y0(4); % d(Astroglia proliferation) : dAp = 0; dM2 = y0(5); % d(Microglia normal) : dM2 = 0; dM1 = y0(6); % d(Microglia reactive) : dM1 = 0; dAb = y0(7); % d(Amyloidbeta molecules) : dAb = 0; Aq = 1e+5; % Preallocate the solution vector: dydt = zeros(7,1); % Definitions of the seven coupled rate equations using % d/d(alpha_1)(d(Population)/dt) : dydt(1) = Aq + alpha(1)*dAq  alpha(2)*dAp  alpha(3)*dM1; dydt(2) = (dydt(1)); dydt(3) = alpha(4)*dM2  alpha(5)*dM1; dydt(4) = (dydt(3)); dydt(5) = (alpha(6) + alpha(11))*dNs  alpha(10)*dNd + (alpha(7) + alpha(12))*dAq ...  alpha(9)*dM1 + alpha(14)*dM2  (alpha(8) + alpha(13))*dAb; dydt(6) = (dydt(5)); %% I/O: Can only run SensitivityMain with either (7a) active and (7b) commented out, or vice versa: % (7a) Original model: %dydt(7) = alpha(15)*dNs  alpha(16)*dM2; % (7b) Adapted model: dydt(7) = alpha(15)*dNs  alpha(16)*dM2  alpha(17)*dAb; % Note: one additional term was required in dydt(7) for the rate at % which Amyloidbeta is removed from the system; ie: alpha(17)*A_beta. return
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