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SA.jl
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using DifferentialEquations, CairoMakie, LinearAlgebra, Distributions, OffsetArrays, Random, LaTeXStrings, GlobalSensitivity, QuasiMonteCarlo, Statistics
function Valve(R, deltaP)
q = 0.0
if (-deltaP) < 0.0
q = deltaP/R
else
q = 0.0
end
return q
end
function ShiElastance(t, Eβα΅’β, Eβββ, Ο, Οββ, Οββ, Eshift)
Οββ = Οββ*Ο
Οββ = Οββ*Ο
#Ο = 4/3(Οββ+Οββ)
tα΅’ = rem(t + (1 - Eshift) * Ο, Ο)
Eβ = (tα΅’ <= Οββ) * (1 - cos(tα΅’ / Οββ * pi)) / 2 +
(tα΅’ > Οββ) * (tα΅’ <= Οββ) * (1 + cos((tα΅’ - Οββ) / (Οββ - Οββ) * pi)) / 2 +
(tα΅’ <= Οββ) * 0
E = Eβα΅’β + (Eβββ - Eβα΅’β) * Eβ
return E
end
function DShiElastance(t, Eβα΅’β, Eβββ, Ο, Οββ, Οββ, Eshift)
Οββ = Οββ*Ο
Οββ = Οββ*Ο
#Ο = 4/3(Οββ+Οββ)
tα΅’ = rem(t + (1 - Eshift) * Ο, Ο)
DEβ = (tα΅’ <= Οββ) * pi / Οββ * sin(tα΅’ / Οββ * pi) / 2 +
(tα΅’ > Οββ) * (tα΅’ <= Οββ) * pi / (Οββ - Οββ) * sin((Οββ - tα΅’) / (Οββ - Οββ) * pi) / 2
(tα΅’ <= Οββ) * 0
DE = (Eβββ - Eβα΅’β) * DEβ
return DE
end
# Model parameter values
Eshift = 0.0
Eβα΅’β = 0.03
Οββ = 0.3
Οββ = 0.45
Eβββ = 1.5
Rmv = 0.06
Ο = 1.0
function NIK!(du, u, p, t)
pLV, psa, psv, Vlv, Qav, Qmv, Qs = u
Οββ, Οββ, Rmv, Zao, Rs, Csa, Csv, Eβββ, Eβα΅’β = p
# 1) Left Ventricle
du[1] = (Qmv - Qav) * ShiElastance(t, Eβα΅’β, Eβββ, Ο, Οββ, Οββ, Eshift) + pLV / ShiElastance(t, Eβα΅’β, Eβββ, Ο, Οββ, Οββ, Eshift) * DShiElastance(t, Eβα΅’β, Eβββ, Ο, Οββ, Οββ, Eshift)
# 2) Systemic arteries
du[2] = (Qav - Qs ) / Csa
# 3) Venous
du[3] = (Qs - Qmv) / Csv
# 4) Left Ventricular Volume
du[4] = Qmv - Qav
# 5) Aortic Valve flow
du[5] = Valve.(Zao, (pLV - psa)) - Qav
# 6) Mitral Valve flow
du[6] = Valve(Rmv, (psv - pLV)) - Qmv
# 7) Systemic flow
du[7] = (du[2] - du[3]) / Rs
nothing
end
##
M = [1. 0 0 0 0 0 0
0 1. 0 0 0 0 0
0 0 1. 0 0 0 0
0 0 0 1. 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 1. ]
Nik_ODE = ODEFunction(NIK!,mass_matrix=M)
u0 = [8.0, 8.0, 8.0, 265.0, 0.0, 0.0, 0.0]
p = [0.3, 0.45, 0.06, 0.033, 1.11, 1.13, 11.0, 1.5, 0.03]
tspan = (0, Ο * 30)
prob = ODEProblem(Nik_ODE, u0, tspan, p)
@time sol = solve(prob, Rodas5P(autodiff = false), adaptive = false, dt = 0.00225, reltol = 1e-12, abstol = 1e-12)
## GSA ##
x = range(start = 15,stop = 16,step = 0.00225)
f1 = function (p) # Para
prob_func(prob,i,repeat) = remake(prob;p=p[:,i])
ensemble_prob = EnsembleProblem(prob,prob_func=prob_func)
sol = solve(ensemble_prob,Rodas5P(autodiff = false), adaptive = false, dt = 0.00225, reltol = 1e-12, abstol = 1e-12, EnsembleThreads();saveat=x,trajectories=size(p,2))
# Now sol[i] is the solution for the ith set of parameters
out = zeros(1335,size(p,2))
for i in 1:size(p,2)
out[1:445,i] = Array(sol[i][1,:]')
out[446:890,i] = Array(sol[i][2,:]')
out[891:1335,i] = Array(sol[i][4,:]')
end
out
end
N = 3000
lb = [0.21, 0.36, 0.042, 0.0231, 0.777, 0.791, 7.7, 1.05, 0.021]
ub = [0.34, 0.585, 0.078, 0.0429, 1.443, 1.469, 14.3, 1.95, 0.039]
bounds = tuple.(lb,ub)
sampler = SobolSample()
A,B = QuasiMonteCarlo.generate_design_matrices(N, lb, ub, sampler)
@time sobol_result_time = gsa(f1,Sobol(),A,B, batch=true)
## Plots for first order Sobol indices
# LV pressure
s11 = sobol_result_time.S1[1:445,:]
# Systemic Artery pressure
s21 = sobol_result_time.S1[446:890,:]
# LV Volume
s31 = sobol_result_time.S1[891:1335,:]
#### Time Averaged Sobol indices #####
sol = solve(prob,Rodas5P(autodiff = false), adaptive = false, dt = 0.00225, reltol = 1e-12, abstol = 1e-12, saveat=x)
### LV.P #####
S1_LVP = s11
S11_LVP = Vector{Float64}(undef,9)
for i in 1:9
for j in 1:length(x)
S11_LVP[i] = (sum(S1_LVP[1:j,i]*var(sol[1,1:j])))/(sum(var(sol[1,1:j])))/length(x)
end
end
### SA.p #####
S1_SAP = s21
S11_SAP = Vector{Float64}(undef,9)
for i in 1:9
for j in 1:length(x)
S11_SAP[i] = (sum(S1_SAP[1:j,i]*var(sol[2,1:j])))/(sum(var(sol[2,1:j])))/length(x)
end
end
### LV.V#####
S1_LVV = s31
S11_LVV = Vector{Float64}(undef,9)
for i in 1:9
for j in 1:length(x)
S11_LVV[i] = (sum(S1_LVV[1:j,i]*var(sol[4,1:j])))/(sum(var(sol[4,1:j])))/length(x)
end
end
### Whole plots
S1 = [S11_LVP S11_SAP S11_LVV]'
## PCA calculation
F = transpose(S1)*S1
e_deomp=eigen(F)
Ξ» = abs.(e_deomp.values)
Q = abs.(e_deomp.vectors)
e_value_sum = sum(Ξ»)
e = Vector{Float64}(undef,9)
for i in 1:9
for j in 1:9
e[i] = sum(Ξ»[j]*Q[i,j])/e_value_sum
end
end
p=sortperm(e,rev=true)
## Orthogonality plot
Orth_heat1 = Matrix{Float64}(undef,9,9)
for j in 1:9
for i in 1:9
if i==j
Orth_heat1[i,j] = 0
else
Orth_heat1[i,j] = sin(acos(((transpose(S1[:,i])*S1[:,j]))/(norm(S1[:,i])*norm(S1[:,j]))-1e-15)) #Slight numerical rounding error without the additional add on
end
end
end
# Histogram calculation
a1 = Orth_heat1[2:end,1]
a2 = Orth_heat1[3:end,2]
a3 = Orth_heat1[4:end,3]
a4 = Orth_heat1[5:end,4]
a5 = Orth_heat1[6:end,5]
a6 = Orth_heat1[7:end,6]
a7 = Orth_heat1[8:end,7]
a8 = Orth_heat1[9:end,8]
a = reduce(vcat, (a1,a2,a3,a4,a5,a6,a7,a8))
begin
f = Figure(resolution = (900, 600),backgroundcolor = RGBf(0.98, 0.98, 0.98));
ax = Axis(f[1,1], xticklabelrotation = Ο / 3, xticklabelalign = (:right, :center), xticks = (1:3, [L"LV.P", L"SA.P", L"LV.V"]), yticks = (1:9, [L"Ο_{es}", L"Ο_{ep}", L"Rmv", L"Zao", L"Rs", L"Csa", L"Csv", L"E_{max}", L"E_{min}"]), title = L"Sobol - First~Order", xlabel = L"Measurements", ylabel = L"Parameters")
hm = CairoMakie.heatmap!(ax,S1, colormap=:plasma)
for i in 1:3, j in 1:9
txtcolor = S1[i, j] < -1000.0 ? :white : :black
text!(ax, "$(round(S1[i,j], digits = 2))", position = (i, j),
color = txtcolor, align = (:center, :center), fontsize = 15)
end
CairoMakie.Colorbar(f[1,2],hm);
ax = Axis(f[1,3],title=L"Parameter~Importance", xticks = (1:9, [L"Ο_{es}", L"Ο_{ep}", L"Rmv", L"Zao", L"Rs", L"Csa", L"Csv", L"E_{max}", L"E_{min}"][p]), xlabel = L"Parameters", ylabel = L"Importance")
CairoMakie.scatter!( e[p], label=L"Condition~Number")
ax1 = Axis(f[2,1], xticks = (1:9, [L"Ο_{es}", L"Ο_{ep}", L"Rmv", L"Zao", L"Rs", L"Csa", L"Csv", L"E_{max}", L"E_{min}"]), yticks = (1:9, [L"Ο_{es}", L"Ο_{ep}", L"Rmv", L"Zao", L"Rs", L"Csa", L"Csv", L"E_{max}", L"E_{min}"]), title = L"S1-Orthogonality~Matrix")
hm1 = CairoMakie.heatmap!(ax1,Orth_heat1, colormap=:plasma)
for i in 1:9, j in 1:9
txtcolor = Orth_heat1[i, j] < -0.0 ? :white : :black
text!(ax1, "$(round(Orth_heat1[i,j], digits = 2))", position = (i, j),
color = txtcolor, align = (:center, :center), fontsize = 15)
end
CairoMakie.Colorbar(f[2,2],hm1, label = L"Orthogonality~Score", ticks = 0.0:0.2:1.0)
ax = Axis(f[2,3], xticks = 0.0:0.1:1.0, title = L"Orthogonality~Spread", xlabel = L"Orthogonality~Score", ylabel = L"Density")
hist!(ax, a, color = :values, bins = 0.0:0.1:1.0, colormap= :plasma, strokewidth = 1, strokecolor = :black)
for (label, layout) in zip(["A", "B", "C", "D"], [f[1,1], f[1,3], f[2,1], f[2,3]])
Label(layout[1, 1, TopRight()], label,fontsize = 18,font = :bold,halign = :right)
end
f
end