% The function third_nac_eval_deriv.m evaluates the right-hand side of a
% set of first-order differential equations which characterize the
% intact circulation with third-order systemic arteries and nonlinear
% large, elastic arterial compliance at a desired time step.
%
% Function arguments:
% P - a 8x1 vector containing the current pressure values
% = [Pl; Pe; Pv; Pr; Ppa; Ppv; Pm; qe]
% Qc - a 2x1 vector containing the current ventricular volume values
% - [Ql; Qr]
% Pbreathe - 4x1 vector containing current respiratory values
% [Qlu; Pth; dPth; Ppac (Palv)]
% th - current parameter values
% sumdeltaT - time surpassed in current cardiac cycle
%
% Function outputs:
% dP - a 8x1 vector containing the derivative of current
% pressure values
% = [dPl; dPe; dPv; dPr; dPpa; dPpv; dPm; dqe]
%
function dP = third_eval_deriv(P,Qc,Pbreathe,th,sumdeltaT);
% Computing flow rates from the current pressure values.
if (P(6) >= P(1))
qli = (P(6)-P(1))/th(20);
else
qli = 0;
end
if (P(1) >= P(2))
qlo = (P(1)-P(2))/th(15);
else
qlo = 0;
end
qla = P(8);
qa = (P(7)-P(3))/th(16);
if (P(3) > P(4) & P(4) >= th(91))
qri = (P(3)-P(4))/th(17);
elseif (P(3) > th(91) & th(91) > P(4))
qri = (P(3)-th(91))/th(17);
else
qri = 0;
end
if (P(4) >= P(5))
qro = (P(4)-P(5))/th(18);
else
qro = 0;
end
h1 = (1330/(1.06*980))*(P(6)-Pbreathe(4));
if (h1 < -10)
h1 = -10;
elseif (h1 > 20)
h1 = 20;
end
h2 = (1330/(1.06*980))*(P(5)-Pbreathe(4));
if (h2 < -10)
h2 = -10;
elseif (h2 > 20)
h2 = 20;
end
qpa = (((P(5)-P(6))/(30*th(19)))*(h1+10)) + (((P(5)-Pbreathe(4))/(30*th(19)))*(h2-h1)) - (((1.06*980)/(2660*(30*th(19))))*(h2^2-h1^2));
% Computing ventricular elastances.
[El,dEl] = var_cap(th(1),th(2),th(24),sumdeltaT);
[Er,dEr] = var_cap(th(5),th(6),th(24),sumdeltaT);
% Computing dP from the above calculations and th parameter vector.
if (P(1) < Pbreathe(2))
g = ((P(1)-Pbreathe(2)))/(El*th(9)*(2/pi));
dPl = El*(qli-qlo)*(1+g^2) + (P(1)-Pbreathe(2))*(dEl/El) + Pbreathe(3);
else
y = (P(1)-Pbreathe(2))/th(31);
x = vent_vol(Qc(1),y,El);
x0 = 1/(El+1);
k = 50;
mbrealscalar(-k*(x-x0));
mbrealscalar(-k*(1-x0));
mbrealscalar(k*x0);
dx = (qli-qlo)/(th(26)-th(9));
dx0 = -dEl/((El+1)^2);
c = 1+exp(-k*(1-x0));
d = 1+exp(k*x0);
g = 1+exp(-k*(x-x0));
dc = k*dx0*exp(-k*(1-x0));
dd = k*dx0*exp(k*x0);
dg = -k*(dx-dx0)*exp(-k*(x-x0));
denb = (1+(1/k)*log(c/d));
b = (1-El)/denb;
db = (-dEl*denb - (1-El)*(1/k)*((d*dc-c*dd)/(c*d)))/(denb^2);
dy = dEl*x + El*dx + db*(x+(1/k)*(log(g/d))) + b*(dx+(1/k)*((d*dg-g*dd)/(g*d)));
dPl = th(31)*dy + Pbreathe(3);
end
if (P(4) < Pbreathe(2))
g = ((P(4)-Pbreathe(2)))/(Er*th(12)*(2/pi));
dPr = Er*(qri-qro)*(1+g^2) + (P(4)-Pbreathe(2))*(dEr/Er) + Pbreathe(3);
else
y = (P(4)-Pbreathe(2))/th(32);
x = vent_vol(Qc(2),y,Er);
x0 = 1/(Er+1);
k = 50;
mbrealscalar(-k*(x-x0));
mbrealscalar(-k*(1-x0));
mbrealscalar(k*x0);
dx = (qri-qro)/(th(27)-th(12));
dx0 = -dEr/((Er+1)^2);
c = 1+exp(-k*(1-x0));
d = 1+exp(k*x0);
g = 1+exp(-k*(x-x0));
dc = k*dx0*exp(-k*(1-x0));
dd = k*dx0*exp(k*x0);
dg = -k*(dx-dx0)*exp(-k*(x-x0));
denb = (1+(1/k)*log(c/d));
b = (1-Er)/denb;
db = (-dEr*denb - (1-Er)*(1/k)*((d*dc-c*dd)/(c*d)))/(denb^2);
dy = dEr*x + Er*dx + db*(x+(1/k)*(log(g/d))) + b*(dx+(1/k)*((d*dg-g*dd)/(g*d)));
dPr = th(32)*dy + Pbreathe(3);
end
alphatau = -th(70)/th(72);
Qamax = th(76)-th(72)^2/th(70);
Qao = th(76)+(th(72)^2/th(70))*exp(-th(70)*th(40)/th(72))*(1-exp(th(70)*th(40)/th(72)));
dP = [dPl; ((qlo-qla)/(alphatau*(Qamax-Qao)*exp(-alphatau*P(2)))); (qa-qri)/th(4); dPr; ((qro-qpa)/th(7))+Pbreathe(3); ((qpa-qli)/th(8))+Pbreathe(3); (qla-qa)/th(73); (P(2)-P(7))/th(69)];