- Author:
- aram148 <a.rampadarath@auckland.ac.nz>
- Date:
- 2021-10-11 21:47:30+13:00
- Desc:
- Peripheral airway pseudo-BG
- Permanent Source URI:
- https://models.physiomeproject.org/workspace/6b0/rawfile/55efde1e181fb28eb5ec12c06d40e63dfe2ca938/USMC/Maggio2012/DM_funcs.m
function [F,q1,q2,p1,p2,Ptm,V,Raw,radius,Pab,vv,tau]=DM_funcs(t,R,lambda,f,rho,N1,N2,Ri_sq,rmax_sq,rmax,P1,P2,gamma,Pawb,P011)%,t1,t2)%,k1,k2)
%% Parameter values used for both ASM-aw and Anafi-Wilson coupled system
a = 60;b = 62; %1st DI
%E = 25;%kPa/ml
E=254.9;%cmH2O/ml %Elastance of acinus
L = 2.6852;%mm %airway length
% L=2*pi*rmax;
aa = 10.19716213; % Conversion factor for kg*(mm)^(-1)*(s)^(-2) to cmH20
moo = 1.9008e-8; % dynamic viscosity kg(mm)^(-1)s^(-1)
k2=0.1;
k1=0.35.*(t<5)+0.06.*(t>=5); %This represents calcium activated rate functions
% k1=0.06;
k20=0.005;
g1=2.*k20;
g2=20.*g1;
g3=3.*g1;
fp1=0.88;
gp1=0.22;
gp2=4.*(fp1+gp1);
gp3=3.*gp1;
% r_ref=sqrt(rmax_sq-(rmax_sq-Ri_sq)*(1-(25./P2)).^-N2);
% R as initial conditions
r(1)=R(1);
r(2)=R(2);
r(3)=R(3);
r(4)=R(4);
r(5)=R(5);
r(6)=R(6);
r(7)=R(7);
r(8)=R(8);
%% Emprical L-T force_l taken from Donovan 2016
% force_l=(sin((pi*r(8))/(2*rmax))).^3.*(r(8)<=2*rmax)+0.*(r(8)>2*rmax);
force_l=1;
%removing the activation of ASM force; possibly simulating varying ASM force activation
% lambda1 = lambda.*(t<=60)+0.02*lambda.*(t>60);
%ASM force
force_a=lambda*(r(2)+r(5));
%% Anafi-Wilson type pressure difference
P01=P011;%.*(t<a)+((20*P011)).*(t>=a).* (t<=b)+P011.*(t>b).*(t<t1)+((20*P011)).*(t>=t1).*(t<=t2)+(P011).*(t>t2);
force=force_l*force_a;
% Pawbb=((P011))*abs(sin(2*pi*f*t));
%Pawbb=P011.*(t<a)+((20*P011)).*(t>=a).* (t<=b)+P011.*(t>b).*(t<t1)+((20*P011)).*(t>=t1).*(t<=t2)+(P011).*(t>t2);
Pawbb = Pawb*abs(sin(2*pi*f*t));%.*(t<a)+((10*Pawb))*abs(sin(2*pi*f*t)).*(t>=a).* (t<=b)+Pawb*abs(sin(2*pi*f*t)).*(t>b).*(t<t1)+((10*Pawb))*abs(sin(2*pi*f*t)).*(t>=t1).*(t<=t2)+(Pawb)*abs(sin(2*pi*f*t)).*(t>t2);
%Vfrc = 1300; %cm^3
Raw = 8*aa*L*moo/(pi*r(8).^4);
Paw = P01 + Pawbb;%pressure in airway
Pab = Pawb*E/(E.^2 + (2*pi*f*Raw).^2)^0.5;
alph = atan(2*pi*f*Raw/E);
Pabb = Pab*abs(sin(2*pi*f*t-alph));%.*(t<a)+((10*Pab))*abs(sin(2*pi*f*t-alph)).*(t>=a).* (t<=b)...
% +Pab*abs(sin(2*pi*f*t-alph)).*(t>b).*(t<t1)+((10*Pab))*abs(sin(2*pi*f*t-alph)).*(t>=t1).*(t<=t2)...
% +(Pab)*abs(sin(2*pi*f*t-alph)).*(t>t2);
% Pabb = ((Pab))*abs(sin(2*pi*f*t-alph));
% Pabb=Pab.*(t<a)+((10*Pab)).*(t>=a).* (t<=b)...
% +Pab.*(t>b).*(t<t1)+((10*Pab)).*(t>=t1).*(t<=t2)...
% +(Pab).*(t>t2);
Pa = P01 + Pabb; %pressure in acinus
vv = 0.2 +0.04*Pa; %Acinus volume
xx = 1-(r(8)/vv.^(1/3));
P_lumen = (Paw + Pa)/2;
%P0=P_lumen + Pa;
tau = Pa + Pa*(1.4*xx+2.1*xx.^2);%parenchymal tethering force
% P0=(((P01))*abs(sin(2*pi*f*t))+Pmin); %to be used for DM ASM coupled system
% tau=1.4*P0*(((r_ref-r(8))/r_ref)+1.5*((r_ref-r(8))/r_ref).^2);
%Transmural pressure as a function of force.
% Ptm=Pmin-((force*r_ref)./r(8))+tau;
Ptm=((P_lumen)-(force./r(8))+tau);
Ten = Ptm*r(8); %tension of ASM
% Ptm=5-(force./rmax)+(P0./rmax);
if Ptm<=0
% a=a0(s).*(1-(Ptm./P1)).^(-N1(s));
rad=sqrt(Ri_sq*(1-(Ptm/P1)).^-N1);
elseif Ptm>=0
% a=(1-((1-a0(s)).*(1-(Ptm./P2)).^(-N2(s))));
rad=sqrt(rmax_sq-(rmax_sq-Ri_sq)*(1-(Ptm/P2)).^-N2);
end
% H=2.223-0.066*Ptm+0.017*Ptm.^2;
% E=H;
radius = rad;
drdt=rho*(rad-(r(8))); % this is to be used for the orginal DM coupled system.
V=-((gamma)/(2*pi*rmax))*drdt;
% dLdt=drdt.*(2*pi);%length change for ASM
%% DM parameters
%Mean for cdf
p1=r(2)/r(1);
p2=r(5)/r(4);
%Standard deviation for cdf
q1=(sqrt((r(3)/r(1))-((r(2)/r(1)).^(2))));
q2=(sqrt((r(6)/r(4))-((r(5)/r(4)).^(2))));
%r,phi and I values for 1st PDE M1_lambda
r0=-p1/q1;
r1=(1-p1)/q1;
% rinf=(Inf-p1)/q1;
% phi0=erf((r0-p1)./q1.^2);
% phi0=normcdf(r0,p1,q1);
phi0=0.5*(1+erf((r0-p1)/(q1*sqrt(2))));
% phi1=normcdf(r1,p1,q1);
phi1=0.5*(1+erf((r1-p1)/(q1*sqrt(2))));
% phi1=erf((r1-p1)./q1.^2);
% phinf=cdf('Normal',rinf,p1,q1);
% phinf=0.5*(1+erf((rinf-p1)/(q1*sqrt(2))));
% phinf=erf((Inf-p1)./q1);
I0=-(exp(-((-p1./q1).^(2))/2))/(sqrt(2*pi));
I1=-(exp(-((((1-p1)./q1)).^(2))/2))/(sqrt(2*pi));
% I2=-(exp(-(((Inf-p1)./q1)).^(2))/2)/(sqrt(2*pi));
%r,phi and I values for 2nd PDE M2_lambda
r20=-p2/q2;
r21=(1-p2)/q2;
% r2inf=(Inf-p2)/q2;
% phi20=erf((r20-p2)/(q2).^2);
% phi21=erf((r21-p2)/(q2).^2);
% phi2inf=erf(r2inf);
% phi20=normcdf(r20,p2,q2);
phi20=0.5*(1+erf((r20-p2)/(q2*sqrt(2))));
% phi21=normcdf(r21,p2,q2);
phi21=0.5*(1+erf((r21-p2)/(q2*sqrt(2))));
% phi2inf=cdf('Normal',r2inf,p2,q2);
% phi2inf=0.5*(1+erf((r2inf-p2)/(q2*sqrt(2))));
%
I20=-(exp(-((-p2./q2).^(2))/2))/(sqrt(2*pi));
I21=-(exp(-((((1-p2)./q2)).^(2))/2))/(sqrt(2*pi));
% I2in=-(exp(-(((Inf-p2)./q2)).^(2))/2)/(sqrt(2*pi));
%Functions for the rhs of the first PDE M1_lambda
J0=phi0;
% J01=phi1;
% J0inf=phinf;
J10=((p1.*phi0)+(q1.*I0));
J11=(p1.*phi1)+(q1.*I1);
% J12=(p1.*phinf)+(q1.*I2);
J20=((p1.^(2)).*phi0)+((2*p1.*q1).*I0)+((q1.^(2)).*(phi0+(r0*I0)));
J21=((p1.^(2)).*phi1)+((2*p1.*q1).*I1)+((q1.^(2)).*(phi1+(r1*I1)));
% J22=((p1.^(2)))+((2*p1.*q1).*I2)+((q1.^(2)));
J30=(p1.^(3).*phi0)+((3.*p1.^(2).*q1).*I0)+((3.*p1.*q1.^(2)).*((phi0)+(r0*I0)))+((q1.^(3).*(2+r0.^(2)).*I0));
J31=(p1.^(3).*phi1)+((3.*p1.^(2).*q1).*I1)+((3.*p1.*q1.^(2)).*(phi1+(r1*I1)))+(q1.^(3).*(2+(r1.^(2))).*I1);
% J32=(p1.^(3))+((3.*p1.^(2).*q1).*I2)+((3.*p1.*q1.^(2)));
r_3 = p1.^3 + 3*p1*q1.^2;
% figure out how to evaluate this between [0,1]
% if X0>=0 && X0<=1
% J_new = fp1*((J11-J10)+(r_3/6)*(3*X0-X0.^3)*(J11-J10));
% % J_new1 = fp1*((J21-J20)+(r_3/6)*(3*X0-X0.^3)*(J21-J20));
% else
% J_new =0;
% end
%Functions defined for the RHS of the second PDE M2_lambda
K0=phi20;
% K01=phi21;
% K0inf=phi2inf;
K10=((p2.*phi20)+(q2.*I20));
K11=(p2.*phi21)+(q2.*I21);
% K12=(p2.*phi2inf)+(q2.*I2in);
K20=((p2.^(2)).*phi20)+((2*p2.*q2).*I20)+((q2.^(2)).*(phi20+(r20*I20)));
K21=((p2.^(2)).*phi21)+((2*p2.*q2).*I21)+((q2.^(2)).*(phi21+(r21*I21)));
% K22=((p2.^(2)))+((2*p2.*q2).*I2in)+((q2.^(2)));
K30=(p2.^(3).*phi20)+((3.*p2.^(2).*q2).*I20)+((3.*p2.*q2.^(2)).*((phi20)+(r20*I20)))+((q2.^(3).*(2+r20.^(2)).*I20));
K31=(p2.^(3).*phi21)+((3.*p2.^(2).*q2).*I21)+((3.*p2.*q2.^(2)).*(phi21+(r21*I21)))+(q2.^(3).*(2+(r21.^(2))).*I21);
% K32=(p2.^(3))+((3.*p2.^(2).*q2).*I2in)+((3.*p2.*q2.^(2)));
%Components for the matrix F that will represent each moment,
%M1_lambda and M2_lambda
A0= ((fp1*(1-r(7)))/2)-(fp1*(J11-J10)*r(1));%-2*(fp1*(K11-K10)*r(4));
A1=((fp1*(1-r(7)))/3)-(fp1*(J21-J20)*r(1));%-2*(fp1*(K21-K20)*r(4));
A2=((fp1*(1-r(7)))/4)-(fp1*(J31-J30)*r(1));%-2*(fp1*(K31-K30)*r(4));
B0=(gp2*J0)+gp1*(J11-J10)+(gp1+gp3)*(p1-J11);
B1=(gp2*J10)+gp1*(J21-J20)+(gp1+gp3)*((p1.^(2)+q1.^(2))-J21);
B2=(gp2*J20)+gp1*(J31-J30)+(gp1+gp3)*((p1.^(3)+3*p1*q1.^(2))-J31);
C0=k1;
C1=k1*p2;
C2=k1*(p2.^(2)+q2.^(2));
D0=k2;
D1=k2*p1;
D2=k2*(p1.^(2)+q1.^(2));
E0=(g2*K0)+g1*(K11-K10)+(g1+g3)*(p2-K11);
E1=(g2*K10)+g1*(K21-K20)+(g1+g3)*((p2.^(2)+q2.^(2))-K21);
E2=(g2*K20)+g1*(K31-K30)+(g1+g3)*(p2.^(3)+(3*p2*q2.^(2))-K31);
% dfdt=-V*lambda*(r(1)+r(4))+lambda*(A1-B1-C1);
% Putting together the matrix F for the RHS of the system of
% equations
F=[A0-B0*r(1)+C0*r(4)-k2*r(1);A1-B1*r(1)+C1*r(4)-k2*r(2)-V*r(1);A2-B2*r(1)+C2*r(4)-k2*r(3)-2*V*r(2);D0*r(1)-E0*r(4)-k1*r(4);D1*r(1)-E1*r(4)-k1*r(5)-V*r(4);D2*r(1)-E2*r(4)-k1*r(6)-2*V*r(5);-k1*r(7)+(1-r(7))*k2; rho*(rad-r(8))];
