// MODEL NUMBER: 0060
// MODEL NAME: Myogenic_Compliant_Vessel
// SHORT DESCRIPTION:
// This model simulates the flow through a passive and actively
// responding vessel driven by a sinusoidal pressure input.
import nsrunit; unit conversion on;
math Myogenic_Compliant_Vessel { realDomain t sec; t.min = 0; t.max = 50; t.delta = 0.01;
// PARAMETERS:
real
Pmean = 30 mmHg, // Mean input driving pressure
Pamp = 2.5 mmHg, // Input driving pressure amplitude
Pc = 30 mmHg, // Control pressure
tnorm = 1 sec, // Normalizing time constant
C1p = 1.719 N/m, // Passive tension parameter
C2p = 14.354, // Passive tension parameter
Dp100 = 255.4 um, // Reference vessel diameter
C1a = 2.306 N/m, // Maximally active tension parameter
C2a = 0.910, // Maximally active tension parameter
C3a = 0.374, // maximally active tension parameter
Cmyo = 4.674 m/N, // Myogenic VSM activation parameter
Cglobal = 2.940, // Basal tone VSM activation parameter
taud = 5 sec, // Passive response time constant
taua = 60 sec, // Actvie VSM response time constant
mu = 3 cP, // Fluid viscosity
L = 12500 um, // Vessel length
Pout = 25 mmHg, // Output pressure
Pext = 0 mmHg, // Extraluminal pressure
Dc um, // Control diameter
Tc N/m, // Control tension
Ac; // Control activation
// VARIABLES:
real
Pin(t) mmHg, // Input driving pressure
T(t) N/m, // Vessel wall tension
Ttarget(t) N/m, // Target SS T for current state
Atarget(t), // Targer SS A for current state
D(t) um, // Current vessel diameter
A(t), // Current VSM activation
R(t) dyne*s/(uL*cm^2), // Vessel resistance to flow
Fout(t) uL/s, // Flow through the vessel
Fin(t) uL/s, // Flow into the vessel
Fcomp(t) uL/s, // Flow attributed to volume change
V(t) uL; // Volume of vessel
// INITIAL CONDITION:
when (t=t.min) {D = Dc; A = Ac; }
// ALGEBRAIC AND ODE EQUATIONS:
Pin = Pmean + Pamp*sin(2*PI*t/tnorm); // Input pressure sine train
Dc > Dp100/4; // Limits put on diameter
Dc < 4*Dp100/3; // so root can be found
// The implicit expression to solve for D as a function of Pin
// This determines the initial conditions for the dynamic response
Dc = (2/Pc) * ( (C1p*exp(C2p*((Dc/Dp100)1)))
+ ( (C1a*exp(1*(((Dc/Dp100)C2a)/C3a)^2))
/ (1+exp((1*Cmyo*(Dc*Pc/2)) + Cglobal)) ) );
Tc = Pmean*Dc/2;
Ac = (1+exp((1*Cmyo*Tc) + Cglobal))^1;
// Law of Laplace and calculation of steady state target values
// for the current vessel state which drive the dynamic response
T = D*Pin/2;
Ttarget = (C1p*exp(C2p*(((2*T/Pin)/Dp100)1)))
+ A * (C1a*exp(1*((((2*T/Pin)/Dp100)C2a)/C3a)^2));
Atarget = (1+exp((1*Cmyo*(D*Pin/2)) + Cglobal))^1;
D:t = (1/taud)*(Dc/Tc)*(TTtarget); // Diameter dynamic response
A:t = (1/taua)*(AtargetA); // Activation dynamic response
R = (128*mu*L) / (PI*D^4); // Vessel resistance to flow
Fout = (Pin  Pout)/R; // Set of equations to solve
Fin = Fout + Fcomp; // for flows and volume
V = ((PI*D^2)/4) * L;
Fcomp = V:t;
} // END OF MML CODE
/*
FIGURE:
Fin
> Pin R Pout
o/\/\/\/\o
 >
  Fout
  Fcomp
 v

C,V =====




o Pext



DETAILED DESCRIPTION:
The model simulates fluid flow, F, through a compliant vessel that
actively responds to the pressure in the vessel given a pressure drop
across the length of the vessel equivalent to Pin  Pout. The previous
compliant vessel models (Compliant vessel, Thin wall compliant vessel and
Nonlinear compliant vessel) have behaved passively to the changes in pressure
but it is known that arterioles in particular respond actively: dilating when
the pressure decreases and constricting when the pressure increases.The active
response of a vessel to pressure is known as the myogenic response. A
misnomer since responses to all different stimulii are effected through
activation or deactivation of the vascular smooth muscle (VSM).
The flows, Fin, Fout and Fcomp, and the vessel volume, V are unknown in this
simulation. Therefore four equations are needed to solve for these four
variables. The expressions for the flows are as presented in the previous
compliant vessel models. To solve for V the myogenic response formulation is
used to evaluate the vessel diameter based on relative contributions of
the passive and myogenically active responses.
The mathematical formulation for the myogenic response is from Carlson
and Secomb (Microcirc 12:327338, 2005). In this study the vessel wall
is represented by a nonlinear spring and a contractile VSM element in
parallel as shown below.
Tp
/\/\/\/\
 
T < > T
  
 VSM 

A*Tma
where T is the total tension, Tp is the passive tension and Tma is
the maximally active tension in the vessel wall. A is the level of
VSM activation with a range of 0 to 1. The passive tension is
described by an exponential function of the form:
Tp = C1p * exp [ C2p * (D/Dp100  1) ]
where C1p and C2p are parameters optimized to fit passive response
experimental data, D is the vessel diameter and Dp100 is the reference
passive diameter of the vessel at a intraluminal pressure of 100 mmHg.
The maximally active tension is described by a gaussian function:
_ _ _ 2 _
  D/Dp100  C2a  
Tma = C1a *exp <     >
_ _ C3a _ _
where C1a, C2a and C3a are parameters optimized to fit myogenically
responding arterioles from experimental data. Finally we have the
VSM activation, A, which is represented by a sigmoidally shaped
function of the form:
1
A = 
1 + exp(Cmyo*T + Cglobal)
where Cmyo and Cglobal are also fit to the myogenically active
arteriolar data and the total tension, T, is given by the law of
Laplace:
Pin*D
T = 
2
where Pin is the pressure at the vessel input and in this case is an
sinusoidal input driving the flow in the vessel. All of these
expressions can be combined to create an implicit function of diameter
as a function of input pressure, Pin.
_
2 
D =  * < C1p * exp [ C2p * (D/Dp100  1) ]
Pc _
_ _ _ 2 _ _
  D/Dp100  C2a   
C1a *exp <     > 
_ _ C3a _ _ 
+  >
1 + exp(Cmyo*(Pin*D/2) + Cglobal) _
This implicit equation is used to solve for the initial diameter and
then this diameter and the VSM activation is updated depending on the
steady state diamter and activation that the vessel would desire to
go to at the current pressure. This update is facilitated by the
ordinary diffeerntial equations for D and A:
dD/dt = (1/taud)*(Dc/Tc)(T  Ttarget)
dA/dt = (1/taua)*(Atarget  A)
where taud and taua are the time constants that determine how fast the
diameter and activation move towards their respective steady state
values, Dtarget and Atarget.
The remaining code is the same as we have developed for the compliant
vessel and the compliant vessel thinwall formulation. The flow out of
the vessel is related to the resistance by the fluid equivalent of
Ohm's Law.
Fout = (Pin  Pout) / R
where Pin  Pout is the difference in pressure between the beginning and
end of the vessel and R is determined from Poiseuille's Law as:
R = 128*mu*L / pi*D^4
where mu is the fluid viscosity.
The flow into the vessel and the flow out of the vessel are different
because of the change in volume which adds or subtracts flow from that
leaving the vessel depending on whether the pressure is increasing or
decreasing in the vessel. So we have:
Fin = Fout + Fcomp
where the flow attributed to the vessel compliance, Fcomp, is given by:
Fcomp = dV/dt
and where V is the vessel volume and is now purely a function of the
diameter
V = PI * D^2 * L / 4
The parameters defining the vessels myogenic response are optimized
to data for ~200 um arteriolar vessels from Liao and Kuo (Am J Physiol,
Heart Circ Physiol 272:H1571H1581, 1997).
SHORTCOMINGS/GENERAL COMMENTS:
 Specific inadequacies or next level steps
KEY WORDS: Resistance, Compliance, Myogenic response, Regulation, Vessel,
Pressure, Flow, Law of Laplace, Ohm's Law, Cardiovascular system, Hemodynamics
REFERENCES:
Carlson BE and Secomb TW; A theoretical model for the myogenic response based
on the lengthtension characteristics of vascular smooth muscle.
Microcirculation 12:327338, 2005.
Liao JC and Kuo L; Interaction between adenosine and flowinduced dilation
in coronary microvascular networks.
Am J Physiol, Heart Circ Physiol 272:H15711581, 1997.
REVISION HISTORY:
Created by Brian Carlson
Modified by Brian Carlson
on 4 September 2007
Modified by Micah Nicholson Date:7Apr2009
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 19992011 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 981955061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
This software was developed with support from NIH grant HL073598.
Please cite this grant in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
*/