This model simulates the flow through a passive and actively responding vessel driven by a sinusoidal pressure input.
Model number: 0060
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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 12:327-338, 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*Tmawhere 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 = -------- 2where 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) / Rwhere 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^4where 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 + Fcompwhere the flow attributed to the vessel compliance, Fcomp, is given by:
Fcomp = dV/dtand where V is the vessel volume and is now purely a function of the diameter
V = PI * D^2 * L / 4The 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:H1571-H1581, 1997).
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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Carlson BE and Secomb TW; A theoretical model for the myogenic response based on the length-tension characteristics of vascular smooth muscle. Microcirculation 12:327338, 2005. Liao JC and Kuo L; Interaction between adenosine and flow-induced dilation in coronary microvascular networks. Am J Physiol, Heart Circ Physiol 272:H1571-1581, 1997.
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[This page was last modified 29Jan20, 1:02 pm.]
Model development and archiving support at physiome.org provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.