4 Region Axially Distributed Multi Path Michaelis-Menten Model applied to analysis of serotonin uptake by lung tissue following injection into pulmonary artery.

Description

        4 Region Axially Distributed Model applied to analysis of serotonin
        uptake by lung tissue following injection into pulmonary artery.
        from article:
	- "The Uptake and Metabolism of Substrate...".by Linehan et al
	- In "Whole Organ Approaches to Cellular Metabolism", chp 17
	- Edited by Bassingthwaighte, Goresky, Linehan
	- Look at serotonin (5-HT) specifically.

	Uptake into the endothelial and parenchymal regions are modeled using a 
        Michaelis-Menten transporter which is two-sided, concentration dependent.
	
	Three curve model used to fit physiological variables to three
	sets of data simulataneously.
	- each separate curve has aaa, bbb, ccc, suffix on the model variable.
	- Example: CMpaaa(t,x): Conc curve one for mother in plasma.
	- 	   CMpbbb(t,x): Conc curve two    "
	- 	   CMpccc(t,x): Conc curve three  "

	- PSg: Passive conductance channel between Plasma and ISF
	- PSecl: COncentration dependent transporter between Plasma and EC
	- PSeca: Concentration dependent trans between ISF and EC
	- PSpc: Conc dependent transporter between ISF and PC
	- Gec: EC consumption, can be set to zero.
	- Gpc: PC consumption

	Multiple Indicator Dilution model.
	4 compartment model that includes a vascular reference and 
        serotonin tracer curves.
	Assumptions:
	- Tracer (14C-5-HT) << then Mother (5-HT)
	- Competition between Mother and Tracer across membranes.
        - No counter transport of 5-HT used in calculations for PSecl, PSeca, and PSpc.
	-  Vascular reference stays within capillary.
	- To shorten computation time no ISF reference is used.
              Sucrose can be used as a ISF reference.
	- Sucrose and Serotonin PSg are equal.
	
	Capillary heterogeneity is taken into account. 
	- This is done by creating separate paths and assigning a relative mass to each, 
	- creating a simple probabilty density function based on flow (Fp).
	- Currently, seven (7) paths used.
	- Example paths for first curve (CMpaaa(t,x)): CMpaa1(t,x), CMpab1(t,x), CMpac1(t,x)
	- Path flows are relative to average plasma flow.

	Constant infusion:
	- Model can accomadate constant infusion of Mother substrate into system.
	- At time t.min the arteriol has a concentration of CMpaaa_init.
	- WHen a bolus is injected then at x1.min = Cin + CMpaaa_init.
	- Can have three different infusion rates, one for each curve.
	- Assumption: THere is no uptake of Mother in Arteriols.
	- Note: if just want whole system at a const Mother(5-HT) conc then modify the BCs
	- so that:  when (t=t.min) { CMvaa1  = 0; }   becomes:
	- 	when (t=t.min) { CMvaa1  = CMpaaa_init; }, etc...

Equations

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.

The Blood tissue exchange model and equations:

Figure 1. Diagram of a four region blood tissue exchange (Bassingthwaighte 1989).D_ec, D_isf, and D_pc are zero. G_p and G_isf are zero.

Figure 1 shows a diagram of the four region model. The four regions are described by the following four equations:

Capillary:

,

Endothelium:

,

Interstitial:

, and

Parenchymal: 

where C_p, C_ec, C_isf, and C_pc are the concentrations (mol/ml) of the substrate in the capillary, endothelial cell (EC), interstitial fluid space (ISF), and parenchymal cell (PC) respectively, dependent on axial position x and time t. *PS indicate saturable transporters. F_p is plasma flow in ml/(g*min), V_p, V_ec, V_isf, and V_pc are volumes of distribution (ml/g) within the plasma, ec, isf, and pc respectively (in this paper V and V' both represent volumes of distribution for the substrate in question). Volumes of distribution are anatomical volumes for a substrate and are concentration dependent. For passive exchanges between regions the volumes of distribution are directly related to the ratio of concentrations between regions. Where steady-state fluxes exist between regions, the volumes are based on concentrations of the substrate and substrate complex at equilibrium. D_p , D_ec, D_isf, D_pc are the effective axial diffusion coefficients (cm²/sec). G_ec and G_pc are consumption terms in ml/(g*min) for the EC and PC. PS_g is the diffusive permeability-surface area term used to describe the rate of diffusion between the plasma and the ISF. The permeability-surface area products (PS) PS_ecl, PS_eca, and PS_pc describe saturable transporters in ml/(g*min) for substrate transport between plasma-EC, EC-ISF, and ISF-PC respectively.

Endothelial Serotonin transporters:

Endothelial uptake of serotonin can be modeled as a facilitated saturable transporter that in normal physiological states obeys Michaelis-Menten kinetics. This is based on previous multiple indicator dilution (MID) experiments that have shown serotonin concentrations that can saturate the transporter (Rickaby 1981 1982; Peeters 1989; Malcorps 1984). The serotonin transporter is modeled using Michaelis-Menten kinetic parameters V_max �and Michaelis constant K_m:

,

where C_p is the total concentration of serotonin in the capillary plasma and the flux through the transporter from capillary to EC is equal to:

Since the labeled serotonin (tracer) concentration is assumed insignificant compared to the unlabeled serotonin (mother) concentration then CM_p can replace Cp:

, reduces to: