# TranspMM.2sol2sided.BolusSw.MMID4

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

Model number: 0033

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## 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 (cm^{2}/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:

_{} _{}

PS_{ecl} and PS_{eca}
are assumed to be identical in kinetic behavior and V_{max_ecl} and K_{m}
are the same for each. The use of a simplified transporter is based on two
assumptions: the serotonin binding and unbinding reactions with the membrane
transporter are fast compared to the movement of the serotonin-transporter
complex from one side to the other side of the membrane itself and the
transporter concentration is small compared to the substrate serotonin. Based
on the first assumption, the transporter is considered a non-capacitance
transporter.

**Capillary flow
heterogeneity**:

The modeling of capillary flow heterogeneity is done using a probability density function (PDF) of relative regional flows, w(f). This is further simplified by using a number of discrete flows, instead of a continuous range, to represent the capillary heterogeneity of the lung. Since we are modeling relative flows, f, to the mean capillary flow rate, by definition, w(f) has a mean of 1 and its area is one (King 1996). With seven capillaries being used this simplifies, for n=7, to:

_{}, reduces to:_{}
and

_{}, reduces to:_{},

where w_{i} is the
frequency of occurrence, within a mass fraction of the organ, in the range
Δf_{i} about relative flow f_{i} . The PDF used has a
relative dispersion of 0.5 based on reported results for the capillary
dispersion within the lung (Knopp et al, 1969). A random walk function is used
to represent the PDF with a RD of 0.5 and skewness of 1.6. The relative flow
range used varied by dataset but was usually between 0.5 and 2.0.

Arteriole and venule heterogeneity are approximated using the same technique. The arterioles and venules are treated as ‘pipes’ in that there is no uptake or metabolism of serotonin. They only add dispersion and a time delay between input curve and output curve.

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## References

- Bassingthwaighte, J.B., C.Y. Wang, and I.S. Chan: "Blood-Tissue Exchange Via Transport and Transformation by Capillary Endothelial-Cells". Circulation Research, 1989. 65(4): p. 997-1020. - Bronikowski, T.A., et al.: "A Mathematical-Model of Indicator Extraction by the Pulmonary Endothelium Via Saturation Kinetics". Mathematical Biosciences, 1982. 61(2): p. 237-266. - Dawson, C.A., et al.: "Kinetics of Serotonin Uptake in the Intact Lung". Annals of Biomedical Engineering, 1987. 15: p. 217-227. - King, R.B., G.M. Raymond, and J.B. Bassingthwaighte: "Modeling blood flow heterogeneity". Annals of Biomedical Engineering, 1996. 24(3): p. 352-372. - Linehan, J.H., S.H. Audi, and C.A. Dawson: "The Uptake and Metabolism of Substrates in the Lung" in Whole Organ Approaches to Cellular Mechanism, J. Bassingthwaighte, C.A. Goresky, and J.H. Linehan, Editors. 1998, Springer-Verlag: New York. p. 427-437. - Linehan, J.H., T.A. Bronikowski, and C.A. Dawson: "Kinetics of Uptake and Metabolism by Endothelial-Cell from Indicator Dilution Data". Annals of Biomedical Engineering, 1986. 14(1): p. 87-87. - Malcorps, C.M., et al." "Lung Serotonin Uptake Kinetics from Indicator-Dilution and Constant-Infusion Methods". Journal of Applied Physiology, 1984. 57(3): p. 720-730. - Rickaby, D.A., C.A. Dawson, and J.H. Linehan: "Influence of Blood and Plasma-Flow Rate on Kinetics of Serotonin Uptake by Lungs". Journal of Applied Physiology, 1982. 53(3): p. 677-684. - Rickaby, D.A., et al.: "Kinetics of Serotonin uptake in the dog lung". J. Appl. Physiol, 1981. 51: p. 405-414.

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## Model History

Get Model history in CVS.## Acknowledgements

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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 19May10, 10:59 am.]

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