# RecircO2CO2BTEX_3CompLung

Four region, recirculating model for the O2-CO2 transport, exchange and metabolism. Based on Dash 2006 paper. Computes dissolved O2 and CO2 from total O2 and CO2 (TO2 and TCO2) using Dash et al. 2016 routine and Christmas 2017 O2/CO2 solubility algorithm. Gas exchange with 3 compartment lung used. Incorporates calculations for temperature changes based on consumption of O2 in parenchymal tissue region.

Use of MPC to build this model.

Model number: 0405

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## Description

Four region, recirculating model for the O2-CO2 transport, exchange and metabolism. Based on Dash et al 2006 and 2010 papers. Computes dissolved O2 and CO2 from total O2 and CO2 (TO2 and TCO2) through numerical inversion method using Dash et al. 2016 routine. O2 and CO2 gas solubility change with time and position based on Christmas et al. 2017 O2/CO2 solubility algorithm that incorporates temperature and density (or water space) into its solubility calculations. Gas exchange with simple 3 compartment lung is used. Incorporates calculations for temperature changes (time and position dependent) based on consumption of O2 generating heat in parenchymal tissue region and releasing heat through lung exchange.

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## Model layout

**Figure 1:** Layout of model and flow of gases (O2, CO2). Flow of gases in the lung is governed by ordinary differential equations (ODEs) with time domain (Air to Alveolus regions). The recirculating blood and tissue exchange equations are governed by partial differential equations (PDEs) with time and axial spatial domains (Lung capillary to Arteries to Tissue to Veins). See Figure 2 below for details of tissue-exchange. Lung capillary sub-region has just blood (RBC and plasma) with a water film region as the interface between it and the Alveolus. The Veins and Arteries just have the blood region (RBC and plasma).

**Figure 2:** Muscle blood-tissue-exchange (BTEX) region. All four sub-regions: RBC, plasma, interstitial fluid (ISF), parenchymal cell (PC) have separate solubility coefficients for O2 and CO2 (αO2, αCO2), and separate O2, CO2, HCO3-, H+ concentrations. All variables have time and one spatial dimension along the axis of the capillary. The consumption of oxygen occurs in the PC which produces carbon dioxide and heat which raises the temperature in the PC. Temperature of RBC and plasma sub-regions are assumed to be the same. Blood flow (Fblood), RBC flow (Frbc), and plasma flow (Fpl) are different based on hematocrit and measured relative velocity differences between plasma and RBCs. Binding of O2 and CO2 to hemoglobin (Hb) occurs in the RBC and O2 to myoglobin occurs in the PC.

## Model Simulation Plots

**Figure 3:** Axial profiles of the partial pressures of O2 and CO2, pH, and HCO3- concentrations in the four sub-regions of the tissue region. Plot occurs at end of simulation run at near steady-state. Note: consumption of O2 as the partial pressure diminishes along the capillary while CO2 increases. **For all plots:** Initial pO2 in blood vessels = 100 mmHg, pCO2 = 40 mmHg, pO2 in isf and pc = 50 mmHg, CO2 in isf and pc = 50 mmHg. pH in the plasma = 7.4. Partial pressure of O2 in air = 150 mmHg, CO2 = 0 mmHg.

**Figure 4:** Axial profiles of the partial pressures of O2 and CO2, pH, and HCO3- concentrations in the three sub-regions of the lung region. pO2Pul_isf and pCO2Pul_isf correspond to the partial pressures of O2 and CO2 in the thin water film sub-region. Plot occurs at end of simulation run at near steady-state. Note: O2 partial pressure increases along the capillary while CO2 decreases as they equilibrate with the lung.

**Figure 5:** O2 and CO2 hemoglobin saturations in each of the four regions (SHbO2, SHbCO2). **Top** two plot Hb saturation as a function of time while the **bottom** plots Hb saturation as a function of distance along the blood vessels.

**Figure 6:** Temperature in the the seven sub-regions as a function of time (Top) and as a function of blood vessel position (Bottom). Heat is produced in the tissue region (TempTis_pc) and released through the veins (TempVen_cap) and lung (TempPul_cap).

## 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.

## Model Feedback

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## References

Allen TH, Krzywicki HJ, Roberts JE. J. Appl. Physiol 1959;14:1005. [PubMed: 13792786] Allen TH, Welch BE, Trujillo TT, Roberts JE. J. Appl. Physiol 1959;14:1009. [PubMed: 13792787] Austin W, Lacombe E, Rand PW, Chatterjee M. Solubility of carbon dioxide in serum from 15 to 38 c. Journal of Applied Physiology 18: 301–304, 1963. Beek JHGM van, Supandi F, Gavai AK, Graaf AA de, Binsl TW, Hettling H. Simulating the physiology of athletes during endurance sports events: Modelling human energy conversion and metabolism. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 369: 4295–4315, 2011. Bassingthwaighte JB, Yipintsoi T, and Harvey RB. Microvasculature of the dog left ventricular myocardium. Microvasc Res 7: 229-249, 1974. Bassingthwaighte JB and Vinnakota KC. The computational integrated myocyte. A view into the virtual heart. In: Modeling in Cardiovascular Systems. Ann. New York Acad. Sci. 1015:, edited by Sideman S. and Beyar R. 2004, pp 391-404. Brozek J, Grande F, Anderston JT, and Keys A. Densitometric analysis of body composition: revision of some quantitative assumptions. Ann NY Acad Sci 110: 113–140, 1963. Christmas KM, Bassingthwaighte JB. Equations for O2 and CO2 solubilities in saline and plasma: Combining temperature and density dependences. Journal of Applied Physiology 122:1313–1320, 2017. Dash RK and Bassingthwaighte JB (2006), "Simultaneous Blood--Tissue Exchange of Oxygen, Carbon Dioxide, Bicarbonate, and Hydrogen Ion", Annals of Biomedical Engineering., Jul, 2006. Vol. 34(7), pp. 1129-1148. Dash RK and Bassingthwaighte JB (2010), "Erratum to: Blood HbO2 and HbCO2 Dissociation Curves at Varied O2, CO2, pH, 2,3-DPG and Temperature Levels", Annals of Biomedical Engineering., Apr, 2010. Vol. 38(4), pp. 1683-1701. Dash RK, Korman B and Bassingthwaighte JB (2016), "Simple accurate mathematical models of blood HbO2 and HbCO2 dissociation curves at varied physiological conditions: evaluation and comparison with other models", European Journal of Applied Physiology., Jan, 2016. Vol. 116(1), pp. 97-113. Effros RM and Chinard FP. The in vivo pH of the extravascular space of the lung. J Clin Invest 48: 1983–1996, 1969. (for water content of RBC) Gonzalez F and Bassingthwaighte JB. Heterogeneities in regional volumes of distribution and flows in the rabbit heart. Am J Physiol Heart Circ Physiol 258: H1012-H1024, 1990. Haurowitz, F. Chemistry and Biology of Proteins. New York: Academic Press; 1950. (for rho protein) Hill EP, Power GG, Longo LD. A mathematical model of carbon dioxide transfer in the placenta and its interaction with oxygen. American Journal of Physiology 224: 283–299,1973. Jardine B, Raymond GM, and Bassingthwaighte JB. Semi-automated modular program constructor for physiological modeling: Building Cell and organ models. F1000 Research 4: 1461 (doi: 10.12688/f1000research.7476.3) 12pp., 2016. PMID:28698795 PMCID:PMC5488124 Kaminsky DA, Whitman T, Callas PW. DLCO versus dlco/va as predictors of pulmonary gas exchange. Respiratory Medicine 101: 989–994, 2007. Kelman G. Digital computer procedure for the conversion of pco2, into blood co2 content. Respiration Physiology 3: 111–115, 1967. Stolwijk J. A mathematical model of physiological temperature regulation in man. Washington, DC, USA: National Aeronautics Space Administration, 1971. Vinnakota K and Bassingthwaighte JB. Myocardial density and composition: A basis for calculating intracellular metabolite concentrations. Am J Physiol Heart Circ Physiol 286: H1742-H1749, 2004. Yipintsoi T, Scanlon PD, and Bassingthwaighte JB. Density and water content of dog ventricular myocardium. Proc Soc Exp Biol Med 141: 1032-1035, 1972. Ypma TJ. Historical development of the newton–Raphson method. SIAM Review 37: 531–551, 1995.

## Related Models

- Dash 2010 Blood HbO2 and HbCO2 Dissociation Curves.
- Dash 2006 O2-CO2 BTEX model
- Dash 2016 Simplified blood HbO2 and HbCO2 Dissociation Curves.
- Lung O2-CO2 transport and exchange between 3 compartment lung and lung capillary.
- Dash compartmental model of recirculating O2-CO2 in the alveolus, pulmonary and tissue capillaries.
- Christmas 2016 O2, CO2 solubility coefficient curves.

## MPC (Modular Program Constructor)

MPC was used to build up this JSim model from smaller MPC annotated model modules. The modules are listed below:

- BTEX model (MPC annotated version of Dash et. al. 2006 model)
- O2/CO2-Hb dissociation equations (MPC annotated version of Dash et. al. 2016 model).
- Three compartment lung model with O2/CO2 exchange into capillaries (MPC annotated version of Lung BTEX with gas exchange.)
- O2, CO2 solubility calculations (MPC annotated version of Christmas and Bassingthwaighte 2017 model).
- Capillary temperature calculations.
- Interstitial fluid (ISF) temperature calculations.
- Parenchymal cell temperature calculations.

The modules where put together using MPC directives stored in MPC text files. First the tissue/muscle region model was built and tested, next the lung region, followed by the veins and arteries. These four sub-models were tested individually before combining together into the full recirculating model. The MPC files used are listed below:

The Tissue/muscle region:

- TissTotO2freeO2Invert.mpc, inverts the O2-Hb dissociation equations for use in the full tissue BTEX module.
- TissTotCO2freeCO2Invert.mpc, Inverts the CO2-Hb dissociation equations for use in the full tissue BTEX module.
- BTEX2006_UpdateTemp.mpc, Incorporates the temperature equations and the O2CO2 dissociation equations into the full tissue BTEX module. This model is used for testing.

The Lung region:

- LungTotO2freeO2Invert.mpc, inverts the O2-Hb dissociation equations for use in the full tissue BTEX module.
- LungTotCO2freeCO2Invert.mpc, Inverts the CO2-Hb dissociation equations for use in the full tissue BTEX module.
- Lung3CompBTEX2006.mpc, Incorporates the temperature equations, 3 compartment lung, and the O2CO2 dissociation equations into the full lung BTEX module. This model is used for testing.

The Arterial region:

- ArtTotO2freeO2Invert.mpc, inverts the O2-Hb dissociation equations for use in the full artery module.
- ArtTotCO2freeCO2Invert.mpc, Inverts the CO2-Hb dissociation equations for use in the full artery module.
- ArtBTEX2006_Temp.mpc, Incorporates the temperature equations and the O2CO2 dissociation equations into the full artery module. This model is used for testing.

The Venous region:

- VeinTotO2freeO2Invert.mpc, inverts the O2-Hb dissociation equations for use in the full venous module.
- VeinTotCO2freeCO2Invert.mpc, Inverts the CO2-Hb dissociation equations for use in the full venous module.
- VeinBTEX2006_Temp.mpc, Incorporates the temperature equations and the O2CO2 dissociation equations into the full venous module. This model is used for testing.

The full recirculating model:

- RecircBTEX_3CompLung.mpc, incorporates all four regions and updates the boundary conditions for each region to be the output of the previous one.

## Key terms

BTEX, PDE, Cardiac grid, Oxygen, CO2, pH, Solubility, tissue exchange, gas exchange, Haldane Hemoglobin binding, MPC

## Model History

Get Model history in CVS.Posted by: BEJ

## Acknowledgements

Please cite **www.physiome.org** 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.

Or send a copy to:

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

[This page was last modified 03May19, 2:00 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.