Simplified Models For Gas Exchange In The Human Lungs
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Structure
Data from several different experiments suggest that complex interactions between the respiratory system and cardiac output exist in humans. Such interactions include the synchronisation between lung ventilation rate and the heart beat, and other examples. Mathematical models to describe the mechanisms underlying these interactions are being developed with the aim of improving our understanding of these physiological phenomena.
Historically, gas exchange in the lungs has been modelled and studied at various different levels of detail. The 2005 paper by Alona Ben-Tal described here presents a hierarchy of models of increasing complexity for gas exchange in the human lungs:
Model A: The first model represents the lung as an inflexible, single compartment (see below);
Model B: The second model represents the lung as a flexible, single compartment (see below);
Model C: The third model represents the lung as a flexible, single compartment with gas exchange (see below); and
Model D: The final model represents the respiratory system as a flexible, multi-compartmental model with gas exchange and gas transport (see below).
The models developed in this paper provide a mathematical framework which is able to link together molecular scale and whole systems scale models. As an integral part of developing this framework, two commonly made assumptions have been re-examined:
1) the hidden assumption that air flow through the mouth is equal to the rate of volume change in the lungs; and
2) the assumption that the process of oxygen binding to hemoglobin is at near equilibrium.
Conditions under which these assumptions are valid have been studied, and the models are sufficiently flexible to be able to qualitatively explain a range of physiological observations.
The models have been described here in CellML (the raw CellML descriptions of the four different Ben-Tal 2005 models can be downloaded in various formats as described in ).
The CellML files provided below represent the 2 different versions of the aforementioned models A,B,C and D.
model diagram
Schematic diagram of inflexible lung model.
model diagram
Schematic diagram of flexible lung model.
model diagram
Schematic diagram of flexible lung model with gas exchange.
model diagram
Schematic diagram of flexible lung model with gas exchange and gas transport.
The complete original paper reference is cited below:
Simplified models for gas exchange in the human lungs, Alona Ben-Tal, 2006.
Journal of Theoretical Biology
, 238(2), 474-495. PubMed ID: 16038941
Full text (HTML) and PDF versions of the article are available to subscribers on the Journal of Theoretical Biology website.
reaction diagram
The lung modelled as a single, inflexible container. Pm represents the mouth pressure, PA is the alveolar pressure, air flow is represented by q, R is the resistance to air flow and V0 is the volume of the container.
reaction diagram
The lung modelled as a single, flexible container. VA - total lung volume, Pm - mouth pressure, PL - pleural pressure, q - air flow, R - overall resistance of the conducting airways, ks - spring constant.
reaction diagram
The lung modelled as a flexible container with gas exchange. All the symbols are the same as in . In addition, fo and fc are the alveolar concentrations of oxygen and carbon dioxide respectively, and pao and pac are the alveolar partial pressures of oxygen and carbon dioxide respectively. po and pc are the blood partial pressures of oxygen and carbon dioxide respectively.
reaction diagram
The lung modelled as a flexible container with gas exchange and gas transport. With each heart beat a new compartment is pushed into the lung. Each compartment consists of a small container representing dissolved gas, and a large container representing the means by which oxygen and carbon dioxide are transported.
$\mathrm{PL}=\mathrm{Pm}-R\mathrm{omega}\frac{\mathrm{VT}}{2.0}\sin (\mathrm{time}\mathrm{omega})+E(2.5-\frac{\mathrm{VT}}{2.0}\cos (\mathrm{time}\mathrm{omega}))\mathrm{omega}=\frac{2.0\times 3.141}{5.0}\mathrm{QA}=q+\mathrm{Dc}(\mathrm{pc}-\mathrm{pac})+\mathrm{Do}(\mathrm{po}-\mathrm{pao})\frac{d \mathrm{PA}}{d \mathrm{time}}=\frac{\mathrm{Pm}E}{\mathrm{PA}}\mathrm{QA}+\mathrm{delta\_PL}\frac{d \mathrm{PL}}{d \mathrm{time}}=\mathrm{delta\_PL}\frac{d \mathrm{fo}}{d \mathrm{time}}=\frac{1.0}{\mathrm{VA}}(\mathrm{Do}(\mathrm{po}-\mathrm{pao})+\mathrm{qi}(\mathrm{foi}-\mathrm{fo})-\mathrm{fo}(\mathrm{Dc}(\mathrm{pc}-\mathrm{pac})+\mathrm{Do}(\mathrm{po}-\mathrm{pao})))\frac{d \mathrm{fc}}{d \mathrm{time}}=\frac{1.0}{\mathrm{VA}}(\mathrm{Dc}(\mathrm{pc}-\mathrm{pac})+\mathrm{qi}(\mathrm{fci}-\mathrm{fc})-\mathrm{fc}(\mathrm{Do}(\mathrm{po}-\mathrm{pao})+\mathrm{Dc}(\mathrm{pc}-\mathrm{pac})))\mathrm{foi}=\begin{cases}\mathrm{fo} & \text{if $\mathrm{VT}< \mathrm{VD}$}\\ \frac{\mathrm{fo}\mathrm{VD}+\mathrm{fom}(\mathrm{VT}-\mathrm{VD})}{\mathrm{VT}} & \text{otherwise}\end{cases}\mathrm{fci}=\begin{cases}\mathrm{fc} & \text{if $\mathrm{VT}< \mathrm{VD}$}\\ \frac{\mathrm{fc}\mathrm{VD}+\mathrm{fcm}(\mathrm{VT}-\mathrm{VD})}{\mathrm{VT}} & \text{otherwise}\end{cases}\frac{d \mathrm{po}}{d \mathrm{time}}=\frac{\mathrm{Do}}{\mathrm{sigma}\mathrm{Vc}(2.0+\frac{4.0\mathrm{Th}}{\mathrm{sigma}}\mathrm{df\_dpo})^{1.0}-(\mathrm{fo}(\mathrm{PA}-\mathrm{pw})\mathrm{po})}\frac{d \mathrm{pc}}{d \mathrm{time}}=\frac{\mathrm{Dc}}{\mathrm{sigma\_c}\mathrm{Vc}}(\mathrm{pac}-\mathrm{pc})+\frac{\mathrm{delta}\mathrm{l2}}{\mathrm{sigma\_c}}hz-\mathrm{delta}\mathrm{r2}\mathrm{pc}\frac{d z}{d \mathrm{time}}=\mathrm{delta}\mathrm{r2}\mathrm{sigma\_c}\mathrm{pc}-\mathrm{delta}\mathrm{l2}hz\frac{d f}{d \mathrm{po}}=\mathrm{df\_dpo}$
cardio-respiratory
gas exchange
lung mechanics
mechanical constitutive laws
Simplified Models For Gas Exchange In The Human Lungs (Model D)
The University of Auckland, Bioengineering Institute
2008-01-11T15:18:05+13:00
Added publication data as the paper has been published since this CellML model was coded up.
Added publication data to the model documentation
Alona Ben-Tal's 2005 mathematical model of a human, flexible lung with gas exchange and gas transport.
2004-12-28T00:00:00+13:00
Catherine
Lloyd
May
16038941
Journal of theoretical biology
2007-06-11
Re-ordered the existing 8 versions as 2 versions of the 4 variants (models A,B,C and D).
2006-01-21
2006-01-21 00:00
Simplified models for gas exchange in the human lungs
238
474
495
Catherine
Lloyd
May
c.lloyd@auckland.ac.nz
Alona
Ben-Tal
Alona
Ben-Tal
keyword
238
Simplified models for gas exchange in the human lungs
2
474
495
Catherine Lloyd
Journal of theoretical biology
Catherine
Lloyd
May
Vignesh
Kumar
16038941
This is the CellML description of Alona Ben-Tal's 2005 mathematical model of a human, flexible lung with gas exchange and gas transport.
The University of Auckland
The Bioengineering Institute
Alona
Ben-Tal
2007-06-11T12:41:57+12:00