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A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries


A mathematical analysis of the process of gas exchange in the lung is presented taking into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion of the species due to haemoglobin. Since the rate at which blood gets oxygenated in the pulmonary capillaries is very fast, it is difficult to set up an experimental study to determine the effects of various parameters on equilibration rate. The proposed study is aimed at determining the effects of various physiological parameters on equilibration rate in pathological conditions.

Among the significant results are that 1. dissolved oxygen takes longer to achieve equilibration across the pulmonary membrane and carbon dioxide attains equilibration faster, 2. the equilibration length increases with increase in blood velocity, haemoglobin concentration, calibre of pulmonary capillaries and fall in alveolar PO2, 3. the alveolar PCO2 and forward and backward reaction rates of haemoglobin with CO2 do not materially affect the equilibration rate or length. 4. At complete equilibration, by the end of the pulmonary capillary 92% of the total haemoglobin has combined with oxygen and 8% free pigment is left which is present as carbamino haemoglobin, met haemoglobin, carboxy haemoglobin etc.

These results are of some importance for anaemic conditions, muscular exercise, meditation, altitude physiology, hypo-ventilation, hyperventilation, etc.

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H+ :

hydrogen ion

O2 :


CO2 :


HbO2 :


HbCO2 :


PO2 :

partial pressure of O2

PCO2 :

partial pressure of CO2

PaO2 :

O2 tension in arterial blood

PaCO2 :

CO2 tension in arterial blood

k1 :

forward rate constant for Eq. (1)

k2 :

backward rate constant for Eq. (1)

m1 :

forward rate constant for Eq. (2)

m2 :

backward rate constant for Eq. (2)


equilibration rate


radius of the capillary


velocity of blood


length of the capillary

D0 :

diffusion coefficient of O2

Dc :

diffusion coefficient of CO2

DH :

diffusion coefficient of Hb


total haemoglobin concentration



c1 :

concentration of dissolved O2 in blood

c2 :

concentration of HbO2 in blood

c3 :

concentration of dissolved CO2 in blood

c4 :

concentration of HbCO2 in blood

c5 :

concentration of haemoglobin

c1alv :

concentration of O2 in the alveolar region

c3alv :

concentration of CO2 in the alveolar region

civen :

concentration of the ith species in venous blood

ciart :

concentrations of the ith species in arterial blood

F is :

concentrations of the species in dimensionless form

J0, I0 :

Bessel's functions

PalvO2 :

tension of O2 in alveolar region

PalvCO2 :

tension of CO2 in alveolar region.


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Singh, M.P., Khetarpal, K. & Sharan, M. A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries. J. Math. Biology 9, 305–330 (1980).

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  • Mathematical Biology
  • Transport Mechanism
  • Matrix Theory
  • Haemoglobin Concentration
  • Molecular Diffusion