Quantification of Ventilation-Perfusion-Ratio by Oxygen Partial Pressure
This additional material provides complementary information concerning the analytical solutions of
Olszowka and Wagner (1) and Rizi, Baumgartner and colleagues (2) to quantify the ventilation-perfusion
ratio by alveolar oxygen partial pressure. Only the final equation is different.
Symbols
C ec CO 2 :
end-capillary carbon dioxide blood content , measured in ml CO2 (STPD) per ml blood
C ec N 2 :
end-capillary nitrogen blood content, measured in ml N2 (STPD) per ml blood.
C ec O 2 :
end-capillary oxygen blood content, measured in ml O2 (STPD) per ml blood
C v CO 2 :
mixed-venous carbon dioxide blood content, measured in ml CO2 (STPD) per ml blood
C v N2 :
mixed venous nitrogen blood content, measured in ml N2 (STPD) per ml blood.
C v O2 :
mixed-venous oxygen blood content, measured in ml O2 (STPD) per ml blood
FA N 2 :
alveolar nitrogen fraction, measured in ml N2 per ml gas (BTPS)
FA O 2 :
alveolar oxygen fraction, measured in ml O2 per ml gas (BTPS)
FI N 2 :
inspiratory nitrogen fraction, measured in ml N2 per ml gas (BTPS)
FI O 2 :
inspiratory oxygen fraction, measured in ml O2 per ml gas (BTPS)
Fv N 2 :
nitrogen gas fraction, equilibrated to mixed venous blood, ml N2 per ml gas (BTPS)
k:
converting factor considering both expansion due to temperature and replacement of gas
fraction by partial pressure, in mmHg
k1 :
converting factor to consider gas expansion by temperature
k2 :
converting factor considering replacement of gas fraction by partial pressure, in mmHg
1
PA CO 2 :
alveolar carbon dioxide partial pressure, measured in mmHg
PA N 2 :
alveolar nitrogen partial pressure, in mmHg
PA O 2 :
alveolar oxygen partial pressure, measured in mmHg
PB :
barometric pressure, in mmHg
PH 2O :
water vapor pressure at body temperature, in mmHg
PI N 2 :
inspiratory nitrogen partial pressure, in mmHg
PI O 2 :
inspiratory oxygen partial pressure, measured in mmHg
Pv N 2 :
mixed venous nitrogen partial pressure, in mmHg
Q:
perfusion, measured in ml blood per minute
TB :
body temperature, in °Kelvin
Ts :
standard temperature, in °Kelvin
VI :
inspiratory ventilation, measured in ml gas (BTPS) per minute
VE :
expiratory ventilation, measured in ml gas (BTPS) per minute
 N2 :
blood gas partitioning coefficient of nitrogen
Basic Equations
The basic assumptions are the steady state mass balance of the gases oxygen, carbon dioxide, and nitrogen.
Ventilated gases are reported under BTPS (body temperature pressure saturated) conditions. It is a
convention that the gases dissolved in blood are calculated under STPD (standard temperature, pressure,
dry) conditions. A converting factor (k1) is used for transforming to BTPS (body temperature pressure
saturated) conditions.
On either side of the equation, the oxygen transport is given as:
VI  FI O 2  VE  FA O 2  k 1  (C ec O 2  C v O 2 )  Q
(1)
2
Rearrangement of this equation results in.
VI
V
 FI O 2  E  FA O 2  k 1  (C ec O 2  C v O 2 )
Q
Q
(2)
The converting factor k1 considers the expansion of the STPD gas by temperature:
k1 
TB
TS
(3)
The gas fractions on the left side of this equation are replaced by partial pressures as follows:
PG  FG  PB  FG  k 2
k 2  PB 
VI 
PG
FG
(4)
PI O 2
P
 VE  A O 2 F  k 1  (C ec O 2  C v O 2 )  Q
k2
k2
(5)
When the conversion factor k is the product of k1 and k2,
k  k 1  k 2  PB 
TB
TS
(6)
the equation can be rearranged to
VI
V
 PI O 2  E  PA O 2  k  (C ec O 2  C v O 2 )
Q
Q
(equation 1)
(7)
Accordingly, a mass balance equation for CO2 is given. In this equation, inspiratory ventilation has been
omitted since inspiratory carbon dioxide fraction is zero.
3
VE
 PA CO 2  k  (C v CO 2  C ec CO 2 )
Q
(equation 2)
(8)
For N2, mass balance can be set up as:
VI  FI N 2  VE  FA N 2  (C ec N 2  C v N 2 )  Q
(9)
Inserting the blood gas partitioning coefficient N2 of the given body temperature and barometric pressure
 N2 
C N2
FN 2
(10)
results in:
VI  FI N 2  VE  FA N 2   N 2 (FA N 2  Fv N 2 )  Q
Inserting the converting factor ( P  k 2
VI 
(11)
 F ) and rearrangement leads to basic equation 3:
P
PI N 2
P
P
 VE  A N 2   N 2 ( A N 2  v N 2 )  Q
k2
k2
k2
k2
VI
V
 PI N 2  E  PA N 2   N 2 (PA N 2  Pv N 2 )
Q
Q
(12)
(equation 3)
(13)
In addition, the alveolar partial pressures of the gases add up to the barometric pressure:
PA O 2  PA CO 2  PA N 2  PH 2O  PB
(equation 4)
(14)
4
Derivation of the (implicit) interdependence between
PA O 2 and PA CO 2
Following Rizi et al. (2) the 4 unknowns of these 4 equations are
PA N 2 and
PA CO 2 ,
VE VI
,
, and PA N 2 .
Q Q
VI
can be eliminated using the linear equations 3 and 4.
Q
According to equation 4, substitute for
PA N 2 is:
PA N 2  PB  PA O 2  PA CO 2  PH 2O
According to equation 3, substitute for
(15)
VI
is:
Q
VI
V
 PI N 2  E  PA N 2   N 2 (PA N 2  Pv N 2 )
Q
Q
(16)
VI
V
 PI N 2  E  PB  PA O 2  PA CO 2  PH 2 O  
Q
Q
 N 2 PB  PA O 2  PA CO 2  PH 2 O  Pv N 2 
(17)
VE
 PB  PA O 2  PA CO 2  PH 2 O    N 2 PB  PA O 2  PA CO 2  PH 2 O  Pv N 2 
VI
Q

Q
PI N 2
In order to eliminate
(18)
VE
equation 1 is rearranged to:
Q
VI
V
 PI O 2  E  PA O 2  k  (C ec O 2  C v O 2 )
Q
Q
(19)
5
VE
 PB  PA O 2  PA CO 2  PH 2O    N 2 PB  PA O 2  PA CO 2  PH 2O  Pv N 2 
Q
 PI O 2
PI N 2

(20)
VE
 PA O 2  k  (C ec O 2  C v O 2 )
Q
P
VE
PB  PA O2  PA CO 2  PH 2O  I O2  VE  PA O 2
Q
PI N 2 Q
  N 2 PB  PA O 2  PA CO 2  PH 2O  Pv N 2 
PI O 2
PI N 2
(21)
 k  (C ec O 2  C v O 2 )

PI O 2
VE 
 PA O 2 
PB  PA O 2  PA CO 2  PH 2 O  
Q 
PI N 2

PI O 2
  N 2  PB  PA O 2  PA CO 2  PH 2 O  Pv N 2  
 k  (C ec O 2  C v O 2 )
PI N 2

FI O 2
VE 
 PA O 2 
PB  PA O 2  PA CO 2  PH 2 O  
Q 
FI N 2

F
  N 2  PB  PA O 2  PA CO 2  PH 2 O  Pv N 2   I O 2  k  (C ec O 2  C v O 2 )
FI N 2
(22)
(23)
Here in (23), the rearrangements of the equation differ from the other authors.
Substitution of
VE
using equation 2 results in:
Q
VE k  (C v CO 2  C ec CO 2 )

Q
PA CO 2
(24)

k  (C v CO 2  C ec CO 2 ) 
F
 PB  PA O 2  PA CO 2  PH 2 O   I O 2  PA O 2 
PA CO 2
FI N 2


FI O 2
  N 2  PB  PA O 2  PA CO 2  PH 2 O  Pv N 2  
 k  (C ec O 2  C v O 2 )
FI N 2
(25)
6
Division by k and rearrangement result in:
(C v CO2  C ec CO2 ) 

F
 PB  PA O 2  PA CO2  PH2O   I O 2  PA O 2 
PA CO2
FI N2


PB  PA O 2  PA CO2  PH2O  Pv N2 FI O 2
  N2 

 (C v O 2  C ec O 2 )  0
k
FI N2
Since the parameters
of
(26)
C ec CO 2 and C ec O 2 are non-linear functions of PA CO 2 , this is an implicit equation
PA CO 2 , which can be solved by numerical analysis. When PA CO 2 is estimated, it will be inserted into
the 4 equations in order to estimate the other unknowns (
VE VI
,
, and PA N 2 ).
Q Q
References
1.
Olszowka AJ, Wagner PD. Numerical analysis of gas exchange. In: West JB, ed. Pulmonary Gas
Exchange New York: Academic Press, 1980:263-306.
2.
Rizi RR, Baumgardner JE, Ishii M, Spector ZZ, Edvinsson JM, Jalali A, Yu J, Itkin M, Lipson
DA, Gefter W. Determination of regional VA/Q by hyperpolarized 3He MRI. Magn Reson Med
2004;52:65-72.
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