ASSIG 3

Problem Set 3: Hemoglobin

Assignment:

1) Use Microsoft Excel* to produce a model of the oxygen-hemoglobin dissociation curve for the

following:

a) Adult hemoglobin

b) Fetal hemoglobin

c) Model the Bohr shift for adult hemoglobin

*To construct this model, you can produce three separate graphs on Excel, using the

mathematics that you have learned in your readings.

2) Write an essay that explains the mathematics and the strategy that you have used to model the

oxygen-hemoglobin dissociation curve in adult and fetus, and how you modeled the Bohr shift.

Alveolar Ven+la+on and Carbon Dioxide

Charles E Wood

How do we predict alveolar CO2 par+al pressure? • Using the same principles of chemistry, physics, physiology, we can calculate the alveolar par;al pressure of carbon dioxide. •  Need to know alveolar ven;la;on rate •  Need to know CO2 produc;on rate

•  Can be calculated from O2 consump;on rate and the respiratory quo;ent

Calcula;on of alveolar CO2 par;al pressure is based on the principle of mass balance, shown in this figure for N2, O2, and CO2

PaCO2 = (k * VCO2)/ VA

At body temperature and at sea level, k = 863

Alveolar Gas Equa.on Charles E Wood

In this assignment…

• We will learn the theore1cal basis that underlies the calcula1on of alveolar par1al pressure of oxygen as a func1on of the following variables: •  Ambient pressure •  Composi1on of ambient inhaled gas •  Alveolar par1al pressure of carbon dioxide •  Respiratory quo1ent

The classic paper of Fenn, Rahn, and O.s that defined the alveolar gas equa.on

The alveolar gas equa.on allows predic.on of alveolar gas pressures at different al.tudes and different levels of alveolar ven.la.on

The second reading presents a simpler deriva.on of the alveolar gas equa.on

This review is wriAen from the perspec1ve of the anesthesiologist, working at sea level and not altering inspired gas composi1on. Fenn, Rahn, and O1s were providing the theore1cal basis of alveolar gas composi1on at al1tude.

Cruickshank and Hirshauer present two forms of the alveolar gas equa.on, a simpler one based on assump.ons made about carbon dioxide tensions

What to learn from these readings…

• Work through the mathema1cs of the alveolar gas equa1on. Understand the predictable and calculable interplay of the variables that determing alveolar oxygen par1al pressure. • Appreciate what this means with regard to high al1tude • Appreciate what this means with regard to diving (hyperbaric condi1ons) • AJer comple1ng the NEXT set of readings…

•  Be able to derive the alveolar gas equa1on.

The alveolar gas equation Steven Cruickshank BA MB BS FRCA

Nicola Hirschauer MD(Munich) FRCA

The alveolar gas equation (AGE) is well known

and relates the alveolar concentration of oxy-

gen FAO2 (or equivalently partial pressure

PAO2) to three variables: FIO2, FACO2 and the

respiratory quotient (R). However, the AGE

predicts an absurdity: if we input a FIO2 suffi-

ciently low, say 0.05 (i.e. a PIO2 of about 5 kPa),

into the equation

FAO2 � FIO2 ÿ FACO2

R

then for typical values of FACO2� 0.05 and R� 0.8,anegativevalueforFAO2ispredicted.There-

fore, it is plain that the AGE is not true for all

conditions. Where does the AGE come from,

and will the derivation explain the problem

and define the conditions under which it is true?

Derivation of the AGE

Carbon dioxide

The metabolic minute production of carbon

dioxide ( _V CO2) is approximately 200 ml (0.20

litre minÿ1) in a resting adult. We will assume

this value is constant. Under steady-state con-

ditions where the PaCO2 is unchanging with

time, this amount must be removed by alveolar

ventilation each minute. The alveolar mem-

brane is thin and carbon dioxide is highly dif-

fusible and we may therefore assume that

PACO2 � PaCO2 to a close approximation

(i.e. that there is a negligible PCO2 gradient

between blood and alveolar space).

Partial pressures

The concentration of a component gas in a

mixture of gases is related to the partial pres-

sure by Dalton's law of partial pressures. This

states that an individual gas such as carbon

dioxide in the mixture of gases in the alveolar

space will be present in a concentration (or

fraction) that is the same proportion as

PACO2 is of the total pressure (i.e. approxi-

mately PI at the end of inspiration). For typical

values of PI� 100 kPa and PaCO2� PACO2� 5

kPa, we have FACO2� 0.05 or 5%. The quantity

of carbon dioxide removed from the alveolar

space in 1 min is simply the alveolar ventilation

( _V A) multiplied by FACO2. Since, for steady-

state conditions, this must be the same as _V CO2, we have

_VCO2 � FACO2� _VA

_VCO2 � PACO2

PI � _VA

Rearranging and using our assumption that

PACO2�PaCO2, we obtain the following (Fig. 1):

PACO2 � _VCO2�PI

_VA

� PaCO2

Oxygen

The input of oxygen into the alveolar space,

and the output from it, must, in steady-state

conditions, be equal.

Input is by alveolar ventilation and using the

same arguments as for carbon dioxide:

Input � _VA�FIO2

Output, on the other hand, is in two directions:

(i) across the alveolar membrane into the

blood -- in steady-state conditions, the

oxygen consumption _V O2; and

(ii) in the expired gas ( _V A � FAO2):

Output � _VO2 � _VA�FAO2

So, for steady-state conditions, we can write

Input � Output

_VA�FIO2 � _VO2 � _VA�FAO2

which rearranges as

_VO2 � _VA��FIO2 ÿ FAO2� We are now ready to complete the derivation

by noting that

R � _VCO2

_VO2

so that

R � _VCO2

_VO2

� FACO2� _VA

_VA��FIO2 ÿ FAO2� which, in turn, with cancellation and rearran-

gement, becomes

R � FACO2

�FIO2 ÿ FAO2�

Steven Cruickshank BA MB BS FRCA

Consultant Anaesthetist Department of Anaesthesia Newcastle General Hospital Newcastle upon Tyne NE4 6BE Tel: 0191 256 3198 Fax: 0191 256 3154 E-mail: stevencruickshank@hotmail.com (for correspondence)

Nicola Hirschauer MD(Munich) FRCA

Specialist Registrar Department of Anaesthesia Newcastle General Hospital Newcastle upon Tyne

Key points

The alveolar gas equation (AGE) relates PAO2 to PIO2, PACO2 and the respiratory quotient (R).

Depending which quantities we fix and which we allow to vary, the AGE describes a linear or non-linear relationship.

The AGE predicts an absurdity (i.e. negative PAO2

at low PIO2).

Derivation of the equation shows how this arises and restricts the validity of the equation to reasonable (non-hypoxic) levels of FIO2.

The full versionof theAGEhas an extra term which is trivial for most clinical purposes. This relates to passive inflow of gas for R less than unity.

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�FIO2 ÿ FAO2� � FACO2

R

FAO2 � FIO2 ÿ FACO2

R

which is almost the familiar version of the AGE. Converting to

partial pressures (Dalton) and using our assumption that

PACO2 � PaCO2, this finally becomes the familiar `short' AGE:

PAO2 � PIO2 ÿ PaCO2

R

What does the AGE say?

We have derived the usual `short' AGE using the assumption of

steady-state conditions. The AGE will only be valid so long as the

assumptions upon which we constructed it remain true. It is evi-

dent that, at low FIO2, the steady-state assumption will be violated

-- the required _V O2 cannot be supplied at the lowered FAO2 -- and

therefore we cannot expect the AGE to describe the resulting FAO2

accurately. In fact, FAO2 then becomes a time-dependent variable

rather than a constant equilibrium quantity, since more is being

taken up than supplied and the AGE relationship breaks down at

some ill-defined (i.e. not defined by our simple model) FIO2; the

patient becomes progressively hypoxic.

FAO2 is dependent upon three variable quantities: PIO2, PACO2

and R. Although all are variables, they are not equally easily

engineered. PIO2 is very easily manipulated, of course. PACO2,

on the other hand, is quite easy to manipulate in the intubated,

ventilated patient but much less so in the spontaneously breathing

subject. R is alterable by manipulation of the diet but not in any

rapid, simple and predictable manner.

PAO2 � PIO2 ÿ PaCO2

R

� With PIO2 as the input variable and PACO2 and R fixed, we have

a linear relationship of the type y � mx� c, where y � PAO2,

m � 1, x � PIO2 and c � [ÿPACO2/R]. The graph of PAO2 vs

PIO2 is linear with unity gradient (Fig. 2). The intercept on the

vertical axis is negative, as we have noted. This is a graph of the

equation, not of the physiology. In the physiological range

(above, say, 8--9 kPa), it will describe the physiology

reasonably well.

� With PIO2 fixed and PACO2 as the input variable, we also have a

linear relationship. However, if we ask how PAO2 is related to _V A -- which, with _V CO2, determines PACO2 -- then the rela-

tionship is non-linear. We have from before

PACO2 � _VCO2�PI

_VA

� PaCO2

Substituting for PaCO2 in the AGE:

PAO2 � PIO2 ÿ _VCO2�PI

_VA�R

PAO2 � PIO2 ÿ 1

_VA

� _VCO2�PI

R

� � This is an equation of the form

y � m� 1

x � c

with y � PAO2 as before,

m � ÿ _VCO2�PI

R

� � x � _V A and c � PIO2. This is an equation of inversely related

quantities whose graph is a rectangular hyperbola, but the negative

signbeforethecoefficientmodifyingthex-termturnsthehyperbola

of Figure 1 upside down so that we subtract it from the (assumed

0 2 4 6 8

20

15

10

5

0

P aC

O 2

(k P

a)

VA (litre min–1) .

Fig. 1 The relationship of alveolar ventilation ( _VA) to the resulting PaCO2

for constant _VCO2 is a non-linear one. It is a rectangular hyperbola.

Fig. 2 The alveolar gas equation predicts a relationship of inspired oxygen partial pressure (PIO2) to the resulting alveolar partial pressure (PAO2) that is linear. It also predicts a nonsensical negative PAO2 at low PIO2. The equation is valid only at values of PIO2 sufficient to allow normal oxygen uptake _VO2.

The alveolar gas equation

Continuing Education in Anaesthesia, Critical Care & Pain | Volume 4 Number 1 2004 25

10 20 30 40 50 60 70 80 90 100

100

90

80

70

60

50

40

30

20

10

0

–10

P A

O 2

(k P

a)

PIO2 (kPa)

0

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constant) PIO2. The hyperbola is also amplified by division by R if it

is less than unity. The effect of the scaling for various values of R is

shown. A PIO2 of 20 kPa is taken to represent breathing air with

allowance for water vapour (Fig. 3). It is apparent from the graph

that a reduction in _V A to levels which allow very high levels of

PaCO2 similarly predict a negative value for PAO2 as before. Once

again, we have to be careful to use the equation only in circum-

stances where it is valid; if the steady-state condition is violated, the

equation does not apply.

It is evident from these graphs that PAO2 is much more effec-

tively raised by increasing the inspired concentration than it is

by hyperventilation. However, in the spontaneously breathing

patient in air, hyperventilation is the only option.

Refining the AGE

In the appendix to West's Respiratory Physiology, and in other

texts, a more elaborate version of the AGE is given:

PAO2 � PIO2 ÿ PaCO2

R � FIO2�PaCO2� 1ÿ R

R

� �� � Where does the fearsome-looking extra term in the expression

come from, and is it important?

In our derivation of the AGE, we obtained expressions for _V O2

and _V CO2 and substituted these into the definition of R. This is

actually not a valid procedure, since we need to look at input and

output volumes in the functional residual capacity (FRC). The

FRC volume is constant in our model, and yet when the R-value is

not unity, more gas is extracted from the FRC by oxygen uptake _V O2 (say 0.25 litre minÿ1) than is added to it by _V CO2 (0.20 litre

minÿ1, when R� 0.8). This is plainly incompatible with a constant

FRC volume; the 50 ml discrepancy must be made up by an

additional passive inflow of gas. Even at end-inspiration or expira-

tion, there must be a small difference in pressure between atmo-

sphere and alveolus such that this volume

_Vp � � _VO2 ÿ _VCO2�

is delivered to the alveolar space each minute. This delivers a small

extra amount of oxygen, which acts to boost the FAO2 a little. This

is a bulk flow of gas caused by pressure difference which is not a

result of diffusion; it follows from the general gas law PV � nRT,

since n (the number of moles of, in this case, oxygen) is being

continuously removed by uptake into the blood. RT and V are

constant. This same process is at work in more dramatic fashion

during apnoeic mass-transfer oxygenation, such as during brain

death testing procedures or bronchoscopy. Details, for those with

a head for algebra, are shown in the box.

Figure 4 shows the effect of taking the extra term into con-

sideration on the linear relationship between PIO2 and PAO2. We

can see that it is a pretty small modification. It is directly propor-

tional to FIO2 and indeed is barely apparent at FIO2< 0.5. Note that

we cannot have both PIO2 and PaCO2 multiplied together in the

Let us modify the AGE derivation according to the refined model:

O2-Input � FIO2 � _VA � FIO2� _Vp � FIO2� �VA � Vp� O2-Output � _VO2 � FAO2� _VA

Since O2-Input � O2-Output,

FIO2� _VA � FIO2� _Vp � _VO2 � FAO2� _VA

Rearranging,

_VO2 � _VA�FIO2 ÿ FAO2� � FIO2� _Vp

Now, using the expression for

_VCO2 � FACO2� _VA

R � _VCO2

_VO2

� FACO2 � _VA

_VA�FIO2 ÿ FAO2� � FIO2� _Vp

_VA�FIO2 ÿ FAO2� � FIO2� _Vp � FACO2� _VA

R

�FIO2 ÿ FAO2� � FIO2� _Vp

_VA

� FACO2

R

FAO2 � FIO2 ÿ FACO2

R � FIO2 � _Vp

_VA

" #

So we have a rather more complicated expression for the AGE. We can tidy this up a little further by manipulating the extra term in brackets:

FIO2 � _Vp

_VA

" # � FIO2� _VO2 ÿ _VCO2�

_VA

But

_VO2 � _VCO2

R

from the definition of R,

FIO2� _VCO2 1 Rÿ 1 � �

_VA

� FIO2 � _VCO2 1ÿR

R

� � _VA

Finally, from before we have

FACO2 � _VCO2

_VA

so the expression in brackets becomes

FIO2� _VCO2 �1ÿ R�=R� � _VA

� FIO2� FACO2� 1ÿ R

R

� � So we obtain the final form of our modified AGE:

FAO2 � FIO2 ÿ FACO2

R � FIO2 � FACO2� 1ÿ R

R

� �� � Put into partial pressure terms, this becomes

PAO2 � PIO2 ÿ PaCO2

R � FIO2� PaCO2� 1ÿ R

R

� �� �

The alveolar gas equation

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extra term in the brackets, since this would give a {pressure

squared}; we must use FIO2 instead.

We can discard the extra term in the full AGE for any usual

clinical purpose; it is complicated and simply too small to make a

difference. By taking the discrepancy in the volumes entering the

FRC into consideration, however, we have understood a minor

detail of the physiology in this context which achieves major

importance in another. In apnoea, PaCO2 increases by 0.3--0.7

kPa minÿ1, say 0.5 kPa minÿ1. If PaCO2 increases at this rate,

then, by our assumption that PaCO2 � PACO2, so does PACO2.

Since this is in the gas phase, we can apply Dalton's law and

know that carbon dioxide is entering the FRC at 0.05 � 2 litre

minÿ1 (i.e. about 10 ml of the 200 ml minÿ1 _V CO2). The remainder

of the _V CO2 remains in the aqueous phase in the body water.

Provided PAO2 is sufficient to saturate the Hb (preoxygenation),

then uptake _V O2 (say 0.25 litre minÿ1) will proceed normally, and

if the airway is filled with oxygen (flowing through catheter in

trachea) this will be drawn into the FRC at a rate of 0.24 litre

minÿ1 by bulk flow: this is apnoeic mass-transfer oxygenation.

Under ideal circumstances (preoxygenation and normal gas

exchange), this technique is limited by carbon dioxide toxicity

and not hypoxia.

Key references

Cruickshank S. Mathematics and Statistics in Anaesthesia. Oxford: Oxford Uni- versity Press, 1998

West JB. Respiratory Physiology: The Essentials, 6th edn. Baltimore: Lippincott Williams and Wilkins, 2000

See multiple choice questions 23±26.

R = 1

0

5

10

15

20

1 2 3 4 5 6 7 8 9

P A

O 2

(k P

a)

VA (litre min–1)

R = 0.8

R = 0.9

.

Fig. 3 The relationship of alveolar ventilation ( _VA) to the resulting PAO2 for constant PIO2 is non-linear.

R = 0.8 PaCO2 = 5 kPA

complete AGE

basic AGE

100

95

90

85

80

75

70

65

60

55

50 50 55 60 65 70 75 80 85 90 95 100

P A

O 2

(k P

a)

PIO2 (kPa)

Fig. 4 The extra term in the complete form of the alveolar gas equation (AGE), where R is less than unity, adds a small amount to the predicted PAO2. This is interesting but clinically trivial and may be neglected for practical purposes.

The alveolar gas equation

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