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Gas exchange in the lungs
Math 392 - Mathematical Models in Biology
Gas exchange in the lungs Math 392 - Mathematical Models in Biology - - PowerPoint PPT Presentation
Gas exchange in the lungs Math 392 - Mathematical Models in Biology - - PowerPoint PPT Presentation
Gas exchange in the lungs Math 392 - Mathematical Models in Biology February, 2014 Introduction Gas transport I = inspired air A = alveolar air E = expired air v = venous blood a = arterial blood . Gas transport I = inspired air A = alveolar
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Gas transport
I = inspired air A = alveolar air E = expired air v = venous blood a = arterial blood.
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Gas transport
I = inspired air A = alveolar air E = expired air v = venous blood a = arterial blood. Quantities: VA = alveolar ventilation (liters/minute) Q = blood flow (liters/minute) c = concentration of particular gas (# of molecules/liter) P = partial pressure of the particular gas (mmHg) σ = solubility of the gas T = absolute temperature (Kelvin)
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Equation of transport
- 1. Steady state - rate in=rate out:
VAcI + QcV = VAcE + Qca
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Equation of transport
- 1. Steady state - rate in=rate out:
VAcI + QcV = VAcE + Qca
- 2. Expired air is a sample of alveolar air:
cE = cA
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Equation of transport
- 1. Steady state - rate in=rate out:
VAcI + QcV = VAcE + Qca
- 2. Expired air is a sample of alveolar air:
cE = cA These assumptions lead to the transport equation VA(cI − cA) = Q(ca − cv).
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Balance of pressures equation
- 3. In the alveolar air the gas behaves like an ideal gas (PV = nkT):
PA = kTcA
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Balance of pressures equation
- 3. In the alveolar air the gas behaves like an ideal gas (PV = nkT):
PA = kTcA
- 4. In the blood the gas forms a simple solution:
ca = σPa
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Balance of pressures equation
- 3. In the alveolar air the gas behaves like an ideal gas (PV = nkT):
PA = kTcA
- 4. In the blood the gas forms a simple solution:
ca = σPa
- 5. The arterial blood and alveolar air are in equilibrium:
Pa = PA
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Balance of pressures equation
- 3. In the alveolar air the gas behaves like an ideal gas (PV = nkT):
PA = kTcA
- 4. In the blood the gas forms a simple solution:
ca = σPa
- 5. The arterial blood and alveolar air are in equilibrium:
Pa = PA These assumptions lead to the balance of pressures equation ca = σkTcA.
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Gas transport system
- VAcA + Qca
= VAcI + Qcv σkTcA − ca = 0
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Gas transport system
- VAcA + Qca
= VAcI + Qcv σkTcA − ca = 0 Solving for cA and ca gives (r = VA/Q is the ventilation/perfusion ratio) CA = VAcI + Qcv VA + QσkT = rcI + cV r + σkT , ca = σkT VAcI + Qcv σkTQ + VA = σkT rcI + cV σkT + r .
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Gas transport system
- VAcA + Qca
= VAcI + Qcv σkTcA − ca = 0 Solving for cA and ca gives (r = VA/Q is the ventilation/perfusion ratio) CA = VAcI + Qcv VA + QσkT = rcI + cV r + σkT , ca = σkT VAcI + Qcv σkTQ + VA = σkT rcI + cV σkT + r . As r → ∞ : cA → cI, ca → σkTcI As r → 0 : cA → cv σkT , ca → cv
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Net gas transport
f = Q(ca − cv) = Qrσ kTcI − 1
σcv
r + σkT = Qrσ PI − Pv r + σkT = Qσ(PI − Pv) r r + σkT
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Net gas transport
f = Q(ca − cv) = Qrσ kTcI − 1
σcv
r + σkT = Qrσ PI − Pv r + σkT = Qσ(PI − Pv) r r + σkT Total gas transport (all alveoli): ri = (VA)i/Qi (cA)i = ricI + cV ri + σkT , (ca)i = σkT ricI + cV ri + σkT .
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Net gas transport
f = Q(ca − cv) = Qrσ kTcI − 1
σcv
r + σkT = Qrσ PI − Pv r + σkT = Qσ(PI − Pv) r r + σkT Total gas transport (all alveoli): ri = (VA)i/Qi (cA)i = ricI + cV ri + σkT , (ca)i = σkT ricI + cV ri + σkT . f =
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fi = σ(PI − Pv)
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Qiri ri + σkT = σ(PI − Pv)E with E = 1 Q0
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