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 air E = expired air v = venous blood a = arterial blood . Quantities: V A = 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)

Equation of transport 1. Steady state - rate in=rate out: V A c I + Qc V = V A c E + Qc a

Equation of transport 1. Steady state - rate in=rate out: V A c I + Qc V = V A c E + Qc a 2. Expired air is a sample of alveolar air: c E = c A

Equation of transport 1. Steady state - rate in=rate out: V A c I + Qc V = V A c E + Qc a 2. Expired air is a sample of alveolar air: c E = c A These assumptions lead to the transport equation V A ( c I − c A ) = Q ( c a − c v ) .

Balance of pressures equation 3. In the alveolar air the gas behaves like an ideal gas ( PV = nkT ): P A = kTc A

Balance of pressures equation 3. In the alveolar air the gas behaves like an ideal gas ( PV = nkT ): P A = kTc A 4. In the blood the gas forms a simple solution: c a = σ P a

Balance of pressures equation 3. In the alveolar air the gas behaves like an ideal gas ( PV = nkT ): P A = kTc A 4. In the blood the gas forms a simple solution: c a = σ P a 5. The arterial blood and alveolar air are in equilibrium: P a = P A

Balance of pressures equation 3. In the alveolar air the gas behaves like an ideal gas ( PV = nkT ): P A = kTc A 4. In the blood the gas forms a simple solution: c a = σ P a 5. The arterial blood and alveolar air are in equilibrium: P a = P A These assumptions lead to the balance of pressures equation c a = σ kTc A .

Gas transport system � V A c A + Qc a = V A c I + Qc v σ kTc A − c a = 0

Gas transport system � V A c A + Qc a = V A c I + Qc v σ kTc A − c a = 0 Solving for c A and c a gives ( r = V A / Q is the ventilation/perfusion ratio) C A = V A c I + Qc v V A + Q σ kT = rc I + c V r + σ kT , c a = σ kT V A c I + Qc v = σ kT rc I + c V σ kT + r . σ kTQ + V A

Gas transport system � V A c A + Qc a = V A c I + Qc v σ kTc A − c a = 0 Solving for c A and c a gives ( r = V A / Q is the ventilation/perfusion ratio) C A = V A c I + Qc v V A + Q σ kT = rc I + c V r + σ kT , c a = σ kT V A c I + Qc v = σ kT rc I + c V σ kT + r . σ kTQ + V A As r → ∞ : c A → c I , c a → σ kTc I c v As r → 0 : c A → σ kT , c a → c v

Net gas transport f = Q ( c a − c v ) = Qr σ kTc I − 1 σ c v = Qr σ P I − P v r + σ kT r + σ kT r = Q σ ( P I − P v ) r + σ kT

Net gas transport f = Q ( c a − c v ) = Qr σ kTc I − 1 σ c v = Qr σ P I − P v r + σ kT r + σ kT r = Q σ ( P I − P v ) r + σ kT Total gas transport (all alveoli): r i = ( V A ) i / Q i ( c A ) i = r i c I + c V ( c a ) i = σ kT r i c I + c V r i + σ kT , r i + σ kT .

Net gas transport f = Q ( c a − c v ) = Qr σ kTc I − 1 σ c v = Qr σ P I − P v r + σ kT r + σ kT r = Q σ ( P I − P v ) r + σ kT Total gas transport (all alveoli): r i = ( V A ) i / Q i ( c A ) i = r i c I + c V ( c a ) i = σ kT r i c I + c V r i + σ kT , r i + σ kT . Q i r i � � f = f i = σ ( P I − P v ) r i + σ kT = σ ( P I − P v ) E i i with E = 1 Q i r i � r i + σ kT Q 0 i 0 < E < 1 measures the efficiency of gas transport.