Portal:Algebra/Selected article/3

The Bohr equation, named after Danish physician Christian Bohr (1855–1911), describes the amount of physiological dead space in a person's lungs. This is given as a ratio of dead space to tidal volume. It differs from anatomical dead space as measured by Fowler's method as it includes alveolar dead space.

Description

The Bohr equation is used to quantify the ratio of physiological dead space to the total tidal volume, and gives an indication of the extent of wasted ventilation. It is stated as follows:

${\frac {\mathbf {V} _{\mathrm {d} }}{\mathbf {V} _{\mathrm {t} }}}={\frac {{\mathrm {P} _{\mathrm {a} }{\mathrm {CO} }_{\mathrm {2} }}-{\mathrm {P} _{\mathrm {e} }{\mathrm {CO} }_{\mathrm {2} }}}{\mathrm {P} _{\mathrm {a} }{\mathrm {CO} }_{\mathrm {2} }}}$

Derivation

Its derivation is based on the fact that only the ventilated gases involved in gas exchange ($V_{a}$) will produce CO₂. Because the Total tidal volume ($V_{t}$) is made up of $V_{a}+V_{d}$ (alveolar volume + dead space volume), we can substitute $V_{a}$ for $V_{t}-V_{d}$.

Initially, Bohr tells us Vt = Vd + Va. The Bohr equation helps us find the amount of any expired gas, Template:CO2, N₂, O₂, etc. In this case we will focus on Template:CO2. Defining Fe as the fraction of expired Template:CO2 and Fa as the fraction of expired alveolar Template:CO2, and Fd as fraction of expired dead space volume Template:CO2, we can say
Vt x Fe = ( Vd x Fd ) + (Va x Fa ). This merely means all the Template:CO2 expired comes from two parts, the dead space volume and the alveolar volume.
If we suppose that Fd = 0 (since carbon dioxide concentration in air is normally negligible), then we can say that:

$V_{t}\times F_{e}=V_{a}\times F_{a}$ Where F_e = Fraction expired CO₂, and Fa = Alveolar fraction of CO₂.

$V_{t}\times F_{e}=(V_{t}-V_{d})\times F_{a}$ Substituted as above.

$V_{t}\times F_{e}=V_{t}\times F_{a}-V_{d}\times F_{a}$ Multiply out of the brackets.

$V_{d}\times F_{a}=V_{t}\times F_{a}-V_{t}\times F_{e}$ Rearrange.

$V_{d}\times F_{a}=V_{t}\times (F_{a}-F_{e})$

$V_{d}/V_{t}={\frac {F_{a}-F_{e}}{F_{a}}}$ Divide by V_t and by F_a.

The above equation makes sense because it describes the total Template:CO2 being measured by the spirometer. The only source of the Template:CO2 we are assuming to measure is from the alveolar space where Template:CO2 and O₂ exchange takes place. Thus alveolar's fractional component, F_a, will always be higher than the total Template:CO2 content of the expired air, F_e, thus we will be always yielding a positive number.

Where Ptot is the total pressure, we obtain:

• $F_{a}\times Ptot=PaCO_{2}$ and
• $F_{e}\times Ptot=PeCO_{2}$

Therefore:

$V_{d}/V_{t}={\frac {FaCO_{2}-FeCO_{2}}{FaCO_{2}}}\times {\frac {Ptot}{Ptot}}$

This is simplified as:

$V_{d}/V_{t}={\frac {PaCO_{2}-PeCO_{2}}{PaCO_{2}}}$

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.