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Equations for Physiology



Symbols stand for:






Starling Reabsorption Forces

NFP = (Pc-Pif)-(pc-pif)

Q= k [(Pc-Pif) - (pc-pif)s]

With s (~1) the reflection coefficient of proteins, and K the hydraulic permeability of the wall

Pif = interstitial fluid , Pc = capillary hydrostatic pressure

pc = capillary osmotic pressure

pif = interstitial fluid os. Pressure

NFP = net filtration pressure

Q = net capillary filtration rate

Increased Pc during exercise results in plasma loss to interstitial fluid

In glomerulus if is bc and pbc is zero, otherwise the same.

Glomerular filtration rate

GFR= Kf (Pcap-Pbc-pcap)


As above except:


I = VR-1 and R-1 = Kf

And GFR = NFP Kf

p = colloid osmotic pressure

GFR = glomerular filtration rate

Kf = capillary filtration coefficient (= area hydraulic conductivity)

P = hydrostatic pressure

capcapillary, bcbowmans capsule

Not exactly true in humans as eqm is not quite reached, but approx is good.

ABP increase increases Pcap and decreases pcap so increases GFR

Blockages Pbc GFR

BP thick basement membrane Kf

Respiratory and Cardiovascular




Henrys Law

[X] = SX PX

[X] = concentration mMoles

SX = solubility Kp mM/mmHg

PX = partial pressure mmHg

Kp solubility at 37degrees ~0.03


Fick Principle


F = removal


Net flux = D[X]A d


. where d MW-1

Q = PX SX A d


Q = net flux

PX = partial pressure of X

SX = solubility coefficient of X

A = exchange surface area

D = diffusion pathway distance

d = diffusion coefficient of X

SX d = k (diffusion constant)


Use to calculate rate of flow of substance down conc. Grad

SX in mmol l-1 mmHg-1


Reynauds Number

Re = rDV


Re = Reynauds number

r = density of fluid

D = diameter of tube

V = velocity of flow

h = viscosity of fluid

If Re > 2000, turbulence certain

If Re < 200, turbulence does not occur


Law of Laplace

P T (capillaries)


P = pressure in tube

T = tension in wall of tube

r = radius of tube

T = PT r if not infinitely thin!

u (PT = transmural pressure, u = wall thickness)


Laplace in a sphere

P = 2T (alveoli)


As surface tenstion = 2 p r T

Opposing = p r2 P

Relevant as surfactant reduces surface T less P so prevents atelectasis


Poiselle in glass tubing


(any fluid)

Flow = DP p r4

8 h l


and Flow = DP


Flow = rate of flow of fluid

DP = pressure gradient

p = 3.14159265358979

r4 = radius to 4th power

h = viscosity (eta)

l = length of tube

Resistance = DP = 8 h l

F p r4

Distributing vessels minimise resistance withour creating turbulence (cf Reynauld). Capillary apparent reduction in viscosity due to bolus flow



R = A v

A = area

v = velocity

Same vol of fluid passes per unit time

Smaller tube faster flow


Bernoulli or Venturi principle

P + rv2 + rgh = c


Pressure, kinetic and potential energy total is always constant

C = constant

P = hydrostatic pressure

r = density

v = velocity

g = 9.8 h = height

Respiratory: during snoring small gap high speed.

Cardiovascular: Kinetic energy rarely significant in the blood, perhaps in pulmonary circulation, during exercise, and at entry to the heart KE helps filling. Pot. energy works against venous return.



v = [2E/m]

Distance = (2 d t) therefore t d2

V = diffusion velocity

E = kinetic energy

m = mass of particle

a) The heavier the particle, the slower the particle diffuses in air.

b) Diffusion only efficient over small d


Dead space



VD = dead space volume

VT = tidal volume

PACO2 = alveolar PCO2

PECO2 = expired PCO2

Derived from


And VA = VT VD


Total lung capacity (Bohr equation)



TLC = total lung capacity (l)

PFRC = pressure after exhaled

DP = change in pressure (Pa)

DV = change in volume (l)

Pressures measured at mouth

DV change in body box volume



Prediction of partial pressures

PACO2 = k VCO2





RQ = CO2 exhaled/O2 taken up

(Normally ~ 0.8)

A = alveolar

I = inspired

V = rate of ventillation

K = a constant

F = small correction factor of 1-3 mmHg


Transport Factor

TF = Q .

t DP


requires end-inspiration alveolar gas sample

TF = transport factor

Q = quantity of CO transferred across epitheliunm (alveoli)

t = time (minutes)

DP = pressure gradient of CO

Single breath of CO and He, as He does not dissolve dilution measures alveolar volume therefore PACO (=DP) and Q/tDP can be calculated



C = DV = 1 .

DP elastance

C = compliance

P = pressure, V = volume

Compliance decreases in fibroses and therefore RV and TLC but FVC


vant Hoff


p = osmotic pressure (Pa)

V = volume (l)

s = relative permeability (1-0)

n = amount of solute (mol)

R = molar gas constant (8.314)

T = temperature (K)

From perfect molar gas equation

PV = nRT


1kPa = 10cmH2O = 7mmHg



pH = pKa + log10 [A-]


pH = -log [H+]

pKa = -log Ka

[A-] = base concentration

[HA] = acid concentration

Blood pH = 6.1 + log [HCO3-]



6.1- composite Pka of both reactions


Bernuille principle

E = PE + KE (or Venturi)

Pressure Energy

Kinetic Energy (mv2)

As airway narrows, kinetic energy increases, pressure drops


Resistance in parallel

1 = 1 + 1 + 1 + 1

RT R1 R2 R3 Rn

Total Resistance of multiple (n) airways or blood vessels

Small airway disease

local R but cannot detect total airway R


D Arcys Law (simplified version)


ABP = arterial blood pressure

CO = cardiac output

TPR = total peripheral resistance

Simplified version of D Arcys Law







Nernst equation

E = RT ln (CO/CI)


or at 37 degrees C:

E = 61mV log (Co/Ci)

E = voltage across membrane

R = gas constant 8.3

T = temperature

n = charge/valency

F = faradays constant 96,500

CO/CI is the concentration gradient across the membrane, from Outside to Inside (for something more concentrated on the inside gradient will have the opposite sign to n)


GHK equation E =

RT ln PK[K]O + PNa[Na]O

F PK[K]O + PNa[Na]O

PX = relative permeability of x

Note: if permeability of only one ion is significant, GHK = Nernst



Q = Q0 (1 - e-T/RC)

T = RC ln2

As RC is time taken to fall to 63%