Equations for Physiology

Name:	Equation:		Symbols stand for:			Use/comments
Renal
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 = V·R^-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, bcbowman’s 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 ÝPbc à ßGFR ÝBP à thick basement membrane ßKf
Respiratory and Cardiovascular
Henry’s 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 [a]-[v]		Net flux = D[X]·A ·d d . where d µ ÖMW^-1 Q = PX · SX · A · d 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
Reynaud’s Number		Re = r·D·V h	Re = Reynaud’s 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 µ T (capillaries) R	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) R	As surface tenstion = 2 p r · T Opposing = p r^2 · P		Relevant as surfactant reduces surface T à less P so prevents atelectasis
Poiselle in glass tubing (any fluid)		Flow = DP · p · r^4 8 · h · l and Flow = DP R	Flow = rate of flow of fluid DP = pressure gradient p = 3.14159265358979… r^4 = radius to 4^th power h = viscosity (eta) l = length of tube		Resistance = DP = 8 · h · l F p · r^4 Distributing vessels minimise resistance withour creating turbulence (cf Reynauld). Capillary apparent reduction in viscosity due to bolus flow
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 + ½ rv^2 + 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.
Einstein		v = Ö [2E/m] Distance = Ö (2 · d · t) therefore t µ d^2	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=VT ·(PACO2 – PECO2) PACO2	VD = dead space volume VT = tidal volume PACO2 = alveolar PCO2 PECO2 = expired PCO2		Derived from VT · PECO2 = PACO2 · VA And VA = VT – VD
Total lung capacity (Bohr equation)		TLC = – PFRC · DP DV	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 FRC = RV + IRV
Prediction of partial pressures		PACO2 = k · VCO2 VA PAO2 = PIO2 – PACO2 +F RQ	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
Compliance		C = DV = 1 . DP elastance	C = compliance P = pressure, V = volume		Compliance decreases in fibroses and therefore Ý RV and TLC but ßFVC
van’t Hoff		pV=s(nRT)	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
Henderson-Hasselbach		pH = pKa + log10 [A^-] [HA]	pH = -log [H+] pKa = -log Ka [A^-] = base concentration [HA] = acid concentration		Blood pH = 6.1 + log [HCO3^-] [H2CO3] 6.1- composite Pka of both reactions
Bernuille principle		E = PE + KE (or Venturi)	Pressure Energy Kinetic Energy (½mv^2)		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’ Arcy’s Law (simplified version)		ABP = CO x. TPR	ABP = arterial blood pressure CO = cardiac output TPR = total peripheral resistance		Simplified version of D’ Arcy’s Law
Nerves
Nernst equation	E = RT · ln (CO/CI) nF 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 = faraday’s 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
Charge	Q = Q0 · (1 - e^-T/RC)		T½ = RC ln2	As RC is time taken to fall to 63%