Equations Flashcards
Ideal gas equation
PV = nRT (or P1V1/T1 = P2V2/T2)
Where n = number of moles of gas present and R = universal gas constant (8.32J per 1C at 0C and 1 atm)
Reynolds’ number
η
Where v = velocity, p = pressure, d = density, η = viscosity
<2000 = laminar
2000-4000 transitional
>4000 turbulent
NB: number is dimensionless
Turbulent flow is ∝ √P ∝ 1/√l ∝ 1/√d
Energy (e.g. stored energy of defibrillator) (5)
1/2 CV2 = 1/2QV = VQ = VIt = Pt
Where C - capacitance, V = voltage, Q = charge, t = time
Resistors
Series: R1 + R2
Parallel: 1/R1 + 1/R2
Wheatstone bridge: R1/R2 = R3/R4
Loading and maintenance doses (e.g. for TIVA without TCI pump)
Loading dose = desired conc x Vd
Maintenance dose = desired conc x clearance
Bolus dose to achieve new conc (in TCI) = [difference btwn current and desired concs] x Vd
Ejection fraction
Stroke volume
EF = SV/EDV
SV = EDV-ESV
Fick principle
Rate of flow to an organ = clearance of substance / A-V difference in substance concentration
e.g. CO = VO2 / (CaO2-CvO2) or RPF = PAH clearance / A-V conc diff
Henderson-Hasselbalch and pKa
pH = pKa + log [base]/[acid]
pKa = pH - log [base]/[acid]
Osmotic pressure (van’t Hoff)
π = RTC
Where π = osmotic pressure, R = universal gas constant, T = absolute temperature, C = osmolality (mosm/kg H2O)
Gibbs-Donnan
[cation]A x [anion]A = [cation]B x [anion]B
Pharmacokinetics
k = 0.693/t1/2 = clearance/Vd
This is the only pharmacokinetics formula to remember; if τ needed, substitute it for k and switch the other two values (because k and τ are reciprocals).
Where k = rate constant, τ = time constant
Also k = rate/quantity
Renal equations (GFR, RPF, RBF, FF)
GFR (inulin or creatinine) = (urine conc x urine flow)/plasma conc
RPF = clearance/[A-V PAH difference]
RBF = RPF/(1-Hct)
Filtration fraction = GFR/RPF (normally 20%)
Osmolarity
2(Na+ + K+) + urea + glucose
Normal = 285-295 mosm/L
Na+ and K+ are doubled to account for the Cl- which accompanies most Na+/K+ ions in the body. Urea reduces the freezing point, although it is not osmotically active. Proteins are osmotically active but not ionic so not included.
SVR and PVR
SVR = (MAP - CVP)/CO x 80 dynes/s/cm to -5 (normal 800-1200)
PVR = (MPAP - PCWP)/CO x 80 (normal 100-200)
MAP
MAP = CO x SVR
MAP = DBP + 1/3 (SBP - DBP)
Alveolar gas equation
PAO2 = PiO2 - (PACO2/R)
Where R (respiratory quotient) = CO2 production/O2 consumption (about 0.8 depending on fuel source)
PiO2 = FiO2 x (Patm - PH2O)
Shunt equation
Qs = CcO2 - CaO2
— ——————
Qt CcO2 - CvO2
Physiological dead space (Bohr)
Vd = PaCO2 - PECO2
— ———————-
Vt PaCO2
Where PECO2 = mixed expired PCO2
Physiological dead space = anatomical dead space (2ml/kg) + alveolar dead space (0 in health)
Paediatric weight formulae
<1y: (age x 0.5) + 4 (age in months)
1-5y: (age x 2) + 8
6-12y: (age x 3) + 7
Therapeutic index
LD50/ED50
Transpulmonary pressure
Alveolar pressure - pleural pressure
Strong ion difference
Apparent SID = (Na+ + K+ + Ca2+ + Mg2+) – (Cl- + lactate)
A ‘strong’ ion is one which completely dissociates at the pH of interest.
Apparent SID is normally about 40 mEq/L.
True or ‘effective’ SID is much more complicated to calculate.
Strong ion gap = difference between apparent and effective SID. Principle is similar to the anion gap.
Bioavailability
AUC (PO)/AUC (IV)
AUC = of a concentration-time curve
Hagen-Pouseille (laminar flow)
Q = πPd4
——–
128ηl
Beer-Lambert
Absorbance = ξcd
Where ξ = molar extinction coefficient, c = molar concentration, d = thickness
Transmission decreases exponentially as the concentration of the medium (Beer) and the thickness of the medium (Lambert) increase.
Fick’s law of diffusion
Rate of diffusion = kp.(A/T).C1-C2
Where kp = permeability constant, A = area, T = thickness, C1-C2 = concentration gradient
Rate is also ∝ 1/√MW (Graham’s law)
Starling forces
Pressure gradient = (P cap + π inst) - (P inst + π cap)
Rate of filtration = k x pressure gradient
Where π = colloid osmotic pressure, P = hydrostatic pressure, inst = interstitial, cap = capillary, k = filtration coefficient
In the kidney, equation changes to hydrostatic forces - osmotic forces:
GFR = kf (P gc - P bc) - (π gc - π bc)
Where kf = glomerular filtration coefficient (permeability x capillary bed surface area)
Cardiac output
Cardiac index
CO = HR x SV
Cardiac index = CO/BSA
Bazett’s formula
QTc = QT
—-
√(R-R)
The QT interval is ‘corrected’ to a HR of 60 bpm.
QTc is prolonged if >440ms in males or >460ms in females.
>500ms is high risk for torsades.
Compliance
- Specific
- Total respiratory system
- Static
- Dynamic
Compliance = ΔV/ΔP
Specific compliance = (ΔV/ΔP) / FRC
1/Cresp = 1/Clung + 1/Ccw
Static compliance = Vt / (Pplat - PEEP)
Dynamic compliance = Vt / (Ppeak - PEEP)
Where V = volume, P = pressure, Cresp = overall respiratory system compliance, Clung = lung compliance, Ccw = chest wall compliance
Oxygen content and delivery (flux)
CaO2 = (Hb x SpO2 x 1.34) + (PaO2 x 0.0225)
DO2 (flux) = CaO2 x CO
Perfusion pressures
- Cerebral
- Spinal cord
- Coronary
CPP = MAP - (ICP + CVP)
SCPP = MAP - CSFP
CoPP = DBP - LVEDP
Closing capacity
CC = CV + RV
Doppler equation
V = 2 Fo
——–
C Fd Cos Theta
V = blood velocity
Fo = original US frequency
Fd = Doppler shift
C = constant (velocity of US in tissue - 1540m/s)
Cos Theta = cosine of angle of incidence (corrects for probe misalignment)
Velocity is then used to calculate flow:
Flow = area x velocity
Stewart-Hamilton equation
Q = I / integral Ci dt
Q = cardiac output
I = indicator amount in moles
Ci dt = integral of indicator conc over time (AUC)
Or:
CO = k(core temp - indicator temp) x vol indicator
———————————————-
Change in blood temp
Inverse square law
Point sources of gravitational force, electric field, light, sound or radiation obey the inverse square law.
Intensity = source strength
———————-
4πr2
i.e. doubling the distance from the source will quarter the intensity.
Anion gap
Correction for albumin
(Na+ + K+) - (Cl- + HCO3-)
Normal = 4-12 mmol/L (older assays 8-16)
High anion gap metabolic acidosis: lactate, ketones, alcohols, renal, salicylate, chronic paracetamol
Normal anion gap metabolic acidosis: Cl- excess, GI losses, diuretics, bicarb loss, ileostomy, RTA, TPN, ileal conduit
Anion gap can be falsely low/negative or ‘normal’ if the albumin is low; low/negative anion gap can also be caused by high Ca/Mg and Li intoxication.
AG (corrected for albumin) = AG + (albumin gap/4)
Where albumin gap = 40 - apparent albumin
Stroke volume variation
SVV = SV(max) - SV (min)
————————-
SV(mean)
SVV >10% suggests fluid responsivenes as SV is sensitive to fluctuations in preload due to the respiratory cycle.
BP falls in inspiration and rises in expiration in spontaneous ventilation.
In PPV, BP rises in inspiration and falls in expiration.
Sodium deficit
Na+ deficit = (0.6 x weight) x (desired Na+ - current Na+)
0.6 indicates 60% TBW being H2O.
Desired Na+ usually taken as 140.
Normal Na+ content is about 60 mmol/kg.
Normal H2O content is about 60% total body weight.
Parkland formula
4ml x kg x %BSA = first 24h requirement
Half over 8h, rest over 16h
Children get maintenance as well
Originally used to include a colloid bolus after the above, no longer used
BMI
Weight in kg / height in m2
Sensitivity
True positives / true positives + false negatives
Specificity
True negatives / true negatives + false positives
Positive predictive value
True positives / all positives
Negative predictive value
True negatives / all negatives
P:F ratio
PaO2/FiO2
Normal is over 60kPa (13.3/0.21)
An alternative is the oxygenation index:
OI = ((FiO2 x mean airway pressure)/PaO2) x 100
Used esp in paeds to determine need for ECMO. OI<25 good, 25-40 = 40% mortality, >40 consider ECMO.
A-a gradient
PAO2 - PaO2
(Former calculated from alveolar gas equation: PAO2 = PiO2 - (PACO2/R) and PiO2 = FiO2 x (Patm - PH2O))
Normal
- On air: 7mmHg young, 14 elderly
- On 100% O2: 31 young, 56 elderly
Rapid Shallow Breathing Index (RSBI)
RSBI = RR/Vt (L)
Energy requirements
REE/BMR + DIT + activity factor + stress factor +/- specific disease state factor
REE/BMR calculated by Harris-Benedict or Schofield equations
HB:
Male: 66.5 + (13.8 x IBW) + (5 x height) - (6.8 x age)
Female: 66.5 + (9.6 x IBW) + (1.7 x height) - (4.7 x age)
95% CI is about +/- 200kCal/day.
1g protein or carb = 4 kCal
1g fat = 9 kCal
RQ
RQ = CO2 produced / O2 consumed
Carbs = 1.0 Protein = 0.8 Fat = 0.7
Osmolar gap
Measured osmolality - calculated osmolarity
Normal = <10
Note the units are different so it doesn’t make mathematical sense - it is just a rough clinical aid.
A high osmolar gap indicates presence of other osmotically active particles e.g. methanol, ethylene glycol, mannitol, sorbitol, polyethylene glycol/propylene glycol (found in IV lorazepam and others), glycine (TURP) and maltose (IGIg).
Free water deficit
H2O deficit = (0.6 x weight) x (current Na+ - desired Na+)/desired Na+
Desired Na+ usually taken as 140.
Normal Na+ content is about 60 mmol/kg.
Normal H2O content is about 60% total body weight.
Corrected Na+ in hyperglycaemia
Corrected Na+ = measured Na+ x (0.3 x (plasma glucose - 5.5))
Basically, for every 5.5mmol/L increase above a standard glucose of 5.5, add 2.4mmol/L to the serum sodium.
Oxygen requirement for transfer
O2 req = MV (L) x FiO2 (as fraction) x time (mins)
Then double it for margin of safety.
Be mindful that basic transport vents may only do 100% O2 and may also use O2 to drive the vent.
Size CD cylinder = 460L (hence at 15L/m a full one will last about 30m) Size E (anaesthetic machine) = 680L (45m) Size F (under trolleys) = 1360L (90m)
Nasal specs FiO2
21% + (O2 flow rate x 3)
e.g. 2L/m
21 + (2 x 3) = 27%
(H)SMR
(Actual deaths/expected deaths) x 100
ABPI
ABPI = SBP at site of interest (e.g. injured/diseased limb) / brachial SBP
Despite the name, can be performed on upper or lower limbs.
ABPI > 0.9 is highly unlikely to have a vascular injury/insufficiency
API < 0.9 indicates possible vascular injury/insufficiency - likely to need CT angiography
Shock index
HR/SBP
Normal = 0.5-0.7
- 8 or more predicts hypotension at induction for emergency intubation, hyperlactataemia in sepsis and 28-day mortality in sepsis
- 0 or more even more strongly associated with sepsis outcomes above
Maddrey’s discriminant function
Bilirubin + (4.6 x (PT - control PT))
> 32 in alc hep suggests poor prognosis + may benefit from steroids
ICU capacity (Hill-Burton formula)
Calculated ICU capacity = (adms/yr x av LoS in days) / (ideal occupancy rate x 365)
Pleural effusion volume estimate
Max depth in mm on US x 20ml
Paediatric fluid maintenance
First 10kg 4ml/kg/h
Next 10kg 2ml/kg/h
All further kg 1ml/kg/h
CPP
CPP = MAP-ICP (or CVP if CVP>ICP)
