The following overview will hopefully improve your understanding of two compartment open model intermittent infusion pharmacokinetics. Calculations are complex and are the results of several equations being chained together to calculate serum levels and AUC.
The initial distribution phase is caused by tissue uptake and is associated with rapid changes in serum levels early after the dose. Pharmacokinetic studies where the independent pharmacokinetic parameters are determined use extensive serum level monitoring with 6 or more levels during the distribution phase and multiple levels during the elimination phase.
In clinical practice serum levels are normally drawn after the initial distribution phase to guide dosing and monitoring. Typically a trough is drawn or a peak and trough post distribution. Due to continued release from the peripheral compartment the terminal elimination rate constant (β) cannot be accurately defined for greater than ten distribution (α) half-lives if only post distribution levels are used.
Key Principle: Just like a one-compartment model, the patient's weight (Vd) and renal function (Cl) have the largest impact on serum levels in a two compartment model.
Cp(t) = A × exp(−α × Time) + B × exp(−β × Time)
Where:
Determination of A, B, α and β is done by curve stripping serum levels after an IV bolus dose and requires numerous levels. Curve stripping will not be discussed here. See a textbook for a description.
A = DoseIV × (α − k21) / (Vc × (α − β))
A is the extrapolated peak from the distribution phase.
B = DoseIV × (k21 − β) / (Vc × (α − β))
B is the extrapolated peak from the elimination phase.
T1/2 distribution = 0.693 / α
T1/2 β = 0.693 / β
Time to steady state is calculated using β as it is the smaller rate constant.
k10, k12, k21 are 1st order micro rate constants that can be used to calculate the macro rate constants α and β.
k10 = Cl / Volume of distribution in central compartment
Rate of elimination from central compartment. For vancomycin clearance is the parameter related to renal function.
k12 = Q / Volume of distribution in central compartment
Rate of transfer to peripheral compartment. Q is clearance between compartments.
k21 = Q / Volume of distribution in the peripheral compartment
Rate of transfer to the central compartment. Q is clearance between compartments.
α = 0.5 × [(k10 + k12 + k21) + ((k10 + k12 + k21)2 − (4 × k21 × k10))0.5]
Distribution rate constant and is larger than β.
β = 0.5 × [(k10 + k12 + k21) − ((k10 + k12 + k21)2 − (4 × k21 × k10))0.5]
Overall elimination rate constant, which is similar to the one-compartment K if levels are analyzed post distribution. Changes in renal function or clearance (Cl) impact β but have little impact on the calculated α, and no impact on k12 and k21.
Vd central = Dose / (A + B)
Independent variable, central compartment, elimination usually occurs from this compartment. For vancomycin the kidney is located in the central compartment.
Vd peripheral = Q / k21
Independent variable, second compartment where drug is distributed into and out of. Acts as a repository with prolonged elimination.
Cl = S × F × D / AUC
Independent variable, this parameter is related to renal function for vancomycin. Clearance is from the central compartment. Clearance is the same regardless of number of compartments and can be calculated without the consideration of the compartment model.
Q = k12 × Vc = k21 × Vperipheral
Independent variable intra-compartment clearance.
Micro constants can be calculated in the following sequence after curve stripping of an IV bolus dose:
k21 = (A × β + B × α) / (A + B)
k10 = α × β / k21
k12 = α + β − k10 − k21
Vcentral = Dose / (A + B)
Vd steady state = Vc × (k12 + k21) / k21
True pharmacokinetic parameter only affected by distribution and not elimination. It is best used when correlating data from one patient to another and can be used when calculating loading doses except for constant infusions where Vc should be used to calculate the loading dose.
Vd steady state = Vc + Vp
Vd peripheral = Vd steady state − Vcentral
Important Relationship: α > β, α > k21, k21 > β
Note: Most two compartment pharmacokinetic studies publish Vc, Vp, Q and Cl. These are the four independent parameters for a two compartment model. The other calculated parameters are dependent on these.
The smaller the value of Vc / Vd extrapolate the greater the degree of multi-compartment characteristic the serum levels display.
Vd extrapolate = Dose / B = Vc × (α − β) / (k21 − β)
Same as one-compartment model Vd derived from curve stripping. As the calculated K calculated, using two levels post-dose, changes during the dosing interval the Vd extrapolate value will change depending on when the levels are drawn. This should not be used for loading doses as it calculates a higher value than Vss and will give an excessive dose.
(Vd extrapolate / Vd β) − 1 = fraction of error in the total clearance when one assumes a one-compartment model instead of a two or higher compartment model
Vd β = Dose × α / (B × α + A × β) = k10 × Vc / β = Cltotal / β = Dose / (β × AUC)
Volume Relationship: Vd extrap > Vd β > Vdss > Vc
Use Vss or Vβ to calculate the loading dose. Due to drug passing into the peripheral compartment the loading dose is normally larger than that calculated for the typical one-compartment model for vancomycin for the Goti model.
Clinical Note: Early levels during therapy without an adequate load may erroneously appear as if a higher maintenance dose is needed if a two compartment model truly applies. Normal early initial levels, 15 mcg/mL after one day of maintenance therapy, without an adequate load will result in supertherapeutic levels. The earlier the levels are drawn the greater the divergence becomes and level predictions appear non-intuitive if the Goti model truly applies. It is better to wait until day three of therapy, two full days of maintenance therapy, to draw levels to minimize the impact of the loading dose on serum levels. This will also help to minimize the affect of a biased Bayesian model on dosage calculations.
LD = (Cp desired × T' × Vc × (β − α) × [1/((k21 − α) × (1 − exp(−α × T')) / α) + 1/((β − k21) × (1 − exp(−β × T')) / β)]
Most accurate method to calculate loading dose as it calculates the loading dose for the desired peak after the first dose is infused assuming no drug is already on board. The dose calculated will be lower than using Vss for the Goti model.
LD = Vd steady state × Desired Serum Concentration
LD = (Vcentral × (k12 + k21) / k21) × Desired Serum Concentration
If a large dose is calculated the dose may be split into several smaller doses given every 4 hours to minimize the chance of red man's syndrome.
Cp(t) = [Dose / ((T' × Vc) × (β − α))] × [((k21 − α) / α) × (1 − exp(−α × Time since start of infusion up to infusion length)) × exp(−α × time since end of infusion or zero if during infusion) + ((β − k21) / β) × (1 − exp(−β × Time since start of infusion up to infusion length)) × exp(−β × time since end of infusion or 0 if during infusion)]
Important Correction: The original posting before 3/12/22 had an error in above equation which was found in two pharmacokinetic textbooks and is now corrected. The equation above was checked using the method of superposition for the two compartment model bolus dose equation. A dose of 1000 mg infused over 2 hours was converted to 120 bolus doses of 8.33 mg each given every minute for 120 minutes and the resulting levels of all doses for 12 hours were calculated and summed for each minute. The calculated peak and trough were then compared to the values calculated using the above equation and they were the same.
Single Dose Short Infusion Simulation Method Of Superposition Excel file
To determine steady state levels, 1/(1 − exp(−α × τ)) is divided into the first part of right side of equation above and 1/(1 − exp(−β × τ)) is divided into the second part of the right side of the equation.
Cp(t) = [Dose / ((T' × Vc) × (β − α))] × [((k21 − α) / α) × (1 − exp(−α × Time into infusion up to infusion length)) × exp(−α × time since end of infusion or 0 if during infusion) / (1 − exp(−α × τ)) + ((β − k21) / β) × (1 − exp(−β × Time into infusion up to infusion length)) × exp(−β × time since end of infusion or zero if during infusion) / (1 − exp(−β × τ))]
Where:
Important Correction: The original posting before 3/12/22 had an error in above equation which was found in two pharmacokinetic textbooks and is now corrected.
AUC per 24 hours = [Dose / ((T' × Vc) × (β − α))] × [((k21 − α) / α2) × (1 − exp(−α × Infusion Length)) × (1 − exp(−α × (τ − Infusion length))) / (1 − exp(−α × τ)) + ((β − k21) / β2) × (1 − exp(−β × Infusion Length)) × (1 − exp(−β × (τ − Infusion Length))) / (1 − exp(−β × τ))] × 24/τ + (Peak + Trough) × (Infusion Length / 2) × (24/τ)
Note: Error in equation corrected 8/15/22
AUC per 24 hours = Dose (mg) / Clearance (L/hr) × (24/τ)
Use the single dosing equation above to calculate the amount of drug remaining in the body at the time of interest for each dose and then sum the individual amounts from all the doses.
The amount of drug in the body from the initial level drawn before the first dose in the series may be added by calculating:
Cp(t) = Cp initial × exp(−β × time)
This assumes the level was drawn in the post distribution phase of any prior dose.
Coming Soon: The following excel spreadsheet demonstrates the method of superposition. It also can be used to demonstrate the impact of changes to the independent pharmacokinetic parameters on serum levels.
If the dosing interval is less than 10 × T1/2, accumulation of drug will occur.
Accumulation ratio = 1 / (1 − exp(−β × τ))