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 (beta) can not be accurately defined for greater than ten distribution (alpha) half lives if only post distribution levels are used. 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.
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Equations
Single Dose IV bolus Two Compartment Open Model
Cp(t) = A*exp(-alpha*Time) + B*exp(-beta*Time) , A and B are the y intercepts of the exponential functions, alpha is the hybrid macro distribution rate constant and beta is the hybrid macro elimination rate constant. Determination of A, B, alpha and beta is done by curve stripping serum levels after an IV bolus dose and requires numerous levels. Curve stripping which will not be discussed here. See a text book for a description.
A = DoseIV*(alpha-k21)/(Vc*(alpha-beta)), A is the extrapolated peak from the distribution phase.
B = DoseIV*((K21-beta)/(Vc*(alpha-beta)), B is the extrapolated peak from the elimination phase.
T1/2 distribution = 0.693/alpha
T1/2 beta = 0.693/beta, time to steady state is calculated using beta 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 alpha and beta.
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.
alpha = 0.5 [(K10+k12+k21) + ( (K10+K12+K21)^2 - (4*K21*K10))^0.5], distribution rate constant and is larger than beta.
beta = 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 beta but have little impact on the calculated alpha, and no impact on K12 and K21.
Vdcenteral = 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 prolong 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 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 using A, B, alpha and beta determined during curve stripping.
K21 = (A*beta + B*alpha) / (A+B), micro rate constant
K10 = alpha*beta / K21, micro rate constant
K12 = alpha+beta-K10-K21, micro rate constant
Vcenteral = Dose / (A+B)
Vdsteady 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.
Vdsteady state = Vc + Vp
Vdperipheral = Vdsteady state - Vcentral
Alpha > beta, alpha > K21, K21 > beta
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/Vdextrapolate the greater the degree of multi-compartment characteristic the serum levels display
Vd extrapolate = Dose/B, = Vc*(alpa-beta)/(K21-beta), 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 exptrapolate 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 / Vdbeta) -1 = fraction of error in the total clearance when one assume an one-compartment model instead of a two or higher compartment model.
Vd beta = Dose*alpha/(B*alpha + A*beta) = K10*Vc/beta = Cltotal/beta, =Dose/(Beta*AUC)
Vd extrap > Vd beta > Vdss > Vc
Two Compartment Intermittent Infusion Open Model Equations
Loading Dose and Initial Levels
Use Vss or Vbeta to calculate the loading dose. Due to drug passing into the peripheral compartment the loading dose is normal larger than that calculated for the typical one-compartment model for vancomycin for the Goti model. 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 super therapeutic 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 drawn 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 = (Cpdesired*T'*Vc*(beta-alpha)* [1/((K21-alpha)*(1-exp(-alpha*T'))/alpha) + 1/((beta-K21)*(-1-exp(-beta*T'))/beta)] is the 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 = VdsteadyState * Desired Serum Concentration.
LD = (Vcentral*(K12+K21)/K21) * Desired Serum Concentration. If an 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.
Single Dose Intermittent Infusion
Level(s) after infusion complete
Cp(t) = [Dose / ((T'*Vc)*(beta-alpha))]* [((K21-alpha)/alpha)*(1-exp(-alpha*Time since start of infusion up to infusion length)) *exp (-alpha *time since end of infusion or zero if during infusion) + ((beta-K21)/beta)*(1-exp(-beta*Time since start of infusion up to infusion length))*exp(-beta*time since end of infusion or 0 if during infusion)]
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 the above equation and they were the same. The attached file has the calculations. Single Dose Short Infusion Simulation Method Of Superposition Excel file
Multiple Doses Intermittent Infusions
To determine steady state levels 1/(1-exp(-alpha*Tau) is divided into the first part of right side of equation above and 1/(1-exp(-beta*Tau) is divided into the second part of the right side of the equation.
Cp(t) = [Dose / ((T'*Vc)*(beta-alpha))]*[((K21-alpha)/alpha)*(1-exp(-alpha*Time into infusion up to infusion length)) *exp (-alpha *time since end of infusion or 0 if during infusion) / (1-exp(-alpha *Tau)) +
((beta-K21)/beta)*(1-exp(-beta*Time into infusion up to infusion length))*exp(-beta*time since end of infusion or zero if during infusion) / (1-exp(-beta*Tau))]
Time into infusion = time since start of infusion to infusion end, then time into infusion is constant at the infusion length.
T'=infusion length
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)*(beta-alpha))]*[((K21-alpha)/alpha2)*(1-exp(-alpha*Infusion Length)))* (1-exp(-alpha*(Tau-Infusion length)))/(1-exp(-alpha*Tau)) + ((beta-K21)/beta2)*(1-exp(-beta*Infusion Length)))*(1-exp(-beta*(Tau-Infusion Length)))/(1-exp(-beta*Tau))]*24/Tau + (Peak+Trough)*(Infusion Length/2)*(24/Tau). Error in equation corrected 8/15/22
AUC per 24 hours = Dose(mg)/Clearance(L/hr) *(24/Tau)
Method of superposition
Use the single dosing equation above to calculated 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) = Cpinitial *exp(-beta*time). This assumes the level was drawn in the post distribution phase of any prior dose.
Not yet available: 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 x T1/2 accumulation of drug will occur. 1/(1-exp(-beta*Tau)) can be used to calculate accumulation ratio.
Beta Half-Life:
Due to redistribution of drug from the central compartment the apparent T1/2 changes during the dosing interval. The following example uses a patient: 70 inches, 70 kg, creatinine clearance 15 ml/min, clearance 1.14 L/hr, Vc 58.4 liters, Vp 38.4 liters, Q 6.5 L/hr, 1000 mg Q36H giving a peak of 30.9 mcg/ml and trough of 18.8 mcg/ml. Depending of the time levels are drawn the apparent T1/2 changes. Levels drawn early give a shorter half life. After 10 alpha half lives calculated beta half life is accurate. This can cause confusion as early levels may be low yet give high predicted state state levels if a two compartment model applies.
If a one-compartment model is used, as long a level are drawn
across the dosing interval (peak 2 hours post-infusion, trough before next dose)
calculations are accurate with minimal prediction error even if the drug
displays two compartment pharmacokinetics. See examples further down the page.
Calculated T1/2 for one-compartment model versus true two compartment Beta.
Note the dependence of time of levels.
Two Compartment Model Serum Level, AUC, and Comparison to one-compartment Model Simulations (Excel downloadable File)
Modified Goti Model Cl (l/hr) = 6.04*(Clcr/120)^0.8, Vdcentral(L) = 58.4*(weight(kg)/70), Vdperipheral = 38.4*(weight(kg)/70), Q(L/hr) =6.5
Creatinine clearance use calculated adjusted body weight when total body weight greater than learn body weight.
Simulations were performed using a modified Goti Model to demonstrate the effects on AUC and serum levels for incremental changes, +10% and -10%, in total clearance, Vd central, Vd peripheral and Q. Each independent parameter was changed incrementally up to 50% while the other parameters were held constant at the baseline values. Clearance caused the most dramatic changes in AUC and is related to renal function. Little change in AUC was noted for other parameters. Magnitude of changes in serum levels were greatest for clearance >> Vd central > Vd peripheral & Q.
The baseline parameter were Clearance (L/hr) = 3.84 , Vcentral (L) = 58.4, Vperipheral (L) = 38.4 Q(L/hr) = 6.5, Baseline AUC = 490. Dose 1000 mg, Tau 12 hours, Infusion Period 2 hours, weight 70 kg, Creatinine Clearance 68 ml/min.
Comparison of one-compartment Model Calculated Levels and AUC with Two Compartment Goti Model Levels and AUC
Most dosing programs use a one-compartment open model for dosing and serum level predictions. This is for several reasons. In clinical practice only one or two levels are measured which is inadequate to describe a two compartment model. A two compartment model requires numerous levels in the distribution and in the elimination phase to fully characterize it. Two compartment model calculation are much more complicated than a one-compartment model. If acceptable predictions can be made with a one-compartment model it is preferred because of ease of use and ease of understanding. It is common clinical practice either to draw only a trough or a peak after the distribution phase and a trough. Drawing a peak post distribution and trough after the same dose allows Vd, K, and AUC to be calculated for the one-compartment model.
In the simulations below the one-compartment volume of distribution was calculated using the level two hours post-infusion from the 2 compartment model. The level 2 hours post-infusion is after most of the distribution to the peripheral compartment has occurred. The elimination rate constant for the one-compartment model was calculated using the two hour post-infusion level and the trough from the 2 compartment model. The calculated parameters, Vd and K, were then placed in a one-compartment open infusion model. The one-compartment model calculated levels and AUCs were compared to the two compartment model's for the dosage interval and percent error in predictions for the one-compartment model were calculated. This was repeated for the clearance decreased and increased by 10% increments up to 50%.
% Error Predicted Level = (one-compartment Level - Two Compartment Level) *100/ Two Compartment Level. Values were calculated for the entire dosing interval from the end of the infusion.
% Error AUC = (one-compartment AUC - Two Compartment AUC) *100/ Two Compartment AUC for the dosing interval.
The baseline parameter were Clearance (L/hr) = 3.84 , Vcentral (L) = 58.4, Vperipheral (L) = 38.4 Q(L/hr) = 6.5, Baseline AUC = 490. Dose 1000 mg, Tau 12 hours, Infusion Period 2 hours, weight 70 kg, Creatinine Clearance 68 ml/min.
The simulations demonstrate that a one-compartment model is adequate for vancomycin dosing when the modified Goti model describes the two compartment pharmacokinetics.
Dose change to 2000 mg Q24H
Dose changed to 3000 mg Q36H
Converting A Two Compartment Open Model Into A Open Compartment Open Model for Varying Weights and Creatinine Clearnaces
A Modified Goti Model was used in the following example. Excel's Solver was used.
A standard dose and frequency such as 1000 mg Q12H was used. Infusion Period 2 hours and height of 70 inches.
Enter a weight range, for example 50 to 100 kg in into the two compartment model in increments of 50 kg, 75 kg and 100 kg
For each weight range enter a creatinine clearance ranges from 15 to 150 ml/min of 15, 30, 60, 90, 120 and 150 ml/min
Calculate the two compartment serum levels and AUC for all of the above categories.
Calculate the one-compartment model K and Vd for for each two compartment categories and place them in a table along with the values noted in the graphic below (Height, BSA, Weight, Clcr, Vd derived, K derived, Cl =(K*Vd). Note the peak two hours post-infusion and trough from the two compartment levels were used in calculations.
Set up a column to calculated the Clearance (L/hr) based on weight (kg), Clcr (ml/min), and Intercept Solver value.
Cl (L/hour) Based on Solver = Weight (kg) * Solver value + Clcr (ml/min) * Solver value *(1.73/Surface Area) + Intercept. This is your open compartment equation for clearance (L/hr)
Square of Error Cl (L/hr) = (Cl (L/hr) Solver - Clearance one-compartment Derived from Two Compartment)^2
SSE (Sum of Square of Errors) = Sum of all Cl (L/hr) errors.
Set up solver to Minimize the SSE for Cl (L/hr) by changing weight solver value, Clcr solver value, and intercept solver value. In the example an intercept of 0 was found so it may be ignored.
Repeat the above procedure for Vd (L)
Vd (l) Solver= Weight (kg) * Solver value + Clcr (ml/min) * Solver value*(1.73/Surface Area) + Intercept Solver value. This is your open compartment equation for Vd (L).
Square of Error Vd (L) = (Vd (L) Solver - Vd one-compartment Derived from Two Compartment)^2
SSE (Sum of Square of Errors) = Sum of all Vd (L) errors.
Set up solver to Minimize the SSE of Vd by changing weight Solver value, Clcr Solver value, and intercept solver value.
Excel's Solver using the Evolutionary fitting was used in the example below.
The results for the simulation are below.
Linear Regression was performed as shown below and the R^2 value was very high for both Clearance and Vd.
The chart below is constructed using a 50 kg, 70 inch patient given 1000 mg Q12H Infusion Period 2 hours with 20% incremental decrease in clearance from baseline. The lines for two compartment and one-compartment levels are superimposed except earlier in the dosing interval.
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