Basic Pharmacokinetic Principles and One-Compartment Model Equations


Bioavailability (F)

The fraction (F) of the administered dose that reaches systemic circulation. Factors affecting bioavailable include route of administration, drug metabolism before reaching systemic circulation, dissolution, dosage form, and absorption. Medication given intravenously is usually 100% bioavailable or F=1.

Equivalent dose new dosage form = Dose current dosage form * F current dosage form / F new dosage form

Chemical form or Salt (S)

S is the fraction of the administered dose that is active drug. Drug that are salts or need to be converted to active forms will have a S of less than 1.

Equivalent dose new dosage form = Dose current dosage form S * F current dosage form / S* F new dosage form

Volume of Distribution (Vd)

The Vd or apparent volume of distribution of a drug is the volume a drug would need to be distributed into to be at the same concentration as the concentration in the plasma (Cp).

Vd (liters) = total amount in body(mg) / Cp(mg/L)

The Vd is affected by lipid solubility, water solubility,  tissue binding, and protein binding. High protein binding and water solubility decrease the Vd. High lipid solubility and tissue binding increase volume of distribution.

Loading dose (LD) is the dose that brings the plasma concentration to that desired at steady state

LD(mg) = Cp(mg/L) * Vd(L)

If drug is already in the system then LD(mg) = (Cpdesired(mg/L) - Cpinitial(mg/L) * Vd(L)

Protein Binding (alpha)

alpha = free drug concentration in plasma / total drug concentration in plasma

alpha = (Cp(mg/L) free / (Cp(mg/L) bound )+ Cp(mg/L) free)

Cp(mg/L) equivalent for normal protein binding = Cp(mg/L) of patient / ((1-alpha)*(albumin(g/dL) of patient /albumin(g/dL) normal)) + alpha)

Phenytoin Protein Binding Calculations Excel Example

Rate of Administration (Ra)

Rate of administration (Ra) is the average rate of a medication entering the systemic circulation

Continuous infusion Ra = mg/hr

Intermittent dosing (mg/hour) = S*F*Dose(mg) / Dosage Interval(hours) or Tau

Intermittent dosing(mg/hour) = S*F*Dose(mg) / Tau

Clearance (Cl)

Clearance (Cl) is the volume of plasma that cleared of a drug during a period of time.

At steady state the rate of clearance and rate of administration are equal.  Rate elimination = Rate administration

Cp(mg/L)*Cl(L/hr) = S*F*D(mg)/Tau

Cl(L/Hr)=S*F*D(mg) / (Cp(mg/L)*Tau)

Maintenance Dose(mg) = Cpssaverage(mg/L)*Tau(hours)*Cl(L/Hr) / (S*F)

Factors affecting clearance: weight, body surface area, plasma protein binding, extraction ratio, renal function, hepatic function, and cardiac output.

Dialyzability of Drugs

Elimination Rate Constant (K)

Drugs following first order-elimination kinetics have an elimination rate that is proportion to the drug concentration and the amount left in the body or plasma reduces logarithmically.

Then natural exponential of the negative of K is the fraction of drug still remaining in th body per unit of time (exp(-K*time)).

Cp(mg/L) = Cp(mg/L)initial * exp(-K(1/hours)*time(hours))

1st Order Elimination Rate Constant Example

One minus the natural exponential of the negative of K is the fraction of drug removed from the body per unit of time.

Amount removed per unit of time = 1- exp(-K(1/hours)*time(hours))

K is a dependent parameter and can be calculated from the independent parameters Vd and Cl.

K(1/hours) = Cl(L/Hr) / Vd(L) or

K(1/hours)= ln(Cp1/Cp2) / time between levels in hours

How K relates to half-life:

(ln(2/1)) / time between levels to decrease by 50% = K

Time between levels to decrease by 50% or T1/2 = (ln(2/1)) /K = 0.693/K

K = 0.693/T1/2 and is best thought of as a proportionality constant related to T1/2.

Half-life (T1/2)

The amount of time it takes for one-half of the drug in the body to be eliminated.

T1/2 =0.693 / K

T1/2= 0.693 *Vd/Cl

Constant Infusion

Cpss = Rate of Infusion(mg/hr) / Cl(L/hr)

Fraction of steady state achieved during infusion = 1-e(-KT')

Cp at time T' = (Rin(mg/hr) / Cl(L/hr) ) * (1-e(-KT')), T' = length of infusion in hours

Fraction of Steady State Achieved During Intermittent Infusion Example

Cp at a time after discontinuation of infusion

Cp2 = (S*F*Rin* (1-e(-KT')) / Cl ) *e(-KT2), where T2 is time post-infusion in hours

Cp During and After Intermittent Infusion Example

Bolus Dosing

Bolus dosing is when the dose is administered or absorbed very rapidly. Time of administration is much shorter than the  T1/2 of medication.  

Accumulation factor during intermittent dosing = 1/(1-e(-KTau))

Cpmaxss maximum steady-state concentration = S*F*D / (Vd*(1-e(-KTau)))

Cpminss minimum steady-state concentration = Cmaxss*e(-K(Tau))

Cpmax maximum after a series of doses, with set dose, and Tau, when N=number of doses given

Cpmax after dose N= S*F*D*(1-e(-N*K*Tau)) / (Vd*(1-e(-KTau)))

Intermittent Infusions

Accumulation factor during intermittent dosing = 1/(1-e(-KTau))

Accumulation Factor During Intermittent Dosing Example

Cpmaxss maximum steady-state concentration = S*F*D*(1-e(-KT')) / (Vd*K*T'*(1-e(-KTau)))

Cpminss minimum steady-state concentration = Cmaxss*e(-K(Tau-T'))

State State Levels Intermittent Infusions Example

Cpmax maximum after a series of doses, with set dose, Tau and T', when N=number of doses given

Cpmax (after N doses) = S*F*D*(1-e(-KT')) *(1-e(-N*K*Tau)) / (Vd*K*T'*(1-e(-KTau)))

Fraction of Steady State Achieved as Determined by Number of Doses Intermittent Infusion

Using the above intermittent infusion equation the effects of changes in Vd and Cl on serum levels is explored. Vd and Cl are independent parameters. K is dependent on Vd and Cl. The graphics demonstrate that changes in clearance have much greater impact of resulting serum levels than a proportional changes in Vd.

Effects of Vd on Serum Levels.  

Effects of Cl on Serum Levels.

AUC Calculation

one-compartment Model

AUC (mg*hour/Liter per Day) = ( (CminSS + CmaxSS)*(T'/2) + (Cpmaxss - Cminss)/K)*24/Tau

AUC (mg*hour/Liter per day) = Dose (mg) / Cl(L/hr) *(24/Tau)

Fraction of Steady State Achieved and AUC Calculation

Method of Superposition or calculating the Cp by summing the contribution to the Cp from individual doses

T1, T2, T3 are the time in hours from the end of infusion 1, 2, 3 respectively to time of level to be determined.

Note doses and dosing intervals do not need to be the same when using this method.

Cp1 = S*F*D1*(1-e(-KT')) *e(-K*T1) / (Vd*K*T')

Cp2 = S*F*D2*(1-e(-KT')) *e(-K*T2) / (Vd*K*T')

Cp3 = S*F*D3*(1-e(-KT')) *e(-K*T3) / (Vd*K*T')

Cpsum = Cp1 +Cp2 + Cp3

Method of Superposition

Method of superposition spreadsheet

Calculating Pharmacokinetic Parameters from Serum Levels

Intermittent Infusion Equations at Steady State

Determine K(1/hours) = (ln(Cmax(mg/L) / Cmin(mg/L)) / T(hours).  Draw both levels after a dose at steady state as accuracy in the K calculation is improved. The peak should be after the distribution phase for medications displaying multiple compartment kinetics. The trough is best drawn just before the next dose.

Cpmaxss =S*F*D (1-e(-KT'))/(Vd*K*T'(1-e(-K*Tau)))

Cpmax measured = S*F*D (1-e(-KT')) *e(-K*(time of level post-dose)) /(Vd*K*T'(1-e(-K*Tau)))

Vd (L) =  S*F*D (1-e(-KT')) *e(-K*(time of level post-dose)) /(Cpmax measured*K*T'(1-e(-K*Tau)))

Calculate a new dosage regimen

Tau (hours) = (ln(Cmaxss Desired/Cminss Desired)/K) + T'

D(mg)= Cpmaxss(Vd*K*T'(1-e(-K*Tau))) / (S*F* (1-e(-KT')))

Compare the K and Vd to population means and new dose and interval to the current dose and interval. If the values are drastically different the calculations or levels should be held in suspicion and levels should be repeated.

Ideal Body Weight (Devine Formula)

IBW(kg) Adult Males (18 years and older): 50 kg + 2.3 *(Height in inches above 60)

IBW(kg) Adult Female (18 years and older): 45.5 + 2.3*(Height in inches above 60)Fat Free Mass (Janmahasatian Formula):
FFM(kg) Adult Males = 9270 * Actual Body Weight(kg) / (668 + (216* BMI))
FFM(kg) Adult Females = 9270 * Actual Body Weight(kg) / (8780 / (244*(BMI))

Creatinine Clearance (ml/min)

Creatinine Clearance (mL/min)= (140-age(years))*(IBW or FFM) /(serum creatinine(mg/dl)*72)

Creatinine Clearnance (mL/min) = 0.85 * males


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