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.
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 or an equation relating creatinine clearance to K.
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.
K - As Vd and Clearance increase at the same rate with increasing weight the value of K is unaffected by weight. The example below assumes aminoglycoside clearance is the same as creatinine clearance.
| Weight (kg) | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | 130 | 140 | 150 | 160 | 170 |
| Age (years) | 70 | 70 | 70 | 70 | 70 | 70 | 70 | 70 | 70 | 70 | 70 | 70 | 70 |
| Creatinine Production (mg/hr) | 29.16667 | 35 | 40.83333 | 46.66667 | 52.5 | 58.33333 | 64.16667 | 70 | 75.83333 | 81.66667 | 87.5 | 93.33333 | 99.16667 |
| Scr (mg/dl) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Crcl (ml/min)=( Rin/60)/(Cp (mg/dl)/100) | 48.61111 | 58.33333 | 68.05556 | 77.77778 | 87.5 | 97.22222 | 106.9444 | 116.6667 | 126.3889 | 136.1111 | 145.8333 | 155.5556 | 165.2778 |
| Crcl L/hr | 2.916667 | 3.5 | 4.083333 | 4.666667 | 5.25 | 5.833333 | 6.416667 | 7 | 7.583333 | 8.166667 | 8.75 | 9.333333 | 9.916667 |
| Vd (L) | 12.5 | 15 | 17.5 | 20 | 22.5 | 25 | 27.5 | 30 | 32.5 | 35 | 37.5 | 40 | 42.5 |
| K (1/hr) | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 | 0.233333 |
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 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)
Males:
Creatinine Clearance (mL/min)= (140-age(years))*(IBW
or FFM) /(serum creatinine(mg/dl)*72)
Females
Creatinine Clearnance (mL/min) = 0.85 * males
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