The fraction (F) of the administered dose that reaches systemic circulation. Factors affecting bioavailability 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 dosenew dosage form = Dosecurrent dosage form × Fcurrent dosage form / Fnew dosage form
S is the fraction of the administered dose that is active drug. Drugs that are salts or need to be converted to active forms will have an S of less than 1.
Equivalent dosenew dosage form = Dosecurrent dosage form × S × Fcurrent dosage form / S × Fnew dosage form
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: LD (mg) = (Cp desired (mg/L) − Cp initial (mg/L)) × Vd (L)
α = free drug concentration in plasma / total drug concentration in plasma
α = Cp free (mg/L) / (Cp bound (mg/L) + Cp free (mg/L))
Cp equivalent for normal protein binding (mg/L) = Cp of patient (mg/L) / ((1−α) × (albuminpatient (g/dL) / albuminnormal (g/dL)) + α)
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 τ
Intermittent dosing (mg/hour) = S × F × Dose (mg) / τ
Clearance (Cl) is the volume of plasma that is 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) / τ
Cl (L/hr) = S × F × D (mg) / (Cp (mg/L) × τ)
Maintenance Dose (mg) = Cpss average (mg/L) × τ (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.
Drugs following first order-elimination kinetics have an elimination rate that is proportional to the drug concentration and the amount left in the body or plasma reduces logarithmically.
The natural exponential of the negative of K is the fraction of drug still remaining in the body per unit of time (exp(−K×time)).
Cp (mg/L) = Cp initial (mg/L) × 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)
K (1/hours) = ln(Cp1/Cp2) / time between levels in hours
(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 and Weight: 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) | 29.17 | 35 | 40.83 | 46.67 | 52.5 | 58.33 | 64.17 | 70 | 75.83 | 81.67 | 87.5 | 93.33 | 99.17 |
| SCr (mg/dL) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| CrCl (mL/min) | 48.61 | 58.33 | 68.06 | 77.78 | 87.5 | 97.22 | 106.94 | 116.67 | 126.39 | 136.11 | 145.83 | 155.56 | 165.28 |
| CrCl (L/hr) | 2.92 | 3.5 | 4.08 | 4.67 | 5.25 | 5.83 | 6.42 | 7 | 7.58 | 8.17 | 8.75 | 9.33 | 9.92 |
| 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) | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 |
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
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')), where T' = length of infusion in hours
Fraction of Steady State Achieved During Intermittent Infusion Example
Cp2 = (S × F × Rin × (1 − e(−KT')) / Cl) × e(−KT2), where T2 is time post-infusion in hours
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(−Kτ))
Cpmax ss maximum steady-state concentration = S × F × D / (Vd × (1 − e(−Kτ)))
Cpmin ss minimum steady-state concentration = Cmax ss × e(−K(τ))
With set dose and τ, when N = number of doses given:
Cpmax after dose N = S × F × D × (1 − e(−N×K×τ)) / (Vd × (1 − e(−Kτ)))
Accumulation factor during intermittent dosing = 1 / (1 − e(−Kτ))
Accumulation Factor During Intermittent Dosing Example
Cpmax ss maximum steady-state concentration = S × F × D × (1 − e(−KT')) / (Vd × K × T' × (1 − e(−Kτ)))
Cpmin ss minimum steady-state concentration = Cmax ss × e(−K(τ−T'))
Steady State Levels Intermittent Infusions Example
With set dose, τ and T', when N = number of doses given:
Cpmax (after N doses) = S × F × D × (1 − e(−KT')) × (1 − e(−N×K×τ)) / (Vd × K × T' × (1 − e(−Kτ)))
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 on resulting serum levels than proportional changes in Vd.
AUC (mg×hour/Liter per Day) = ((Cmin SS + Cmax SS) × (T'/2) + (Cpmax ss − Cmin ss)/K) × 24/τ
AUC (mg×hour/Liter per day) = Dose (mg) / Cl (L/hr) × (24/τ)
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
Step 1: Determine K
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.
Step 2: Calculate Vd
Cpmax ss = S × F × D × (1 − e(−KT')) / (Vd × K × T' × (1 − e(−Kτ)))
Cpmax measured = S × F × D × (1 − e(−KT')) × e(−K×(time of level post-dose)) / (Vd × K × T' × (1 − e(−Kτ)))
Vd (L) = S × F × D × (1 − e(−KT')) × e(−K×(time of level post-dose)) / (Cpmax measured × K × T' × (1 − e(−Kτ)))
τ (hours) = (ln(Cmax ss Desired / Cmin ss Desired) / K) + T'
D (mg) = Cpmax ss × (Vd × K × T' × (1 − e(−Kτ))) / (S × F × (1 − e(−KT')))
Important: 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.
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)
FFM (kg) Adult Males = 9270 × Actual Body Weight (kg) / (6680 + (216 × BMI))
FFM (kg) Adult Females = 9270 × Actual Body Weight (kg) / (8780 + (244 × BMI))
Creatinine Clearance (mL/min) = (140 − age (years)) × (IBW or FFM) / (serum creatinine (mg/dL) × 72)
Creatinine Clearance (mL/min) = 0.85 × males