Pharmacokinetics

“What the body does to the drug”

Table 1: Pharmacokinetic symbols

Symbol	Description
$\mathrm{Dose}$	Dose
$\tau$	Dosing interval (interdose interval)
$\mathrm{CL}$	Clearance
${\mathrm{V}}_{\mathrm{d}}$	Volume of distribution
${\mathrm{k}}_{\mathrm{e}}$	Elimination rate constant
${\mathrm{k}}_{\mathrm{a}}$	Absorption rate constant
$\mathrm{F}$	Bioavailability
${\mathrm{R}}_{\mathrm{in}}$	Infusion rate
$T_{\text{inf}}$	Duration of infusion
${\mathrm{C}}_{\mathrm{pl}}$	Plasma concentration
${\mathrm{R}}_{\mathrm{ac}}$	Accumulation ratio

Intravenous bolus

Single i.v. dose

Initial concentration Equation 1.

$$\begin{array}{}\text{(1)}& {\mathrm{C}}_0=\frac{\mathrm{Dose}}{{\mathrm{V}}_{\mathrm{d}}}\end{array}$$

Plasma concentration Equation 2.

$$\begin{array}{}\text{(2)}& \mathrm{C}(t)={\mathrm{C}}_0\times e^{-{\mathrm{k}}_{\mathrm{e}}\times t}\end{array}$$

Multiple i.v. doses

$${\mathrm{C}}_n(t)=\frac{\mathrm{Dose}}{{\mathrm{V}}_{\mathrm{d}}}\times \frac{(1-e^{-n\times {\mathrm{k}}_{\mathrm{e}}\times \tau })}{(1-e^{-{\mathrm{k}}_{\mathrm{e}}\times \tau })}\times e^{-{\mathrm{k}}_{\mathrm{e}}\times t}$$

at peak: $t=0$; at steady state $\mathrm{n}\to \mathrm{\infty }$; at trough: $t=\tau$

$$\begin{array}{}\text{(3)}& {\mathrm{C}}_{\text{max, ss}}=\frac{\mathrm{Dose}}{\mathrm{V}}\times \frac{1}{(1-e^{-{\mathrm{k}}_{\mathrm{e}}\tau })}\end{array}$$

$$\begin{array}{}\text{(4)}& {\mathrm{C}}_{\text{min, ss}}={\mathrm{C}}_{\text{max, ss}}\times e^{-{\mathrm{k}}_{\mathrm{e}}\times \tau }\end{array}$$

Plasma concentration Equation 5.

$$\begin{array}{}\text{(5)}& \mathrm{C}(t)={\mathrm{C}}_{\max }\times e^{-{\mathrm{k}}_{\mathrm{e}}\times t}\end{array}$$

Peak Equation 6.

$$\begin{array}{}\text{(6)}& {\mathrm{C}}_{\max }=\frac{{\mathrm{C}}_0}{1-e^{-{\mathrm{k}}_{\mathrm{e}}\times \tau }}\end{array}$$

Trough Equation 7.

$$\begin{array}{}\text{(7)}& {\mathrm{C}}_{\min }={\mathrm{C}}_{\max }\times e^{-{\mathrm{k}}_{\mathrm{e}}\times \tau }\end{array}$$

Average concentration (steady state) Equation 8.

$$\begin{array}{}\text{(8)}& {\overline{\mathrm{C}}}_{\text{pl, ss}}=\frac{\mathrm{Dose}}{\mathrm{CL}\times \tau }\end{array}$$

Oral administration

Single p.o. dose

Plasma concentration Equation 9.

$$\begin{array}{}\text{(9)}& \mathrm{C}=\frac{\mathrm{F}\times \mathrm{Dose}\times {\mathrm{k}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{d}}({\mathrm{k}}_{\mathrm{a}}-{\mathrm{k}}_{\mathrm{e}})}(e^{-{\mathrm{k}}_{\mathrm{e}}\times t}-e^{-{\mathrm{k}}_{\mathrm{a}}\times t})\end{array}$$

Time of maximum concentration Equation 10.

$$\begin{array}{}\text{(10)}& t_{\text{max}}=\frac{\ln \left({\mathrm{k}}_{\mathrm{a}}/{\mathrm{k}}_{\mathrm{e}}\right)}{{\mathrm{k}}_{\mathrm{a}}-{\mathrm{k}}_{\mathrm{e}}}\end{array}$$

Multiple p.o. doses

Plasma concentration Equation 11.

$$\begin{array}{}\text{(11)}& \mathrm{C}=\frac{\mathrm{F}\times \mathrm{Dose}\times {\mathrm{k}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{d}}({\mathrm{k}}_{\mathrm{a}}-{\mathrm{k}}_{\mathrm{e}})}(\frac{e^{-{\mathrm{k}}_{\mathrm{e}}\times t}}{1-e^{-{\mathrm{k}}_{\mathrm{e}}\times \tau }}-\frac{e^{-{\mathrm{k}}_{\mathrm{a}}\times t}}{1-e^{-{\mathrm{k}}_{\mathrm{a}}\times \tau }})\end{array}$$

Time of maximum concentration Equation 12.

$$\text{t\_text{max} = frac{ ln left( frac{ ka times left(1 - e^{-ke times ii} right) }{ ke times left(1 - e^{-ka times ii} right) } right) }{ left(ka - ke}right) } tag{12}}$$

Average concentration (steady state) Equation 13.

$$\begin{array}{}\text{(13)}& \overline{\mathrm{C}}=\frac{\mathrm{F}\times \mathrm{Dose}}{\mathrm{CL}\times \tau }\end{array}$$

Clearance

$$\begin{array}{}\text{(14)}& \mathrm{CL}=\frac{\mathrm{Dose}\times \mathrm{F}}{{\mathrm{AUC}}_{0\mathrm{-}\mathrm{\infty }}}={\mathrm{k}}_{\mathrm{e}}\times {\mathrm{V}}_{\mathrm{d}}\end{array}$$

Intravenous infusion

$${\mathrm{R}}_{\mathrm{in}}=\frac{\mathrm{Dose}}{T}$$

Plasma concentration (steady state) Equation 15

$$\begin{array}{}\text{(15)}& {\mathrm{C}}_{\mathrm{pl},\mathrm{ss}}=\frac{{\mathrm{R}}_{\mathrm{in}}}{\mathrm{CL}}\end{array}$$

Plasma concentration (during infusion) Equation 16

$$\begin{array}{}\text{(16)}& \mathrm{C}(t)=\frac{{\mathrm{R}}_{\mathrm{in}}}{\mathrm{CL}}(1-e^{-{\mathrm{k}}_{\mathrm{e}}\times t})\end{array}$$

Calculated clearance (Chiou equation) Equation 17

$$\begin{array}{}\text{(17)}& \mathrm{CL}=\frac{2{\mathrm{R}}_{\mathrm{in}}}{({\mathrm{C}}_1+{\mathrm{C}}_2)}+\frac{2{\mathrm{V}}_{\mathrm{d}}({\mathrm{C}}_1-{\mathrm{C}}_2)}{({\mathrm{C}}_1+{\mathrm{C}}_2)(t_2-t_1)}\end{array}$$

Single infusion

Since $\tau =t$ for ${\mathrm{C}}_{\max }$

Peak Equation 18

$$\begin{array}{}\text{(18)}& {\mathrm{C}}_{\max }=\frac{{\mathrm{R}}_{\mathrm{in}}}{\mathrm{CL}}(1-e^{-{\mathrm{k}}_{\mathrm{e}}T})\end{array}$$

Trough Equation 19

$$\begin{array}{}\text{(19)}& {\mathrm{C}}_{\min }={\mathrm{C}}_{\max }\times e^{-{\mathrm{k}}_{\mathrm{e}}(\tau -T)}\end{array}$$

Multiple infusions

Peak Equation 20.

$$\begin{array}{}\text{(20)}& {\mathrm{C}}_{\max }=\frac{{\mathrm{R}}_{\mathrm{in}}}{\mathrm{CL}}\times \frac{(1-e^{-{\mathrm{k}}_{\mathrm{e}}\times T})}{(1-e^{-{\mathrm{k}}_{\mathrm{e}}\times \tau })}\end{array}$$

Trough Equation 21.

$$\begin{array}{}\text{(21)}& {\mathrm{C}}_{\min }={\mathrm{C}}_{\max }\times e^{-{\mathrm{k}}_{\mathrm{e}}(\tau -T)}\end{array}$$

Calculated parameters

Calculated elimination rate constant (1-compartment case) Equation 22.
With C^*_max = measured peak and C^*min = measured trough, measured over the time interval $\mathrm{\Delta }t$ $$\begin{array}{}\text{(22)}& {\mathrm{k}}_{\mathrm{e}}=\frac{\ln \left(\frac{{{\mathrm{C}}_{\max }}^{\ast }}{{{\mathrm{C}}_{\min }}^{\ast }}\right)}{\mathrm{\Delta }t}\end{array}$$

Calculated peak Equation 23.
With C^*_max = measured peak, measured at time t^* after the end of the infusion

$$\begin{array}{}\text{(23)}& {\mathrm{C}}_{\max }=\frac{{{\mathrm{C}}_{\max }}^{\ast }}{e^{-{\mathrm{k}}_{\mathrm{e}}{t}^{\ast }}}\end{array}$$

Calculated trough Equation 24.
With C^*min = measured trough, measured at time t^* before the start of the next infusion

$$\begin{array}{}\text{(24)}& {\mathrm{C}}_{\min }={{\mathrm{C}}_{\min }}^{\ast }\times e^{-{\mathrm{k}}_{\mathrm{e}}\times t^{\ast }}\end{array}$$

Calculated volume of distribution Equation 25.

$$\begin{array}{}\text{(25)}& {\mathrm{V}}_{\mathrm{d}}=\frac{{\mathrm{R}}_{\mathrm{in}}}{{\mathrm{k}}_{\mathrm{e}}}\times \frac{(1-e^{-{\mathrm{k}}_{\mathrm{e}}\times T})}{[{\mathrm{C}}_{\max }-({\mathrm{C}}_{\min }\times e^{-{\mathrm{k}}_{\mathrm{e}}\times T})]}\end{array}$$

Calculated recommended dosing interval for infusion start Equation 26.

$$\begin{array}{}\text{(26)}& \tau =\frac{\ln \left(\frac{{\mathrm{C}}_{\text{max, desired}}}{{\mathrm{C}}_{\text{min, desired}}}\right)}{{\mathrm{k}}_{\mathrm{e}}}+T\end{array}$$

Calculated recommended dose Equation 27.

$$\begin{array}{}\text{(27)}& \mathrm{Dose}={\mathrm{C}}_{\text{max, desired}}\times {\mathrm{k}}_{\mathrm{e}}\times \mathrm{V}\times T\frac{(1-e^{-{\mathrm{k}}_{\mathrm{e}}\tau })}{(1-e^{-{\mathrm{k}}_{\mathrm{e}}T})}\end{array}$$

Two-compartment PK model

$$\begin{array}{}\text{(28)}& \mathrm{C}=a\times e^{-\alpha t}+b\times e^{-\rho t}\end{array}$$

$$\begin{array}{}\text{(29)}& {\mathrm{AUC}}_{0\mathrm{-}\mathrm{\infty }}=a/\alpha +b/\beta \end{array}$$

$$\begin{array}{}\text{(30)}& {\mathrm{V}}_{\text{d, area}}>{\mathrm{V}}_{\text{ss}}>{\mathrm{V}}_{\text{c}}\end{array}$$

Vd_area = V beta?

Dosage regimens

$${\mathrm{C}}_{\text{inf}}={\mathrm{C}}_{\text{ss}}\times (1-e^{-{\mathrm{k}}_{\mathrm{e}}t})$$ (1-compartment)

$${\mathrm{C}}_{\text{ss, avg}}=\mathrm{F}\times \mathrm{Dose}/(\mathrm{CL}\times \tau )$$ (variation of eq. 24)

$${\mathrm{R}}_{\mathrm{ac}}=1/(1-e^{-{\mathrm{k}}_{\mathrm{e}}\tau })$$ (1-compartment)

$${\mathrm{C}}_{\text{ss, max}}=\mathrm{F}\times \mathrm{Dose}\times {\mathrm{R}}_{\mathrm{ac}}/\mathrm{V}$$ (1-compartment)

$${\mathrm{C}}_{\text{ss, min}}={\mathrm{C}}_{\text{ss, max}}\times e^{-k\times \tau }$$ (1-compartment, combination of eq. 22 and 28)

$${\tau }_{\text{max}}=\ln \left({\mathrm{C}}_{\text{ss, max}}/{\mathrm{C}}_{\text{ss, min}}\right)/{\mathrm{k}}_{\mathrm{e}}$$ (1-compartment, variation of eq. 23)

$$\text{Corresponding dose}\times \mathrm{F}=\mathrm{V}\times ({\mathrm{C}}_{\text{max, ss}}-{\mathrm{C}}_{\text{min, ss}})$$ (1-compartment)

Exposure (AUC)

$${\mathrm{AUC}}_{\text{iv}}={\mathrm{C}}_0/{\mathrm{k}}_{\mathrm{e}}$$ (1-compartment)

$${\mathrm{AUC}}_{\text{iv}}={\mathrm{C}}_1/{\lambda }_1+{\mathrm{C}}_2/{\lambda }_2$$ (2-compartment)

$$\mathrm{AUC}=\mathrm{\Sigma }(\text{trapezoids})+{\mathrm{C}}_{\text{last}}/{\mathrm{k}}_{\mathrm{e}}$$ (Always applicable)

Ideal Body Weight

Male

IBW = 50 kg + 2.3 kg for each inch over 5ft in height

Female

IBW = 45.5 kg + 2.3 kg for each inch over 5ft in height

Obese

ABW = IBW + 0.4 * (TBW-IBW)