Table 2.

Exact solutions to pertinent single-compartment and two-compartment models under equi-dosing regimen inputs

Singl-compartment IV equi-bolus dosing (M1,Beq):

$\begin{array}{c}{a}_c(t)=F{D}_0{\sum }_{j=1}^{M}H(t_j)e^{-k_e{t}_j},\hfill \end{array}$       (3.1)

$\begin{array}{c}{a}_c^M(t_M)=F{D}_0\left(\frac{1-e^{-M{k}_{e}T}}{1-e^{-k_{e}T}}\right)e^{-k_e{t}_M},\text{for}0\le t_M<T,\hfill \end{array}$       (3.2)

$\begin{array}{c}{a}_c^{\infty }(t_{\infty })=\left(\frac{F{D}_0}{1-e^{-k_{e}T}}\right)e^{-k_e{t}_{\infty }},\text{for}0\le t_{\infty }<T.\hfill \end{array}$      (3.3)

Single-compartment IV equi-bolus dosing with loading dose (M1,BeqL), with $a_c^{\infty }(t_{\infty })$ given by (3.3):

$\begin{array}{c}{a}_c(t)=F\left\{D_0,\left({\sum }_{j=1}^M,H,(,t_j,),e^{-k_e{t}_j}\right),+,(D_L-D_0),e^{-k_{e}t}\right\},\hfill \end{array}$      (3.4)

$\begin{array}{c}{a}_c^M(t_M)=F\left\{D_0,\left(\frac{1-e^{-M{k}_{e}T}}{1-e^{-k_{e}T}}\right),e^{-k_e{t}_M},+,(D_L-D_0),e^{-k_e(t_M+(M-1)T)}\right\}.\hfill \end{array}$      (3.5)

Single-compartment IV equi-infusion dosing (M1,Ieq):

$\begin{array}{c}{a}_c(t)=\frac{F{k}_{\mathit{in}}}{k_e}{\sum }_{j=1}^{M}H(t_j)(1-e^{-k_e{t}_j})-H(t_j-t_f)(1-e^{-k_e(t_j-t_f)}),\hfill \end{array}$      (3.6)

$\begin{array}{c}{a}_c^M(t_M)=\frac{F{k}_{\mathit{in}}}{k_e}\left\{(1-e^{-k_e{t}_M}),-,H,(t_M-t_f),(1-e^{-k_e(t_M-t_f)}),+,(e^{k_e{t}_f}-1),\left(\frac{e^{-M{k}_{e}T}-e^{-k_{e}T}}{e^{-k_{e}T}-1}\right),e^{-k_e{t}_M}\right\}.\hfill \end{array}$      (3.7)

$\begin{array}{c}{a}_c^{\infty }(t_{\infty })=\frac{F{k}_{\mathit{in}}}{k_e}\left\{1,-,\frac{e^{k_e{t}_f}-e^{k_{e}T}}{1-e^{k_{e}T}},e^{-k_e{t}_{\infty }},-,H,(t_{\infty }-t_f),(1-e^{-k_e(t_{\infty }-t_f)})\right\},\text{for}0\le t_{\infty }<T.\hfill \end{array}$       (3.8)

Two-compartment oral equi-bolus dosing (M2,Beq):

$\begin{array}{c}{a}_c(t)=\frac{k_a}{k_e-k_a}F{D}_0{\sum }_{j=1}^{M}H(t_j)\left[e^{-k_a{t}_j},-,e^{-k_e{t}_j}\right],\hfill \end{array}$      (3.9)

$\begin{array}{c}{a}_c^M(t_M)=\frac{k_a}{k_e-k_a}F{D}_0\left[\left(\frac{1-e^{-M{k}_{a}T}}{1-e^{-k_{a}T}}\right),e^{-k_a{t}_M},-,\left(\frac{1-e^{-M{k}_{e}T}}{1-e^{-k_{e}T}}\right),e^{-k_e{t}_M}\right],\hfill \end{array}$       (3.10)

$\begin{array}{c}{a}_c^{\infty }(t_{\infty })=\frac{k_a}{k_e-k_a}F{D}_0\left[\frac{e^{-k_a{t}_{\infty }}}{1-e^{-k_{a}T}},-,\frac{e^{-k_e{t}_{\infty }}}{1-e^{-k_{e}T}}\right].\hfill \end{array}$      (3.11)

Two-compartment oral equi-bolus dosing with loading dose (M2,BeqL), with $a_c^{\infty }(t_{\infty })$ given by (3.11):

$\begin{array}{c}{a}_c(t)=\frac{k_a}{k_e-k_a}F\left\{D_0,{\sum }_{j=1}^M,H,(,t_j,),\left[e^{-k_a{t}_j},-,e^{-k_e{t}_j}\right],+,(D_L-D_0),\left[e^{-k_{a}t},-,e^{-k_{e}t}\right]\right\},\hfill \end{array}$      (3.12)

$\begin{array}{c}{a}_c^M(t_M)=\frac{k_{a}F}{k_e-k_a}\{D_0\left[\left(\frac{1-e^{-M{k}_{a}T}}{1-e^{-k_{a}T}}\right),e^{-k_a{t}_M},-,\left(\frac{1-e^{-M{k}_{e}T}}{1-e^{-k_{e}T}}\right),e^{-k_e{t}_M}\right]\hfill \\ +(D_L-D_0)\left[e^{-k_a(t_M+(M-1)T)},-,e^{-k_e(t_M+(M-1)T)}\right]\}.\hfill \end{array}$       (3.13)

Models and dosing inputs are as in Table 1, H is the Heaviside function, and $t_j$ is given by (3.14)