(1) Derive imipenem concentration at an infusion time after
multiple infusions.
When Tin <t≤τ , the steady state concentration can
be assessed by formula (1):
\(C=\frac{\text{Dose}}{\text{kV}T_{\text{in}}}\bullet\frac{1-e^{-k\bullet Tin}}{1-e^{-k\bullet\tau}}\bullet e^{-k(t-Tin)}\)(1)
where Tin is the time of infusion; C, the plasma concentration;t , the time after-dose; τ , dosing interval; k , the
elimination rate constant; and V , apparent distribution volume.
When 0≤t≤Tin , the imipenem plasma concentration at time t after
the Mth dosage was the summed concentration from the first dosage to the
Nth dosage. The concentration at time t after the Mth dosage can be
calculated by (2)(3)(4):
\(C=\frac{\text{Dose}}{\text{kV}T_{\text{in}}}\bullet\left(1-e^{-k\bullet t}\right)+C1+C2\cdots+C_{N}\)(2)
\(Cn=Cmax\bullet e^{-k(n\tau-Tin+t)}\) (3)
Then,
\(C=\frac{\text{Dose}}{\text{kV}T_{\text{in}}}\bullet\left(1-e^{-k\bullet t}\right)+\frac{\text{Dose}}{\text{kV}T_{\text{in}}}\bullet\left(1-e^{-k\bullet
Tin}\right)\bullet e^{-kt}\bullet e^{k\bullet Tin}\sum_{n=1}^{N}e^{-nk\tau}\)(4)
where M is the number of times doses were given and N is the number of
times doses were given after steady state was reached.
In formula (4),
1+\(\sum_{n=1}^{N}e^{-\text{nkτ}}\)=\(\frac{1-e^{-nk\bullet\tau}}{1-e^{-k\bullet\tau}}\)
When n→∞, then \(e^{-nk\bullet\tau}\)→0
Therefore, we conclude the following:
\(\sum_{n=1}^{N}e^{-nk\tau}=\frac{e^{-k\tau}}{1-e^{-k\tau}}\)(5)
Applying formula (5) to (4), the concentration at the time of infusion
at steady state could be assessed by formula (6):
\(C=\frac{\text{Dose}}{\text{kV}T_{\text{in}}}\bullet\left(1-e^{-k\bullet t}\right)+\frac{\text{Dose}}{\text{kV}T_{\text{in}}}\bullet\left(e^{k\bullet Tin}-1\right)\bullet e^{-kt}\bullet\frac{e^{-k\bullet\tau}}{1-e^{-k\bullet\tau}}\)(6)
(2) CalculatefT>MICafter multiple infusions.
When the free plasma concentration equals the MIC,f T>MIC is the D-value of
te_MIC (in elimination time) and
tin_MIC (in infusion time).
Take \(e^{-kt}\)from (6),
\(e^{-kt}=\frac{1-e^{-k\tau}}{e^{k\bullet(Tin-\tau)}-1}\bullet(\frac{ckV\bullet
Tin}{\text{Dose}}-1)\)(7)
The formula for elimination time was:
\(e^{-kt}=\frac{1-e^{-k\tau}}{e^{k\bullet Tin}-1}\bullet\frac{ckV\bullet Tin}{\text{Dose}}\)(8)
Take (7)/(8),
\(e^{k(te\_MIC-t\text{in}\_MIC)}=\frac{e^{k\text{Tin}}-1}{e^{k\bullet(Tin-\tau)}-1}\bullet(1-\frac{\text{Dose}}{\text{fu}\bullet\text{MIC}\bullet kV\bullet Tin}\)(9)
f T>MIC can be assessed by the
following formula:
ƒT>MIC=\(\frac{1}{k}\bullet ln\frac{(e^{\text{kTin}}-1)\bullet(1-\frac{fu\bullet\text{Dose}}{\text{MIC}\bullet kV\bullet Tin})}{e^{k\bullet(Tin-\tau)}-1}\ \)(10)