Worked Dose Selection for Time Above MIC

Why Calculate Time Above MIC Patient by Patient?

For antimicrobials whose activity is related to time above the minimum inhibitory concentration (MIC), a useful dose-selection quantity is the fraction of a dosing interval during which the concentration exceeds the patient's MIC. This is an individual-level quantity: two patients given the same regimen can have different fractions because their pharmacokinetic parameters or MICs differ. A population summary, such as the proportion of simulated patients meeting a target, then helps compare candidate regimens.

This tutorial works through a deliberately simple concentration model so that every crossing time can be calculated and checked. The numerical patients and decision thresholds are hypothetical and are not clinical recommendations. In a realistic population simulation, PK parameters, covariates, random effects, and MIC values should be generated or sampled according to the intended population and design.

Definition: For patient $i$, dose interval $\tau$, concentration profile $C_i(t)$, and MIC $M_i$, the individual fraction of the interval above MIC is $$ fT>MIC_i=\frac{1}{\tau}\int_0^\tau I\{C_i(t)>M_i\}\,dt, $$ where $I\{\cdot\}$ equals 1 when the condition is true and 0 otherwise. For a target fraction $q$, the simulated probability of target attainment (PTA) is the proportion of simulated patients for whom $fT>MIC_i\ge q$.

The time fraction is between 0 and 1. For example, $fT>MIC_i=0.60$ means the simulated concentration for that patient is above the patient's MIC for 60% of the interval. A PTA calculated from $N$ simulated patients is an empirical proportion, not a guarantee about every patient in the intended population.

Concentration Profile and Crossing Times

Assume a one-compartment model with linear elimination, constant clearance $CL_i$, volume $V_i$, and elimination rate $k_i=CL_i/V_i$. The drug is infused at a constant rate over $T_{\mathrm{inf}}$ hours, once every $\tau$ hours. Assume dosing has continued long enough to reach steady state, the MIC is constant over the interval, and the concentration at the site represented by the model is the relevant concentration for the target. These assumptions make it possible to calculate exact crossing times; they are not universally appropriate for every drug or infection site.

For a given patient, write $R_0=D/T_{\mathrm{inf}}$ for the infusion rate, where $D$ is the dose. At steady state, let $C_{\max}$ be the concentration at the end of infusion and $C_{\min}$ the concentration immediately before the next infusion. With time $t=0$ at the start of infusion, the concentrations during and after infusion are:

C_{\max}=\frac{R_0}{CL}\frac{1-e^{-kT_{\mathrm{inf}}}}{1-e^{-k\tau}}, \qquad C_{\min}=C_{\max}e^{-k(\tau-T_{\mathrm{inf}})},

C(t)= \begin{cases} C_{\min}e^{-kt}+\dfrac{R_0}{CL}(1-e^{-kt}), & 0\le t\le T_{\mathrm{inf}},\\[6pt] C_{\max}e^{-k(t-T_{\mathrm{inf}})}, & T_{\mathrm{inf}}<t\le\tau. \end{cases}

If $C_{\min}<M<C_{\max}$, concentration crosses the MIC once while rising during infusion and once while declining after infusion. Solving the infusion equation $C(t)=M$ gives the rising crossing time; solving the elimination equation gives the descending crossing time:

t_{\mathrm{up}}=-\frac{1}{k}\ln\left(\frac{R_0/CL-M}{R_0/CL-C_{\min}}\right), \qquad t_{\mathrm{down}}=T_{\mathrm{inf}}+\frac{1}{k}\ln\left(\frac{C_{\max}}{M}\right).

The fraction above MIC is then $(t_{\mathrm{down}}-t_{\mathrm{up}})/\tau$, provided these crossings lie in the interval. If $C_{\max}\le M$, the fraction is zero, apart from equality at an isolated time, which contributes no duration. If $C_{\min}\ge M$, the fraction is one. In the remaining crossing case, use the two crossing times. Equality at a single crossing point does not change the integral definition.

Worked Calculation for One Simulated Patient

Worked Example: A 1,000 mg Dose Every 8 Hours

Consider one hypothetical patient receiving a 1,000 mg IV infusion over 1 hour every 8 hours. Suppose $CL=25$ L/h, $V=50$ L, and $M=1$ mg/L. The model assumptions above are stipulated for this calculation: linear one-compartment PK, constant parameters, and steady state. The infusion rate is $R_0=1{,}000/1=1{,}000$ mg/h and $k=25/50=0.5$ h$^{-1}$.

First calculate the end-infusion concentration:

C_{\max} =\frac{1000\ \mathrm{mg/h}}{25\ \mathrm{L/h}} \frac{1-e^{-(0.5\ \mathrm{h}^{-1})(1\ \mathrm{h})}} {1-e^{-(0.5\ \mathrm{h}^{-1})(8\ \mathrm{h})}} =40\ \mathrm{mg/L}\times\frac{0.393469}{0.981684} \approx16.03\ \mathrm{mg/L}.

The pre-infusion concentration is $C_{\min}=16.03e^{-(0.5)(7)}\approx0.4841$ mg/L. Thus $C_{\min}<1<C_{\max}$, confirming that the profile crosses the MIC once during infusion and once during decline. The infusion asymptote $R_0/CL$ is 40 mg/L, which is also greater than the MIC, so the rising-crossing expression is valid.

Substitute into the crossing equations:

t_{\mathrm{up}} =-\frac{1}{0.5}\ln\left(\frac{40-1}{40-0.4841}\right) \approx0.0263\ \mathrm{h}, \qquad t_{\mathrm{down}} =1+\frac{1}{0.5}\ln\left(\frac{16.03}{1}\right) \approx6.549\ \mathrm{h}.

The above-MIC duration is approximately $6.549-0.0263=6.523$ h, so $fT>MIC=6.523/8\approx0.8154$. Under the stipulated model, this patient's concentration is above 1 mg/L for about 81.5% of the 8-hour interval. This is a model-based individual calculation, not a claim about a real patient.

Comparing Regimens Across a Simulated Set

A dose-selection comparison requires both an individual target and a population target. For illustration, define success as $fT>MIC_i\ge0.50$ for a simulated patient, and compare the empirical PTA with a hypothetical requirement of at least 80%. Neither threshold is asserted to be a universal antimicrobial target. To keep the arithmetic transparent, use five hypothetical patients with the same stipulated $CL=25$ L/h and $V=50$ L, but different MICs. This isolates the effect of MIC; it does not represent all relevant sources of patient variability.

Worked Example: Comparing Dose and Interval Candidates

The five simulated MIC values are 0.5, 1, 2, 4, and 8 mg/L. For each candidate, calculate the steady-state profile and each patient's crossing times as above, then test the individual fraction against 0.50. The following rounded fractions result; values at least 0.50 meet the hypothetical individual target.

Regimen	MIC 0.5	MIC 1	MIC 2	MIC 4	MIC 8	PTA for $fT>MIC\ge0.50$
1,000 mg every 8 h	0.992	0.815	0.636	0.449	0.246	3/5 = 60%
1,000 mg every 12 h	0.657	0.539	0.419	0.295	0.160	2/5 = 40%
2,000 mg every 8 h	1.000	0.992	0.815	0.636	0.449	4/5 = 80%

For example, for 1,000 mg every 8 hours and MIC 4 mg/L, the same profile has $C_{\min}\approx0.4841$ mg/L and $C_{\max}\approx16.03$ mg/L. Substitution gives $t_{\mathrm{up}}=-2\ln[(40-4)/(40-0.4841)]\approx0.1864$ h and $t_{\mathrm{down}}=1+2\ln(16.03/4)\approx3.777$ h. Thus $fT>MIC\approx(3.777-0.1862)/8=0.449$, below 0.50.

For 2,000 mg every 8 hours, concentrations double under the linear PK assumption: $C_{\max}\approx32.06$ mg/L and $C_{\min}\approx0.9683$ mg/L, and the infusion asymptote $R_0/CL$ is 80 mg/L. For MIC 4 mg/L, $t_{\mathrm{up}}=-2\ln[(80-4)/(80-0.9683)]\approx0.0782$ h and $t_{\mathrm{down}}=1+2\ln(32.06/4)\approx5.163$ h. Therefore $fT>MIC\approx(5.163-0.0783)/8=0.636$. This patient now meets the 0.50 target; the patient with MIC 8 mg/L still does not, with a fraction of approximately 0.449.

Under this five-patient illustration, the 2,000 mg every-8-hour candidate reaches the stipulated 80% empirical PTA threshold, while the other two candidates do not. The result is only a comparison within this constructed set. Five simulations provide a coarse estimate, and the example omits potentially important PK variability, clinical constraints, and uncertainty in model inputs.

Why Interval and Dose Both Matter

In this example, shortening the interval from 12 to 8 hours at the same dose increased the calculated fractions, particularly for the higher MICs. Increasing the dose at the same interval also increased the fractions because the assumed PK is linear. Those patterns follow from the specified concentration profiles, but they should not be turned into a general claim that one regimen is clinically preferable. Dose and interval can also affect peak or trough concentrations, feasibility, toxicity, and other endpoints.

The crossing-time calculation is an analytic alternative to evaluating concentration on a time grid. Analytic crossings are efficient and avoid errors caused by a grid that is too coarse, but require a profile with solvable crossings and appropriate assumptions. For complex models, nonlinear PK, time-varying parameters, or profiles with multiple crossings, one can instead evaluate $C_i(t)$ on a sufficiently fine grid and estimate the fraction of time above MIC. Grid-based calculations should be checked by refining the grid and confirming that the estimated fractions are stable.

Assumptions, Diagnostics, and Common Pitfalls

Before interpreting a comparison, confirm that each simulated concentration profile corresponds to the intended population, regimen, and time horizon. The formulas here assume a one-compartment, linear model at steady state with an infusion shorter than the dosing interval. If the intended question concerns the first dose, loading doses, changing clearance, nonlinear elimination, or a different concentration target site, this calculation does not directly apply without adapting the model.

Check profile boundaries. Confirm $C_{\max}$, $C_{\min}$, $t_{\mathrm{up}}$, and $t_{\mathrm{down}}$ are physically consistent and within the intended interval for the crossing case.
Check units. $R_0/CL$, $C_{\max}$, $C_{\min}$, and MIC must use compatible concentration units; $k$ is in inverse time and crossing times use the same time unit as $\tau$.
Check the simulation design. Verify that simulated patient-level PK parameters and MIC values reflect the intended population and are paired as intended. The five-person example varies MIC only, so it cannot quantify PK variability.
Check numerical resolution when using a grid. Recalculate with a finer time step and inspect profiles near MIC crossings; otherwise a short above- or below-MIC period may be missed.

Common mistake: A learner may compare only end-infusion peaks, conclude that a higher peak guarantees target attainment, and report PTA from the number of peaks above MIC. That is not time-above-MIC attainment: a peak does not reveal how long the concentration remains above MIC. The correction is to calculate the full within-interval profile, determine the duration above each patient's MIC, divide by the interval, and only then apply the individual target and summarize the proportion attaining it.

Key takeaway: Time-above-MIC dose selection proceeds from each patient's concentration-time profile to an individual fraction, and from those fractions to PTA. State the model assumptions and both decision thresholds before interpreting a simulated regimen comparison.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In one sentence, distinguish an individual's $fT>MIC$ from population PTA.
For a patient with $C_{\min}<M<C_{\max}$, which two crossing times determine the above-MIC duration in the infusion model?
In the five-patient comparison, how many patients meet the 0.50 individual target on 1,000 mg every 8 hours, and what is the empirical PTA?
Why can a regimen with a higher end-infusion concentration still fail to meet a time-above-MIC target for a high-MIC patient?
Name one assumption of the worked crossing-time calculation that should be reconsidered for a first-dose prediction.

Solutions. (1) Individual $fT>MIC$ is the fraction of one patient's interval above that patient's MIC; PTA is the proportion of simulated patients whose fractions meet a specified target. (2) The rising crossing $t_{\mathrm{up}}$ and descending crossing $t_{\mathrm{down}}$; the duration is their difference when both are inside the interval. (3) Three of five meet the target, so the empirical PTA is $3/5=60\%$. (4) The concentration could decline below MIC early enough that the above-MIC fraction remains too small; the peak alone gives no duration information. (5) The calculation assumes steady state; a first-dose profile is not at steady state and needs its own concentration-time calculation.