Dose Selection Across MIC Distributions

Why the MIC Distribution Matters

A dose can produce adequate exposure for an organism with a low minimum inhibitory concentration (MIC) but fail to reach the relevant pharmacodynamic target for an organism with a higher MIC. Thus, a probability of target attainment (PTA) calculated at one MIC does not by itself describe the chance of target attainment across a population of infecting organisms. To make that population-level calculation, combine simulated patient exposures with the frequencies of MIC values in the specified organism distribution.

The result depends on the population being described. MIC frequencies from one organism, setting, or time period need not represent another. If different organisms have different pharmacodynamic targets, retain those targets when calculating PTA; averaging MICs first or applying one target to all organisms can change the question being answered.

Definition: For dose \(d\), organism \(s\), and MIC value \(m\), conditional PTA is the proportion of the target population whose simulated exposure under \(d\) meets the specified target for organism \(s\) at MIC \(m\). PTA across an MIC distribution is the weighted average of these conditional PTAs, where each weight is the frequency of that MIC in the specified distribution. If several organisms are included, an overall PTA is the weighted average of organism-specific PTAs using explicitly defined organism weights.

Calculation Across an MIC Distribution

Let \(E_i(d)\) denote the simulated exposure for patient \(i\) under dose \(d\), and let \(g_s(E_i(d),m)\) be the pharmacodynamic index evaluated for organism \(s\) at MIC \(m\). For example, \(g_s\) could be an AUC/MIC ratio or the fraction of time that concentration exceeds the MIC. Let \(T_s\) be the target for organism \(s\). Patient \(i\) attains the target at MIC \(m\) when \(g_s(E_i(d),m)\geq T_s\).

Suppose there are \(N\) simulated patients and MIC values \(m_1,\ldots,m_K\), with frequencies \(w_{sk}\) for organism \(s\). The weights must be nonnegative and sum to one within that organism. With equally weighted simulated patients, the calculation is:

\widehat{\mathrm{PTA}}_s(d) = \sum_{k=1}^{K} w_{sk} \left[ \frac{1}{N}\sum_{i=1}^{N} I\{g_s(E_i(d),m_k)\geq T_s\} \right].

Here \(I\{\cdot\}\) equals 1 when its condition is true and 0 otherwise. The expression in square brackets is the simulated conditional PTA at MIC \(m_k\). If organisms occur with specified proportions \(q_s\), where \(\sum_s q_s=1\), the overall estimate is:

\widehat{\mathrm{PTA}}_{\mathrm{overall}}(d) = \sum_s q_s\,\widehat{\mathrm{PTA}}_s(d).

This overall value describes an average over the stated organism mixture and MIC distributions. It is not a guarantee of adequate attainment for every organism or every MIC. Report the organism-specific values as well when the overall average could hide a poorly covered subgroup.

Why the Weighted Average Is the Right Quantity

The calculation follows directly from averaging an indicator over the joint distribution of patients and MIC values. For a fixed organism \(s\), define \(A\) as the event that the target is attained. If simulated patients and MICs are sampled independently from their specified distributions, then conditioning on MIC gives:

P(A\mid s) = \sum_{k=1}^{K} P(A\mid s,\mathrm{MIC}=m_k)\, P(\mathrm{MIC}=m_k\mid s).

Replacing the conditional probabilities by their simulated proportions and MIC probabilities by the specified frequencies gives the PTA estimator above. This justification also clarifies the assumptions: the MIC weights must represent the intended population, the simulated exposures must represent the intended patients under the candidate dose, and the target and pharmacodynamic index must be appropriate for the organism. If patient characteristics and infecting organism are associated in the intended population, a simple independent combination may not represent their joint distribution; the simulation or weighting scheme should preserve the relevant association.

Worked Example: AUC/MIC PTA for One Organism

Hypothetical data. For a candidate regimen, five simulated patients have free-drug AUC values of 400, 300, 250, 180, and 120 mg·h/L. The target for this organism is \(f\mathrm{AUC}/\mathrm{MIC}\geq 50\) h. The specified MIC distribution is 1 mg/L with frequency 0.50, 2 mg/L with frequency 0.30, and 4 mg/L with frequency 0.20. Assume the same simulated exposure distribution applies across these MIC categories.

At MIC 1 mg/L, the target requires free-drug AUC of at least \(50\times1=50\) mg·h/L. All five simulated values meet it, so conditional PTA is \(5/5=1.00\).

At MIC 2 mg/L, the required AUC is \(50\times2=100\) mg·h/L. Again, all five values meet it, giving \(5/5=1.00\).

At MIC 4 mg/L, the required AUC is \(50\times4=200\) mg·h/L. The values 400, 300, and 250 meet the target; 180 and 120 do not. Conditional PTA is \(3/5=0.600\).

Weight these results by the MIC frequencies:

\widehat{\mathrm{PTA}}(d) = (0.50)(1.00)+(0.30)(1.00)+(0.20)(0.600) =0.920.

The estimated PTA across this specified distribution is 92.0%. It is lower than the 100% conditional PTA at MIC 1 mg/L because the distribution includes organisms at MIC 4 mg/L, for which two of the five simulated patients miss the target. These five patients are a small, illustrative simulation; an actual analysis would use a simulation size adequate for its intended precision.

Organism-Specific Targets and Population Weights

The MIC distribution and target should be paired at the organism level. A target derived for one organism or pharmacodynamic endpoint should not automatically be transferred to another. If the requested summary is across several organisms, first calculate PTA within each organism using its own target and MIC distribution. Only then combine those values with organism weights that correspond to the intended population.

Worked Example: Combining Two Organisms

Hypothetical data. Organism A accounts for 60% of the specified infections and organism B for 40%. For organism A, the target is \(fT>\mathrm{MIC}\geq 50\%\). Its MIC distribution is 1 mg/L with weight 0.70 and 2 mg/L with weight 0.30. At those two MIC values, respectively, 2 of 4 and 2 of 4 simulated patient profiles attain the target. Thus:

\widehat{\mathrm{PTA}}_A(d) =(0.70)(2/4)+(0.30)(2/4) =0.500.

For organism B, the target is \(f\mathrm{AUC}/\mathrm{MIC}\geq 40\) h. Its MIC distribution is 2 mg/L with weight 0.50 and 4 mg/L with weight 0.50. At MIC 2 mg/L, all 4 of 4 simulated profiles attain the target; at MIC 4 mg/L, 2 of 4 do. Therefore:

\widehat{\mathrm{PTA}}_B(d) =(0.50)(4/4)+(0.50)(2/4) =0.750.

The overall PTA for the specified mixture is:

\widehat{\mathrm{PTA}}_{\mathrm{overall}}(d) =(0.60)(0.500)+(0.40)(0.750) =0.600.

The overall estimate is 60.0%, but the organism-specific results differ: 50.0% for A and 75.0% for B. The combined value is meaningful only for the stated organism weights and target definitions. It does not imply that either organism individually has 60.0% PTA.

MIC Summaries, Diagnostics, and Alternatives

A common shortcut is to select a representative MIC, such as the median MIC, calculate PTA there, and treat that result as PTA across the distribution. This generally is not valid. Target attainment is an indicator, so averaging or selecting MICs before evaluating attainment need not give the same result as evaluating each MIC and then averaging its attainment probability.

Worked Example: Why a Median MIC Can Mislead

Hypothetical data. Suppose every simulated patient has free-drug AUC of 300 mg·h/L, the target is \(f\mathrm{AUC}/\mathrm{MIC}\geq 50\) h, and the MIC distribution has values 1, 2, 4, and 8 mg/L with weights 0.20, 0.40, 0.30, and 0.10. The required AUCs are, respectively, 50, 100, 200, and 400 mg·h/L. The fixed simulated AUC meets the target at the first three MIC values but not at 8 mg/L.

Therefore, PTA across the distribution is:

$$ (0.20)(1)+(0.40)(1)+(0.30)(1)+(0.10)(0)=0.900. $$

The median MIC is 2 mg/L, where all simulated patients attain the target. Using only that MIC would report 100% and miss the 10% of the specified distribution at which target attainment fails. In this example, even the MIC90 is 4 mg/L, which also yields 100% conditional PTA, while the distribution-wide PTA remains 90.0%.

Useful checks include confirming that MIC weights sum to one within each organism, the organism weights sum to one, and the exposure and MIC units are compatible. For an AUC/MIC target, for example, AUC in mg·h/L and MIC in mg/L give a ratio in hours. Check that target comparisons use the intended boundary, including whether equality counts as attainment. Review conditional PTA by MIC and organism: abrupt or implausible changes may indicate a coding, unit, indexing, or weighting error. A dose-response curve that behaves unexpectedly should prompt checks of simulated exposure inputs and the target calculation, rather than being hidden by a pooled average.

The distribution-wide PTA is an alternative to reporting conditional PTA at a single MIC, but it is not the only useful summary. A median-MIC calculation is simple to communicate, yet it suppresses tails. A worst-case-MIC calculation is conservative for the chosen range but can overemphasize rare MIC values. Reporting the full conditional PTA profile alongside the weighted average makes these trade-offs visible. Clinical outcome analyses answer a different question: a model-based PTA is a prediction based on simulated exposures, targets, and MIC frequencies, not direct proof of clinical effectiveness.

Interpret the estimate only for its specified population and assumptions. A changed MIC distribution can change PTA even when the dose and patient exposure model remain unchanged. Likewise, a different target, organism mixture, or exposure simulation can change the result. An overall PTA may be useful for comparing regimens against a prespecified decision threshold, but examine organism-specific performance before relying on it for dose selection.

Common Mistakes

Using one MIC as if it represented the whole distribution. A conditional PTA is not necessarily a distribution-wide PTA. Calculate at each MIC and weight by its frequency.
Averaging MIC values before evaluating the target. Because target attainment is a threshold event, the result at an average MIC generally does not equal the average attainment across MICs.
Combining organisms before applying their targets. Calculate each organism-specific PTA with its own target and MIC distribution, then apply the stated organism weights.
Reporting an overall value without its population definition. State which organism proportions and MIC distributions were used; a different mixture can yield a different PTA.
Ignoring units or the attainment boundary. Check the MIC and exposure units and specify whether the target is met at equality.

Key takeaway: Estimate conditional PTA across MIC values first, weight those values by the specified organism-specific MIC distribution, and combine organisms only with explicit population weights. The resulting PTA describes that defined mixture, not every possible pathogen population.

Check Your Understanding

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

Conditional PTA is 0.80 at MIC 1 mg/L and 0.40 at MIC 4 mg/L. The MIC frequencies are 0.75 and 0.25. What is PTA across this distribution?
Explain why calculating PTA only at the median MIC can give a different result from calculating it across all MIC categories.
Two organisms have PTAs of 0.70 and 0.50, and their specified population weights are 0.30 and 0.70. Calculate overall PTA.
Name two checks that can identify an error in a distribution-wide PTA calculation.

Solutions. (1) \( (0.75)(0.80)+(0.25)(0.40)=0.60+0.10=0.70\), so PTA is 70.0%. (2) Target attainment is evaluated against a threshold at each MIC; the median discards the other MIC categories and their frequencies, including any tail where attainment differs. (3) \( (0.30)(0.70)+(0.70)(0.50)=0.21+0.35=0.56\), so overall PTA is 56.0% for that organism mixture. (4) Examples include checking that MIC weights sum to one within each organism, organism weights sum to one, exposure and MIC units are compatible, and conditional PTA values change plausibly across MICs.