A clinical pharmacologist is analyzing the steady-state trough concentrations of a newly a… — USMLE Step 3

A clinical pharmacologist is analyzing the steady-state trough concentrations of a newly approved monoclonal antibody targeted for severe refractory eosinophilic asthma. In a post-marketing surveillance cohort of $5,000$ adult patients, the serum trough concentrations follow a normal, symmetric bell-shaped distribution.

The variance of the steady-state trough concentrations in this population is calculated to be $9.0 \text{ (mcg/mL)}^2$ . Furthermore, statistical analysis reveals that the $84^\text{th}$ percentile for the trough concentration is $17.0 \text{ mcg/mL}$ .

The clinical trial protocol dictates that any patient with a trough concentration falling at or below the $2.5^\text{th}$ percentile is at extremely high risk for subtherapeutic exposure and should undergo immediate dose escalation.

Below what trough concentration (in $\text{mcg/mL}$ ) will a patient qualify for this dose escalation?

Answer: 8 mcg/mL

Answer

The threshold for the dose escalation is 8.0 mcg/mL.

In a perfectly normal distribution, percentiles directly map to standard deviations away from the mean based on the 68-95-99.7 rule. The standard deviation is the square root of the variance (

9.0

), which is

3.0

. Because

68\%

of values lie within

1

standard deviation, the upper boundary of this central area (

50\% + 34\% = 84\%

) represents the

84^\text{th}

percentile. This means

\mu + 1\sigma = 17.0

. Given

\sigma = 3.0

, the mean

\mu = 14.0

. The protocol requires finding the

2.5^\text{th}

percentile. Because

95\%

of values lie within

2

standard deviations,

5\%

lie outside this range, leaving

2.5\%

symmetrically in the lower tail. Therefore, the

2.5^\text{th}

percentile is

\mu - 2\sigma

, which evaluates to

14.0 - 2(3.0) = 8.0 \text{ mcg/mL}

Step-by-Step Solution

Calculate the standard deviation (

\sigma

) by taking the square root of the population variance.

\sigma = \sqrt{9.0} = 3.0 \text{ mcg/mL}

The empirical rule (68-95-99.7) requires the use of standard deviation, not variance.

Map the

84^\text{th}

percentile to its position on the normal distribution using the empirical rule.

The

84^\text{th}

percentile corresponds to

+1\sigma

above the mean.

In a normal distribution,

50\%

of the data lies below the mean. The first standard deviation above the mean contains

34\%

of the data. Thus,

50\% + 34\% = 84\%

, meaning the

84^\text{th}

percentile is exactly at

\mu + 1\sigma

Calculate the population mean (

\mu

) using the value of the

84^\text{th}

percentile and the standard deviation.

\mu + 1(3.0) = 17.0 \implies \mu = 14.0 \text{ mcg/mL}

Establishing the mean is required to find any other specific percentile value in the distribution.

Map the

2.5^\text{th}

percentile to its position on the normal distribution.

The

2.5^\text{th}

percentile corresponds to

-2\sigma

below the mean.

According to the empirical rule,

95\%

of data falls within

\pm 2\sigma

. This leaves

5\%

in the combined tails, meaning exactly

2.5\%

lies in the lower tail (below

-2\sigma

Calculate the exact trough concentration for the

2.5^\text{th}

percentile threshold.

14.0 - 2(3.0) = 14.0 - 6.0 = 8.0 \text{ mcg/mL}

This is the final application of the derived mean and standard deviation to identify the specific clinical threshold requested.

Key Concept

Application of the empirical (68-95-99.7) rule to a normal distribution, including the conversion of variance to standard deviation to calculate specific clinical percentiles.

Estimated Time:3m 0s

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