A tertiary care transplant center is validating a new rapid biomarker assay to detect an i… — USMLE Step 3

A tertiary care transplant center is validating a new rapid biomarker assay to detect an invasive opportunistic fungal infection in post-operative patients. According to internal hospital data, the baseline prevalence of this fungal infection among their solid organ transplant recipients is $25\%$ . During the validation phase, the rapid assay correctly identified $90\%$ of patients who had biopsy-proven fungal tissue invasion. The assay also correctly returned a negative result in $90\%$ of transplant recipients who were proven to be free of the infection. A 58-year-old man who recently received a renal allograft is tested as part of a routine post-operative monitoring protocol, and his biomarker assay returns a positive result. Based on the test's performance characteristics in this population, what is the probability (expressed as a percentage) that he actually has the invasive fungal infection?

Answer: 75 %

Answer

The patient's probability of actually having the infection given a positive test result is defined as the Positive Predictive Value (PPV). To calculate PPV, we can construct a

2 \times 2

contingency table using a hypothetical cohort of

1,000

solid organ transplant recipients. Based on the

25\%

baseline prevalence, there are

250

patients with the disease and

750

without it. The assay's

90\%

sensitivity correctly identifies

225

true positive cases from the diseased group (

250 \times 0.90

). The assay's

90\%

specificity correctly identifies

675

true negative cases, which leaves

75

false positive cases from the disease-free group (

750 \times 0.10

). The PPV is the ratio of true positives to the total number of positive results. Therefore,

\text{PPV} = \frac{225}{225 + 75} = \frac{225}{300} = 0.75

, which equals

75\%

Step-by-Step Solution

Extract the provided statistical parameters from the clinical vignette

Prevalence =

25\%

, Sensitivity =

90\%

, Specificity =

90\%

These are the core variables required to calculate the positive predictive value (PPV) of the diagnostic assay.

Set up a hypothetical population cohort

Assume a population size of

1,000

patients

Using a hypothetical round number population simplifies the calculation of absolute numbers for a

2 \times 2

contingency table.

Calculate the total number of patients with and without the infection

Infected =

250

(

1,000 \times 0.25

); Uninfected =

750

(

1,000 \times 0.75

)

The baseline prevalence splits the total population into true positive and true negative base groups.

Calculate True Positives (TP) and False Positives (FP)

\text{TP} = 250 \times 0.90 = 225

;

\text{FP} = 750 \times (1 - 0.90) = 75

TP represents infected patients correctly identified by sensitivity. FP represents uninfected patients incorrectly identified as positive based on the false positive rate (which is

1 - \text{Specificity}

Calculate the Positive Predictive Value (PPV)

\text{PPV} = \frac{225}{225 + 75} = \frac{225}{300} = 0.75

PPV is the proportion of true positives out of all positive test results, which answers the patient's probability of actually having the disease given a positive result.

Convert the resulting PPV probability into a percentage

0.75 \times 100 = 75\%

The question specifically requests the probability expressed as a percentage.

Key Concept

The positive predictive value (PPV) of a test depends heavily on the prevalence of the disease in the tested population, in addition to the test's inherent sensitivity and specificity.

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