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Question 41Question →

A 68-year-old male is admitted to the hospital with acute decompensated heart failure. He has a history of ischemic cardiomyopathy with a left ventricular ejection fraction of $25\%$ , stage 2 chronic kidney disease with a baseline serum creatinine of $1.1 \text{ mg/dL}$ , and type 2 diabetes mellitus. Upon admission, he is started on a continuous intravenous infusion of furosemide. On day 3 of hospitalization, his urine output decreases to $25 \text{ mL/hour}$ . His physical examination reveals elevated jugular venous pressure, bibasilar crackles, and $2+$ bilateral pitting lower extremity edema.

Laboratory results on day 3 are as follows:

Parameter	Result	Reference Range
Serum
Sodium	$135 \text{ mEq/L}$	$136 - 145 \text{ mEq/L}$
Potassium	$3.6 \text{ mEq/L}$	$3.5 - 5.0 \text{ mEq/L}$
Chloride	$96 \text{ mEq/L}$	$95 - 105 \text{ mEq/L}$
Bicarbonate	$29 \text{ mEq/L}$	$22 - 28 \text{ mEq/L}$
Blood urea nitrogen (BUN)	$56 \text{ mg/dL}$	$7 - 18 \text{ mg/dL}$
Creatinine	$2.4 \text{ mg/dL}$	$0.6 - 1.2 \text{ mg/dL}$
Urine
Sodium	$54 \text{ mEq/L}$	Variable
Urea nitrogen	$280 \text{ mg/dL}$	Variable
Creatinine	$48 \text{ mg/dL}$	Variable

Given the patient's current pharmacotherapy, calculate the value (in percentage) of the most appropriate fractional excretion index to determine whether his acute kidney injury is intrinsic or hemodynamically mediated.

Show answer & explanation

Answer: 25

Answer

25

The correct calculation is

25\%

. In evaluating acute kidney injury in a patient actively receiving loop diuretics, the Fractional Excretion of Sodium (FENa) is unreliable due to pharmacologically induced natriuresis. The Fractional Excretion of Urea (FEUrea) is the correct index to use, as urea reabsorption occurs predominantly in the proximal tubule and is unaffected by loop diuretics. Using the provided values,

FE_{Urea} = \frac{U_{Urea} \times S_{Cr}}{S_{BUN} \times U_{Cr}} \times 100 = \frac{280 \times 2.4}{56 \times 48} \times 100 = 25\%

. Because

25\%

is less than

35\%

, it correctly points to hemodynamically mediated (prerenal) AKI driven by cardiorenal syndrome.

Step-by-Step Solution

1

Identify the clinical context and the physiological impact of the patient's medications on laboratory indices.

The patient has acute kidney injury (AKI) secondary to decompensated heart failure (cardiorenal syndrome) and is on a continuous furosemide infusion, which artificially increases urinary sodium excretion.

Loop diuretics inhibit the Na-K-2Cl cotransporter in the thick ascending limb, rendering sodium-based fractional excretion indices inaccurate.

2

Determine the most appropriate fractional excretion index to use.

Fractional Excretion of Urea (FEUrea) must be used instead of Fractional Excretion of Sodium (FENa).

Unlike sodium, urea is primarily reabsorbed in the proximal tubule. Its reabsorption is coupled with water and is unaffected by loop diuretics, making it a reliable marker of hemodynamically mediated (prerenal) AKI.

3

Recall the formula for calculating FEUrea.

FE_{Urea} = \frac{U_{Urea} \times S_{Cr}}{S_{Urea} \times U_{Cr}} \times 100

This formula normalizes the urea clearance to the creatinine clearance to determine the fractional excretion.

4

Substitute the patient's laboratory values into the formula and calculate.

FE_{Urea} = \frac{280 \times 2.4}{56 \times 48} \times 100 = \frac{672}{2688} \times 100 = 0.25 \times 100 = 25\%

.

Mathematical execution of the clinical index. A value

< 35\%

confirms prerenal azotemia.

Key Concept

In patients receiving diuretic therapy, the Fractional Excretion of Urea (FEUrea) should be utilized instead of the Fractional Excretion of Sodium (FENa) to differentiate between prerenal azotemia and intrinsic acute tubular necrosis. An FEUrea

< 35\%

indicates a prerenal etiology.

Alternative Method

While calculating the BUN-to-Creatinine ratio (

56 / 2.4 \approx 23.3

) provides a rapid bedside clue for prerenal azotemia, it is not a formally derived fractional excretion index. Determining the FEUrea provides diagnostic certainty when differentiating the exact mechanism of AKI in patients with complex hemodynamics on heavy diuresis.

Estimated Time:3m 0s

Question 42Question →

A 32-year-old woman is evaluated in the emergency department for sudden-onset severe shortness of breath and right-sided chest pain that worsens with deep inspiration. She recently returned from a 14-hour international flight. Her medical history is unremarkable, and her only medication is an oral contraceptive pill. On physical examination, her lungs are clear to auscultation bilaterally. Her right calf is mildly swollen and tender.

Vital signs:
Temperature: $37.1^\circ\text{C}$ ( $98.8^\circ\text{F}$ )
Blood pressure: $118/74\text{ mmHg}$
Pulse: $115/\text{min}$
Respiratory rate: $26/\text{min}$

An arterial blood gas is obtained while the patient is breathing room air at sea level:
pH: $7.52$
$PaCO_2$ : $28\text{ mmHg}$
$PaO_2$ : $56\text{ mmHg}$
$HCO_3^-$ : $22\text{ mEq/L}$

To further evaluate the pathophysiology of her hypoxemia, the physician decides to calculate the alveolar-arterial (A-a) oxygen gradient. Assume the partial pressure of inspired oxygen ( $PiO_2$ ) is $150\text{ mmHg}$ and the respiratory quotient (R) is $0.8$ .

Based on the laboratory results, what is the patient's calculated A-a gradient?

Show answer & explanation

Answer: 59

Answer

59

The correct calculation of the A-a gradient requires two steps. First, the alveolar oxygen tension (

P_AO_2

) is determined using the alveolar gas equation:

P_AO_2 = PiO_2 - \frac{PaCO_2}{R}

. Given a

PiO_2

of

150\text{ mmHg}

and an R of

0.8

, the equation becomes

150 - \frac{28}{0.8} = 150 - 35 = 115\text{ mmHg}

. Second, the measured arterial oxygen tension (

P_aO_2

of

56\text{ mmHg}

) is subtracted from the calculated

P_AO_2

. Thus,

115 - 56 = 59\text{ mmHg}

. This significantly elevated A-a gradient, combined with the clinical presentation of sudden-onset pleuritic chest pain, tachypnea, and a swollen right calf following prolonged immobilization, is highly indicative of a pulmonary embolism causing ventilation-perfusion (V/Q) mismatch.

Step-by-Step Solution

1

Calculate the partial pressure of oxygen in the alveoli (

P_AO_2

).

P_AO_2 = 115\text{ mmHg}

The alveolar gas equation estimates

P_AO_2

as

PiO_2 - \frac{PaCO_2}{R}

. Substituting the given values:

P_AO_2 = 150 - \frac{28}{0.8} = 150 - 35 = 115\text{ mmHg}

.

2

Calculate the alveolar-arterial (A-a) oxygen gradient.

59\text{ mmHg}

The A-a gradient is the difference between the alveolar and arterial oxygen tensions (

P_AO_2 - P_aO_2

). Substituting the calculated and measured values:

115 - 56 = 59\text{ mmHg}

.

Key Concept

The alveolar-arterial (A-a) oxygen gradient helps differentiate causes of hypoxemia; an elevated gradient (> 10-15 mmHg in young adults) in the setting of respiratory alkalosis strongly suggests ventilation-perfusion (V/Q) mismatch, commonly seen in pulmonary embolism.

Estimated Time:1m 30s

Question 43Question →

A 35-year-old woman comes to the clinic reporting progressive fatigue, diffuse muscle weakness, and excessive thirst for the past three weeks. Her medical history is notable for Sjögren syndrome and two episodes of calcium phosphate nephrolithiasis over the past five years. She currently takes no medications. Her temperature is 37.0°C (98.6°F), blood pressure is 110/70 mmHg, pulse is 88/min, and respiratory rate is 16/min. Physical examination reveals dry mucous membranes but is otherwise unremarkable. Laboratory studies are obtained as shown below:

Serum Laboratory Results:

Test	Result	Reference Range
Sodium (Na⁺)	139 mEq/L	136-145 mEq/L
Potassium (K⁺)	2.7 mEq/L	3.5-5.0 mEq/L
Chloride (Cl⁻)	112 mEq/L	98-106 mEq/L
Bicarbonate (HCO₃⁻)	15 mEq/L	22-28 mEq/L
Blood urea nitrogen (BUN)	14 mg/dL	7-20 mg/dL
Creatinine	0.9 mg/dL	0.6-1.2 mg/dL
Glucose	88 mg/dL	70-100 mg/dL

Urine Laboratory Results:

Test	Result
pH	6.5
Sodium (Na⁺)	40 mEq/L
Potassium (K⁺)	25 mEq/L
Chloride (Cl⁻)	50 mEq/L
Urea Nitrogen (UUN)	420 mg/dL
Glucose	0 mg/dL
Measured Osmolality	310 mOsm/kg H₂O

Based on the laboratory data provided, what is the patient's calculated urine osmolar gap? (Enter the numerical value only)

Show answer & explanation

Answer: 30

Answer

30

The patient has a normal anion gap metabolic acidosis (NAGMA) secondary to distal (type 1) renal tubular acidosis, which is strongly associated with Sjögren syndrome and calcium phosphate nephrolithiasis. In the workup of NAGMA, it is critical to determine if the kidneys are appropriately secreting acid in the form of ammonium (

NH_4^+

). The urine osmolar gap (UOG) is the most accurate indirect measure of urinary ammonium. The gap is the difference between the measured urine osmolality and the calculated urine osmolality.

The formula for calculated urine osmolality is:

\text{Calculated U}_{\text{osm}} = 2 \times (\text{Na}^+ + \text{K}^+) + \frac{\text{UUN}}{2.8} + \frac{\text{Glucose}}{18}

Plugging in the patient's values:

\text{Calculated U}_{\text{osm}} = 2 \times (40 + 25) + \frac{420}{2.8} + 0

\text{Calculated U}_{\text{osm}} = 2 \times 65 + 150 = 130 + 150 = 280 \text{ mOsm/kg}

The UOG is the measured osmolality minus the calculated osmolality:

310 - 280 = 30 \text{ mOsm/kg}

A UOG of less than 40-50 mOsm/kg indicates low urinary ammonium excretion, confirming a renal etiology for the acidosis (RTA), whereas a UOG > 100 mOsm/kg would indicate adequate ammonium excretion (suggesting gastrointestinal bicarbonate loss).

Step-by-Step Solution

1

Calculate the serum anion gap.

Serum AG = 139 - (112 + 15) = 12 mEq/L.

To classify the metabolic acidosis as a normal anion gap metabolic acidosis (NAGMA), which dictates the next step in laboratory interpretation.

2

Apply the calculated urine osmolality formula.

Calculated U_osm = 2 * (40 + 25) + (420 / 2.8) + (0 / 18) = 130 + 150 = 280 mOsm/kg.

To estimate the osmolar contribution of the measured major urinary solutes (sodium, potassium, urea, and glucose).

3

Calculate the urine osmolar gap.

Gap = 310 (Measured) - 280 (Calculated) = 30 mOsm/kg.

The difference between measured and calculated osmolality reflects unmeasured osmoles, which in the urine primarily consist of ammonium and its accompanying anions.

Key Concept

The urine osmolar gap is used to indirectly measure urinary ammonium excretion in the evaluation of a normal anion gap metabolic acidosis.

Question 44Question →

A 45-year-old man is brought to the emergency department after being rescued from a remote hiking trail. He was lost for two days without access to food or water. On presentation, he is lethargic and confused. Vital signs show a temperature of $37.8^\circ\text{C}$ ( $100.0^\circ\text{F}$ ), pulse of $118/\text{min}$ , blood pressure of $92/58\text{ mmHg}$ , and respiratory rate of $16/\text{min}$ . Physical examination reveals dry mucous membranes, decreased skin turgor, and delayed capillary refill. His estimated premorbid weight is $70\text{ kg}$ .

Initial laboratory evaluation reveals the following:

Test	Result	Reference Range
Serum Sodium	$154\text{ mEq/L}$	$136-145\text{ mEq/L}$
Serum Potassium	$4.1\text{ mEq/L}$	$3.5-5.0\text{ mEq/L}$
Serum Chloride	$118\text{ mEq/L}$	$95-105\text{ mEq/L}$
Serum Bicarbonate	$25\text{ mEq/L}$	$22-28\text{ mEq/L}$
Blood Urea Nitrogen	$50\text{ mg/dL}$	$7-18\text{ mg/dL}$
Serum Creatinine	$1.9\text{ mg/dL}$	$0.6-1.2\text{ mg/dL}$
Serum Glucose	$98\text{ mg/dL}$	$70-110\text{ mg/dL}$

Based on the patient's presentation and laboratory results, what is his estimated free water deficit in liters? (Assume a target normal serum sodium of $140\text{ mEq/L}$ .)

Show answer & explanation

Answer: 4.2

Answer

4.2

The patient is experiencing severe hypovolemic hypernatremia due to unreplaced sensible and insensible free water losses in the setting of environmental exposure. To properly direct intravenous fluid resuscitation (typically with

5\%

dextrose in water or enteral free water), the free water deficit must be calculated. The standard formula is:

\text{Free Water Deficit} = \text{TBW} \times \left( \frac{\text{Measured } Na^+}{\text{Target } Na^+} - 1 \right)

. First, calculate Total Body Water (TBW), which is approximately

60\%

(

0.6

) of lean body weight in a young or middle-aged adult male.

\text{TBW} = 0.6 \times 70\text{ kg} = 42\text{ L}

. Next, insert the measured sodium (

154\text{ mEq/L}

) and the target normal sodium (

140\text{ mEq/L}

) into the formula:

42 \times \left(\frac{154}{140} - 1\right) = 42 \times (1.1 - 1) = 42 \times 0.1 = 4.2\text{ Liters}

.

Step-by-Step Solution

1

Determine the patient's estimated Total Body Water (TBW).

For an adult male,

\text{TBW} = 0.6 \times \text{body weight (kg)}

. Therefore,

0.6 \times 70\text{ kg} = 42\text{ L}

.

Free water deficit calculations rely on the patient's total body water, which varies by sex and age (

0.6

for adult men,

0.5

for adult women and elderly men,

0.45

for elderly women).

2

Calculate the ratio of measured serum sodium to the target normal serum sodium.

\frac{154\text{ mEq/L}}{140\text{ mEq/L}} = 1.1

.

This ratio determines the proportional deficit of free water relative to the body's solute concentration.

3

Apply the free water deficit formula.

\text{Deficit} = \text{TBW} \times (1.1 - 1) = 42 \times 0.1 = 4.2\text{ Liters}

.

Multiplying the TBW by the excess sodium fraction yields the absolute volume of free water that has been lost.

Key Concept

Calculating free water deficit in hypernatremia

Question 45Question →

A 28-year-old woman comes to the clinic complaining of chronic, watery diarrhea for the past 4 months. She states she has 3 to 5 loose, large-volume bowel movements daily, accompanied by significant flatulence and abdominal bloating. The patient notes that her symptoms improve significantly when she fasts or restricts her diet. Her past medical history is unremarkable, and she takes no medications. Vital signs are within normal limits. Physical examination reveals a non-tender abdomen with hyperactive bowel sounds.

A fresh stool sample is sent for chemical and electrolyte analysis, and the results are as follows:

Measured stool osmolality: $295\text{ mOsm/kg}$
Stool sodium: $30\text{ mEq/L}$
Stool potassium: $45\text{ mEq/L}$
Stool chloride: $20\text{ mEq/L}$
Stool pH: $5.4$

Based on the laboratory results provided, what is the calculated stool osmotic gap in mOsm/kg?

Show answer & explanation

Answer: 145

Answer

145

The stool osmotic gap is correctly calculated by subtracting twice the sum of stool sodium and potassium from the measured stool osmolality (

295 - 2 \times (30 + 45) = 145

). A gap greater than 50 mOsm/kg is characteristic of osmotic diarrhea, consistent with this patient's history of symptom improvement with fasting and consumption of a dairy-rich diet.

Step-by-Step Solution

1

Identify the formula for calculating the stool osmotic gap.

Stool Osmotic Gap = Measured Stool Osmolality - 2 × (Stool Na + Stool K)

This formula estimates the contribution of unmeasured osmoles (such as unabsorbed carbohydrates) in the stool.

2

Extract the relevant laboratory values from the clinical vignette.

Measured stool osmolality = 295 mOsm/kg, Stool Na = 30 mEq/L, Stool K = 45 mEq/L.

Only sodium, potassium, and measured osmolality are needed; chloride and pH are distractors for the calculation.

3

Calculate the sum of the stool cations (sodium and potassium).

30 + 45 = 75 mEq/L

Sodium and potassium are the primary measured cations in the stool fluid.

4

Multiply the sum of the cations by 2.

2 × 75 = 150 mOsm/kg

Multiplying by 2 accounts for the osmotic contribution of the unmeasured accompanying anions.

5

Subtract the calculated value from the measured stool osmolality.

295 - 150 = 145 mOsm/kg

This yields the final stool osmotic gap, which is greater than 50 mOsm/kg, confirming an osmotic etiology for the diarrhea.

Key Concept

Calculating the stool osmotic gap helps differentiate between osmotic and secretory diarrhea. A gap > 50 mOsm/kg (and specifically > 100 mOsm/kg) indicates osmotic diarrhea, while a gap < 50 mOsm/kg indicates secretory diarrhea.

Question 46Question →

A pharmacy and therapeutics (P&T) committee at a health maintenance organization is evaluating a newly approved oral anticoagulant for addition to the formulary. The new medication is being compared to the current standard of care, warfarin, for stroke prevention in patients with nonvalvular atrial fibrillation. The committee reviews the annual costs and health outcomes associated with both treatments.

The current standard of care (warfarin, including the cost of the drug, routine INR monitoring, and associated clinic visits) costs $800$ dollars per patient annually and provides an average of $0.75$ Quality-Adjusted Life Years (QALYs) per year.

The new oral anticoagulant costs $4,800$ dollars per patient annually (including drug costs and minimal required monitoring) and provides an average of $0.80$ QALYs per year due to a lower incidence of major bleeding events and improved patient adherence.

Based on this data, what is the incremental cost-effectiveness ratio (ICER) of the new oral anticoagulant compared to the standard of care? (Express your answer in dollars per QALY gained as a whole number).

Show answer & explanation

Answer: 80000

Answer

80000 dollars per QALY gained

The Incremental Cost-Effectiveness Ratio (ICER) is a fundamental metric in healthcare economics used to determine the value of a new intervention compared to a standard intervention. It is calculated by dividing the difference in cost between two interventions by the difference in their effect (usually measured in QALYs). In this scenario, the incremental cost is

4,800 - 800 = 4,000

dollars. The incremental effect is

0.80 - 0.75 = 0.05

QALYs. Dividing

4,000

by

0.05

yields an ICER of

80,000

dollars per QALY gained. This metric helps health systems decide if the additional clinical benefit of a new therapy justifies its additional cost.

Step-by-Step Solution

1

Calculate the incremental cost (difference in cost between the two interventions).

4,800 - 800 = 4,000

dollars

ICER requires knowing how much extra the new intervention costs compared to the baseline standard of care.

2

Calculate the incremental effect (difference in QALYs between the two interventions).

0.80 - 0.75 = 0.05

QALYs

ICER requires knowing the additional clinical benefit provided by the new intervention compared to the baseline.

3

Divide the incremental cost by the incremental effect.

4,000 / 0.05 = 80,000

This yields the cost per each additional QALY gained, which represents the formula for the Incremental Cost-Effectiveness Ratio.

Key Concept

Incremental Cost-Effectiveness Ratio (ICER)

Estimated Time:1m 30s

Question 47Question →

A health system's population health committee is reviewing strategies to reduce 30-day hospital readmissions for patients with acute exacerbations of chronic obstructive pulmonary disease (COPD), which currently incur significant financial penalties under the Hospital Readmissions Reduction Program. Under standard discharge planning, the 30-day readmission rate for this population is 22%.

The committee is considering implementing an intensive transitional care program that provides home visits by a respiratory therapist, home delivery of new maintenance inhalers, and a telemedicine follow-up within 48 hours of discharge. A multi-center randomized controlled trial has shown that this program reduces the 30-day readmission rate in this specific population to 17%. The incremental cost to provide this intensive transitional care program is 750 dollars per patient.

Based on the trial data, what is the estimated cost (in dollars) to prevent one 30-day hospital readmission using this transitional care program?

Show answer & explanation

Answer: 15000

Answer

15000

To determine the cost of preventing a single clinical event (in this case, a 30-day hospital readmission), one must first determine how many patients need to be treated to prevent one event. This metric is the Number Needed to Treat (NNT), calculated as the inverse of the Absolute Risk Reduction (ARR). The ARR is the difference in event rates between standard care and the intervention (

0.22 - 0.17 = 0.05

). Thus,

NNT = \frac{1}{0.05} = 20

. This means 20 patients must receive the transitional care program to prevent one readmission. Since the program costs 750 dollars per patient, treating a cohort of 20 patients will cost

20 \times 750 = 15000

dollars.

Step-by-Step Solution

1

Calculate the Absolute Risk Reduction (ARR).

ARR = 0.22 - 0.17 = 0.05

ARR represents the absolute difference in readmission rates between standard care and the intervention, determining the actual clinical benefit magnitude.

2

Calculate the Number Needed to Treat (NNT).

NNT = \frac{1}{0.05} = 20

The NNT identifies how many patients must receive the transitional care program to prevent exactly one additional hospital readmission.

3

Calculate the cost to prevent one readmission.

20 \times 750 = 15000

Multiplying the NNT (20 patients) by the cost per patient (750 dollars) yields the total financial investment required to prevent one event.

Key Concept

The economic cost to prevent a single adverse health event is calculated by multiplying the Number Needed to Treat (NNT) by the cost of the intervention per patient. NNT is derived as the inverse of the Absolute Risk Reduction (ARR).

Question 48Question →

A $52$ -year-old woman with severe, seropositive rheumatoid arthritis is evaluated in the rheumatology clinic. Her disease has progressed despite optimized therapy with methotrexate. The physician decides to initiate a tumor necrosis factor (TNF) inhibitor. The patient expresses concern about the financial burden of this new therapy, as she recently switched to an employer-sponsored high-deductible health plan.

The clinic's financial counselor reviews her insurance benefits for the new calendar year:

* Annual deductible: USD $2,500$
* Coinsurance: $20\%$ (patient responsibility after the deductible is met)
* Annual out-of-pocket maximum: USD $6,500$

The patient has not utilized any healthcare services yet this year, meaning USD $0$ has been applied to her deductible and out-of-pocket maximum. The negotiated rate for a one-month supply of the TNF inhibitor is USD $4,500$ .

What is the exact out-of-pocket amount (in dollars) the patient will be required to pay for her first month's supply of this medication?

Show answer & explanation

Answer: 2900

Answer

2900

The correct calculation follows the standard sequence of health insurance benefits: first, the patient pays 100% of costs until the annual deductible is met. The patient pays the first USD 2,500 of the USD 4,500 medication cost. The remaining USD 2,000 is shared between the patient and the insurer according to the coinsurance rate. The patient pays 20% of USD 2,000, which is USD 400. Adding the deductible (USD 2,500) and the coinsurance (USD 400) gives a total out-of-pocket cost of USD 2,900. This amount does not exceed the annual out-of-pocket maximum of USD 6,500, so no further adjustments are needed.

Step-by-Step Solution

1

Determine the amount of the drug cost applied to the annual deductible.

USD

2,500

The patient has not used any healthcare services this year, so she must meet the entire deductible before the insurance plan begins to share costs.

2

Calculate the remaining balance of the medication cost after the deductible is met.

USD

4,500

- USD

2,500

= USD

2,000

This is the portion of the cost that is subject to the coinsurance rate.

3

Calculate the patient's coinsurance responsibility.

USD

2,000

\times 0.20 = USD

400

The patient is responsible for 20% of the costs remaining after the deductible is met.

4

Calculate the total out-of-pocket cost for this encounter.

USD

2,500

+ USD

400

= USD

2,900

The total financial responsibility is the sum of the deductible paid and the coinsurance paid.

5

Verify the total against the annual out-of-pocket maximum.

USD

2,900

is less than USD

6,500

Because the total is below the maximum, the calculated amount is the exact amount the patient owes.

Key Concept

Calculation of patient financial responsibility including deductibles, coinsurance, and out-of-pocket maximums, which is critical for advising patients on high-value care and mitigating financial toxicity.

Question 49Question →

A health system's pharmacy and therapeutics committee is reviewing a cost-utility analysis for a novel immunotherapy being considered for the formulary to treat advanced non-small cell lung cancer. The novel agent will be compared to the current standard of care, a conventional chemotherapy regimen.

The health economist presents the following data regarding the total lifetime costs and clinical outcomes per patient for each treatment pathway:

Treatment Pathway	Total Lifetime Cost	Average Life Expectancy	Health-Related Quality of Life (Utility)
Conventional Chemotherapy	$40,000	2.5 years	0.40
Novel Immunotherapy	$190,000	4.0 years	0.75

The committee mandates that new therapies must not exceed a willingness-to-pay threshold of $100,000 per quality-adjusted life year (QALY) gained.

What is the incremental cost-effectiveness ratio (ICER) of the novel immunotherapy compared to conventional chemotherapy, expressed in dollars per QALY gained?

Show answer & explanation

Answer: 75000

Answer

75000

The correct answer is derived by calculating the difference in costs between the two interventions and dividing it by the difference in their clinical benefits (measured in QALYs). The ICER formula is

ICER = \frac{\Delta Cost}{\Delta QALY}

. By substituting the given values, the resulting ICER is 75,000, which falls well below the committee's $100,000 threshold, indicating it is a highly cost-effective intervention.

Step-by-Step Solution

1

Calculate the Quality-Adjusted Life Years (QALYs) for the standard of care.

2.5 \text{ years} \times 0.40 = 1.0 \text{ QALY}

QALY represents disease burden, factoring in both the quality and quantity of life lived. It is calculated by multiplying the life expectancy by the utility weight.

2

Calculate the Quality-Adjusted Life Years (QALYs) for the novel therapy.

4.0 \text{ years} \times 0.75 = 3.0 \text{ QALYs}

This determines the total health benefit provided by the new intervention.

3

Determine the incremental health benefit (

\Delta QALY

).

3.0 - 1.0 = 2.0 \text{ QALYs gained}

The denominator of the ICER equation requires the net difference in health outcomes between the two therapies.

4

Determine the incremental cost (

\Delta Cost

).

190,000 - \$40,000 = \$150,000

The numerator of the ICER equation requires the net financial difference between the two therapies.

5

Calculate the Incremental Cost-Effectiveness Ratio (ICER).

\frac{\$150,000}{2.0 \text{ QALYs}} = 75,000

The ICER represents the additional cost required to generate one additional year of perfect health.

Key Concept

The Incremental Cost-Effectiveness Ratio (ICER) is a fundamental metric in healthcare economics used to assess the value of a new medical intervention compared to the standard of care by measuring the additional cost per additional Quality-Adjusted Life Year (QALY) gained.

Question 50Question →

A multicenter, randomized controlled trial is conducted to evaluate the safety of a novel oral anticoagulant (NOAC) compared to standard-of-care warfarin for stroke prevention in patients with nonvalvular atrial fibrillation and concurrent moderate chronic kidney disease. The study enrolls 2,400 patients, who are randomly assigned in a 1:1 ratio to receive either the NOAC or warfarin. All patients are followed for exactly two years. During the follow-up period, the primary safety endpoint of major gastrointestinal bleeding is assessed.

The investigators report the following safety data:
- NOAC group ( $n = 1,200$ ): 54 patients experienced a major gastrointestinal bleed.
- Warfarin group ( $n = 1,200$ ): 30 patients experienced a major gastrointestinal bleed.

Based on these study results, what is the number needed to harm (NNH) for major gastrointestinal bleeding associated with the NOAC compared to warfarin? (Enter your answer as a whole number).

Show answer & explanation

Answer: 50

Answer

50

The Number Needed to Harm (NNH) represents the number of patients who need to be exposed to a risk factor or experimental treatment to cause one additional adverse event compared to the control group. It is calculated as the inverse of the Absolute Risk Increase (ARI). First, the incidence of major gastrointestinal bleeding in the NOAC group is calculated as

54 / 1200 = 0.045

(

4.5\%

). The incidence in the warfarin group is

30 / 1200 = 0.025

(

2.5\%

). The ARI is the absolute difference between these two rates:

0.045 - 0.025 = 0.020

(

2.0\%

). The NNH is calculated as

1 / \text{ARI}

, which is

1 / 0.020 = 50

. Therefore, for every 50 patients treated with the NOAC instead of warfarin, one additional major gastrointestinal bleed will occur.

Step-by-Step Solution

1

Calculate experimental event rate

0.045 (4.5%)

Needed to determine the absolute risk increase associated with the new intervention.

2

Calculate control event rate

0.025 (2.5%)

Needed to determine the baseline risk of the adverse event under standard care.

3

Calculate Absolute Risk Increase (ARI)

0.020 (2.0%)

ARI represents the excess risk of the adverse event directly attributable to the NOAC.

4

Calculate Number Needed to Harm (NNH)

50

NNH is the inverse of the ARI (

1 / 0.020

).

Key Concept

Calculation and interpretation of the Number Needed to Harm (NNH)

Question 51Question →

A randomized controlled trial is conducted to evaluate the efficacy and safety of a novel P2Y12 inhibitor added to aspirin, compared to aspirin alone, in patients with a recent transient ischemic attack (TIA) or minor ischemic stroke. The primary efficacy endpoint is the 1-year incidence of recurrent stroke, and the primary safety endpoint is the 1-year incidence of major bleeding.

The 1-year event rates in the aspirin-alone (control) group were:
- Recurrent stroke: $16.0\%$
- Major bleeding: $2.0\%$

In the group receiving the novel P2Y12 inhibitor plus aspirin, the relative risk reduction (RRR) for recurrent stroke was $25.0\%$ compared to the control group. However, this intervention group also had a relative risk (RR) of $2.25$ for major bleeding compared to the control group.

Based on these results, what is the number needed to harm (NNH) for major bleeding? (Enter the exact integer).

Show answer & explanation

Answer: 40

Answer

40

To calculate the Number Needed to Harm (NNH), the Absolute Risk Increase (ARI) must first be determined. For major bleeding, the baseline Control Event Rate (CER) is

2.0\%

. The Experimental Event Rate (EER) is the CER multiplied by the Relative Risk (RR):

2.0\% \times 2.25 = 4.5\%

. The ARI is the difference between the EER and CER:

4.5\% - 2.0\% = 2.5\%

, or

0.025

. The NNH is the reciprocal of the ARI (

1 / 0.025 = 40

), meaning 40 patients need to be treated with the novel P2Y12 inhibitor to cause one additional major bleeding event.

Step-by-Step Solution

1

Identify the relevant data for major bleeding.

Control Event Rate (CER) for bleeding is 2.0% and Relative Risk (RR) is 2.25.

The question asks for the Number Needed to Harm (NNH), which applies to the adverse safety endpoint (major bleeding), so efficacy data must be filtered out.

2

Calculate the Experimental Event Rate (EER) for major bleeding.

EER = 2.0% × 2.25 = 4.5%.

EER must be determined to find the absolute difference in risk between the two groups.

3

Calculate the Absolute Risk Increase (ARI).

ARI = 4.5% - 2.0% = 2.5% (or 0.025).

ARI represents the excess risk of bleeding attributable to the novel therapy.

4

Calculate the Number Needed to Harm (NNH).

NNH = 1 / 0.025 = 40.

NNH is the reciprocal of the Absolute Risk Increase.

Key Concept

Number Needed to Harm (NNH) calculation using Control Event Rate and Relative Risk.

Question 52Question →

Investigators recently concluded a Phase III clinical study assessing a new monoclonal antibody against interleukin-5 (IL-5). The trial enrolled adults with severe, uncontrolled eosinophilic asthma to determine the drug's efficacy in preventing disease exacerbations. A total of $2,000$ participants were randomly assigned to receive either the IL-5 inhibitor or a matching placebo. The primary endpoint was the incidence of at least one severe asthma exacerbation requiring an emergency department visit or hospitalization over a $12$ -month period.

Data analysis revealed the following:
- Among the $1,200$ participants who received the IL-5 inhibitor, $108$ experienced a severe exacerbation.
- Among the $800$ participants who received the placebo, $112$ experienced a severe exacerbation.

Based on these findings, calculate the number needed to treat (NNT) to prevent one additional severe asthma exacerbation. (Enter your answer as a whole number)

Show answer & explanation

Answer: 20

Answer

20

The Number Needed to Treat (NNT) is calculated as

1 / \text{Absolute Risk Reduction (ARR)}

. First, calculate the event rates for both groups. The Control Event Rate (CER) is

112 / 800 = 0.14

(

14\%

). The Experimental Event Rate (EER) is

108 / 1200 = 0.09

(

9\%

). The ARR is the difference between these two rates:

0.14 - 0.09 = 0.05

(

5\%

). Finally, the NNT is the inverse of the ARR:

1 / 0.05 = 20

. This means that

20

patients must be treated with the IL-5 inhibitor for one year to prevent one additional severe asthma exacerbation.

Step-by-Step Solution

1

Calculate the Control Event Rate (CER)

CER = 112 / 800 = 0.14

CER represents the baseline risk of an exacerbation in the placebo group.

2

Calculate the Experimental Event Rate (EER)

EER = 108 / 1200 = 0.09

EER represents the risk of an exacerbation in the treatment group.

3

Calculate the Absolute Risk Reduction (ARR)

ARR = 0.14 - 0.09 = 0.05

ARR measures the absolute difference in risk between the control and experimental groups.

4

Calculate the Number Needed to Treat (NNT)

NNT = 1 / 0.05 = 20

NNT is the inverse of the ARR, representing how many patients must receive the intervention to prevent one adverse event.

Key Concept

Number Needed to Treat (NNT) Calculation

Question 53Question →

An investigational noninvasive urine biomarker is being evaluated for the detection of urothelial carcinoma in patients with asymptomatic microscopic hematuria. Epidemiological data indicates that the prevalence of urothelial carcinoma in the target demographic (adults aged 50-70 with isolated microscopic hematuria) is $4\%$ .

Validation studies of the new biomarker demonstrate a sensitivity of $80\%$ and a specificity of $90\%$ for detecting urothelial carcinoma.

A 62-year-old man with asymptomatic microscopic hematuria undergoes testing with this new urine biomarker, and the result is positive.

Based on these test characteristics, what is the probability (in percentage) that this patient actually has urothelial carcinoma?

Show answer & explanation

Answer: 25

Answer

The probability that the patient actually has urothelial carcinoma is 25%.

The positive predictive value (PPV) is heavily influenced by the prevalence of the disease in the population being tested. Despite a seemingly high sensitivity (80%) and specificity (90%), the low prevalence (4%) means that the absolute number of healthy individuals is vastly larger than the number of diseased individuals. Consequently, the 10% false positive rate generates a large number of false positive results (960) that outnumber the true positive results (320). Therefore, the probability that a positive test indicates true disease is only 320 / 1280, or 25%.

Step-by-Step Solution

1

Assume a hypothetical population of 10,000 patients from the target demographic.

Total population = 10,000.

Using a hypothetical population size makes calculating the exact number of true and false positives simpler than using probabilities directly.

2

Calculate the number of patients with and without the disease based on the 4% prevalence.

Patients with disease = 400. Patients without disease = 9,600.

Prevalence dictates the baseline risk (pretest probability) in the specific demographic being tested.

3

Calculate True Positives (TP) using the 80% test sensitivity.

TP = 400 × 0.80 = 320.

Sensitivity is the proportion of patients with the disease who correctly test positive.

4

Calculate False Positives (FP) using the 90% test specificity.

FP = 9,600 × (1 - 0.90) = 9,600 × 0.10 = 960.

Specificity is the proportion of healthy patients who correctly test negative. The remainder (10%) represents healthy patients who incorrectly test positive.

5

Calculate the Positive Predictive Value (PPV).

PPV = 320 / (320 + 960) = 320 / 1,280 = 0.25 (or 25%).

PPV represents the probability that a positive test result is a true positive.

Key Concept

Positive Predictive Value (PPV) dependence on disease prevalence

Alternative Method

This can also be solved using Likelihood Ratios (LR). Pretest Probability = 0.04, so Pretest Odds = 0.04 / 0.96 = 1/24. The Positive Likelihood Ratio (LR+) = Sensitivity / (1 - Specificity) = 0.80 / 0.10 = 8. Post-test Odds = Pretest Odds × LR+ = (1/24) × 8 = 1/3. Converting back to probability: Post-test Probability = Odds / (1 + Odds) = (1/3) / (4/3) = 1/4 = 25%.

Estimated Time:1m 30s

Question 54Question →

A maternal-fetal medicine network is evaluating the impact of a newly proposed sequential screening protocol for maternal autoantibodies associated with fetal heart block. The prevalence of these autoantibodies in their patient population is $2\%$ . The proposed protocol evaluates all pregnant women with a primary serological assay. Patients who test positive on the primary assay subsequently undergo a secondary advanced functional assay. Patients who test negative on the primary assay undergo no further testing.

Performance characteristics of the assays are as follows:
- Primary Assay: Sensitivity = $100\%$ , Specificity = $75\%$
- Secondary Assay: Sensitivity = $90\%$ , Specificity = $96\%$

If a cohort of $25,000$ pregnant women is screened using this protocol, how many women are expected to receive a false-positive final diagnosis?

Show answer & explanation

Answer: 245

Answer

245

To find the number of final false-positive diagnoses in a sequential screening protocol, we must track the disease-free population through each testing stage. First, identify the number of true negatives in the cohort: a

2\%

prevalence in

25,000

individuals means there are

500

true positives and

24,500

true negatives. The primary assay has a specificity of

75\%

, which corresponds to a false positive rate of

25\%

. When the

24,500

disease-free women are tested,

24,500 \times 0.25 = 6,125

will incorrectly test positive. In a sequential testing algorithm, only those who test positive on the initial screen proceed to the confirmatory screen. Thus, only these

6,125

false-positive women (along with the

500

true positives) will undergo the secondary assay. The secondary assay has a specificity of

96\%

, which corresponds to a false positive rate of

4\%

. When the

6,125

healthy women take this test,

6,125 \times 0.04 = 245

will receive a second false-positive result. These

245

women represent the final false-positive diagnoses for the protocol.

Step-by-Step Solution

1

Calculate the total number of disease-free individuals in the screened cohort.

24,500

women

False-positive results can only occur in individuals who do not actually have the condition. With a

2\%

prevalence in a cohort of

25,000

, there are

25,000 \times (1 - 0.02) = 24,500

disease-free women.

2

Determine the number of false positives generated by the primary assay.

6,125

women

The primary assay has a specificity of

75\%

, meaning it correctly identifies

75\%

of healthy individuals. The remaining

25\%

will receive a false-positive result (

24,500 \times 0.25 = 6,125

).

3

Calculate the number of false positives generated by the secondary assay.

245

women

Because this is sequential testing, only those who tested positive on the primary assay (

6,125

false positives) are subjected to the secondary assay. The secondary assay has a specificity of

96\%

, meaning

4\%

of these healthy women will incorrectly test positive again (

6,125 \times 0.04 = 245

).

Key Concept

In sequential (series) diagnostic testing, the overall false positive rate is determined by applying the false positive rate (1 - specificity) of the subsequent test only to the false positives identified by the preceding test.

Question 55Question →

An independent medical advisory board is reviewing a newly developed oral agent for the prevention of major adverse cardiovascular events (MACE) in patients with familial hypercholesterolemia. They are analyzing $5$ -year outcome data from a randomized, placebo-controlled trial to evaluate the drug's net clinical benefit.

The trial provides the following cumulative incidence data over the $5$ -year study period:
- Baseline risk (placebo group) of MACE: $25\%$
- Relative risk (RR) of MACE with the new agent: $0.60$
- Baseline risk (placebo group) of drug-induced hepatotoxicity: $2\%$
- Relative risk (RR) of drug-induced hepatotoxicity with the new agent: $2.0$

Based on these findings, what is the absolute difference between the Number Needed to Harm (NNH) for hepatotoxicity and the Number Needed to Treat (NNT) to prevent one MACE over the $5$ -year period?

(Enter the exact whole number.)

Show answer & explanation

Answer: 40

Answer

The absolute difference between the NNH and NNT is 40.

To find the NNT and NNH, you must first convert the given Relative Risks (RR) and baseline risks (CER) into absolute risks for the treatment group (EER). For MACE, the CER is

0.25

and the RR is

0.60

, yielding an EER of

0.15

. The ARR is

0.25 - 0.15 = 0.10

, meaning the NNT is

\frac{1}{0.10} = 10

. For hepatotoxicity, the CER is

0.02

and the RR is

2.0

, yielding an EER of

0.04

. The ARI is

0.04 - 0.02 = 0.02

, meaning the NNH is

\frac{1}{0.02} = 50

. The absolute difference between the NNH (

50

) and the NNT (

10

) is

40

.

Step-by-Step Solution

1

Calculate the Experimental Event Rate (EER) and Absolute Risk Reduction (ARR) for MACE.

EER for MACE =

0.25 \times 0.60 = 0.15

(

15\%

). ARR = Control Event Rate (CER) - EER =

0.25 - 0.15 = 0.10

(

10\%

).

Relative Risk (RR) is the ratio of EER to CER. Therefore,

EER = CER \times RR

. ARR is the absolute difference in risk between the control and experimental groups for a beneficial outcome.

2

Calculate the Number Needed to Treat (NNT) for MACE.

NNT = \frac{1}{ARR} = \frac{1}{0.10} = 10

.

NNT represents the number of patients who need to receive the intervention to prevent one adverse event.

3

Calculate the Experimental Event Rate (EER) and Absolute Risk Increase (ARI) for hepatotoxicity.

EER for hepatotoxicity =

0.02 \times 2.0 = 0.04

(

4\%

). ARI = EER - CER =

0.04 - 0.02 = 0.02

(

2\%

).

Just like with beneficial outcomes, the EER for an adverse effect is calculated as

CER \times RR

. ARI is the absolute increase in risk caused by the intervention.

4

Calculate the Number Needed to Harm (NNH) for hepatotoxicity and find the absolute difference.

NNH = \frac{1}{ARI} = \frac{1}{0.02} = 50

. The absolute difference is

|50 - 10| = 40

.

NNH represents the number of patients who need to be exposed to the risk factor to cause one additional adverse event. The question specifically asks for the absolute difference between NNH and NNT.

Key Concept

Calculation of Absolute Risk Reduction (ARR), Absolute Risk Increase (ARI), Number Needed to Treat (NNT), and Number Needed to Harm (NNH) using Relative Risk (RR) and baseline incidence.

Question 56Question →

Researchers at an academic medical center investigate the use of a novel microbiome restorative therapy to prevent recurrent *Clostridioides difficile* infection (CDI). The study enrolls adults who have recently completed standard antibiotic therapy for a first episode of CDI. Participants are randomly assigned in a 1:1 ratio to receive either the novel therapy or a visually matched placebo.

Over a 6-month follow-up period, the following data are recorded:

Group	Total Patients	Recurrent CDI Events
Placebo	400	80
Novel therapy	400	32

Based on the study results, what is the number needed to treat (NNT) to prevent one additional episode of recurrent CDI? (Provide your answer as a whole number, applying standard epidemiological rounding principles).

Show answer & explanation

Answer: 9

Answer

9

To find the number needed to treat (NNT), first calculate the absolute risk reduction (ARR). The risk of recurrent CDI in the placebo group (CER) is

80/400 = 0.20

. The risk in the treatment group (EER) is

32/400 = 0.08

. The ARR is the difference between these two rates:

0.20 - 0.08 = 0.12

. The NNT is the reciprocal of the ARR:

1 / 0.12 = 8.33

. In biostatistics, the NNT must always be rounded up to the next whole number (ceiling function) to provide a conservative estimate. Therefore,

8.33

rounds up to

9

.

Step-by-Step Solution

1

Calculate the Control Event Rate (CER).

CER =

80 / 400 = 0.20

(

20\%

).

This establishes the baseline risk of the outcome (recurrent CDI) without the new intervention.

2

Calculate the Experimental Event Rate (EER).

EER =

32 / 400 = 0.08

(

8\%

).

This establishes the risk of the outcome when using the novel therapy.

3

Calculate the Absolute Risk Reduction (ARR).

ARR = CER - EER =

0.20 - 0.08 = 0.12

(

12\%

).

ARR represents the actual difference in risk between the two groups, showing how much the intervention reduces the absolute risk of the event.

4

Calculate the Number Needed to Treat (NNT).

NNT =

1 / \text{ARR} = 1 / 0.12 = 8.33

.

NNT is the reciprocal of the Absolute Risk Reduction. It indicates the theoretical number of patients that need to be treated to prevent one bad outcome.

5

Apply conservative rounding rules for NNT.

Round

8.33

up to

9

.

Because treating

8

patients would only prevent

8 \times 0.12 = 0.96

events (less than one full event), you must treat the

9^{\text{th}}

patient to guarantee the prevention of at least one event. NNT is always rounded up to the nearest whole number (ceiling function) to avoid overstating clinical benefit.

Key Concept

Number Needed to Treat (NNT) calculation and conservative rounding principles

Question 57Question →

A 62-year-old man presents to the emergency department with sudden-onset left-sided weakness and slurred speech. He has a history of hypertension and type 2 diabetes. A new point-of-care biomarker assay is being evaluated in the emergency department to aid in the diagnosis of acute ischemic stroke. The prevalence (pre-test probability) of acute ischemic stroke in patients presenting to this department with similar symptoms is estimated to be $30\%$ . The biomarker assay has a reported sensitivity of $90\%$ and a specificity of $85\%$ . The patient undergoes the assay, and the result is positive.

Based on these parameters, what is the probability (in percentage) that this patient actually has an acute ischemic stroke?

Show answer & explanation

Answer: 72

Answer

72

The correct answer is

72

. The positive predictive value (PPV) calculates the probability that a patient has the disease given a positive test result. This value heavily depends on the sensitivity, specificity, and prevalence (pre-test probability) of the disease. Using a

2 \times 2

table for a hypothetical cohort of

1000

patients:

300

have the disease and

700

do not. The test correctly identifies

270

of the

300

diseased patients (

90\%

sensitivity). It incorrectly identifies

105

of the

700

healthy patients (

100\% - 85\%

specificity). The total number of positive tests is

270 + 105 = 375

. The PPV is the number of true positives divided by the total positives:

\frac{270}{375} = 72\%

.

Step-by-Step Solution

1

Determine the disease and non-disease populations

Disease = 300, No Disease = 700

Using a hypothetical cohort of 1000 patients and a prevalence of 30% simplifies the calculation.

2

Calculate True Positives (TP)

TP = 270

Sensitivity is 90%, so 90% of the 300 patients with the disease will correctly test positive.

3

Calculate False Positives (FP)

FP = 105

Specificity is 85%, so 15% of the 700 patients without the disease will incorrectly test positive.

4

Calculate the Positive Predictive Value (PPV)

PPV = 72%

PPV is the proportion of true positives out of all positive tests: TP / (TP + FP) = 270 / 375 = 0.72.

Key Concept

Positive Predictive Value calculation using pre-test probability (prevalence), sensitivity, and specificity.

Question 58Question →

A team of epidemiologists conducts a case-control study to investigate a potential link between the use of a specific pre-workout dietary supplement and the occurrence of new-onset atrial fibrillation in adults under the age of 40. The researchers identify 150 patients diagnosed with new-onset atrial fibrillation (cases) and 300 healthy matched individuals without a cardiac history (controls).

Detailed dietary history interviews indicate that 90 of the patients with atrial fibrillation regularly consumed the pre-workout supplement in the month preceding their diagnosis. Among the healthy controls, 120 reported regular consumption of the same supplement.

Based on these findings, what is the odds ratio for the association between regular consumption of the pre-workout supplement and the development of new-onset atrial fibrillation? (Provide the exact value).

Show answer & explanation

Answer: 2.25

Answer

2.25

In a case-control study, the odds ratio (OR) is the appropriate measure of association. It is calculated by dividing the odds of exposure among cases (

a/c

) by the odds of exposure among controls (

b/d

), which simplifies to the cross-product

(a \times d) / (b \times c)

. Using the values

a=90

,

c=60

,

b=120

, and

d=180

, the calculation is

(90 \times 180) / (120 \times 60) = 16200 / 7200 = 2.25

.

Step-by-Step Solution

1

Determine the exposure status among the cases (atrial fibrillation group).

Exposed cases = 90. Unexposed cases = 150 - 90 = 60.

To find the numerator and denominator for the odds of exposure in the case group.

2

Determine the exposure status among the controls (healthy group).

Exposed controls = 120. Unexposed controls = 300 - 120 = 180.

To find the numerator and denominator for the odds of exposure in the control group.

3

Calculate the odds of exposure for both groups.

Odds for cases = 90 / 60 = 1.5. Odds for controls = 120 / 180 = 0.667.

The odds ratio compares the odds of exposure in cases to the odds of exposure in controls.

4

Calculate the final odds ratio.

Odds Ratio = 1.5 / 0.667 = 2.25.

To quantify the strength of the association between the supplement and atrial fibrillation.

Key Concept

Odds Ratio (OR) calculation in a case-control study

Question 59Question →

A public health task force is evaluating a new rapid serologic assay for a tick-borne illness endemic to a specific region. During initial clinical validation at a tertiary care center, the assay was tested on a cohort of $2,000$ individuals, exactly half of whom had PCR-confirmed infection. In this validation cohort, the assay correctly identified $900$ of the infected individuals and yielded a positive result in $25$ of the uninfected individuals.

The task force now plans to deploy this assay to a rural outpatient clinic where epidemiological data indicates that $10\%$ of the tested patients actually have the infection.

If a patient at this rural clinic tests positive using the rapid assay, what is the probability (expressed as a percentage) that they truly have the infection?

Show answer & explanation

Answer: 80

Answer

80

The positive predictive value (PPV) represents the probability that a patient with a positive test result actually has the disease. Because predictive values are highly dependent on disease prevalence, the PPV from the validation cohort (

50\%

prevalence) cannot be generalized to the rural clinic (

10\%

prevalence). However, the intrinsic properties of the test—sensitivity (

90\%

) and specificity (

97.5\%

)—remain constant. By applying these constant values to a hypothetical population representing the rural clinic (e.g.,

10,000

patients, yielding

1,000

infected and

9,000

uninfected), we determine the new proportion of true positives (

900

) to all positive test results (

900 + 225 = 1125

). Calculating

900 / 1125

yields exactly

80\%

.

Step-by-Step Solution

1

Calculate the test's sensitivity and specificity using data from the validation cohort (prevalence = 50%).

Sensitivity = 90% (900 TP / 1000 infected). Specificity = 97.5% (975 TN / 1000 uninfected, since FP = 25).

Intrinsic test properties remain constant regardless of disease prevalence and are required for downstream calculations.

2

Determine the number of True Positives (TP) and False Positives (FP) for the rural clinic population (prevalence = 10%). Assume a cohort of 10,000 patients for simpler math.

Infected = 1,000; Uninfected = 9,000. TP = 1,000 × 0.90 = 900. FP = 9,000 × 0.025 = 225.

Because PPV is prevalence-dependent, we must calculate the absolute numbers of individuals testing positive in this specific clinical setting.

3

Calculate the Positive Predictive Value (PPV) using the newly calculated TP and FP values.

PPV = 900 / (900 + 225) = 900 / 1125 = 0.80, or 80%.

PPV represents the probability that a positive test result indicates a true infection.

Key Concept

The relationship between disease prevalence and predictive values; while sensitivity and specificity are fixed properties of a test, Positive Predictive Value (PPV) decreases significantly as disease prevalence decreases.

Question 60Question →

A 48-year-old man is brought to the emergency department by emergency medical services after being found unresponsive in a park. His medical history is unknown. On physical examination, he is unarousable and his breath has a faint, sweet chemical odor. Vital signs show a temperature of $36.2^\circ\text{C}$ ( $97.2^\circ\text{F}$ ), blood pressure of $110/70\text{ mm Hg}$ , heart rate of $105/\text{min}$ , and respiratory rate of $22/\text{min}$ .

Initial laboratory studies are obtained:

Laboratory Parameter	Result	Reference Range
Sodium	$138\text{ mEq/L}$	$136-145\text{ mEq/L}$
Potassium	$4.5\text{ mEq/L}$	$3.5-5.0\text{ mEq/L}$
Chloride	$102\text{ mEq/L}$	$98-106\text{ mEq/L}$
Bicarbonate	$14\text{ mEq/L}$	$22-28\text{ mEq/L}$
Blood urea nitrogen (BUN)	$14\text{ mg/dL}$	$8-20\text{ mg/dL}$
Glucose	$126\text{ mg/dL}$	$70-110\text{ mg/dL}$
Measured osmolality	$318\text{ mOsm/kg H}_2\text{O}$	$275-295\text{ mOsm/kg H}_2\text{O}$

Arterial blood gas testing on room air shows a pH of 7.24, $p\text{CO}_2$ of $28\text{ mm Hg}$ , and $p\text{O}_2$ of $92\text{ mm Hg}$ . A serum ethanol level is undetectable.

What is this patient's serum osmolal gap?

Show answer & explanation

Answer: 30

Answer

30

The correct answer is 30. The patient's presentation of unresponsiveness, high anion gap metabolic acidosis (anion gap of 22), and an elevated osmolal gap is highly suspicious for a toxic alcohol ingestion (such as methanol or ethylene glycol). To find the osmolal gap, one must first determine the calculated osmolality using the formula:

2[\text{Na}^+] + \frac{\text{Glucose}}{18} + \frac{\text{BUN}}{2.8}

. Using the patient's lab values, this is

2(138) + \frac{126}{18} + \frac{14}{2.8} = 276 + 7 + 5 = 288\text{ mOsm/kg H}_2\text{O}

. The osmolal gap is then obtained by subtracting the calculated osmolality from the measured osmolality:

318 - 288 = 30

. A normal osmolal gap is typically

\le 10\text{ mOsm/kg H}_2\text{O}

. A gap of 30 indicates the presence of unmeasured, osmotically active substances in the blood.

Step-by-Step Solution

1

Identify the formula for calculated serum osmolality.

\text{Calculated Osmolality} = 2[\text{Na}^+] + \frac{\text{Glucose}}{18} + \frac{\text{BUN}}{2.8}

This standard formula converts the major serum solutes (sodium and its accompanying anions, glucose, and urea) into their milliosmole equivalents.

2

Input the patient's laboratory values into the formula.

2(138) + \frac{126}{18} + \frac{14}{2.8}

Sodium is 138 mEq/L, glucose is 126 mg/dL, and BUN is 14 mg/dL.

3

Calculate the individual components and sum them up.

276 + 7 + 5 = 288\text{ mOsm/kg H}_2\text{O}

Completing the arithmetic provides the expected osmolality based on normal physiological solutes.

4

Determine the osmolal gap.

\text{Osmolal Gap} = 318 - 288 = 30\text{ mOsm/kg H}_2\text{O}

The osmolal gap is the difference between the directly measured osmolality and the calculated osmolality.

Key Concept

The serum osmolal gap is calculated by subtracting the calculated serum osmolality from the measured serum osmolality. A normal gap is typically less than 10 mOsm/kg H2O; an elevated gap in the setting of a high anion gap metabolic acidosis strongly suggests toxic alcohol ingestion (e.g., methanol, ethylene glycol).

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