A 52-year-old man is evaluated in the hospital for several episodes of hemoptysis that developed over the past day. He reports feeling well until about 3 weeks ago when he developed myalgia, arthralgia, occasional epistaxis, and diminished hearing. One week ago he developed a rash and weakness in his right hand. History is otherwise unremarkable, and he takes no medications.

On physical examination, temperature is 38.0 °C (100.4 °F), blood pressure is 152/100 mm Hg, pulse rate is 72/min, and respiration rate is 24/min. Conjunctivitis is present in both eyes. Decreased hearing in both ears is noted. Bilateral maxillary sinus tenderness is present. Chest examination reveals diffuse rhonchi. The right hand has decreased grip strength. Palpable purpura of the bilateral lower extremities is present.

Laboratory studies:

Erythrocyte sedimentation rate

84 mm/h

Leukocyte count

12,300/µL (12.3 × 109/L), eosinophils <2%

Complements (C3, C4)

Normal

Creatinine

2.1 mg/dL (185.6 µmol/L)

Urinalysis

3+ protein; 50 erythrocytes/hpf; 20 leukocytes/hpf; several mixed cellular casts

Sinus radiograph shows bony erosion of the septum and turbinates. Chest radiograph shows diffuse infiltrates.

Which of the following is most likely to establish the diagnosis?