The most appropriate management is to increase the positive end-expiratory pressure (PEEP). This patient is hypoxic due to acute respiratory distress syndrome (ARDS). Her gas exchange has worsened to the point of an arterial PO₂/FIO₂ ratio of 54 mm Hg (7.2 kPa), which indicates very severe ARDS. PEEP improves lung compliance, ventilation-perfusion (V/Q) matching, and oxygenation. A 2013 systematic review found no difference in survival using higher PEEP (approximately 15 cm H₂O) versus lower PEEP (approximately 8 cm H₂O); however, the patients with the most severe ARDS did show improvement in oxygenation when treated with higher PEEP, but there was no improvement in the number of ventilator-free days. Current recommendations are to use a PEEP level that achieves adequate oxygenation with an FIO₂ of less than 0.6 and does not cause hypotension.

Decreasing the tidal volume could be considered a lung-protective measure, but this patient's tidal volume is already at the target of 6 mL/kg of ideal body weight. Reducing the tidal volume further will likely result in greater alveolar hypoventilation and increasing hypoxemia.

Prone positioning is a maneuver for improving oxygenation, and may be helpful in patients with ARDS in whom it is difficult to maintain adequate levels of oxygenation. However, its effect on oxygenation is slow, and it would not be the best next step toward improving this patient's gas exchange.

Increasing the set respiration rate on the ventilator probably would not improve this patient's oxygenation unless it induced auto-PEEP, which would raise the effective PEEP level indirectly. This would not be the best way to recruit alveoli, however, because the auto-PEEP in this situation is impossible to regulate as lung mechanics change over time. It is safer and more reliable to raise the set level of PEEP.