To adjust respiratory rate to meet a target PaCO2, the formula for RR is:

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The essential idea is that PaCO2 is governed by alveolar ventilation: with constant CO2 production, PaCO2 changes inversely with minute ventilation. To move toward a target PaCO2 by adjusting the respiratory rate while keeping tidal volume the same, you scale the current rate by the ratio of the current PaCO2 to the desired PaCO2. The correct expression is RR_new = (current PaCO2 / desired PaCO2) × current RR. This means if the current PaCO2 is higher than the target, the factor is greater than one and RR increases to blow off more CO2; if the current PaCO2 is lower than the target, the factor is less than one and RR decreases. This approach assumes CO2 production is stable and tidal volume remains unchanged, and it’s a simplified way to approximate how to adjust RR to reach a target PaCO2. The other forms either misplace the ratio or treat PaCO2 as directly proportional instead of inversely related to minute ventilation, which wouldn’t reliably steer PaCO2 toward the target.