The main aim of this study is to evaluate the PK, safety, tolerability and immunogenicity of subcutaneous (SC) administration of TAK-881 in adult and pediatric participants with PIDD and compare them to HYQVIA in participants 16 years old and older. The participants will be treated with TAK-881/HYQVIA or HYQVIA/TAK-881 with the same dose and dosing interval of immunoglobulin for up to 51 weeks (for participants greater than or equal to \[\>=\]16 years) and only with TAK-881 for up to 27 weeks (for participants aged 2 to less than \[\<\]16 years) as they were treated with another immunoglobulin before enrollment. Participants will need to visit the clinic every 3 or 4 weeks during the duration of the study.
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Area Under the Curve During the Dosing Interval at Steady-State (AUC0-tau;ss) of total IgG with TAK-881 and HYQVIA in Participants Aged >=16 Years with PIDD
Timeframe: 3-Week dosing: Day 1 (pre and post-infusion), 24, 48, 72 hours and 7, 14, 21 days (post-infusion); 4-Week dosing: Day 1 (pre and post-infusion), 24, 48 and 72 hours and 7, 14, 21, 28 days (post-infusion) of last infusion in each cross-over epoch