Genome-wide association study (GWAS) data are used to explore the associations between blood metabolites and 5 respiratory diseases: asthma, tuberculosis (TB), chronic obstructive pulmonary disease (COPD), cor pulmonale, and bronchitis. The main method of analysis used is the inverse-variance weighted (IVW) approach, complemented by several sensitivity analyses, including MR-Egger regression, the weighted median, the weighted mode, Cochran's Q test, and the pleiotropy test. Additional directional tests, Meta-analysis and metabolic pathway analyses are conducted for deeper insights. 3 metabolites showing significant causal relationships are identified. Catechol glucuronide levels as a protective factor have a positive causal relationship with asthma; the creatine to carnitine ratio has a negative causal relationship with COPD as a risk factor; and the adenosine 5'-diphosphate (ADP) to N-acetylglucosamine to N-acetylgalactosamine ratio as a protective factor has a positive causal relationship with bronchitis. Additionally, 13 metabolites demonstrate strong causal relationships. Furthermore, we delineate 14 metabolic pathways related to the outcomes, including 6 associated with asthma, 2 with TB, 1 with COPD, 4 with cor pulmonale, and 1 with bronchitis. A causal relationship between blood metabolites and 5 respiratory diseases has been established. The identified metabolites and pathways offer new insights into the underlying mechanisms of these diseases, necessitating further experimental validation.

Aiming at parabolic optimal control problem with integral control constraints, a fully discrete finite element method based on Crank-Nicolson scheme is proposed. The state is discretized by piecewise linear finite elements for the space discretization, Crank-Nicolson scheme for time discretization, and the control variables are approximated by piecewise linear function approximation, so as to obtain a discrete optimal system. The error estimates of state variables, adjoint state variables and control variables are proved, and the theoretical results are verified by numerical examples.