I don’t have time to look into this in any detail, but this quote suggests that milk thistle won’t reach high enough concentrations to influence rapamycin metabolism in humans unless perhaps in those that suffer from liver damage:
" Both SLM and GLZ formulations, taken by a relatively large number of patients (15 and 19, respectively), were shown to reduce sirolimus CL/F significantly by about 34%. Silymarin, a flavonolignan from the ‘milk thistle’ plant, has been used almost exclusively for hepatoprotection for hundreds of years all over the world . It is a complex mixture of four flavonolignan isomers, namely silybin, isosilybin, silydianin and silychristin . In vitro studies have shown that silymarin, in higher concentrations, has a strong inhibitory effect on both P450 3A4 [56–58] and P-gp [59–61]. The concentration tested in vitro exceeded physiologically attainable levels [56–58, 60, 61] and hence previous studies in vivo have shown no to mild inhibition effects when co-administered with ranitidine , nifedipine , midazolam  and indinavir [65, 66] in healthy volunteers. Recently, Schrieber et al. reported that AUC0–24 h for the sum of total silymarin flavonolignans was 2.4-, 3.3- and 4.7-fold higher for hepatitis C virus (HCV) noncirrhosis, non-alcoholic fatty liver disease, and HCV cirrhosis cohorts, respectively, compared with healthy volunteers . It is a common practice in China as well as in the current study that SLM or GLZ is added by clinicians when patients have abnormal liver function indices. Therefore, the remarkable inhibition effect of SLM on sirolimus CL/F could be explained by accumulated high concentrations of SLM in these patients with hepatic impairment."