Try spectrophotometer based dpph and abts free radical scavenging assays for estimation of antioxidants especially phenolics
but if sample contains lyophilic antioxidants with lower polarity than you can use dpph assay estimated by electron paramagnetic resonance (epr) following
optimization of the extraction of antioxidative constituents of six barley cultivars and their antioxidant properties. J. Agric. Food chem. 2006, 54, 8048–8057.
Indeed, none of the above methods measures bioavailability of any antioxidant. You could estimate the impact of food antioxidant on the organism antioxidative defence system only. If you want to quantify bioavailability of some antioxidant you actually need to use compound-specific pharmacokinetic protocol(s).