How to determine the actual flux in the human metabolic model of a cell line?
I am no expert on the field, but quantifying gene expression isn't a simple problem, although it is a possibility.

A solution I know to be used is using radioactive markers. See for example this review and particularly "The cell-cell lactate shuttle" section, which might give some insights.

This can include radioactive tracers (such as [14C]lactate), non-radioactive tracers (with [13C] or [2H]), "net exchange measurements" (I don't really know what this means here, perhaps the kind of exchange calculations they present here), and muscle biopsies (it is then possible to measure the metabolite concentration by enzymatic reaction or spectrophotometry for example).