Clinicians generally assume that Medical Compression Stockings (MCS) work by reducing vein luminal diameter and, in this way, help to prevent blood pooling. Conflicting results have been reported however in the case of lower leg deep veins which call into question this hypothesis. The purpose of this contribution is to study the biomechanical response of the main lower leg deep veins to elastic compression and muscle contraction with the objective of improving our current understanding of the mechanism by which MCS convey their benefits. The development of a finite-element model of a slice of the lower leg from MR images is detailed. Analysis of the finite-element model shows that the contribution of the MCS to the deep vein diameter reduction is rather small, and in fact negligible, compared to that of the contracting muscle (3 and 9% decrease in the vein cross-sectional area with a grade II compression stocking in the supine and standing positions respectively, while complete collapse was obtained at the end of muscle activation). A more accurate representation of the muscle activation is eventually proposed to study the effect of muscle contraction on a vein wall. The impact on the venous blood draining is discussed.