Resistance to venous flow: A mathematical description and implications for compartment syndromes.

Background: The object of this work was to describe resistance to flow within a vein in a closed compartment.

Methods: A vein is mathematically modeled as a collapsible cylinder with fixed perimeter exposed to extraluminal hydrostatic pressure within a closed compartment of the body. The principle of minimization of energy is used to determine the cross-sectional area and resistance to flow through such a cylinder in various states of collapse.

Results: A mathematical expression for the cross-sectional area of a partially collapsed tube is derived. Resistance to flow is calculated within the tube in various states of collapse and compared with the resistance to flow in an annular tube of identical cross-sectional area. Resistance increases very rapidly in the first 5% of collapse and remains greater than that in an annular vessel of identical cross-sectional area through all further collapse. Resistance to flow closely follows a logarithmic gain as the tube undergoes collapse from extraluminal pressure until opposite sides of the vein make contact.

Conclusion: Within a closed compartment in which there is rising pressure, resistance to flow through a vein is predicted to increase as a logarithmic function of the vein's cross-sectional area. This rapid rise in resistance and hence decline in flow are consistent with the position that in compartment syndromes of all anatomic locations, the venous contribution to resistance of flow is of paramount importance to the pathophysiology.