Erratum—Fluid Flow in the VRDP Model
In our recent paper, we studied the joint contact mechanics between two symmetric, spherical, cortical bones covered with biphasic, linearly elastic cartilage layers. We analyzed the effects of the fluid boundary conditions at the contacting surfaces (permeable or impermeable) on the fluid flow patterns using a finite element model.
The condition in Eq. (9) is correct only in regions where no contact occurs. In the contact region, fluid flow is controlled by the difference in fluid pressure between the contacting surfaces. This result agrees with earlier findings by Ateshian and Wang 1 and Kelkar and Ateshian 2, who showed that the biphasic jump condition [Eq. (11)] must be enforced in the region of contact, while free-draining conditions should be used anywhere else at the articular surface.
The load sharing between elastic stress and fluid pressure in the VRDP model was recalculated using the corrected fluid boundary conditions [Eq. (9, corrected)] and were compared to the results obtained using the SEAL model which had impermeable surfaces (Fig. 3, corrected). The predictions from the VRDP model show a stress relaxation that is physically realistic, whereas those predicted by the SEAL model are not. The stress is carried mainly by the fluid phase in the VRDP model (which is in contrast with the results presented in our original manuscript).
Furthermore, we plotted the correct fluid flow at (end of the loading ramp; Fig. 4, corrected) and at (end of the simulation; Fig. 5, corrected).
At the end of the loading ramp (Fig. 4, corrected), the fluid flow in the VRDP model is qualitatively similar to that of the SEAL model.
At the end of the simulation the fluid flow predicted by the VRDP model is one-dimensional in the contact region [Fig. 5(b), corrected]. Therefore, the biphasic jump condition is preserved. Outside the contact region, the fluid flow is two-dimensional, with an outward component.
The corrected solution presented here confirms conceptually the results in our original paper. The permeable surface model (VRDP) predicts the stress relaxation behavior of the tissue properly, whereas the impermeable surface model (SEAL) does not. Moreover, the fluid flow predicted by the VRDP model is two-dimensional, and the fluid velocity at the free surface has outward components, which is consistent with the experimental observations on fluid exudation.
Dr. Gerard A. Ateshian, for the technical suggestions and the references provided.
Corresponding author: Walter Herzog Human Performance Laboratory, Faculty of Kinesiology, The University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4.