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Research Papers

Flow-Induced Deformation of Poroelastic Tissues and Gels: A New Perspective on Equilibrium Pressure-Flow-Thickness Relations

[+] Author and Article Information
Thomas M. Quinn

Department of Chemical Engineering,
McGill University Montreal,
Quebec, CanadaH3A 2B2
e-mail: thomas.quinn@mcgill.ca

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received July 27, 2012; final manuscript received November 15, 2012; accepted manuscript posted November 29, 2012; published online December 27, 2012. Assoc. Editor: James C. Iatridis.

J Biomech Eng 135(1), 011009 (Dec 27, 2012) (8 pages) Paper No: BIO-12-1326; doi: 10.1115/1.4023095 History: Received July 27, 2012; Revised November 15, 2012; Accepted November 29, 2012

Hydrostatic pressure-driven flows through soft tissues and gels cause deformations of the solid network to occur, due to drag from the flowing fluid. This phenomenon occurs in many contexts including physiological flows and infusions through soft tissues, in mechanically stimulated engineered tissues, and in direct permeation measurements of hydraulic permeability. Existing theoretical descriptions are satisfactory in particular cases, but none provide a description which is easy to generalize for the design and interpretation of permeation experiments involving a range of different boundary conditions and gel properties. Here a theoretical description of flow-induced permeation is developed using a relatively simple approximate constitutive law for strain-dependent permeability and an assumed constant elastic modulus, using dimensionless parameters which emerge naturally. Analytical solutions are obtained for relationships between fundamental variables, such as flow rate and pressure drop, which were not previously available. Guidelines are provided for assuring that direct measurements of hydraulic permeability are performed accurately, and suggestions emerge for alternative measurement protocols. Insights obtained may be applied to interpretation of flow-induced deformation and related phenomena in many contexts.

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Figures

Grahic Jump Location
Fig. 1

One-dimensional fluid flow through a soft tissue block is considered under two situations. (a) In the free surface situation, the tissue is compressed solely by flow-induced deformation due to a steady fluid velocity U in the z direction. This forces it against a rigid, porous support at its downstream surface (where pressure p = 0), while its upstream surface [where pressure p = p(0)] contacts free fluid. (b) In the fixed thickness situation, the tissue is held between two rigid, porous supports at a thickness d less than its uncompressed free-swelling thickness. Flow-induced deformation nevertheless affects its internal strain field.

Grahic Jump Location
Fig. 2

Relationship between normalized fluid velocity (Pe) and applied pressure (p¯) for the free surface situation. Trends are shown for different strain dependencies of hydraulic permeability characterized by M = 0 (solid line), M = 1 (long-dashed line), and M = 4 (short-dashed line).

Grahic Jump Location
Fig. 3

(a) Normalized tissue thickness (d¯) and (b) mean permeability (km/kf) versus applied pressure (p¯) for the free surface situation. Trends are shown for different strain dependencies of hydraulic permeability characterized by M = 0 (solid lines), M = 1 (long-dashed lines), and M = 4 (short-dashed lines).

Grahic Jump Location
Fig. 4

(a), (c), and (e) Strain (ε) and (b), (d), and (f) normalized permeability (k/kf) fields for the free surface situation, plotted versus normalized position z/d (note that d depends upon flow rate). Trends are shown for a range of different strain dependencies of hydraulic permeability including (a) and (b) M = 0, (c) and (d) M = 1, and (e) and (f) M = 4. In all cases, results are shown for normalized applied pressures (p¯) of 0.1 (short-dashed lines), 0.3 (long-dashed lines), and 0.5 (solid lines).

Grahic Jump Location
Fig. 5

Relationship between normalized fluid velocity (Pe) and applied pressure (p¯) for the fixed thickness situation with normalized tissue thicknesses of (a) d¯=0.9, (b) d¯=0.8, and (c) d¯=0.7. Trends are shown for different strain dependencies of hydraulic permeability characterized by M = 0 (solid lines), M = 1 (long-dashed lines), and M = 4 (short-dashed lines). Black dots represent points at which flow-induced deformation reduces tissue thickness to less than the imposed fixed thickness.

Grahic Jump Location
Fig. 6

Normalized mean permeability (km/kf) versus applied pressure (p¯) for the fixed thickness situation with normalized tissue thicknesses of (a) d¯=0.9, (b) d¯=0.8, and (c) d¯=0.7. Trends are shown for different strain dependencies of hydraulic permeability characterized by M = 0 (solid lines), M = 1 (long-dashed lines), and M = 4 (short-dashed lines). Black dots represent points at which flow-induced deformation reduces tissue thickness to less than the imposed fixed thickness.

Grahic Jump Location
Fig. 7

(a), (c), and (e) Strain (ε) and (b), (d), and (f) normalized permeability (k/kf) fields for the fixed thickness situation, plotted versus normalized position z/d for the case of normalized tissue thickness d¯=0.7. Trends are shown for a range of different strain dependencies of hydraulic permeability including (a) and (b) M = 0, (c) and (d) M = 1, and (e) and (f) M = 4. In all cases, results are shown for normalized applied pressures (p¯) of zero (dotted lines), 0.1 (short-dashed lines), 0.3 (long-dashed lines), and 0.5 (solid lines).

Grahic Jump Location
Fig. 8

Trends for (a) normalized fluid velocity (Pe), (b) normalized thickness (d¯), (c) normalized mean permeability (km/kf), and (d) normalized mean permeability calculated naively from imposed fixed thickness (kmx/kf) versus normalized applied pressure (p¯) for a transition from the fixed thickness situation to the free surface situation. Black dots represent points at which flow-induced deformation becomes large enough to reduce tissue thickness to less than the fixed thickness imposed at relatively small p¯, inducing the transition as p¯ increases. Trends are shown for different strain dependencies of hydraulic permeability characterized by M = 0 (solid lines), M = 1 (long-dashed lines), and M = 4 (short-dashed lines).

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