Poroelastic Modeling of Fluid Flow at the Haversian and Lacunar Scales

Transcription

1 Poroelastic Modeling of Fluid Flow at the Haversian and Lacunar Scales Colby C. Swan, Ph.D, Associate Professor, Civil Eng., UI research collaborators: Roderic S. Lakes, Ph.D, Professor, Eng. Mech., UWM Richard A. Brand, M.D. Professor, Ortho. Surg., UI CEE Professional Seminar 11 October 2001

2 World Congress on Medical Physics and Biomedical Engineering, July 2000 Length Scales & Microstructure Cortical Bone Ω s Periosteum Haversian canals Osteons Trabecular bone λ 3 Medullary canal Whole Bone Scale λ 1 λ 2 Osteonal Scale

3 Phenomena of Bone Adaptation 1) With increased mechanical stimulus, bones thicken and densify; 2) With lack of the same, the body resorbs bone mass; Bones become porous, and dimensions thin out; Consequently, bones lose their strength. 3) The fact that bones adapt/respond to mechanical stimulus has been recognized for over 100 years (Wolff s Law). 4) However, the mechanisms by which bones adapt to mechanical stimulus remains very poorly understood! 5) Furthermore, the ability of orthopaedists/engineers to quantitatively predict bone adaptations is presently very poor. Our Research Objectives: a) Explore the potential role of fluid flow in bone adaptation; b) Improve ability to quantitatively predict bone adaptation;

4 Experimental Observations... a) Bone adaptation depends upon the cyclic frequence of applied stimulus. [Hert et al, 1969, 1971, 1972; Chamay and Tschantz, 1972; Churches et al, 1979, 1982;Lanyon and Rubin, 1984; Turner and Forwood, 1993; McLeod et al, 1990, 1992;] b) Bone adaptation shows selective dependence on the amplitude of mechanical stimulus. [Rubin and Lanyon, 1984, 1985; Frost, 1987] c) Bone adaptation responds to stimulus in a trigger like fashion. [Rubin and Lanyon, 1984, 1985; Turner et al, 1995; Brighton et al, 1992;Neidlinger Wilke et al, 1995; Stanford et al, 1995]

5 Bone Adaptation Hypotheses 1) Damage in Bone: Under mechanical stimulus, bone accumulates damage. In repairing itself, bone adapts. [Burr et al, 1985; Martin and Burr, 1989; Prendergast and Taylor, 1994] 2) Strain Energy Density: Bone seeks to stay within a certain strain energy densities under applied loadings. [Beaupre et al, 1990; Cowin, 1993; Luo et al, 1995; Mullender et al, 1994; Jacobs et al, 1995;] 3) Mechanically Generated Electric Fields: a) piezoelectricity; [Spadaro, 1977; Binderman et al, 1985; Gjelsvik, 1973] b) streaming potentials; [Eriksson, 1974; Pollack et al, 1984] 4) Fluid Flow in Bone: Fluid flow provides stimulus to bone cells either by pressures, or shearing. [Weinbaum et al, 1994; Mak et al, 1996; Keanini et al, 1995; Turner et al, 1995]

6 Constitutive Assumptions for Cortical Bone 1) Both bone matrix and fluid are compressible; 2) On microscale, fluid has no shear viscosity; 3) Both fluid and bone matrix are linear elastic; 4) Bone is fully saturated.

7 Whole bone specimen Osteonal unit cell Canalicular/lacunar network Bone mineral matrix Hierarchical Structure of Whole Bone Unit cell of bone matrix Osteons and cement lines Canalicular/lacunar unit cell Canaliculus unit cell

8 UNIT CELL MODELING ASSUMPTIONS Bone matrix is homogeneous, isotropic, elastic (E = 12 GPa, ν = 0.38). (Lamellar structure neglected) Fluid is elastic, with no shear viscosity on microscale (K = 2.1 GPa). The bone is fully saturated. Canal matrix in unit cell denotes: Haversian canal at osteonal scale; canaliculus at lacunar scale; UNIT CELL MODELLING RESULTS 4% fluid filled porosity All parameters in the poroelastic model are computed. Physical observation: Loading along longitudinal canal axis generates "small" fluid pressures. Loading transversely to canal axis generates larger fluid pressures.

9 World Congress on Medical Physics and Biomedical Engineering, July 2000 FLUID CONDUCTIVITY PROPERTIES Experiments of Rouhana et al (1980) k longitudinal = 5 *10 13 m 2 k transverse = 5* m 2

10 CORTICAL BONE SPECIMEN Experiments: Dynamic bending/torsion excitation. Measure viscoelastic damping characteristic tan(δ). Air Dry and Saturated Analysis: Pressure relaxation under step loading. Compute viscoelastic damping characteristic tan(δ). Fully Saturated Specimen 4.31cm 0.64cm

11 Computed Haversian Pressure Relaxation Behaviors

12 Computed Viscoelastic tan(δ) Behaviors Associated with Fluid Flow in Haversian System.

13 Shear Induced Fluid Flow at Lacunar/Canalicular Scale 28 µm 8 µm Applied Loading is 1% Shear Strain Bone matrix is poro elastic, with anisotropy due to canaliculi. Induced fluid pressures in the canaliculi dissipate on the order microseconds. 20 µm 28 µm p=6.8mpa p=0 t =.01 µs p max = 6.8MPa t = 1.0 µs p max = 4.2MPa t = 100 µs p max = 0.59 MPa p= 6.8MPa Space/Time Fluid Pressure Distributions in Lacunar Unit Cell t = 0.01 s p max = 0.031MPa

14 FINDINGS: The Haversian and Volksman canals function as freely draining conduits under mechanical excitation applied well below 1 MHz. On the lacunar scale, load induced fluid pressures in canaliculi relax quickly [O(1 100µs)] into lacunae. Fluid pressure relaxation frequencies on both the whole bone and the lacunar length scales are on the order of 1 10 MHz. These are much larger than what are thought to the physiologically meaningful frequencies (.1 Hz 1 khz). Our extensive experimental measurements of viscoelastic energy dissipation in cortical bone (f <= 10 khz) show no evidence of a Debye peak associated with pressure driven fluid flow.