Permeability of Musculoskeletal Tissues and Scaffolding Materials: Experimental Results and Theoretical Predictions

Edward A. Sander¹ & Eric A. Nauman¹²

1Department of Biomedical Engineering, 2Department of Orthopaedic Surgery, Tulane University, New Orleans, Louisiana, USA

Address all correspondence to Eric A. Nauman, Suite 500, Lindy Boggs Center, Tulane University, New Orleans, LA 70118; enauman@tulane.edu

ABSTRACT: Fluid movement produces a wide range of phenomena in musculoskeletal tissues, including streaming potentials, hydrodynamic lubrication, nutrient transport, and mechanical signaling, all of which are governed by tissue permeability. Permeability is a measurement of the ease with which a fl uid passes through a material and is described by Darcy’s law. An appreciation of the fl ow-related structure–function relationships that have been found for musculoskeletal tissues as well as the materials used to engineer substitutes is important for clinicians and engineers alike. In addition, fl uid transport phenomena is one of the most often neglected, but important, aspects of developing functional musculoskel-etal tissue replacements. Mathematical models provide the means to relate permeability to microstructure and enable one to span multiple hierarchical length scales. In this article, we have summarized the experimentally determined permeability for a range of musculoskeletal tissues. In addition, we have included a summary of the microstructural models that are available to relate bulk permeability to microstructural fl ow profi les. Th ese models have the potential to predict cellular level fl uid shear stresses, nutrient and drug transport, degradation kinetics, and the fl uid–solid interactions that govern mechanical response.

KEYWORDS: bone, cartilage, Darcy’s law, scaff olds, microstructure, coralline hydroxy-apatite

I. INTRODUCTION

Fluid movement produces a wide range of phenomena in musculoskeletal tissues. Streaming potentials in cortical bone are generated by the movement of the ionic interstitial fl uid past negatively charged bone surfaces.¹³ Hydrodynamic lubrica-tion at articulating surfaces is a result of high-pressure loads driving fl uid out of the cartilage tissue interior.⁴ Similarly, fl uid shear stresses caused by mechanical load-

Critical Reviews™ in Biomedical Engineering, 31(1):1–26 (2003)

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. . . .

ing have been identifi ed as one potential input that drives functional adaptation.⁵⁶ Furthermore, nutrient transport may be enhanced by convection of bulk interstitial fl uids, particularly in unvascularized tissues.⁷⁹

Permeability is perhaps the most central concept applied in modeling the in-teraction between solid and fl uid phases in biomechanics. Permeability also plays an important role in graft integration and implant fi xation. It is a measurement of the ease with which a fl uid passes through a porous solid and is based on the work of Henri Philibert Gaspard Darcy. In 1856, he published a report that included the observation that the fl ow rate of water through a bed of sand was proportional to the applied pressure and the cross-sectional area of the medium and inversely proportional to the thickness of the medium.¹⁰

From Darcy’s law several proportionality constants have been defi ned, including hydraulic conductance, hydraulic permeability, and intrinsic permeability. Darcy’s law is expressed as:

QA

u c PD= = ∆ (1)

where Q is the fl ow rate, A is the sample cross-sectional area in the direction of fl ow, uD is the volume-averaged Darcy velocity, ∆P is the pressure diff erence across the sample, and c is a proportionality constant called hydraulic conductance [L²T / M]. Hydraulic conductance is often used when comparisons are made between samples of equal length or in instances in which sample thickness is diffi cult to quantify, such as in monolayer culture.¹¹¹² When sample thickness L is separated from c, such that c = κ / L, the resulting term, c = κ / L, is referred to as hydraulic permeability and has dimensions of [L³T / M]. Hydraulic permeability is dependent on the viscosity of the fl uid and makes comparisons diffi cult between experiments that use diff erent fl uids. For structures in which the pores are large relative to the characteristic length scale of bulk fl uid movement, it is possible to separate the fl uid viscosity from the hydraulic permeability. Th e resulting constant is no longer dependent on the properties of the fl uid and is solely dependent of the properties of the solid. Th is constant, k = µκ, is the intrinsic, or specifi c, permeability and has dimensions of [L²].

Each of these proportionality constants quantifi es the average or bulk fl ow behavior of the material under consideration. Th e choice of which proportionality constant to use for Darcy’s law is dependent upon the application. For most soft tissue applications it is not possible to separate the properties of the fl uid from the proportionality constant because these values are not well established. At the level of the soft tissue microporosity, the properties of interstitial fl uid, or water, diff er from those of the bulk fl uid. Here, the fl uid takes on an organized structure that results in an eff ective viscosity. Consequently, the permeability of soft tissues is usu-ally reported in terms of its hydraulic permeability.

Experimentally determined values of permeability are diffi cult to obtain, especially in soft tissues. Th is is in part due to tissue heterogeneity, particularly at the scale at which practical sample sizes are obtained, and requires careful evaluation of the data with regard to the microstructure of the tissue. Nevertheless, it is fruitful to examine the fl ow-related structure–function relationships that have been found for musculoskeletal tissues as well as porous biomaterials and put these data in context. Mathematical models provide the means to relate permeability to microstructure and enable one to span multiple hierarchical length scales. Th ese are especially useful because many con-stitutive laws incorporate some form of permeability, usually to account for frictional losses from fl uid and solid phase interactions.¹³¹⁶ Unfortunately, there are few models that relate the tissue permeability to its microstructure. Consequently, our objective is to review the experimentally determined permeability of musculoskeletal tissues and tissue engineered constructs in relation to the structure-based models available.

II. MODELS OF FLOW THROUGH POROUS MEDIA

Several strategies that attempt to relate some feature of the medium to the fl ow behavior have been employed to model fl ow through porous media.¹⁷¹⁹ Th e most obvious feature to consider is the porosity, but connectivity, shape, pore size distribu-tion, arrangement, orientation, and spacing are also important. Phenomenological models rely on porosity and are not capable of estimating fl uid behavior within the porous medium. Models that incorporate some aspect of pore structure or material microarchitecture can be divided into two broad classifi cations based on whether the fl ow is internal or external to the solid structure. Hydraulic radius theories model Poiseuille fl ow through channels and are usually appropriate for low- to medium-porosity media. Drag theories apply Stokes fl ow equations around objects and are more suitable for high-porosity materials. Both approaches relate the governing equations to Darcy’s law and remain valid when a linear relationship between the pressure drop and the fl ow rate is observed and the Reynolds number is low.¹⁸ Th ese models improve upon phenomenological models by examining the eff ect of structure on the physics of fl uid fl ow. Many avenues of biomedical research seek to relate macroscale mechanics to some cellular response. Any eff ort to model the interplay between the two will require structural models so that the translation of tissue-level forces to the cellular level environment can be better approximated.

II.A. Hydraulic Radius Theories

Th e most basic methods for describing fl uid fl ow through porous media are those based on hydraulic radius. Hydraulic radius is used to estimate the resistance to fl ow

. . . .

of irregularly shaped channels in terms of a representative length. Typically, the ratio of the pore volume to the surface area in contact with the fl uid is used. Th is concept is important because it allows for the porosity to be described by an idealized geometry in place of the complicated microstructure. Th e most widely used hydraulic radius theory is the Kozeny–Carman equation. In this case, the pore space is equated to a bundle of parallel tubes, each of which has a diameter equivalent to four times the hydraulic radius. Th is is the diameter necessary to create a tube that will have the same amount of frictional losses as the pores in the material. Because this situation assumes fl ows where inertial forces are negligible, the intrinsic permeability can be determined in terms of the Hagen–Poiseuille equation:

kSo

=−( )ϕ

ϕ

3

2 21

1

Κ (2)

where ϕ is the pore volume fraction (porosity), So is the specifi c surface (the ratio between the surface area and the volume of the solid structure), and K is the Kozeny constant.¹⁸ So arises from manipulating the relationship for hydraulic radius to refl ect pore volume fraction. Th e Kozeny constant incorporates both a shape factor and the tortuosity, which is a correction to account for the actual path length the fl uid travels. Carman²⁰ recommended K = 5 to fi t most experimental data, but good approxima-tions are generally only attainable for porosities below 0.8. Further improvements on this model have incorporated orientation factors, pore nonuniformities, and pore size distribution functions.¹⁷ While hydraulic radius theories do incorporate some features of structure, only averaged fl uid behavior is attainable, and no description of the internal environment is possible.

II.B. Drag Theories

Drag theories are more commonly applied to fi brous materials and assume that viscous losses result from fl ow past objects.¹⁸²¹ Th ese models are usually based on idealized Stokes fl ow around solid cylinders grouped in periodic arrays of diff ering geometries.²²²⁵ Th e governing equations and boundary conditions for a given unit cell geometry can be solved in terms of the fl ow rate and integrated to yield a form equivalent to Darcy’s law. Th e relationships derived are dependent on the direction of fl ow and can be found for fl ow parallel and transverse to the cylinder axis.

Building on the early work of Emersleben,¹⁸ Iberall²⁶ summed the frictional losses for one cylinder axially oriented with the direction of fl ow with two cylinders oriented transversely to the fl ow and equated the total frictional loss to the pressure drop across the material. Th e total pressure drop was used to determine the bulk

intrinsic permeability and resulted in a relationship which can be expressed in terms of the pore volume fraction, ϕ, the fi ber radius, a, and the Reynolds number,

ka Re

Re=

−−−

34 1

24

2ϕϕ( )

ln( )ln( )

(3)

where the Reynolds number of the fl ow is given by Re = (2aρν / µ), ν = (uD / ϕ) is the pore velocity, ρ is the fl uid density, and µ is the fl uid viscosity. Th ere are two diffi -culties with Iberall’s formulation. Th e fi rst is that the permeability is dependent on the fl ow rate, which implies a nonlinear relationship between fl ow rate and pressure gradient. Th e second is the assumption that fi bers have no infl uence on the fl ow fi elds around neighboring fi bers. Consequently, this theory overpredicts the permeability for three-dimensional structures.

Th e simplest unit cell confi guration for unidirectional, evenly spaced fi bers is that of an inner solid cylinder with radius a and a rectangular outer fl ow fi eld. Because the equations of motion are easier to solve on a circular domain, an eff ective radius,

bwh=π

, is often used to approximate the unit cell geometry. Th is geometry was

investigated by Happel,²² who developed equations for permeability both parallel, kp, and transverse, kt, to the cylinder. Th ese relations are given as

ka

ka

p

t

=−

− − − − + −

=−

−

2 2

2

4 12 1

12

38

1

8 11

( )( )

( )ln( )

( )( )

ϕϕ ϕ ϕ

ϕϕ 22

21

1 11

−− +

− −

( )ln( )

ϕϕ (4)

where a no slip condition on the cylinder surface and no friction (zero shear stress) at the fl uid outer boundary was assumed.

Additional work by Sparrow and Loeffl er²³ considered longitudinal fl ow between cylinders arranged in triangular and square arrays and used numerical methods to obtain the fl ow rate through the unit cell. Th e best approximations to date have been obtained by Drummond and Tahir,²⁵ who examined fl ow both parallel and transverse to the axes of cylinders arranged in triangular, square, rectangular, and hexagonal arrays.

Drag theories work well for high-porosity materials, especially for unidirectional fi ber beds, but decrease in accuracy as the pore space decreases. For three-dimensional

. . . .

networks, drag theories of the type proposed by Iberall overpredict permeability.²⁷ To address this issue, Fourie and Du Plessis²⁸ developed a three-dimensional model of struts with square cross-sections and superposed the pressure drop due to fi ber drag with that due to Poiseuille fl ow between parallel rods. Th eir model provides a good estimate of the permeability in three-dimensional networks and is a useful framework for future models of fl uid fl ow in porous media. Experimental measure-ments of permeability and microstructural analysis must be used in conjunction with these or similar models to understand the role of permeability in musculoskeletal tissues and tissue-engineered scaff olding materials.

III. MUSCULOSKELETAL TISSUES

III.A. Bone

Th e importance of quantifying cancellous and cortical bone permeability is far-reach-ing. Th e fl uid conductance of cancellous bone grafts has been correlated with their integration and overall clinical success.²⁹ It is thought that a highly permeable graft promotes blood fl ow and vascular invasion, followed by bone ingrowth and long-term remodeling. Similarly, the fi xation of joint replacement hardware and vertebral body reconstruction is dependent on the depth of bone cement infi ltration into cancellous bone pores and therefore intimately related to the permeability of the underlying structure.³⁰³³ In addition, fl uid-solid interactions govern the mechanical response of both cancellous and cortical bone, making permeability an integral parameter in multiphase, mixture theory, or poroelastic models. It also plays a role in ultrasound wave propagation,³⁴³⁵ infl uences nutrient and calcium transport,⁷⁹ and governs electrokinetic phenomena such as streaming potentials.¹³

III.B. Cancellous Bone

Primarily because of its role in bone graft success and implant integration, cancel-lous bone from a variety of anatomic sites and a wide range of volume fractions has been tested (Table 1). Reported values range over four orders of magnitude between 0.002 and 20 × 10–⁹ m² (Fıg. 1). In general, permeability increases with increasing porosity (or decreasing apparent density), but porosity alone cannot account for all the observed variability. Beaudoin et al.³⁰ found a strong negative correlation between permeability and apparent density, but Lim and Hong³⁶ reported that a power-law fi t best described this relationship. Neither porosity nor apparent density is capable of distinguishing heterogeneity arising from architectural diff erences

Tab

le 1

. E

xper

imen

tally

Det

erm

ined

Can

cello

us B

one

Per

mea

bili

ty

Aut

hor

Ana

tom

ic s

ite

Dir

ecti

on

Test

ing

met

hod

No

. o

f sa

mp

les

Per

mea

bili

ty

rang

e(1

0–9

m2)

Per

mea

bili

ty

aver

age

(10

–9 m

2)

Gri

mm

and

Will

iam

s35

Hum

an c

alca

neus

Med

ial–

late

ral

Co

nsta

nt p

ress

ure

220.

4 –

11.0

3.5

± 2

.6

Ko

hles

et

al.3

8 B

ovi

ne d

ista

l fem

urM

edia

l–la

tera

lC

ons

tant

pre

ssur

e20

0.05

9 –

0.73

0.

23 ±

0.1

6

Ant

erio

r–p

ost

erio

r0.

092

– 1.

00.

45 ±

0.2

7

Sup

erio

r–in

feri

or

0.02

3 –

1.30

0.47

± 0

.35

Och

ia a

nd C

hing

44H

uman

ver

teb

ral b

od

ySu

per

ior–

infe

rio

rC

ons

tant

fl o

w10

0.13

– 0

.97

0.49

± 0

.44

Lim

and

Ho

ng36

Bo

vine

ver

teb

ral b

od

yC

epha

lad

–cau

dal

Co

nsta

nt p

ress

ure

620.

0027

– 0

.29

0.16

± 0

.08

Hui

et

al.2

9P

orc

ine

fem

ora

l hea

dLo

ngit

udin

alC

ons

tant

pre

ssur

e77

0.00

4 –

0.13

30.

049

± 0

.03

Tran

sver

se81

0.00

2 –

0.04

50.

02 ±

0.0

1

Nau

man

et

al.3

7H

uman

ver

teb

ral b

od

yLo

ngit

udin

alC

ons

tant

fl o

w11

3.89

– 2

0.0

8.05

± 4

.75

Hum

an v

erte

bra

l bo

dy

Tran

sver

se9

1.57

– 7

.29

3.59

± 1

.90

Hum

an p

roxi

mal

fem

urLo

ngit

udin

al6

0.30

8 –

7.03

2.76

± 2

.74

Hum

an p

roxi

mal

fem

urTr

ansv

erse

60.

03 –

0.3

00.

12 ±

0.1

1

Bo

vine

pro

xim

al t

ibia

Long

itud

inal

122.

00 –

5.1

33.

17 ±

1.0

2

Bo

vine

pro

xim

al t

ibia

Tran

sver

se12

0.08

– 3

.06

0.74

± 0

.83

Beu

ado

in e

t al

.30

Hum

an p

roxi

mal

tib

iaSu

per

ior–

infe

rio

rC

ons

tant

fl o

w22

3.2

– 16

.27.

8 ±

3.6

. . . .

between samples. It is clear that structural diff erences play a role in the measured permeability because cancellous bone samples have been found to have a strong directional dependence.³⁷³⁸ Nauman et al.³⁷ conducted permeability experiments on human vertebral bodies and femoral necks and bovine proximal tibia in both the longitudinal (along the principal trabecular orientation) and transverse direc-tions. Permeability was dependent on the direction of fl ow relative to the principle trabecular orientation, anatomic site, and, to a lesser extent, volume fraction. Th eir values ranged from 0.03 to 20 × 10–⁹ m². Kohles et al.³⁸ measured the anisotropic permeability of cubic bovine distal femur samples and found a directional depen-dence within each sample. Th e medial–lateral direction was signifi cantly lower than the superior–inferior and anterior–posterior directions. Th e measured anisotropy within samples refl ects complex structure–function relationships that are diffi cult to interpret because they did not relate their results to the specimen’s principal trabecular orientation

Th e role of bone marrow, which behaves like a fl uid in vivo,³⁹ has also been a

Grimm and Williams35

Kohles et al.38 ML

Kohles et al. 38 AP

Kohles et al. 38 SI

Hui et al.29 Perpendicular

Hui et al. 29 Parallel

Nauman et al.37 HF-Trans

Nauman et al.37 HF-Long

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10-12

10-11

10-10

10-9

10-8

10-7

FIGURE 1. Intrinsic permeability of cancellous bone from different anatomical sites and species. Filled

data points indicate human specimens.

Per

mea

bili

ty (

m2)

Porosity

source of much investigation.⁴⁰ Several studies have indirectly assessed permeability by investigating the potential for hydraulic stiff ening of bone as a mechanism for mechanical signaling.⁴¹⁴⁴ Th ese studies measured hydraulic resistance, which is inversely proportional to permeability, to ascertain whether or not suffi cient resis-tance to fl uid fl ow exists to allow hydraulic stiff ening. Simkin et al.⁴¹ investigated the hydraulic resistance of trabecular bone in matched convex and concave canine shoulders. Downey et al.⁴² conducted a similar study with human hips. Both re-ported a linear relationship between intraosseous pressure and fl ow, suggesting that Darcy’s law is valid for these more complicated fl ow conditions. Hydraulic resistance was found to be higher in the convex humerus and femoral head than in the con-cave scapula and acetabulum. Th is result was attributed to the compartmentalized, honey comb microstructure of the convex bone as compared to the open plate and strut confi guration of the concave bone. It is interesting that when the specimens in Simkin’s study were cut down to cubes 1 cm in length, the hydraulic resistance decreased 9.6 times for the humerus and 4.3 times for the scapula. Th is suggests that hydraulic resistance is highly dependent on sample boundary conditions and that specimens should be selected to minimize disruptions in the trabecular architecture that reduce resistance to fl uid exudation.⁴¹ Subsequent work by Ochia and Ching⁴⁴ investigated the hydraulic resistance and permeability of intact human vertebral bodies and arrived at similar conclusions.

Beaudoin et al.³⁰ modeled the fl ow of non-Newtonian fl uid through a parallel array of capillaries, each fi lled with equidistant torroidal elements to simulate the fl ow of bone cement through cancellous bone. Morphological parameters, such as porosity, trabecular spacing, and surface-to-volume ratios, were investigated by altering the geometry of the model elements. Th eir model eff ectively combined internal and external fl ows but did not reproduce the actual geometry of cancellous bone. For a given model porosity, it was found that a nonlinear pseudoplastic fl uid demonstrated a higher permeability than a linear Newtonian fl uid. Th is observa-tion was attributed to shear thinning near the no-slip boundary. Furthermore, for a constant applied pressure, the depth of penetration increased with decreasing surface-to-volume ratio.

Sander et al.⁴⁵ proposed a cellular solid model, based on histomorphological data,⁴⁶ to investigate the structure–function relationships in the cancellous bone of human vertebral bodies. In particular, they modeled the changes in microstructure that occur as a consequence of aging. Fluid fl ow between trabeculae was estimated by combining a simple hydraulic radius model with a fi ber drag theory.²⁸ Th e model predicted the experimental permeability well but demonstrated only small diff er-ences between longitudinal and transverse permeability (Fıg. 2). Adding plates, thickening the nodes, creating irregularities in trabecular geometry, and gathering more experimental data will improve directional distinctions for permeability that are predicted by the model.

. . . .

III.C. Cortical Bone

It is well established that mechanical forces infl uence the growth and maintenance of bone. However, the nature of these forces and the mechanisms through which they act remain unresolved, particularly with regard to the role of fl uid fl ow. Corti-cal bone is pervaded by interstitial fl uid, and the lacunar–canalicular porosity in particular is thought to transport nutrients and infl uence cellular level mechanical stimuli.⁴⁷⁴⁹ Several studies have examined the role of fl uid in mechanotransduc-tion in the form of pressure and shear stress⁴⁹⁵¹ and in molecular transport.⁸ In elucidating these mechanisms, the permeability of cortical bone is often considered. Poroelastic models can be used to understand the role of both solid and fl uid behavior

10-10

10-9

10-8

10-7

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Per

mea

bili

ty (

m2 )

Vf

kV

kT

Per

mea

bili

ty (

m2)

Vf

FIGURE 2. Permeability vs. volume fraction. The predicted permeability in the transverse, kT, and

vertical, kV, directions fi t the experimental data well.37 Experimentally determined permeabilities are

indicated by fi lled circles (transverse direction) and open circles (vertical direction). Both curves decrease

nonlinearly with increasing volume fraction and at a given volume fraction; transverse permeability is

always lower than vertical permeability.

in the mechanics of bone. To refi ne the predictive capabilities of these models and to further our understanding of bone physiology, a better understanding of perme-ability is necessary.

Th ere are few studies that have experimentally measured the permeability of cortical bone (Table 2).⁵²⁵³ Th e early work of Johnson et al.⁵² measured cortical bone permeability at the level of vascular channels rather than the lacunar and canalicular porosity. Th ey found an average value of 2.5 × 10 –¹⁴ m². Li and coworkers⁵³ found that juvenile cortical bone from the canine tibia was six times more permeable than that of adult specimens and was well correlated to the bone porosity. Furthermore, perme-ability was dependent on anatomic location, with posterior specimens demonstrating higher permeability than anterior, medial, and lateral specimens, possibly because of greater vascularization in that region of the tissue. It was also determined that fl uid did not fl ow through adult specimens with intact periosteum. When the periosteum was removed, fl uid movement through Haversian and Volkmann’s canals and to a lesser extent through the canaliculi and lacunae was verifi ed with India ink.

A phenomenological model of in vivo fl uid behavior proposed by Johnson et al.⁵⁴ incorporated permeability into a pressure relaxation function that estimated relaxation time on the order of milliseconds within vascular channels. Johnson also pointed out that in vitro measurements will overpredict in vivo permeability as a result of the vasculature and presence of cells. Numerous eff orts have been made to incorporate the lacunar–canalicular porosity into fl uid fl ow models of cortical bone,⁵⁵⁵⁷ but only Weinbaum et al.⁴⁹ have attempted to model the fl ow profi le within the canaliculi and relate it back to the bulk tissue permeability. To date, little experimented work has been done to determine the actual permeability of the lacunar–canalicular porosity.

Table 2. Experimentally Determined Cortical Bone Permeability

AuthorAnatomic

siteDirection

Testing method

No. samples

Permeability range

(10–14 m2)

Permeability average

(10–14 m2)

Johnson52 Bovine femur Longitudinal Constant pressure

4 0.9–3.4

Radial 2 2.3, 3.9

Li et al.53 Adult canine tibia

NA Constant pressure

4.01–7.85 5.6 ± 0.48

Juvenile canine tibia

NA 2.74–4.42 3.44 ± 0.19

. . . .

III.D. Cartilage

Th e mechanical properties of cartilage are a consequence of interstitial fl uid and extracellular matrix interactions.¹³⁵⁸ Joint loading deforms the extracellular matrix, pressurizes the interstitial fl uid, and forces fl uid out of the tissue and into the joint capsule. During this cycle, electrical, physicochemical, and mechanical signals⁵⁸⁵⁹are produced that depend on the permeability of the extracellular matrix. Tissue permeability follows both from the interaction between phases and from tissue morphology.

Cartilage consists of an ion-rich fl uid phase and a collagen–proteoglycan-dense solid phase. Th e organization and composition of the solid phase is depth dependent and is divided into four distinct zones (Fıg. 3).⁵⁸ Th e superfi cial zone is an acellular layer at the articulating surface that extends to a sublayer in which both fl attened chondrocytes and collagen fi brils lie parallel to the surface.⁶⁰ In the transitional zone chondrocytes are generally spherical, collagen fi brils larger in diameter and more random in orientation, and water content lower. Th e deep zone has the most proteoglycan, the least water, the largest diameter collagen fi brils, and rounded cells stacked in columns perpendicular to the surface. Th e calcifi ed cartilage zone is just below the deep zone and integrates with the subchondral bone.

Early interest in the permeability of cartilage developed from investigations into the fl uid mechanics of joint lubrication,⁶¹⁶² nutrient transport,⁶³ and the etiol-ogy of osteoarthritis. In the course of this work, permeability was found to range between 0.1 and 2.0 × 10–¹⁵ m⁴/Ns and was dependent on the composition⁶⁴⁶⁵ and organization (depth) of the sample⁶¹⁶⁴⁶⁶ as well as the state of mechanical load-

FIGURE 3. Zonal organization of articular cartilage.

ing (Table 3).⁶²⁶⁷ Permeability is approximately 40% lower in the superfi cial zone than in the neighboring middle zone, and this appears to be due to collagen fi bril organization and spacing.⁶⁶ Permeability decreases with increasing tissue depth until just before the calcifi ed layer.⁶¹⁶⁶ Th is decrease was shown to correlate well with the concomitant rise in proteoglycan content⁶⁴ and to a lesser extent with increased collagen content.⁶⁵

Th ere have been several eff orts to determine which molecules of the extracellular matrix are most responsible for the tissue’s resistance to fl ow.⁶⁸⁶⁹ Jackson measured the permeability for hyaluronic acid solutions ranging from 0.05% to 2.5% by weight and found that the permeability was 50 times greater than for a tissue with the same concentration.⁷⁰ Th is fi nding suggests that collagen also plays a considerable role in tissue permeability and may amplify the eff ect glycosaminoglycans have on permeability.⁶⁹ In addition, they also developed a structural drag theory model based on cubic lattices that agreed well with their experimental data. Values of perme-ability decreased with concentration from 1 × 10–¹⁵ m² (×10–² m⁴/Ns) to 1 × 10–¹⁷ m² (×10–¹⁴ m⁴/Ns).

An elegant combination of experiments⁶² and mathematical models¹³⁷¹⁷² have helped elucidate the form and source of the strain-dependent permeability exhibited by normal cartilage. As the tissue is compressed, the spacing between collagen fi brils decreases, and the fi xed charge density increases, both of which result in a greater resistance to fl uid fl ow. In particular, cartilage permeability obeys an empirical rela-tion of the form

Table 3. Articular Cartilage Permeability Obtained Directly from Experiments

Author Anatomic Site Testing MethodPressures

Tested (MPa)

No. Samples

Permeability x10–15

(m4/Ns)

McCuthen61 Bovine shoulder joint

Constant pressure 0.6 NA 0.43–0.765

Maroudas64 0.1 44 0.14–0.55

Maroudas63 Human femoral condyle

Constant pressure NA 6 0.14–0.38

Muir65 Human femoral condyle

Constant pressure NA 25 0.18–0.64

Maroudas67 Human femoral condyle

Pressure/compression

0.172–1.72 NA 0.011–0.22*

Mansour62 Bovine femoral condyle

Pressure/compression

0.14–2.76 14 0.25–2.0

Mow13 Bovine femoral condyle

Pressure/compression

0.069–1.72 6 0.13–1.6

* Tissue hydration between 50 and 100% of initial weight

. . . .

κ = A (PA )exp[-α (PA) εc] (5)

where κ is the hydraulic permeability, εc is the clamping strain, and A(PA) and α(PA) are functions of the applied pressure, PA, and are determined from curve-fi tting the experimental data.¹⁴⁵⁹ Th e limit as PA approaches zero results in an expression for the permeability which is only a function of the applied strain. A decrease in permeability with decreasing tissue hydration has also been observed and indirectly linked to tissue compression.⁶⁷⁷³

In contrast to the relatively small number of experimental studies of cartilage permeability, numerous researchers have used mixture-theory-based computational models to estimate the tissue permeability from confi ned and unconfi ned compres-sion experiments.¹³¹⁵⁷⁴⁷⁷ In these models, permeability is either assumed or obtained from curve-fi tting (Table 4) but is not itself linked to a direct measurement or de-rived from structural considerations. Th e diffi culty with using these values to draw comparisons between studies is that mixture theory models of cartilage have evolved considerably from the relatively simple biphasic model¹³⁷⁸⁷⁹ to electrokinetic,⁷⁴⁷⁵ triphasic,⁸⁰⁸¹ and biphasic poroviscoelastic formulations.⁸² In addition, special care must be taken to match the boundary conditions between the model and the experi-ment¹⁵ in order to accurately predict the undeformed permeability.

III.E. Intervertebral Disc

Th e intervertebral disc carries compressive loads that are distributed through the tissue as a result of the interaction between interstitial fl uid and tissue. Fluid trans-port and the mechanical properties of the tissue are governed by the permeability that results from the tissue microstructure and composition. Th e vast majority of the

Table 4. Articular Cartilage Permeability Obtained from Curve Fitting

Author Anatomic site ModelNo.

samplesPermeability

x10–15 (m4/Ns)

Mow13 Bovine femoral condyle

Biphasic poroelastic theory (KLM) 10 7.6 ± 4.2

Armstrong73 Human patella Biphasic poroelastic theory (KLM) 103 0.5 – 19.5

Holmes78 Human patella Biphasic poroelastic theory (KLM) 1 3.2

Frank75 Bovine femoro-patellar groove

Electrokinetic theory 8 3.0

Sah76 Bovine femor-patellar groove

Electrokinetic theory 6 2.0 ± 1.0

literature reports hydraulic permeability obtained from multiphase or poroelastic models for the annulus fi brosus on the order of 10–¹⁶ m⁴/Ns.⁸³⁸⁶ In contrast, Gu et al.⁸⁷ investigated the hydraulic permeability of the annulus fi brosus through direct permeation experiments as a function of tissue integrity, orientation, and age. Th rough experimentation, they found a value of permeability that was an order of magnitude higher at 10–¹⁵ m⁴/Ns. For healthy, non-degenerated discs a directional dependence on the tissue permeability was also found. Tissue permeability was highest in the radial (1.53 ± 0.05 × 10¹⁵ m⁴/Ns) direction and lowest in the circumferential direction (1.15 ± 0.06 × 10–¹⁵ m⁴/Ns) with the axial direction between (1.92 ± 0.05 × 10–¹⁵ m⁴/Ns). As the annulus fi brosus degenerates, the directional dependence decreases and the permeabilities converge to 1.6 ± 0.1 × 10–¹⁵ m⁴/Ns. Th is is suggestive of a loss of microstructure but may also refl ect a decrease in tissue hydration (which also reduces permeability). It is likely that both mechanisms are responsible because permeability in the radial direction decreases with tissue quality, whereas it increases in the axial and circumferential directions. In addition, permeability was positively correlated with age,⁸⁷ even though tissue hydration decreases with age.⁸⁵ Curve-fi t confi ned compression studies have reported that permeability is lower at the periphery of the annulus fi brosus⁸⁵ where collagen content is higher and proteoglycan and water content is lower.⁸³ Th ese results indicate that permeability can be used to measure degenerative changes in the intervertebral disc and may help to explain the eff ects of morphological changes that occur with age and disease.

III.F. Meniscus

Th e permeability of the meniscus has also been investigated through direct per-meation experiments⁸⁸ and model-based approximations.⁸⁸⁹⁰ Reported values of permeability span three orders of magnitude between 0.01 and 6.78 × 10–¹⁵ m⁴/Ns, and this range may be attributed to diff erences in testing method and curve fi tting procedures (Table 5).⁸⁹⁹⁰ It is interesting that no signifi cant diff erences in perme-ability have been shown with anatomical location⁸⁸ or specimen orientation,⁹⁰ but a permeability diff erence between species has been demonstrated.⁸⁹

III.G. LIGAMENT AND TENDON

Some direct measurements of permeability have been done on the human medial collateral ligament.⁹¹ Th e permeability in the transverse direction (× 10–¹⁶ m⁴/Ns) is an order of magnitude lower than that of cartilage (× 10–¹⁵ m⁴/Ns) and qualitatively demonstrates similar dependence on both pressure and strain. Chen et al.⁹² performed

. . . .

a parametric study on a microstructural model of ligaments and tendons based on a fi ber-bed porous material. In this model, collagen fi bers were physiologically spaced in square arrays of parallel collagen fi bers, and Stokes fl ow was considered in both the transverse and longitudinal directions. For an unstrained porosity of 0.32, the transverse permeability, kt, was 2.3 × 10–¹³ m⁴/Ns, and the longitudinal permeability, kl, was 3.8 × 10–¹³ m⁴/Ns. Permeability was found to be greater in the longitudinal direction, highly anisotropic, and dependent on porosity, but the predicted perme-ability overestimated the experimentally measured permeability⁹¹ by three orders of magnitude. It should be noted that this model did not include the eff ects of ground substance or a distribution of collagen diameters and interstitial spacing, assumptions that should be included in future models.

IV. SCAFFOLDING MATERIALS FOR TISSUE ENGINEERING

With the advent of tissue engineering, a growing number of materials are being examined for their capacity to facilitate tissue regeneration.⁹³⁹⁷ One of the most popular material groups under investigation is biodegradable polymer scaff olds. Th e predominate focus in evaluating these materials has been the characteriza-tion of mechanical properties,⁹⁸⁹⁹ degradation behavior,¹⁰⁰¹⁰³ and long-term biocompatibility.¹⁰⁴¹⁰⁵ Microstructure has also been cited as a major design consideration¹⁰⁶¹⁰⁷ and is frequently described in terms of the surface area, void volume, or porosity. Several studies have determined that high-porosity scaff olds

Table 5. Experimental and Model-Based Permeability Values for the Meniscus

AuthorAnatomic

siteMethod Condition

No. samples

Permeabilityx10–15 (m4/Ns)

Proctor Bovine Pressure/compression 0.17 MPa 12 0.64

Biphasic Compression 63 0.81 ± 0.45

Joshi Primate Biphasic Compression 5 6.78 ± 1.12

Canine 5 3.62 ± 0.52

Bovine 5 3.19 ± 0.15

Human 5 1.99 ± 0.79

Ovine 5 1.91 ± 0.46

Porcine 5 1.74 ± 0.19

LeRoux Canine 3D biphasic FEM Tension 19 0.01

are essential for suffi cient molecular transport,¹⁰⁷ cellular and tissue infi ltration,¹⁰⁸ and vascularization.²⁹ Furthermore, the degradation kinetics of several biodegrad-able polymers are dependent on the surface-to-volume ratio of the polymer, a parameter that is directly related to porosity. In the case of poly(α-hydroxy) acids, high porosity foams degrade at a slower rate than their bulk counterparts because acidic byproducts formed during degradation are more easily evacuated from the environment. Th is prevents the pH from dropping and accelerating the rate of degradation.¹⁰⁹ However, a measurement of porosity does not necessarily refl ect the permeability that ultimately governs these events. In two separate studies, Agrawal et al.¹¹⁰¹¹¹ compared porosity and permeability for 50:50 PLGA scaf-folds and found that, as a result of degradation over time, changes in permeability did not follow those for porosity. Permeability in these studies ranged between 2 × 10–¹⁰ and 2 × 10–¹² m² and was dependent on the fabrication method and the polymer:salt ratio. Similar values were reported by Spain et al.¹¹² As the scaff old degrades, the porosity may change such that the connectivity between pores and the ability of fl uid to infi ltrate the scaff old is also aff ected. In static conditions, foams with diff erent permeabilities degraded at the same rate.¹¹⁰ In a fl ow model of scaff old degradation, it was found that higher permeability scaff olds degraded at a slower rate than those with lower permeability.¹¹¹ Direct measurements of scaff old permeability provide better assessments of the transport properties of a scaff old than porosity measurements alone. To date, there has been little eff ort to develop structural models, but they may help isolate additional factors that contribute to permeability changes over time.

Coralline hydroxyapatite has been examined as a potential bone substitute¹¹³¹¹⁵ because its mechanical and material properties are similar to the cancellous bone it is designed to replace. It is less porous than cancellous bone and exhibits an orientation-dependent permeability ranging from 1.7 to 8.6 × 10–¹⁰ m².¹¹⁵ Th e di-rectional dependence of permeability is indicative of a heterogeneous microstructure that should be considered as another factor in implant and scaff old design. Relating the directional permeability to the predominant fl ow paths in the damaged tissue may be an important aspect of tissue integration that is often overlooked. Although the permeability of a scaff old or implant will change over time with tissue ingrowth, the initial cellular infi ltration and vascularization necessary for long-term success is dependent on a threshold permeability.²⁹

Gels are popular scaff olding materials because they are well hydrated and are often composed of biological based materials, such as polysaccharides, or extracellular matrix materials, such as collagen. Agarose gels (polysaccharides) have been inves-tigated as 3D matrices for many cell types, including neural cells¹¹⁶ and chondro-cytes.¹¹⁷¹¹⁸ Johnson and Deen¹¹⁹ measured the intrinsic permeability of agarose gels ranging from 2.0 to 7.3% (w/v) and found them to be between 3.5 × 10–¹⁶ m² and 1.5 × 10–¹⁷ m². A drag theory model based on a random array of cylinders

. . . .

under predicted the measured permeability, particularly at lower concentrations. Th e authors postulated that, at lower concentrations, the microstructure may diff er or is more heterogeneous than the homogeneous structure assumed in the model.

Collagen gels have been used to tissue engineer cartilage¹²⁰ and bone,¹²¹ but few studies have reported permeability. Ramanujan et al.¹²² fabricated collagen type I gels between 1 and 4.5 % (w/v) and found intrinsic permeability decreased from 1 × 10–¹⁵ to 1 × 10–¹⁶ m² in cell free gels. A Brinkman (eff ective medium)-based structural model provided theoretical predictions that also underpredicted the experimentally measured values for collagen gels with solid volume fractions greater than 1%. Th is is likely due to idealizations of the gel structure and composition.

One important diffi culty presented by collagen gels is that, when seeded with cells, they contract over time as a result of cell-matrix interactions. Th is contraction decreases the permeability and likely aff ects cell viability in the scaff old interior. In an eff ort to prevent gel contraction, Gentleman et al.¹²³ introduced 1–2-mm-long collagen fi bers with diameters on the order of 125 µm. Collagen gels with randomly oriented short collagen fi bers were 100 to 1000 times more permeable (×10–¹¹ m²) than those without (×10–¹⁴ m²) and maintained their initial shape throughout the culture period.

It should be noted that permeability has been found to be velocity dependent for some gels. Th is is only a concern when the gel does not possess suffi cient rigidity or the velocities used are high enough to induce deformation and should be considered on a case-by-case basis.¹²⁴

V. CONCLUSION

Permeability is dependant on tissue composition and morphology and is an impor-tant parameter for understanding structure–function relationships within musculo-skeletal tissues. It governs both diff usive and convective transport, electrokinetic events, and tissue mechanics, and may play a role in mechanotransduction. Th e fact that permeability varies with location and direction in many materials underscores the importance of developing mathematical models that include structural param-eters beyond porosity or apparent density. Without the inclusion of architectural information, only the bulk permeability of a material can be predicted, which may overshadow potentially important structural variations that impact cellular activity. Future theoretical investigations should include idealized structures, high-resolution digital reconstructions, and combinations of the two in order to develop more ac-curate and effi cient models. In addition to building more representative geometries, it may also be necessary to include compositional properties, particularly charge eff ects from proteoglycan molecules and the like.

Mixture-theory-based models of musculoskeletal tissues typically use Darcy’s law to model the interplay between the fl uid and solid phases. To date, few attempts have been made to integrate structure-dependent permeability into these types of models. One notable success was developed in a series of articles by Weinbaum et al.⁴⁹¹²⁵¹²⁶ wherein an idealized microstructural model of the canalicular glycocalyx was used to estimate the permeability of lamellar bone tissue.

Because it is diffi cult to make accurate force measurements at the cellular or subcellular level, models must be used to estimate them from the macroscale forces and the tissue structure. Consequently, the ability to predict permeability as a result of structural parameters may also provide insight into the character of mechanical stimuli at the cellular level and may help to pinpoint signaling sources. Elucidating the structure–function relationships of musculoskeletal tissues in this manner may also have the added benefi t of providing data that can be used to improve constitu-tive laws for the solid phase.

In addition to providing a better understanding of tissue biomechanics, it should be noted that transport plays a strong role in the integration of graft tissues and tissue-engineered scaff olds. Hui et al.²⁹ showed that permeability plays an important role in the vascularization, integration, and ultimate success of cancellous bone grafts implanted into a cortical defect. Consequently, permeability may serve as a useful parameter to optimize the structural design of engineered scaff olds. In particular, the structure of the scaff old material could be designed to yield a fl uid conductance higher than some critical minimum value below that needed to promote vascular and cellular invasion. Unfortunately, there are few studies of this kind, and more work needs to be done to determine the variability of this minimum fl uid conductance, especially with regard to tissue type, anatomic site, and species. Furthermore, it may be of benefi t to design a scaff olding material to possess a permeability that mimics that of the natural tissue to replicate mechanical signals that may be necessary for cell diff erentiation and maintenance of phenotype. It should be noted that although it may be advantageous to design a highly porous, interconnected scaff old, the need to maintain suffi cient strength often limits the porosity that can be used, especially in musculoskeletal applications.

An appreciation of the fl uid transport properties in musculoskeletal applications is an important aspect of developing functional musculoskeletal tissue replacements. In this article, we have summarized the experimentally determined permeability for a range of musculoskeletal tissues including bone, cartilage, meniscus, intervertebral disc, ligaments, and a variety of scaff olds commonly used for tissue engineering (Fıg. 4). In addition, we have included a summary of the microstructural models that are available to relate the bulk permeability to the microstructural fl ow profi les. Th ese models have the potential to predict cellular-level fl uid shear stresses, nutrient and drug transport, degradation kinetics, and the fl uid–solid interactions that govern mechanical response.

. . . .

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FIGURE 4. Permeability ranges for various musculoskeletal tissues. Hard tissues are reported in terms

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