Dear BIOMCH-L,

Does anyone have a reference that would explain the definition of

the hydraulic permeability coefficient used to express fluid flow through

cortical or trabecular bone?

I read in several articles different units for this permeability

coefficient and some of them don't seem to be right.

1) In [Mow 1991] the rate of volume discharge Q across an area A is

related to the

hydraulic permeability coefficient k by Darcy's law:

Q = (k*A*dP) / h

where h is the thickness of the specimen and dP the pressure gradient.

As results form permeation experiments for normal cartilage and meniscus, k

ranges

form 10^-15 to 10^-16 m^4/(N*s).

[Mow 1991]

Mow V.C., Hayes W.C.: Basic Orthopaedic Biomechanics. Library of Congress

Cataloging-in-Publication Data. Raven Press New York, pp. 158-159.

2) On the other hand [Grimm 1997] explains Darcy's law by using following

equation:

K = (Q*l*µ) / (dP*A)

where

K = permeability

Q = rate of volume discharge

l = specimen length

µ = kinematic viscosity

dP = pressure gradient

A = specimen cross sectional area

Out of his model the permeability coefficient ranges from 0.4 to 11*10^-9

with the unit [m^2]. In this case the hydraulic permeability coefficient of

trabecular bone is proportional to the area and a function of the porosity

factor.

[Grimm 1997]

Grimm M. J., Williams J.L.: Measurements of permeability in human calcaneal

trabecular bone. Journal of Biomechanics, Vol. 30, No. 7, pp. 743-745,

1997.

3) [Keanini 1995] used the time averaged form of Darcy's law with:

u = -k*w*P / µ

where

µ = fluid viscosity

k = permeability

P = time averaged fluid pressure

w = gradient operator

u = time averaged velocity component

To do their calculations they used a permeability for cortical bone ranging

form 10^-14 to 10^-11 m^2. And here again the permeability coefficient has

the unit [m^2].

[Keanini 1995]

Keanini R.G., Roer R.D., Dillaman R.M.: A theoretical model of circulatory

interstitial fluid flow and species transport within porous cortical bone.

Journal of Biomechanics, Vol. 18, No. 8, pp. 901-914, 1995.

And that's where I have a problem, how can it be that three different

references have used the same law but with different units in some

parameters?

Isn't it true that the kinematic viscosity is already taken into account by

explaining the hydraulic permeability coefficient, so that the variable for

the kinematic viscosity is not part of Darcy's law?

Does anyone has any further information on Darcy's law and the definition

of it's parameters?

Thanks in advance.

Best regards,

Michael L.

************************************************** **********

Dipl.-Ing. Michael Liebschner E-mail: mliebsch@emba.uvm.edu

University of Vermont

Mechanical Engineering Phone: (802) 656-1432

119 Votey Building

Burlington, VT 05405 Fax: (802) 656-4441

************************************************** **********

You can not solve problems with the same level of thinking

that existed when the problems were created.

--Dr. Albert Einstein

Does anyone have a reference that would explain the definition of

the hydraulic permeability coefficient used to express fluid flow through

cortical or trabecular bone?

I read in several articles different units for this permeability

coefficient and some of them don't seem to be right.

1) In [Mow 1991] the rate of volume discharge Q across an area A is

related to the

hydraulic permeability coefficient k by Darcy's law:

Q = (k*A*dP) / h

where h is the thickness of the specimen and dP the pressure gradient.

As results form permeation experiments for normal cartilage and meniscus, k

ranges

form 10^-15 to 10^-16 m^4/(N*s).

[Mow 1991]

Mow V.C., Hayes W.C.: Basic Orthopaedic Biomechanics. Library of Congress

Cataloging-in-Publication Data. Raven Press New York, pp. 158-159.

2) On the other hand [Grimm 1997] explains Darcy's law by using following

equation:

K = (Q*l*µ) / (dP*A)

where

K = permeability

Q = rate of volume discharge

l = specimen length

µ = kinematic viscosity

dP = pressure gradient

A = specimen cross sectional area

Out of his model the permeability coefficient ranges from 0.4 to 11*10^-9

with the unit [m^2]. In this case the hydraulic permeability coefficient of

trabecular bone is proportional to the area and a function of the porosity

factor.

[Grimm 1997]

Grimm M. J., Williams J.L.: Measurements of permeability in human calcaneal

trabecular bone. Journal of Biomechanics, Vol. 30, No. 7, pp. 743-745,

1997.

3) [Keanini 1995] used the time averaged form of Darcy's law with:

u = -k*w*P / µ

where

µ = fluid viscosity

k = permeability

P = time averaged fluid pressure

w = gradient operator

u = time averaged velocity component

To do their calculations they used a permeability for cortical bone ranging

form 10^-14 to 10^-11 m^2. And here again the permeability coefficient has

the unit [m^2].

[Keanini 1995]

Keanini R.G., Roer R.D., Dillaman R.M.: A theoretical model of circulatory

interstitial fluid flow and species transport within porous cortical bone.

Journal of Biomechanics, Vol. 18, No. 8, pp. 901-914, 1995.

And that's where I have a problem, how can it be that three different

references have used the same law but with different units in some

parameters?

Isn't it true that the kinematic viscosity is already taken into account by

explaining the hydraulic permeability coefficient, so that the variable for

the kinematic viscosity is not part of Darcy's law?

Does anyone has any further information on Darcy's law and the definition

of it's parameters?

Thanks in advance.

Best regards,

Michael L.

************************************************** **********

Dipl.-Ing. Michael Liebschner E-mail: mliebsch@emba.uvm.edu

University of Vermont

Mechanical Engineering Phone: (802) 656-1432

119 Votey Building

Burlington, VT 05405 Fax: (802) 656-4441

************************************************** **********

You can not solve problems with the same level of thinking

that existed when the problems were created.

--Dr. Albert Einstein