00221 - University of Delawaregilbert/papers/comput_acoust_june-04.pdf · 2004. 7. 18. · Title: 00221.dvi Created Date: 6/30/2004 11:24:25 AM

June 23, 2004 9:26 WSPC/130-JCA 00221

Journal of Computational Acoustics, Vol. 12, No. 2 (2004) 99–126c© IMACS

DETERMINATION OF THE PARAMETERS OF

CANCELLOUS BONE USING LOW FREQUENCY

ACOUSTIC MEASUREMENTS

JAMES L. BUCHANAN

Mathematics Department, United States Naval Academy Annapolis MD, USA

ROBERT P. GILBERT

Department of Mathematical Sciences, University of Delaware, Newark DE, USA

KHALDOUN KHASHANAH

Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken NJ 07030, USA

Received 29 May 2002

The Biot model is widely used to model poroelastic media. Several authors have studied its ap-plicability to cancellous bone. In this article the feasibility of determining the Biot parameters ofcancellous bone by acoustic interrogation using frequencies in the 5–15 kHz range is studied. It isfound that the porosity of the specimen can be determined with a high degree of accuracy. Thedegree to which other parameters can be determined accurately depends upon porosity.

Keywords: Osteoporosis; cancellous bone; poroelastic media; Biot model; inverse problem, finiteelements; simplex method.

1. Introduction

Cancellous bone is a two component material consisting of a calcified bone matrix with

interstial fatty marrow. Hence mathematical models of poroelastic media are applicable.

McKelvie and Palmer,6 Williams,7 and Hosokawa and Otani5 discuss the application of

Biot’s model for a poroelastic medium to cancellous bone. Use of this model requires deter-

mination of the parameters upon which it depends. This can be an expensive process. In this

article we investigate whether these parameters can be ascertained by acoustic interrogation.

2. The Biot Model Applied to Cancellous Bone

The Biot model treats a poroelastic medium as an elastic frame with interstial pore fluid.

Cancellous bone is anisotropic, however, as pointed out by Williams, if the acoustic waves

passing through it travel in the trabecular direction an isotropic model may be acceptable.

We will simulate a two dimensional version of the experiments described in McKelvie and

Palmer and Hosokawa and Otani. The motion of the frame and fluid within the bone are

99

June 23, 2004 9:26 WSPC/130-JCA 00221

100 J. L. Buchanan, R. P. Gilbert & K. Khashanah

tracked by position vectors u = [u, v] and U = [U, V ]. The constitutive equations used by

Biot are those of a linear elastic material with terms added to account for the interaction

of the frame and interstial fluid

σxx = 2µexx + λe+Qε ,

σyy = 2µeyy + λe+Qε ,

σxy = µexy, σyx = µeyx ,

σ = Qe+Rε

(1)

where the solid and fluid dilatations are given by

e = ∇ · u =∂u

∂x+∂v

∂y, ε = ∇ ·U =

∂U

∂x+∂V

∂y. (2)

The stress-strain relations are

exx =∂u

∂x, exy = eyx =

∂u

∂y+∂v

∂x, eyy =

∂v

∂y. (3)

The parameter µ, the complex frame shear modulus is measured. The other parameters

λ, R and Q occurring in the constitutive equations are calculated from the measured or

estimated values of the parameters given in Table 1 using the formulas

λ = Kb −2

3µ+

(Kr −Kb)2 − 2βKr(Kr −Kb) + β2K2

r

D −Kb

R =β2K2

r

D −Kb

Q =βKr((1 − β)Kr −Kb)

D −Kb

.

(4)

where

D = Kr(1 + β(Kr/Kf − 1)) . (5)

Table 1. Parameters in the Biot model.

Symbol Parameter

ρf Density of the pore fluidρr Density of frame materialKb Complex frame bulk modulusµ Complex frame shear modulus

Kf Fluid bulk modulusKr Frame material bulk modulusβ Porosityη Viscosity of pore fluidk Permeabilityα Structure constanta Pore size parameter

June 23, 2004 9:26 WSPC/130-JCA 00221

Determination of the Parameters of Cancellous Bone 101

The bulk and shear moduli Kb and µ are often given imaginary parts to account for frame

inelasticity. Equations (1), (2) and (3) and an argument based upon Lagrangian dynamics

are shown in Ref. 2 to lead to the following equations of motion for the displacements u, U

and dilatations e, ε,

µ∇2u + ∇[(λ+ µ)e+Qε] =∂2

∂t2(ρ11u + ρ12U) + b

∂

∂t(u−U)

∇[Qe+Rε] =∂2

∂t2(ρ12u + ρ22U) − b

∂

∂t(u−U) .

(6)

Here ρ11 and ρ22 are density parameters for the solid and fluid, ρ12 is a density coupling

parameter, and b is a dissipation parameter. These are calculated from the inputs of Table 1

using the formulas

ρ11 = (1 − β)ρr − β(ρf −mβ)

ρ12 = β(ρf −mβ)

ρ22 = mβ2

b =F (a

√

ωρf/η)β2η

k

where

m =αρf

β

and the multiplicative factor F (ζ), which was introduced in Ref. 3 to correct for the inva-

lidity of the assumption of Poiseuille flow at high frequencies, is given by

F (ζ) =1

4

ζT (ζ)

1 − 2T (ζ)/iζ(7)

where T is defined in terms of Kelvin functions

T (ζ) =ber′(ζ) + ibei′(ζ)

ber(ζ) + ibei(ζ).

The bone specimen is assumed to oscillate harmonically in time: u(x, y, t) = u(x, y)eiωt,

U(x, y, t) = U(x, y)eiωt. Substituting these representations into (6) gives

µ∇2u + ∇[(λ+ µ)e+Qε] + p11u + p12U = 0

∇[Qe+Rε] + p12u + p22U = 0(8)

where

p11 := ω2ρ11 − iωb , p12 := ω2ρ12 + iωb , p22 := ω2ρ22 − iωb . (9)

The article of McKelvie and Palmer contains estimates of the Biot parameters of can-

cellous bone in the human os calcis (heel bone) for the normal (β = 0.72) and severely

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 2. Estimated values of some Biot parameters at different porosities taken fromMcKelvie and Palmer or Hosokawa and Otani. The second set of values for the perme-ability were calculated from the indicated value of the pore size parameter using theKozeny–Carmen equation.

β k a α Re Kb Re µ

0.72 5 × 10−9, 7.99 × 10−9 4.71 × 10−4 1.10 3.18 × 109 1.30 × 109

0.75 7 × 10−9, 2.40 × 10−8 8.00 × 10−4 1.08 2.69 × 109 1.10 × 109

0.81 2 × 10−8 1.20 × 10−3 1.06 1.80 × 109 7.38 × 108

0.83 3 × 10−8, 7.56 × 10−8 1.35 × 10−3 1.05 1.55 × 109 6.27 × 108

0.95 5 × 10−7, 2.30 × 10−7 2.20 × 10−3 1.01 2.57 × 108 1.05 × 108

osteoporotic (β = 0.95) cases while the article of Hosokawa and Otani has estimates for

bovine femoral bone for porosities of β = 0.75, 0.81 and 0.83. The question we shall address

is whether these parameters can be recovered by acoustic interrogation of the specimen.

Table 2 contains estimates of these six Biot parameters for five bone specimens. In obtain-

ing them we have followed the estimation procedures used by McKelvie and Palmer and

Hosokawa and Otani:

• The real parts of Kb and µ were calculated using the formulas of Williams

ReKb =E

3(1 − 2ν)V n

f

Reµ =E

2(1 + ν)V n

f

(10)

used by Hosokawa and Otani. Here Vf = 1−β is the bone volume fraction. Theoretically

n = 1 for waves travelling in the trabecular direction and is between 2 and 3 for transverse

waves, however there is enough randomness in the trabecular direction in bone that

authors have empirically adjusted the exponent to agree better with experiment. Williams

arrived at a value of n = 1.23 based on comparing the Biot predictions for Type I

compressional and shear wave velocity assuming the form (10) to the measured speeds

obtained from experiments conducted on samples taken from bovine tibia. Hosokawa

and Otani found that n = 1.46 agreed well with their data from experiments on bone

specimens from bovine femora. We shall use the exponent of Hosokawa and Otani and

also their values E = 2.2 × 1010, ν = 0.32 for the Young’s modulus and Poisson ratio of

solid bone.

• The imaginary parts of Kb and µ were calculated using a log decrement ` : ImK ∗

b =

`ReK∗

b /π, Imµ = `Reµ∗/π with a value ` = 0.1 which is typical of that used in under-

water acoustics. There appears to be little sensitivity to these parameters, however.

• The structure factor was calculated using the formula of Berryman α = 1 − r(1 − 1/β)

with r = 0.25, again following Williams and Hosokawa and Otani.

• The pore size parameter was estimated by McKelvie and Palmer using electron microscopy

and by Hosokawa and Otani using x-ray examination. Figure 1 shows that their estimates

indicate that pore size is approximately a linear function of porosity.

June 23, 2004 9:26 WSPC/130-JCA 00221


0.65 0.7 0.75 0.8 0.85 0.9 0.95 10

0.5

1

1.5

2

2.5

3x 10−3

Porosity

Por

e si

ze

Fig. 1. Estimated values of the pore size parameter (∗) for five bone specimens along with the regressionline.

• Permeability is a difficult parameter to estimate. Figure 2 shows the values for the five

specimens of McKelvie and Palmer and Hosokawa and Otani. The estimates indicate that

permeability is approximately a log-linear function of porosity. However McKelvie and

Palmer characterize their values without elaboration as “estimates” and Hosokawa and

Otani state that their estimates are based upon those of McKelvie and Palmer. Hence

the apparent log-linear relation should be regarded circumspectly. Indeed according to

the Kozeny–Carmen formula

k =βa2

4K, (11)

where K ≈ 5 is an empirical constant, the relation is not log-linear. Figure 2 shows that if

pore size is indeed a linear function of porosity as indicated by Fig. 1, then permeability,

as predicted by (11), will deviate significantly from log-linearity.

Table 3 gives the values we shall use for the other Biot parameters. The value for ρf

is from McKelvie and Palmer, but the value used by Hosokawa and Otani, 930, is similar.

Likewise both sets of authors used about the same value for viscosity η. The fluid bulk

modulus Kf is from Hosokawa and Otani. The frame material densities used by McKelvie

and Palmer and Hosokawa and Otani were somewhat different. Williams reports a range of

estimates for bovine cortical bone of ρr = 1930 → 2000. We follow Williams and Hosokawa

and Otani in using ρr = 1960. The frame material bulk modulus was calculated from (10)

with Vf = 1.

June 23, 2004 9:26 WSPC/130-JCA 00221


0.65 0.7 0.75 0.8 0.85 0.9 0.95 110−9

10−8

10−7

10−6

10−5

Porosity

Per

mea

bilit

y

Fig. 2. Estimated values of permeability (∗) for five bone specimens. Dashed line: regression line. Solid line:Value of premeability predicted by the Kozeny–Carmen equation assuming a linear relation between poresize and porosity.

Table 3. Parameters for cancellous bone to be used for allspecimens.

Parameter Symbol Value

Pore fluid density ρf 950

Fluid bulk modulus Kf 2.00 × 109

Pore fluid viscosity η 1.5Frame material density ρr 1960

Frame material bulk modulus Kr 2.00 × 1010

The question we shall address is whether it is feasible to recover some of the Biot

parameters by measuring the acoustic field arising from a point source placed in a tank

of water containing a specimen of bone. Based on the discussion above the parameters we

shall seek to recover are the ones concerning which there is the most uncertainty: porosity

β, permeability k, pore size a, structure factor α and the real parts of the bulk and shear

frame moduli Kb and µ.

3. Finite Element Formulation of the Problem

A bone specimen is placed in a water tank. The region occupied by the bone specimen

and the water are Ωb and Ωw respectively. In Ωw we have in the two-dimensional case the

June 23, 2004 9:26 WSPC/130-JCA 00221


differential equations for fluid pressure P and displacement [Uw, Vw]

−∇2P − k20P = S(x, y, x0, y0)

∇P + ρwω2[Uw, Vw] = 0 ,

(12)

assuming a source S located at (x0, y0). Multiplying by a test function and applying the

divergence theorem gives∫∫

Ωw

(∇P · ∇ψ − k20Pψ)dA−

∫

∂Ωw

nw · ∇Pψds =

∫∫

Ωw

s(x, y, x0, y0)ψdA

where the unit normal vector nw points into the bone.

In two dimensions the Eq. (8) are

(λ+ 2µ)∂xxu+ µ∂yyu+ (λ+ µ)∂xyv +Q∂xxU +Q∂xyV + p11u+ p12U = 0

(λ+ 2µ)∂yyv + µ∂xxv + (λ+ µ)∂xyu+Q∂yyV +Q∂xyU + p11v + p12V = 0

Q∂x(∂xu+ ∂yv) +R∂x(∂xU + ∂yV ) + p12u+ p22U = 0

Q∂y(∂xu+ ∂yv) +R∂y(∂xU + ∂yV ) + p12v + p22V = 0

(13)

The finite elements package FEMLAB was used for the computations in this article. In

FEMLAB systems of partial differential equations are written

−∂xj(c`kji∂xi

uk + α`kjuk + γ`j) + β`ki∂xiuk + a`kuk = f` (14)

with the summation notation convention in effect. For the Biot equations without a source

α, β, γ, f = 0 which gives

−∂xj(c`kji∂xi

uk) + a`kuk = 0 . (15)

Multiplying (15) by a test function φ and integrating over Ωb gives∫∫

Ωb

(−∂xj(c`kji∂xi

uk) + a`kuk)φdA = 0 .

Applying the divergence theorem gives∫∫

Ωb

(c`kji∂xiuk∂xj

φ+ a`kukφ)dA −

∫

∂Ωb

nbj(c`kji∂xiuk)φds = 0 , ` = 1, 2, 3, 4

where nb = (nbj) is the outward unit normal from Ωb. We consider the two dimensional case

x1 = x, x2 = y and take u1 = u, u2 = v, u3 = U, u4 = V . The stress tensor Tj` = c`kji∂xiuk

must be chosen appropriately for the interface conditions. At the water-bone interface the

following conditions are required (cf. Ref. 4)

• continuity of flux: [Uw, Vw] = β[U, V ] + (1 − β)[u, v]

• continuity of stress: P = σxx + σ, P = σyy + σ, σxy = σyx = 0

• continuity of pore fluid pressure: βP = σ.

June 23, 2004 9:26 WSPC/130-JCA 00221


This suggests we take

T =

[

σxx + σ σxy σ 0

σyx σyy + σ 0 σ

]

.

It then follows from (1), (2) and (3) that

∂xjTj1 = ∂x(σxx + σ) + ∂yσyx

= ∂x(2µ∂xu+ λ(∂xu+ ∂yv) +Q(∂xU + ∂yV ) +Q(∂xu+ ∂yv)

+R(∂xU + ∂yV )) + ∂yµ(∂yu+ ∂xv)

= 2µ∂xxu+ λ∂xxu+ λ∂xyv +Q∂xxU +Q∂xyV +Q∂xxu+Q∂xyv

+R∂xxU +R∂xyV + µ∂yyu+ µ∂xyv

= (λ+ 2µ+Q)∂xxu+ (λ+Q+ µ)∂xyv + (R+Q)∂xxU + (R+Q)∂xyV + µ∂yyu

∂xjTj3 = ∂xσ

= Q(∂xxu+ ∂xyv) +R(∂xxU + ∂xyV )

and similarly for ∂xjTj2 and ∂xj

Tj4. Adding (13)3 to (13)1 and (13)4 to (13)2 gives the

desired form of the equations

(λ+ 2µ+Q)∂xxu+ µ∂yyu+ (λ+ µ+Q)∂xyv + (R+Q)∂xxU

+(R+Q)∂xyV + (p11 + p12)u+ (p12 + p22)U = 0

(λ+ 2µ+Q)∂yyv + µ∂xxv + (λ+ µ+Q)∂xyu+ (R +Q)∂yyV

+(R+Q)∂xyU + (p11 + p12)v + (p12 + p22)V = 0

Q∂x(∂xu+ ∂yv) +R∂x(∂xU + ∂yV ) + p12u+ p22U = 0

Q∂y(∂xu+ ∂yv) +R∂y(∂xU + ∂yV ) + p12v + p22V = 0 .

Thus we want

T11 = c1k1i∂xiuk = c1111∂xu+ c1112∂yu+ c1211∂xv + c1212∂yv

+ c1311∂xU + c1312∂yU + c1411∂xV + c1412∂yV

= σxx + σ = (λ+ 2µ+Q)∂xu+ (λ+Q)∂yv + (Q+R)∂xU + (Q+R)∂yV

⇒ c1111 = λ+ 2µ+Q, c1212 = λ+Q, c1311 = c1412 = Q+R


+ c1321∂xU + c1322∂yU + c1421∂xV + c1422∂yV

= σyx = µ(∂yu+ ∂xv)

⇒ c1122 = c1221 = µ

June 23, 2004 9:26 WSPC/130-JCA 00221



+ c2311∂xU + c2312∂yU + c2411∂xV + c2412∂yV

= σxy = µ(∂xv + ∂yu)

⇒ c2112 = c2211 = µ


+ c2321∂xU + c2322∂yU + c2421∂xV + c2422∂yV

= σyy + σ = (λ+ 2µ+Q)∂yv + (λ+Q)∂xu+ (Q+R)∂xU + (Q+R)∂yV

⇒ c2121 = λ+Q, c2222 = λ+ 2µ+Q, c2321 = c2422 = Q+R


+ c3311∂xU + c3312∂yU + c3411∂xV + c3412∂yV

= σ = Q(∂xu+ ∂yv) +R(∂xU + ∂yV )

⇒ c3111 = c3212 = Q, c3311 = c3412 = R


+ c4321∂xU + c4322∂yU + c4421∂xV + c4422∂yV

= σ = Q(∂xu+ ∂yv) +R(∂xU + ∂yV )

⇒ c4121 = c4222 = Q, c4321 = c4422 = R

This gives

c11.. =

(

λ+ 2µ+Q 0

0 µ

)

, c12.. =

(

0 λ+Q

µ 0

)

c13.. =

(

Q+R 0

0 0

)

, c14.. =

(

0 Q+R

0 0

)

c21.. =

(

0 µ

λ+Q 0

)

, c22.. =

(

µ 0

0 λ+ 2µ+Q

)

c23.. =

(

0 0

Q+R 0

)

, c24.. =

(

0 0

0 Q+R

)

c31.. =

(

Q 0

0 0

)

, c32.. =

(

0 Q

0 0

)

June 23, 2004 9:26 WSPC/130-JCA 00221


c33.. =

(

R 0

0 0

)

, c34.. =

(

0 R

0 0

)

c41.. =

(

0 0

Q 0

)

, c42.. =

(

0 0

0 Q

)

c43.. =

(

0 0

R 0

)

, c44.. =

(

0 0

0 R

)

.

Also

a11 = a22 = −(p11 + p12) , a13 = a24 = −(p12 + p22)

a31 = a42 = −p12 , a33 = a44 = −p22 .

In FEMLAB interface conditions are of the form n · (c∇u)+q ·u = 0 where for this problem

u = [P, u, v, U, V ]T . The interface conditions are

• Continuity of flux: From (12)2

nw · ∇P = −nw · ρwω2[Uw, Vw] = −nw · ρwω

2(β[U, V ] + (1 − β)[u, v]) .

and thus

nw · ∇P + nw · ρwω2(β[U, V ] + (1 − β)[u, v]) = 0 .

Here nw points into the bone.

• Continuity of stress:

nbjTj` + nb`P = 0 , ` = 1, 2

since an expansion of the bone induces a compression (P < 0) in the water. Here nb

points into the water.

• Continuity of pore pressure:

nbjTj` − nb`βP = 0 , ` = 3, 4 .

This gives

q =

0 ρwω2(1 − β)nwx ρwω

2(1 − β)nwy ρwω2βnwx ρwω

2βnwy

nbxP 0 0 0 0

nbyP 0 0 0 0

−nbxβP 0 0 0 0

−nbyβP 0 0 0 0

.

June 23, 2004 9:26 WSPC/130-JCA 00221


4. Preliminary Explorations

We want to simulate the following experiment: a rectangular bone specimen of the approx-

imate dimensions of the ones used in the experiment described in Hosokawa and Otani,

0.01 × 0.03 m is placed in a tank of water. A time-harmonic point source is located at the

origin. The tank is open at the top whence we use a pressure release condition P = 0.

The sides of the tank are assumed to be perfectly reflecting ∂P/∂n = 0. The dimensions of

the tank will be based upon a distance s: The bone specimen is centered on the x-axis a

distance 2s to the right of the origin. The left edge of the tank is at x = −s. The surface of

the water is a distance 2s above the top of the bone, the bottom of the tank a distance 2s

below the bottom of the bone and the right edge of the tank a distance 2s from the right

edge of the bone. We will experiment with different values of s.

We shall assume that the acoustic field generated by the point source has values P ∗

ij

at points (xi, yj), i = 1, . . . , L, j = 1, . . . , M located in the tank. A set of Biot parame-

ters which produces trial values Pij will be compared to the “measured” values using the

objective function

f(Pij, P∗

ij) =

L∑

i=1

‖Pi. − P ∗

i.‖2

L∑

i=1

‖P ∗

i.‖2

. (16)

Thus the problem is minimize the objective function. The minimization will be carried

out using the Nelder–Mead simplex algorithm. The process is complicated by the various

errors involved: (1) in an actual experiment there would be errors in the measurements P ∗

ij,

(2) the trial values Pij are calculated from the Biot model and thus will be in poor agreement

with the measured values to the extent that the Biot model is a poor approximation to

cancellous bone, and (3) since the trials values will be calculated using the finite element

method, there will be discretization error as well. Lacking experimental data, we are not in

a position to assess the effects of the first two types of error. Consequently we will focus

on the third type. To assess it we will calculate the “measured” data P ∗

ij assuming the Biot

model and using a finite elements mesh which is the third refinement of FEMLAB’s initial

mesh, but use only two refinements when calculating the trial fields Pij .

Figure 3 shows the result of calculating the objective function f(Pij, P∗

ij) for the size

parameters s = 0.03, 0.07 and s = 0.10 m with measurements taken at 10 points in the

middle 80% of the tank along two vertical lines, one midway between the source and the

bone specimen, the other midway between the bone and the right edge of the tank. Both Pij

and P ∗

ij were calculated using the Biot parameters given for porosity β = 0.72 in Tables 2

and 3, but Pij was calculated using two refinements of the initial finite element mesh whereas

P ∗

ij was calculated with three refinements. As can be seen the agreement between the two

fields varies considerably with frequency, suggesting that the expediency and the success

in recovering the Biot parameters of the bone specimen may depend on the interrogating

frequency. The values of the objective function are large above about 7 kHz for s = 0.10 and

June 23, 2004 9:26 WSPC/130-JCA 00221


0 5000 10000 1500010−4

10−3

10−2

10−1

100

101

Frequency (Hz)

s = 0.10s = 0.07s = 0.03

Fig. 3. Objective function values for three tank sizes. The target pressure data was calculated using the finiteelement method with three refinements, the trial data with two refinements. The porostiy of the specimenwas 0.72.

about 10 kHz for s = 0.07 suggesting that calculating the trial data with two refinements of

the initial mesh may be inadequate for tanks with dimensions on the order of a half meter

on a side. While the agreement is good at frequencies in the low kilohertz range the question

arises as to whether this is simply due to little interaction with the bone specimen.

To investigate the influence of the interrogating frequency and the effect of using two

refinements of the finite elements mesh for the trial data, but three refinements for the

simulated data, we attempted to recover six Biot parameters starting with guesses that were

in fact the correct values for the target specimen. Table 4 shows the results of attempting

to recover the six parameters β, k, a, α, ReKb and Reµ using the Nelder–Mead simplex

method to perform a multivariate minimization on the objective function (16) at different

frequencies when the tank size parameter was s = 0.03. As can be seen the real part of

the bulk modulus became negative at 1 kHz, suggesting that interaction with the bone at

low frequencies may be insufficient to determine some parameters. The three frequencies

6500, 8000 and 9000 were chosen because they represent respectively the cases where for

β = 0.72 the agreement in pressure between two and three refinements of the finite element

mesh was good, intermediate and poor (cf. Fig. 3). For these three frequencies between 5

and 10 kHz there was no easily discernible pattern as to which frequency produced the best

results for a particular parameter. It may be noted however that the porosity estimates were

somewhat more accurate at 6500 Hz where the agreement for two and three refinements

of the finite element mesh was best. Also noteworthy is that the pore size parameter was

in three instances off by a factor of more than two, indicating this parameter may not be

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 4. Results of an application of the simplex method when the initial guess for the parameters was thecorrect one for the target specimen. Three refinements of the initial finite element mesh were used for thetarget data, two for the trial data.

Frequency f(Pij , P ∗

ij ) β k a α ReKb Reµ

Guess/Target 0.72 5.00 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

1000 0.0022 0.747 4.29 × 10−9 0.000211 1.30 −7.53 × 105 2.89 × 109

6500 0.0052 0.741 7.40 × 10−9 0.00104 0.0005 3.48 × 109 1.45 × 109

8000 0.0103 0.760 5.03 × 10−9 0.000121 1.87 3.08 × 109 8.48 × 108

9000 0.1036 0.740 5.32 × 10−9 0.000436 1.09 3.04 × 109 1.35 × 109

Guess/Target 0.75 7.00 × 10−9 0.0008 1.08 2.69 × 109 1.10 × 109

6500 0.0054 0.769 7.21 × 10−9 0.000812 1.11 2.57 × 109 1.07 × 109

8000 0.0105 0.789 7.09 × 10−9 0.000798 0.67 2.69 × 109 7.47 × 108

9000 0.1064 0.773 6.01 × 10−9 0.000512 1.70 2.58 × 109 9.11 × 108

Guess/Target 0.83 3.00 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

6500 0.0058 0.853 4.30 × 10−8 0.00160 1.20 1.04 × 109 4.36 × 108

8000 0.0102 0.854 2.69 × 10−8 0.00118 0.93 1.72 × 109 5.30 × 108

9000 0.1132 0.856 3.87 × 10−8 0.00141 1.09 1.17 × 109 4.90 × 108

Guess/Target 0.95 5.00 × 10−7 0.0022 1.01 2.57 × 108 1.05 × 108

6500 0.0052 0.951 2.66 × 10−7 0.0145 0.98 1.93 × 108 8.08 × 107

8000 0.0087 0.963 7.57 × 10−7 0.00276 0.99 1.93 × 108 8.05 × 107

9000 0.0678 0.953 4.92 × 10−7 0.00226 1.00 2.54 × 108 1.07 × 108

Table 5. Number of iterations and evaluations of the objective function and finalvalue of the objective function resulting from an application of the simplex method,the results of which are shown in Table 4.

Frequency β Iterations Evaluations f(Pij , P ∗

ij ) fmin

0.72

6500 0.741 229 373 5.2 × 10−3 1.61 × 10−3

8000 0.760 255 411 1.2 × 10−2 6.16 × 10−4

9000 0.740 91 165 1.0 × 10−1 2.21 × 10−3

0.75

6500 0.769 60 113 5.4 × 10−3 2.25 × 10−3

8000 0.789 434 694 1.1 × 10−2 6.21 × 10−4

9000 0.773 392 619 1.1 × 10−1 1.29 × 10−3

0.83

6500 0.853 309 492 5.8 × 10−3 1.86 × 10−3

8000 0.854 479 742 1.0 × 10−2 4.81 × 10−4

9000 0.856 396 621 1.1 × 10−1 3.30 × 10−3

0.95

6500 0.951 771 1200 5.2 × 10−3 1.49 × 10−3

8000 0.963 523 823 8.7 × 10−3 4.68 × 10−4

9000 0.953 192 324 6.8 × 10−2 2.08 × 10−3

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 6. Results of an application of the simplex method when the initial guesses for theparameters were those for cancellous bone of porosity 0.81 (see Table 2). Three refinements ofthe initial finite element mesh were used for the target data, two for the trail data.

Frequency β k a α ReKb Re µ

Guess 0.81 2.00 × 10−8 0.0012 1.06 1.80 × 109 7.38 × 108

Target 0.72 5.00 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

6500 0.739 6.39 × 10−9 0.000832 −0.57 3.50 × 109 1.73 × 109

8000 0.749 4.91 × 10−9 0.000720 0.20 4.27 × 109 1.72 × 109

9000 0.784 1.52 × 10−8 0.00157 1.26 1.37 × 109 8.66 × 108

Target 0.75 7.00 × 10−9 0.0008 1.08 2.69 × 109 1.10 × 109

6500 0.767 1.68 × 10−8 0.00234 −0.39 2.75 × 109 1.15 × 109

8000 0.781 7.08 × 10−9 0.000937 0.63 2.63 × 109 1.22 × 109

9000 0.790 1.25 × 10−8 0.00140 1.13 1.89 × 109 8.37 × 108

Target 0.83 3.00 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

6500 0.854 4.26 × 10−8 0.00155 1.22 1.03 × 109 4.31 × 108

8000 0.862 3.33 × 10−8 0.00139 0.91 1.45 × 109 4.19 × 108

9000 0.841 2.26 × 10−8 0.00116 0.73 1.60 × 109 7.24 × 108

Target 0.95 5.00 × 10−7 0.0022 1.01 2.57 × 108 1.05 × 108

6500 0.838 5.20 × 10−8 −0.000958 0.84 −1.66 × 104 1.34 × 109

8000 0.883 5.92 × 10−8 −0.000389 1.20 −2.81 × 107 8.96 × 108

9000 0.896 7.14 × 10−8 −0.000646 1.02 8.72 × 108 3.62 × 108

strongly influential and that, except for bone of porosity 0.95 the structure factor α drifted

far from its initial (correct) value indicating that this parameter is not influential for bone

at the lower porosities.

Table 5 indicates that there was wide variance in the number of iterations and evaluations

of the objective function the simplex method required to converge. There was no strong

correlation between frequency and efficiency for the three frequencies tested. It may be

noted however that at 6500 Hz where the initial agreement was best, the reduction of the

value of the objective function was least. Since reduction of the objective function below the

value produced by the correct values of the parameter when two refinements of the finite

element mesh are used represents uncertain progress, this may be a virtue.

Another preliminary test that we conducted was to see if a simple scheme for supplying

initial guesses for the six parameters to the simplex minimization method would suffice.

Table 6 shows the results obtained when the initial guesses were the parameters for cancel-

lous bone of porosity 0.81 and the targets were bone of porosities 0.72, 0.75, 0.83 and 0.95.

This simple approach worked well for the three lower porosities, but poorly for β = 0.95.

Hence if parameter recovery is to be successful over the entire range of porosities that are

expected, a more sophisticated algorithm is required.

June 23, 2004 9:26 WSPC/130-JCA 00221


5. An Algorithm for Recovering the Biot Parameters

As indicated in the last section the recovery of the Biot parameters of a bone specimen by

a minimization may be successful if sufficiently good initial guesses for the parameters can

be found. We will generate these guesses by first formulating the problem as a univariate

minimization problem for the single parameter porosity. This is feasible because of the

various formulas relating other Biot parameters to porosity discussed in Sec. 2. Since these

formulas were used in creating the target Biot parameters that we are trying to recover,

variations based on the uncertainties mentioned were incorporated where possible. For a

given value of porosity β

• ReKb(β) and Reµ(β) are determined from the formulas (10). Since the values for the

target specimens of Table 2 were determined using Hosokawa and Otani’s value n =

1.46 we used Williams value n = 1.23 in calculating the trail data in the univariate

minimization process.

• Pore size a(β) is determined by the regression line in Fig. 1. Since this parameter was

measured independently by McKelvie and Palmer and Hosokawa and Otani for their bone

specimens and the results indicate an approximately linear relation between pore size and

porosity, this is justifiable.

• Permeability k(β) is determined from the regression line shown in Fig. 2. Since, as in-

dicated earlier, there is some uncertainty about the log-linear relationship indicated by

Fig. 1, we shall also test the algorithm when the permeability of the target specimen

is calculated from the Kozeny–Carmen equation (11) using the pore size parameter in

0 5000 10000 150000.7

0.75

0.8

0.85

0.9

0.95

1

Por

osity

Frequency (Hz)0 5000 10000 15000

10−4

10−3

10−2

10−1f(P

,P*)

Fig. 4. Estimated porosity (solid line) and objective function value (dashed line) for bone of porosity 0.72.

June 23, 2004 9:26 WSPC/130-JCA 00221


0 5000 10000 150000.75

0.8

0.85

0.9

0.95

1

Por

osity

0 5000 10000 1500010−3

10−2

10−1

100

f(P,P

*)

Frequency (Hz)

Fig. 5. Estimated porosity (solid line) and objective function value (dashed line) for bone of porosity 0.83.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x 104

10−3

10−2

10−1

100

Frequency (Hz)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

x 104

10−3

10−2

10−1

100

Fig. 6. Solid line: values of f(Pij , P ∗

ij ) when Pij and P ∗

ij are calculated for bone specimens of porosity 0.72(top) and 0.83 (bottom) using finite elements with two and three mesh refinements respectively. Dashed line:values of f(Pij (βmin), P ∗

ij).

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 2. As indicated in Fig. 2 this gives only modest agreement with the estimates of

McKelvie and Palmer and Hosokawa and Otani.

• The structure factor α(β) is determined from the formula of Berryman.1 The value of

the parameter r used was 0.227 suggested by the finite element analysis of Yavari and

Bedford,8 rather than the value 0.25 used by Hosokawa and Otani and Williams, but this

makes little difference in the predicted value.

• The values of all other Biot parameters are taken from Table 3.

For any given porosity β a set of Biot parameters can be constructed as described above

and from this a pressure field Pij(β) calculated using finite elements with two refinements of

the initial mesh. At a given frequency the “measured” data P ∗

ij can be calculated using three

mesh refinements and the value of β which minimizes the objective function f(Pij(β), P ∗

ij)

found. Figures 4 and 5 show the results of this univariate minimization procedure over

the frequency range 1–15 kHz when the target specimens had porosities 0.72 and 0.83

respectively. The minimum was sought in the range 0.60 ≤ β ≤ 0.99 and the tank size

Table 7. Results of two applications of the simplex method. Three refinements of the initial finite elementmesh were used for the simulated data, two for the trial data. The values for permeability were the firstset in Table 2.

Frequency β k a α ReKb Reµ fmin

6000

Target 0.72 5.00 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

Guess 1 0.74 5.87 × 10−9 0.000602 1.08 3.89 × 109 1.59 × 109

Result 1 0.735 5.31 × 10−9 0.000583 0.36 3.17 × 109 1.12 × 109 2.93 × 10−4

Guess 2 0.735 5.31 × 10−8 0.000566 1.08 3.17 × 109 1.12 × 109

Result 2 0.734 5.10 × 10−9 0.000379 1.23 2.97 × 109 1.22 × 109 2.98 × 10−4

6000

Target 0.75 7.00 × 10−9 0.000800 1.08 2.67 × 109 1.10 × 109

Guess 1 0.76 8.81 × 10−9 0.000748 1.07 3.52 × 109 1.44 × 109

Result 1 0.767 8.22 × 10−9 0.000870 1.63 2.76 × 109 1.15 × 109 1.80 × 10−3

Guess 2 0.767 1.01 × 10−8 0.000798 1.07 2.76 × 109 1.15 × 109

Result 2 0.767 1.08 × 10−8 0.00134 0.93 2.76 × 109 1.15 × 108 1.80 × 10−3

6500

Target 0.83 3.00 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

Guess 1 0.83 3.66 × 10−8 0.00126 1.05 2.30 × 109 9.42 × 108

Result 1 0.848 4.69 × 10−8 0.00214 0.98 1.18 × 109 4.91 × 108 1.96 × 10−3

Guess 2 0.848 5.29 × 10−8 0.00139 1.04 1.18 × 109 4.91 × 108

Result 2 0.853 4.27 × 10−8 0.00158 1.21 1.05 × 109 4.36 × 108 1.86 × 10−3

6500

Target 0.95 5.00 × 10−7 0.00220 1.01 2.57 × 108 1.05 × 108

Guess 1 0.94 3.42 × 10−7 0.00206 1.01 6.40 × 108 2.62 × 108

Result 1 0.932 3.44 × 10−7 0.00245 0.97 6.83 × 108 2.86 × 108 1.17 × 10−3

Guess 2 0.932 2.88 × 10−7 0.00200 1.02 6.83 × 108 2.86 × 108

Result 2 0.933 4.90 × 10−7 0.00363 0.96 4.99 × 108 2.09 × 108 1.03 × 10−3

June 23, 2004 9:26 WSPC/130-JCA 00221


parameter was s = 0.03. Except at a few frequencies the estimated porosities were close

to the target values. Also shown are minimum values of the objective function at each

frequency. As Fig. 6 shows these minimum values follow the trend of those for s = 0.03 in

Fig. 3, and thus we have a means of finding frequencies at which the agreement between

trial and “measured” pressure fields may be good without knowing the Biot parameters for

the target specimen.

Tables 7–14 show the results of using the simplex method to perform a multivariate

minimization when the pressure data and Biot parameters for the target specimens were

calculated different ways. The algorithm used was as follows: The univariate minimization

procedure described above was used to generate initial estimates, labeled “Guess 1” in the

Tables, for the six Biot parameters for which values were sought. The frequency used was the

one that produced the best agreement between trial and “measured” data in the frequency

range 5.5–10 kHz (cf. Figs. 4 and 5). The Nelder–Mead simplex method was initialized

with these values. As can be seen in Tables 7 and 9 the initial estimates for porosity and

hence the other parameters were good when the target permeabilities were the first set in

Table 8. Wave speeds and attenuations resulting from two applications ofthe simplex method. Three refinements of the initial finite element meshwere used for the simulated data, two for the trial data. The values forpermeability were the first set in Table 2.

Frequency β cp1 cp2 cs γp1 γp2 γs

6000Target 0.72 2365 594 1043 1.27 40.01 3.06Guess 1 0.74 2546 611 1169 1.24 35.18 2.64Result 1 0.735 2329 601 974 1.01 40.83 2.78Guess 2 0.735 2332 573 977 0.97 36.94 2.63Result 2 0.734 2324 605 1016 1.00 41.72 2.77




June 23, 2004 9:26 WSPC/130-JCA 00221


Table 9. Results of two applications of the simplex method. Three refinements of the initial finite elementmesh were used for the simulated data, two for the trial data. The values for permeability were the secondset in Table 2.


6500

Target 0.72 7.99 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

Guess 1 0.74 5.87 × 10−9 0.000602 1.08 3.89 × 109 1.59 × 109

Result 1 0.719 7.10 × 10−9 0.000247 1.49 2.75 × 108 1.60 × 109 1.19 × 10−3

Guess 2 0.719 3.80 × 10−9 0.000446 1.09 2.75 × 108 1.60 × 109

Result 2 0.719 7.10 × 10−9 0.000246 1.49 2.74 × 108 1.60 × 109 1.19 × 10−3

6500

Target 0.75 2.40 × 10−8 0.000800 1.08 2.69 × 109 1.10 × 109

Guess 1 0.90 1.52 × 10−7 0.00177 1.03 1.20 × 109 4.91 × 108

Result 1 0.769 3.55 × 10−8 0.00128 0.93 1.51 × 109 7.45 × 108 1.26 × 10−3

Guess 2 0.769 1.07 × 10−8 0.000816 1.07 1.51 × 109 7.45 × 108

Result 2 0.769 2.64 × 10−8 0.000819 1.11 1.43 × 109 8.29 × 108 1.22 × 10−3

6500

Target 0.83 7.56 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

Guess 1 0.92 2.28 × 10−7 0.00192 1.02 9.12 × 108 3.73 × 108

Result 1 0.846 1.75 × 10−7 0.00341 0.95 9.30 × 108 3.88 × 108 9.81 × 10−4

Guess 2 0.846 5.03 × 10−8 0.00137 1.04 9.30 × 108 3.88 × 108

Result 2 0.842 6.28 × 10−8 0.000913 1.17 7.95 × 108 3.94 × 108 1.19 × 10−3

6500

Target 0.95 2.30 × 10−7 0.00220 1.01 2.57 × 108 1.05 × 108

Guess 1 0.99 9.46 × 10−7 0.00243 1.00 7.06 × 107 2.89 × 107

Result 1 0.977 6.62 × 10−7 0.00272 1.02 8.40 × 107 3.53 × 107 1.33 × 10−3

Guess 2 0.977 7.32 × 10−7 0.00233 1.01 8.40 × 107 3.53 × 107

Result 2 0.977 8.16 × 10−7 0.00355 1.01 7.46 × 107 3.14 × 107 1.27 × 10−3

Table 2, but sometimes were poor when the second (Kozeny–Carmen) set of permeabilities

was used. The simplex method was always fairly successful in determining the porosity, but

was sometimes less successful in determining the other parameters. The result of the first

application are referred to as “Result 1” in the Tables. To see whether a better initial guess

for the parameters would improve the accuracy of the other parameters, a new initial guess,

labeled “Guess 2”, for the simplex method was constructed using the porosity, bulk and

shear moduli determined by the first application, but with the pore size and permeability

calculated from the value of porosity found by the first application using the regression lines

shown in Figs. 1 and 2. The result of this second application is referred to as “Result 2” in

the Tables. Tables 8 and 10 express the results in terms of an alternative set of parameters,

the speeds cp1 and cp2 of type I and II compressional waves, the speed cs of shear waves, and

the attenuations γp1, γp2 and γs, measured in decibels per wavelength, of the three types of

waves. See Ref. 4 for the details of how these quantities are computed. Tables 11–14 give the

results of applying the guess and refine procedure at frequencies 8 and 9 kHz (cf. Table 5).

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 10. Wave speeds and attenuations resulting from two applications ofthe simplex method. Three refinements of the initial finite element meshwere used for the simulated data, two for the trial data. The values forpermeability were the second set in Table 2.

Frequency β cp1 cp2 cs γp1 γp2 γs





6. Conclusions

In discussing the results of our simulations we consider the specimens of porosities 0.72, 0.75

and 0.83 separately from that of porosity 0.95 since different parameters were influential

in the latter case. Tables 15 and 16 give the percentage relative errors made by the algo-

rithm described in the preceding section in determining the five Biot parameters porosity,

permeability, pore size, and real parts of the bulk and shear moduli. The structure factor

is not included since was rarely determined with much accuracy and often the value found

was well below its theoretical minimum of 1.00. The algorithm was uniformly successful in

finding the porosity to within 3%. Percentage errors for all of the remaining parameters

were often higher, but the target values of these parameters varied over at least one order

of magnitude. In the case when the permeability of the specimen was the first one given in

Table 2 the second application of the simplex method did not improve the results, indeed

as indicated by the averages in the last rows of Table 15, they were slightly worse. This is

not surprising since the regression lines used in estimating the permeability and pore size

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 11. Results of two applications of the simplex method at 8 kHz. Three refinements of the initialfinite element mesh were used for the simulated data, two for the trial data. The values for permeabilitywere the first set in Table 2.


8000

Target 0.72 5.00 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

Guess 1 0.73 4.79 × 10−9 0.000529 1.08 4.07 × 109 1.66 × 109

Result 1 0.758 4.95 × 10−9 0.000298 1.36 2.99 × 109 9.21 × 108 6.13 × 10−4

Guess 2 0.758 8.45 × 10−9 0.000733 1.07 2.99 × 109 9.21 × 108

Result 2 0.755 4.83 × 10−9 0.000575 0.66 4.11 × 109 1.28 × 109 1.83 × 10−3

8000

Target 0.75 7.00 × 10−.9 0.000800 1.08 2.69 × 109 1.10 × 109

Guess 1 0.75 7.19 × 10−9 0.000675 1.08 3.70 × 109 1.51 × 109

Result 1 0.784 7.66 × 10−9 0.000970 0.30 2.44 × 109 9.44 × 108 6.13 × 10−4

Guess 2 0.784 1.43 × 10−8 0.000921 1.06 2.44 × 109 9.44 × 108

Result 2 0.797 1.07 × 10−8 0.00128 0.00 3.04 × 109 3.91 × 108 9.50 × 10−4

8000

Target 0.83 3.00 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

Guess 1 0.97 6.30 × 10−7 0.00228 1.00 2.73 × 108 1.12 × 108

Result 1 0.960 2.36 × 10−7 0.000766 1.50 4.40 × 108 1.86 × 108 6.98 × 10−3

Guess 2 0.960 5.20 × 10−7 0.00221 1.01 4.40 × 108 1.86 × 108

Result 2 0.853 2.59 × 10−7 0.0150 0.49 1.40 × 109 5.74 × 108 5.45 × 10−4

8000

Target 0.95 5.00 × 10−7 0.00220 1.01 2.57 × 108 1.05 × 108

Guess 1 0.96 5.14 × 10−7 0.00221 1.01 3.89 × 108 1.59 × 108

Result 1 0.949 5.08 × 10−7 0.00267 0.99 3.56 × 108 1.49 × 108 2.37 × 10−3

Guess 2 0.949 4.09 × 10−7 0.00213 1.01 3.56 × 108 1.49 × 108

Result 2 0.975 2.03 × 10−6 0.00519 1.00 1.60 × 108 6.72 × 107 1.18 × 10−3

for Guess 1 yielded good estimates, assuming the estimate for porosity was accurate. When

the target specimen had the second (Kozeny–Carmen) value for permeability the situation

was different. The second application of the simplex method substantially improved poor

results for permeability and pore size for the specimens of porosity 0.75 and 0.83. At the

other two frequencies tested, 8 and 9 kHz, the determination of porosity was on the whole

slightly less accurate (Tables 17–20). Also the procedure failed to find a reasonable value

for permeability for the specimen of porosity 0.83 at 8 kHz (Table 17).

The percentage errors made by the algorithm in determining the parameters of the

specimen of porosity 0.95 are given in Table 21. The algorithm was successful in determining

the porosity and the estimate for the structure factor was much more accurate than for the

lower porosity specimens. Determination of the pore size parameter and the real parts of

the bulk and shear moduli was less accurate than at the lower porosities. Determination of

permeability was very inaccurate when the Kozeny–Carmen values were used and therefore

the initial guesses given to the simplex method were poor. Thus only the porosity seemed

to be reliably ascertainable.

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 12. Results of two applications of the simplex method at 8 kHz. Three refinements of the initialfinite element mesh were used for the simulated data, two for the trial data. The values for permeabilitywere the second set in Table 2.


8000

Target 0.72 7.99 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

Guess 1 0.75 7.19 × 10−9 0.000675 1.08 3.70 × 109 1.51 × 109

Result 1 0.747 7.80 × 10−9 0.000418 0.90 2.92 × 109 1.12 × 109 6.20 × 10−4

Guess 2 0.747 6.83 × 10−9 0.000656 1.08 2.92 × 109 1.12 × 109

Result 2 0.744 7.93 × 10−9 0.000260 1.28 2.59 × 109 1.08 × 109 4.83 × 10−4

8000

Target 0.75 2.40 × 10−8 0.000800 1.08 2.69 × 109 1.10 × 109

Guess 1 0.75 7.19 × 10−9 0.000675 1.08 3.70 × 109 1.51 × 109

Result 1 0.763 2.55 × 10−8 0.00105 0.78 2.95 × 109 1.28 × 109 8.12 × 10−4

Guess 2 0.763 9.37 × 10−9 0.000770 1.07 2.95 × 109 1.28 × 109

Result 2 0.766 2.08 × 10−8 0.000641 1.04 2.87 × 109 1.20 × 108 7.89 × 10−4

8000

Target 0.83 7.56 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

Guess 1 0.91 1.86 × 10−7 0.00184 1.02 1.05 × 109 4.31 × 108

Result 1 0.862 1.95 × 10−7 0.00358 0.94 1.52 × 109 3.51 × 108 8.31 × 10−4

Guess 2 0.862 7.04 × 10−8 0.00149 1.04 1.52 × 108 3.51 × 108

Result 2 0.858 7.41 × 10−8 0.00112 1.05 1.56 × 109 4.02 × 108 8.30 × 10−4

8000

Target 0.95 2.30 × 10−7 0.00220 1.01 2.57 × 108 1.05 × 108

Guess 1 0.99 9.46 × 10−7 0.00243 1.00 7.06 × 107 2.90 × 107

Result 1 0.979 5.18 × 10−7 0.00202 1.02 3.46 7.58 × 107 1.37 × 10−3

Guess 2 0.979 7.58 × 10−7 0.00235 1.00 3.46 7.58 × 107

Result 2 0.979 6.33 × 10−7 0.00258 1.01 3.68 7.58 × 107 1.37 × 10−3

References

1. J. G. Berryman, Confirmation of biot’s theory, Appl. Phys. Lett. 37 (1980) 382–384.2. M. A. Biot, Theory of propogation of elastic waves in a fluid-saturated porous solid. I. Lower

frequency range, J. Acoust. Soc. Am. 28 (1956) 168–178.3. M. A. Biot, Theory of propogation of elastic waves in a fluid-saturated porous solid. II. Higher

frequency range, J. Acoust. Soc. Am. 28 (1956) 179–191.4. J. L. Buchanan and R. P. Gilbert, Transmission loss in a shallow ocean over a two-layer seabed,

Int. J. Solids Structures 35(34–35) (1998) 4779–4801.5. A. Hosokawa and T. Otani, Ultrasonic wave propagation in bovine cancellous bone, J. Acoust.

Soc. Am. 101 (1997) 558–562.6. M. L. McKelvie and S. B. Palmer, The interaction of ultrasound with cancellous bone, Phys.

Med. Biol. 10 (1991) 1331–1340.7. J. L. Williams, Prediction of some experimental results by biot’s theory, J. Acoust. Soc. Am. 91

(1992) 1106–1112.8. B. Yavari and A. Bedford, Comparison of numerical calculations of two biot coefficients with

analytical solutions, J. Acoust. Soc. Am. 90 (1991) 985–990.

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 13. Results of two applications of the simplex method at 9 kHz. Three refinements of the initial finiteelement mesh were used for the simulated data, two for the trial data. The values for permeability were thefirst set in Table 2.

Frequency β k a α Re Kb Re µ fmin

9000

Target 0.72 5.00 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

Guess 1 0.75 7.19 × 10−9 0.000675 1.08 3.70 × 109 1.51 × 109

Result 1 0.743 6.03 × 10−9 0.000703 0.0002 2.92 × 109 1.15 × 109 1.32 × 10−3

Guess 2 0.743 6.26 × 10−9 0.000625 1.08 2.92 × 109 1.15 × 109

Result 2 0.754 6.01 × 10−9 0.000622 1.07 2.95 × 109 1.19 × 109 4.67 × 10−3

9000

Target 0.75 7.00 × 10−9 0.000800 1.08 2.67 × 109 1.10 × 109

Guess 1 0.75 7.19 × 10−9 0.000675 1.08 3.70 × 109 1.51 × 109

Result 1 0.760 6.67 × 10−9 0.000673 1.10 3.78 × 109 1.55 × 109 6.43 × 10−3

Guess 2 0.760 8.80 × 10−9 0.000748 1.07 3.78 × 109 1.55 × 109

Result 2 0.768 8.14 × 10−9 0.000926 0.97 3.78 × 109 1.39 × 109 4.89 × 10−3

9000

Target 0.83 3.00 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

Guess 1 0.84 4.48 × 10−8 0.00133 1.04 2.14 × 109 8.75 × 108

Result 1 0.853 5.19 × 10−8 0.00234 0.84 1.23 × 109 5.25 × 108 1.22 × 10−3

Guess 2 0.853 5.83 × 10−8 0.00143 1.04 1.23 × 109 5.25 × 108

Result 2 0.856 4.73 × 10−8 0.00181 1.05 1.04 × 109 5.29 × 108 4.15 × 10−3

9000

Target 0.95 5.00 × 10−7 0.00220 1.01 2.57 × 108 1.05 × 108

Guess 1 0.94 3.42 × 10−7 0.00206 1.01 6.40 × 108 2.62 × 108

Result 1 0.927 2.70 × 10−7 0.00200 0.98 3.84 × 108 1.89 × 108 5.32 × 10−3

Guess 2 0.927 2.62 × 10−7 0.00197 1.02 3.84 × 108 1.89 × 108

Result 2 0.938 3.74 × 10−7 0.00218 1.00 3.16 × 108 1.56 × 108 3.35 × 10−3

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 14. Results of two applications of the simplex method at 9 kHz. Three refinements of the initialfinite element mesh were used for the simulated data, two for the trial data. The values for permeabilitywere the second set in Table 2.


9000

Target 0.72 7.99 × 10−9 0.000471 1.10 3.18 × 109 1.30 × 109

Guess 1 0.76 8.81 × 10−9 0.000748 1.07 3.52 × 109 1.44 × 109

Result 1 0.749 1.01 × 10−8 0.000541 1.12 3.88 × 109 1.47 × 109 6.11 × 10−3

Guess 2 0.749 7.02 × 10−9 0.000666 1.08 3.88 × 109 1.47 × 109

Result 2 0.736 8.63 × 10−9 0.000440 1.04 3.94 × 109 1.57 × 109 4.00 × 10−3

9000

Target 0.75 2.40 × 10−8 0.000800 1.08 2.69 × 109 1.10 × 109

Guess 1 0.92 2.28 × 10−7 0.00192 1.02 9.12 × 108 3.73 × 108

Result 1 0.798 2.87 × 10−8 0.000648 1.21 2.59 × 109 3.14 × 108 1.73 × 10−3

Guess 2 0.798 1.92 × 10−8 0.00103 1.06 2.59 × 109 3.14 × 108

Result 2 0.779 2.44 × 10−8 0.000732 1.06 2.98 × 109 3.34 × 108 4.58 × 10−3

9000

Target 0.83 7.56 × 10−8 0.00135 1.05 1.53 × 109 6.27 × 108

Guess 1 0.92 2.28 × 10−7 0.00192 1.02 9.12 × 108 3.73 × 108

Result 1 0.908 2.34 × 10−7 0.00187 1.05 9.22 × 108 3.76 × 108 8.39 × 10−3

Guess 2 0.908 1.78 × 10−7 0.00183 1.02 9.22 × 108 3.76 × 108

Result 2 0.861 9.90 × 10−8 0.00147 1.03 1.33 × 109 3.67 × 108 1.52 × 10−3

9000

Target 0.95 2.30 × 10−7 0.00220 1.01 2.57 × 108 1.05 × 108

Guess 1 0.98 7.72 × 10−7 0.00235 1.00 1.66 × 108 6.78 × 107

Result 1 0.975 7.41 × 10−7 0.00340 1.06 1.30 × 108 6.71 × 107 6.60 × 10−3

Guess 2 0.975 7.02 × 10−7 0.00232 1.01 1.30 × 108 6.71 × 107

Result 2 0.980 7.58 × 10−7 0.00310 1.01 8.42 × 107 5.68 × 107 7.43 × 10−3

June 23, 2004 9:26 WSPC/130-JCA 00221


Table 15. Percentage errors in determining five Biot parameters. Three refinementsof the initial finite element mesh were used for the simulated data, two for the trialdata. The values for permeability were the first set in Table 2.

Frequency β k a ReKb Reµ

6000

Target 0.72 5.00 × 10−9 0.000471 3.18 × 109 1.30 × 109

Result 1 2.1% 6.2% 23.8% 0.3% 13.8%Result 2 1.9% 2.0% 19.5% 6.6% 6.2%

6500

Target 0.75 7.00 × 10−9 0.000800 2.67 × 109 1.10 × 109

Result 1 2.3% 17.4% 8.8% 3.4% 4.5%Result 2 2.3% 44.3% 67.5% 3.4% 4.5%

6500

Target 0.83 3.00 × 10−8 0.00135 1.53 × 109 6.27 × 108

Result 1 2.2% 56.3% 58.5% 22.9% 21.7%Result 2 2.8% 42.3% 17.0% 31.4% 30.5%

Average ErrorResult 1 2.2% 26.7% 30.3% 8.9% 13.4%Result 2 2.3% 29.5% 34.7% 13.8% 13.7%

Table 16. Percentage errors in determining five Biot parameters. Three refinementsof the initial finite element mesh were used for the simulated data, two for the trialdata. The values for permeability were the second set in Table 2.


6500

Target 0.72 7.99 × 10−9 0.000471 3.18 × 109 1.30 × 109

Result 1 0.1% 11.1% 47.6% 13.5% 23.1%Result 2 0.1% 11.1% 47.8% 13.8% 23.1%

6500

Target 0.75 2.40 × 10−8 0.000800 2.67 × 109 1.10 × 109

Result 1 2.5% 47.9% 60.0% 43.4% 32.3%Result 2 2.5% 10.0% 2.4% 46.4% 24.6%

6500

Target 0.83 7.56 × 10−8 0.00135 1.53 × 109 6.27 × 108

Result 1 1.9% 131.5% 152.6% 39.2% 38.1%Result 2 1.4% 16.9% 32.4% 48.0% 37.2%


June 23, 2004 9:26 WSPC/130-JCA 00221


Table 17. Percentage errors in determining five Biot parameters at 8 kHz. Threerefinements of the initial finite element mesh were used for the simulated data, twofor the trial data. The values for permeability were the first set in Table 2.

Frequency β k a Re Kb Re µ

8000

Target 0.72 5.00 × 10−9 0.000471 3.18 × 109 1.30 × 109

Result 1 5.3% 1.0% 36.7% 6.0% 29.2%Result 2 4.9% 3.4% 22.2% 29.2% 1.5%

8000

Target 0.75 7.00 × 10−9 0.000800 2.67 × 109 1.10 × 109

Result 1 4.5% 9.4% 21.3% 8.6% 14.2%Result 2 6.3% 52.9% 60.0% 13.9% 64.5%

8000

Target 0.83 3.00 × 10−8 0.00135 1.53 × 109 6.27 × 108

Result 1 15.7% 686.7% 43.3% 72.1% 70.3%Result 2 2.8% 763.3% 1011.1% 8.5% 8.5%


Table 18. Percentage errors in determining five Biot parameters at 8 kHz. Threerefinements of the initial finite element mesh were used for the simulated data, twofor the trial data. The values for permeability were the second set in Table 2.


8000

Target 0.72 7.99 × 10−9 0.000471 3.18 × 109 1.30 × 109

Result 1 3.8% 2.4% 11.3% 8.2% 13.8%Result 2 3.3% 0.8% 45.6% 18.6% 16.9%

8000

Target 0.75 2.40 × 10−8 0.000800 2.67 × 109 1.10 × 109

Result 1 1.7% 6.3% 31.3% 10.5% 16.4%Result 2 2.1% 12.9% 19.9% 7.5% 9.1%

8000

Target 0.83 7.56 × 10−8 0.00135 1.53 × 109 6.27 × 108

Result 1 3.9% 157.9% 165.2% 0.7% 44.0%Result 2 3.4% 2.0% 14.1% 2.0% 35.9%


June 23, 2004 9:26 WSPC/130-JCA 00221


Table 19. Percentage errors in determining five Biot parameters at 9 kHz. Threerefinements of the initial finite element mesh were used for the simulated data, twofor the trial data. The values for permeability were the first set in Table 2.


9000

Target 0.72 5.00 × 10−9 0.000471 3.18 × 109 1.30 × 109

Result 1 3.2% 20.6% 49.3% 8.2% 11.5%Result 2 4.7% 20.2% 32.1% 7.2% 8.5%

9000

Target 0.75 7.00 × 10−9 0.000800 2.67 × 109 1.10 × 109

Result 1 1.3% 4.7% 15.9% 41.6% 40.9%Result 2 2.4% 16.3% 15.8% 41.6% 26.4%

9000

Target 0.83 3.00 × 10−8 0.00135 1.53 × 109 6.27 × 108

Result 1 2.8% 73.0% 73.3% 19.6% 16.3%Result 2 3.1% 57.7% 34.1% 32.0% 15.6%


Table 20. Percentage errors in determining five Biot parameters at 9 kHz. Threerefinements of the initial finite element mesh were used for the simulated data, twofor the trial data. The values for permeability were the second set in Table 2.


9000

Target 0.72 7.99 × 10−9 0.000471 3.18 × 109 1.30 × 109

Result 1 4.0% 26.4% 14.9% 22.0% 13.1%Result 2 2.2% 8.0% 6.6% 23.9% 20.8%

9000

Target 0.75 2.40 × 10−8 0.000800 2.67 × 109 1.10 × 109

Result 1 6.4% 19.6% 19.0% 3.0% 71.5%Result 2 3.9% 1.7% 8.5% 11.6% 69.6%

9000

Target 0.83 7.56 × 10−8 0.00135 1.53 × 109 6.27 × 108

Result 1 9.4% 209.5% 38.5% 39.7% 40.0%Result 2 3.7% 31.0% 8.9% 13.1% 41.5%


June 23, 2004 9:26 WSPC/130-JCA 00221


Table 21. Percentage errors resulting from two applications of the simplex method forspecimens of porosity 0.95. Three refinements of the initial finite elements mesh wereused for the target data, two for the trial data.

Frequency β k a α ReKb Reµ

Target 0.95 5.00 × 10−7 0.0022 1.01 2.57 × 108 1.05 × 108

6500Result 1 1.9% 31.2% 11.4% 4.0% 165.8% 172.4%Result 2 1.8% 2.0% 65.0% 5.0% 94.2% 99.0%

8000Result 1 0.1% 1.6% 21.4% 2.0% 38.5% 41.9%Result 2 2.6% 306.0% 135.9% 1.0% 37.7% 36.0%

9000Result 1 2.4% 46.0% 9.1% 3.0% 49.4% 80.0%Result 2 1.3% 25.2% 0.9% 1.0% 23.0% 48.6%

Target 0.95 2.30 × 10−7 0.0022 1.01 2.57 × 108 1.05 × 108

6500Result 1 2.8% 187.8% 23.6% 1.0% 67.3% 66.4%Result 2 2.8% 254.8% 61.4% 1.0% 71.0% 70.1%

8000Result 1 3.1% 124.8% 8.2% 1.0% 100% 27.8%Result 2 3.1% 175.2% 17.3% 0.0% 100% 27.8%

9000Result 1 2.6% 222.2% 54.5% 5.0% 49.4% 36.1%Result 2 3.2% 229.6% 40.9% 0.0% 67.2% 45.9%

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00221 - University of Delawaregilbert/papers/comput_acoust_june-04.pdf · 2004. 7. 18. · Title: 00221.dvi Created Date: 6/30/2004 11:24:25 AM