be-tween different bone properties and ultrasound parameters. More-over, simulations can also help to interpret experimental findings between tissue structure, composition and acoustic properties. A nu-merical simulation is essentially a nunu-merical approach for solving a mathematical problem. Therefore, it is essential to formulate a math-ematical description of wave propagation in bone. For a body with a small volume and surface area, Newton’s second law dictates that the sum of the surface stresses and force (e.g., gravitational, electri-cal, magnetic) acting on the body is equal to the acceleration field experienced by the mass of the body. When the total stress on the surface of the body is considered to be equal to the net force per unit volume of the body, Newton’s law for all continua undergoing small deformations can be expressed as:
∂σij
∂xj +Fi = ρ∂2wi
∂t2 i,j=1, 2, 3. (3.2) In the expressionσij are components of the stress tensor [N/m2], xj
is a component of the displacement vector, Fi represents the compo-nents of the body force (N.B.this is equal to zero for freely vibrating media), ρ is the mass density [kg/m3] of the body, ∂ denotes the partial differential operator, t is time [s] and wi is a component of the body’s displacement field.
In acoustics, the linear ‘spring constants’ pertaining to elastic vibrations with small amplitude can be used to relate the stress and strain present in a particular material by Hooke’s law. In prac-tice, six independent stresses are related to six independent strains through 21 independent constants in a homogeneous (i.e., constants do not depend on space coordinates) anisotropic medium. As a first approximation, the calcified matrix may be considered as a homoge-neous isotropic material. This means that for a given calcified matrix, wave propagation in cortical bone will depend on its microstructure.
This assumption can be useful for determining important trends observed in experimental measurements for wavelengths of the or-der of a millimeter [35, 36]. For an isotropic material, Hooke’s law may be expressed in terms of the first and second Lamé constants [N/m2] λ and µ, respectively. Replacing the stresses in Newton’s law of motion with strains according to Hooke’s law rewrites the equation in terms of displacements
ρ∂2w
∂t2 = µ∇2w+ (λ+µ)∇(∇ ·w), (3.3) where ∇ is the gradient operator, ∇ · is the divergence operator and vectorw(x,y,z,t)is the body’s three-dimensional displacement field.
When the wavelength is smaller than the dimensions of the spec-imen in which it propagates, the wave will not “see” the boundaries of the specimen. Therefore, the wave (i.e., bulk wave) may be de-scribed as propagating in an infinite or unbounded medium. Then the SOS in both longitudinal and transverse directions is estimated as listed in Table 3.1.
The finite difference time domain (FDTD) method can be ap-plied to resolve wave propagation problems in continuous media,
Ultrasound assessment of bone
like bone [93, 124, 153, 163]. In the FDTD method, a material is rep-resented by isotropic grid elements consisting of nodes that are dis-placed, with respect to time, relative to their neighboring nodes. The value at these nodes at a certain time is approximated by replacing the derivatives in eq. 3.3 by finite differences. Three finite difference forms are generally considered in the FDTD method. For a function
f(x)with a continuous derivative, the forward difference is:
d f
dx(x0) = f(x0+h)− f(x0)
h , (3.4)
the backward difference is d f
dx(x0) = f(x0)− f(x0−h)
h , (3.5)
and the central difference is d f
dx(x0) = f(x0+h)− f(x0−h)
2h , (3.6)
wherex0 is the node andhis the spatial increment. Equations eq. 3.4 and eq. 3.5 are first-order approximation of the first derivative, while eq. 3.6 is a second-order approximation of the second derivative. As time increments become smaller, the spatial increments will also decrease until the difference between the partial differential and the difference approximation vanishes.
4 Aims of the present study
It has been speculated that changes in porosity would influence speed of sound (SOS) in cortical bone and thus influence the assess-ment of cortical thickness if this is to be determined by a time of flight method. The main aim of this thesis was to improve the reliabil-ity of pulse-echo thickness measurements of cortical bone. This was approached by investigating bone properties that affect radial SOS, by determining the relationship between radial SOS and porosity, and then by applying the SOS-porosity relationship as a correction factor in thickness measurements.
The specific aims of this thesis were to:
• Investigate how concurrent changes in the elastic properties and microstructure affect radial SOS in cortical bone.
• Characterize the relationship between radial SOS and cortical bone porosity and describe its effect on the PE assessment of cortical thickness.
• Estimate cortical porosity from ultrasound backscatter informa-tion and apply the porosity to determine SOS which can be applied in the assessment of cortical thickness.
5 Materials and methods
This thesis comprises three independent studies (I-III). The mate-rials and methods used in these studies are summarized in Table 5.1.
Table 5.1:Summary of the materials and methods used in studiesI - III.
Study Samples (Human) n Method Parameters
I Embedded femoral diaphysis 6 24 nominal and Radial SOS cylindrical VOIs 24 24 sample specific 3D microstructure from micro-CT images FDTD simulations
II Femoral diaphysis 18 in vitroPE and Radial SOS, Porosity (blocks) 44 2D FDTD simulations
III Femoral diaphysis 17 in vitroPE and IRC, AIB, Porosity
(blocks) 43 PLS analysis Thickness
VOI= volume of interest,FDTD= finite difference time domain,PE= pulse echo measurements,SOS= speed of sound, 3D= three-dimensional, 2D= two-dimensional,IRC= integrated reflection coefficient,AIB= apparent integrated backscatterPLS= partial least squares.
5.1 SAMPLE PREPARATION
In studyI, six ring shaped cortical bone cross sections were collected from the femoral diaphysis of cadavers (male: 17–82 yr). From each cross section, four cylindrical volumes of interest (VOIs) were seg-mented (Fig. 5.1). The cylindrical VOIs, oriented perpendicular to the long axis of the femoral shaft (periosteum to endosteum), were segmented from four quadrants [37] (anterior, posterior, medial, lat-eral) of the cross sections using a custom MATLAB-based program (v7.12, The MathWorks, Natick, MA).
In all studies, human femurs from cadaveric donors, with no pre-existing conditions that might have affected bone metabolism, were
Figure 5.1:Sample preparation. In studyI, volumes of interest (VOIs) were seg-mented from four quadrants (figure shows only anterior) in a micro-CT image of the diaphysis. In studies IIandIIIblock cut out and shaped from the anterior section of the diaphysis because this is the side assessed during in vivo measure-ments.
acquired from Kuopio University Hospital. Cortical bone samples from the femoral shafts were harvested. The same samples were used in studies II andIII. However, one sample used in study II was too difficult to be measured reliably in studyIIIand thus was omitted. In studiesIIandIII, the femurs (3 female, 39-58 yr in both and 15 male, 26-78 yr in study II and 14 male, 26-74 yr in study III) were shaped into blocks (4 ×2× 5 mm3) with parallel sides (Fig. 5.1) using a low-speed saw (Isomet Buehler Ltd., Lake Bluff, IL, USA). To ensure a consistent surface roughness for ultrasound measurements, parallel surfaces of the samples were ground and polished with a plane grinder (EXAKT 400CS, Exakt Apparatebau, Norderstedt, Germany) using polish grits of successively decreasing grain sizes (ISO/FEPA grit: P1200 and P4000, Struers A/S, Ballerup, Denmark). The distance between parallel periosteal and endosteal walls of each processed sample was measured with a digital
mi-crometer (Mitutoyo No 293-561-30, Mitutoyo Corporation, Kawasaki, Japan). During the measurements, the samples were kept in
phos-Materials and methods
phate buffered saline (PBS) solution. The permissions to collect the samples required for all the studies were granted by the National Au-thority for Medicolegal Affairs (TEO), permission 5783/04/044/07.