Recent Developments of DSPI for Bone Strain Measurement
Dr. Lianxiang Yang1 and Dr. Hiroki Yokota2, 3
1
Department of Mechanical Engineering, Oakland University, Rochester, MI 48309, USA,
Email: [email protected]
2
Department of Anatomy and Cell Biology, and 3Department of Biomedical Engineering,
Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202, USA
Email: [email protected]
ABSTRACT
It has been proposed that mechanical strain is a principal determinant of bone formation and a minimum effective strain
threshold may be surpassed to activate new bone formation. It is therefore critical to evaluate local and global strain
distributions at high resolution. Strain gauges, frequently used in bone mechanics, do not provide sufficient resolution or 3D
components in strain distributions. DSPI (Digital Speckle Pattern Interferometry), originally called ESPI (Electronic Speckle
Pattern Interferometry), can achieve whole-field, non-contacting measurement of microscopic three deformation components
and, thus, for high-resolution determination of strain distributions. It does have been applied for bone strain measurement. This
paper presents recent developments of DSPI for bone strain measurement, which include measurement of 3D-deformations
based on determining absolute deformation, measurement of 3D-deformations and strain on a curve surface (bone has usually
a curve rather than flat surface) and so on. The theory and methodology of DSPI for deformation and strain measurement are
described. In particular, the potentials and limitations for bone strain measurement are discussed. The applications are
demonstrated by examples of strain measurements for different materials and bones.
KEYWORDS: Digital Speckle Pattern Interferometry (DSPI), Electronic Speckle Pattern Interferometry (ESPI), deformation
measurement, strain measurement, bone.
1. INTRODUCTION
Mechanical loading is a potent osteogenic stimulator, and animal and clinical studies support that mechanical loading
strengthens bone, and prevent bone loss [1-3]. Despite those outcomes, the mechanism of load-induced bone formation
needs further investigation. In order to examine the role of in situ strain in bone formation, it is therefore critical to evaluate
local and global strain distributions at high resolution. Strain gauges, frequently used in bone mechanics, do not provide
sufficient resolution or 3D components in strain distributions. Therefore, a whole-field strain measurement method with high
measuring sensitivity is always desired for bone strain measurement.
Whole field strain measurement can be usually conducted by digital image correlation (DIC) or digital speckle pattern
interferometry (DSPI) techniques [4-6]. Though DIC is a robust technique for whole filed measurement of 3D-deformations,
DSPI is usually more desired for measuring a small deformation from submicron to a few microns, because its high sensitive.
So far, DSPI techniques have been used to measure bone strains without considering any significant contour change and
measurement sensitivity variation. However, the measurement error could be significant if the sample to be tested has a
curved surface, such as a bone surface, and its size is not very small in comparison with the distance from the sample to
camera [7].
DSPI is an acronym of digital speckle pattern interferometry, it was originally called electronic speckle pattern interferometry
(ESPI). DSPI is based on speckle pattern interferometry in the aspect of optics, but technically, DSPI is able to digitize phase
value at each pixel. Therefore, the phase distribution of the speckle interferogram, which is directly related to deformation, can
be determined quantitatively. After an explanation of measurement principle of DSPI, the paper presents recent developments
of DSPI for strain measurement, especially for bone strain measurement, which include measurement of 3D-deformations
based on determining absolute deformation, measurement of 3D-deformations and strain on a curve surface, and so on. The
theory and methodology of DSPI for deformation and strain measurement are described. In particular, the potentials and
limitations for bone strain measurement are discussed. The applications are demonstrated by examples of strain
measurements for different materials and bones.
2. MEASURING PRINCIPLE OF DSPI
Figure 1 shows the optical layout of DSPI. The laser beam is split into two beams: one as the object beam and the other as a
reference beam. A piezoelectric actuator (PZT) mirror set up in the path of reference beam is utilized for the phase shift
technique for quantitative determination of phase distribution. The rays reflected from the object surface recombine with the
reference beam after they pass through a beam combiner. The result is an interference pattern called speckle interferogram,
which is captured by a CCD camera.
non deformed
(Reference state)
φ
Illumination 1
Illumination 2
deformed
φ+∆
Fig. 1
The intensity of the speckle interferogram captured by the CCD camera is given by:
I(x,y) = IO(x,y) + IR(x,y) + 2
I O ( x , y) I R ( x , y) cos [θO(x,y) - θR(x,y)]
(1)
where: IO, IR and θO, θR are the intensities and phases of object and reference beams, respectively. For simplification, the
intensity equation is usually rewritten by:
I(x,y) = a(x,y) + b(x,y) cos φ(x,y)
where: a(x,y) =IO(x,y)+IR(x,y) is the background intensity, b(x,y) = 2
(2)
I O ( x , y) I R ( x , y) is the modulation amplitude of the
speckle interferogram and φ(x,y) = θO(x,y) - θR(x,y) is the phase difference between the object beam and reference beam as
shown in Fig.1. After the object is deformed, the phase difference φ(x,y) becomes φ’(x,y) which can be expressed by [φ(x,y) +
∆(x, y)], Now the light intensity becomes:
I’(x, y) = a(x, y) + b(x, y) cos [φ(x, y) + ∆(x, y)]
(3)
∆(x, y) is a change of the phase difference caused by the deformation. Therefore, a relationship between the phase change
∆(x, y) and deformation exists and is given by [8]:
∆(x, y) = S(x,y) d(x,y) = S(x,y) [ i u(x,y) + j v(x,y) + k w(x,y)]
(4)
where d is the deformation vector, u(x,y), v(x,y) and w(x,y) are the components of the deformation vector and i, j, and k are
the unity vectors in x, y and z directions, respectively. S(x,y) is measuring sensitivity vector which is indicated as measuring
direction in Figure 1. The sensitivity vector S(x,y) lies along the bisector of the angle α created by the illumination direction, i.e.
from the laser to the point investigated, and the viewing direction, i.e. from the investigated point to the camera, and the angle
α is usually called the illumination angle (cf. Fig. 1). The magnitude of the sensitivity vector is equal to (4π/λ) cos (α/2), where
λ is the laser wavelength.
Equation 4 is the fundamental equation of speckle pattern interferometry for deformation measurement. What DSPI can
measure is relative values of phase change ∆(x, y) (related to a reference point) at each pixels by applying a phase shift
method [9]. The deformation components u(x,y), v(x,y), and w(x,y) are what one wants to measure.
3. MEASUREMENT OF THREE COMPONENTS OF DEFORMATION
The fundamental equation of speckle pattern interferometry as presented in equation 4 builds a relationship among the phase
change ∆(x, y), the measuring sensitivity S(x,y), and the three deformation components u(x,y), v(x,y), and w(x,y). DSPI
enables an automatic and quantitative measurement of relative values of phase change ∆(x, y) at each pixel. The sensitivity
vector is not same at each point on the object surface due to a variation of illumination and observation directions, but it is
determinable according to the coordinates of the point investigated and the locations of laser and CCD-camera. The three
components u(x,y), v(x,y), and w(x,y) are what one wants to measure, and they are unknown. It is obvious that one can not
measure them from a single equation with three unknowns.
3.1. Measuring Three Deformation Components Based on a Constant Sensitivity Vector
Usually, the components in x and y directions; u(x,y) and v(x,y), are called in-plane deformations or displacements, and the
component in z direction; w(x,y), i.e. in the direction perpendicular to object surface, is called out-of-plane displacement. The
existing method to measure the three components; u(x,y), v(x,y), and w(x,y), is based on a constant sensitivity vector. If the
dimensions of the object to be investigated are much smaller than distances from the object to the camera and to the laser, the
sensitivity vector S(x,y) can be assumed to be approximately same. With this assumption, three components u, v, and w can
be measured by three individual setups, i.e. a setup with a xoz plane dual-beam illumination, a setup with a yoz plane dualbeam illumination, and a setup with a perpendicular illumination, respectively [10]:
U(x,y) =
λ
∆ u ( x , y)
4 π sin α xoz
(for a setup with a xoz-plane dual-beam illumination),
(5)
V(x,y) =
λ
∆ v ( x , y)
4 π sin α yoz
(for a setup with a yoz-plane dual-beam illumination),
(6)
W(x,y) =
λ
∆ w ( x , y)
4π
(for a setup with a perpendicular illumination),
(7)
where αxoz and αyoz are dual-beam illumination angles in xoz and yoz planes, respectively. They are built from laser through
middle point of sample to camera, λ is laser wavelength, and ∆u, ∆v, and ∆w, are the phase changes of setups for measuring u,
v and w, respectively. Because the laser wavelength λ is given and the illumination angle α can be determined from the optical
setup, the three components of a deformation u(x,y), v(x,y), and w(x,y) can be quantified as long as the phase changes
∆u(x,y), ∆v(x,y), and ∆w(x,y) are determined by the phase shift technique.
Equations (5) to (7) are so far the most popular equations for 3D-deformation measurement. These equations are suited well
for applications of deformation and strain measurement if the illumination angles from the laser through any point on object
surface and then to the camera are approximately same. However, a measurement error is no longer negligible if the
illumination angles are varied. The problem becomes worse if a micro-lens system which has a very small working distance
will be utilized. For such kind of application, a measurement method based on variable sensitivity vector should be adopted.
The following chapter will present new development for measuring three deformation components based on variable sensitivity
vector and absolute phase value.
3.2. Measuring Three Deformation Components Based on Variable Sensitivity Vectors and Absolute Phase Values
As mentioned above, the phase shift techniques can determine only a relative phase value related to a reference point rather
than an absolute phase value. If the reference point is P0 and its coordinates on object surface are (x0,y0), the absolute phase
value at the reference is unknown, say ∆0(x0,y0). Now we look at any point investigated, say P1 and its coordinates on object
surface are (x1, y1). The relative phase change (related to the reference point) ∆(x1,y1) can be determinable by applying the
phase shift technique. Now the absolute phase at the point P1 can be expressed by [∆0(x0,y0)+∆(x1,y1)]. For point P1, the
fundamental equation 4 becomes:
[∆0(x0,y0)+∆(x1,y1)] = S(x1,y1) [ i u(x1,y1) + j v(x1,y1) + k w(x1,y1) ]
(8)
In this equation, the sensitivity vector S(x1,y1) at the point (x1, y1) is determinable according to the coordinates of the point
investigated and the positions of laser and CCD-camera. Obviously, there are four unknowns in the equation, they are the
three components of the deformation; u(x1,y1), v(x1,y1), w(x1,y1), at the investigated point and the absolute phase value at the
reference point ∆0(x0,y0). Determination of absolute phase value in the speckle interferometry has been a challenge task for
the researchers of this area in the last two decades years. Some methods work with the change of single parameters of the
measurement setup, like wavelength of illumination or geometrical parameter, in order to control the sign and number of
fringes arising [11, 12]. These methods of course demand a precise control of the parameters and require the acquisition of
multiple images where the setup should not change.
Another approach based on the changes of the sensitivity vector [13]. The measurements are done using three speckle
interferometers with different directions of illumination which results in three measuring results, i.e. three equations. As these
three equations still consist of six unknowns; three are the three components of displacement and the other three are the
absolute phase values at the reference point of the three images, these equations cannot be solved without additional
information.
A practical and effective method to measure absolute phase value was reported by the literatures [14] and [15]. In this new
method, measurements using four speckle interferometers with different laser illuminations, i.e. with different sensitivity
vectors, were adopted. Each laser illumination direction has its own reference beam as shown in Fig. 1. Thus, the four
interferometers with four different sensitivity directions generate four interference patterns. At the investigated point P1(x1, y1)
from the four interference images, one can get four equations:
[∆0-S1 (x0,y0)+∆S1 (x1,y1)] = S1 (x1,y1) [ i u(x1,y1) + j v(x1,y1)
[∆0-S2 (x0,y0)+∆S2 (x1,y1)] = S2 (x1,y1) [ i u(x1,y1) + j v(x1,y1)
[∆0-S3 (x0,y0)+∆S3 (x1,y1)] = S3 (x1,y1) [ i u(x1,y1) + j v(x1,y1)
[∆0-S4 (x0,y0)+∆S4 (x1,y1)] = S4 (x1,y1) [ i u(x1,y1) + j v(x1,y1)
+ k w(x1,y1) ]
+ k w(x1,y1) ]
+ k w(x1,y1) ]
+ k w(x1,y1) ]
(9)
where ∆0-S i (x0,y0), i = 1, 2, 3, 4, represent the absolute phase values at the reference point P0(x0,y0) for the i th image.
∆S i (x1,y1) stand for the relative phase values (related to the reference point) at the investigated point P1(x1,y1) of the i th
image, which can be determined by existing phase shift method. Si(x1,y1) are the sensitivity vectors at the point P1(x1, y1) for
ith images. Both the directions and magnitudes of the four sensitivity vectors can be determined according to the coordinates
of the investigated point P1(x1, y1), the four laser positions, and the CCD-camera. Thus, in the above four equations, there are
totally seven unknowns, they are the four absolute phase values at the reference point in each image; and the three
displacement components u(x1,y1), v(x1,y1), and w(x1,y1) at the point P1(x1,y1). Obviously, seven unknowns can not be solved
from four equations, because there are three more numbers of unknowns than equation numbers.
However, if another point of the four images is investigated, say P2(x2,y2), the absolute phase values ∆0-S i (x0,y0) at the
reference point are the same as in point P1(x1, y1). Only three displacement components at P2(x2,y2) are additional unknowns.
So one can obtain another four equations, but only with additional three unknowns:
[∆0-S1 (x0,y0)+∆S1 (x2,y2)] = S1 (x2,y2) [ i u(x2,y2) + j v(x2,y2)
[∆0-S2 (x0,y0)+∆S2 (x2,y2)] = S2 (x2,y2) [ i u(x2,y2) + j v(x2,y2)
[∆0-S3 (x0,y0)+∆S3 (x2,y2)] = S3 (x2,y2) [ i u(x2,y2) + j v(x2,y2)
[∆0-S4 (x0,y0)+∆S4 (x2,y2)] = S4 (x2,y2) [ i u(x2,y2) + j v(x2,y2)
+ k w(x2,y2) ]
+ k w(x2,y2) ]
+ k w(x2,y2) ]
+ k w(x2,y2) ]
(10)
The combination of equations 9 and 10 generates 8 equations with ten unknowns (7+3), it is still impossible to determine the
absolute fringe orders, but now only two more number of unknowns than equation numbers.
If additional two points of the four images will be investigated, say P3(x3,y3) and P4(x4,y4), the other 8 equations with 6
unknowns will be obtained. And totally 16 equations with 16 unknowns are obtained, they are the four absolute phase value at
the reference point of the four images and the twelve displacement components, i.e. [u(x1,y1), v(x1,y1), w(x1,y1)] at the point
P1(x1,y1), [u(x2,y2), v(x2,y2), w(x2,y2)] at the point P2(x2,y2), [u(x3,y3), v(x3,y3), w(x3,y3)] at the point P3(x3,y3), and [u(x4,y4),
v(x4,y4), w(x4,y4)] at the point P4(x4,y4). Because the unknowns’ number is same as the equation number, the absolute phase
value can now be determined. After the absolute phase values of the four images: ∆0-S1 (x0,y0), ∆0-S2 (x0,y0), ∆0-S3 (x0,y0), and
∆0-S4 (x0,y0), are determined, three deformation components at any point in the measuring area can be determined using any
three of four equations.
In conclusion, measurement of 3D-defdormations based on variable sensitivity vectors and absolute phase values can be
conducted by using 4 illumination directions. Usually, two illumination directions are located in-xoz plane and the other two inyoz plane. The four illuminations are even distributed from right and left, and from top and bottom. A laser beam is usually split
into 5 directions; 4 for illuminations and one for a reference beam. Each illumination interferences with the reference beam, so
that 4 speckle pattern interferometers can be formed. Fig. 2 shows a schematic setup of the 3D-DSPI measurement system. In
order to display the optical layout clearly, the figure shows only two illumination directions located in yoz-plane. Unlike the wellknown dual-beam illuminations which interference with each other, each illumination beam interferences with the reference
beam, thus, two speckle pattern interferometers are formed. Note that the labels S1, S2 and S5 indicate three path-controlling
switches. An interferometer is produced by opening the switches S1 and S5, and the other is generated by opening S2 and
S5. Similarly, there are other two illumination directions and the corresponding switches are S3 and S4, which are not
indicated in the figure. The speckle interferometers in the third and fourth directions are generated by opening switches S3
and S5 as well as S4 and S5, respectively. While measuring, the phase distributions of four speckle interferograms before
loading are successively recorded and calculated by controlling the switches. After the sample is loaded, the phase
distributions of other four speckle interferograms are calculated in turn again. Subtraction of the corresponding phase
distributions before and after loading generates four phase maps. Therefore, the equation group (9) or (10) for any point
studied can be obtained. Figure 3 presents a measurement of 3D-displacements and strains of a flat aluminum sample with
one groove and three holes and the components of deformation are calculated based on variable sensitivity vectors and
absolute phase values. The loading is located at the top of right by driving in a screw (cf. the live image). Because the 3Ddisplacements are calculated from four phase maps, the patterns of the evaluated 3D-diaplacements are completely different
from the distributions of the four phase maps. The new method explained above enables to measure an absolute phase value
and, thus, enable to determine an absolute deformation on the surface without knowing sample boundary condition and
without an assumption of constant measuring sensitivity.
M2
Illumination 1
Object
Reference beam
S1
Out-of-plane
α
S5
Camera
y
α
Laser
z
o
S2
In-plane
x
Illumination 2
M1
Figure 2. The schematic setup of 3D-DSPI measurement system.
Loading location
and direction
Live image of the sample investigated
Phase map of ∆1
Calculated u(x,y)
Phase map of ∆2
Phase map of ∆3
Calculated v(x,y)
Phase map of ∆4
Calculated w(x,y)
Strain εxx
Strain εyy
Shear Strain γxy
Figure 3. Measurement of 3D-deformations and strains of a flat aluminum sample with one groove and three holes, three
components of deformation are calculated based on variable sensitivity vectors and absolute phase values, the strain are
obtained by differentiating the deformation data.
4. MEASUREMENT OF CONTOUR
So far, DSPI has successfully been applied for measurement of 3D-deformations and strains. However, most of current
measuring systems do not evaluate any contour changes of the sample surface. Therefore, the local coordinate on the sample
surface, particularly, a complex surface of biological samples, such as a bone surface, can be significantly different from the
sensor-fixed coordinate. Namely, if deformations and strains on a bone surface are determined without considering alteration
of the sample contour, the result could be misleading because of deviations between the local coordinate and sensor-fixed
coordinate [16-19]. Some researchers do measure both contour and 3D-deformations, but use different measuring systems,
such as measuring 3D-deforemation with DSPI and measuring contour with 3D-computer vision or laser scanning, or
measuring both 3D-deformation and contour with DSPI, but with two sensors. To match both results from different sensors is,
however, difficult and time-consuming. Therefore, it is always desired that both contour and 3D-deormation will be measured
by DSPI technique and by a single sensor. Here, a measuring system has a capability to measure both 3D-deformations and
contour will be introduced.
Let’s see Fig. 2 again, while measuring contour, a dual-beam illumination method is needed. In this case, S3, S4 and S5 are
all closed and the S1 and S2 are opened at same time. This optical layout is identical to the speckle interferometer for an inplane deformation measurement. The difference is that the two exposures are recorded when tilting the illumination beams by
moving the mirrors M1 and M2 rather than loading the object as in deformation measurements. The mirrors M1 and M2 shown
at the top and bottom in Fig.1 are driven by the piezoelectric actuator. The amount for movement is from submicron to a few
microns depending on different setups. The phase change due to the movement of mirrors M1 and M2 is linked to a surface
contour of the sample. If two mirrors shown in Fig. 2 have a same movement amount δ in the same direction, the relationship
among the change of phase difference; ∆, the contour data; Z, and the movement amount δ is shown as follows [20-21]:
Z=
( 2π / λ )
∆
( δ / L ) sin 2α
(11)
where λ is the wavelength of laser used, L is a distance from one of two mirrors to the center point of the object, α is the
illuminating angle. Obviously, the contour data can be determined by measuring ∆ using the phase shift technique.
5. DETERMINATION OF STRAINS
Strain analysis supplies needed information for designing and dimensioning products as well as providing a scientific basis for
optimizing a component’s shape and for quality control and assurance.
Like other experimental techniques, DSPI measure also only surface information and the measured results are functions of
only x and y. That means only in-plane strains εxx = ∂u/∂x, εyy = ∂v/∂y, and γxy = ∂u/∂y + ∂v/∂x can be determined and the outof-plane strain related to z can not be measured. As long as the in-plane deformations u(x,y) and v(x,y) have been measured,
the in-plane strains can be determined by numerically differentiating the data of u(x,y) and v(x,y). This is the standard
technique for strain measurement by DSPI if a sample to be studied has a flat surface.
If a sample has an extensive curved surface, however, a coordinate system on the object surface varies with the contour.
Usually, the deformations on object surface are different from deformations measured in a sensor coordinate system,
however, the deformations measured in the sensor coordinate system can be transformed into the object coordinate system
by applying coordinate transformation equation [22]:
 u '   l1 , m1 , n1   u 
 
  
 v'  =  l 2 , m 2 , n 2   v 
 w'   l , m , n   w 
3
3   
  3
(12)
where u, v, and w are the measured deformations in the sensor coordinate system denoted as a xyz set of coordinates, u’, v’
and w’ are the deformations in the coordinate system on the object surface denoted as a x’y’z’ set of coordinates. Both xyz
and x’y’z’ are the Cartesian coordinates. li, mi and ni are the direction cosines values between x’, y’ and z’ axes and x, y and z
axes, which can be determined by the contour data measured. After the contour of the object and the deformations in the
sensor coordinate system have been measured, the three components of deformation u’(x,y), v’(x,y) and w’(x,y) on the curved
surface in the x’y’z’ coordinate system can be determined by Eq. 12. Now the in-plane strain on the curved surface can be
determined as follows:
ε x' x' =
∂ u'
∂ u' ∂ x
∂ u'
=
=
l1,
∂ x'
∂ x ∂ x'
∂x
∂ v'
∂ v'
m2 ,
ε y' y' =
=
∂ y'
∂y
(13)
∂ u' ∂ v'
∂ u'
∂ v'
m2 +
l1 .
γ x' y' =
+
=
∂ y' ∂ x'
∂y
∂x
The principal strains on object surface can also be determined by the equation:
 ε x' x' + ε y' y'
ε p1 p2 = 
2


 ±


 ε x' x' + ε y' y'


2

2

 + γ x' y' 2 .


(14)
6. APPLICATIONS FOR BONE STRAIN MEASUREMENT
Bone bears forces and moments with limited amount of deformation, while a joint provides a smooth, flexible linkage of
motions. In the presence of varying loads in a daily life, relatively soft bone such as the epiphysis underneath the joint may
deform larger than hard cortical bone in the diaphysis. In this application, we employed an axial loading modality (ALM). In
ALM loads are applied axially to cortical bone through the knee and the hip joints.
6.1 Bone samples
Mouse femurs were harvested from C57/BL6 female mice (a body weight, ~ 20 g) (Harlan Sprague-Dawley, Inc.) after
anesthetizing using isoflurane. The isolated femurs were carefully cleaned out of muscles and connective tissues, and stored
at 4 ˚C in minimum essential medium. All procedures, performed in this study, were in accordance with the Indiana University
Animal Care and Use Committee guidelines.
6.2 Mechanical loading and measurement setups
In ALM, an axial load was applied to the femur through the knee and hip joints (cf. Fig. 4). A custom-made piezoelectric
mechanical loader was employed to applied loads to a mouse femur ex vivo. The loader was driven by a voltage signal
generated with a piezoelectric driver. Fig. 5 shows the measurement setup including the loading device and the 3D-DSPI
sensor based on the new measurement principle described in the paper. A 3D-DSPI sensor, called Q-100, from DantecDynamic GmbH, former called Dantec-Ettemeyer GmbH, Germany, which has capability to measure both contour and
absolute deformations, has been adopted in the investigation.
Fixture
Femur
Connect to Load Sensor
Figure 4. Schematic layouts of the loading setups for an axial loading modality
Control system for loading
3D-DSPI sensor with functionality to
measure contour and deformation
(from Dantec-Ettemeyer GmbH)
Position to fix the bone
Fig. 5. Experimental setup utilized in the investigations
6.3 Measurement Results
The combination of shape and deformation provides all necessary data for an accurate and quantitative determination of true
strains on the mouse bone surface with an extensive curved surface. The in-plane strains on the bone surface (εx’x’ and εy’y’)
and the shear strain (γx’y’) components on the bone surface can be determined by Eq. 13, and the principle strains are
calculated by Eq. 14. The strains vary with direction in which the first derivatives are made. To avoid the inconsistency due to
the loading axis and measurement axis, the principal strains have been displaced, which are unique and independent of the
sensor orientation instead of strains along the co-ordinate axis.
Figure 6 shows the measurement results for a femur with an axial loading modality (an ALM loading). In the ALM, one end of
the femur movement is constrained, while the other end is loaded axially using the piezo-mechanical loader. The circular
region in the femur represents trabecular bone in the epiphysis while the rectangular region along the x-axis is cortical bone in
the diaphysis. Fig. 7 displayed the calculated principal strains 1 and 2. The loading for the measurements was about 1 N. In
these diagrams, the color codes represent the contour data in millimeter, the deformation in micrometer, the strain information
in millistrain, and while the black scale indicated on the square frame represents the dimensions in X and Y directions,
expressed in millimeter.
Fig. 6. The contour and deformation data measured for a femur with an ALM loading; from left to right and from top to bottom:
contour data; in-plane deformation in horizontal direction u; in-plane deformation in vertical direction v; and out-of-plane
deformation w.
Principal Strain 1
Principal Strain 2
Fig. 67. Principal strains for the ALM loading
6.4. Discussions
The principal strain distributions of the femur in response to an axial loading modality have been determined using a
developed 3D-DSPI system. In contrast to the traditional technique we have incorporated absolute deformation and the
contour changes in the strain calculations. With the axial loading modality, the results clearly demonstrated the two distinctive
strain regions in the femur in the image of principal strain 2 (because of a compression loading, the principal strain 2 is more
interested). In the distal epiphysis where the mechanical loads were directly applied, the strain reached a significant level of
approximately 1-1.4 milli strain. The cortical bone in the femoral midshaft away from the loading site, on the other hand,
presented a smaller level of strain (compressive strain). Because of the small size of mouse femurs and the complex
morphology of the epiphysis, these results were unable to be obtained by the conventional measurements with strain gauges.
Obviously, 3D-DSPI technique opens new possibilities for accurate, high sensitive and high spatial resolution measurement of
bone strain.
Limitations: Application of the current DSPI system for characterization of biomaterials is yet to be improved. First, one of
the requirements for measurements is specimen stabilization. Sample to be measured should be immobilized tightly to the
sample table but this immobilization procedure may alter original strain distributions. Second, coating a sample surface with a
white powder is necessary to increase surface reflectance. However, coating may alter the mechanical behavior of the
sample. Third, the current technique has difficulty in measuring stains of a wet sample, since the wet surface tends to
decorrelate speckle patterns. In spite of those limitations, potential applications of DSPI in characterization of biomaterials,
particularly, of bones, are innumerable. With further developments and improvements, it will be a powerful tool for
characterizing biomaterials and understanding the role of strains in biomedical processes.
7. CONCLUTIONS
The recent developments of 3D-DSPI for strain measurement have been presented. A method for measuring absolute phase
distribution, and thus for determining absolute deformation components; u, v and w, has been explained in detail. An optical
layout for measuring both deformation and contour data by using a single DSPI system is displayed. Applications of the new
developments for strain measurement have been demonstrated by example for a metal sample and a mouse bone. A strain
distribution on a mouse femur in response to the ALM loading, applied to the distal epiphysis, was quantified using the new
3D-DSPI technique and the piezoelectric mechanical loader. The results revealed that the reproducible pattern of high-level
strains appeared at the loading site with ALM. Unlike the traditional ESPI method, the strain data has been calculated taking
into account the contour changes of femur. Therefore, the measured results represent the true strain on the mouse femoral
surface rather that the stain on the sensor co-ordinate system. The potentials and limitations of current 3D-DSPI systems for
bone strain measurement are discussed. The range for the strain measurement using 3D-DSPI is usually from a few dozen to
-6
a few hundred micro strain (10 ) in a one step measurement. However a step-by-step measurement method, the
-3
measurement range for strain can be extended to the milli-strain (10 ) or bigger level.
ACKNOWLEDGEMENTS
This study was in part supported by NIH R01AR52144.
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