J. Biomedical Science and Engineering, 2011, 4, 362-374
doi:10.4236/jbise.2011.45046 Published Online May 2011 (http://www.SciRP.org/journal/jbise/
Published Online May 2011 in SciRes. http://www.scirp.org/journal/JBiSE
A doublet mechanics model for the ultrasound
characterization of malignant tissues
Francesco Gentile1,2, Jason Sakamoto3, Raffaella Righetti4, Paolo Decuzzi1,3, Mauro Ferrari3,5
1Center of Bio-Nanotechnology and Engineering for Medicine, Magna Graecia University, Catanzaro, Italy;
2Italian Institute of Technology, Genova, Italy;
3Department of Nanomedicine and Biomedical Engineering, The Methodist Hospital Research Institute, Houston, USA;
4Department of Electrical and Computer Engineering, Texas A&M, 214 Zachry Engineering Center, College Station, USA;
5Anderson Cancer Center and Rice University, Houston, USA.
Email: francesco.gentile@iit.it
Received 30 October 2010; revised 4 November 2011; accepted 4 January 2011.
Non invasive ultrasound-based imaging systems are
being more commonly used in clinical bio-microscopy
applications for both ex vivo and in vivo analysis of
tissue pathological and physiological states. These
modalities usually employ high-frequency ultrasound
systems to overcome spatial resolution limits of con-
ventional clinical diagnostic approaches. Biological
tissues are non continuous, non homogeneous and
exhibit a multiscale organization from the sub-cellu-
lar level (1 mm) to the organ level (1 cm). When
the ultrasonic wavelength used to probe the tissues
becomes comparable with the tissue’s microstructure
scale, the propagation and reflection of ultrasound
waves cannot be fully interpreted employing classical
models developed within the continuum assumption.
In this study, we present a multiscale model for ana-
lyzing the mechanical response of a non-continuum
double-layer system exposed to an ultrasound source.
The model is developed within the framework of the
Doublet Mechanics theory and can be applied to the
non-invasive analysis of complex biological tissues.
Keywords: Nanomechanics; Doublet Mechanics;
Ultrasound; Spectroscopy; Biopsy; Microscopic
Elastography; Ultrasound Biomicroscopy
In the last few years, there has been an increased interest
to extend the limits of conventional clinical approaches
to the level of microscopic resolution [1-5]. The goal
here is to optimize the imaging of small tissue structures
and, in general, to obtain information not available from
the corresponding conventional macroscale applications.
The ability to quantitatively and non invasively differen-
tiate living tissues based upon their biological and
physical properties would enable major breakthroughs in
the early detection and diagnosis of diseases and in
monitoring therapeutic effects.
While surgical biopsy remains the ‘gold standard’ for
the clinical screening of tissues and the assessment of
pathologic conditions, there is a concerted effort to de-
velop new imaging modalities that non-invasively visu-
alize tissues providing information previously only
available from biopsy [6-8]. Morphologic presentation
of tissues together with the microbiologic, immunologic
and molecular analysis are still critical for determining
personalized medical treatments. However, the analysis
of surgical biopsies suffers from being inherently opera-
tor dependent and, ultimately, the quality of diagnoses is
entrusted with the pathologist’s experience and knowl-
edge [9-11]. Availability of quantitative imaging tech-
niques capable of probing tissues at the microscale level
may improve the accuracy of the diagnosis of biological
samples and, most importantly, provide a new non-in-
vasive means for the assessment of living tissues in situ.
Recent developments in the fields of optics, nuclear
medicine, computed tomography, magnetic resonance
and ultrasound have suggested the feasibility of obtain-
ing tissue information at the micrometer-scale level with
high accuracy, sensitivity and contrast-to-noise ratios.
For example, optical in vivo biopsy is a growing area in
optical computed tomography applications, which
promises the assessment of tissue morphology and cell
function as well as the detection of early-stage tissue
abnormalities associated with diseases [7,12]. Similarly,
new magnetic resonance microscopy techniques now
permit imaging tissue’s fine architecture in applications
that range from assessing neural tissues [13] to imaging
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374 363
angiogenesis and gene expression in cancers [14,15].
Among the various imaging techniques, ultrasound
methods have always offered distinctive characteristics,
which make them particularly suitable for clinical
screening of tissues. These include: cost-effectiveness,
portability, non-invasiveness, and the ability to provide
in vivo tissue clinical information in real-time, at high
resolutions and relatively large depths.
Although the first acoustic microscope was proposed
in the early 1930s [16], it was only in the late 1980s that
the use of pulse-echo imaging systems operating at fre-
quencies higher than the frequency range conventionally
used in diagnostic imaging began to be experimented [17,
18]. Today, the ultrasonic visualization of tissues at mi-
croscopic resolution is usually referred to in the litera-
ture as ultrasound biomicroscopy or, more simply, high
frequency ultrasound [4,6]. Important clinical applica-
tions of high frequency ultrasound techniques include:
ophthalmology [19], dermatology [20], intravascular
ultrasound [18], cartilage imaging [21], and obstetrics
[22]. While the imaging performance of the ultrasound
system is ultimately determined by the frequency of the
ultrasonic transducer, its geometry and the tissue acous-
tic properties, the choice of the imaging system to be
used in a given application is highly dependent on the
nature of the application itself. For example, ophthalmic
applications usually employ transducers with frequent-
cies in the 40 to 60 MHz range [19]. These are used for
imaging glaucoma, scleral and corneal diseases, and
melanomas of the iris among the various clinical appli-
cations. The typical frequency range for investigating
skin and assess skin tumor markers is from 20 to 40
MHz [20], with possible extension up to 100 MHz [23].
For intravascular applications, ultrasound systems usu-
ally operate in the range of 20 to 30 MHz, allowing high
resolution imaging of vessel walls and coronary arteries
[24]. Finally, a new and promising application of high
frequency ultrasound systems relates to the development
of microscopic elasticity imaging and intravascular
elastography imaging techniques, which use pre- and
post-excitation high frequency ultrasound data to recon-
struct maps of the microscopic mechanical properties of
tissues [25-28].
A common denominator for all the aforementioned ul-
trasound applications is the use of high-frequency sys-
tems to probe the tissues so that high spatial resolutions
can be achieved. However, biological tissues are non
continuous, non homogeneous and exhibit a multiscale
organization from the sub-cellular level (1 mm) to the
organ level (1 cm). When the ultrasonic wavelength
becomes comparable with the tissue’s microstructure
scale, the propagation and reflection of ultrasound waves
cannot be fully interpreted employing classical models
developed within the continuum assumption. In these
regards, we have recently proposed the use of ultra-
sound-based Non Destructive Evaluation (NDE) tech-
niques in conjunction with multiscale mathematical
models as an integrated tool to automatically screen tis-
sue biopsy specimens with high accuracy and resolution
[29-32]. Biopsy samples are exposed to an ultrasound
source and the tissue response and physical properties
can be interpreted employing multiscale mathematical
models. Normal and malignant tissues are expected to
provide different responses that could be readily de-
Several techniques have been proposed to model the
mechanical behavior of materials at the nano/micro scale.
This is the case for instance of the Lattice Dynamics (LD)
and Molecular Dynamics [33], non-local Micromechan-
ics theories of the differential (CGT) [34] and integral
type (INT) [35-36]. In addition to these somehow clas-
sical approaches, the theory of Doublet Mechanics (DM)
has been developed over the last twenty years as a mul-
tiscale field theory that allows to bridge the gap between
Continuum Mechanics (CM) and discrete meso scale
models without contradiction [37-39].
In this paper, we develop a mathematical model based
on the DM approach to study the response of multilay-
ered non continuum solids to ultrasounds. While the
model is formulated for the analysis of biopsy samples,
it may also be applied to the detection of malignant tu-
mors developing in natural multi-layered systems such
as the skin or the eye and might become an important
tool for the further development of novel high frequency
ultrasound elastography techniques.
The theory of Doublet Mechanics is a multi-scale theory
which recapitulates Lattice Dynamics at the nanoscale
limit and is fully compatible with the continuum me-
chanics framework at the macroscale limit. In the pre-
sent paragraph, the governing equations for a linear elas-
ticity problem are briefly recapitulated. A more detailed
description can be found in [37-39].
Within the DM framework, a solid is considered as a
spatial array of points (nodes) at finite distances. Any
pair of adjacent nodes is termed a doublet comprising a
reference node X and a node Yα located at a separation
distance o
and aligned along the doublet axis with
unit vector o
(Figure 1). A doublet is univocally
identified by specifying its reference node (X),
orientation (o
) and separation distance (o
). The
superscript o means initial configuration. The node X is
surrounded by other m adjacent nodes, which form a
number m of doublets with corresponding Yα nodes.
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374
Figure 1. A doublet comprising a reference
node X and a node Y, at a separation distance
η aligned along the doublet axis indicated by
the unit vector τ.
Such a set of nodes constitutes a bundle for the reference
node X. If the nodes are arranged as to form one of the
fourteen Bravais lattices, m would coincide with the
coordination number of the lattice and the internodal
distances would coincide with the lattice constants.
Under externally applied loads, the nodes of a doublet
are displaced giving rise to microdisplacements and
microstrains. In general, if the node are separated along
, the doublet undergoes an elongation
, if the
nodes are rotated about o
or separated normally to
, the doublet undergoes a torsion μα or a shear γα,
respectively. In the sequel, it is assumed that each
doublet can only undergo elongations (central interac-
tions) which would be associated with the build up of a
microstress pα along o
An orthogonal Cartesian frame of reference is intro-
duced with unit vectors ei(i = 1, 2, 3), and each node X is
associated to a position vector x = x
iei, where the clas-
sical convention of the repeated Roman indices is used.
A displacement vector uα(x; t) can be introduced and for
each doublet; the increment displacement vector Δuα(x; t)
can be defined as the difference between the displace-
ment of the node γα and that of the node X at time t
aYX t
 uu uuxux
,t (1)
As in linear elasticity, it is assumed that the relative
displacement (;)
atux is small compared to the dou-
blet separation distance
ux so that
the initial and final configuration of the system can be
as- sumed to coincide. Expanding a in a convergent
Taylor series in a neighborhood of the reference point X,
it follows, in scalar form
 
where each of the subscript 1,,kk
runs through 1 to
3. The order M at which the series is truncated defines
the degree of approximation employed by the DM theory.
For M = 1, the continuum theory of elasticity is recov-
The small elongation of the doublet
can be de-
fined as
and, using (2), Equation (3) can be rephrased as
 
 
which can be interpreted as the compatibility equation
within the linear DM theory. The relationship between
the doublet microstress pa and microstrain
is given
 
in the case of linear and homogeneous internodal central
interactions. Equation (5) can be interpreted as the con-
stitutive equation in the linear and homogeneous DM
theory, and Aαβ is the matrix of the homogeneous micro
modulii of the doublet. Finally, static equilibrium is im-
posed as
/2 1
! ...
1, 2, 3;
kk i
 (6)
where Fi are the scalar components of the volume force
F. The boundary conditions expressed in terms of
stresses take the form
/2 1
1, 2, 3;
kk ir
 
where nkr are the scalar components of the unit vector n
normal to the body surface, and the subscript r = 1, ,
M – 1 for M 2 and r = 1 for M = 1. Ti are the scalar
components of the vector force T. The term dr1 is the
Kronecker delta function. The Equations (4), (5) and (6)
together with (7) give a boundary value problem within
DM for a linear elastic body. Notice that Equations (6)
and (7) are general to the extent that they are written as a
function of the doublet unit vectors
The relationship between the micro- and the macro-
stresses has been derived in [37] as
 
1! ...
 
 
at a generic level of approximation M.
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374 365
A thin slice of biological tissue is considered (histology-
cal sample) embedded between two glass slides. An aux-
iliary continuous layer is placed upon the biological slice,
as in Figure 2 to simulate a multilayered complex tissue
system. An ultrasound transducer is used to probe this
sandwich-like structure, generating mechanical waves of
both shear and longitudinal type. The initial train of
waves (0) travels unperturbed in the θ0 direction (Figure
3, see also Appendix A). Upon the interaction of (0) with
the first glass layer four more waves are generated (as-
suming specular reflection): two are forward scattered
into the system (waves (3) and (4)) and two are back
scattered into the glass substrate (waves (1) and (2)).
Similarly, other waves are generated at the interface be-
tween the auxiliary layer and the tissue layer at the in-
terface between the tissue and the second glass layer.
Under the approximation of neglecting multiple reflec-
tions higher than the second order, the total number of
waves travelling within the system is thirteen for the
present configuration (Figure 3). Theoretically, a com-
parison between the interacting waves at each interface
would allow to deduce the reflection coefficients, given
by the ratio between the amplitudes of the incident and
reflecting waves.
In the sequel, the DM theory is used to model the het-
erogeneous biological tissue and the auxiliary continu-
ous layer, for which a scaleless approximation (M = 1:
Continuum Mechanics) would be sufficient. In particular,
the reflection coefficients are derived. Details of these
derivations are given in Appendix A. According to the
theory derived in this study, the reflection coefficient at
the first interface, R1, is a complex function of the inter-
nodal distance η, the doublets configuration embodied
by t’s, the elastic microconstants Aαβ. Thus, estimation of
the reflection coefficients can provide important infor-
mation regarding the tissue microstructure and me-
chanical properties. Since different tissues are expected
to exhibit different responses, spectral analysis of the
reflection coefficients may allow differentiation between
different tissue types as well as monitoring changes oc-
curring in the microstructural properties of a tissue due
to pathology. In the next section, we analyze two distinc-
tive cases where the application of the developed theory
is used for the ultrasonic characterization of a biological
tissue: a mono-layer model with a single tissue, and a
double-layer model comprising a biological slice and an
auxiliary continuum layer.
3.1. The Mono-Layer Model
In the limit that the thickness of the auxiliary layer is
going to zero (h1 0) a mono-layer, discrete model is
Figure 2. The sandwhich like structure comprising an interme-
diate continuum layer, a biological discrete tissue substrate and
two external glass dishes. The system is feasible to be tested
through ultrasounds.
Figure 3. Elastic waves propagating in the system.
obtained as in [29-32]. Figure 4 shows a plot of the re-
flection coefficient R1 as a function of the frequency f of
the ultrasound transducer. For comparison, a plot of the
reflection coefficient as obtained using the CM theory as
opposed to the DM theory is also shown. These plots are
obtained by imposing η = 0, λ =1.805 GPa, μ = 0.04875
GPa, and the DM solution, with η = 5 μm, and A11 = 2
GPa, A44 = 0.195 GPa as derived using λ and μ in equa-
tion (15). These parameters are set based on previous
studies retrievable in the literature [29-32,40]. These
results show that at low frequencies, smaller than about
10 MHz, the CM and DM solutions tend to overlap
without any significant difference; whereas at higher
frequencies, significant discrepancies emerge. This may
be explained by observing that, at low frequencies, the
probing waves have a finite length that is greater than
the characteristic length scale of the system η (tens of
microns against η = 5 μm). Consequently the micro
structure of the layer is averaged out within the ultra-
sonic wavelength, which ultimately limits the spatial
resolution, and the predictions of CM and DM coincide.
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374
Figure 4. The reflection coefficient R1 shown against the fre-
quency f and comparison between the CM solution.
Notice that η has the same order of magnitude of a cell
size. Conversely, as f increases, and the ultrasonic wave-
lengths become comparable to the scale length of the
small components of the material (cells), the difference
between the discrete and continuum approach becomes
more and more evident.
The relation between the angular frequency ω, the
wave number ki and the phase velocity ci in the frame-
work of DM is embodied by the quite complex equations
(18) (Appendix A). Interestingly, these relations reduce
to the non scale, classical dispersion ones (where simple
relations ki = ω/ci, ci = (E/ρ)1/2 hold true) via two differ-
ent assumptions: 1) the constituent granules of the bio-
logical substrate are material points whose sizes are in-
finitesimal η = 0 and all the waves may have arbitrary
but finite length, 2) the constituent particles of the bio-
logical substrate may have an arbitrary but finite size
(η > 0) and all the waves have an infinite length (k 0).
The diagrams of Figure 4 show the occurrence of a
number of minima and maxima. These can be explained
considering phenomena of interference occurring at each
interface between the reflecting and transmitting waves.
3.2. The Double Layer Model
Herein the results stemming out from the double layer
model are presented, where both h1 and h2 are different
from zero. For the analysis, it is assumed that the auxil-
iary layer is made up of a polymeric material commonly
used in biomedical applications (Ta b le 1 ). The biologi-
cal tissue has the properties listed in Table 2. These pa-
rameters are set based on previous studies retrievable in
literature [29-32,40]. The CM and DM frameworks are
used in describing the auxiliary and biological substrate,
respectively. In Figures from 5 to 9, the reflection spec-
tra of the system under study are shown for different
auxiliary layer materials, with varying Young’s modulus
E and for wave frequencies ranging from 4 to 12 MHz. It
can be concluded that as E increases, the number of
Table 1. The CM properties of the intermediate layer.
Material ρ
[103 kg/m3]
[GPa] ν h
PDMS 0.97
4.8 10–4 0.49 0.133
Polyurethane 1.10
5.3 10–3 0.48 0.133
Polyethylene 0.955 0.70 0.41 0.133
Polypropylene0.91 1.30 0.42 0.133
Polycarbonate 1.23 2.40 0.41 0.133
Table 2. The CM properties of the glass substrates, and the
DM parameters of the biological tissue.
Material ρ
[103 kg/m3]
[GPa] ν h
CROWN glass2.50 71.8 0.23 0.133
Material ρ
[103 kg/m3]
Bio tissue 0.95 2.20 0.36 0.1335.6
minima decreases within the frequency range considered,
as it does the difference in frequency between two suc-
cessive minima. Most importantly, at high values of E,
the minimum values of the reflection coefficient do not
vary significantly. Since increased tissue stiffness is of-
ten associated with changes in tissue pathology, spectral
analysis of reflection coefficients as interpreted within
the DM theory may contain important markers for the
assessment of a tissue state and its changes due to the
onset of diseases. These simulations also show that ma-
terials with a high compliances, such as PDMS (Figure
5), generate a very complex response, with several min-
ima. Such complexity would spoil the reflection signal,
inducing more noise and making more difficult the ac-
curate interpretation of the spectra. Stiffer intermediate
layers would be more convenient. Additional simulation
(Figure 10) shows the effects of varying the Young’s
modulus (ΔE = 10%) about the mean value for the
Polyethylene case. As E increases, the spectrum under-
goes a rigid translation towards higher frequencies (see
Table 3 for the complete list of values).
The effect of thickness h1 of the continuum, interme-
diate layer has been also investigated. Starting from a
thickness of 134 μm, a 10% variation has been imposed
(Polyethylene, Δh = 10%) and the reflection spectra
evaluated. Results are shown in Figure 11 and Table 4.
It is observed that as h1 increases, the spectra rigidly
move towards lower frequencies, with no change in the
number of minima. The thickness h1 also affects the
damping of the system, the thicker is the intermediate
layer and the stronger is the attenuation for the spectra.
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374 367
Figure 5. The reflection spectrum of the system assuming that
the intermediate layer is PDMS.
Figure 6. The reflection spectrum of the system assuming that
the intermediate layer is PU.
Figure 7. The reflection spectrum of the system assuming that
the intermediate layer is PE.
Figure 8. The reflection spectrum of the system assuming that
the intermediate layer is PP.
Figure 9. The reflection spectrum of the system assuming that
the intermediate layer is PC.
Figure 10. The effect of the variation of the Young’s modulus
upon the overall response of the system. E
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374
Copyright © 2011 SciRes.
Table 3. Characterist minima points for different plastic substrates.
f [MHz] R f [MHz] R f [MHz] R f [MHz] R f [MHz] R
4.595 0.333 4.534 0.334 5.348 0.224 6.301 0.342 6.602 10.418
4.946 0.282 5.269 0.186 8.031 0.487 9.930 0.322 0.181 0.187
5.298 0.311 6.005 0.192 10.681 0.347
5.650 0.391 6.739 0.363
6.002 0.411 7.457 0.684
6.354 0.409 8.109 0.921
6.705 0.446 8.456 0.949
7.055 0.510 9.091 0.738
7.405 0.677 9.806 0.431
7.753 0.835 10.538 0.235
8.101 0.940 11.274 0.211
8.501 0.946 12.009 0.347
8.847 0.850
9.193 0.710
9.543 0.578
9.894 0.421
10.246 0.327
10.597 0.252
10.949 0.200
11.300 0.240
11.652 0.315
12.003 0.365
Table 4. Characterist minima points for different plastic substrates.
Initial E +10% –10%
1 2 3 1 2 3 1 2 3
f [MHz] 5.348 8.031 10.681 5.514 8.214 10.919 5.160 7.883 10.315
R 0.224 0.487 0.347 0.252 0.445 0.415 0.203 0.503 0.221
Notice that the thickness h1 and the stiffness E have op-
posite effects.
In this study, we have presented a multiscale model for
analyzing the mechanical response of a non-continuum
system exposed to an ultrasound source. The proposed
model may be interpreted as a quantitative, non-invasive
ultrasound spectroscopy method, which is based on the
analysis of the reflection coefficients as a function of the
ultrasonic wavelength (or conversely the frequency) for
estimating changes in the tissue microstructure and me-
chanical properties. The model was developed within the
framework of the Doublet Mechanics theory, which led
to the determination of the reflection coefficients as a
function of the internodal distance, the doublets con-
figuration and the elastic microconstants. In its present
form, the model does not take into consideration scatter-
ing effects at the various interfaces or within the tissue
layer. Coupling this effect with the DM theory would
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374 369
Figure 11. The effect of the variation of the intermediate layer
thickness h1 upon the overall response of the system.
require large mathematical efforts, which are considered
outside the scope of the presented study and left for fu-
ture investigations. In Appendix B, we provide a detailed
analysis of the conditions under which this approxima-
tion is valid.
As a test platform, the DM model developed in this
study was applied to the ultrasonic characterization of
two tissue models: a mono-layer solid simulating a tissue
specimen for histological analysis and a double-layer
system simulating a more complex tissue. For the
mono-layer case, the results obtained using the DM the-
ory were statistically compared with the results obtained
from the same tissue model using the classical CM the-
ory. The results reported in this study demonstrate that at
frequencies as high as 10 MHz and above, the CM re-
sults significantly deviate from the DM results. This
would suggest that the use of CM-based approaches to
characterize the ultrasonic behavior of tissues exposed to
high frequency (>10 MHz) insonications may lead to
incorrect or incomplete interpretation of the ultrasonic
parameters of interest with respect to the tissue micro-
structural properties. In such experimental conditions, a
more complicated model should be considered. The
model described in this paper offers an attractive alterna-
tive, which is based on the spectral analysis of the re-
ceiving signals to differentiate between tissue types. In
addition, the analysis of the single layer tissue model
carried out in this paper may prove as a useful example
on how to practically apply the proposed DM model to
ultrasound-based histological applications.
The present work with the development of a double
layer mathematical model constitutes the first step to-
wards a more realistic representation of natural multi-
layered systems as the human skin and the growth and
spread of malignancies such as melanoma. Human skin
is a unique organ that permits life by regulating heat and
water loss from the body while preventing the ingress of
noxious chemicals or microorganisms [41]. Skin mem-
branes may be examined at various levels of complexity.
While the membrane is regarded sometimes as a simple
physical barrier, more complexity may be introduced by
considering skin as various layers in series, namely 1)
the innermost subcutaneous fat layer (hypodermis); 2)
the overlying dermis; 3) the viable epidermis; 4) the
outermost layer of the tissue (the stratum corneum). All
the cited layers posses, at different extents, a certain de-
gree of heterogeneity due to sebaceous glands, hair folli-
cles, fat lobules, blood vessels, nucleii and desmosomal
junctions [41]. With such a scenario, a simple single-
layer model would not be sufficient for interpreting the
system response. In addition, skin cancer is associated
with localized changes in the tissue microstructure and
morphological modification at the interfaces between
different skin layers making single-layer models poten-
tially inaccurate in predicting the onset and spread of
tumor masses.
As a demonstration of the practical applicability of
some of the concepts exposed in this paper, we have
previously developed an ex vivo apparatus that uses ul-
trasound technology and the Doublet Mechanics theory
to obtain information about tissue pathological states
[31,40]. For the purpose of illustration, a sketch of such
apparatus and set-up is provided in Figure 12. The ul-
trasonic wand delivers an acoustical wave of known
frequency into the tissue, and then detects the reflected
wave fractions with separate transducers. As described in
this study, a characteristic reflection spectrum can be
plotted depicting the reflection coefficient versus the
Figure 12. A sketch of an ex vivo apparatus based on ultra-
sound technology and the Doublet Mechanics theory for the
determination of the malignancy potential of a cutaneous or
mucousal growth such as melanoma.
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374
excitation frequency. The spectrum can be related to
tissue physical properties such as density, microelastic
constants, attenuation, and internodal distance via the
DM theory. Through comparative analysis of reflection
spectra from normal and diseased tissue, it may be pos-
sible to determine the malignancy potential of a cutane-
ous or mucousal growth such as melanoma. This would
allow physicians to screen patients on an annual or
semi-annual basis for the presence of dysplastic nevi
(pre-cancerous) or early stage cancerous skin lesions.
In addition to dermatology applications, the proposed
model may have significant potentials in several other
clinical applications that employ high frequency ultra-
sound systems for the assessment of tissue pathological
and physiological states. It may be particularly useful in
ophthalmology, where the anatomy of the eye resembles
a multilayered system and where the use of high fre-
quency ultrasound techniques has already shown sig-
nificant potentials for the diagnosis and treatment of eye
diseases [19]. Other possible applications may include
cartilage assessment, intravascular applications, and high
frequency elastography techniques, which ultimately
long to the quantitative estimation of the tissue me-
chanical behavior at the microscopic scale.
The Doublet Mechanics theory has been employed to
model the propagation of elastic waves within a biologi-
cal tissue. The present case is characterized by hetero-
geneous and not continuous materials that, consequently,
may be conveniently analyzed within the multiscale DM
It has been observed that an operating frequency of
the ultrasound generator increases, the difference be-
tween the DM solution and the classical elasticity solu-
tion becomes larger, as a consequence of the intimate
interaction among waves and microscopic components
of the biological tissue. It has been discussed that a DM
multilayer model could be effective in describing the
skin multi-layer structure. The design requirements for a
characterization-mode ultrasound skin cancer detection
system should provide primary care physicians with a
rapid, noninvasive, screening tool for malignant mela-
noma that would assign a quantitative malignancy po-
tential for specific cutaneous lesions.
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F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374
The propagation of the waves in the media can be un-
ambiguously identified specifying thirteen angles (Fig-
ure 3)
108 110 122
   
 (9)
where the identities in (9) stem from the second law of
optics of reflection. Given the mechanical properties of
the media and θ0, Snell’s law may be used to derive six
further independent equations having the form
where θi and θt are the incident and transmitted wave
directions, and the ci the wave phase velocities. Using
Equations (9) and (10) it is thus straightforward ex-
pressing any characteristic parameter θi in terms of the
sole incident angle θ0. The displacements u(n) can be
expressed as
 
exp ;
nn n
with (n) an index indicating the (n)th wave, An the am-
plitude of the waves, x is the position vector, d(n) the
particle motion unit vector, p is propagation unit vector
and kn = 2π/λ the wave number.
The most relevant coefficient in the system is the re-
flection coefficient R1 = A1/A0 which can be determined
imposing at each interface suitable boundary conditions
for either the displacements and the stresses
 
 
22 12
here, m is the number of perturbations insisting upon the
same boundary, u1, u2, σ22 and σ12 are the displacements
and the stresses, of tangential and normal type respect-
tively. Relations (12) generate a total of twelve equations
meaning that the problem of solving for the twelve re-
flection coefficients (four per interface) is completely
The reflection coefficients Ri depend upon the waves
amplitude An and thus, through Equation (11), also upon
the displacements u(n). On the other hand, displacements
may be conveniently rephrased in terms of stresses, pro-
vided that appropriate constitutive equations are used.
The waves (0), (1), (2), (3), (4), (5), (6), (11) and (12)
can be handled using the conventional theory of elastic-
ity in that travel in continuum media. On the contrary
DM is necessary in describing waves (7), (8), (9) and (10)
which propagate in the biological, non continuum sub-
strate. In this perspective, M has been chosen as M = 2,
that is the smallest value of M that retains the scale fea-
tures of DM. It is assumed that the dynamic process is
isothermal and the volume forces vanish. The particle
interactions are assumed to be longitudinal (central), so
that the shear and torsion microstresses vanish every-
where in the body. Recalling (4), for M = 2 the elonga-
tion microstrains are derived as
ijij ijkijk
 
 
 ,
u (13)
while, recalling (5), the micro moduli matrix Aαβ takes
the form
1112 121515
11 121515
11 1515
Sym A
11 4444
 A (15)
where solely the two constants A11 and A44 are inde-
pendent. Under these conditions, relation (8) is simpli-
fied into
ijijij kk
 
The unit vectors τ are chosen as follows
0,1, 0,
  
  
  
  
  
and correspondingly the doublets distribute in space as
in Figure 13. Notice that the problem is fully three-di-
Substituting back within relations (12) the stresses and
displacements written in terms of the amplitudes An ac-
cording to Equation (11) and Equations from (13) to (16),
a subset of 12 explicit equations is derived, which may
be clothed in matrix form as to obtain M·R = B, where
M is the [12 × 12] coefficients matrix, B the [12 × 1]
vector comprising the known terms and R represents the
[12 × 1] vector enclosing of the unknown reflection co-
efficients Ri. Both M and B depend upon a number of
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374 373
Figure 13. The doublets distributing in space as to form a 3D
parameters, namely: the micro mechanical properties of
the biological substrate (density, nodal separation dis-
tance ηα, micro moduli Aαβ); the classical continuum
mechanics properties of the glass substrates and of the
auxiliary layer (density, constants of Lamè λ and μ); the
thickness of the layers (h1, h2); the initial set-up angle θ0;
the phase velocities ci; the time harmonic waves’ angular
frequency ω = 2πf. Noticeably, while conventionally the
relation between ω and ki is represented by the cele-
brated relation ki = ω/ci (ci = (E/ρ)1/2) in the framework
of DM it assumes a far more elaborated form, for either
pressure (P) or shear (S) waves
44 1144
1144 44
44 44
,( )
1,( )
 
 
 
and we shall call the latter equations of dispersion in that
demonstrate that propagation in discrete-media is dis-
persive, and strongly influenced by the micro moduli
and by the scale length.
In its present form, the model presented in the paper
does not take into account diffraction or scattering. Cou-
pling these effects with the DM theory would require
considerable mathematical efforts which are considered
outside the scope of this study and left for future invest-
tigations. While there are several practical ways to re-
duce speckle (such as the use of spatial and frequency
compounding techniques [42], it is important to under-
stand that, under certain circumstances a simple reflec-
tion/refraction model is still sufficient for describing
tissue/US interaction.
A practical proof for this statement is the fact that ul-
trasound reflection techniques are commonly employed
to accurately measure the thickness of different tissues
or tissue layers and for non destructive testing of materi-
als. These circumstances are insonicating from different
angles and averaging the results. This sentence is further
substantiated in the following.
A speckle field arises when a wave impinges on a
rough surface, generating a scattered wave radiating in
all directions. Each echo signal is the combination of the
many signals coming from a group of scatterers within
the resolution cell. At each point, the amplitude of the
echo signal depends on whether constructive or destruct-
tive interference predominates. Since the scattered wave
emanates from numerous contributors, it is appropriate
to characterize it in statistical terms. If this field is inte-
grated over a finite detector area (that would be the US
receiver) the probability distribution of the integrated
intensity I is [43]
 
mI m
pI I
Im I
where Γ(m) is the Gamma function, the operator <·>
stands for spatial average and thus <I> would corre-
spond to the “true” mean intensity in the limit of a per-
fectly flat reflective surface, and m is, to a first approxi-
mation, the number of correlation cells or speckles fal-
ling onto the detection area. Figure 14 illustrates Equa-
tion (18) for different m’s. Notice that for increasing m
Figure 14. The probability distribution of the speckle inte-
grated intensity I for different m’s.
opyright © 2011 SciRes. JBiSE
F. Gentile et al. / J. Biomedical Science and Engineering 4 (2011) 362-374
Copyright © 2011 SciRes.
the diagrams turn into quasi-Gaussian with most prob-
able intensity <I>. Equivalently, in the limit of large de-
tector area, the speckles would just represent a back-
ground noise that would have negligible influence on the
response of the sample. Founding on these considera-
tions, scattering effects can be reasonably neglected.
Note that by large detector area we simply mean an area
containing a large number of scatterers compared to the
resolution cell, which is equivalent to state that the tissue
scatterer density satisfies the requirement for obtaining
fully developed Raleigh backscatterers and thus m1.