IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS,...

IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 1, NO. 1, JANUARY 2011 1

Probabilistic inverseproblem to characterize

tissue-equivalent material mechanical properties

Nicolas Bochud1 and Guillermo Rus1

Special Issue: Novel Embedded Systems for Ultrasonic Imaging and Signal

Processing - Medical Imaging

Abstract

The understanding of internal processes that affect the changes of consistency of soft tissue is

a challenging problem. An ultrasound monitoring Petri dishhas been designed to control in real-

time the evolution of relevant mechanical parameters during engineered tissue formation processes.

A better understanding of the measured ultrasonic signalsrequired the use of numerical models of the

ultrasound-tissue interactions. The extraction of relevant data and its evolution with sufficient sensitivity

and accuracy is addressed by applying well-known signal processing techniques to both the experimental

and numerically-predicted measurements. In addition, a stochastic model-class selection formulation is

used to rank which of the proposed interaction models are more plausible. The sensitivity of the system

is verified by monitoring a gelation process.

Index Terms

Tissular mechanics, Inverse Problem, Model-class selection, Ultrasound, Signal processing, Tissue

characterization.

I. INTRODUCTION

The rational principles of continuum mechanics are proposed together with a formal signal

processing framework to address the problem of characterizing mechanical properties of tissue

cultures based on noninvasive and non-ionizing ultrasonicmeasurements.

1Departmentof Structural Mechanics, University of Granada,Politécnico de Fuentenueva,18071 Granada,Spain. Email:

{nbochud,grus}@ugr.es, www.ugr.es/~grus

October 17, 2011 DRAFT


In the non-destructive evaluation (NDE) community, signalprocessing techniques have played

an increasing role over the last decades [1]. An essential element in NDE systems is the analysis

of the captured signal, by means of a noise-free parameter extraction, in order to obtain relevant

information from the tested specimen. Among the methods of noise impact minimization, it is

worth to note filtering techniques: Rodríguezet al. [2] selected the frequency range corresponding

to the detected echoes location. Thus, by making use of the Wigner-Ville transform, the echoes

from the crack can be distinguished from the noisy echoes generated by the scattered ultrasonic

waves through the material grains. Bilgutayet al. [3] proposed a transformation based on filter-

bank techniques by applying the split spectrum processing (SSP) method to obtain a reconstructed

signal less affected by noise.

The first proposals to solve the deconvolution problem in NDEuse classical techniques, such as

the Wiener filter, the spectral extrapolation, deconvolution of minimal least variance, estimation

of the least squares [4], or homomorphic deconvolution by computation of the cepstrum [5]. The

cepstrum also has been used as an efficient parametrization tool for ultrasonics signals, since

the cepstral coefficients involve deconvolutionated signal information [7]. Higher-order statistics

(HOS) enables the developing of blind deconvolution techniques, avoiding any prior information

on the signal noise or defect [8]. In some cases, the extracted signal parameters can be processed

again to reduce the dimensionality of the feature vectors, by applying classical linear discriminant

analysis (LDA) [9] orprincipal components analysis (PCA) [10].

Recently, signal processing methods have been increasingly used to provide suitable features

extraction from biomedical imaging, especially for elastography imaging (B-scan), to diagnose

pathologies. Ultrasound data were captured from postmortem coronary arteries to develop radio

frequency analysis techniques for the characterisation ofatherosclerotic plaque [11]. A system

was proposed by Scheiperset al. [12] for prostate diagnostics based on multifeature tissue

characterization. Their Neuro-fuzzy inference system combine spectral features and textural

features of first and second order with clinical variables and morphological descriptors for

diseases classification. Abeyratneet al. [13] proposed the use of a wavelet transform based

technique to estimate the inter-scatterer-distribution (ISS) in diagnosing focal diseases of the liver.

Maggio et al. [14] employed a multi-feature kernel classification model based on generalized

discriminant analysis to support prostate cancer diagnosis. Siebers et al. [15] presented an

ultrasound based system for computer aided characterization of biologic tissue and its application



to differential diagnosis of parotid gland lesions, based on a supervised classification of malignant

and benign cases using tissue-describing features derivedfrom ultrasound radio-frequency (RF)

echo signals and image data.

However, most of the above-mentioned applications deal with determination of the presence of

damages/pathologies in the evaluated specimen. Hence, thefinal goal of the system is limited to a

classification of the states damage/no-damage or malignant/benign, respectively. These techniques

require a huge amount of experimental data and an expensive training process, without providing

any information at the physical level, nor using knowledge about the physiology to aid the

decision.

Several laboratory-scaleimaging modalitieshavebeenproposedby otherauthors,thatprovide

limited information aboutshearelastic propertiesof tissue.Shearwavescanbegeneratedwithin

tissuesusing focused,impulsive, acoustic radiation force excitations [16]–[18] acoustic remote

palpation, sonoelastography or shearwave dispersion ultrasoundvibrometry, and the resulting

displacementresponsecanbeultrasonically trackedthroughtime.An alternative way to measure

shearmodulus is elastography [19]–[21], wherethe displacement is inducedby indentation and

the strain is computedby a speckle-tracing method.The impulsive force excitation combined

with ultrasoundvibrometry developedby Sandrin et al. [22] hasbeencommercializedby Hitachi

under thenameof Fibroscanr asa liver disorderdiagnosisdevice. Only a few works incorporate

mechanical models for the identification of clinical processes. Moultonet al. [23] solved an

inverse boundary value problem todetermine the unknown material parameters for a nonlinear,

nonhomogeneous material law, using ap-version finite element model of the heart. Hanet al. [24]

presented a finite element based nonlinear inverse scheme toreconstruct the elastic properties

of soft tissues subjected to an external compression. Recently, Guo et al. [25] developed a

novel finite element method-based direct method for the material reconstruction in soft tissue

elastography.

Some recent experimental observation may tangentially suggest that nonlinear mechanical

properties may be a key signature to quantify tissueconsistencychanges,evenbetter than just

bulk or shearmoduli. Barannik et al. [26] revealedthemechanical relaxationprocessesappearing

at higherfrequenciesthanin thequasi-staticregime.Theappearanceof theserelaxationdynamics

wasassociatedto the presenceof inhomogeneitiesin tissue.In the pastyears,numerousmodels

havebeendevelopedin the NDE community that investigatethe possibility of using nonlinear



techniquesto detectdamagedinterfacein solids [27], [28]or dynamicnonlinearstress-strainfea-

turesin micro-inhomogeneousmaterials [29]–[31].Recently, someauthorshaveextendedsome

of theseexperimental techniques(NRUS, NEWS,etc.) for application to damageassessment in

cortical bone [32]and to measureacoustic nonlinearity in trabecular bone [33].

Theproposedsystemavoidstheneedof stepssuchascross-correlation of echographicimages,

by a moredirect procedure,henceincreasing the precision.Second, it allows the useof a com-

plex propagation model in order to quantify modelparameters of interestbeyond modulus and

attenuation. Nonlinearphenomenahavenot beenconsideredin this study,but this is a research

topic currently underdevelopmentat our laboratory. This void in the mechanical characterization

and interpretation of material defects may be overcome by adopting model-based inverse problem

strategies. This is the main goal of the present work, which is successfully applied for tissue

characterization.

Many types of uncertainties involved in the modeling of interaction between ultrasonic waves

and tissue, such as excitation, material viscosity, and material heterogeneity are responsible for

noise in the output. In this paper, several models of ultrasound-tissue interaction are proposed,

implemented and contrasted against experimental observations. All assume homogeneous media

with varying moduli and energy-dissipation forms that are expressed as attenuation models. High

frequency ultrasound is adopted for excitation and measurement in order to analyze and interact

with small-scale specimens. The minimization between experimental and numerically predicted

measurements is addressed by solving a probabilistic inverse problem whichincludes advanced

signal processing techniques.

This approach allows to obtain not only the optimal parameters in a model class, but also

the uncertainty associated with the parameter estimates. Some recent developments and civil

engineering applications of Bayesian model class selection have been carefully reviewed by

Yuen [34]. The model-class selection is formulated following Becket al. [35]. Finally, a simple

formulation of the joint probability is proposed, from which either the inverse problem or the

model-class selection can be derived just by extracting specific marginal probabilities, thus

unifying all the approaches.

A model-classselection algorithm is useful to understandunknown propagation models in

complex materials such as engineeredtissue,which is the specimen this device will be used

for. With the purposeof validating the model-classselection algorithm, a gelation processis



Tissue

Monitoring dish

Transmitter Receiver

Simplified

US paths

Waveform

generatorAmplifier

A/D

converterAmplifier

Computer

Internet

SyncTransmitted

signal

Received

signal

u1 u u2 3

Fig. 1. Schematic experimental and electronic setup.

monitoredandanalyzed,of which thebehavior is controlled andassumedby themajority of the

literature to be viscoelastic [36]–[38].

II. M ETHODOLOGY

The proposed methodology combines four elements. (1) The signal acquisition of the ultrasonic

signals obtained from the waves interaction with a sample oftissue, a (2) set of alternative

attenuation models that simulate the ultrasound-tissue interaction, which is numerically solved

by the transfer matrix formalism, a (3) stochastic model-class selection formulation used to rank

which of the models parametrization are more plausible, anda (4) NDE oriented signal processing

framework that extract relevant features from both the experimental and numerically predicted

signals. The latter is used to reconstruct the evolution of the relevant mechanical parameters

during the culture reaction time.

A. Experimental setup

A Petri dish with a specifically designedsub-wavelength thickness ultrasonic transmitter

and receiver in angle position was manufactured for real-time measurement of mechanical and

geometrical properties of thin layers of tissue culture (ofthe order of 100 [µm]). The monitored

Petri dish is connected to the electronic setup detailed in Fig 1.

Thetransmitting andreceiving transducersaredesignedto bein angleposition (45◦) in orderto

avoidreverberation echoesinsidethepetri dishplateparts. The transmitted signal is generated as



a 1-cycle burst composed by a 20 [MHz] sine of amplitude that amounts to 5 [V] with a repetition

rate of 1 [ms], using anarbitrary wavegenerator (Agilent 33220). The recording is digitizedwith

a high resolution A/D converter with a base of 10 [V] after 40 [dB] preamplification, during a

period of 5 [µs] and a sampling rate of 400 [MHz]. The gelation process is monitored during

half an hour at 5 [s] intervals, resulting in a database of 350measurements. Each measurement

corresponds to the average of 300 captures of the signal, providing an effective reduction of

noise according to the signal-to-noise ratio (25 dB).Only compressional wavesare generated

by the transducers and no modeconversion wavesare measuredin the presentcase,although

the methodology is extensible to shearor otherwaves.

The materials and the concentration for the gel culture werechosen according to Ortegaet al.

[39]: 92.5 % of water, 5 % of glycerol and 2.5 % of agar. In orderto obtain an homogeneous

solution, water has been first heated and then the remaining materials have been added. The

material properties of the Petri dish material (PMMA, polymetylmetacrilate) and the gel culture

are summarized in Table I.

TABLE I

MECHANICAL PROPERTIES OF MATERIALS(GOLOMBECK [40]).

Material Modulus Poisson Density Speed

[Pa] ν ρ [Kg/m3] cp [m/s]

PMMA E = 2960 · 106 0.43 1180 2672

Gel (initial) unknown 0.5 1000 unknown

B. Propagation models

The experimental system is idealized by a mathematical model of the propagation and in-

teraction of the transmitted ultrasonic waves with all the parts of the system until they are

received by the sensor. The relevant ultrasonic paths alongthe Petri dish material (PMMA,

polymetylmetacrilate) and the gel culture are illustratedin Fig 1.

Several models are tested to idealize the removal of energy by dissipation or radiation.Three

alternative damping models are used: viscous, hysteretic,and proportional to integer time deriva-

tives of the particle movement, based on their fractional time derivatives.Thedamping is defined



in termsof the wavemodulus M , which is modified from the undampedoneM0 to generatea

dispersiveone,which is a frequency-dependentcomplex modulusM∗(ω), whereω is theangular

frequencyif the modulus dispersion is representedby its frequencydomain.The viscousmodel

is defined in termsof the frequency-dependent lossfactor η, obtainedas the ratio betweenloss

and storagemoduli [41]. In this context, a specific view of hysteretic damping is taken,where

it is expressedas a frequency-independent damping [42]. The last model, basedon fractional

time derivatives, leadsto a damping function that may be expressedas a power law, and thus

improvescurve-fitting properties for relaxation [41], [42]. These models are selected according

to their performance demonstrated in a previous study [43].The viscoelastic η andhysteretic ζ

models aredefined according to Maia et al. [44],

M∗(ω) = M0 (1− iωη − iζ) (1)

whereη and ζ are the viscoelastic and hysteretic damping coefficientsof tissue,respectively.

The fractional time derivative damping is defined as,

M∗(ω) = M0 1 + b(iω)β

1 + a(iω)α(2)

The three models are summarized hereafter, highlighting the combination of the considered

parameters,

TABLE II

COMBINATION OF MODELS.

Tag Size Parameters

1 4 K tissue ζ zampl ztime

2 4 K tissue η zampl ztime

3 6 K tissue a b α = β zampl ztime

where K tissue denotes the Bulk modulus of tissue. The fractional derivative constants are

defined asa, b, andα = β. Two additional parameterszampl and ztime are introduced to control

the correction of the amplitude and the time-shift of the input in the culture, which corrects

effects of temperature and other phenomena on the sensors, that affect attenuation and delay

on the path from the electronics to the arrival of the signal at the culture specimen.The factor



zampl correctsvariations of the amplitudeand phaseof the excitation over the reaction process

time, that may suffer the influenceof several simultaneous factors including the temperature.

Nonetheless, we assume that thesefactors can be summarized as a whole by introducing a

phenomenological factor, labeledasztime.

The mathematical model is approximated by a semi-analytical model of the wave interactions

within multilayered materials based on the transfer matrixformalism (TMF) [45], describing the

ultrasonic waves interactions between the Petri dish and the culture.

C. Probabilistic inverse problem

Following the probabilistic formulation of the model reconstruction inverse problem estab-

lished by Tarantolaet al. [46], the solution is not a single-valued set of model parametersM.

On the contrary, the solution is provided by probability density functions (PDF)p(M) of the

values of the model parametersM within the manifoldM of possible values. The probability

density is assigned the sense of plausibility of the model values to be true.

Statistical inference theory is used to incorporate to thea priori information about the measured

observationsO, the model parametersM and the model classC, the information of idealized

relationship between themO = O(M) computed by a numerical model pertaining to a model

classC. The former are defined by the probability densities to prior(labeled 0) data p0(O),

p0(M) andp0(C) respectively, whereas the additional information about relationship (labeledm)

between observations and model provided by the model classC is given by the PDFpm(O,M|C).

The a posteriori probability p(O,M, C) of the hypothetical modelM is obtained jointly with

the observationsO and classC,

p(O,M, C) = k1p0(O,M, C)pm(O,M, C)

µ(O,M, C)(3)

whereµ(O,M, C) is the noninformative density function andk1 is a normalization constant.

Assuming first thatO, M and C are independenta priori allows to split the joint prior infor-

mation and the uniform distribution. Secondly, the probabilistic model can be represented by a

computation ofO depending onM. Finally, the model is not assumed to provide conditional

information between model and class. This simplifies the expression to,

p(O,M, C) = k1p0(O)p0(M)p0(C)pm(O|M, C)

µ(O)(4)



The posterior probability of the modelM is obtained from the joint probability by extracting

the marginal probabilityp(M)∣

∣

C=Cifor all possible observations and provided the model class

Ci ∈ C is assumed to be true,

p(M)∣

∣

C=Ci=

∫

C=Ci

∫

Op(O,M, C)dOdC

= k2∫

O

p0(O)p0(M)pm(O|M,C)µ(O)

dO(5)

As a last simplification, we assume to have no prior information about the modelp0(M),

which is therefore represented by the noninformative distribution p0(M) = µ(M), which can

in turn be dropped in the case of Jeffreys parameters are used,

p(M)∣

∣

C=Ci= k3

∫

O

p0(O)pm(O|M, C)dO (6)

where k3 is a normalization constant that replaces the dropped uniform distributions, and is

needed forp(M)∣

∣

C=Cito fulfill the theorem of total probability.

The observations are assumed to follow a Gaussian distribution O ∼ N (E[Oexp], Cexp)

with mean of the experimental observationsOexp and covariance matrixCexp standing for

the measurement noise. Additionally, the observations areassumed to be a Gaussian process

O ∼ N (O(M), Cnum) centered at the numerically computed onesE[Onum] = O(M) with

covariance matrixCnum.

The probabilistic observationsO are in our case a vector of functions of timeO = oi(t) at

every measuring timet ∈ [0, T ] and repetitioni ∈ [1...Ni], and the assumptions made above

are valid for every instantt and sensori. Considering that the compound probability of the

information from all sensors and time instants is the productory of that of each one individually,

and that this productory is equivalent to a summation withinthe exponentiation, the Gaussian

distribution allows for an explicit expression of the probability densities,

J(M) =1

2

Ni∑

i,j=1

∫ t=T

t=0

Oi

(

cexpij + cnumij

)−1Ojdt (7)

whereOk = (ok(t,M)− oexpk (t)), ∀k = i, j. The termJ(M) corresponds to a misfit function

between model and observations,

p(M)∣

∣

C=Ci= k6e

−J(M) (8)



and the constantk6 is derived from the theorem of total probability applied over all possible

modelsM, which is integrated by Quasi Montecarlo using a Sobol sequence with 218 points.

The best-fitting model is found by minimizingJ(M) instead of maximizingp(M) since,

M = argmaxM

{

p(M)∣

∣

C=Ci= k6e

−J(M)}

= argminM

{J(M)} (9)

Let model classC ∈ C denotes an idealized mathematical model hypothesized to simulate the

experimental system, whereas modelM denotes the set of physical parameters that the model-

class depends on. Different model classes can be formulatedand hypothesized to idealize the

experimental system, and each of them can be used to solve theprobabilistic inverse problem,

yielding different values of model parameters but also physically different sets of parameters.

To select among the infinitely many possible model classes that can be defined, a probabilistic

criteria can be defined based on their compatibility betweenprior information about observations

O, model parametersM and model classC, and probabilistic model information [35].

The goal is to find the probabilityp(C), understood as a measure of plausibility of a model

classC [47]. It can be derived as the marginal probability of the posterior probabilityp(O,M, C)

defined in Equation 4,

p(C) =∫

O

∫

Mp(O,M, C)dMdO

= k1p0(C)

∫

O

∫

M

p0(O)p0(M)pm(O|M,C)µ(O)

dMdO(10)

The plausibility of a residue definition given a measurementdomain and model classCi in the

sense of matching between model and experimental observations can be computed by introducing

the residue spaceR ∈ R, and deriving the corresponding marginal probability as,

p(R) =∫

O

∫

Mp(R,O,M, C)|C=CidMdO

= k7∫

O

∫

M

p0(O)p0(M)pm(O|R,M,Ci)µ(O)

dMdO(11)

where the constantk7 = k7p0(C)p0(R).

The minimization ofp(M) for monitoring the evolution of the culture is carried out bytwo

sequential algorithms: When an initial guess is not available, which is the case at the beginning of

the process to be monitored, genetic algorithms are used as afull-range random search technique

[48]. Since the change between consecutive measurements ofthe process is expected to be small,

the BFGS-algorithm is employed thereon as a local search based on Hessian update [49], assisted

by finite differentiation and line search.



In the case the model parametersm are Jeffreys constants [50], they are replaced by unitary

logarithmic parametersm,

mm = m0me

mlnm1

m

m0m (12)

which maps the dimensional parametersmm from the preferential rangemm ∈ [m0m, m

1m] to a

nondimensional Jeffreys parameterm ∈ [0, 1]. This furthermore stabilizes the search algorithms.

In this case, the noninformative distributionµ(M) can just be replaced by a constant.

D. NDE oriented signal processing

This part aims to provide a suitable representation of the ultrasonic signals, appropriate for

tissular reaction process identification. The signals are first preprocessed, by means of a temporal

windowing. Then, different parametrization approaches can be applied, and the obtained spectral

parameters are usually transformed, to provide a more uncorrelated and dimensionally reduced

representation. It results from the applied analysis that each signal can be represented by a

feature vector containing the analysis parameters. Finally, some methods are defined to improve

the calculation of the discrepancy between the experimental and numerically predicted feature

vectors.

1) Preprocessing: After acquisition, the signals have been decimated at a sample frequency

of 40 MHz, in order to reduce part of the noise and focus on the frequency band of interest.

Secondly, a normalization with respect to the peak amplitude has been applied, considering that

the system must be unsensitive to changes in signals amplitude. Finally, the signals have been

multiplied by a Hamming window. Since ultrasonic signals are finite by nature, the window is

foremost used to weight the signal samples over the time, andthus to show off the signal echoes

[51].

2) Feature extraction and spectral-domain: The proposed methodology includes a nonpara-

metric technique approach, that directly estimates the spectrum features from the signal itself,

providing a sufficiently accurate representation for many types of signals in many different

applications [52], where as in NDE systems, the pursued information is hidden in a complex

signal. The spectral analysis is performed by determining the magnitude spectrum of the detected

signals, easily obtained by applying the discrete Fourier transform (DFT).



3) Homomorphic transformations: Among the homomorphic transformations, the basic idea

of the cepstrum consists of converting a convolution into a sum [53], and thus to obtain

a decorrelated representation of the signals. The cepstrumof a discrete signals(n), whose

corresponding spectrum is denoted byS(ω), is defined as the inverse Fourier transform of the

logarithmic spectrum:

c(n) = F−1[log(S(ω))] =1

2π

∫ π

−π

log(S(ω)) dω (13)

Generally, the spectrumS(ω) is a complex and even function obtained by applying the Fourier

transform. Alternatively, areal cepstrum can be obtained by considering the magnitude spectrum

|S(ω)| as,

c(n) =1

2π

∫ π

−π

log(|S(ω)|) dω (14)

In practice, thereal cepstrum can be easily obtained by applying the fast Fourier transform

(FFT) as,

c(n) = IFFT[log(|FFT(s(n))|)] (15)

where IFFT denotes the inverse fast Fourier transform. In this case, thereal cepstrum is usually

calledcepstrum FFT. In an algebraic sense, the associatedcomplex cepstrum could be obtained

similarly. However, computing thecomplex cepstrum is usually cumbersome due to the unwrap-

ping of the digital phase. In the cepstral-domain, due to theharmonic nature of the ultrasonic

signals the wave components appear as equidistant peaks at higherquefrencies, rightly separated

by a value that corresponds to the fundamental period of the analyzed signal echoes. Thus, this

cepstral representation allows to decompose the spectrum in its two main characteristics: The

spectral envelope and the fine spectrum.

4) Dimensionality reduction and deconvolution property: It can be useful to restrict the

number of cepstral coefficients, by applying a window to ruleout lower and/or higherquefrencies.

This process is calledliftering, and is applied as,

c(n) = c(n)w(n) (16)



wheren = 1, ..., L. Several windowing schemes can be directly derived from therectangular win-

dow in the cepstral-domain. This study focuses on theshort-pass liftering [54], that corresponds

to a smoothing of the spectrum, preserving its spectral envelope while removing the fine spectrum

information. Although it has been less studied, it is interesting to point out that the application of

a liftering on the complex cepstrum allows to observe the effects of removing quefrencies on the

recovered signal in the time-domain. Moreover, applying windows different from the rectangular

one allows to weight the cepstral coefficients depending on their discriminative performance.

E. Discrepancy between the experimental and numerical feature vectors

The residue definitions used in the probabilistic inverse problem are generalized to the vectors

obtained from the feature extraction. Thus, the residue is defined as a likelihood measure between

two feature vectors. The classical residuer0 has been defined as,

r0 = sx − s(p) (17)

wheresx ands(p) denote the vectors obtained from the features extraction, corresponding to the

experimental and numerically predicted signals, respectively. Some enhancements are proposed

by defining weighted residues. First, a weighted residue is defined, which includes the variance

of the measurementsσx over the temporal evolution of the reaction process. Thus, the goodness

of fit of the model predictions to the experimental values is assessed with the weighted least

squares criterion [55]:

r1 = σx(sx − s(p)) (18)

Additionally, a weighted residue is defined, which includesthe variance of the measurements

σx

Rover the reaction time of the process.

r2 =sx − s(p)

σx

R

(19)

Finally, a weighted residue is defined, including both aforementioned variances:

r3 = (sx − s(p))σx

σx

R

(20)

It is worth to note that the varianceσ of the coefficients from the experimental signals amounts

to a value close to zero when the corresponding signal intensity is low. Thus, a slightly biased



variance valueσ is used in the above-mentioned residue definitions by makinguse of theroot

mean square of the varianceσ, in order to avoid that the residue tends to infinity:

σx = σx + 0.1

√

√

√

√

1

L

L∑

n=0

(σx

n)2 (21)

Equations 18-20 allows to (i) reduce the uncertainties due to the measurements noise, by

assuming statistical independence of the errors, and (ii) enhance parts of the signals that may

contain information of the reaction process. In a probabilistic sense, residue definitions that take

into account variance information can be understand as a prior knowledge on the measurement

quality and/or evolution of the reaction process. Thus, they may give some enhancements on the

interpretation of the gelation process.

III. RESULTS

The recorded signals by the ultrasound-monitored Petri dish every 250 seconds are shown in

Fig 2, without and with specimen, respectively. No clear evolution is detectable by bare visual

inspection of the signals.Therecordedsignalsaremainly composedof threedifferentwaveforms

(simplified pathsof Fig. 1), namely(1) the wavefront that propagatesonly throughthe PMMA

layer (labeledasu1), (2) a wave that crossesboth the PMMA layer and the specimen(labeled

as u2), and (3) a wave echo producedby the former wave after crossing twice the specimen

(labeledasu3).

It is noteworthy thatwhenthespecimenis on place,themajority of theexcitation signal (reg-

isteredwithout specimenfor calibration) is transmittedinsteadof reflected. Since the wavelength

in gel is compatible with the layer thickness, the individual echoes generated by the multiple

reflections inside the gel layer can be analyzed separately by signal processing.

A. Signal simulation

The transfer matrix formalism is used to generate sample signals, after calibrating the estimated

parameters using the inverse problem, for the first signal (initial evolution time). Time-domain

signals and magnitude spectra are shown in Fig. 3-6 for the viscous model (case 2)at the initial

time of the reaction process, respectively. In the lower figures, an analysis window(Hamming)



has been applied to the signals, and classical and weighted residue definitions are considered,

denoted asr0 andr3, respectively.

A significative ability to simulate the system can be observed visually. The influence of the

signal windowing yields to the following observations for the magnitude spectrum (Fig. 5-6): The

envelope, that corresponds to the redundant character of the signal, remains almost unchanged. In

contrast, the fine spectrum presents accentuatedpeakiness due to the enhanced echoes of the time-

domain signals. In the time-domain, the classical residuer0 is strongly correlated with the signals

themselves (Fig. 3), highlighting higheramplitudewherethe signalsenergy is higher.However,

the weighted residuer3 allows to remove some variability due to the measurements uncertainties

(Fig. 4). Additionally, it enhances the signal parts containing information of the reaction process

(here mainly the wave front), while allowing to remove the parts which are invariant to the

process evolution(scattering parts). In the frequency-domain, the weighted residue allows to

remove the frequency range being unsensitive to the reaction process, or being erroneous due to

measurements noise.Thus,the resulting magnitudespectra of Fig. 6 showconcentratedprocess

information at certain frequencies, in contrast to the magnitudespectrum of Fig. 5.

B. Posterior probability of the model

The probability density function is computed for the viscous attenuation model, and some

relevant samples issued from the results obtained in the previous section are shown in Figs. 7-

10. Since the PDF is a multidimensional function, without loss of generality, only a slice along

two parameters is represented, namely the Bulk modulus of the tissue and the viscous damping

coefficient.

The inspection of these plots reveals several local minima,valleys in the probability density

function and variations of several orders of magnitude fromgood to bad model parameters.

This implies a bad conditioning of the reconstruction inverse problem and justifies the use of

advanced search algorithms such as genetic algorithms. Nonetheless, the use of a weighted

residue definition enhance the slope of these local minima, and thus speed up the convergence

of the search algorithm. It is noteworthy that the other attenuation models present similar trends.

Additionally, some irrelevant samples issued from the cepstral analysis are illustrated in Figs. 11-

12.



The inspection of the plots obtained from the cepstral analysis reveals many local minima that

approximately have the same values, leading to an ill-conditioned solution space.

C. Model class plausibility

The posterior probabilityp(C) of every proposed model classC ∈ C is computed by Quasi

Montecarlo integration using218 Sobol sampling points. Additionally, the estimation of Occam’s

factor, as well as the certainty metricσ are summarized in Tables III-V for time-domain signals,

magnitude spectra and real cepstra, respectively.

TABLE III

PLAUSIBILITY OF MODEL CLASSES. TIME -DOMAIN .

Windowing Residue Model class 1 2 3

w0

r0

p(C) [%] 31.55 32.39 36.06

Occam [−log10] 2.24 4.10 1.90

Certainty [log10] 0.11 0.75 0.37

r1

p(C) [%] 33.33 33.33 33.33

Occam [−log10] 8.11 7.93 11.93

Certainty [log10] 1.69 1.66 1.97

r3

p(C) [%] 15.53 63.10 21.37

Occam [−log10] 7.34 9.95 14.78

Certainty [log10] 2.23 2.74 2.97

w1

r0

p(C) [%] 32.45 32.99 34.57

Occam [−log10] 7.14 6.50 7.44

Certainty [log10] 1.40 1.16 1.02

r1

p(C) [%] 33.33 33.33 33.33

Occam [−log10] 7.93 7.90 13.08

Certainty [log10] 1.83 1.95 ∞

r3

p(C) [%] 34.81 31.57 33.62

Occam [−log10] 4.25 6.17 8.72

Certainty [log10] 1.39 1.79 1.45



TABLE IV

PLAUSIBILITY OF MODEL CLASSES. MAGNITUDE SPECTRUM.


w0

r0

p(C) [%] 29.79 29.39 40.82

Occam [−log10] 0.09 -0.42 -5.17

Certainty [log10] 0.40 0.38 0.71

r1

p(C) [%] 49.94 50.06 -

Occam [−log10] 7.71 8.82 -

Certainty [log10] 1.52 1.63 -

r3

p(C) [%] 0.02 56.47 43.51

Occam [−log10] 4.84 5.88 22.37

Certainty [log10] 1.93 1.89 4.10

w1

r0

p(C) [%] 26.90 26.11 46.99

Occam [−log10] 2.47 1.82 -8.55

Certainty [log10] 0.08 0.08 -1.57

r1

p(C) [%] 49.98 50.02 -

Occam [−log10] 8.68 7.81 -

Certainty [log10] 1.80 1.49 -

r3

p(C) [%] 31.98 44.95 23.07

Occam [−log10] 5.11 4.77 16.15

Certainty [log10] 1.61 1.54 2.85

The most plausible model class is shown to be2, involving K tissue, viscoelastic damping, and

temperature and amplitude corrections. It is closely followed by class1 (hysteretic), whereas class

3 does not provide results for all proposed signal processingtechniques. The magnitude spec-

trum computed with weighted residue definitionsr1 and r3 show significantly higher posterior

probabilityp(C) than the other domains of representation. The real cepstrumprovides bad results

as well for the posterior probability, which is consistent with the observations in the previous

section. This evidence further supports the validity of theprobabilistic formulation. Hence, the

obtained equiprobable values demonstrate its insensitiveness with respect to the selected mode



TABLE V

PLAUSIBILITY OF MODEL CLASSES. REAL CEPSTRUM.


w0

r0

p(C) [%] 33.33 33.33 33.34

Occam [−log10] 1.99 2.91 2.67

Certainty [log10] 0.53 0.45 0.08

r1

p(C) [%] 33.33 33.33 33.34

Occam [−log10] 8.23 8.51 10.95

Certainty [log10] 2.38 2.42 1.94

r3

p(C) [%] 33.33 33.33 33.34

Occam [−log10] 0.21 2.21 3.28

Certainty [log10] 0.12 0.31 0.32

classes.

The posterior probabilityp(R) of every consistent residue definitionR ∈ R is computed

accordingly to the posterior probabilityp(C). Tables VI-VII summarized the obtained values

together with the Occam’s factor estimation and certainty metric, for the hysteretic and viscous

damping, respectively.

The most plausible residue definition appears to ber1, which involves the inclusion of some

prior information on the variance of the measurements over the temporal evolution of the reaction

process, when the inverse problem is achieved in the frequency-domain. This ranking remains

consistent independently of the model class. Signal windowing has a feeble influence on the

results improvement.

D. Monitoring of evolution

The evolution of the relevant reconstructed mechanical parameters during the reaction process

is shown in Fig 13-14 for the most relevant model class and residue definitions, respectively.The

valueof the reconstructedBulk modulus at thebeginning of the processapproximatelyamounts



TABLE VI

PLAUSIBILITY OF RESIDUE DEFINITIONS. TIME -DOMAIN .

Model Window Residue r0 r1 r2 r3

1

w0

p(R) [%] 22.27 26.57 0 0

Occam 2.24 8.11 -22.96 -7.34

Certainty 0.11 1.68 -6.30 -2.23

w1

p(R) [%] 24.56 26.57 0 0

Occam 7.14 7.93 -19.54 -4.25

Certainty 1.40 1.83 -5.40 -1.39

2

w0

p(R) [%] 22.64 26.31 0 0

Occam 4.10 7.93 -23.77 -9.95

Certainty 0.74 1.66 -6.25 -2.74

w1

p(C) [%] 24.72 26.31 0 0

Occam 6.50 7.90 -20.30 -6.17

Certainty 1.16 1.95 -5.38 -1.79

to 2.385 [GPa]. By making useof the following formula [42],

cp =(K + 4

3G)

ρ(22)

andunder the hypothesis that the shearmodulus G andthe density ρ of the material amount

to 1000[Pa] and1000[kg/m3], the wavevelocity is found to be 1544[m/s]. The latter is close

to the valuedepictedby otherauthors for similar materials, amongthemNorisuyeet al. [37].

Some parameter evolutions reconstructed using residues and models with low plausibility (not

shown in this paper for space constraints) show larger scattering and instabilities, consistently

with the results in the twoprevious subsections. Arriving at the same conclusionthan crossed

observations further supports the validity of the formulation and conclusions.

IV. CONCLUSION

A numerical method to determine the elastic and dynamic energy dissipation properties during

a gelation process has been developed by combining the solution of a probabilistic inverse



TABLE VII

PLAUSIBILITY OF RESIDUE DEFINITIONS. MAGNITUDE SPECTRUM.

Model Window Residue r0 r1 r2 r3

1

w0

p(R) [%] 5.87 38.87 0 0

Occam 0.09 7.71 -14.49 -4.84

Certainty -0.40 1.52 -3.98 -1.93

w1

p(R) [%] 15.64 39.61 0 0

Occam 2.47 8.68 -11.81 -5.11

Certainty 0.08 1.80 -3.40 -1.61

2

w0

p(R) [%] 5.82 39.12 0 0

Occam -0.42 8.82 -14.87 -5.88

Certainty -0.38 1.63 -4.00 -1.89

w1

p(C) [%] 15.25 39.81 0 0

Occam 1.82 7.81 -12.02 -4.77

Certainty 0.08 1.49 -3.35 -1.55

problem with signal processing techniques, applying genetic algorithms to minimize a cost

functional, and using a semi-analytical model of the interaction between ultrasonic waves and

tissue.

The proposed model-class and residue selection and their subjacent class plausibility have

enabled to rank both the models and the suitable residue definitions according to their com-

patibility with the observations. The resulting trade-offbetween model simplicity and fitting to

observations have demonstrated that the viscous damping models, combined with some prior

information on the measurements variance over the reactionprocess evolution, are feasible to

characterize the complex evolution of the process.

The reconstructed model parameters highlight the following statements. For the viscoelastic

models, the Bulk modulus consistently decreases while increasing the damping coefficient.

Therefore, both parameters may be associated to the same phenomena, but a careful interpretation

has not been carried out at that time. The evolution of the model parameters has a stronger slope

during the first 200-300 seconds of the reaction process, andremain almost constant afterwards.



This trend validates the observations donein situ where the gelation occurred during the first

3− 5 minutes. Consequently, the proposed methodology demonstrates capability to discriminate

the process during its early solidification phase. For a better understanding of the ultrasonic tissue

monitoring,in vitro studies on real tissue combined with histological studies may be conducted.

ACKNOWLEDGMENT

The authors would like to thank the Ministry of Education of Spain through Grant No. PI2010-

17065 (MICINN), the Servicio Andaluz de Salud, Junta de Andalucía, through Grant No. PI-0308

(SAS), and the Junta de Andalucía through Grant No. P08-TIC-03911.

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Nicolas Bochud holds a Master degree in Mechanical Engineering from the Swiss Federal Institute of

Technology (ETH), Zurich, 2008 and a Master degree in Multimedia System at the University of Granada,

2010. He is currently doing his PhD at the Department of Structural Mechanics of the University of

Granada, focusing on the identification of pathologies in advanced and bio-materials using ultrasonics.

His research field includes non-destructive evaluation, signal processing and nonlinear ultrasound.

Guillermo Rus started his research on computational mechanics at the University of Granada (UGR,

1995), where he disputed the PhD thesis on Numerical Methodsfor Nondestructive Identification of

Defects (2001), providing defect search algoritms and sensitivity computation with Boundary Elements.

He applied these experimentally at the NDE Lab at MIT (USA) asa Fulbright Postdoctoral Fellow,

rendering novel robust quantitative approaches to ultrasonics and impact testing. He started up the NDE

Lab at the UGR (www.ugr.es/~grus) in 2003 to focus on bioengineering applications like bone implant

debonding in collaboration with University College London, bone quality diagnosis with ultrasound with Université Paris VI

or biomaterials characterization with the Nanomaterials Technology Laboratory and Institute of Bioengineering, Alicante. He is

also transferring this diagnosis technology to civil engineering for monitoring structural health of advanced materials like CFRP,

in collaboration with Andong National University, Korea. Rus tenured as associate professor in 2009 at UGR, is the author of

20 SCI papers and 7 books, in addition to 40 international conference contributions and 11 invited presentations. His research

career has been awarded by the Juan Carlos Simo prize for young researchers (Spain, 2007), the Honorary Fellowship of the

Wessex Institute of Technology (UK, 2005), Fulbright Fellowship (USA, 2002) and the Excellence PhD award (Granada, 2001).



0 0.5 1 1.5 2 2.5

−0.5

0

0.5

1

Time, t [µs]

Am

plitu

de

[arb

itra

ryunits]

0 0.5 1 1.5 2 2.5

−0.4

−0.2

0

0.2

0.4

Time, t [µs]

Am

plitu

de

[arb

itra

ryunits]

u1

u2

u3

Fig. 2. Signals sample: sequenceof signals without specimen (above);signal with specimen registered every 250 seconds

(below).



1.2 1.4 1.6 1.8 2 2.2 2.4

−0.06

−0.04

−0.02

0

0.02

0.04

Sig

nal [

mV

]

US time, t [µs]

ExperimentalSyntheticGuess

Fig. 3. Example of fitting of experimental and simulated observations.Viscous model. Time-domain. Residuer0.



1.2 1.4 1.6 1.8 2 2.2 2.4

−50

−40

−30

−20

−10

0

10

20

Sig

nal [

mV

]

US time, t [µs]


Fig. 4. Example of fitting of experimental and simulated observations. Viscous model. Time-domain. Residuer3.



0 5 10 15 20 25 30 35 40 45

0.2

0.4

0.6

0.8

1

1.2

Mag

nitu

de S

pect

rum

Number of spectral coefficients


Fig. 5. Example of fitting of experimental and simulated observations.Viscous model. Magnitude-spectrum. Residuer0.



0 5 10 15 20 25 30 35 40 45

20

40

60

80

100

120

Mag

nitu

de S

pect

rum

Number of spectral coefficients


Fig. 6. Example of fitting of experimental and simulated observations.Viscous model. Magnitude-spectrum. Residuer3.



−1−0.8−0.6−0.6

−0.6

−0.6

Bulk modulus, K [Pa]

Vis

cous

dam

ping

Probability Density, P(m) [log10

−scale]

1.33e+09 1.63e+09 1.99e+09 2.43e+09 2.97e+09 3.63e+095e−11

1.36e−10

3.69e−10

1.00e−09

2.73e−09

7.42e−09

Fig. 7. Posterior probability of the model. Slice along two parameters. Viscous model. Time-domain. Residuer0.



−3000

−2000

−2000

−100

0

−1000

−5 −5 −5−5−5−5


Vis

cous

dam

ping


−scale]

1.33e+09 1.63e+09 1.99e+09 2.43e+09 2.97e+09 3.63e+095e−11

1.36e−10

3.69e−10

1.00e−09

2.73e−09

7.42e−09

Fig. 8. Poste1ior probability of the model. Slice along two parameters. Viscous model. Time-domain. Residuer3.



−800−400−400

−400

−5

−5

−5

−5


Vis

cous

dam

ping


−scale]

1.33e+09 1.63e+09 1.99e+09 2.43e+09 2.97e+09 3.63e+095e−11

1.36e−10

3.69e−10

1.00e−09

2.73e−09

7.42e−09

Fig. 9. Posterior probability of the model. Slice along two parameters. Viscous model. Magnitude-spectrum. Residuer0.



−6000000−4000000

−2000000

−2000000

−5−5


Vis

cous

dam

ping


−scale]

1.33e+09 1.63e+09 1.99e+09 2.43e+09 2.97e+09 3.63e+095e−11

1.36e−10

3.69e−10

1.00e−09

2.73e−09

7.42e−09

Fig. 10. Posterior probability of the model. Slice along twoparameters. Viscous model. Magnitude-spectrum. Residuer3.



−0.4348

−0.4346

−0.4346

−0.4

346

−0.4346

−0.4346

−0.4344

−0.4344

−0.

4344

−0.4344−0.4344

−0.4344

−0.4344

−0.4344

−0.4344

−0.4344

−0.4342

−0.4342

−0.4342−0.4342

−0.4

342

−0.434

−0.434

−0.434−0.434−0.434

−0.434


Vis

cous

dam

ping


−scale]

1.33e+09 1.63e+09 1.99e+09 2.43e+09 2.97e+09 3.63e+095e−11

1.36e−10

3.69e−10

1.00e−09

2.73e−09

7.42e−09

Fig. 11. Posterior probability of the model. Slice along twoparameters. Viscous model. Real cepstrum. Residuer0.



−0.4348

−0.4344

−0.4344

−0.

4344

−0.4344−0.4344

−0.4344

−0.4344

−0.4344

−0.4

344

−0.434

−0.4

34

−0.434

−0.434−0.434

−0.434


Vis

cous

dam

ping


−scale]

1.33e+09 1.63e+09 1.99e+09 2.43e+09 2.97e+09 3.63e+095e−11

1.36e−10

3.69e−10

1.00e−09

2.73e−09

7.42e−09

Fig. 12. Posterior probability of the model. Slice along twoparameters. Viscous model. Real cepstrum. Residuer3.



0 200 400 600 800 1000 1200 1400 1600 1800−10

−5

0

5

10

15

Reaction time, t [s]

Evo

lutio

n [%

]

Bulk modulus, K [Pa] = 2.385e+09Viscous damping = 2.092e−10Amplitude correction = 0.7878Temperature correction = 0.9895

Fig. 13. Evolution of model parameters during reaction. Viscoelastic model. Time-domain. Classical residue. Withoutsignal

windowing.



0 200 400 600 800 1000 1200 1400 1600 1800−8

−6

−4

−2

0

2

4

6

8

10

Reaction time, t [s]

Evo

lutio

n [%

]

Bulk modulus, K [Pa] = 2.384e+09Viscous damping = 2.146e−10Amplitude correction = 0.7836Temperature correction = 0.9967

Fig. 14. Evolution of model parameters during reaction. Viscoelastic model. Magnitude spectrum. Weighted residue. Signal

windowing.


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