Nonlinear shear wave propagation to assess biomechanical properties in softtissue

Juan MELCHOR(1,2,3), Guillermo RUS(1,2,3), Antonio CALLEJAS(1,2), Inas FARIS(1,2), Javier NARANJO-PÉREZ(4), Miguel

RIVEIRO(1)

AbstractUltrasonic-based characterization of soft tissue is an emerging technology with great potential as a clinical di-agnostic tool. There is evidence that abnormalities in the structural architecture of soft tissues are linked to abroad range of pathologies including tumors, liver fibrosis, preterm birth...The present work derives differenttheories of non-linear formulation of wave propagation using generalized and particular coordinates on a mate-rial characterized by different strain energy functions. Afterwards, these formulations are adapted to simulatethe propagation of shear waves on nonlinear elastic materials to explain the properties of soft tissue when ismeasured under these conditions. The resultant system of equations could be solved under a finite differencesframework and probabilistic inverse problem to explore how improves the understanding of soft tissues. Theresponse of the system varying the linear/non-linear parameters of the strain energy is obtained via Finite Dif-ferences Time Domain simulation showing significant variations among the different models. Nonlinear wavepropagation approach implies a new paradigm to model the nonlinear biomechanical behavior of soft tissueimproving the plausibility versus previous elastic or viscoelastic predictions.Keywords: Sound, Insulation, Transmission

1 INTRODUCTIONThe nonlinear elastic field has introduced a great interest in the study of hard and soft tissues mechanics in thelast decades [17, 16]. The research in shear and longitudinal waves propagation in tissue starts at the beginningof the second part of the twentieth century. The ultrasonic imaging in soft tissues begins when the use ofdynamic tests to ultrasonically estimate the compressibility and mobility of breast tumors was analyzed [25].Ultrasonic shear elastography have been applied by in order to measure the linear and nonlinear elastic responseof hyperelastic tissues. The Static Elastography, originally proposed by Ophir et al. [18], has been successfullyused to measure nonlinear properties of vascular tissues or malingnant and beningn tumors of breast tissues[7, 9]. Nevertheless, the results of this method are not satisfactory for organs or tissues which are deep ordifficult to compress as the strain profile may be uncertain. The Transient Elastography introduced by StefanCatheline [4] has been used, for example, to diagnose cirrhosis [5] and for assessment of hepatic fibrosis [21].The Supersonic Shear Imaging (SSI) technique, presented in 2004 by Bercoff and Tanter [2], has also beenused to obtain the third and fourth order constants, A and D, respectively, of pig brain tissue or human breasttissue. However, these studies were limited to ex vivo experiments [6, 20, 9]. Nowadays, the consideration oftorsional waves (they propagate radially and in depth) is a potential key due to the limitations of shear andcompressional waves [12]. Firstly, this typology of wave can propagate by quasi-compressible material and,it is more sensitive to quantify the consistency changes, for example, in the case of tumors [23]. Secondly,the variations of biomechanical parameters are more sensitive in the regime of low energy where this wave isgenerated. Finally, torsional movement does not generate secondary interfering P-waves at the boundary of thetransducer where pure shear waves are difficult to create [12, 11]. A torsional ultrasonic transducer has beenused to measure nonlinear parameters of ligament tissue and the shear modulus of cervical tissue in pregnantwomen [3].The aim of this work is to elaborate a set of nonlinear formulations of a torsional and shear wave propagatingon a hyperelastic tissue defined from the Hamilton?s strain energy function and others [26, 13]. While thequadratically nonlinear propagation of the torsional wave on a hyperelastic material has been studied [19], the

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consideration of new nonlinear terms in the equations is an open issue to consider experimentally en the future.

2 MATERIAL AND METHODS2.1 FundamentalsThe Green-Lagrange strain tensor that governs the elasticity law is defined in terms of displacements in indexnotation as,

εi j =12(ui, j +u j,i +uk,iuk, j) (1)

where the third term is related with geometrical nonlinearity. The physical nonlinearity is here focused onhyperelastic materials, in which the strain energy function W is defined per unit reference (undeformed) volumeand acts as potential of the stress. The strain energy function is defined following the expression by Landauand Lifshitz [10],

W =λ

2I21 +µI2 +

A3

I3 +BI1I2 +C3

I31 (2)

where µ and λ are the Lame constants and A, B and C are the TOEC. I1, I2 and I3 are the invariants of theGreen-Lagrange strain tensor defined by Eringen et al. in 1974 [15],

I1 = trε = εii, I2 = trε2 = εi jε ji, I3 = trε

3 = εi jε jlεli (3)

The expansion to fourth order of the energy density is necessary when the nonlinear effects in shear wavesare considered. This is because for incompressible media, nonlinearity at third order is missing in the particledisplacement [8]. This expansion is characterized by four new terms, yielding,

W =λ

2I21 +µI2 +

A3

I3 +BI1I2 +C3

I31 +DI2

2 +EI1I2 +FI21 I2 +GI4

1 (4)

where D, E, F and G are the Fourth Order Elastic Constants (FOEC).When the case of quasi-incompressible soft tissue media is analyzed, where λ >> µ , the strain energy functionis simplified to,

W = µI2 +A3

I3 +DI22 (5)

where A and D are the third and fourth order elastic constants, respectively [8].The motion equation in the contravariant basis in the Lagrangian configuration is obtained in general coordinatesas,

∇i

[(δ

ij +∇ jui)S jk

]= ρ

∂uk

∂ t2 (6)

This is the compact form of the momentum equation expressed in the Lagrangian configuration. To expand theequation, several previous steps are required. The tensor including these derivatives multiplied by S jk gives asa result another second kind tensor, ∇ jui ·S jk = C ik. Hence, Equation 6 can be reformulated as the divergenceof a tensor Bik which results,

∂Bik

∂ξ i +BmkΓ

imi +Bim

Γkmi = ρ

∂uk

∂ t2 (7)

Note that for this particular case, the general coordinates are ξ 1 = r, ξ 2 = θ and ξ 3 = z, the Christoffel symbolsthe configuration depends only on the velocity of the angular coordinate θ .From other point of view, following where the series expansion concept put forth by Landau [10], only volu-metric part is detailed in terms of the nonlinear acoustic parameter β as was explored recently [13].There are four combinations of nonlinear acoustic parameters β that may explain a different scenario of experi-mental calculations as in this case where the exploration of quasi-fluids nonlinearity is considered. The concept

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of the non uniqueness for the classical acoustic parameter β have been recently studied by Bender [1] but with-out a physical explanation. The aforementioned combinations could be expanded as exploring the whole set ofcombinations by quadratic terms as follows,

σi j =−3Kvδi j︸ ︷︷ ︸pressure

+2µDi j︸ ︷︷ ︸shear︸ ︷︷ ︸

σLi j (Linear)

−3ηvvδi j︸ ︷︷ ︸

pressure

+2ηDi j︸ ︷︷ ︸shear︸ ︷︷ ︸

σVi j (Viscous)

++9Kβvpv2

δi j +9Kβd pDkpDpkδi j︸ ︷︷ ︸

pressure

+4µβdsDikDk j +4µβ

csvDi j︸ ︷︷ ︸shear︸ ︷︷ ︸

σNLi j (Nonlinear)

(8)

where v and Di j are the volumetric and deviatoric parts of the stress respectively. Four nonlinear parametersof first order have been defined as β vp,β d p as the volumetric, the deviatoric and the compound denoting thepressure components and β ds,β cs as the deviatoric, and compound denoting the shear components. However inthe case of cuasi-compressible, when compressibility is much higher than shear moduli, it means K >> µ whenthe volumetric nonlinear terms could be neglected,

σi j =−3Kvδi j︸ ︷︷ ︸pressure

+2µDi j︸ ︷︷ ︸shear︸ ︷︷ ︸

σLi j (Linear)

−3ηvvδi j︸ ︷︷ ︸

pressure

+2ηDi j︸ ︷︷ ︸shear︸ ︷︷ ︸

σVi j (Viscous)

+9Kβd pDkpDpkδi j︸ ︷︷ ︸pressure

+4µβdsDikDk j︸ ︷︷ ︸shear︸ ︷︷ ︸

σNLi j (Nonlinear)

(9)

With the aim to find a relationship between this nonlinear expansion of beta parameters and TOEC, we assumingnow that the strains are separated in volumetric and deviatoric part to the second order, so ε yields,

εikεk j = DikDk j; εi j = Di j; εk j = Dk j; εkp = Dkp; εpk = Dpk

By making use of Cauchy stress described in Equation 9, in nonlinear regime, an equivalence is deducted interms of Third Order Elastic Constants TOEC,

σNLi j = DikDk j(A +4µ)+DkpDpkBδi j (10)

Since the relationship is established in the nonlinear constitutive equation, nonlinear acoustic parameters of firstorder are explicitly deducted as follows in terms of Third Order Elastic Constants,

βd p =

B

9K; β

ds =A +4µ

4µ(11)

2.2 Experimental setupA schematic view of the experimental setup is shown in Figure 1. The transmitter and receiver are located inthe same transducer as was explained in the previous sections. The measurements was taken as in transmission,with guarantees perfect alignment along the propagation direction of sound. The response signals were sampledwith a high resolution A/D converter with amplification Fonestar without preamplification stage and saved forpost-processing. The transmitter was a torsional transducer manufactured in our END lab at University ofGranada.

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to the resulting average of 490 captures of the signal, providing an effective

reduction of noise for the detected response signal, increasing the signal-to-

noise ratio around 25 dB. Figure 3.2 depicts the experimental setup used to

record the ultrasonic signals.

Acquisition cardA/D converter

�� AA

- -���HHH

6

--

Wave generator

Excitation

signal

Res

pons

esi

gnal

Transmitter

Receiver

Amplifier

Specimen

Computer

Synchronized signal

3.3. Semi-analytical approach

The measurements procedure consists of four steps. Initially, the response

signal was measured in water at two different positions corresponding to

locations ahead of and at the back of the specimen (steps 1 and 2 ).

Then, the response signals were measured with the specimen in situ along

the scanning area at both frequencies (steps 3 and 4 ).

In order to reconstruct the linear and nonlinear material’s properties of

the specimen under inspection, a semi-analytical approach is proposed by

considering the following assumptions: (i) the attenuation in the water layers

is negligible, (ii) the thickness a3 of water layer 2 is small, and thus the

nonlinearity can be assumed not to accumulate over this distance (i.e. �(3)w ⇡

0) [10], (iii) only second harmonics are considered, and thus frequency-mixing

12

Figure 1. Experimental configuration for a nonlinear torsional transducer

We used an arbitrary generator (Agilent 33500) to drive the transmitter at different frequencies. The ultrasonicexcitation frequencies were arbitrarily chosen. Different frequency combinations were used: f1 = [600,800,1000]Hz. The amplitude of the input signal at the frequency was increasing, [5,10] V. To avoid unwanted interferenceeffects in the propagation medium, even the risk of burning, we used short trains of pulses for excitation and a 8ms pulse duration and 3 cycles. We investigated the generated waves as a function of distance. The transmitter-receiver distance was a total distance of 2.43 mm. At the distance x from the transmitter, the pressure of thefundamental, u(0)1 , and that of the generated harmonics, u(1)1 , were determined, for the different frequencies andexcitation levels.

Figure 2. Connective tissue and silicone mold during measurements and oscilloscope view

The material used in this experiment was the silicone mold as a first step. Secondly, a connective tissue wasexplored with the transducer prototyped as is shown in Figure 2, with the characteristic amplitudes as in thelongitudinal waves.

3 RESULTSBy adopting the acoustic nonlinear constitutive equation presented above Equations 9 in terms of deviatoric avolumetric parts, is possible to establish the three dimensional nonlinear equation of motion up to first-ordernonlinearity in terms of four parameters β . This formulation, implies that beta parameter is not unique and canbe defined as the separation between pressure and shear waves and it is solved under perturbation scheme as,

Bac3s

x1A21ω1

= 3(9βd pK +2β

dsµ) (12)

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This results can be compared with the Zarembo ones [26].

βτZ =

µ

ρ0+

1ρ

(A

2+B

)(13)

So, converting β d p and β ds in the corresponding TOEC is posible to obtain a similar expression as,

βT =

(B+A /2+2µ)

4(3K +4µ)(14)

In order to validate experimentally the theoretical background, is necessary to design an experimental proce-dure, for this purpose these results should be convert into a pressure magnitudes. Then, by making use of theconversion to amplitudes of fundamental and first harmonics into transversal stresses, it is posible to calculatethe nonlinear shear parameters in terms of transversal stresses.Then, torsional beta, β T could be defined by in terms of pressures and disaggregated in two parts, one due tothe liquid or water part and the other referred to collagen or fiber undulation part,

βT =

(B+A /2+2µ)

4(3K +4µ)=− ρc3

s T (1)12

π f1x(T (0)12 )2

(15)

where the relation 2π f = ω1 has been introduced and the relationship between pressure and displacements.

3.1 Experimental resultsTo obtain the experimental results in the case on torsional nonlinearity coefficient, is necessary to use thefollowing equation calculated from PZT-5 Piezoelectric material where elastic and electric field are coupled,

T12 =CE44S12 + e24E2 (16)

Neglecting elastic field to found a conversion factor between voltage and Pascal, 122.9 Pa was the multiplicationfactor for the case of 10V and silicon mold. The conversion factor is formulated from the analytical simplifiedmodel where piezoelectric and design material effects were considered interacting with two layers of tissue. Forthis reason, the stress on the piezoelectric ceramic is approximately the stress on the tissue by a correctionfactor αc,

T piezo12 = αcT tissue

12 (17)

Parallel to this, to obtain αc transmission coefficient should be calculated, by making use of the energy of thetransmitter and receiver.

T =4z1z2

(z1 + z2)2 =energy,tenergy,r

(18)

where energy, t is the energy of the transmitter and energy,r is the energy of the receiver. Knowing the differentlayers of the receiver, the stress and displacement ratios could be deducted as,

Tt = Ti2zt

zi + ztor ut = ui

2zi

zi + zt(19)

The final experimental results are detailed in Table 1 and in Figures 3 where nonlinear torsional acoustic pa-rameter β T was extracted for silicon mold, ligament tissue and liver tissue through the speed of sound of shearwaves cs, density ρ , shear modulus µ and transmission coefficient Z.

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Sample Energy Frequency ρ cs Z µ β T

(V) (Hz) (kg/m3) (m/s) (KPa)Silicon mold 10 800 1100 13.2 13200 0.16 -400±3

Connective tissue 10 800 1000 90 90000 8.1 -8000±0.0Liver tissue 10 800 1000 4 4000 0.016 -88±20

Table 1. Nonlinear torsional results for input frequency and amplitude f1 = 800 Hz and A= 5,10 V, respectively,for silicone mold, connective and liver tissue.

Time [µ s]-5 0 5 10 15

Signal[kPa]

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Frequency, f, [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Pressure

[Pa]

10-4

10-2

100

102

104

Time [µ s]-5 0 5 10 15

Signal[kPa]

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Frequency, f, [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Pressure

[Pa]

10-4

10-2

100

102

104

Time [µ s]-5 0 5 10 15

Signal[kPa]

×10-3

-8

-6

-4

-2

0

2

4

6

Frequency, f, [kHz]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Pressure

[Pa]

10-4

10-2

100

102

104

Figure 3. Nonlinear torsional signal and harmonics from silicon, connective and liver tissue for 10V.

3.2 Numerical resultsThe difficulty of finding the close form solution makes it necessary to the numerical method Finite DifferenceTime Domain [22]. The boundary conditions are expressed in terms of velocities and stresses and geometricaland mechanical symmetries have been taken into account. The displacements are permitted without any con-straint and no stresses are created. In these boundaries the components of the second Piola-Kirchhoff stresstensor are nil. The velocity is uθ = 0. The propagation of the signal cannot be simulated indefinitely along thespace. An issue arises when analyzing open space regions where the simulation domain must be limited. Inthis work, to ensure that the boundary does not reflect the wave, an Absorbing Boundary Conditions ABC hasbeen considered in those boundaries where the stress is equal to zero and the wave can be reflected.It can be noticed that the nonlinear curve does not have a purely sinusoidal behavior as the linear curve does.Conversely, the ridges and valleys are slightly sloping, which is normal because of its nonlinear nature. TheFast Fourier Transform (FFT) algorithm has been used to obtain the solution in the frequency domain. The mainremark about the frequency domain nonlinear response is that only odd harmonics are relevant. For instance,for an excitation frequency of 500Hz, the first (500Hz), the third (1500Hz) and the fifth (2500Hz) harmonicsare clearly observable [14].

a)0 5 10 15 20 25

Time [ms]

-5

0

5

Dis

pla

cem

ent

uθ [µ

m]

nonlinear

linear

b)0 2 4 6 8 10

Time [ms]

-10

-5

0

5

10

Dis

pla

cem

ent

uθ [µ

m]

nonlinear

linear

c)0 2 4 6 8 10

Time [ms]

-5

0

5

Dis

pla

cem

ent

uθ [µ

m]

nonlinear

linear

a)0 5 10 15 20

Frequency [kHz]

10-12

10-10

10-8

10-6

Po

wer

sp

ectr

um

nonlinear

linear

b)0 1 2 3 4 5 6

Frequency [kHz]

10-10

10-8

10-6

Pow

er s

pec

trum

nonlinear

linear

c)0 5 10 15 20

Frequency [kHz]

10-10

10-8

10-6

Po

wer

sp

ectr

um

nonlinear

linear

Figure 4. Displacement uθ at the receiver (left column) and Fourier transform (right column) for a frequencyof: a) 500Hz, b) 1500Hz and c) 2000Hz. The parameters of the hydrogel are: A = 40kPa and D = 3000kPa.

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4 CONCLUSIONSIn this work, a set of formulations haven been developed from classical theories of nonlinear elasticity andits derivation to cylindrical coordinates of the propagation of the torsional wave on a hyperelastic material.For this reason, a breakthrough has been achieved in the field of ultrasound.The nonlinear classical acousticsextension may be also applied to derive a solution that separates the nonlinear terms that are interpreted asnonlinearity from the matrix and the nonlinearity from the fibers. This is an important conclusion with a mainnovelty in tissue microstructure. The results suggest that these nonlinear terms that from now can be obtainedat real time describe the behavior of the tissue in a new scale. Although this works have contributed in someway to advancing the understanding of ultrasonic nonlinearity, there are several aspects to be studied in thefuture with potential impacts on biomedical research and understanding the mechanics of tissues. However, softtissues should be modeled as transverse isotropic, orthotropic or, more accurately, anisotropic materials. Anotherimportant consideration is the viscosity of soft tissue both in linear and nonlinear regime.

ACKNOWLEDGEMENTSThis research was supported by Ministry of Education DPI2017-83859- R, DPI2014-51870-R, and UNGR15-CE-3664, Ministry of Health PI16/00339 Carlos III/FEDER and Junta de Andalucia PI-0107-2017, PIN-0030-2017.

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