1. Introduction
Modeling of soft tissue implies new perspectives that carry several clinical applications. It could be used, for example, in tissue engineering [
1,
2,
3], for finite element modeling [
4,
5,
6,
7], to analyze virtual reality in clinical practice [
8,
9] and for surgery planning [
10,
11]. To simulate those applications, the theory of linear elasticity has been employed to understand the results of mechanical tests on soft tissues [
12,
13]. However, surgical procedures lead to consider large displacements and linear elasticity is a simplification when considering small strains. There is a need among researchers to use simplified models which can represent the nonlinear behavior of soft tissues. The simplicity of the proposed model in conjunction with a good correlation with the experimental data can be presented as an accurate and simple model in computational solid mechanics field.
Although the nature of soft tissue behaviour is viscoelastic [
14], a simplification of hyperelasticity allows a reasonable characterization of the mechanical properties, specifically when the loss of strain energy is small (low loading rates). Veronda and Westmann [
15] and Fung [
16] were the first works that used hyperelasticity for soft tissue modeling. The hyperelastic approach postulates the existence of the strain energy function which relates the displacement of the tissue to the corresponding stress values [
17]. The most common strain energy functions for the modeling of soft tissues are polynomial forms, such as Mooney–Rivlin and Ogden models. Many authors have modeled the behaviour of soft tissues such as, porcine spleen, porcine kidney, porcine liver, rat or human brain [
18,
19,
20,
21,
22]. Regarding cervical tissue, uniaxial tension tests [
23,
24,
25,
26,
27] and compression [
25,
28,
29] have been studied in rat tissue and human tissue using load-relaxation protocols. A nonlinear stress–strain response has been shown in the tension and compression tests and the response of the tissue was noticeably stiffer in tension than in compression. It was observed that tissue from pregnant patients was one to two orders of magnitude more compliant than tissue from nonpregnant patients [
25,
29]. In a work carried out by Yoshida et al. [
27], load relaxation ring tests were performed on pregnant and nonpregnant rat cervices. The pregnant tissue showed a very large stress-relaxation compared to the nonpregnant tissue. Myers et al. observed that the cervix stiffness changes along its length in the uniaxial tensile test, where the external os had a stiffer response than the internal os [
25]. The relationship between stiffness and gestational age was studied by Poellmann et al. and Jayyosi et al. [
26,
30]. The works concluded that stiffness decreased as gestational age increased. In the works mentioned above were uniaxial, compression and traction tests were performed, the mechanical properties of the tissues have been obtained. However, in those works the nonlinear elastic properties of ex vivo human cervical tissue, using the Fourth Order Elastic Constants (FOECs), Ogden, and Mooney-Rivlin models have not been obtained through uniaxial tensile tests yet.
Soft tissues are composed of several layers; each one of these layers has different compositions, for instance, cervical tissues have an epithelial outer layer and a connective layer. The connective layer is composed by an extracellular matrix (ECM) that ensures the strength and integrity of the cervix, resisting shear deformation, through a fibrous scaffold [
31]. The main component of the ECM is fibrillar collagen, which determines a cross-linked network interlaced with the elastin protein, enclosed by a ground substance of proteoglycans and glycosaminoglycans [
32,
33,
34]. Researchers have identified three zones of structured collagen in the connective layer: the innermost and outermost rings of stroma contain collagen fibers preferentially aligned in the longitudinal direction, and the middle layer contains collagen fibers preferentially aligned in the circumferential direction [
35,
36]. Regarding the collagen content, the middle zone had higher levels of collagen content when compared with the inner and the outer zones [
36]. According to the mechanical studies on soft tissues, the connective layer is often considered as the most important from a mechanical point of view [
37,
38,
39]. However, other studies, based on Torsional Wave Elastography, consider the epithelial layer as a key apart from the connective one [
40,
41,
42]. The reason is that torsional waves not only propagate in depth but along the surface before being registered by the receiver. One of the purposes of this work is to study the differences in stiffness between the epithelial and connective layers of ex vivo human cervical tissue that comes from the hyperelastic models employed.
The main aim of this work is to propose a new hyperelastic model based on FOEC in the sense of Landau’s theory. The evaluation of the agreement between classical hyperelasticity theory and acoustoelasticity theory, in particular third and fourth order elastic constants, in predicting the mechanical response has potentially high impact. Recent advances in medical imaging techniques for non-invasive evaluation of rheological behavior of soft tissues provides new perspectives, in particular for in vivo quantification of nonlinear shear wave-based elastographic biomarkers, which could be diagnostic predictors of a broad spectrum of labor disorders.
This work has been divided into several sections. First, a brief introduction to the core content has been presented. Second, a section of materials and methodology details the methods used in this work. Uniaxial tensile tests of human ex vivo cervical tissues were performed to fit the hyperelastic models. Third, the results of the comparison between the experimental test and the hyperelastic models were analyzed.
4. Discussion
According to the evidence found in literature, Myers et al. [
29] investigated the nonlinear time-dependent stress response of cervical samples from different human hysterectomy specimens. Results showed the nonlinear response of cervical stroma, which was dependent on obstetric history. However, to our knowledge, the nonlinear elastic properties of ex vivo human cervical tissue, using the aforementioned hyperelastic models, have not been obtained by uniaxial tensile tests yet.
This work aims at representing a first step toward a nonlinear characterization of human cervical tissue. The nonlinear elastic properties of ex vivo cervical tissue have been obtained for the first time by uniaxial tensile tests.
The first contribution of this study is to propose a new hyperelastic model (nonlinear model) based on the FOEC in the sense of Landau’s theory [
56] and compare the obtained results with the most used hyperelastic models in the literature, Mooney–Rivlin, and Ogden models [
49,
51].
As a second contribution, the differences in shear modulus between epithelial and connective layers of ex vivo human cervical tissue are analyzed. The TWE technique proposed in the literature aims to locally measure the mechanical parameters of the cervix that can be correlated with the different stages of the cervix maturing during pregnancy [
41]. These waves interact not only with the superficial layer, the epithelial layer, but also with the deeper layers, i.e., connective layer. Therefore, a validation of the mechanical parameters reconstructed in both layers is necessary through uniaxial tensile tests of ex vivo samples from hysterectomies of healthy women.
The mechanical behavior of the cervix is nonlinear, as are most of the soft biological tissues. The nonlinear parameters of the FOEC proposed model were obtained through a fit of the experimental data measured in the uniaxial tensile tests with the theoretical law that governs the hyperelastic model. The same procedure was carried out in order to get the nonlinear parameters from the two most employed hyperelastic models to characterize soft biological tissue in the literature, Mooney–Rivlin and Ogden models.
The proposed FOEC hyperelastic model, Equation (
9), presents three parameters, the shear modulus
, the third order elastic constant
A and the fourth order elastic constant
D. In this study, the proposed theoretical stress-strain relationship has been simplified with the aim of comparing with the two-term Mooney–Rivlin and Ogden models. The two parameters that have adjusted the experimental data were the shear modulus
and the third order elastic constant
A. Analyzing the results of the shear modulus, there was no significant variation in both, the epithelial 1.29 ± 0.15 MPa and the connective layer 3.60 ± 0.63 MPa for each of the hysterectomy samples, see
Table 5. The values of the shear stiffness are highly dependent on the strain ramp rate used, the larger strain rate, the larger shear modulus [
57,
58]. The values obtained for cervical tissue agree with those found in the literature [
23,
29]. However, regarding the nonlinear parameter
A, large variation was observed, varying the parameter from positive to negative values (see
Table 2). Previous studies in the literature have obtained the value of parameter A in tissue-mimicking phantoms and breast tissue [
59,
60]. Gennisson et al. [
59] investigated the nonlinear behavior of quasi-incompressible agar-gelatin-based phantoms using supersonic shear imaging technique. The study concluded that, for different samples of tissue-mimicking phantoms, the value of parameter A varied from a negative value to a positive one. Similar results were obtained by Bernal et al. [
60] in breast tissue, parameter A presents high variability due to internal changes in tissue consistency. According to the present study, it is worth pointing out that, to our knowledge, this is the first work that studies the nonlinear parameters of the FOEC hyperelastic model in cervical tissue. The variability found in the third elastic constant parameter
A could be associated with the heterogeneity of the tissue, like the two previous studies [
61], also depends on the particular epidemiological aspect of each patient, as reported by Myers et al. [
62]. Despite the variability of parameter A, and the low number of samples, the quadratic regression studied in the connective tissue samples showed a high correlation with women’s age (
) (
Figure 8). On the other hand, the cubic regressions in the connective layer of the infinitesimal shear modulus
from the Ogden model, the
parameter from the Mooney–Rivlin model, and the
parameter from the Mooney–Rivlin model against the women’s age have smaller correlations,
Figure 9,
Figure 10 and
Figure 11 (
,
, and
respectively). This result is very preliminary because only seven samples have been considered, so future studies to explore other relationships with physiological markers require a greater number of patients, despite the difficulty of obtaining ex vivo samples.
On the one hand, the shear modulus is one of the most used parameters in the characterization of soft biological tissues by various techniques. This value can be extracted directly by means of the
parameter of the FOEC proposed model, through the slope of the stress–strain curve in the linear region or also through a combination of the two parameters of the Ogden model, the infinitesimal shear modulus
and the stiffening parameter
. The values of the shear modulus for each reconstruction technique and for each sample are shown in
Table 5. The parameters that govern the Mooney–Rivlin model have no physical sense and, therefore, the shear modulus value can not be extracted from them. The results of the shear modulus for each reconstruction technique and for each cervical layers were compared by using a Student’s
t-test.
t-test results are shown in
Figure 13, in terms of
p-values. All of these values were below 0.001, which can be considered to represent a significant difference in the shear modulus between the epithelial and connective layers. In conclusion, the shear modulus was dependent on the anatomical location of the cervical tissue as shown in
Table 5 and
Figure 13. On the other hand, the infinitesimal shear modulus (
) is one of the parameters that governs the hyperelastic Ogden model. The p-value obtained for the comparison between the epithelial and connective layers (
p-value = 0.0016) shows that there is a significant difference (see
Figure 14), therefore, this parameter could also indicate differences in stiffness for both layers.
In general, by the obtained results, it can be concluded that the nonlinear parameter A could be an important biomarker in the characterization of connective cervical tissue. In addition, the calculated shear modulus depended on the anatomical location of the cervical tissue. The protocol of measurements is applicable to other tissues and it is possible to explore a set of new nonlinear measurements from different procedures, for example in vivo measurements in women using ultrasound wave propagation by employing Hamilton’s formulation, one step further from the diagnostic point of view.