BioMed Research International

Volume 2017 (2017), Article ID 7852971, 7 pages

https://doi.org/10.1155/2017/7852971

## The Elasticity Coefficients Measurement of Human Dentin Based on RUS

Key Laboratory of Ministry of Education for Biomechanics and Mechanobiology, School of Biological Science and Medical Engineering, Beihang University, Beijing, China

Correspondence should be addressed to Haijun Niu

Received 4 November 2016; Accepted 5 April 2017; Published 30 April 2017

Academic Editor: Andrea Scribante

Copyright © 2017 Fan Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposed to take advantages of resonant ultrasound spectroscopy (RUS) to measure the mechanical properties of human dentin specimen. The resonant spectroscopy of the dentin specimen was obtained between the frequency bands 155 and 575 kHz, and resonant frequencies were extracted by linear predictive filter and then by Levenberg-Marquardt method. By inverse problem approach, 13 experimental resonant frequencies progressively matched to the first 30 orders of theoretical resonant frequencies calculated by Lagrangian variational method. The full second-order elastic tensor of dentin specimen was adjusted. The whole set of human dentin engineering moduli, including Young’s moduli ( GPa, GPa), shear moduli ( GPa, Gpa), and Poisson’s ratios (, ), were finally calculated. This study demonstrates that RUS can be successfully adapted to measure the mechanical properties of low quality factor biomaterials.

#### 1. Introduction

As the most abundant mineralized tissue in human teeth, dentin is composed of about 50% inorganic components (basically calcium hydroxyapatite), 30% organic components (mainly type I collagen fibers), and 20% water [1, 2]. Compared with enamel, dentin has less inorganic components and thus is softer and more elastic. These characteristics ensure dentin being indispensable to cushion chewing force and protect internal pulp [3].

The researches about dentin mechanical properties began from 19th century and never stopped since then [3]. To date, the main methods of dentin mechanical properties measurement mainly included tensile and compression test [2, 4], acoustics method (pulse echo method, bulk wave method, elasticity imaging, etc.) [4–8], and macroscopic indentation and nanoindentation method [9–11]. These methods helped not only promote the understanding of macromechanical properties of dentin and micromechanical properties of dentin tubule, but offer important significance for the design, development, and evaluation of clinical dental restorative materials as well. Nevertheless, these methods mentioned above have their own limitations. For example, the macrotest methods required a relative larger size of the specimen, which was difficult to obtain from human dentin. Besides, although the nanoindentation method could be used to measure the elastic moduli in different directions of small-sized specimen, it mainly reflected that the mechanical properties deeply rely on the scope of indentation, which led to the differences from the macroscopic mechanical properties.

Since the 1990s, resonant ultrasound spectroscopy (RUS) has been developed as an accurate and efficient method to characterize the material properties [12–15]. The basic mechanism of RUS is to obtain a series of mechanical resonant frequencies by generating free vibrations with ultrasound excitations, then predict theoretical model frequencies with Lagrangian variational method, and finally get the material elastic properties by comparing the measured frequencies with the predicted ones (inverse problem approach). RUS has been regarded as the gold standard for measuring the elastic moduli of high (quality factor) solid materials. And it beats other methods by the following three advantages: ① The full elastic tensor could be assessed from a single sample in a single nondestructive experiment over other elasticity measurement methods; ② RUS was well-adapted to small-sized samples (a few millimeters or less); ③ The measurement results were more repeatable and accurate [12, 14, 16]. In recent years, Pascal’s group were dedicated to measuring the elastic coefficients of cortical bone by RUS, which made it possible for breaking the limitations when measuring low- materials by RUS [12, 13, 17].

In this paper, the method of human dentin elasticity extraction based on RUS was studied. Firstly, the resonant spectroscopy of dentin specimen was obtained by ultrasound experiment. Then, the resonant frequencies were extracted through signal processing of linear predictive filter and then by nonlinear least squares method (Levenberg-Marquardt method). Combined with the theoretical resonant frequencies calculated by Lagrangian variational method, inverse problem approach was introduced to obtain the complete second-order elastic tensor of dentin samples. The engineering moduli, including Young’s moduli, shear moduli, and Poisson’s ratios, were finally calculated.

#### 2. Materials and Methods

##### 2.1. Specimen Preparations

The tooth used in this paper was complete, fresh, and caries-free. It was a left upper third molar from a 25-year-old male, which was collected from Beihang Hospital. The donor provided consent to donate his tooth for this study. The entire experimental procedures were approved by the Institutional Animal Care and Use Committee of Beihang University and performed under the guidelines of the National Institutes of Health.

Before experimentation began, the specimen was stored in saline solution at room temperature (22°C). After cleaning the entire tooth surface by removing calculus and granulation, the dentin part was cut into a rectangular parallelepiped by a low speed diamond cutting machine (SYJ-150, Shenyang Kejing Auto-Instrument Co., Ltd., Shenyang, China) with 0.01 mm positioning accuracy and 25–300 rad/min rotational speed and a whole sintered diamond saw blade (0.3 mm thick). The six faces of the specimen were polished with P500, P800, and P1000 abrasive paper progressively. With repeated protractor measurements of each corner of the specimen and polish, a standard rectangular parallelepiped human dentin specimen was obtained (accuracy: ) [12]. The mass of the specimen was 111 mg, and the dimensions were mm.

##### 2.2. Theoretical Resonant Frequencies Calculation

The resonant frequencies of solid material are related to many factors such as density, dimensions, elastic tensor, and boundary conditions. Moreover, the relationship among these factors is nonlinear and no analytical solutions exist either. To figure out the approximate numerical solutions, Lagrange variational method was imported here. As (1) shows, the resonant angular frequencies could be calculated by searching for the stationary points of the Lagrangian under free-surface boundary condition [14, 18–21]. where and are kinetic energy and potential energy, respectively, given by

In (2), and are the specimen’s density and volume, respectively, is the component of the displacement field in Cartesian coordinates, and are the stiffness constants of solid material.

Kinney et al. found the elastic constants of hydrated dentin exhibited as transverse isotropy, with five independent constants: , , , , and , as (3) shows [1, 22]:

To find the stationary point of the Lagrangian , (4) was calculated:

To solve the numerical solutions of (4), by expanding the displacement field to a set of polynomial functions, Rayleigh-Ritz method was introduced:where the choice of is rather arbitrary.

Historically, Visscher et al. found that there was none simpler than powers of the Cartesian coordinates when expanding the displacement field [18], so a set of power exponent functions were chosen for simplifying computing:where .

When , the solutions of (4) are the exact solutions. Considering a good compromise between computational accuracy and computing time, was chosen as 12.

Based on (5) and (6), (4) was transferred to the generalized eigenvalue problem:where and are expressed as follows:

In the end, the theoretical values of solid resonant frequency could be calculated by solving (7).

Before the resonance experiment, a set of initial elastic tensors combined with the density and dentin specimen dimensions were needed to be set; then the theoretical resonant frequencies could be calculated later. The initial elastic tensor set plays a decisive role in iterative efficiency and accuracy, so it is indispensable to find an exact-value-closest set as shown in matrix (9) [18, 22]. In this paper, the first 30 frequencies’ range was chosen as the experimental frequency sweep range reference.

##### 2.3. RUS Experiment

The RUS experiment platform is shown in Figure 1. The platform was made just to fit the free-surface boundary condition. The dentin specimen was mounted on opposing corners between two shear wave contact transducers (V154RM, Panametrics Inc., Waltham, US) in the RUS system. A network analyzer (Bode 100, Omicron electronics GmbH, Klaus, Austria) was used to output a swept-frequency signal between 155 kHz and 575 kHz (frequency resolution: 30 Hz) as the excitation of the transmit transducer. The frequency response of the specimen was received by the other transducer, amplified by a broadband charge amplifier (HQA-15M-10T, Femto Messtechnik GmbH, Berlin, Germany), sent back to the network analyzer, and recorded [12].