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COMPUTER METHODS IN BIOMECHANICS AND BIOMEDICAL ENGINEERING: IMAGING & VISUALIZATION, 2015 VOL. XX, NO. X, 111 http://dx.doi.org/10.1080/21681163.2015.1079505
Using visual image measurements to validate a novel finite element model of crack propagation and fracture patterns of proximal femur Awad Bettamera, Ridha Hamblib, Samir Allaouib and Ahmad Almhdie-Imjabberc Department of Mechanical Engineering, University of Benghazi, Benghazi, Libya; bPrisme Laboratory, Orléans, France; cDepartment of Electrical and 5AQ1 Electronic Engineering, University of Sebha, Libya
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ABSTRACT
ARTICLE HISTORY
In this paper, a simple and practical two-dimensional finite element (FE) model coupled to a quasi-brittle damage law has been developed to describe the initiation and progressive propagation of damage of human proximal femur under quasi-static load until complete fracture. In order to validate the model, ten human proximal femurs were tested till complete fracture under one-legged stance quasi-static load. During each load step, visual image measurements of full field real time strain was achieved using a digital image correlation technique consisting in an optical image system with recording cameras linked to a computer with image-processing software. Two-dimensional FE femur models were derived by the projection of micro computed tomography scans and the specimen fractures were simulated using the same loads and boundary conditions as in the experimental tests. The predicted and optically measured strain field magnitudes and distributions were compared for the ten specimens. Three femurs were used for calibration of the model and the remaining seven femurs were used for validation. The numerical calibration phase was used to establish the relationship between the finite element density and the strain at fracture needed for description of the damage growth. Very good agreement (R2 = 0.89) was obtained between predicted and visualized measured results, indicating that the proposed FE proximal femur fracture model in the quasi-static regime can capture the initiation and propagation of cracks within femurs till complete organ failure. In addition, we show that full-field visual strain measurement provides a much more general and accurate validation than traditional methods based on strain gauges or simple force–displacement curves. The FE model developed here, based on two-dimensional representations of proximal femur geometry and areal bone mineral density distributions, could be applied by clinicians to predict the femur fracture risk of patients using simple and rapid modeling combined with 2D radiographs.
Received 14 June 2015 Accepted 31 July 2015
1. Introduction 10
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In order to predict human proximal femur fracture, linear and non-linear isotropic and anisotropic finite element (FE) models have been developed by several authors (Lotz et al. 1995; Ford et al. 1996; Cody et al. 1999; Ota et al. 1999; Keyak 2001; Crawford et al. 2003; Keyak & Falkinstein 2003; Taddei et al. 2006; Bessho et al. 2007; Schileo et al. 2008; Dragomir-Daescu et al. 2011; Juszczyk et al. 2011, Koivumäki et al. 2012). For simplicity and due to our limited knowledge of the anisotropic behavior of bone, most FE models for femur fracture simulation consider the bone as an inhomogeneous and isotropic material. Empirical density–elasticity relationships are generally applied to assign a single isotropic elastic modulus to every FE of the mesh driven by CT scans. Indeed, the determination of material trajectories related to the trabecular orientations from clinical quantitative computed tomography (QCT) scans remains an open question (Taylor et al. 2002; Wirtz et al. 2003; Tabor & Rokita 2007). Previous FE models applied different uncoupled fracture criteria in order to predict the onset of human proximal femur fracture under excessive load. These criteria are limited in general
CONTACT Ridha Hambli © 2015 Taylor & Francis
[email protected]
KEYWORDS
Visual image measurements; proximal femur; fracture; finite element; experimental validation; digital image correlation
to the prediction of the initiation of local bone failure only. They do not take into consideration the complete quasi-brittle fracturing process of proximal femur and the loss of bone material stiffness generated by progressive damage accumulation prior to fracture. Recently, several authors investigated the fracture of cortical bone based on fracture mechanics concepts (Malik et al. 2003; Vashishth et al. 2003; Ural & Vashishth 2007; Abdel-Wahab & Silberschmidt 2011) but failed to predict the complete fracture pattern of bone since these methods are restricted to the problem of a single dominant idealized planar crack. In spite of the large number of FE studies dealing with bone fracture under monotonic load, there is still a lack of practical and simple FE models that simulate the complete and realistic behavior of bone from the elastic stage till complete fracture. Such models can be developed by incorporating the concept of continuum damage mechanics (CDM) in order to predict the progressive initiation and propagation of cracks, leading to complete fracture of the bone organ. In addition, the techniques used to validate these FE models are generally based on the comparison of the predicted force–
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displacement curves (Keyak 2001; Keyak & Falkinstein 2003; Koivumäki et al. 2012; Hambli 2013) and/or the measurement of the displacement/strain of single points using strain gauges (Vollmer et al. 2000; Marinescu et al. 2005; Al-Sukhun et al. 2007; Rayfield 2011). Such approaches have major limitations: the strain distribution across the proximal surface is not captured, the surface preparation can be time-consuming and the accurate determination of strain gauge positions on the model surface is difficult if the 3D surface topography of the bone surface is not measured. The full-field strain measurement technique based on digital image correlation (DIC) can overcome these problems in the study of surface strain in bone organs (Yang & Ettemeyer 2003; Yang & Yokota 2007; Gröning et al. 2009, 2012). The approach forms an alternative to measuring and visualizing displacements and strains on the surface of bones and the advantage lies in the lack of contact with the investigated specimen. Nevertheless, the potential of this technique has not yet been fully exploited in validation studies in the field of ex-vivo human proximal femur fracture experiments. In general, proximal femur fracture is governed by bone mechanical properties, bone geometry and boundary conditions which can be captured partially by dual X-ray absorptiometry (DXA) measurements. Therefore, 2D FE models are still promising tools for predicting hip fracture. The clinical implementation of three dimensional computed tomography finite element methods is still limited due to the requirement of expensive computer hardware to achieve solutions of 3D FE models within a clinically acceptable time, as well as, the need for robust 3D segmentation and meshing techniques. Segmentation, meshing and FE analysis of a two dimensional (2D) geometry can be accomplished fast and are potentially more robust than of 3D CT/FE (Langton et al. 2009; Op Den Buijs & Dragomir-Daescu 2011). The objectives of this study are twofold. The first aim is to develop a simple and practical two-dimensional FE model coupled to a quasi-brittle damage law in order to simulate the initiation and progressive propagation of damage of human proximal femur under quasi-static load till complete fracture. The second aim is to validate the 2D FE model method using experimental data based on visual DIC full-strain field measurements. To achieve these two aims, we performed in vitro fracture experiments on ten human proximal femurs under one-leg stance configuration in the quasi-static regime. Generally, it is more important to predict the side fall configuration since hip fractures often occur during sideways falls (Greenspan et al. 1998; Schwartz et al. 1998; Kannus et al. 2006; de Baker et al. 2009). However, predicting hip fractures under the one-legged stance configuration is necessary to study features such as the occurrence of spontaneous fractures (Cristofolini et al. 2007; Viceconti et al. 2012). Besides, the effect of pathologies on bone strength (Keyak et al. 2007) and the incidence of atypical hip fractures (Abrahamsen et al. 2009; Brauer et al. 2009; Thompson et al. 2012), primary care of spontaneous fracture prevention should be aware of and alert to the possibility of spontaneous fractures in osteoporotic bone patients. In the present study, two-dimensional FE femur models were derived by projection of micro computed tomography (µ-CT) scans and the specimen fractures were simulated using the same loads and boundary conditions as in the experimental tests. The predicted and optically measured full-strain fields were
compared for the ten specimens. Calibration of the method was performed on a set of three of the ten specimens, and validated on the remaining seven specimens. The numerical calibration phase was used to establish the relationship between the finite element density and the strain at fracture needed for description of the damage growth. Very good agreement was obtained between predicted and measured results, indicating that the proposed FE proximal femur fracture model in the quasi-static regime can capture the initiation and propagation of cracks within femurs till complete organ failure. In addition, we showed that full-field strain optical image measurement can provide a much more general and accurate validation than traditional methods based on strain gauges or on simple force–displacement curves. The FE model developed here, based on two-dimensional representations of proximal femur geometry and areal bone mineral density distributions, could be applied by clinicians to predict the femur fracture risk of patients using simple and rapid modeling combined with 2D radiographs.
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2. Methods and materials 2.1. Specimen preparation Ten human cadaveric femurs (seven females and three males) were provided by the Institute of Anatomy (University of Paris V, Paris, France). Individuals with bone diseases other than osteoporosis or osteopenia were excluded from the study. To cover a wide range of strength values, the femurs were selected from three groups consisting in: three ‘normal’, four ‘osteopenic’ and three ‘osteoporotic’ femurs as classified by femoral neck BMD according to the standards of the World Health Organization (Kanis et al. 2008). The length of the ten femurs ranged from 167 to 207 mm. Each femur was cut transversely to a maximum length of 120 mm beyond the lesser trochanter to the femoral head. This dimension was chosen with respect to the space available in the different devices used (see below). Between each processing step, the freshly harvested femurs were wrapped in a cloth soaked in saline (NaCl 0.9%) in order to avoid dehydration. All the samples were carefully defatted, without causing any damage to the bone, by submerging each of them 2 h in hot bleach at 85 °C. This operation was repeated twice for each specimen after cooling between two stages. The samples were stored in a refrigerator at a temperature of 4 °C. Then, in vitro DXA scans of the femurs were obtained with a Hologic QDR 4500 scanner (Hologic Inc., Waltham, USA) using the standard protocol for the proximal femur. Standard positioning was used across all specimens, and the proximal femoral areal bone mineral density aBMD was evaluated with the software provided by the manufacturer (Table 1). In addition, X-Ray tomographic images were obtained using a high resolution Computed Tomography (CT) scanner (Nanotom Phoenix). Each specimen was placed on a rotating stage and projections at different angles were obtained. Each image was reconstructed using an implementation based on Feldkamp’s cone beam reconstruction algorithm (Feldkamp et al. 1984). Each specimen was imaged with an isotropic pixel size of 50 µm. Each image set took approximately 8 h to acquire and 1 h to reconstruct off-line.
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Table 1. Study sample characteristics. Specimen A B C D E F G H I J
Donor age 85
Gender F
91
F
80
M
98 100 62 87
F F M F
Side Right Left Right Left Right Left Left Left Left Left
Neck aBMD (g cm−2) 0.41 0.45 0.50 0.48 0.75 0.79 0.49 0.43 0.74 0.51
Total aBMD (g cm−2) 0.64 0.72 0.75 0.68 0.83 0.86 0.62 0.53 0.99 0.68
Load
rint p r o Fixture o f ne Mon ur onli colo CCD
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Figure 1. (a) Experimental set-up for the proximal femur specimens test and the optical devices. (b) The region of interest for the DIC post-processing.
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In order to minimise the image noise inherent to the acquisition, a filtering step was necessary to segment the 3D images. This was a simple closed-open filtering with a digital ball as structuring element with a diameter of 3 voxels (Le Trong et al. 2008). After this filtering step, two distributions were easily distinguished on the histogram: one for the porous phase and another for the bone phase. Segmentation was then done by direct thresholding of the phases. The threshold was the grey value corresponding to the local minima of the histogram. Hence, the value 0 was assigned to the porous phase, and the value 1 was assigned to the solid phase. The characteristics of the selected femurs are presented in Table 1. After acquisition, the ten human proximal femurs were prepared for mechanical testing till complete fracture, under a one-legged stance quasi-static load. The femurs were fixed on the experimental apparatus designed and built for this purpose. The distal portion of each femur was fixed on the holder with epoxy resin (SICOMIN Epoxy Systems, France) which was prepared by mixing 100 g of resin (SR 1500) with 33 g of hardener (SD2505). A length of 80 mm was kept for testing and the remaining 40 mm of the distal end was embedded in the resin. Once the resin had been cured, the human femurs were installed in the single limb set up (Figure 1(a)). The holder maintained the orientation of the bone such that the neck was internally rotated 20° within the coronal plane following previous studies performed on proximal femurs (Ota et al. 1999; Keyak 2001; Bessho et al. 2007; van der Steenhoven et al. 2009).
30 2.2. Stance configuration fracture testing Ten human proximal femurs were placed in an INSTRON testing machine (model 4411, Instron Corp., Canton, USA) and
quasi-static compressive loads were applied from zero till complete fracture of the specimens with an increased force of 50 N steps (Figure 1.a). The load cell accuracy is about ±0.2% of the reached load. A constant low loading speed equal to 2 mm min−1 was applied for every test and the displacement of the crosshead was measured using a displacement sensor (LVDT). Charge coupled device (CCD) cameras were used to continuously record images of the sample during the compressive test. The maximum image resolution of these cameras is 1380 × 1024 pixels with an 8-bit digitization for grey levels (28 = 256 grey levels). The cameras were linked to a computer with image-processing software (Deftac 2D, Deftac 3D and 7D) (Vacher et al. 1999). Based on these displacements and the surface topography measurements, full-strain fields were calculated. Overall ten measurements were taken from the beginning till complete fracture of the femurs.
2.3. DIC strain field measurements As stated previously, the current study involves 2D FE models. Therefore, for direct validation, we used 2D DIC measurements to obtain the 2D displacement and strain mapping of each specimen. Consequently, only numerical images taken by one of the two cameras were used to generate the 2D measurements and were post-processed using a 2D DIC method. The principle of DIC method is to assess the displacement fields, and thus strain fields (if required), over the surface of a deforming material by comparing two images acquired at different stages of deformation. The first image is referred to as the ‘reference image’ and the second, acquired after some increment of deformation, as the ‘deformed image’.
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int r pr e o f o n Mon ur onli colo (a)
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Figure 2. (a) Example of a tested femur. (b) CT scan. (c) 2D projection. (d) 2D FE mesh obtained from projected image of CT scan. (e) Young’s modulus distribution in MPa using Eq. (8) and boundary conditions which consisted in (i) application of nodal displacements of a set of selected nodes located at the top surface of the femur head (red line) with an orientation of 20° from the shaft axis till complete fracture and (ii) the distal portion of the model was restrained. Movement perpendicular to the applied displacements was permitted.
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Prior to applying the measuring process, a region of interest (ROI) is manually defined on the initial image, by tracing the bone contour (Figure 1(b)) and, (ii) the DIC method used was validated using calibrated typical images of bone pattern samples, which were strained to various levels with all the data points analyzed. Moreover, before measuring the 2D strain and displacement, the stability of the experimental setup during the measurements was checked by the steady and gradual increase of the strains with increasing applied loads.
bone toughness) and the load testing speed. In general, at a low load rate (quasi-static regime), the proximal femur behaves as 45 a quasi-brittle material with a non-linear behavior till complete fracture (Keyak 2001; Keyak & Falkinstein 2003; Bessho et al. 2007; Dragomir-Daescu et al. 2011). In the quasi-static regime, the isotropic stress–strain relation of elasticity based damage mechanics is expressed by (Lemaitre 50 1985; Hambli 2011; Hambli et al. 2013):
2.4. Model creation After segmentation of the specimens, a binarized medium was obtained and was represented by only two phases. All these images were stacked in Simpleware software (trial version 5.1, Simpleware Ltd) to produce 3D continuum femur models by coarsening the image resolution until a continuum model was obtained. Finally, projected 2D continuum heterogeneous FE models (for the Abaqus code) based on density-based isotropic material properties were generated from the coarsened 3D models (Figure 2). The models are composed of about 29,400 quadratic fournode plane stress elements with a constant thickness. We applied the same loads and boundary conditions as in the previously described experiments. For each element, the average element bone mineral density value ρele (g cm−2) was calculated by 4-point Gaussian integration using the pixel values closest to the midpoint of each side of the quadratic elements. The following power law was assumed between the Young’s modulus E and ρele which was shown to provide the closest results when compared to in-vitro experiments (Op Den Buijs & Dragomir-Daescu 2011):
where D denotes the damage variable, σij the stress components, ɛkl the strains and Cijkl are the elasticity tensor components. 55 The damage law can be expressed in the following general form (Hambli et al. 2012, 2013):
E = 29.8 𝜌1.56 ele (GPa)
(1)
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The Poisson’s ratio was kept constant ν = 0.3
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2.5. Finite element prediction Human femur can experience brittle behavior (Link et al. 2003; Schileo et al. 2008; Juszczyk, et al. 2011) to quasi-brittle failure behavior (Keyak 2001; Keyak & Falkinestein 2003; Bessho et al. 2007; Dragomir-Daescu et al. 2011) depending mainly on bone organ geometry and intrinsic properties, viscosity, specimen preparation (fresh frozen, embalmed), aging (decrease in the
𝜎ij = (1 − D)Cijkl 𝜀kl
⎧ D=0; 𝜀 ≤𝜀 eq y ⎪ n ⎨ D = 𝛼 𝜀eq ; 𝜀y < 𝜀eq < 𝜀f ⎪ D=D ; 𝜀 ≥𝜀 c eq f ⎩
(2)
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ɛeq, α, n, 𝜀y and 𝜀f are respectively the equivalent strain, the damage coefficient, the damage exponent, yield strain (damage strain threshold when damage starts) and the strain at fracture. 65 The equivalent strain εeq is expressed by: √ 2 (4) 𝜀eq = 𝜀𝜀 3 ij ij εij are the strain tensor components. To ensure the objectivity of the numerical model in relation 70 with the physical cracking process and the mesh dependence problem, a weighting (linear form) of the strain at fracture as a function of a characteristic FE length (LFE) and the crack length (Lfrx) is applied in the form (Hambli 2013): ( ) 75 Lfrx 𝜀f = 𝜀true (5) f LFE where 𝜀true denotes the true measured strain at fracture which f can be assessed based on experimental results. LFE is computed automatically and provided by the Abaqus 80 code (Abaqus v.6.11, 2012) at every numerical iteration. The average crack lengths found in bones are typically about 50–100 μm (Burr & Stafford 1990; Taylor & Lee 2003; Sobelman et al. 2004). The characteristic length Lfrx was therefore set to (Lfrx = 0.075 mm).
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Table 2. Material properties and model parameters for bone used for the fracture simulation. Parameters Elastic modulus Poisson ratio Damage factor Damage exponent Threshold strain value for damage initiation Strain at fracture in tension Strain at fracture in compression Critical damage at fracture in tension Critical damage at fracture in compression Mesh characteristic length
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Notation E (GPa) ν α n 𝜀y (%) 𝜀true−T f 𝜀true−C f DcT DcC Lfrx (mm)
Value E = 29.8 𝜌1.56 ele 0.3 650 1.25 0.01% 0.0157 (cortical) 0.025 (trabecular) 0.025 (cortical) 0.04 (trabecular) 0.95 0.5 0.075
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Wolfram et al. (2011) Hambli (2013) and Wolfram et al. (2011) Hambli (2013) and Wolfram et al. (2011) Hambli (2013) and Haddock et al. (2004) Taylor and Lee (2003) and Thurner et al. (2007)
Niebur et al. 2000; Keaveny et al. 2001; Morgan & Keaveny 2001; Arthur Moore & Gibson 2002; Bayraktar et al. 2004; Nagaraja et al. 2005; Nalla et al. 2005; Wolfram et al. 2011). Therefore, the experimentally measured damage (Dexp) was computed by:
𝜀true = 𝜀ftrue−T and Dc = DcT in tension f
𝜀eq−exp is the experimentally measured equivalent strain by the DIC method. In the yielding stage (displacement = 1.33 mm), the damage is initiated locally at the inferior cortex located at the maximum shearing strain generated by the gradient displacement between the femur head and the subcapital region. After the yielding phase, the damage continues to grow rapidly, following a perpendicular path to the superior cortex (displacement = 1.66 mm), leading to complete separation of the proximal femur (displacement > 1.8 mm). The predicted and experimentally measured equivalent strain corresponding to the initiation of the cracks is given in Figure 5 for the seven specimens used for validation. Predicted and experimental results showed that catastrophic failure occurred in the form of crack bands located in the subcapital region. The current FE proximal femur fracture model provided excellent agreement between predicted and experimentally (DIC) measured equivalent strain values and distributions. Both analyses revealed that equivalent strain was concentrated in the inferior cortex of the subcapital region. In normal gait, the greatest stresses occur in the subcapital and mid-femoral neck regions (Lotz et al. 1995). Within these regions, maximum compressive and shear stresses occur inferiorly and smaller magnitude tensile stresses occur superiorly (Lotz et al. 1995).
(6-a)
𝜀true = 𝜀ftrue−C and Dc = DcC in compression and shearing f (6-b)
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Fitted based on Wolfram et al. (2011) and Parsamian (2002)
Numerous studies showed that the damage threshold strains and stresses of trabecular and cortical bone tissue are different in tension and compression (Reilly & Burstein 1974, 1975; Currey 1990; Keaveny et al. 1994, 1999, 2001; Kotha & Guzelsu 2003; Wolfram et al. 2011). Therefore, to account for the asymmetrical bone yields, the strain at fracture is given by:
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References Op Den Buijs and Dragomir-Daescu (2011)
𝜀ftrue−T and 𝜀ftrue−C are the true measured tensile and compressive strain at fracture respectively. DcT and DcC are critical damage values at fracture in tension and compression. A summary of the model parameters is given in Table 2.
3. Results 3.1. Calibration results A mesh convergence study was performed considering different FE models with different mesh sizes (coarse to fine meshes) to evaluate the influence of the mesh size effect on the damage growth. The proposed mesh dependency analysis showed that the ultimate force at fracture was not affected by the mesh size, suggesting that the proposed mesh regularization technique is reliable. Nevertheless, fine meshes (size
