INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. 2014; 30:845–856 Published online 18 February 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2631
3D finite element modeling and analysis of dynamic force in bone drilling for orthopedic surgery Lin Qi*,† , Xiaona Wang and Max Q. Meng Department of Electronic Engineering, the Chinese University of Hong Kong, Hong Kong
SUMMARY Three-dimensional finite element modeling and analysis are made to simulate the dynamic process of bone drilling during the orthopedic surgery. This study is proposed to evaluate the performance of various surgical tools and the possible pre-operative biohazard. In the simulation, the strain–stress curve of the bone is divided into linear elastic region and nonlinear plastic region according to the strain range. Rigid-plasticity and elasto-plasticity are used as bone material. The performances of twist drill bit and hollow drill bit are evaluated. The results of finite element analysis give different patterns of stress distribution on the two types of bone models and drill bits. The FE simulations show dynamic drilling process that the drill bit penetrates through the bone model. In vitro drilling experiment on porcine femur is conducted to measure the drilling force for the validation of the FEM. Copyright © 2014 John Wiley & Sons, Ltd. Received 22 August 2013; Revised 27 December 2013; Accepted 19 January 2014 KEY WORDS:
finite element analysis; orthopedic surgery; surgical drilling; constitutive equation; porcine femur
1. INTRODUCTION In orthopedic surgery, bone machining operations such as drilling, milling and sawing are similar to those in industrial manufacturing application. However, because of the requirements of critical safety, low damage, minimal invasion and short time cost, various surgical tools with novel functionality have been introduced in the orthopedic surgery, neuro-surgery and dental implant surgery. Engineers and surgeons are being devoted to instrument design and analysis on dynamic process intro- and post-operative procedures. Because the simulation based on finite element analysis (FEA) can act as a possible substitute for high-cost and complex experimental work, especially as a useful tool for validation of analytical results, FEA has been widely used for medical instrument design, evaluation of tissue biomechanics and various surgery processes. In maxillofacial surgery, three-dimensional (3D) FEA was used to evaluate post-operative stress distribution in the fixation plates and screws in the surrounding bone. The biomechanical behavior of implant and bone was simulated under vertical, horizontal and oblique loading forces [1–3]. In oral surgery, FEA has been extensively used as a tool of functional assessments for values or gradient distribution of stress and strain in the field of oral osseointegration [4]. In orthopedic surgery, Battula [5] utilized a 3D FE model to examine the effect of axial push-out and pull-out of self-tapping cortical bone screws, which were inserted to various depths in normal and osteoporosis bone materials. Keyak et al. [6, 7] applied and compared linear and nonlinear finite element models to identify proximal femoral fracture load. Static load was applied on the model with fracture criterion derived from the in vitro experiment. *Correspondence to: Lin Qi, Department of Electronic Engineering, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong. † E-mail:
[email protected] Copyright © 2014 John Wiley & Sons, Ltd.
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However, those previous works mainly focused on small-scale deformation of bone with static load. Elastic model was usually adopted for bone materials. In practice, the biomechanics of bone performed elastic under small strain and plastic under large strain. When the stress applied on bone beyond ultimate stress, fracture will occur. For bone machining operation, such as high speed and high load cutting or milling, the surgical region of bone performs large-scale deformation, local fracture and heat increase with chip separation. Therefore, nonlinear plastic models shall be involved in the finite element method. Paszenda et al. [8] proposed a 3D model in ANSYS workbench to simulate the stress distribution on the drill bit during femur drilling process. The objective of simulation was to evaluate the effect of different geometries of drill bits for optimizing the medical instrument design. The FE model was based on elastic model for simulation on drilling process so the deformation of bone was far from the real operation. Alam et al. [9] took bone as elasto-plastic material on finite element modeling (FEM) for plane bone cutting simulation. Because the proposed model was two-dimensional, the drilling process had to be approximated to the orthogonal plane cutting Chip generation; temperature and stress distribution of simulation were given and compared with the cutting experiment on frozen and dry bovine bone. In this work, several FE based computational studies were made to simulate and predict the mechanical performance and the stress states for bone drilling process. Mechanical analysis was made by applying parameters associated with specific porcine bone and two types of drill bits. For two material models, the constitutive characteristic was based on the rate-dependent stress–strain curve. Drilling experiments on porcine bone were conducted. Drilling force, which was significant surgical information, was recorded for verification of the FEM results.
2. MATERIAL AND METHOD 2.1. Material properties of bone and drill bit for FEM Stress–strain ( ") curve is used to describe the whole biomechanical property of compact bone. For elastic deformation of bone, external load does not cause permanent deformation, but once the yield stress (y ) is exceeded, some deformation is permanent. The drill bit, which we used, is high quality 3Cr13 stainless steel, which is assumed to undergo elastic deformation. Basic elastic parameters of drill bit and femur were listed in Table I [10–13]. In this FE simulation, we used these purely elastic parameters for the elastic deformation. After the yield stress is reached, bone exhibits extensive unrecovered deformation before failing, as indicated by the plastic region on the stress–strain curve. When ultimate stress (ult / is reached along the plastic curve, the bone will be broken 14]. When strain " < 0:006, cortical bone undergoes linear elastic deformation. When strain " > 0:006 with yield stress y (80–100 MPa), cortical bone undergoes nonlinear plastic deformation. When " exceeds 0.025 and > ult (160–180 MPa), the cortical bone fracture occurs [14]. Besides stress and strain, biomechanical behavior of bone varies with the rate at which it is loaded. When loads are applied at higher strain rates ("P), the bone is stiffer and sustains a higher load before failure. This rate-dependent inelastic behavior is called visco-plasticity. Table I. Elastic parameters of drill bit and cortical bone.
Density (kgm3 / Young’s modulus (GPa/ Poisson’s ratio Yielding strength (MPa/ Tension strength (MPa/ Copyright © 2014 John Wiley & Sons, Ltd.
Drill bit
Femur cortical bone
7840 220 0.3 608 1000
2100 17 0.35 110 148
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Figure 1. Fitted curve and the reported experimental data in plastic region. Solid lines are fitted curves, dot points are experimental data reprinted from [13].
Melnis and Knets [13] measured and recorded the tensile stress with strain range [0, 0.025] under five strain rates (105 , 103 , 102 , 101 and 1 s1 / on the middle portion of left tibia of nine male cadavers. The raw rate-dependent stress–strain points of the plastic region were plotted in Figure 1. In this FE simulation, the visco-plasticity model was adopted for plastic deformation. The empirical power law as r T "p ; "Pp ; T D c "np "Pm Cy (1) p T0 was adopted to estimate the stress–strain curve under different strain rates in plastic region, where "p is the plastic strain, "Pp is the plastic strain rate, T is the temperature and T0 is the reference room temperature. c; n; m; r and y are coefficients [15]. In this research, T was considered unchanged as T0 for the plastic deformation, then fitted parameters were calculated as c D 408:796, m D 0:0629271, n D 0:305785 and y D 32:7191 through simplex method. The fitted curves were drawn in Figure 1. Bone drilling was a large-scale deformation process; thus, two types of material properties of bone were studied in FEM [15]. 1. Rigid-plastic objects are modeled depending only on the plastic characteristic, which assumes that stress ( ) increased with strain rate (P") until a threshold strain rate beyond which the objects deformed plastically. The rigid-plastic model performs good simulation of real material behavior and response of the strain rate sensitivity. 2. Elasto-plastic objects are treated as elastic objects until the yield stress is reached. Any portions of the object that reach the yield point are treated as plastic, while the remainder of the object as elastic. The total strain in the object is a combination of elastic strain and plastic strain. Elasto-plastic model provided a realistic simulation for elastic recovery and variation of strain. However, the simulation for elasto-plastic objects always takes extremely long solving time and difficult for convergence. 2.2. Finite element modeling We employed two kinds of drill bits, twist drill bit and hollow drill bit. The twist drill bit, which consisted of cutting lips and flutes (Figure 2(a)), was commonly used for orthopedic surgery. In the experiment and FE simulation, the diameter and lips angle of twist drill bit were 3.2 mm and 120ı . The hollow drill bit (Figure 2(b)) was particularly used to remove bone block or take core samples of bone in post-surgical procedure [16] or in vitro experiment [17]. The hollow drill bit consists of three cutting blades with flutes and inner cored grinder with diameter of 3.2 mm. For analytical computation of drilling force, the process of drilling bone by a twist-bit was always approximated to orthogonal cutting operation [18]. However, different from the twist-bit, the process Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 2. Solid CAD models in SolidWorks®. (a) Hollow drill bit, (b) Twist drill bit (c) Femur shaft model, (d) Hollow cylinder for rigid-plastic material of bone, (e) Disk for elasto-plastic material of bone.
of drilling bone by hollow-bit contained cutting, milling and punching. It is difficult to set up analytical analysis for this complex tool, but FEA is a sufficient tool instead. The effects of these two drill bits were evaluated in FEA and in vitro experiment. The femoral shaft, on which the drilling region located, has a simpler structure; thus, the segment of porcine femur was modeled as a hollow cylinder (Figure 2(d)). Because the thickness of the fresh porcine femur samples varied from 1 to 5 mm, the thickness of the drilling region on 3D model was set as 2 mm for trade-off between accuracy and feasibility. Models of the bone and drill bits were assumed to be homogenous and isotropic. To shorten the simulation time for solving the elasto-plastic model, the bone was modeled as a thin piece of disk with 2 mm height and 8 mm diameter (Figure 2(e)). Geometry models of drill bits and bone were constructed in CAD software SolidWorks® (Dassault Systemes S.A., France), and the CAD models were imported into a FE package DEFORM3D™ (SFTC, USA). Ten-node tetrahedral elements were chosen for mesh generation with dual mesh density technology in order to improve accuracy and reduce solving time. High mesh density was applied on the region of interest (ROI) in order to predict where steep stress or temperature gradient would arise, while low mesh density was chosen for the regions away from ROI with low stress gradient. The length of element edge was set to 0.08 mm for high mesh density and 0.8 mm for low mesh density. The bone model consisted of 273,374 tetrahedral elements for rigid-plastic model and 25,407 tetrahedral elements for elasto-plastic model. The movement configuration (MC) of drill bit included feed speed (Vf ) and spindle speed (!) along Z-axis. For twist drill bit, the MCs were Vf D 12:5 mm/s (MC1) and 1.25 mm/s (MC2) with fixed ! D 600 rpm. For hollow drill bit, the MC was Vf D 12:5 mm/s with fixed ! D 600 rpm (MC1). Because the bone model kept static during the simulation of drilling process, the distal edge of the bone model was fixed on the configuration of boundary condition (Figures 3 and 4). Copyright © 2014 John Wiley & Sons, Ltd.
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Figure 3. Bone was modeled as hollow cylinder with rigid-plastic model. (a) and (b) showed the movement of twist-bit and hollow-bit and boundary configuration on distal edges of bone model.
Figure 4. Bone was modeled as piece of cylinder with elasto-plastic model. (a) and (b) showed the movements of twist-bit and hollow-bit and boundary configuration on circle edge of bone model.
During drilling process, since the contact area between bone’s chip and drill bit undergone ‘stick-slide’ region, it was more complicated to use hybrid contact model including both Coulomb friction and shear friction for the computation [19]. For fast machining and large-scale deformation process, high contact stress between drill bit and bone was generated so that the normal Coulomb model did not exactly reflect the contact friction [9]. In this study, a simple linear shear friction model was used as fs D m k
(2)
where fs is the shear friction, k is the shear yield stress of bone’s chip and m is the friction factor(assumed as 0.3 [9]). Nalla et al. [20] reported that mechanistic fracture of human cortical bone was consistent with strain-controlled failure, which could be taken as ductile damage. The normalized Cockroft and Latham fracture criterion was adopted to predict and determine the time and site of ductile damage initiation and propagation according to the accumulation law: Z "eff max d "N (3) Df D eff 0 Copyright © 2014 John Wiley & Sons, Ltd.
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where Df is the damage value of the material, eff is the effective stress, max is the maximum stress, "eff is the effective strain of fracture and d "N is the effective strain increment [21]. According to Equation (3), when the damage value Df reached the critical value, the material failure occurred, which would cause element deletion and separation from the object in the simulation, and then the chip would be generated while drilling. 3. IN VITRO EXPERIMENTAL VALIDATION 3.1. Experiment setup In this part, we reported on the experiment designed to test the effectiveness of the proposed method. As the object of the experiment, the bone’s biomechanics depended on its composition as well as size and shape of the testing specimens. Ref.[12] and [22] reported that porcine bone best resembled human’s bone on density, geometry size and biomechanics compared with several mammal’s specimens. For validation of FEM, two pieces of fresh porcine femur was chosen for study. Before experiment, muscle and fat were cleaned off, and then 12 holes were drilled on the femurs. Figure 5 showed the experimental setup based on the robot-assisted orthopedic surgery system. A 6 degrees of freedom robot arm TX60 (Stäubli Ltd, France) for hyper-clean application was adopted in this system. The robot can approach 0.1 mm accuracy along three dimensions of translation and rotation. The robot’s movement can be controlled by remote PC via Ethernet. Six-axis Force/Torque transducer mini85 (ATI Industrial Automation Inc., Apex, NC, USA) was mounted on the endeffector of robot arm for measuring drilling force. Force and torque were recorded on real-time through ATI NetBox via Ethernet. Surgical drill with high rotation torque under fixed spindle speed of 600 rpm was used for drilling bone. The extra weight of surgical drill could be removed through Bias function of Force transducer before measurement; the drill bit moved along one degree of freedom (Z-axis), and drilling force FZ was recorded. 3.2. Evaluation of results from experiment and FEA The magnitude and dynamic change of drilling force are the significant information during bone surgery. For FE analysis, the drilling force (FZ in Figures 3 and 4) is the integration of stress distribution of elements near the contact zone. The drilling force from FEA was compared with the results of measurement in experiment. In addition, mean stress (or hydrostatic stress) mean responsible for volumetric changes was also evaluated. The mean is negative as compressive stress and positive as tensile stress.
Figure 5. Force sensing experiment setup based on the robot-assisted orthopedic surgery system. Copyright © 2014 John Wiley & Sons, Ltd.
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4. RESULTS 4.1. Drilling force of FEA and experimental results with twist drill bit Figures 6 and 7 illustrated the recorded drilling force on twist-bit in vitro experiments and from FEA under two movement configurations and two material models. The experiment and FEA results showed that the drilling force increased whereas contact and penetration finally decreased while breakthrough. In the drilling experiment on porcine femur, the thickness of specimen varies from 1 to 5 mm; thus, the time scale of recorded drilling force varies with respect to different thickness and feed speed (Vf /. The fluctuations on force profile of FEA results were due to remeshing and interpolation by finite element solver. Under MC1, the maximum experimental drilling force was
Figure 6. Comparison of recorded drilling force and finite element analysis (FEA) results under feed speed of 12.5 mm/s and spindle speed of 600 rpm. (a) Recorded drilling force on twist-bit along Z-axis in experiment (enlarged time axis). (b) Drilling force on twist drill bit along Z-axis from FEA with rigid-plastic model. (c) Drilling force on twist drill bit along Z-axis from FEA with elasto-plastic model.
Figure 7. Comparison of recorded drilling force and finite element analysis (FEA) result under feed speed of 1.25 mm/s and spindle speed of 600 rpm. (a) Recorded drilling force on twist-bit along Z-axis in experiment (enlarged time axis). (b) Drilling force on twist drill bit along Z-axis from FEA with rigid-plastic model.
Figure 8. Comparison of recorded force and finite element analysis (FEA) result under feed speed of 12.5 mm/s and spindle speed of 600 rpm. (a) Recorded force on hollow-bit along Z-axis in experiment (enlarged time axis). (b) Drilling force on hollow-bit along Z-axis from FEA with rigid-plastic model. (c) Drilling force on hollow-bit along Z-axis from FEA with elasto-plastic model. Copyright © 2014 John Wiley & Sons, Ltd.
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220˙30 N, while maximum forces of FEA results varied from 150 to 170 N. By contrast, under MC2, the maximum experimental drilling force was 20˙5 N, while maximum forces of FEA results varied from 15 to 20 N. Force profile of FEA result under MC1 with elasto-plastic material was shown in Figure 6(c). At the beginning of drilling, the bone underwent elastic deformation with smoother force curve. While the bone underwent plastic deformation, large force fluctuations were caused by remeshing and interpolation of FE solver.
Figure 9. Chip separation and stress distribution of twist-bit drilling simulation from finite element analysis. The feed speed was 12.5 mm/s and the spindle speed was 600 rpm. (a), (b) and (c) showed chip generation of drilling in simulation, stress distribution inside and around the drilled hole from the top view and stress distribution on twist-bit with rigid-plastic model. (d), (e) and (f) showed the respective results with elasto-plastic model. Copyright © 2014 John Wiley & Sons, Ltd.
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4.2. Drilling force of FEA and experimental results with hollow drill bit Figure 8(a) showed the recorded drilling force on hollow-bit in vitro experiment and from FEA for two models. Under MC1, the maximum experimental drilling force was 420˙40 N, while the maximum forces of the FEA varied from 450 to 550 N. For elasto-plastic model, sparse solver engine was always applied to improve solution speed, and accordingly, the simulation process was more sensitive to time-step. The whole process of simulation would require extreme long time and large memory allocation for remeshing and interpolation. Figure 8(c) showed more frequent remeshing with small time-step, while the hollow-bit stepped deeper into the bone model.
Figure 10. Chip separation and stress distribution of hollow-bit drilling simulation from finite element analysis. The feed speed was 12.5 mm/s and the spindle speed was 600 rpm. (a), (b) and (c) showed chip generation of drilling in simulation, stress distribution inside and around the drilled hole from the top view and stress distribution on hollow-bit with rigid-plastic model. (d), (e) and (f) showed the respective results with elasto-plastic model. Copyright © 2014 John Wiley & Sons, Ltd.
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4.3. Preliminary FEA with twist drill bit Three-dimensional simulation was shown in Figure 9 as twist-bit penetrating into the bone model. The main stress distributions were similar for the rigid-plastic and elasto-plastic model, and the chips were separated along the cutting flutes of twist-bit. 4.4. Preliminary FEA with hollow drill bit Figure 10 showed 3D simulation as hollow-bit penetrating into the bone model. The main stress distributed not only on the cutting blades of the hollow-bit but also on the crosssections of the inner grinder (Figure 10(c and f)). The chips were separated along the outer cutting flutes, and the bone block was extracted by inner hollow core of the hollow-bit. 5. DISCUSSION 5.1. Drilling force Because it is difficult to place strain gauges either inside the drilled object or on cutting surface of drill bit, the dynamic stress distribution is hard to be directly measured through experiment. However, from experiment and FEA, it is easier to record and estimate the drilling force, which represents the volumetric sum of shear stress that produced gross deformation. For normal twist-bit, if the thickness of bone is assumed infinite, after penetrating into the single layer workpiece, the drilling force would keep unchanged and linear with material constants(Ks ) and feed rate (a D 2 vf =!) [23] as FZ D Ks a
D ˇ si n 2 2
(4)
where D is the diameter of twist drill bit and ˇ is the angle of cutting lips. As shown in Figures 6–8, the profile of drilling force performed a gradual rise at the beginning of penetration, then arrival of a peak and finally, a gradual fall while the drill bit broke through the bone. Because of the complex structure of hollow drill bit, to the knowledge of the authors, no research report has given the analytical approach of the drilling force. Only one set of movement configurations was applied in FEM and compared with the results of experiment. Because hollow-bit had much more contact area with bone, the magnitude of drilling force was much larger than that of twist-bit. 5.2. Implication of the preliminary FEA Because the drilling process of twist-bit was equivalent to orthogonal cutting, the main stress distributed inside and around drilled hole. From Figure 9(b and c), the largest stress distributed on the contact surface between drill bit and bone. Although there existed differences on the magnitude, the distribution of stress was consistent with the results by research [9]. For twist drill bit, in Figure 9(c and f), the main stress, similar to the results by research [24], distributed on the cutting lips and chisel edges. Because the drilling process of hollow-bit consisted of cutting, milling and punching operations, the main stress distributed both on the contact surface inside the drilled hole and on the whole surface of the bone segment (Figure 10(a, b, d and e)). mean > 0 meant tensile stress, and mean < 0 meant compressive stress. From the FEA results, the magnitude of stress for the elasto-plastic model was larger than that of rigid-plastic model. This was because the yield stress of rigid-plastic material was taken as y D y in Equation (1) as the extrapolation result by the FE solver. However, for elasto-plastic material, the yield stress, which was calculated through Equation (1) by designated yield strain ("y =0.006), was always larger than y. Copyright © 2014 John Wiley & Sons, Ltd.
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5.3. Limitation of the research In this preliminary FEM, because of the long solving time, the number of tests under different movement configurations was limited for further statistical analysis. The complex structure of the selected porcine femurs with varied thickness also caused much fluctuation on the experimental measurements. In this study, the bone model in simulation was assumed as homogenous and isotropic material; the thermal effect between drill bit and bone was omitted either. However, in practice, bone consists of multi-layer osseous tissue such as cortical bone and cancellous bone; in addition, the real bone’s biomechanics varies along different directions. The micro-structure of bone shows porous property, which is compressible visco-plastic. In future work, multi-layers and anisotropic model with thermal effect shall be considered for the simulation, although this will cost tremendous computation time and involve complex experiments for validation in which phantom material shall be applied. The bone-like phantom material, which is taken as the substitute of bone, can be precisely machined and shaped through computer-aided manufacturing. In addition, because phantom material can eliminate unknown interference caused by animal specimens, it is suitable to set up an ideal experiment for testing validation of the FEM. In the further study, it is necessary to construct the object with complex shape and structure, such as the head and neck parts of femur, in FEM based on CT images and 3D reconstruction technology [25]. 6. CONCLUSION In this paper, a general FE based dynamic simulation and analysis of drilling on femur is implemented. In the simulation, the properties of bone material are taken as rigid-plasticity and elasto-plasticity, respectively. The model of simple middle portion of femur is constructed through FEM. As the simulation results show different stress distributions on bone model and two types of drill bits during the processes of drilling, it is helpful to evaluate the performance of surgical tools instead of difficult theoretical work. The dynamic process of drilling and change of stress distribution for orthopedic surgery are visualized. The drilling forces from FEA are compared with the results of in vitro experiment to verify the FEM. The magnitude and variation tendency of drilling force from the FEA are in accordance with those of experimental results. For high speed machining operation and large-scale deformation, the profile of drilling force and the stress distribution of rigid-plastic model are similar to those of elasto-plastic model; in this case, the rigid-plastic model can be used in FEM as approximation for the real process. ACKNOWLEDGEMENTS
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Int. J. Numer. Meth. Biomed. Engng. 2014; 30:845–856 DOI: 10.1002/cnm