Acta of Bioengineering and Biomechanics Original paperVol. 20, No. 2, 2018 DOI: 10.5277/ABB-01080-2018-02

Three-dimensional biomechanical modeling and simulationof trephine cutting cornea for keratoplasty

PENG SU1, 2, DA LU1, SHIJING DENG3, LEIYU ZHANG4, YUXIN HAO1, YANG YANG5*

1 School of Electromechanical Engineering, Beijing Information Science and Technology University, Beijing, China.2 Key Laboratory of Rehabilitation Aids Technology and System of the Ministry of Civil Affairs,

National Research Center for Rehabilitation Technical Aids, Beijing, China.3 Beijing Institute of Ophthalmology, Beijing Tongren Eye Center, Capital Medical University, Beijing, China.

4 College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing, China.5 School of Mechanical Engineering and Automation, Beihang University, Beijing, China.

Purpose: Trephination is one of the basic operations of keratoplasty, and the biomechanical mechanism of the operation can be re-vealed based on three-dimensional modeling and simulation of trephine cutting cornea. Methods: Based on the analysis of the physicaland biomechanical characteristics of corneal trephination, a three-dimensional numerical model of corneal trephination is built, where thecornea can be simplified to two layers structure including stroma and epithelium, and the trephine cuts the cornea under the verticalmotion load and the rotational motion load. A three-dimensional failure criterion of corneal material is proposed based on the yieldstrength theory. On this basis, trephination simulation is carried out, and the units of corneal material are removed from the model whenthey meet the defined failure criterion. Results: Under the given parameters including the velocity, the angle and the angular velocity, thetrephine force curves, include the linear cutting force and the rotary cutting force are obtained, and show the change of the forces withdisplacement during the process of trephination simulation. The maps of the equivalent stress show the destruction and deformation ofthe cornea. Then, the experiment of robotic trephination is carried out under the same parameters and the effectiveness of the simulationis evaluated. Conclusions: Based on mechanics theory and finite element method, the process of trephine cutting cornea has been repro-duced, and the interaction mechanism is revealed, which lays the foundation for the development of real-time simulation and virtualsystem of the corneal surgery.

Key words: cornea, trephine, biomechanics, failure criterion, finite element simulation

1. Introduction

Keratoplasty as a delicate microsurgical surgeryis using the healthy corneal graft to replace lesionpart of the cornea in order to achieve the purpose ofimproving vision or treating corneal diseases. Thesurgical procedures can be divided into three steps,i.e., trephination, transplantation, and suture. Amongthem, trephination is the most basic operation inmaking corneal bed and graft in keratoplasty, whichplays an important role.

Nowadays, some surgical procedures for trephi-nation are manual operations. The manual operations

have the advantages of controllability, simple opera-tion and low-cost, but the effect of the operationsdepends on experience of surgeon and technologyused and is limited by physiological limitations ofsurgeon. Particularly, manual operation will lead toa loss of endothelial cells at the cutting edge [5], [22],and it leads to non-uniform cutting force, and irregulargraft or bed, which induces postoperative astigmatismand complications [2], [9]. Because of this, the re-searchers evaluated the effects of various trephinationinstruments and methods on the surgery [12], andexplored the use of advanced technologies, such asrobotics technology, laser technology and so on [12],[14], [27].

______________________________

* Corresponding author: Yang Yang, School of Mechanical Engineering and Automation, Beihang University, XueYuan Road No. 37,HaiDian District, Beijing 100191, China. Phone: +86 01082339698, e-mail: yang_mech@126.com

Received: January 5th, 2018Accepted for publication: March 19th, 2018

P. SU et al.24

Robot-assisted keratoplasty is a surgical procedurethat is similar to the traditional manual operation, andit has the advantages of stable operation and high reli-ability. Based on the signals from micro-force sensorand position sensor, the robot manipulator of trephi-nation can judge the organizational state informationand cutting depth, which can effectively improve thecutting precision and make up for the shortcomings ofmanual operation [14], [27]. In the field of surgery,many scholars are also exploring the trephination ro-bot which has the ability to interact with biologicaltissues in real-time [10], [26]. In addition, laser tech-nologies are also increasingly used in resecting cor-neal lesion and making corneal grafts, such as theexcimer laser technology, the femtosecond laser tech-nology and so on [25], [29]. As a new technology, thefemtosecond laser technology has many advantagesover traditional corneal surgery, such as accurate con-trol of the cutting depth and corneal shape, and higherverticality in the edge of the cornea [15]. The studyfound that the open eyes and corrected visual acuitywere significantly improved after surgery. Althoughthe surgery is better using femtosecond lasers, thetechnology still has many problems and limitations,such as insecurity, instability, limitations, economicfactors or other issues [22], [25].

With the development and progress of medicaltechnology, in addition to improving surgical tech-niques and developing surgical manipulator, we shouldalso make clear the biomechanical mechanism of thesurgery [18]. The surgical mechanism can reveal in-teraction of instruments with tissue radically and beused to guide the design of the surgical manipulator[4]. Some researchers simulated the biomechanicaldeformation of corneal tissue using the finite elementmethod, and simulated the damaged mechanism of thetissues of the eyeball in the case of external force act-ing on the eyeball [11], [28]. In addition, the finiteelement simulation on interaction of instruments withtissue can be using to the development of virtual sur-gery training system, and the numerical simulationcan not only be used as a tool for surgical planning,evaluation and judgment of the operation, but alsoestablish the mechanical condition that cannot beachieved in experiments [17], [21].

Because the cornea is a complicated viscoelasticbody, there are many difficulties in researching inter-action mechanism of instruments with corneal tissue,including the deformation of biological tissues inloading, the relaxation effects of tissue penetration,and simulation setting in numerical simulation. In thepreliminary study, a corneal suture test system wasbuilt and the biomechanical model of needle insertion

into cornea was established, and the biomechanicalcharacteristics of was analyzed based on insertion ex-periment and finite element simulation [29]. In most ofthe simulation on the interaction of the instrumentswith biological soft tissue, such as insertion simula-tion [6], [17], [24], the soft tissue is simplified to bea two-dimensional model. Horever, the surgery of cor-neal trephination has a more complex biomechanicalbehavior, and some parameters may influence theresults including linear velocity, rotation angle androtation velocity [22]. It is necessary to establisha three-dimensional numerical model of trephinationin order to reveal the biomechanical response of theoperation effectively.

The interaction mechanism between trephine andthe corneal tissue should be discussed due to the im-portance of corneal transplantation. The finite elementmodel of corneal trephination is established based onthe operation, and the failure criterion of corneal ma-terial is derived using the fourth strength theory. Onthis basis, three-dimensional biomechanical simula-tion on interaction of trephine with corneal tissue isperformed, and the effectiveness of the simulation isevaluated through comparing cutting force curveswith robotic experiment and analyzing the maps ofequivalent stress.

2. Materials and methods

2.1. Finite element modelingof corneal trephination

Considering the operation requirements and char-acteristics of finite element simulation, geometricmodel of eyeball is built according to the physiologi-cal structure [11], [13], as shown in Fig. 1a. In thetrephination experiment, disposable corneal trephineof 7.5 mm diameter was chosen [22], and its geomet-ric model is shown in Fig. 1b.

According to the structure and biomechanical prop-erties of the cornea, the cornea can be divided into fivelayers, including epithelium, lamina elastica anterior,stroma, lamina elastica posterior, and endothelium,and they have different geometric thickness and dif-ferent mechanical properties, such as elastic modulusand Poisson’s ratio. But in a lot of research on cornealmechanical behavior under different loads the corneais simplified as the biological soft tissue that has onlyone layer structure [20]. In the cornea surgical opera-tion, the mechanical response caused by interaction of

Three-dimensional biomechanical modeling and simulation of trephine cutting cornea for keratoplasty 25

rigid instrument and flexible corneal tissue can beused as a reference for accurate control of the cuttingdepth [1], so the model of corneal trephination is es-tablished considering the different corneal layers. Thestroma constitutes about 90% of corneal thickness,which is the main part of the cornea and determinesbiomechanical properties of the cornea. Epithelium,the second thickest layer of the cornea, is generallyconsidered to be a corneal barrier because it is hardand located at the outwards. The other layers are sothin that they may be ignored in finite element simu-lation of corneal trephination, so in order to reduce thedifficulty of calculation, the cornea is simplified totwo layers structure, including stroma and epithelium,as shown in Fig. 1a. Mooney–Rivlin hyperelastic ma-terial parameters of stroma and neo-Hookean hyper-elastic material parameters of epithelium are set up inthe simulation model [13], [23], and the sclera wassimplified as an elastic body that the elastic modulusE = 3.08 MPa and Poisson’s ratio = 0.49 [3].

In the process of cutting the corneal tissue witha trephine, there is vertical and rotational movement.Three-dimensional finite element model of trephinecutting cornea is established as shown in Fig. 2. In themodel, the difference between vertical and horizontalof eye model is ignored, and the eyeball model andthe trephine model are established by the method in-cluding 3D, deformable, solid, and rotation. For ob-servation and analysis, the rotation angle of the corneais 180 degrees, and the rotation angle of the trephineis 360 degrees. Based on the physiological structureof the eyeball, the sclera and cornea are used as thewhole, and the biomechanical response of othereyeball tissues such as the choroid and the retina areignored in the finite element analysis. The intraocularpressure is produced by chamber water of the eyeball,and normal intraocular pressure is from 11 mmHg to21 mmHg. In order to ensure the effectiveness of thesimulation, intraocular pressure is imitated by impos-ing uniform pressure on the inner wall of the eyeball,

and the pressure value is 17.5 mmHg [24]. In addition,in order to improve the computational efficiency, it isassumed that the cornea is an isotropic symmetricstructure on the basis of not affecting the simulationresult [16]. The trephine is set as rigid body.

Fig. 2. Three-dimensional finite element modelof cutting cornea with trephine

In the eyeball model, the lower hemisphere ishinged, and the vertical motion load v is applied onthe center axis of the trephine, and the rotational mo-tion load is applied at the same time including rotationvelocity ω and rotation angle . In order to describethe rotational motion under the constraint condition,a periodic curve function is established. The functionis expressed by Fourier series,

)](sin)(cos[)( 001

00 ttBttnAAAta nn

N

n

(1)

where is the circular frequency, and t0 is the startingtime, and A0 is the initial amplitude, and An and Bn

(a) (b)

Fig. 1. The geometric model (dimension are given in mm).(a) geometric model of eyeball, (b) geometric model of trephine

P. SU et al.26

are cosine and sine coefficients. Setting N = 1, Bn = , = and other parameters are 0. The dynamic ex-plicit analysis step is set in the analysis, and the targettime increment is 1.0 10–5 s.

(a)

(b)

(c)

Fig. 3. The mesh of corneal model. (a) umbrella mesh,(b) vertical mesh; (c) horizontal mesh

Because the cornea is thin and irregular, in orderto change and simplify the topology, three mesheshave been examined based on mesh partition method,as shown in Fig. 3, including the umbrella mesh, thevertical mesh, and the horizontal mesh. As shown inTable 1, the simulation results show that the umbrella

mesh can obtain better convergence, with the simula-tion time of about 68 s, and in variation trends anddata areas, trephine force has better similarity with theexperimental results. Contact algorithm is surface-to-surface contact, in which a rigid material is chosen asthe principal surface to obtain the best contact simula-tion, and the subordinate surface based on node areselected. So, the first contact surface is the trephine,and the second contact surface is the cornea, i.e., thesubdivision region in the Fig. 3a. The contact action istangential contact friction, and the normal pressure ishard contact.

2.2. The three-dimensionalfailure criterion of cornea material

In order to establish the strength conditions incomplex stress states, the researchers have proposedthe assumption and calculation method for the failureof materials under various stress conditions, i.e.,strength theory. The yield or fracture failure of a ma-terial is caused by a factor in stress, strain or strainenergy density, and independent of the stress state.Strength theory can also be divided into fracturestrength theory of brittle material and yield strengththeory of plastic material. In the yield strength theoryincluding the third strength theory and the fourthstrength theory, a failure criterion is defined, and theunit loses the load-carrying capacity and is deletedwhen it meets the failure criterion. The equivalentstress failure criterion, based on the fourth strengththeory, is one of the commonly used failure criteria,and is commonly used in small strain polymeric mate-rials [24], [30].

As a typical biomaterial, cornea has hyperelastic-ity and viscoelasticity. A three-dimensional failurecriterion of cornea material have been studied topropose for interaction simulation of trephine withcorneal tissue based on the yield strength theory.According to the model of corneal trephination, thereare three-dimensional features in the operation, andthe material deformation gradient tensor of the cor-nea, i.e., the stretching tensor F, contains 9 compo-

Table 1. The comparison about different meshes of corneal model

Number of meshes Trephine force comparedwith experimental dataMesh partition

methodtotal trephine area

Simulationtime [s]

variation trend data areaThe umbrella mesh 17215 13008 68 similar similarThe vertical mesh 21064 11709 165 similar smallerThe horizontal mesh 19377 15560 283 difference larger

Three-dimensional biomechanical modeling and simulation of trephine cutting cornea for keratoplasty 27

nents in the three-dimensional state, so it can be de-scribed as

333231

232221

131211

FFFFFFFFF

F (2)

For symmetric tensors, F11, F22, F33 are direct vari-ables, and F12, F13, F21, F23, F32, F31 are indirect vari-ables, and F21 = F12, F32 = F23, and F13 = F31. TheMooney–Rivlin constitutive model of corneal materialcan be described as [23]

220111021 )1(1)3()3(),,( J

DICICJIIWW

(3)

In the above formula, the volume ratio J = det(F).1I and 2I are the first and second invariants of the sym-

metric modified Cauchy–Green tensor [7], wherein

1I = 132

IJ

, 2I = 234

IJ

.Hyperelastic material is characterized by the pres-

ence of a potential strain function W, and it is the po-tential energy of second Piola–Kirchhoff stress tensorS, which exists in

CJ

JW

CI

IW

CI

IWS 2

2

1

12 (4)

In the formula, the right Cauchy–Green deforma-tion tensor C = FT F. Through the derivation, the ex-pression of Piola–Kirchhoff stress tensor can be ob-tained, which is

1

132

10 312 CIJC IS

CCIIIJC 1111

34

01 31

342 I

1)1(2 JCJD

(5)

where I is Unit matrix. Cauchy stress componentis described by the Cauchy–Green strain tensor, andthere is

IBBTTITB

TIBFSFσ

)1(231

342

3121

01

10

JD

CJ

CJJ

T

(6)

In the formula, T is the trace of the right Cauchy–Green deformation tensor matrix, i.e., T = trace ).(BThe corrected right Cauchy–Green deformation tensor

,FFB T in which the modified deformation gradient

can be described as .31

FF

J In three-dimensionalstate, is the 3 3 order matrix, that is,

333231

232221

131211

σ (7)

where 11, 22, 33 are normal stress components, and12, 13, 21, 23, 31, 32 are shear stress components,and 21 = 12, 32 = 23 and 13 = 31.

By calculating the Cauchy–Green stress invariants,the existence of normal stress h and shear stress tcan be expressed as

)3(231

3

121

1

II

I

t

h

(8)

Based on the static equilibrium equations, h andt can be indicated by stress components, i.e.,

)(6)()()(31

3

231

223

212

21133

23322

22211

332211

t

h

(9)

In the above formula, the 11, 22, 33, 12, 23 and 31 are six stress components in the three-dimensionalstate. The equivalent stress can be determined by any stress state directly, and by combining (8) and (9), thefollowing formula can be obtained, which is

)(3])()()[(21 2

31223

212

21133

23322

22211 s (10)

P. SU et al.28

The energy stored due to deformation of an elas-tic body under the action of applied forces is calledstrain energy, and the energy stored in a unit volumeis called strain energy density. According to thefourth strength theory, the strain energy density isthe major factor causing the destruction of materials.No matter what the stress state of the material, thematerial will yield failure as long as the distortionenergy density d reaches its limit value d0. d0 canbe determined by the uniaxial tensile test.

20 ])([

6)1(2

Ed

(11)

In the above-mentioned formula, [] is the ulti-mate stress obtained by the axial tensile test, so theyield limit of the material can be indicated as

0dd (12)

Define the state variable as , and the yield failurecondition can be expressed as

][ s (13)

The formula (13) is the yield condition of thecornea under three-dimensional conditions. Basedon the condition, the failure criterion of cornealmaterial is defined and described in the subroutineVUMAT of ABAQUS. On the basis of the knownstrain increment, the stress increment of the mate-rial is obtained, and the equivalent stress of thematerial is calculated. When the unit of the materialmeets the defined failure criterion in simulation, itloses the carrying capacity and is removed from themodel. The state variable is defined as a constantbefore being compared with the limit stress []. Incalculations, if < [], the state variable is de-fined as 0, which means that the material failure,otherwise, = 1 means that the material is normal.In integral point of material of each time step, themain program will call the subroutine for failurecalculation. Then, in the next time step, if the initialstate variable 0 0, the program will continue toperform the operation, and the judgement of thematerial failure is proceeded.

Problems occuring in the numerical simulationof corneal trephination, are great deformation andchanging contact surface, besides the failure anddestruction of the corneal structure layer. At pres-ent, an explicit solver can be used to deal with thiskind of problem [8], and the material failure andseparation can be achieved using the unit deletiontechnology.

3. Results

The process of corneal trephination involves theinstantaneous three-dimensional status of the interac-tion between rigid body and soft tissue, and it is morecomplex and difficult to calculate, compared to thesimulation of needle insertion into soft tissue. Basedon the established numerical model of corneal trephi-nation and the three-dimensional failure criterion ofcorneal material, the numerical simulation of trephinecutting cornea is carried out, and the changes of tre-phine forces are discussed. Comparison of the resultsof corneal trephination, which is operated by robotswith the same parameters is made and the validity ofthe simulation is verified.

The simulation parameters are set as [, , ] =[20.0, 1.25, 33.5], where is the velocity of the verti-cal motion and the unit is m/s, is the angle of therotary motion and the unit is rad, is the angularvelocity of the rotary motion and the unit is r/min(i.e., revolutions per minute). The simulation is car-ried out in ABAQUS, where simulation time is 70 s,and the change curves of trephine forces with dis-placement are shown in Fig. 4. Figure 4a describesrotating load in the simulation, and the peak value,A point, shows the rotation angle = 1.25 rad. Fig-ures 4b and 4c, present the change curves of the linearcutting force Fz and the rotary cutting force Fc, re-spectively. In the figures, the AB interval is the effec-tive interval for trephination simulation, and thesimulation time is about 60 s. In Fig. 4b, the curve a isthe sampling data, and the curve b is the fitting curveobtained by fitting the sampling data by Gauss inter-polation, and the fitting function g(x) is shown in theformula (14). The curve a shows a fluctuation growthand the stability can be improved by changing theprecision of mesh division and the setting of timeincrement step. At point C, the linear cutting force Fz

reaches the maximum value which is 14.40 10–2 N,and the puncture force Fp is about 2.76 10–2 N. Theepithelium has been punctured at point D, and pointE shows the corresponding linear cutting force at thetime of 38.5 s. From the fitting curve b, the changeregulation of the linear cutting force Fz can beshown, and it shows an obvious decline when thestroma has been punctured at point B. In Fig. 4c,rotating cutting force Fc shows a periodic change andan obvious increasing trend, and the maximum valueof the force Fc is about 3.55 10–2 N. After a stablestate, the force Fc reaches the punctured position,i.e., point B, and it shows the reducing trend, periodic

Three-dimensional biomechanical modeling and simulation of trephine cutting cornea for keratoplasty 29

variation and kept constant. The fitting function ofsampled data is

]70,0[exp)(22

1

xc

bxaxgi

i

ii (14)

in which,

63.1886.3306.004.2236.6210.0

222

111

cbacba

.

(a)

(b)

(c)

Fig. 4. Simulation results of trephination.(a) the relationship between time and rotary angle,(b) simulation curves of the linear cutting force,

(c) simulation result of rotary cutting force

The equivalent stress maps of trephination simula-tion are shown in Fig. 5, which reflects the destructionand deformation of the cornea in the process oftrephination. Figures 5a–e correspond to the points O,A, D, E, and B of Fig. 4, respectively. Figure 5a de-picts the initial state of the simulation. Figure 5b

shows that the trephine contacts and extrudes the cor-neal surface when the simulation time is about 3.5 s,and the obvious stress changes are shown in the wholecornea due to extrusion force and intraocular pressure.Figure 5c shows that the epithelium has been punc-tured when the simulation time is about 7.7 s, and dueto the failure stress and material are different, the twocell layers can be clearly identified from the figure.Figure 5d shows the deformation and destruction ofthe cornea when the simulation time is about 38.5 s.The shade is the initial state of the simulation, and theentity is the instantaneous state of trephination. Bycomparing two statues, it can be shown that the wholecornea and the unconstrained sclera will have oc-curred apparent elastic deformation in the process.Figure 5e shows the equivalent stress state where thestroma has been punctured when the simulation timeis about 62.5 s, and it can be regarded that the wholecornea was completely punctured, and the deforma-tion of the cornea is marked as m. Figure 5f is thestress state when the cornea is penetrated and thesimulation time is about 65 s, and it can be seen thatthe current state of corneal deformation marked as n ismore apparent than the deformation m in Fig. 5e. Itillustrates that the cornea is affected by the intraocularpressure after it has been punctured in the simulation,and will also produce reaction force to the trephine, sothe trephine force will not disappear immediately,which is consistent with the analysis of the trephineforce above.

In order to verify the simulation, the experiment ofrobotic trephination is carried out [22], as shown inFig. 6, where the operating parameters are set up withreference to the simulation parameters. The experi-ment results of robotic trephination are obtained, asshown in Fig. 7. In Fig. 7a, the AB interval is the ef-fective interval of trephination experiment, and thetime is about 90 s, which is longer than the simulationtime. That shows that the deformation of the eyeballin the experiment is greater than the simulation. Atpoint C, The linear cutting force Fz reached the maxi-mum value of about 14.92 10–2 N, and the maximumvalue of the puncture force Fp is about 2.30 10–2 N.These two values are basically close to the values ofthe simulation. The trephine force drops rapidly tonear the zero point after point B, indicating that thecornea has been punctured and has lost carrying ca-pacity of the trephine. In Fig. 7b, the rotary cuttingforce Fc, which is less than 2.40 10–2 N, shows anobvious trend of increase, and it is similar to thesimulation results. In the simulation and the experi-ment, the trephine forces are shown in Table 2, wherethe data correspond to the force curves. Before the

P. SU et al.30

Fig. 5. The equivalent stress maps of trephination simulation, where t is trephination time.(a) corneal pre-deformed status, (b) cutting status of epithelium at t = 3.5 s, (c) corneal epithelium has been punctured,

(d) cutting status of stroma at t = 38.5 s, (e) the stroma has been punctured,(f) broken status of the cornea at t = 65 s

Three-dimensional biomechanical modeling and simulation of trephine cutting cornea for keratoplasty 31

cornea is punctured, it shows corneal deformation thatthe displacements are greater than the corneal thick-ness. Comparing the results of experiment with that ofsimulation, it is shown that the change and data area ofthe trephine forces are in agreement. The modeling andsimulation on interaction of trephine with cornea arecorrect and effective, and the simulation can reflect thebiomechanical characteristics of the experiment.

Fig. 6. The experiment of robotic trephination.(a) the robot of corneal trephination,

(b) experiment of robotic thephination

(a)

(b)

Fig. 7. The experimental results of the robotic trephination.(a) experimental curves of the linear cutting force,

(b) the result of rotary cutting force

Table 2. Statistics on the trephine forces, where simulation dataof the linear cutting force Fz is the fitting value

and experimental data of the force Fz is the mean value

Simulation ExperimentTime[s]

Displacement[m] Fz (N) Fc (N) Fz (N) Fc (N)

0 0 0 0 0 03.5 70 0.72 0 0 0.017.7 154 0.81 0.16 0.78 0.0330 600 7.31 2.77 5.70 1.07

38.5 770 8.80 3.43 5.75 1.3050 1000 10.04 3.55 8.82 1.62

58.1 1162 11.05 3.50 9.05 1.6562.5 1250 14.40 3.28 10.00 2.1265 1300 10.78 3.05 10.05 2.31

90.3 1806 9.65 3.12 12.83 2.4092.5 1850 9.53 3.08 11.21 2.3295 1900 9.59 3.16 1.31 0.01100 2000 9.60 3.10 1.06 0.01

4. Discussion

A three-dimensional model of interaction of tre-phine with corneal tissue is established, and it is a newattempt to confirm that the cornea consists of differentstructural layers that have different geometric thick-ness and different mechanical properties. On this ba-sis, the finite element model is established where themesh of the cornea is refined based on partitionmethod of umbrella mesh that has better performancethan vertical or horizontal mesh. Simulation resultsdemonstrate that the umbrella mesh improves the effi-ciency and convergence of the finite element simula-tion. In the simulation, the motion of the trephinecutting cornea is defined by a periodic curve function,i.e., Fourier series, and these loads generate the verti-cal velocity, the rotation velocity and the rotation an-gle. It simulates the surgeon’s operation, including thevertical motion and the rotational motion.

In the research of trephine cutting cornea, it is es-sential to explore the assumption and calculationmethod for the failure of materials under mechanicstheory. A three-dimensional failure criterion of cornealmaterial is proposed based on the yield strength theory,where the three-dimensional form of the equivalentstress can be evaluated on the basis of mathematicdeduction. Based on the criterion, the units of cornealmaterial are removed from the model due to the lossof the carrying capacity when they meet the definedfailure criterion. And the results show that the crite-rion may be used for corneal material within a certainrange of accuracy.

P. SU et al.32

By three-dimensional simulation of corneal trephi-nation under determined simulation parameters, thechange curves of trephine forces with cutting displace-ment are obtained during the process. Among them, thelinear cutting force continues to increase due to thereaction force caused by squeezing the eyeball. Al-though the punctured cornea lose the carrying capacitydue to the disappearance of constraints in the simula-tion, the failure unit and the normal unit still give tre-phine friction force and reaction force in the process ofsimulation, so the force did not fall to zero positionafter the cornea has been punctured. The rotary cuttingforce shows a periodic change and an obvious increas-ing trend. Then, the equivalent stress maps of the finiteelement simulation are analyzed, which correspond tothe marked points of the trephine forces curves, andthey show the failure process of the corneal material.

The experiment of robotic trephination is carriedout under the same parameters to verify the simula-tion. The results show that the simulation results arebasically consistent with the experimental results invariation trend and data area of trephine forces, andthe simulation can reflect the biomechanical behaviorof the operation.

The research has reproduced the process of theinteraction of trephine with cornea, and the biome-chanical model of corneal trephination is the basis forthe surgery simulation and training system based onforce feedback. Further studies are necessary to ex-plore the real-time simulation of the surgery based onfinite element methods and sensor technology.

Acknowledgements

The author wishes to thank Young Elite Scientists Sponsor-ship Program by CAST (Grant No. YESS20160065), TrainingPlan of Young Top-notch Talent by Beijing Municipal EducationCommission (Grant No. CIT&TCD201704063), and the Fund ofBeijing Key Laboratory for Photoelectric Measurement Technology(Grant No. GDKF2017001) that supported this work.

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