The Development of Nonlinear Viscoelastic Model for the Application ...

Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, Oct 29 - Nov 2, 2007

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The Development of Nonlinear Viscoelastic Model for the Application of Soft Tissue Identification Hongbin Liu*, Student Member IEEE, David P. Noonan, Student Member IEEE, Yahya H. Zweiri, Member IEEE, Kaspar A. Althoefer, Member IEEE, Lakmal D. Seneviratne, Member IEEE

Abstract—This paper proposes a novel nonlinear viscoelastic soft tissue model generated from ex vivo experimental results on ovine liver using a force sensitive probe. In order to study the bio-mechanics of soft tissue, static indentation tests were applied on ovine liver. An empirical constitutive equation was extracted from the examined data. A mechanical model combining linear viscoelasticity with a nonlinear function of strain–stress is proposed. The developed model has been evaluated both statically and dynamically with different strain rates – i.e. where the velocity of indentation is varied. By comparing simulation results and measured experimental data, it has been concluded that the proposed model is robust for modelling both static and dynamic indentation conditions. The effect of changing boundary conditions on the parameters in the proposed model has been studied by choosing test sites with different underlying tissue thicknesses. The results indicate that for small strain, the effect of the thickness condition is reasonable to be neglected.

I. INTRODUCTION

T

HIS paper presents research conducted to develop a novel mechanical soft tissue model capable of precisely simulating the nonlinear stress relaxation of liver tissue. The purpose is to implement the model for tissue identification purposes to provide a surgeon with the ability to classify tissue properties during Minimally Invasive Surgery. Over recent years, there has been a tremendous development in the field of surgical robotic devices. The system most widely integrated into clinical practice is the daVinciTM system from Intuitive Surgical. With its enhanced end-effector motion, magnified stereoscopic vision and EndowristTM technology, it allows surgeons conduct delicate operations within a constrained space [6, 7]. However, while these represent significant improvements over existing techniques, the surgeon is still severely hindered by a complete loss of tactile and haptic feedback. For example, the ability of a surgeon to palpate tissue when searching for tumours, diseased tissue, arterial blockages or other tissue *H. Liu is with the Department of Mechanical Engineering, Kings College London, UK (phone: +44 20 7848 1862; e-mail: [email protected]). D. Noonan is with the Department of Mechanical Engineering, Kings College London, UK ( e-mail: [email protected]). Y. H. Zweiri is with Deparment of Mechanical Engineering, University of Mu΄tah, Karak, Jordan (e-mail: [email protected]). K. Althoefer is with the Mechanical Engineering Department, King’s College London, Strand, London, WC2R 2LS, UK (e-mail: [email protected]). L. D. Seneviratne is with the Mechanical Engineering Department, King’s College London, Strand, London, WC2R 2LS, UK (e-mail: [email protected]).

1-4244-0912-8/07/$25.00 ©2007 IEEE.

abnormalities is absent. A major area of research to overcome the lack of tactile feedback in Robotic Minimally Invasive Surgery is that of soft tissue property identification. Such information can be utilized for improving surgical simulations [8, 9], haptic modeling [10] and the identification of tissue abnormalities in solid organs. Models which can accurately represent the biomechanical characteristics of soft tissue are essential for soft tissue identification. In this paper, in the footsteps of the forerunners in Biomechanics, we present work for the further development of a simple mechanical model which integrates linear viscoelasticity theory with a nonlinear strain-stress incremental law to describe the mechanical properties of biological tissue. II. BACKGROUND A. Biomechanics of Biological Soft Tissue Nonload-bearing biological soft tissues are well known for their highly nonlinear characteristics and viscoelasicity. Many soft tissues are anisotropic, heterogeneous, nearly incompressible, have porous internal structure, and variable mechanics depending on the environment such as pH, temperature, health etc. Due to their viscoelastic nature, when held at constant strain, they show stress relaxation. When held at constant stress, they show creep. Their stress-strain relationship is incrementally nonlinear with strain. They exhibit hysteresis loops in cyclic loading and unloading. Under repeated cycles, they show preconditioning which is a steady state where the stiffness and hysteresis stabilize in successive cycles. The biomechanics of soft tissue is time and strain rate dependent [2, 5]. They are difficult to be characterized due to their inherent complexity, the degradation of mechanical properties after death and poorly known boundary conditions [1, 2]. B. Linear Viscoelasticity The development of the mathematic theory of linear viscoelasticity is based on a “superposition principle” [4]. This implies that the strain at any time is directly proportional to the stress. The general differential equation for linear viscoelasticity is expressed as follows in [4]: (1+ α1

∂ ∂2 ∂n ∂ ∂2 ∂m + α2 2 + αn n )σ = (β0 + β1 + β2 2 + βm m )γ ∂t ∂t ∂t ∂t ∂ t ∂t

(1)

where n=m or m-1, γ is strain, σ is stress, αi, βi are constants. Linear viscoelastic mechanical models are popularly used to describe the behavior of biological tissues. In mechanical

208

models, Hookean elasticity is represented by a spring and Newtonian viscosity by a dashpot. The basic models include the Voigt (spring and dashpot in series), Maxwell (spring and dashpot in parallel), and Kelvin (spring in parallel with a Maxwell) models [1-4]. By adding more elements to basic models, more complicated models can be obtained. Mechanical linear viscoelastic models have been used to model biological soft tissue by numerous researches over recent years. Liu and Bilston [14] implemented the Maxwell model when examining ex vivo bovine liver; Farshad et al. utilized the Kelvin model [16] for ex vivo kidney; Kerdok et al. used a Genelarized Voigt model for perfused ex vivo porcine liver [12, 13]. Kim et al. conducted in vivo tests on porcine liver and modeled using the Kelvin model [17]. In rheological theory, Roscoe described that all models, irrespective of their complexity, can be reduced to two canonical forms as shown in Fig.1 (without spring k1) [11]. Subsequently, Fung added a spring to each of the canonical forms to correct these models for biological soft tissue (shown in Fig.1). They are named as generalized Kelvin body and generalized Maxwell body [2].

uniaxial force-displacement characteristics of soft tissue. Han et al. [20] presented an exponential equation to describe deformations in human breast tissue, extending earlier work proposed by Fung. Introduced by Fung [2], the quasilinear viscoelasticity mathematical form is the most popular form of constitutive model found in the literature for biological tissues to date [8], despite its limitation with respect to low strain-rates (0.06-0.75% strain/sec) and inability to capture the effects of preconditioning [5, 15]. III. DEVELOPMENT OF NONLINEAR VISCOELASTIC MODEL A. Ex-Vivo Test Rig The stress relaxation of soft tissue was evaluated using a force sensitive cylindrical probe (6mm in diameter) to measure mechanical responses of the tissue post-indentation. Ovine liver has been chosen as test sample, and a stainless steel probe of diameter 6mm as the instrument interacting with the liver. A testing facility was required to ensure that the probe was under accurate and repeatable control at all times. To facilitate this, the probe was attached to the distal tip of a Mitsubishi RV-6SL 6-DOF robotic manipulator. An ATI MINI40 Force/Torque sensor (calibration SI-20-1, resolution 0.01N with 16-bit DAQ) was mounted at the interface between the probe and the manipulator end-effector as shown in Fig.2. This allows for the measurement of the interaction force imparted by the tissue onto the probe.

Fig. 1. The generalized Maxwell body (1) and Kelvin body (2). The

If d/dt is written by the symbol D, then the differential equation of generalized Kelvin body of order n+1 is [2]: f n+1 (D)F = gn+1 (D)u,

(2)

where f n+1 (D) = f n (D)(1+

kn+1 D), bn+1

gn+1 (D) =g n (D)(1+

kn+1 D) + kn+1 f n (D)D. bn+1

Fig. 2. Schematic of ex-vivo probe test rig

The generalized Maxwell Model of order n+1 is expressed as n+1

F =∑ i =1

D u, D / ki +1/ bi

(3)

where F is the force, u is the deformation, ki, bi are the elasticity and viscosity respectively (the coefficients of dashpots start from b2). Because the linear theory is applicable only to small changes in strain [1-4], numerous researchers have developed empirical formulae to predict the nonlinear stress-strain characteristics for large deformation of soft tissue [18-20]. Okamura et al. [18] proposed a second order polynomial function to fit experimental data for bovine liver, while Ahmadian et al. [19] developed empirical equations to predict

B. Static Indentation Tests and Curve Fitting For measuring the viscoelastic properties of ovine liver, static indentation tests were conducted. In order to obtain consistent results without pre-conditioning the tissue, each test was completed in a different position. It was ensured (as much as was possible) that the boundary conditions were identical for each test. Three ovine livers overlaid on top of each other and a homogenous region (approximately 20mm in diameter, average thickness 62mm) on the left lobe of the top liver was chosen as the test site. For simulating the situation when soft tissue is subjected a step constant deformation, the probe indented into the liver with high speed (50mm/s). The indentation depth was varied from 1.0mm to 10mm in 0.5mm increments and the indentation depth was kept constant for 20 seconds.

209

The measured stress curve at each indentation depth has been fitted in Matlab using low order formulae of Eq.2 and Eq.3 (the meaning of all symbols in this paragraph are shown Eq.2, Eq.3 and Fig.1). It was found that the generalized Kelvin body of 2nd order (Kelvin Model) typically fitted the measured curve, but the error becomes considerable at deep indentation depths. It was found that a 3rd order generalized Kelvin body without spring k1 (Dual Maxwell Model) was the most suitable model in terms of maximum accuracy with minimal complexity. The stress relaxation forms of the Maxwell, Kelvin and Dual Maxwell models are expressed as Eq.4, Eq.5 and Eq.6 respectively.

F = k 2u ⋅ e

−

k2 t b2

F = k 2u ⋅ e

k2 − t b2

k3

k2

(4)

.

F = k1u + k 2 u ⋅ e

The curve fitting results are shown in Table II. Additionally, it was also found that the ratios of k2/b2, k3/b3 remain constant as the indentation depth changes (k2/b2 = 0.71, k3/b3= -0.014).

k2 − t b2

+ k 3u ⋅ e

k2/k3

(5)

. k3 − t b3

(6)

where t is time, F is force, u is the deformation, ki, bi are the elasticity and viscosity respectively shown in Fig.1. Fig. 3. Spring coefficient k2 and k3. Dashed line: the fitted curve of k3; solid line: the fitted curve of k2. The ratio of k2/ k3 is a nonlinear decay

TABLE I SSE OF FITTED CURVE USING KELVIN MODEL EQUATION AND DUAL MAXWELL MODEL EQUATION Indent Kelvin Model Dual Maxwell Depth 10mm 0.9658 0.7339 8mm

1.024

0.928

6mm

0.6108

0.5857

4mm

0.2116

0.2014

D. Constitutive Equation & Dual Maxwell Model Based on curve fitting results, an empirical constitutive equation which describes the tissue stress relaxation under constant strain was extracted from the examined data. The stress relaxation behaviour of ovine liver is expressed as:

Table I shows the SSE of curve fitting for the Kelvin Model equation and the Dual Maxwell Model equation. The results show that Dual Maxwell Model is superior to Kelvin Model in terms of representing ovine liver stress relaxation.

C. Analysis of Curve Fitting Results As previously described, the stress relaxation of liver can be expressed using Eq.6. Analysis of coefficients at each indentation depth showed that k2 and k3 increment nonlinearly with indentation, as shown in Fig.3. The ratio of k2 over k3 is not constant but undergoes a nonlinear decay as the indentation increases. Fig.3 indicates that the nonlinear elastic response of biological tissue is significant for large deformations. Therefore a formula which accounts for this nonlinearity is required. The polynomial P(y) = P0+P1y+P2y2+P3y3 (where y represents indentation depth) has been applied to fit both the response of k2 and k3 for increasing indentation depths. TABLE II POLYNOMIAL CURVE FITTING RESULTS Curve fitting k2-depth k3-depth

Equation:

P(y) = P0+P1y+P2y2+P3y3

P0

P1

P2

P3

0 0

12.24 24.56

0 52.55

251000 590800

σ ( y, t ) =

(a p1 y + a p 2 y 2 + a p 3 y 3 )e bt + (c p1 y + c p 2 y 2 + c p 3 y 3 )e dt (7) πr 2

where σ is stress, y is the indentation depth, t represents time (s), r is the probe radius (m), ap1=12.24, ap2=0 ap3=251000, cp1=24.56, cp2=52.55, cp3=590800, b=-0.71 , d=-0.014 . r=0.003.

Fig. 4. Dual Maxwell Model with nonlinear stress-strain functions

According to Eq.7, the stress relaxation of ovine liver can be described as the sum of two exponentials at any indentation depth. To obtain this two Maxwell models are arranged in parallel. To account for realistic biomechanical behaviour, two nonlinear elasticity factors (P(u), Q(u)), are added to each linear Maxwell Model to cope with large deformations. Consequently a new model has been developed, as shown in Fig.4. The differential equation of the nonlinear Dual Maxwell Model has been deduced from Eq.3 and is expressed as:

210

f +(

b1 b2 & b1b2 && bb bb + )f + f = [ P(u)b1 + Q(u)b2 ]u& + [ P(u) 1 2 + Q(u) 1 2 ]u&& k1 k 2 k1k 2 k2 k1

(8)

capable of simulating the stress relaxation behaviour of ovine liver accurately.

where f is the force, u is the deformation, ki, bi are the elasticity and viscosity respectively, k1 =12.24 N/m, k2=24.56 N/m. b1=15.92 Ns/m, b2=1787.8 Ns/m. P(u) and Q(u) are nonlinear elasticity factors and expressed as: P(u)=P1u+P2u2+P3u3 (P1=12.24, P2=0, P3=25100); Q(u)=Q1u+Q2u2+Q3u3(Q1=24.56,Q2=52.55,Q3=5.9*105).

TABLE III RMSE OF THE MODELING RESULTS FOR DIFFERENT INDENTATION DEPTHS

Under constant deformation, u=y, the stress relaxation of the nonlinear viscoleastic model is expressed as:

f = P( y)k1 ⋅ y ⋅ e

−

k1 t b1

+ Q( y )k 2 ⋅ y ⋅ e

−

k2 t b2

(9)

Under linear deformation (u=Ht), the predicted tissue response is given as: k1

f = P( Ht )k1 ⋅ H

k2

− t − t b1 b (1 − e b1 ) + Q( Ht )k 2 ⋅ H 2 (1 − e b 2 ) k1 k2

(10)

E. Error Analysis for Static Indentation In order to investigate the accuracy of the Dual Maxwell Model, the simulation results of each indentation level were compared with the corresponding measured test results. The input to the simulation is a constant which corresponds to the specific indentation depth being examined.

Fig. 5. Dual-Maxwell model with nonlinear function of deformation. The solid lines represent the simulated curve.

Fig.5 shows the measured force signals from 2mm to 10mm indentation and the corresponding simulation results for each indentation. All simulations and tests were run for 20 seconds. Table III shows the RMSE of modelled curves. It can be seen that the developed Dual-Maxwell Model is

10mm

9mm

8mm

7mm

6mm

5mm

4mm

3mm

0.030

0.033

0.038

0.040

0.034

0.020

0.016

0.019

IV. DUAL MAXWELL MODEL FOR DYNAMIC INDENTATION

A. The Modalities of Dynamic Indentation Experiments The biomechanical properties of soft tissue are time and strain rate dependent. The history of the strain will affect the stress. In order to examine the capability of the mechanical nonlinear model in terms of modeling the strain rate dependency, dynamic indentation experiments were carried out. Two modalities have been used: 1. Stress vs. different strain-rate, i.e. the probe indented ovine liver at different speeds. 2. Stress vs. cyclic loading/ unloading, i.e. the probe repeatedly indented the same location with a constant speed. B. Stress vs. Different Strain-Rate To avoid the change of mechanical properties of the liver due to lost water content and the preconditioning of soft tissues, all dynamic experiments were carried out on new ovine liver. In order to obtain comparable results, the boundary condition and location of the test sites on the liver were chosen as similar as was possible for each indentation test. Consequently, a recalibration of parameters was not required. The protocol of the experiments was to indent the probe into the liver to a depth of 4.5mm and hold at that depth for 20 seconds. This was repeated at different speeds. The five different indentation speeds used were: A=2mm/s, B=4mm/s, C=9mm/s, D=14mm/s and E=18mm/s. To validate the Dual Maxwell Model for the strain-rate dependency, experimental data and modelling results have been analysed.

Fig. 6. Modeling of different indentation speed. A=2mm/s, B=4mm/s, C=9mm/s, D=14mm/s and E=18mm/s.

Fig.6 shows the simulation results at the five different speeds of indentation. Fig.7 shows the comparison between the modelling results and experimental data when the testing protocol employed was that the probe indented 4.5 mm at a

211

speed of 2mm/s then was left for 20 seconds. Table IV shows the RMSE of modelling results for dynamic indentation.

shows the comparison between the modeling results and experimental data (RMSE=0.005). The error analysis shows that the simulated force for each cycle is in good agreement with the experimental data. Thus the developed model is able to, at least, predict the beginning stage of the process of preconditioning the liver. V. THE INVESTIGATION OF THE BOUNDARY CONDITION

Fig.7. the comparison between the simulated data and experimental data when the probe indented to a depth of 4.5 mm at speed of 2mm/s and left for 20 seconds. TABLE IV RMSE OF THE DUAL MAXWELL DYNAMIC MODELLING RESULTS Speed

SpeedA

SpeedB

SpeedC

SpeedD

SpeedE

RMSE

0.031

0.031

0.029

0.033

0.033

The error analysis demonstrates that the predicted force curve at each speed level is in good agreement with the corresponding measured force curve. Thus it can be concluded that the proposed model is able to model the effect of varying strain-rate on stress.

The parameters of the developed mechanical model are dependent on the probe dimensions and the boundary condition of the soft tissue under examination, especially the non-linear strain-stress relationship which would be significantly affected by the thickness of the test sites. Therefore, the set of parameters proposed in this paper are only applicable to experiments conducted in a similar manner. In order to investigate the effect of soft tissue thickness on the non-linear strain-stress relationship, experiments using the static indentation protocol have been conducted on 6 test sites which have different thickness. The thicknesses are: T1=28.8mm, T2=38.4, T3=49.9, T4=53.5, T5=53.8, T6=61.7. Another two livers have been used for this set of tests to demonstrate the generality of the developed model.

C. Stress vs. Cyclic Loading and Unloading If a biological tissue is tested by imposing a cyclically varying strain, the stress response will show a hysteresis loop with each cycle, but the loop decreases with succeeding cycles, rapidly at first, then tending to a steady state after a number of cycles [2]. The aforementioned process is known as ‘preconditioning’. In order to examine the capability of the developed model to account for the preconditioning of liver, cyclic loading/unloading tests were conducted.

Fig.9.The plot of k2 of each different thickness condition

Fig.8. the comparison of the modeling results (dashed line) and experimental data (gray) for the first 10 cycles.

The protocol employed was to indent the probe to a depth of 9mm at a speed of 4.5mm/s repeatedly for 40 cycles. The experimental data shows that steady state has not been reached even after 40 cycles. For the purposes of this paper only the first 10 cycles were taken into consideration. Fig.8

Fig.10.The plot of k3 of each different thickness condition

Curve fitting which was described in Section III C was repeated. It was found that the ratios of k2/b2, k3/b3 were similar to values shown in Table II (the average of k2/b2 = 0.73, the average of k3/b3= -0.015). In order to better understand the effect of k2 and k3 on the strain-stress relationship, further investigation was conducted on them.

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[5]

Fig.9 and Fig.10 show the nonlinearity of k2 and k3 for each thickness condition. It can be seen from Fig.9 and Fig.10 that, for shallow indentations (below 4mm), the plot of both k2 and k3 are similar for all the thickness conditions. However, the plots increasingly diverge as the indentation depth increases. With regards to large indentations, no conclusions can be drawn from this experiment. This demonstrates the difficulties associated with changing boundary conditions when attempting to develop soft tissue models. This is especially pertinent when considering that in-vivo boundary conditions are unknown. Consequently, a generic soft tissue model is extremely difficult to be developed. While this preliminary study on the effect of thickness does not lead directly to generic model, it does indicate that for small strain (< 6%), the effect of the thickness condition is negligible.

[11]

VI. LIMITATIONS OF THE DUAL MAXWELL MODEL

[12]

There are several limitations to the proposed mechanical model. Firstly, the model is not able to completely recover its original state after the compressive load releases. This means that although, the Dual Maxwell Model could precisely model the tissue behaviour under compressive deformation, it is not suitable in terms of modelling the tissue behaviour after the compressive load is released. For completeness, a specific model to deal with deformation recovery is required. Secondly, this model is not a generic physical-based model but an empirical curve fitted model. Therefore recalibration is required whenever the boundary conditions change. The approach of calibrating the model could potentially be time consuming. Thirdly, all testing was carried out ex-vivo when the tissue is not in its natural physiological state. Since the mechanical properties would clearly change after death, it is anticipated that all the parameters in this model would have to be recalibrated using a similar test protocol in-vivo. However, the developed simple mechanical model does show the capability of modelling the mechanical properties of soft tissue such as nonlinear elasticity, stress relaxation, strain-rate dependency and preconditioning. By fine-tuning the parameters in vivo, this model could be potentially used in the application of surgical simulation and tissue identification. The limitation of model regarding changing boundary conditions can be overcome by identifying healthy tissue during preliminary tests, and then calculating model parameters at that location for subsequent comparison to suspected diseased tissue [21]. A surgeon could potentially utilise this model during MIS to probe organs which were previously unreachable due to operating constraints

[6] [7] [8] [9] [10]

[13] [14] [15]

[16]

[17] [18] [19] [20] [21]

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