An Efficient Soft Tissue Characterization Method for Haptic Rendering of Soft Tissue Deformation in Medical Simulation Bummo Ahn and Jung Kim School of Mechanical, Aerospace & Systems Engineering, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, Korea
[email protected],
[email protected] Abstract The modeling of soft tissue behavior is essential in haptic rendering for virtual reality based medical simulators, which provide a safe and objective medium for training medical personnel. This paper presents a measuring device, modeling and effective characterization method of soft tissue’s material properties. Experiments were conducted on five porcine livers using an indentation device and ramp-and-hold trajectory. The soft tissue was assumed to be a continuous, incompressible, homogeneous and isotropic solid and a nonlinear constitutive model based on the observation of the force-displacement behavior of soft tissues. The constitutive model was fitted to the experimental data using the LevenbergMarquardt optimization algorithm combined with a three-dimensional nonlinear finite element simulation. Scripting software was used to automatically characterize the mechanical properties and to repeat the FE simulation until the simulated and experimental responses matched.
Schwartz et al. estimated the Young’s modulus of cervine livers with an in vitro indentation device [6]. Ottensmeyer et al. obtained the in vivo mechanical properties of porcine abdominal organs with TeMPeST 1-D [7]. Valtorta et al. investigated the characteristics of the porcine liver using a torsional resonator device (TRD) [8]. Kalanovic et al. introduced ROSA-2 (rotary shear applicator-2), which can measure the in vivo and in vitro mechanical properties of porcine and bovine abdominal organs [9]. Mazza et al. characterized the material properties of the human uterine cervix with an aspiration pipette [10]. This paper developed an efficient characterization algorithm of the soft tissues, reducing the number of the target parameters in order to reduce the computing time for characterization. The method iteratively updates the parameters by minimizing the errors between finite element (FE) model predictions and experimental responses.
2. Methods 2.1. Experimental setup and experiments
1. Introduction A Virtual Reality (VR)-based medical simulation system [1, 2], which provides an innovative tool for training medical personnel, requires accurate organ geometry, fast computation algorithms, and mechanical properties of the soft tissue. While accurate organ geometry and computation algorithms have been widely studied [3, 4], the characterization of material properties has not been sufficiently investigated due to the difficulty of testing and modeling the complexities of nonlinearity and time dependency of the soft tissue. Several research groups have performed soft tissue characterization and modeling. Carter et al. measured the in vivo Young’s modulus of the porcine liver and spleen with two different indentation devices [5].
Figure 1 shows the experimental setup to record the force-displacement-time data from the tissue response. The device is composed of an electromechanical motor, a linear guide, a force transducer and an indenter. The force transducer's resolution is 0.7 mN and its measurement range is from 0.7 mN to 9.8 kN. The maximum positional errors of the system are less than 5 µm after careful calibration of the motor position. The indenter's displacement range is up to 41.5 mm, and the maximum speed is 8 mm/s. The device can induce unit step, sine, rectangular and saw tooth wave inputs of 10 µm ~ 41.5 mm displacements. The porcine liver was selected as a target organ because of its similarity in structure and function to the human liver. Fresh livers were obtained from a local abattoir, which were preserved in an icebox and delivered to the laboratory within a few hours after
G(t) is a scalar function of time and often can be expressed by the Prony series, N P t G(t ) = G0 (1 − ∑ g i (1 − exp(− ))) , G (0) = G0 (2)
PC
i =1
Indenter
P i
Controller
Motor Linear guide
Force transducer Indenter
Liver
Figure 1. Experimental setup and indenter extraction to avoid dehydration. The experiments were conducted on five subjects using a one-dimensional indentation device and three data sets were used to characterize the liver model’s parameters. The indenter was brought into contact with the liver’s surface, and induced the ramp-and-hold input to the liver. Then the liver’s dynamic response was measured with the data acquisition board.
2.2. Material models Although the haptic rendering is based on linear elastic models with small deformation, most medical procedures induce large deformation by instruments. Therefore, the tissue behavior under large strains should be described by the nonlinear elasticity theory and the quasi-linear viscoelasticity (QLV) framework proposed by Fung [11] was used for the modeling. This QLV model has been successfully applied to a variety of tissue types, such as lower limb tissue [12], ligaments /tendons [13] and a porcine aortic valve [14]. This approach assumes that material behavior can be decoupled into two parts: a linear viscoelastic stressrelaxation response and a time-independent elastic response. These behaviors can be determined separately from the experiments. The stresses in each tissue, which may be linear or nonlinear, are linearly superposed with respect to time. The three dimensional constitutive relationship in the QLV framework is given by t ∂ S e ( E ( λ )) S (t ) = G (t ) S e ( 0 ) + ∫ G (t − τ ) d τ (1) τ ∂ 0 where S(t) is the second Piola-Kirchhoff stress tensor and G(t) is known as the reduced relaxation function. λ is the stretch ratio and E(λ) is the strain tensor. Se(E(λ)) is termed the material’s pure elastic response and can be nonlinear or linear. The reduced relaxation function
τi
where g and τi are the Prony series’ parameters and reduced relaxation time constants respectively. For the time-independent elastic response, the hyperelastic material representation works was selected. Usually, its parameters can be determined by the strain energy function (W). Ideally, W is defined with only the parameters that are required to make an FE model. The incompressible neo-Hookean model was selected for use here, which is widely used in soft tissue simulations. The strain energy function of the three dimensional incompressible neo-Hookean model is given by [15] W = C10 ( I1 − 3) (3) where C10 is a mechanical parameter having a unit of stress, and I1 is a principal invariant. To take into account the boundary conditions and estimate the material parameters to the modeling, the inverse FE model parameter optimization algorithm has been applied. The algorithm uses FE simulation iteratively to find the parameter fitted to the experimental results. The entire characterization process for all parameters takes several iterations to converge, and each FE simulation requires much time for the simulated results to match the experimental results; for this reason, this method’s computational time is expensive. Instead of estimating all required material parameters in a single step, the characterization process was separated into two steps. The first step determined the viscoelastic parameters from the normalized force response against the rampand-hold indentation. With the viscoelastic parameters estimated in the first stage, the inverse FE model parameter optimization algorithm was used to determine the remaining elastic parameters. The validity of estimating these parameters was separately investigated by Richard et al. [16]. They found that the stress relaxation response could be determined without regard to the particular form of the constitutive law. This separation of parameters is also similar to the work by Kauer et al [17]. However, they fixed the stress relaxation’s time constants (0.1s, 1s, 10s and 100s), fit their magnitudes, and then determined the rest of the parameters using an inverse FEM calculation. The present study estimated the parameters in the viscoelastic model directly from the normalized forcedisplacement data in the experiments. Characterization procedures in this study are illustrated in Figure 2.
2.4. Estimation of the hyperelastic parameters
Normalized viscoelastic parameters Initial hyperelastic parameters
Write the FE model input file
New parameters
The characterization algorithm compares the simulated forces from the FE simulation to the experimental forces. Hence, it is possible to approach the coefficient of determination (R2) to one given by
R2 = 1−
FE model Optimization algorithm
simulation
Experimental results
∑
e i2
Close enough? No
Yes
Final result
Figure 2. Flow chart for the Inverse FE model parameter optimization algorithm
2.3. Estimation of the viscoelastic parameters This section develops a three dimensional viscoelastic model of the liver from the forcedisplacement experimental data. This method has been successfully used to model a human fingerpad [18]. The second order standard linear solid model as a linear viscoelastic model is
G (t ) = (k0 + k1e −t / τ 1 + k 2e −t / τ 2 )
(4) where ki, τi and t are the rigidity modulus, reduced relaxation time constant and given time respectively, which can be determined by using a nonlinear least square method. This approach allows the rigidity modulus to be expressed as a Prony series expansion in the time domain in Eq. (2).
G (t ) = G0 (1 − g1P (1 − e − t / τ1 ) + g 2P (1 − e − t / τ 2 )) = (k 0 + k1e − t / τ1 + k 2e − t / τ 2 )
(5) From the G(t = 0), G(t = ∞) conditions, the Prony series parameters are ki , (i = 1, 2) (6) g iP = k0 + k1 + k2 Table 1 lists the organ’s computed Prony series parameters, which can be used to directly represent viscoelastic modeling in many FEM codes. Table 1. The Prony series parameters from the normalized experimental data Exp. No.
τ1 (sec)
τ2 (sec)
g1p
g 2p
1
1.376
49.369
0.348
0.358
2
1.087
52.545
0.353
0.365
3
0.944
51.698
0.408
0.288
∑
m
(( Fs ( t i ) − Fe ( t i )) 2 /( Fe (t i )) 2 ) (7) t i = (t i , t 2 , " , t m ) where Fe, Fs, ti and m are measured forces, simulated forces, time and total number of data, respectively. Among several optimization algorithms that could be used, the nonlinear least square optimization known as the Marquardt-Levenberg algorithm [19] is adopted. It updates the parameters iteratively depending on the norm of JTJ and the Marquardt parameter λ. H = ( J T J + λI ) (i ) ∂F ∂F (8) J = [ s , s "] ∂p1 ∂p2 G G G p ( i+1) = p ( i ) + H −1[ J T ⋅ ( f e − f s ( p (i ) ))] G Here, H, p( i ) and J are the Hessian, estimated parameters and Jacobian vectors for the corresponding iteration (i) respectively. Because the parameter is contained implicitly in the FEM simulation, the Jacobian vector J should be computed numerically with respect to each parameter variation. This requires perturbing one parameter, running the entire FEM simulation, and measuring the perturbation’s effect. The entire characterization process takes several iterations to converge, and each FEM simulation requires much time for the simulated results to match the experimental results; for this reason, although the Levenberg-Marquardt algorithm has been used successfully in finite strain applications [20, 21], the entire process is computationally expensive. Therefore, we focus on only the hyperelastic parameters, which can reduce both the number of parameters and the estimation time. This approach also avoided potential numerical errors, due to poor scaling of the Jacobian vector, which would have arisen as a result of the 10,000 to 100,000-fold difference between the relaxation time constants and the elastic constants. The model was assumed to be a continuous, isotropic, homogeneous and incompressible solid, and the deformations imposed were small compared to the organ’s size under an axisymmetric condition. The model was built in Altair HyperMesh 7.0 and simulated with ABAQUS/Standard 6.5.1. The inverse FE model parameter optimization algorithm was implemented in the MATLAB script. After selecting the initial parameters and the first simulation, the program reads the output files and computes the next parameter set. It i =1
then generates new input files with the updated parameters and starts a new simulation. The program is synchronized with the FEM simulation, hence it repeats automatically until the algorithm determines the parameters that satisfy the relation between the simulated and experimental forces as (9) R 2 > 0 .99
3. Results The viscoelastic parameters were estimated from the normalized force–displacement profiles in the experiments. These computed parameters were fed into the ABAQUS database for the nonlinear elastic parameters. The initial hyperelastic parameters were estimated from a prior knowledge of the experiments. The parameter reached convergence after two or three iterations, as shown in Figure 3. Table 2 presents the initial and estimated parameters for the neo-Hookean model which was 287.5 ± 53.7 Pa. Figure 4 shows the stress and deformation contours of the developed FE model. Figure 5 shows the predicted forces from the FE simulation with the estimated parameter and experimental results. The force responses of the hyperelastic model and experimental data are almost the same from the value of R2.
Figure 3. Convergence of the material parameter (MPa)
Indenter
Indenter
(mm)
4. Discussion This paper shows an algorithm for effectively characterizing soft tissue’s material parameters for haptic rendering of medical simulations. The viscoelastic and hyperelastic material parameters were estimated in two stages in the QLV framework. To characterize the parameters to the experimental results, an FE model was developed to simulate the forces at the indenter, and an optimization program was also developed that updates new parameters and runs the simulation iteratively. The inverse FE model parameter optimization algorithm can be extended easily to include more general features of soft tissue behavior. The approach may be extended to increasingly complex organs by building layered organ models and estimating each layer’s material properties. The material model presented in this study offers two basic uses in haptic rendering for VR-based medical simulations. First, it can be used directly in the simulations to compute visual deformations and interaction forces that are displayed in real time. Second, the mathematical model presented here can be used as a standard for the evaluation of new real-time algorithms for computing deformation.
Figure 4. Simulation of the FE model in ABAQUS
Data 2 Data 3
Data 1
Figure 5. Force responses of the simulation and experiments
Table 2. The initial and estimated parameters for the neoHookean model, and coefficient of determinant for each experimental data Estimated parameters Exp. Initial parameters R2 No. C10(Pa) C10(Pa)
1 2 3
207.9
246.1 320.7 295.7
0.9951 0.9967 0.9937
Acknowledgements This work has been supported by a Basic Research Fund of the Korea Institute of Machinery and Materials (KIMM).
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