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The point collocation method of finite spheres (PCMFS) is used to model the hyperelastic response of soft biological tissue in real time within the framework of virtual surgery simulation. The proper orthogonal decomposition (POD) model order reduction (MOR) technique was used to achieve reduced-order model of the problem, minimizing computational cost. The PCMFS is a physics-based meshfree numerical technique for real-time simulation of surgical procedures where the approximation functions are applied directly on the strong form of the boundary value problem without the need for integration, increasing computational efficiency. Since computational speed has a significant role in simulation of surgical procedures, the proposed technique was able to model realistic nonlinear behavior of organs in real time. Numerical results are shown to demonstrate the effectiveness of the new methodology through a comparison between full and reduced analyses for several nonlinear problems. It is shown that the proposed technique was able to achieve good agreement with the full model; moreover, the computational and data storage costs were significantly reduced.

Minimally invasive surgery (MIS) is becoming the method of choice for most surgical procedures with its many advantages of reduced postoperative pain, shorter hospital stays, quicker recoveries, less scarring, and better cosmetic results [

The success of flight simulators to train pilots has fuelled the enthusiasm for computer-based training systems for surgeons. The goal of surgical simulation is to produce a realistic virtual environment where a trainee can explore in real time a medical procedure of a three-dimensional organ model using his or her sense of vision and touch through specialized haptic interface devices [

Several methods have been proposed for fast computation of mechanical deformation of soft tissues. Early attempts were based on a nonphysical approach, focusing on the visualization aspect of deformation and operation. In [

Mass-spring model is one of the most widely used physical methods in which the material is represented by an array of nodes connected by elastic springs [

However, both models inaccurately simplify the governing equations and offer a very unrealistic behavior [

Linear elastic models are only applicable for small deformations; experimental characterization of soft biological tissue indicates that the behavior of tissues is rather nonlinear. Therefore, a viscoelastic constitutive model is more evocative [

The finite volume method was used in [

Although computerized skill trainers and VR training systems have been developed, none of them has been integrated officially into a medical curriculum or any other official training program or course, and the current technology is inadequate to address the issues of realistic simulation and rendering in such simulators.

Meshfree collocation-based methods [

The goal of this research is the development of physics-based simulation techniques for the modeling of surgical tool-soft tissue interactions, such as deformation, incision, and cutting as well as the reaction forces on the surgical tools, in real time. In [

Different model order reduction techniques were applied to reduce the computational time. The results were promising, but more investigations need to be done. This method has the potential to develop into the de facto standard in future surgical simulators. However, in [

The technology developed in this work is a significant step towards the development of VR-based surgical training systems which will enable medical students and residents to train and practicing surgeons to retrain on complex surgical procedures. The PCMFS is combined with the POD to produce a fast physics-based virtual environment; this will significantly reduce the computational cost which allows for more nonlinear phenomena to be modeled in real time.

The point collocation method of finite spheres is presented in Section

The PCMFS is a computationally efficient technique proposed in [

Schematic of the PCMFS for dynamic surgery simulation. Discretization of a domain

During surgical simulation, the surgical tool interacts with the portion

In the point collocation method [

In PCMFS, the moving least squares [

Biological soft tissues are complicated; they are anisotropic, viscoelastic, and inhomogeneous, and they allow large deformation. Therefore, there is no known constitute model that can capture the exact mechanical and thermodynamical behavior of all tissues. In this work, we are concerned with certain organ, that is, the liver. Liver can be considered to be homogeneous, isotropic, and incompressible because liver tissue is highly consistent with a high water content. A hyperelastic constitute model for liver is widely used [

Most surgical simulations focus on linear elastic models for soft tissue as in [

In [

The Mooney-Rivlin material is a special case of this function where a polynomial form of the strain energy function is used as follows:

Hookean model is the simplest hyperelastic model with

Model order reduction (MOR) methods have been developed for large-scale dynamical systems [

It reduces the number of variables significantly relative to the full-order model.

It is controlled by a limited number of relevant inputs.

It is relatively inexpensive to solve and store in computer’s memory.

Krylov subspace-based methods,

Hankel norm and truncated balancing realization (TBR-) based methods,

Karhunen-Lóeve expansion or proper orthogonal decomposition (POD) methods.

Krylov subspace-based methods are numerically robust algorithms since they preserve a certain number of moments of the transfer function in the reduced model. Therefore, the reduced system approximates well the original transfer function around a specified frequency or collection of frequency points [

The second group of methods is based on the Hankel norm and truncated balancing realization (TBR). Unlike the Krylov subspace methods, these methods have provable error bounds and guarantee that stability of the original system will be preserved in the reduced-order model [

Karhunen-Lóeve expansion or proper orthogonal decomposition (POD) method offers yet another alternative [

Generating a reduced-order model of the high-fidelity original partial differential equation consists of the following two steps.

The first step is to transform the kinematic information, that is, in our case, the displacement field, to a smaller number of modes.

Then, the full-system is reduced to the dynamics implied by the reduced modes.

Let

The main idea of POD is to obtain a basis

The reduced-order model solution

Using direct simulation of the initial value problem in (

Using the POD modes equation (

In order to demonstrate the effectiveness of the proposed technique, we will show 2 examples of hyperelastic models.

In this example, we apply an axial force on the right side of the beam, whereas the left side of the beam is fixed. The beam is 200 mm long with a square cross-sectional area with sides of length 20 mm, as shown in Figure

Beam under traction: concentrated force is applied to the right side, whereas the other side is fixed.

The force ^{3}.

Explicit time integration was used in the full model, and the simulation period was 1 sec where a snapshot was taken every 0.01 sec.

Figure

Average relative error in the solution of the displacement of the beam as the number of basis modes in the POD increases.

It is shown from Figure

Figure

CPU time used in seconds for the solution of the beam problem using the PCMFS with POD model order reduction method.

As shown in the figure, it is noticed that the time scales almost linearly with increasing the number of basis modes for the beam problem.

Figure

Time consumption relative to the full model as the number of POD basis modes increases.

Here, we will consider a circular hyperelastic membrane of a radius ^{3}, Figure

Hyperelastic circular membrane with fixed boundary conditions and concentrated force at the center of the top surface.

Figure

Relative error in the displacement field of the hyperelastic circular membrane as the number of POD basis modes increases.

The CPU time consumption is shown in Figure

CPU time consumed in the solution of the hyperelastic circular membrane as the number of POD basis modes increases.

Figure

Solution of the displacement field of the hyperelastic circular membrane using POD and PCMFS.

In this paper, we have developed a point collocation-based method of finite spheres (PCMFS) approach with POD model order reduction technique for the solution of nonlinear hyperelastic problems that arise in the simulation of soft tissue response. In this technique, an approximation is generated using the moving least squares method, while the point collocation method is used as the weighted residual technique. The advantage of this method is that numerical integration is not necessary which allows for efficient computations necessary for the simulation of applications such as virtual surgery.

POD model order reduction technique, used extensively in the field of large-scale dynamical systems, has been applied to reduce the computational cost associated with nonlinear hyperelastic problems found in soft tissue modeling. POD technique reduces the computational cost by generating an offline set of snapshots used to generate POD basis functions that is used during online computation to reduce the large-order original system by a smaller-order system reducing the computational cost.

The combination of the two methods was able to solve the boundary value problem in real time with relatively acceptable margin of error. The results showed that the computational time for the evaluation of the system matrices increases linearly with the number of nodes. For example, we were able to achieve relatively negligible error with only 8 basis functions with a time consumption of almost 3% of the time used in the full-order model.

The proposed technique can play a vital role in the development of physics-based virtual surgery environment that can account for nonlinear behavior of soft biological tissue; this method can be extended to more complicated material models in which more complicated phenomena such as deformation, incision, and cutting, as well as the reaction forces on the surgical tools, are modeled in real time.