Pneumoperitoneum simulation based on mass-spring-damper models for laparoscopic surgical planning

Abstract.

Laparoscopic surgery, which is one minimally invasive surgical technique that is now widely performed, is done by making a working space (pneumoperitoneum) by infusing carbon dioxide (${\mathrm{CO}}_2$) gas into the abdominal cavity. A virtual pneumoperitoneum method that simulates the abdominal wall and viscera motion by the pneumoperitoneum based on mass-spring-damper models (MSDMs) with mechanical properties is proposed. Our proposed method simulates the pneumoperitoneum based on MSDMs and Newton’s equations of motion. The parameters of MSDMs are determined by the anatomical knowledge of the mechanical properties of human tissues. Virtual ${\mathrm{CO}}_2$ gas pressure is applied to the boundary surface of the abdominal cavity. The abdominal shapes after creation of the pneumoperitoneum are computed by solving the equations of motion. The mean position errors of our proposed method using 10 mmHg virtual gas pressure were $26.9\pm 5.9\mathrm{mm}$, and the position error of the previous method proposed by Kitasaka et al. was 35.6 mm. The differences in the errors were statistically significant ($p<0.001$, Student’s $t$-test). The position error of the proposed method was reduced from $26.9\pm 5.9$ to $23.4\pm 4.5\mathrm{mm}$ using 30 mmHg virtual gas pressure. The proposed method simulated abdominal wall motion by infused gas pressure and generated deformed volumetric images from a preoperative volumetric image. Our method predicted abdominal wall deformation by just giving the ${\mathrm{CO}}_2$ gas pressure and the tissue properties. Measurement of the visceral displacement will be required to validate the visceral motion.

1. Introduction

Laparoscopic surgery, which is a minimally invasive surgical technique that is now widely performed,^1–3 is done through small incisions as opposed to the larger incisions needed in abdominal laparotomy. Laparoscopic surgery is performed by making a working space (pneumoperitoneum) by infusing carbon dioxide (${\mathrm{CO}}_2$) gas into the abdominal cavity.⁴ A laparoscope is inserted through a trocar to view the surgical site in the abdominal cavity after the pneumoperitoneum process. Laparoscopic surgery provides patients many advantages against open surgery, including less pain, shorter recovery times, and improved quality of life due to the smaller incisions. Furthermore, laparoscopic surgery reduces the risk of infection by reducing the exposure of internal organs. It has clear advantages for patients.^1–3,5 However, the procedure is more difficult than traditional open surgery,^6–8 because surgeons must operate in the limited space of the abdominal cavity. A laparoscope also strongly limits the viewing space of the surgical site. There are several additional disadvantages, such as poor depth perception and indirect tactile sense with forceps. Therefore, substantial experience and skill are required to perform laparoscopic surgery.^9–11

Surgical simulators are generally used to train surgeons in specific types of procedures without using animals or cadavers before working on live patients.^12–14 Although such surgical training systems enable surgeons to simulate surgery on the virtual organs of virtual or generic patients, it is impossible to perform patient-specific simulation, which needs to reproduce anatomical variations. The next generation of surgical simulation is expected to use a volumetric image of an actual patient to enable surgeons to rehearse a procedure and understand the anatomy, the disease, or the injury site of a specific patient.^15–17 Surgical planning is usually performed on a preoperative volumetric image. Determining trocar position is one important part of laparoscopic surgical planning to provide a good triangulation between tools as well as a good laparoscope viewpoint to perform laparoscopic surgery reliably and safely.^18–20 However, the preoperative volumetric images are not deformed by ${\mathrm{CO}}_2$ gas infusion. Since the pneumoperitoneum distends and separates the abdominal wall from the surgical site, the trocar position determined during laparoscopic surgical planning becomes inconsistent with the actual position in the operating room. This inconsistency limits the viewing space and decreases accessibility of the surgical site.

Only two groups have proposed simulation methods of the pneumoperitoneum for laparoscopic surgical planning.^21,22 Kitasaka et al.²¹ proposed a method based on node-spring models to perform a virtual pneumoperitoneum. Their method also generates a volumetric image deformed by a virtual pneumoperitoneum from the preoperative volumetric image. The deformed volumetric image is quite useful to check the trocar positioning and to understand the patient’s anatomy, since the volumetric image contains such internal tissue information as adjacent organs that may be involved with the disease and the anatomical variations of blood vessels. However, their method ignores masses, external forces, and motion dynamics. The shape of the deformed model depends on the various parameters that are empirically chosen by visual inspection. Moreover, their method does not take into account the visceral movements. Bano et al.²² also proposed a pneumoperitoneum simulation method based on a SOFA simulation engine.^23,24 SOFA is an open-source framework whose primary target is real-time simulation. Their method can successfully simulate the movements of the abdominal wall and the viscera. However, it only deforms a volumetric mesh, so the surgeon does not obtain a volumetric image deformed by the virtual pneumoperitoneum. Their method cannot provide internal tissue information or detailed information of patient anatomy, such as small blood vessels. Although this work is validated by porcine data, there is no evaluation using human data. Oktay et al.²⁵ utilized their method to align pre- and intraoperative three-dimensional images.

In this paper, we propose a virtual pneumoperitoneum method that simulates the abdominal wall and the viscera motion by the pneumoperitoneum based on mass-spring-damper models (MSDMs)^26,27 with mechanical properties. The preliminary version of this method was proposed by our group.²⁸ Our proposed method can predict abdominal wall deformation by just giving ${\mathrm{CO}}_2$ gas pressure and tissue properties based on anatomical knowledge. Additionally, we generate a deformed volumetric image from a preoperative volumetric image. In Sec. 2, we describe our proposed method in detail. Our experimental results and discussion are described in Secs. 3 and 4.

2. Method

2.1. Overview

The purpose of our proposed method is the generation of a volumetric image after the pneumoperitoneum from a preoperative volumetric image. The generated volumetric image enables surgeons not only to determine a trocar position that has high tool accessibility and a good viewpoint but also to understand the patient’s anatomy after the pneumoperitoneum by volumetric image visualization. The proposed method simulates the pneumoperitoneum based on MSDMs and Newton’s equations of motion that describe the behavior of the abdomen. MSDMs are widely used to simulate the mechanical behavior of deformable objects in structural dynamics.^29,30 Figure 1 shows a MSD system. The proposed method consists of four procedures: (a) volumetric mesh generation, (b) volumetric mesh deformation, (c) volumetric image reconstruction, and (d) volumetric image visualization. In the following subsections, we describe them in detail.

Fig. 1.

Mass-spring-damper system: Mass $m$ attached to a spring with spring stiffness $k$ and damper with damping coefficient $c$. $f$ denotes an external force.

2.2. Volume Mesh Generation

2.2.1. Segmentation

It is necessary to segment the abdominal wall and the visceral regions from a preoperative volumetric image to create volumetric meshes. The abdominal wall region consists of skin, muscle, and fat. The visceral region consists of various organs, fat, blood vessels, etc. Figure 2 shows some manually segmented major abdominal organs, most of which are soft tissues and have different mechanical properties. These regions have very similar intensity on a volumetric image. Automatic segmentation of them is a crucial task, not only in the simulation of the pneumoperitoneum but also in other medical applications. In this paper, we manually extract a boundary surface that divides the abdominal wall and the visceral regions. To create volumetric meshes, we utilize these two regions instead of all the abdominal organs for simplification.

Fig. 2.

Example of abdominal wall segmentation: (a) abdominal wall and viscera. Yellow (Y) and blue (B) regions indicate abdominal wall and viscera, respectively. (b) Some major organs are included in abdominal wall and viscera.

2.2.2. Volumetric mesh

We create volumetric meshes in this step. First, we divide the volumetric image into a set of cubes by equal grid intervals. Each cube consists of $L\times L\times L$ voxels. Next, we remove the cubes that do not contain abdominal walls or viscera that were extracted in the previous step to reduce computational cost, since these cubes are beyond the scope of deformation. We assign an additional cube, if a cube contains both an abdominal wall and viscera. Figure 3 shows an example of cube allocation. The yellow and blue cubes include the abdominal wall and the viscera, respectively. The green cubes include the boundary surface of the abdominal cavity and are doubly allocated. We create fixed cubes colored purple. None of the nodes in the purple cubes moves. MSDMs are constructed from these cubes. We consider the vertices of a cube as the nodes having mass. The springs and dampers are assigned to the edges and the diagonals on the surfaces of a cube. One cube is divided into five tetrahedrons by this tessellation (Fig. 4). Each tetrahedron consists of four nodes and six springs and dampers. The natural length of a spring is the distance between two connected nodes. These mechanical parameters will be detailed in the following subsections.

Fig. 3.

Example of cube allocation: Yellow (Y) and blue (B) cubes include abdominal wall and viscera, respectively. Green (G) cubes include boundary surface of abdominal cavity. Therefore, they are doubly allocated. We created fixed cubes colored purple (P). No nodes included in purple cubes move.

Fig. 4.

Mass, spring, and damper allocation: there are two types of allocation patterns (a) and (b) since the diagonals are shared by adjacent cubes.

2.3. Volumetric Mesh Deformation

2.3.1. Equation of motion

Our proposed method estimates the abdominal shape after the pneumoperitoneum by the numerical integration of the equations of motion of all the nodes. The equation of motion of node $i$ at time $t$ is given by

$$m_i{\ddot{\mathit{x}}}_i(t)=\sum _{j\in N_i}{c}_{ij}{\dot{\mathit{x}}}_{ij}(t)+\sum _{j\in Ni}{k}_{ij}[{\mathit{x}}_{ij}(t)-{\mathit{x}}_{ij}(0)]+{\mathit{f}}_i(t),$$ (1)

where $m_i$, $k_{ij}$, and $c_{ij}$ are the mass, the spring stiffness, and the damping coefficient, respectively. Determination of these mechanical parameters is described in Sec. 2.3.2. ${\mathit{x}}_{ij}(t)$ and ${\dot{\mathit{x}}}_{ij}(t)$ denote the relative position and velocity of nodes $i$ to $j$ at time $t$, respectively. ${\ddot{\mathit{x}}}_i(t)$ means the absolute acceleration of node $i$ at time $t$. $N_i$ is a set of nodes connected to node $i$ and ${\mathit{x}}_{ij}(0)$ is the initial relative position. ${\mathit{f}}_i(t)$ is an external force added to node $i$ at time $t$. For simplification, we assume that all nodes have the same mechanical properties in this paper. In the following part, we simply denote mass $m_i$, spring stiffness $k_{ij}$, and damping coefficient $c_{ij}$ as $m$, $k$, and $c$.

2.3.2. Mechanical parameters

Although MSDMs quickly compute structural dynamics in general, the deformation accuracy is low. The major difficulties in building realistic MSDMs are the estimation of the spring stiffness and the damping coefficient. Therefore, our proposed method computes these parameters based on the following mechanical properties: density of tissues $\rho$, Young’s modulus $E$, Poisson’s ratio $\nu$, and damping ratio $\zeta$.

We define mass $m$ of a node as

$$m=\rho V,$$ (2)

where $\rho$ and $V$ are the density of the tissues and the volume based on the local region of a node, respectively. Volume $V$ can be computed using grid interval $L$ and the acquisition parameters of the CT image. The paper assumes that the tissue Young’s modulus $E$ is uniform. Therefore spring stiffness $k$ is computed by a method proposed by Lloyd et al.,³¹ which estimates spring stiffness $k$ from the Young’s modulus $E$ and Poisson’s ratio $\nu$ based on a comparison between MSDMs and finite element models (FEMs) with tetrahedron elements. According to their paper, when Poisson’s ratio $\nu$ is 0.25, spring stiffness $k$ is given by

$$k=\frac{8\sqrt{2}}{105}lE,$$ (3)

where $E$ is the Young’s modulus of the deformable objects and $l$ is the average edge length that is estimated from the volume of a tetrahedron element. See their paper for a detailed explanation of the above equations.³¹ Damping coefficient $c$ is given by

$$c=2\zeta \sqrt{mk},$$ (4)

where $\zeta$ means the damping ratio that characterizes the oscillation decay. These parameters can be determined by the anatomical knowledge of the mechanical properties of the human body.^32–34

2.3.3. External force

The external forces, which are applied to the boundary surface of the abdominal cavity, are calculated based on the virtual ${\mathrm{CO}}_2$ gas pressure and the area of the boundary surface. However, we need the external forces applied to each node to solve the equations of motion. In this paper, we compute external force ${\mathit{f}}_i(t)$ applied to node $i$ using external forces applied to the boundary surface.

External force ${\mathit{f}}_i(t)$ on node $i$ is given by

$${\mathit{f}}_i(t)=\sum _{j\in B_i}\frac{1}{3}{\mathit{F}}_j(t),$$ (5)

where ${\mathit{F}}_j(t)$ is the external force that is applied to triangle patch $j$. Here, the triangle patch is one of the surfaces of a tetrahedron and faces the abdominal cavity. $B_i$ shows a set of triangle patches that have node $i$ as a vertex. ${\mathit{F}}_j(t)$ is given by

$${\mathit{F}}_j(t)=P{S}_j(t){\mathit{n}}_j(t),$$ (6)

where $P$ is the virtual ${\mathrm{CO}}_2$ gas pressure and $S_j(t)$ and ${\mathit{n}}_j(t)$ are the area and the normal vector of triangle patch $j$. $S_j(t)$ and $n_j(t)$ are computed by the positions of the three nodes included in the triangle patch $j$. The external force needs to be updated when the positions of the nodes on the boundary surface change. Note that the external forces of the nodes that are not on the boundary are zero.

2.3.4. Dynamic analysis

The position of each node is determined by solving the equations of motion for all the nodes. We utilize the Newmark-$\beta$ method³⁵ to solve them. The Newmark-$\beta$ method, which is widely used in the numerical evaluations of the dynamic response of structures, solves the equations of motion by dual loops.²⁷ The inner loop is the computation of the acceleration at a certain time; the outer loop represents the numerical integration of the equations of motion over the integration time. First, we explain the inner loop of the Newmark-$\beta$ method, which calculates the absolute position and the velocity of node $i$ at time $t+\mathrm{\Delta }t$ by

$${\mathit{x}}_i(x+\mathrm{\Delta }t)={\mathit{x}}_i(t)+{\dot{\mathit{x}}}_i(t)\mathrm{\Delta }t+(0.5-\beta ){\ddot{\mathit{x}}}_i(t)\mathrm{\Delta }{t}^2+\beta {\ddot{\mathit{x}}}_i(t+\mathrm{\Delta }t)\mathrm{\Delta }{t}^2,$$ (7)

$${\dot{\mathit{x}}}_i(t+\mathrm{\Delta }t)={\dot{\mathit{x}}}_i(t)+(1-\delta ){\ddot{\mathit{x}}}_i(t)\mathrm{\Delta }t+\delta {\ddot{\mathit{x}}}_i(t+\mathrm{\Delta }t)\mathrm{\Delta }t,$$ (8)

where $\mathrm{\Delta }t$ means the time interval. $\beta$ and $\delta$ are the parameters of influence for stability and accuracy. Since absolute acceleration ${\ddot{\mathit{x}}}_i(t+\mathrm{\Delta }t)$ is unknown at time step $t+\mathrm{\Delta }t$, we use ${\widetilde{\ddot{\mathit{x}}}}_i(t+\mathrm{\Delta }t)$ instead of ${\ddot{\mathit{x}}}_i(t+\mathrm{\Delta }t)$ in these two equations. At the beginning of this time step, ${\widetilde{\ddot{\mathit{x}}}}_i(t+\mathrm{\Delta }t)$ is initialized to ${\ddot{\mathit{x}}}_i(t)$. ${\widetilde{\ddot{\mathit{x}}}}_i(t+\mathrm{\Delta }t)$ is updated by

$${\widetilde{\ddot{\mathit{x}}}}_i(t+\mathrm{\Delta }t)=\frac{\sum _{j\in N_i}c{\dot{\mathit{x}}}_{ij}(t+\mathrm{\Delta }t)+\sum _{j\in N_i}k[{\mathit{x}}_{ij}(t+\mathrm{\Delta }t)-{\mathit{x}}_{ij}(0)]+{\mathit{f}}_i(t+\mathrm{\Delta }t)}{m}.$$ (9)

These updates are repeated until ${\widetilde{\ddot{\mathit{x}}}}_i(t+\mathrm{\Delta }t)$ converges. The converged value is the solution of ${\ddot{\mathit{x}}}_i(t+\mathrm{\Delta }t)$.

Furthermore, this internal loop is continued to the next time step until the accelerations of all the nodes become nearly zero. This is the outer loop of Newmark-$\beta$.

2.4. Volumetric Image Reconstruction

Our proposed method deforms a volumetric image based on the geometric relationship between the tetrahedrons of before and after deformation. Homogeneous coordinate ${\mathit{p}}^{\prime }$ of the inner point $\mathit{p}(t)$ of tetrahedron $j$ at time $t$ is given by

$${\mathit{p}}^{\prime }=\left[\begin{array}{c}\mathit{p}(t)\\ 1\end{array}\right]=\sum _{i\in C_j}{w}_i{\mathit{x}}_i^{\prime },$$ (10)

where $w_i$ is the weight of the linear interpolation to compute inner point ${\mathit{p}}^{\prime }$ from the vertices of the tetrahedron and $C_j$ is a set of four nodes consisting of tetrahedron $j$. These weights satisfy $\sum _{i\in C_j}{w}_i=1$ and $0\le w_i\le 1$. ${\mathit{x}}_i^{\prime }$ is a homogeneous coordinate of vertex $i$ and is given by

$${\mathit{x}}_i^{\prime }=\left(\begin{array}{c}{x}_{1i}^{\prime }\\ x_{2i}^{\prime }\\ x_{3i}^{\prime }\\ 1\end{array}\right)=\left[\begin{array}{c}{\mathit{x}}_i(t)\\ 1\end{array}\right],$$ (11)

where $x_{ki}^{\prime }$ are the elements of ${\mathit{x}}_i^{\prime }$ and ${\mathit{x}}_i(t)$ is the position of vertex $i$ at time $t$. The equation for inner point ${\mathit{p}}^{\prime }$ can be represented as

$${\mathit{p}}^{\prime }=\left(\begin{array}{cccc}{x}_{11}^{\prime }& x_{12}^{\prime }& x_{13}^{\prime }& x_{14}^{\prime }\\ x_{21}^{\prime }& x_{22}^{\prime }& x_{23}^{\prime }& x_{24}^{\prime }\\ x_{31}^{\prime }& x_{32}^{\prime }& x_{33}^{\prime }& x_{34}^{\prime }\\ 1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{w}_1\\ w_2\\ w_3\\ w_4\end{array}\right)=M^{\prime }\mathit{w}.$$ (12)

Therefore, $\mathit{w}$ is given by

$$\mathit{w}=M^{\prime -1}{\mathit{p}}^{\prime }.$$ (13)

The corresponding homogeneous coordinate $\mathit{p}$ of inner point $\mathit{p}(0)$ in the same tetrahedron of the preoperative volumetric image is represented using the same weights. The intensity of the volumetric image after deformation can be easily computed from the preoperative volumetric image by

$$\mathit{p}=\left[\begin{array}{c}\mathit{p}(0)\\ 1\end{array}\right]=\sum _{i\in C_j}{w}_i{\mathit{x}}_i=\left(\begin{array}{cccc}{x}_{11}& x_{12}& x_{13}& x_{14}\\ x_{21}& x_{22}& x_{23}& x_{24}\\ x_{31}& x_{32}& x_{33}& x_{34}\\ 1& 1& 1& 1\end{array}\right)\left(\begin{array}{c}{w}_1\\ w_2\\ w_3\\ w_4\end{array}\right)=M\mathit{w}=M{M}^{\prime -1}{\mathit{p}}^{\prime },$$ (14)

where ${\mathit{x}}_i$ is a homogeneous coordinate of vertex position ${\mathit{x}}_i(0)$. Consequently, we obtain a deformed volumetric image at time $t$.

2.5. Visualization

To assist the determination of the trocar positioning in laparoscopic surgical planning, we display a two-dimensional projection of the volumetric image, which is obtained by the above virtual pneumoperitoneum process from any arbitrary viewpoint using a volume rendering technique.³⁶ Volume rendering is useful to check trocar positioning and understand a patient’s anatomy, since the volumetric image contains internal tissue information and surgeons can observe the disease and blood vessels within the translucent organs.

2.6. Evaluation Scheme

We evaluate our method by an evaluation scheme proposed by Oda et al.³⁷ Human validation is performed using a real patient and its preoperative CT image. In this work, abdominal wall movement between real and virtual pneumoperitoneums is compared. Thirteen points on the abdominal surface are defined to evaluate different deformation according to the positions on the abdominal surface. These 13 corresponding points are measured at four different stages; (a) preoperative real patient, (b) intraoperative pneumoperitoneum real patient, (c) preoperative CT image, (d) virtual pneumoperitoneum CT image. The value of deformation of real pneumoperitoneum is obtained using (a) and (b). The value of deformation of virtual pneumoperitoneum is obtained using (c) and (d). In addition, to evaluate the point displacement, we apply the point-based rigid registration to match real patient coordinate system and preoperative CT image coordinate system using (a) and (c). Here, the point displacement refers to how much each point is offset to the right or the left. We describe the detail of the evaluation scheme in the following part.

First, we measure the thirteen corresponding points on the abdominal surface of a real patient and the preoperative volumetric image. We locate thirteen marks on the body surface to define these points. Figure 5 shows the location definition of the thirteen points utilized in our experiments. These points are utilized in the registration and evaluation processes described below. We utilize an optical positioning sensor, Polaris Spectra (Northern Digital Inc., Waterloo, ON, Canada), to measure 13 points ${\mathit{q}}_i$ on a real patient in the operating room. Corresponding points ${\mathit{p}}_i$ on the volumetric image are measured by in-house software. Figure 6 shows the examples of the quantitative measurements of a real pneumoperitoneum obtained for actual patients in operating rooms. Since the two point sets represent the same points on the abdominal surface, these points satisfy

$${\mathit{q}}_i=\mathit{R}{\mathit{p}}_i+\mathit{t},$$ (15)

where $\mathit{R}$ and $\mathit{t}$ are a rotation matrix and a translation vector that project a point from the volumetric image coordinate system to the optical positioning sensor coordinate system. Rotation matrix $\mathit{R}$ and translation vector $\mathit{t}$ can be obtained using point-based rigid registration proposed by Horn.³⁸

Fig. 5.

Thirteen corresponding points measured in the evaluation: Points of a real patient are measured by optical position sensor. Corresponding points of a preoperative volumetric image are measured by in-house application.

Fig. 6.

Measurements of real pneumoperitoneum in the operating room. (a) Surgeon measures 13 points corresponding on abdominal surface using an optical positioning sensor. (b) and (c) Measurement before and after pneumoperitoneum.

After registration of the two coordinate systems, a pneumoperitoneum is performed to elevate the abdominal wall of the real patient in the operating room. We deform the preoperative volumetric image by the proposed method.

We also measure the same 13 points ${\mathit{q}}_i^{\prime }$ on the abdominal surface of a real patient after pneumoperitoneum by the optical positioning sensor in the operating room. Their corresponding points ${\mathit{p}}_i^{\prime }$ are easily computed by the geometric relationship between the tetrahedrons of the before and after deformation described in Sec. 2.4. In the experiments, we evaluate two types of errors. One is the positional error that shows the average distance between the true and simulated positions. The other is the displacement error that shows the average displacement to the simulated positions from the true positions. Position error $E_{\text{Position}}$ is defined as

$$E_{\text{Position}}=\frac{1}{N}\sum _i^N{\Vert {\mathit{q}}_i^{\prime }-(\mathit{R}{\mathit{p}}_i^{\prime }+\mathit{t})\Vert }_2,$$ (16)

where ${\Vert ·\Vert }_2$ means a Euclidean norm. $N$ is the number of the corresponding points, which is thirteen in this paper (Fig. 5). Displacement error ${\mathit{E}}_{\text{Displacement}}$ is given by

$${\mathit{E}}_{\text{Displacement}}=\frac{1}{N}\sum _i^N[{\mathit{q}}_i^{\prime }-(\mathit{R}{\mathit{p}}_i^{\prime }+\mathit{t})].$$ (17)

3. Experiments and Results

3.1. Materials and Parameters

We used 16 laparoscopic surgical cases to validate the deformation simulation by the proposed method. Additionally, we compared the deformation accuracy between the proposed method and that proposed by Kitasaka et al.²¹ The study was approved by the institutional review board in Aichi Cancer Center. The following are the acquisition parameters of the CT image: slice size: $512\times 512\text{pixels}$, number of slices: 538 to 1097 slices, pixel spacing: 0.654 to 0.783 mm, slice thickness: 1.0 mm, and reconstruction pitch: 0.5 to 0.8 mm. The parameters used in the experiment were $L=8$, $\beta =0.25$, $\delta =0.5$, and $\mathrm{\Delta }t=0.1\mathrm{ms}$. The mechanical parameters were $\rho =1.096\times 103\mathrm{kg}/{\mathrm{m}}^3$, $E=60.0\mathrm{kN}/{\mathrm{m}}^2$, $\nu =0.25$, and $\zeta =0.4$. They were determined by the mechanical properties of the human body.^29–31 Gas pressure $P$ was 10.0 mmHg, which is the general value of a real pneumoperitoneum in the operating room. To evaluate the previous method, we adjusted the parameters that obtain the best result in each case.

3.2. Results

The position and displacement errors are shown in Table 1. The mean position errors of previous and proposed methods are 35.6 mm and 26.9 mm, respectively. Figures 7–9 show the position and displacement errors. Table 2 shows the errors of each point measured for evaluation. Examples of the volumetric images generated by the proposed method and their visualizations are shown in Figs. 10–12. The average of computation time for deformation process was about 150 min by a PC that equips dual Intel Xeon T5500 3.33 GHz processors.

Table 1.

Mean position and displacement errors. Mean position errors of proposed method using 10 and 30 mmHg were 26.9 and 23.4 mm, respectively. Mean position error of the previous method was 35.6 mm. We adjusted parameters of the previous method to minimize the errors, because it does not have the parameter about gas pressure.

Method	Position error (mm)	Displacement error (mm)
		Horizontal (right to left)	Ventrodorsal (front to back)	Cephalocaudal (head to tail)
Kitasaka et al.	$35.6\pm 6.4$	$-0.4\pm 7.1$	$-33.0\pm 11.3$	$-1.1\pm 8.3$
Proposed (10 mmHg)	$26.9\pm 5.9$	$-0.2\pm 7.6$	$-23.2\pm 10.0$	$0.5\pm 7.6$
Proposed (30 mmHg)	$23.4\pm 4.5$	$0.1\pm 8.3$	$-10.0\pm 17.9$	$2.9\pm 8.3$

Fig. 7.

Boxplot of errors of previous and proposed methods. (a) Mean position errors of previous and proposed methods are 35.6 and 26.9 mm, respectively. These two error results are significantly different ($p=6.0\times {10}^{-8}$, Student’s $t$-test). (b) Displacement errors of ventrodorsal direction of previous and proposed methods were $-32.6$ and $-23.2\mathrm{mm}$, respectively. Displacement errors of ventrodorsal direction are significantly different ($p=1.3\times {10}^{-29}$, Student’s $t$-test).

Fig. 8.

Relationship of errors and strength of virtual gas pressure: (a) position and (b) displacement errors.

Fig. 9.

Relationship of errors and strength of Young’s modulus: (a) position and (b) displacement errors.

Table 2.

Position and displacement errors of each point. Virtual gas pressure is 10 mmHg.

Point #	Position error (mm)	Displacement error (mm)
		Horizontal (right to left)	Vetrodorsal (front to back)	Cephalocaudal (head to tail)
1	27.1	$-4.4$	$-21.1$	$-9.2$
2	19.1	$-3.4$	$-15.7$	$-3.8$
3	20.4	3.4	$-16.7$	$-2.2$
4	21.8	1.2	$-19.6$	$-1.2$
5	25.6	1.2	$-23.9$	$-3.1$
6	22.7	$-5.9$	$-18.7$	4.2
7	30.0	$-6.4$	$-27.8$	3.5
8	32.7	0.5	$-31.7$	1.5
9	28.5	0.3	$-26.9$	$-1.4$
10	20.8	1.6	$-16.9$	7.2
11	30.0	5.2	$-27.3$	5.0
12	31.1	2.7	$-29.5$	4.3
13	27.5	1.9	$-25.9$	1.7
Overall	$26.9\pm 5.9$	$-0.2\pm 7.6$	$-23.2\pm 10.0$	$0.5\pm 7.6$

Fig. 10.

Examples of volumetric images generated by proposed method: Left side of each subfigure shows a preoperative volumetric image. Right side shows a volumetric image generated by proposed method.

Fig. 11.

Visceral deformation: Surface of visceral region moves to back direction by virtual gas pressure.

Fig. 12.

Examples of visualization of volumetric images generated by proposed method: (a) left half of body from right side. (b) Virtual laparoscopic view. Subfigures in middle and right columns show that visualized images are obtained by superimposing organ labels within the volumetric image.

4. Discussion

4.1. Overview

As shown in Table 1 and Fig. 7(a), the mean position error of the proposed method was 26.9 mm, while the mean position error of the previous method was 35.6 mm. These two errors are significantly different ($p=6.0\times {10}^{-8}$, Student’s $t$-test). In addition, the results of the previous method were obtained to minimize the errors by manually adjusting the parameters, because the previous method does not have the parameter about gas pressure. It is hard to predict the true abdominal wall deformation by the previous method since it requires that the parameters be adjusted by visual inspection. In contrast, our method generates deformed volumetric images automatically without adjusting the parameters and predicts abdominal wall deformation by just giving ${\mathrm{CO}}_2$ gas pressure and the tissue properties, which can be fixed by anatomical knowledge. Therefore, our proposed method is more useful for laparoscopic surgical planning.

4.2. Parameters

Table 1 also shows the displacement errors of the previous and proposed methods. There is insufficient elevation in the virtual pneumoperitoneums of both methods, because the displacement errors of the ventrodorsal directions of the previous and proposed methods were $-33.0$ and $-23.2\mathrm{mm}$, respectively. Figure 7(b) shows boxplots of the displacement errors of each direction. The displacement errors of the ventrodorsal direction had a statistically significant difference ($p=1.3\times {10}^{-29}$, Student’s $t$-test). The MSDMs in this paper were constructed by assuming tissue uniformity. Therefore, perhaps the optimal virtual gas pressure differs from the actual gas pressure. Figure 8(a) shows the relationship of the position error and the strength of the virtual gas pressure in the virtual pneumoperitoneum process of the proposed method. The position error was slightly reduced from 26.9 to 23.4 mm (the error using 30 mmHg) by increasing the virtual gas pressure. Accordingly, our method needs virtual gas pressure that is greater than the actual gas pressure. The position error remains large even though the virtual gas pressure was increased. Figure 8(b) shows the relationship between the displacement error and the virtual gas pressure. The elevation of the abdominal wall is insufficient, although the position error was minimized by adjusting the virtual gas pressure. The errors varied with individual cases due to the different physiques among patients. For example, the abdominal wall of each patient has different thickness. This problem will be solved by considering heterogeneity of tissue.

We also investigated the relationship of the position error and Young’s modulus in the virtual pneumoperitoneum process of the proposed method. The results are shown in Fig. 9. The position error can be slightly reduced from 26.9 to 23.9 mm by adjusting Young’s modulus, which is $20.0\mathrm{kN}/{\mathrm{m}}^2$; this is one-third of a previously described value.³³ However, the ventrodorsal displacement error increased, probably due to tissue uniformity. One solution is the introduction of a volumetric mesh that can represent heterogeneity of tissue. Therefore, we must segment various tissues from a preoperative volumetric image and set up mechanical properties based on actual tissues. We must also validate Eq. (3) in an actual environment.

In Table 2, the displacement errors of the ventrolateral points (8 and 12 in Fig. 5) are greater than the errors of the other points. We obtained a result having the same tendency even if the virtual gas pressure increases. This indicates that one of the causes of insufficient elevation is the limitation of the deformation region (Fig. 3).

The average of computation time for deformation process was about 150 min. The computation time of our method is still long. It depends on the time step $\delta t$ in Newmark-$\beta$ method. We utilized $\delta t=0.1\mathrm{ms}$ in this experiment. We need to evaluate and optimize the time step to reduce the computation time.

4.3. Visualization

Figure 10 shows examples of the volumetric images generated by the proposed method. The left side of each subfigure shows a slice of the original volumetric image. The right side shows a slice of the volumetric image generated by the proposed method, which applied the virtual pneumoperitoneum process to all the cases. Furthermore, our method considers visceral deformation. Figure 11 shows that the surface of the visceral region moves to the back direction by virtual gas pressure. However, we cannot evaluate the deformation accuracy of the viscera by measurement points on the abdominal surfaces. For further validation of the proposed method, we need to evaluate the displacement of the visceral regions by measuring the interior of the abdominal cavity after pneumoperitoneum.

Figure 12 shows examples of the visualization of the volumetric image deformed by the proposed method. We can observe the abdominal surface and cavity deformed by pneumoperitoneum from any arbitrary viewpoints. Changing the transfer function of volume rendering enables us to easily understand the internal structures of the abdominal wall and the viscera. Furthermore, our method can apply the deformation computed in the virtual pneumoperitoneum process to other types of images, such as organ labels generated by a medical image segmentation technique.³⁹ The subfigures in the middle and right columns show that the visualized images are obtained by superimposing organ labels within the volumetric image. Therefore, our proposed method can sufficiently utilize the preoperative information. To provide more useful information to surgeons about trocar positioning in laparoscopic surgical planning, a system must be developed that can suggest the optimal trocar positions based on a patient’s anatomical structure, disease, and surgical instruments.

4.4. Limitation

This paper utilized the MSDMs for computing abdominal wall deformation caused by pneumoperitoneum due to the consideration of computation time. Also, pneumoperitoneum deformation is a large deformation, which was difficult to directly apply to the FEM. However, recently several papers or software platform (especially SOFA²³) have tried to solve such limitations of FEM with respect to medical applications. Utilization of such FEM techniques in pneumoperitoneum simulation is our primary future work.

The elevation of the abdominal wall is insufficient, although the position error was minimized by adjusting the virtual gas pressure. However, excessive virtual gas pressure increased the displacement error in the cephalocaudal direction. This result was caused by the process of applying gas pressure. Our proposed method applies a maximum gas pressure value to the MSDMs from the beginning of deformation. Therefore, the deformation process converges soon by the high reactions of the dampers. To solve this problem, we must develop a new process of applying gas pressure by considering time variations. For example, the strength of the virtual gas pressure may gradually increase over time. This is one of our future works.

Our proposed method deforms the anterior portion of the body and fixes the posterior portion (Fig. 3). This limitation reduces the range of movement of the nodes in the ventrolateral regions. Moreover, the propagation of the movement restriction reduces the range of movement of all the nodes. To solve this problem, we need to extract the whole abdominal cavity and generate a volumetric mesh from the whole abdomen.

We think that tissue homogeneity is an appropriate assumption with respect to deformation of abdominal wall. Introducing inhomogeneity makes the model very complex. Furthermore, detailed tissue segmentation is a time-consuming task. However, each patient has different physique, thickness of abdominal wall, and amount of muscle. Optimal physical property is different even if we assume that the tissue is uniform. We think the tissue uniformity was excessive assumption. To facilitate better planning for surgical procedures, we are considering specific organ deformation and movement by organ modeling. Tissue segmentation and modeling with different physical properties will be our next work.

In the experiment, the point correspondence was determined by using the location definition shown in Fig. 5. These points were explicitly marked on the body surface of a patient before the pneumoperitoneum. The points of the real patient were measured by the surgeons on the basis of the umbilicus and ensiform cartilage. The points of the CT volume were measured by the engineer on the same basis. Therefore, there is the error based on the positional gap caused by this measurement scheme. Additionally, the position error includes the error caused by the patient’s posture because the CT scan was performed several days before surgery. The average of fiducial registration error of point-based rigid registration was 10.9 mm. In order to improve the reliability of the evaluation of our method, we require a more robust evaluation scheme, such as intraoperative CT or MRI.

5. Conclusion

This paper proposed a method of virtual pneumoperitoneum based on MSDMs and the direct time integration of Newton’s equations of motion. Our proposed method simulated the abdominal wall and visceral motions by infused gas pressure and generated deformed volumetric images from a preoperative volumetric image. The virtual pneumoperitoneum process needed only virtual gas pressure and tissue properties based on anatomical knowledge. The experimental results revealed that a method using 10 mmHg virtual gas pressure elevated the abdominal wall within position error of $26.9\pm 5.9\mathrm{mm}$ and reduced the position error of the proposed method to $23.4\pm 4.5\mathrm{mm}$ using 30 mmHg virtual gas pressure. Future work includes a nonuniformity parameter setting, validation of visceral displacement, application to preoperative surgical planning, and intraoperative surgical navigation.

Biographies

Yukitaka Nimura is a researcher at Nagoya University. He received his BS and MS degrees in engineering from Nagoya University, in 1999 and 2001, respectively. His current research interests include medical image processing, analysis, and visualization.

Biographies for the other authors are not available.