Telemedicine plays an important role in Corona Virus Disease 2019 (COVID-19). The virtual surgery simulation system, as a key component in telemedicine, requires to compute in real-time. Therefore, this paper proposes a real-time cutting model based on finite element and order reduction method, which improves the computational speed and ensure the real-time performance. The proposed model uses the finite element model to construct a deformation model of the virtual lung. Meanwhile, a model order reduction method combining proper orthogonal decomposition and Galerkin projection is employed to reduce the amount of deformation computation. In addition, the cutting path is formed according to the collision intersection position of the surgical instrument and the lesion area of the virtual lung. Then, the Bezier curve is adopted to draw the incision outline after the virtual lung has been cut. Finally, the simulation system is set up on the PHANTOM OMNI force haptic feedback device to realize the cutting simulation of the virtual lung. Experimental results show that the proposed model can enhance the real-time performance of telemedicine, reduce the complexity of the cutting simulation and make the incision smoother and more natural.

The outbreak of COVID-19 in China at the end of 2019 has attracted extensive attention in the world. COVID-19 spreads rapidly and can spread from person to person. Once people are exposed to COVID-19, it will cause respiratory tract infection or even pneumonia. At the beginning of 2020, COVID-19 has spread quickly from Wuhan. The epidemic resulted in a serious shortage of medical resources in Wuhan and other hardest-hit areas, which makes it difficult to get a sickbed for patients. Therefore, the Chinese government allocated all kinds of resources, such as food, clothing, medicine, and assembled a large number of medical personnel in Wuhan and other hardest-hit areas. Many other cities in China have also formed a close relationship with Wuhan to provide strong support. However, due to the uneven distribution of medical resources, the aid does not work immediately, and the traditional medical model is facing challenges. So far, more than 100 million people have been diagnosed with COVID-19, which demonstrates that COVID-19 is raging and spreading all over the world. Therefore, in order to reduce the risk of cross-infection, and improve the cure rate of COVID-19, there is the increasingly urgent need for telemedicine.

Telemedicine means doctors can make a comprehensive and elaborative analysis of a patient’s condition without the presence of patients. Therefore, doctors can put forward correct diagnosis and develop scientific treatment schemes. During the treatment process of telemedicine, doctors need to perform force-tactile interactive operations on virtual diseased organs. The key technical difficulty is to reduce the calculation amount while ensuring real-time performance during surgery simulation [

In addition, when the virtual surgical instrument exerts an external force on the surface of the soft tissue, the soft tissue will be deformed, and then as the external force gradually increases, the amount of deformation of the soft tissue will gradually increase and stress concentration will appear at the deformed area. When the external force exceeds the stress threshold, the deformation reaches the limit, and the soft tissue starts to break from there, thereby forming a cut. When the soft tissue is cut, it will produce an incision on its surface. The traditional method of drawing the surface incision needs to be described by standard equations, such as the cutting model based on B-spline [

To address the issues mentioned above, this paper proposes a new real-time cutting model based on finite element and order reduction. The proposed model simulates the deformation of the virtual lung and the cutting operation in the lesion area. During the process of soft tissue simulation, the finite element model is used to construct model for the virtual lung deformation. Moreover, the model order reduction algorithm, which combines proper orthogonal decomposition and Galerkin projection, is added to the deformation model to reduce the calculation amount. Then, the cutting simulation is operated after the deformation simulation. According to the collision intersection position of the surgical instrument and the lesion area on the virtual lung, the cutting path is formed. Finally, the incision outline is drawn based on the Bezier curve.

The rest part of the paper is organized as follows. Section 2 elaborates on the cutting model. Then, experimental results and analysis to verify the performance of the cutting model are presented and discussed in Section 3. Finally, Section 4 concludes the paper.

The use of telemedicine to treat patients with COVID-19 first needs to model the deformation of the soft tissues of the lungs to facilitate the simulation of subsequent cutting operations [

First, according to the finite element model, the soft tissue is divided into several non-overlapping tetrahedral elements, and the motion behavior of the tetrahedral elements is mechanically analyzed. Assuming that the displacements of the four vertices of a given tetrahedral element are _{1}, _{2}, _{3}, _{4}]^{T}, respectively, the displacement of any node in the tetrahedral element can be obtained by linear combination of the displacement of the vertices of the element and the corresponding shape function:_{1}, Φ_{2}, Φ_{3}, Φ_{4}] represents the shape function matrix, and

In order to describe the strain produced by the soft tissue under external forces, a linear Cauchy strain matrix is introduced:

_{s} represents the strain energy generated by the tetrahedral element, _{f} represents the total work done by the external and body forces acting on the tetrahedral element,

where Ω represents the deformation solution domain, _{t} represents the stress boundary. According to the variational principle,

Then substituting

Define

Substituting

Substituting

Substituting

Since ^{T} has arbitrary, eliminate this term and simplify ^{b} + ^{t} represents the resultant force on the node. The resultant force on the node includes the external force exerted by the virtual surgical instrument and the internal force generated by the soft tissue to resist the external force. The internal force includes elastic force and damping force, and

Among them,

Secondly, according to the model reduction method [^{T}

Discrete the continuous iteration time into an interval _{n}, _{n+1}]:

_{n+1} − _{n}, and

_{n+1}.

Since the internal force _{n}.

Finally, the solutions of a set of finite element full-order models are obtained. The specific numerical calculation equation is:

Subsequently, a set of deformation displacement solution vectors are obtained by solving _{1}, _{2}, …, _{N}) represents the obtained orthogonal basis function based on the eigen-orthogonal decomposition, and _{i} represents the corresponding coefficient of the basis function _{i}. Next, in order to obtain a low-order approximation of the original model while minimizing the dimensionality of the full-order model, the obtained orthogonal basis function is truncated, and the error function

Finally, the space formed by the orthogonal basis function of the low-order approximation of the original model obtained by

The cutting path in virtual surgery can be understood as a continuous fold line drawn by the surgical instrument on the surface of the soft tissue in a discrete time. In order to solve the cutting path, this study uses a straight-line model as the basis to simulate the cutting tool, that is, the surgical instrument is simplified into a line-segment at each discrete moment, and the triangular unit is used to simulate the tissue structure. Then the collision detection between the surgical instrument and the soft tissue is transformed into the detection of the intersection of the line-segment and the surface triangle unit, as shown in

First, determine the spacial linear equation of the surgical instrument segment at a discrete time. Assuming that the two end-points of the line-segment are _{A}(_{A}, _{A}, _{A}) and _{B}(_{B}, _{B}, _{B}), respectively, the spacial straight-line equation of the line segment

Then the coordinates of any node on the line-segment are:

Secondly, determine the plane equation of the triangular element that intersects with the line-segment of surgical instrument. Assuming that the vertices of a triangular element on the surface of the soft tissue are _{O}(_{O}, _{O}, _{O}), _{P}(_{P}, _{P}, _{P}), and _{Q}(_{Q}, _{Q}, _{Q}), respectively, and its normal vector is _{x}, _{y}, _{z}). Let the triangle element plane be:

According to the method of calculating the plane equation, the plane equation can be obtained by the coordinates of the point

Then, combine _{s}(_{s}, _{s}, _{s}) between the surgical instrument line-segment and the triangular element plane at a discrete time. The specific calculation equation is as follows:

Finally, connect the discrete intersection points, and the formed line-segment is the desired cutting path.

After the soft tissue is cut, an incision will be made on its surface. In this paper, a Bezier curve is used to draw the surface incision. Bezier curve is a kind of approximation spline curve. It does not need to be described by standard algebraic equations. It can be constructed only by a given number of control points [

First, suppose that the positions of _{i} = (_{i}, _{i}, _{i}), _{0} and _{n} that approximates the Bessel polynomial function:_{i,n}(

Secondly, the quadratic Bezier curve is generated by three control points, and

Then, the surface incision drawing equation based on the quadratic Bezier curve is obtained:

Finally, the process of drawing the incision based on the quadratic Bezier curve is shown in

Determine the start position _{10}, _{11}, _{12}, _{13}, _{14}, _{15}}, and the _{20}, _{21}, _{22}, _{23}, _{24}, _{25}, _{26}, _{27}, _{28}, _{29}, _{30}, _{31}} point set is the nodes around the cutting path, which are distributed on both sides of the cutting path. Select nodes

The computer and the force-tactile interaction device PHANTOM OMNI hand controller were used for force-tactile interaction, and the soft tissue force-tactile interaction system was built through the finite element real-time cutting model based on the model reduction method, which realized the deformation and cutting simulation of virtual lung soft tissue under interactive action, the simulation environment is shown in

In order to verify the validity and authenticity of the model proposed in this paper, the stress threshold was set to two N, the number of nodes was 1,445, and the number of tetrahedral units was 6,738 to model the virtual lung soft tissue, and the virtual finger was used to apply pressure to the lung soft tissue to simulate the simulation effect of its compression deformation, as shown in

It can be seen from

In order to verify the real-time performance of the model proposed in this article, the virtual lung soft tissue was discretized into the following six different types of units: 4,279 units, 5,540 units, 11,467 units, 14,607 units, 25,971 units, and 39,148 units. Then, only the original finite element model and the finite element model based on the model reduction method were used to simulate the deformation and cutting of the lung soft tissue, and the two models were calculated under the time step

Number of units | Total calculation time (mm) | Deformation time (mm) | Cutting time (mm) |
---|---|---|---|

4,279 | 989.1 | 433.9 | 555.2 |

5,540 | 1,362.1 | 563.5 | 798.6 |

11,467 | 2,607.8 | 1,106.1 | 1,501.7 |

14,607 | 3,632.1 | 1,531 | 2,101.1 |

25,971 | 6,033.4 | 2,552.4 | 3,481 |

39,148 | 9,595 | 4,281.7 | 5,313.3 |

Number of units | Total calculation time (mm) | Deformation time (mm) | Cutting time (mm) |
---|---|---|---|

4,279 | 400.5 | 175.7 | 224.8 |

5,540 | 551.6 | 228.2 | 323.4 |

11,467 | 1,266.9 | 523.1 | 703.8 |

14,607 | 1,710.4 | 717.9 | 992.5 |

25,971 | 2,896.2 | 1,220.6 | 1,675.6 |

39,148 | 4,641.6 | 2,068.1 | 2,573.5 |

In order to verify the accuracy of the model proposed in this paper, the standard modeling software ANSYS was used to establish a reference model [

Then, in order to quantitatively reflect the offset degree of the node displacement between the model proposed in this article and the reference model, the mean square error was defined:_{m} represents the number of marked nodes, _{i} represents the displacement of the marked nodes in the three different models mentioned above. It can be seen from

Model | Mean square error |
---|---|

The model mentioned in this article | 2.05 |

Co-rotating finite element model [ |
4.01 |

Finite element model based on neural network [ |
3.28 |

In order to evaluate the effect of the incision after the cutting operation, three models were used to compare the cutting simulation of virtual gastric soft tissue, including the proposed model, the cutting model based on B-spline [

Stability and fluency are important indicators to measure whether the virtual surgery simulation system is running well, and response time is the most intuitive manifestation of system fluency. We were fortunate to invite 20 doctors from the First Affiliated Hospital of Nanjing Medical University, including six interns, five residents, five deputy chief physicians, and four chief physicians to use the model proposed in this article to perform cutting experiments on virtual lung, stomach and spleen soft tissues. Set the internal stress threshold of the soft tissue to two N. When the stress of the soft tissue exceeds the threshold, a cutting operation is generated and the response time of the system is recorded, as shown in

Cutting object | Average response time/ |
Slowest response time/ |
Failure rate (carton, error report)/% |
---|---|---|---|

Lung | 0.08 | 0.15 | 0.0 |

Stomach | 0.11 | 0.19 | 4.3 |

Spleen | 0.04 | 0.1 | 1.5 |

Generally speaking, the response time of the virtual surgery simulation system should be within 0.28 s to meet the requirements of operational fluency [

In order to improve the real-time performance under the premise of keeping the accuracy of finite element model to the maximum extent, this paper proposes a real-time cutting model based on finite element and model reduction method. This model is used to simulate the deformation and cutting operations of the diseased lung area in the virtual surgery simulation system. The deformation solution vector obtained from the finite element solution of the soft tissue is used as a sample, and the model reduction method combining eigen-orthogonal decomposition and Galerkin projection is introduced to reduce the order of the full-order finite element model with high dimension, and then the solution vector is projected to the reduced order space, which can reduce the amount of deformation calculation. In order to reduce the computational complexity of the cutting simulation, this paper uses the line model and the straight-line between triangular units to describe the cutting path formed by the collision between the surgical instrument and the soft tissue of the diseased area, and uses the Bezier curve to draw the surface incision generated after the soft tissue is cut, which makes the incision more natural and smooth. Finally, the effectiveness of the proposed model is proved by a number of experiments in terms of the computational efficiency, accuracy, incision effect and system fluency. In the future, we will pay more attention to the study of deformation and cutting models of other human soft tissues, such as liver tissue.