Simple human chest impact response modeling method with efficient calculation

Abstract

The invention relates to a simple human chest impact response modeling method with efficient calculation, and mainly solves the problems of high calculation cost and low efficiency of a high-precision finite element model in the traditional explosive shock wave lung injury research. The method comprises the following steps: geometrically simplifying the chest of a human body, simplifying ribs into equal-height cylinders, and simplifying lungs into solid spheres; the method comprises the following steps of: performing hybrid grid division, performing adaptive grid (dynamic encryption based on a Zienkiewicz-Zhu error estimator and a Hessian matrix anisotropic size field) on muscular tissues, and performing hexahedral structured grid (generated through a parameterized mapping method to ensure unit quality) on bones and lungs; material setting: endowing muscles, bones and lungs with standardized linear elastic material parameters, and simulating the interaction between an explosion flow field and a human body through Euler-Lagrange coupling. The model only contains 2676 units, the calculation time is shortened to half an hour from several weeks, and the memory demand is reduced to 14 MB from 200 MB.

Description

Technical Field

[0001] This invention relates to the field of modeling technology, and in particular to a computationally efficient and simple method for modeling the impact response of the human chest. Background Technology

[0002] Lung injury caused by blast shockwaves is a leading cause of death in battlefields, terrorist attacks, and industrial accidents. Traditional research methods on lung damage from blast shockwaves largely rely on animal experiments (pigs, sheep, rabbits, etc.) or clinical autopsy data, which suffer from extrapolation errors due to species differences and a lack of transparency regarding dynamic damage mechanisms. Finite element human models, through computer simulations of the mechanical interaction between shockwaves and biological tissues, provide a new approach to quantifying lung injury.

[0003] Existing high-precision finite element human body models (such as GHBMC and THUMS) suffer from significant operational complexity and computational efficiency issues in mesh generation. Furthermore, existing models generally employ global uniform refinement or fixed-partition meshing strategies, which directly lead to over-meshing in non-critical areas, wasting computational resources. These models typically contain 2-5 million elements, requiring high computer memory configurations, and single simulations can take tens of hours. In addition, existing models already have complete parameter and connectivity settings for various parts of the human body; deleting other parts of the model to retain only the chest area would damage the model and make mesh generation impossible. Existing simplified models (such as the Axelsson model) can only evaluate the overall chest response and cannot predict local overpressure distribution. Summary of the Invention

[0004] The technical solution of this invention to solve the above-mentioned technical problems is to provide a simple and computationally efficient method for modeling the impact response of the human chest, comprising the following steps:

[0005] Step 1, Geometric simplification of the human chest: Simplify the ribs into cylinders of equal height and the lungs into solid spheres;

[0006] Step 2, Hybrid Mesh Generation:

[0007] a) For external muscle tissue, an adaptive mesh is used for subdivision, and the Zienkiewicz-Zhu error estimator is used to locate high-error regions:

[0008]

[0009] In the formula, η K σ*: Error estimate of each element; σh: Smooth recovery value of nodal stress; L: Stress at each part in the simplified model; 2 (K): L on unit K 2 Norm;

[0010] Anisotropic size fields are generated based on the eigenvalues ​​of the Hessian matrix:

[0011]

[0012] In the formula, h i new : The size of the new mesh in direction i; h i current λi: Dimension of the current mesh in direction i; Q: Eigenvector matrix; τtol: Error tolerance;

[0013] If |λi|>τtol, it means the curvature is too large, and it is necessary to reduce hinew, that is, refine the mesh; if |λi|<τtol, it means the curvature is too small, and it is necessary to keep hinew, that is, keep the mesh unchanged.

[0014] b. For bones and lungs, a hexahedral structured mesh is used, and the hexahedral structured mesh is generated using a parametric mapping method:

[0015]

[0016]

[0017] In the formula, x: coordinates in physical space; (ξ,η,ζ): normalized coordinates of the parameters, the natural coordinate system of the hexahedral element, with values ​​ranging from [-1,1]; Ni: trilinear shape function; xi: physical coordinates of the i-th node of the hexahedral element;

[0018] Using trilinear shape functions to ensure that the unit Jacobian determinant > 0:

[0019]

[0020] In the formula, J: Jacobian determinant, which detects whether the cell is distorted. If J < 0, the mesh needs to be re-divided.

[0021] Step 3, Material Setup: Muscle tissue with a density of 1 g / cm³ 3 A linear elastic model with a Young's modulus of 1.28 GPa and a Poisson's ratio of 0.4 was used, with the skeleton having a density of 1.561 g / cm³. 3 A linear elastic model with Young's modulus of 7.92 GPa and Poisson's ratio of 0.379 was used, and the lung density was 0.2 g / cm³. 3 A linear elastic model with Young's modulus of 0.001 GPa and Poisson's ratio of 0.3.

[0022] The explosive and air are processed using S-ALE element meshes with a mesh size of 20mm×20mm×20mm, a mesh count of 1,000,125, and a node count of 1,033,076. The S-ALE elements and Lagrangian elements are defined as Eulerian-Lagrangian couplings.

[0023] The technical solution of this invention uses a simplified human chest model with 2676 elements based on adaptive and structured mesh generation, and a high-precision THUMS finite element model with 1.9 million elements. While ensuring the accuracy of macroscopic mechanical response, the number of meshes is reduced to 0.27% of that of traditional million-level mesh models. Computation time is shortened from one week to half an hour, demonstrating a significant advantage in computational efficiency. Memory requirements are reduced from 200MB to 14MB, and storage space is compressed by 20 times. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0025] Figure 1 This is a flowchart illustrating the steps of a computationally efficient and simplified method for modeling the impact response of the human chest, as described in this invention.

[0026] Figure 2 This is a flowchart illustrating the overall scheme of a computationally efficient and simplified human chest impact response modeling method described in this invention.

[0027] Figure 3 A human model diagram established for the present invention's computationally efficient and simplified human chest impact response modeling method;

[0028] Figure 4 This is a model diagram of the explosion flow field at a 1m detonation distance according to the present invention;

[0029] Figure 5 This is a pressure cloud diagram of the explosion flow field of the present invention;

[0030] Figure 6 This is a peak chest overpressure diagram of the two individuals in the first case of this invention;

[0031] Figure 7 This is a diagram of the Axelsson chest model of the present invention;

[0032] Figure 8 The peak chest overpressure values ​​of the first two individuals in the second case of this invention. Detailed Implementation

[0033] This invention proposes a computationally efficient and simple method for modeling the impact response of the human chest, aiming to solve the problem of excessively high computational costs caused by overly fine meshing in high-precision finite element simulation models.

[0034] The following will describe a computationally efficient and simplified method for modeling the human chest impact response proposed in this invention, using specific embodiments:

[0035] Example 1:

[0036] In the technical solution of this embodiment, such as Figure 1 As shown, a computationally efficient and simple method for modeling the impact response of the human chest includes the following steps:

[0037] Step 1, Geometric simplification of the human chest: Simplify the ribs into cylinders of equal height and the lungs into solid spheres;

[0038] Step 2, Hybrid Mesh Generation:

[0039] a) For external muscle tissue, an adaptive mesh is used for subdivision, and the Zienkiewicz-Zhu error estimator is used to locate high-error regions:

[0040]

[0041] In the formula, η K : The error estimate of each element. If the error is large, the mesh needs to be refined; if the error is small, the mesh needs to be coarsened. σ*: The smooth recovery value of nodal stress, i.e., the nodal averaging of σh. σh: The stress in each part of the simplified model. L2(K): The L2 norm on element K.

[0042] Anisotropic size fields are generated based on the eigenvalues ​​of the Hessian matrix:

[0043]

[0044] In the formula, hinew: the size of the new mesh in direction i; hicurrent: the size of the current mesh in direction i; λi: the curvature in direction i; Q: the eigenvector matrix, representing the principal curvature direction, which determines the reference direction for anisotropic mesh adjustment; τtol: error tolerance, which controls the mesh density (the smaller the value, the denser the mesh).

[0045] If |λi|>τtol, it means the curvature is too large, and it is necessary to reduce hinew, that is, refine the mesh; if |λi|<τtol, it means the curvature is too small, and it is necessary to keep hinew, that is, keep the mesh unchanged.

[0046] This meshing strategy automatically increases mesh density in regions with abrupt curvature changes, such as the muscle-bone interface, reducing redundant elements by more than 50% compared to global uniform refinement. Maintaining elongated elements along the muscle fiber direction ensures that the calculation error for muscle bundle tensile stress is less than 5%, while isotropic meshes have an error of up to 15%.

[0047] b. For bones and lungs, a hexahedral structured mesh is used, and the hexahedral structured mesh is generated using a parametric mapping method:

[0048]

[0049]

[0050] In the formula, x: coordinates in physical space; (ξ,η,ζ): normalized coordinates of the parameters, the natural coordinate system of the hexahedral element, with values ​​ranging from [-1,1]; Ni: trilinear shape function; xi: physical coordinates of the i-th node of the hexahedral element;

[0051] Using trilinear shape functions to ensure that the unit Jacobian determinant > 0:

[0052]

[0053] In the formula, J: Jacobian determinant, which detects whether the cell is distorted. If J < 0, the mesh needs to be re-divided.

[0054] With the same number of nodes, the stiffness matrix condition number of hexahedral elements is 1-2 orders of magnitude lower than that of tetrahedral elements, and the iterative solution speed is 40% faster. The stress oscillation amplitude of hexahedral meshes can be controlled within 1%, and stress calculation is more accurate than other meshing methods.

[0055] Step 3, Material Setup: Muscle tissue with a density of 1 g / cm³ 3 A linear elastic model with a Young's modulus of 1.28 GPa and a Poisson's ratio of 0.4 was used, with the skeleton having a density of 1.561 g / cm³. 3 A linear elastic model with Young's modulus of 7.92 GPa and Poisson's ratio of 0.379 was used, and the lung density was 0.2 g / cm³. 3 A linear elastic model with a Young's modulus of 0.001 GPa and a Poisson's ratio of 0.3. The linear elastic model is made of MATELASTIC material.

[0056] The explosive and air were simulated using S-ALE element meshes with a mesh size of 20mm × 20mm × 20mm, a mesh count of 1,000,125, and 1,033,076 nodes. The S-ALE and Lagrangian elements were defined as Eulerian-Lagrangian coupled. To avoid the influence of boundary reflections on the simulation results, the boundary conditions were set to non-reflective boundary conditions using the keyword BOUNDARY_SALE_MESH_FACE to simulate an infinite air domain.

[0057] Example 2:

[0058] A computationally efficient and simplified method for modeling the impact response of the human chest, such as... Figure 2 As shown, it includes the following steps:

[0059] Human Chest Geometry Simplification: The human body model was established using a male with a height of 1.75m and a weight of 70kg as the reference standard. The outer solid shell element model was adjusted to a length of 400mm, a width of 240mm, a height of 650mm, and a muscle layer thickness of 10mm. The human sternum and internal organs were modeled using the DesignModeler geometry editor in ANSYS Workbench. The rib model was simplified to be parallel to the horizontal plane and of uniform size. If the influence between ribs is not considered, it can be simplified to a cylinder of the same height. By drawing a 180mm*110mm rectangle and rotating it 360 degrees around its long side, a cylinder with a base radius of 110mm and a height of 180mm was obtained, with a thickness of 10mm. Since the key indicators of lung injury are peak pressure and impulse, which are not significantly related to the geometry of the organ, a solid sphere with a radius of 100mm was drawn within the skeletal model to represent the lung tissue.

[0060] Hybrid mesh generation: Since the external muscle tissue of the human body is not a regular shape, an adaptive mesh is used for generation, and the Zienkiewicz-Zhu error estimator is used to locate high-error regions.

[0061]

[0062] ηK: Error estimate of each element. If the error is large, the mesh needs to be refined; if the error is small, the mesh needs to be coarsened. σ*: Nodal stress smoothing recovery value, i.e., the nodal averaging of σh. σh: Stress of each part in the simplified model. L2(K): L2 norm on element K.

[0063] Anisotropic size fields are generated based on the eigenvalues ​​of the Hessian matrix:

[0064] H(u h )=QΛQ T ;

[0065] hinew: Size of the new mesh in direction i; hicurrent: Size of the current mesh in direction i; λi: Curvature in direction i; Q: Eigenvector matrix, representing the principal curvature direction, which determines the reference direction for anisotropic mesh adjustment; τtol: Error tolerance, controlling the mesh density (the smaller the value, the denser the mesh).

[0066] If |λi|>τtol, it means the curvature is too large, and it is necessary to reduce hinew, that is, refine the mesh; if |λi|<τtol, it means the curvature is too small, and it is necessary to keep hinew, that is, keep the mesh unchanged.

[0067] This meshing strategy automatically increases mesh density in regions with abrupt curvature changes, such as the muscle-bone interface, reducing redundant elements by more than 50% compared to global uniform refinement. Maintaining elongated elements along the muscle fiber direction ensures that the calculation error for muscle bundle tensile stress is less than 5%, while isotropic meshes have an error of up to 15%.

[0068] The skeleton and internal organs are simplified into regular basic shapes, and a hexahedral structured mesh is generated using a parametric mapping method.

[0069]

[0070]

[0071] x: Coordinates in physical space; (ξ,η,ζ): Normalized coordinates of the parameters, natural coordinates of the hexahedral element, with values ​​ranging from [-1,1]; Ni: Trilinear shape function; xi: Physical coordinates of the i-th node of the hexahedral element.

[0072] Using trilinear shape functions to ensure that the unit Jacobian determinant > 0:

[0073]

[0074] J: Jacobian determinant, used to detect whether the cell is distorted. If J < 0, the mesh needs to be re-divided.

[0075] With the same number of nodes, the stiffness matrix condition number of hexahedral elements is 1-2 orders of magnitude lower than that of tetrahedral elements, and the iterative solution speed is 40% faster. The stress oscillation amplitude of hexahedral meshes can be controlled within 1%, and stress calculation is more accurate than other meshing methods.

[0076] To ensure coupling stability, a uniform 20mm mesh size was used for all elements, resulting in a total of 2676 elements for the human body model. The meshed model is shown below. Figure 3 As shown.

[0077] For muscles, bones, and lungs, the linear elastic material model MAT_ELASTIC was used.

[0078] The explosion flow field includes the explosive and air. The air domain uses an Eulerian grid and measures 2m × 2m × 2m, completely encompassing the human model. The explosive is cubic TNT, positioned 0.5m above the ground, at the same height as the human sternum. Figure 4 As shown.

[0079] The TNT explosive was simulated using material MAT_HIGH_EXPLOSIVE_BURN (MAT_008), and the equation of state was EOS_JWL.

[0080]

[0081] In the formula, A1 = 371 GPa, B1 = 3.23 GPa, R1 = 4.15, R2 = 0.95, ω = 0.3, E1 = 7 GPa, and V is the relative volume. The density of TNT explosive is 1.63 g / cm3, the detonation velocity is 8.0 km / s, and the detonation pressure is 21 GPa.

[0082] Air was simulated using material MAT_009, with the equation of state EOS_LINEAR_POLYNOMIAL:

[0083] p = D0 + D1μ + D2μ 2 +D3μ 3 +(D4+D5μ+D6μ 2 E2;

[0084] In the formula, D0=D1=D2=D3=D6=0, D4=0.4, D5=0.4, E2=253kPa; μ=ρ / ρ0 -1 ρ is the air density, ρ0 is the initial air density, and ρ0 = 1.25 kg / m3.

[0085] The explosive and air were modeled using S-ALE element meshes with dimensions of 20mm × 20mm × 20mm, a mesh count of 1,000,125, and 1,033,076 nodes. The S-ALE elements and Lagrangian elements (human body model) were defined as Eulerian-Lagrangian coupled. To avoid the influence of boundary reflections on the simulation results, the boundary conditions were set to non-reflective boundary conditions using the keyword BOUNDARY_SALE_MESH_FACE to simulate an infinite air domain.

[0086] The typical working condition is the explosion of 500g TNT at a distance of 0.5m from a human body model. Figure 5The pressure cloud diagram of the frontal explosion flow field shows the propagation process of the shock wave at different times (a is 0.3ms, b is 0.5ms, c is 0.7ms, d is 1ms). It can be seen that the explosion shock wave can penetrate soft tissues such as skin and muscles, and undergo reflection, refraction and energy deposition at the muscle-bone tissue interface, causing an increase in flow field pressure near the chest wall. Subsequently, it diffracts to the back of the human body. That is, both the chest and back of the human body are affected by the explosion shock wave, but the explosion load on the back is much lower than that on the chest. The peak pressure on the front of the human chest is about 3.5MPa, and the pressure on the back is about 2.4MPa.

[0087] This invention utilizes a simplified human chest model with 2676 elements based on adaptive and structured mesh generation, compared to a high-precision THUMS finite element model with 1.9 million elements. While maintaining accuracy in macroscopic mechanical response, the number of meshes is reduced to 0.27% of traditional million-level mesh models. Computation time is shortened from one week to half an hour, demonstrating a significant computational efficiency advantage. Memory requirements are reduced from 200MB to 14MB, resulting in a 20-fold reduction in storage space.

[0088] Real-world examples:

[0089] Case 1: Hamit et al. reported an explosion in South Korea involving 1.6 kg of explosives. Three people were present: Person 1, working 1 meter away from the explosives, died instantly; Person 2, 3.05 meters away, lost consciousness immediately and sustained significant injuries to the eyes, ears, hands, head, and right forearm. He regained consciousness upon arrival at the hospital, but the duration of his coma was unknown. The patient was almost completely deaf and blind, with a depressed fracture of the right parietal bone, complex fractures of the third, fourth, and fifth metacarpals and phalanges of the right hand, a complex fracture of the right ulna, and a simple fracture of the right scaphoid bone. Chest X-rays showed diffuse infiltration in both hilum of the lungs, extending to the right middle lobe and both lower lobes. Upon arrival at the hospital, the patient received primary treatment for external injuries and a blood transfusion, but no treatment was administered specifically for the lungs. The patient stopped coughing up blood 24 hours after the accident, and the pulmonary infiltration disappeared 13 days later, indicating spontaneous healing of the lung damage. Luo Wei et al. argued that, under certain principles, the provision in the "Guidelines for Forensic Identification of the Degree of Bodily Injury" that "a ruptured lung treated surgically is considered a serious injury" could be applied. Based on the fact that Person No. 2 healed on their own without surgery, it can be determined that Person No. 2 does not meet the criteria for serious injury. According to the "Standards for the Identification of Personal Injuries" regarding chest injuries, it can be inferred that Person No. 2's injury level falls between minor injury level one and moderate injury.

[0090] Second case: Through medical analysis of typical victims in the 2005 London bombings, when the TNT equivalent was 4.5 kg, survivors within 1.5 meters of the blast center mostly suffered severe injuries, such as bilateral pulmonary contusions, hemopneumothorax, and embedded metal fragments; those at a distance of about 2 meters mainly suffered moderate injuries, including mild pulmonary contusions with localized fragmentation injuries and burns; and victims at a TNT equivalent of 2.5 kg who were very close to the blast source (blast distance < 1 meter) suffered fatal injuries such as ruptured hearts and skull fractures due to the excessively high intensity of the shock wave.

[0091] Axelsson damage model validation:

[0092] An experiment was conducted using a simplified human body constructed according to Embodiment 2 of the present invention under the same working conditions, and the overpressure results obtained were the same as those experienced by the two injured persons in the first case. Figure 6 The experimental results show that the maximum overpressure peak experienced by personnel No. 1 was approximately 1.44 MPa, and the maximum overpressure peak experienced by personnel No. 2 was approximately 110.23 kPa.

[0093] To predict the chest's response to the shockwave of an explosion, Axelsson et al. developed a simplified mechanical model to simulate the human lung. The mathematical expression of the model is as follows:

[0094]

[0095] In the formula: M is the equivalent mass of the lung; x is the chest wall displacement; J is the energy dissipation characteristic of the chest tissue; K is the equivalent stiffness of the chest wall and lung tissue; A is the effective area of ​​the shock wave; p(t) is the overpressure of the explosion shock wave; P0 is the ambient pressure (usually 101.3 kPa); V is the initial volume of lung gas at rest; g is the lung gas polyvariance index. Table 1 shows the parameters of the Axelsson injury model.

[0096] Table 1. Axelsson damage model parameters:

[0097]

[0098] The explosive load p(t) acting on the chest under different explosion conditions was obtained through simulation. Substituting this load into the above formula, the chest wall velocity can be calculated. The peak chest wall velocity was selected as the evaluation index for chest and abdominal injuries. Since Axelsson et al. did not verify the relationship between this model and the actual response of the human chest cavity, the peak chest wall velocity calculated based on the above formula is usually called the chest wall velocity predictor (vcwp). Based on animal experimental data, Axelsson et al. established the relationship between vcwp and chest and abdominal blast injuries, as shown in Table 2.

[0099] Table 2 Relationship between predicted chest wall velocity and injury grade

[0100]

[0101]

[0102] Currently, most studies on human lung injury using finite element simulation employ this method to verify the reliability of the human body model. Therefore, this invention also utilizes this method for verification.

[0103] Based on the above experimental results, the overpressure peak value p(t) is substituted into the equation. By solving the second-order nonlinear differential equation, the vcwp of person number 1 was found to be 13.24 m / s², indicating a mortality rate exceeding 50% and a high risk of death. The vcwp of person number 2 was 6.59 m / s², indicating an injury between minor and moderate. The experimental results show that the injury assessment of persons number 1 and 2 using the Axelsson injury model is consistent with the actual situation.

[0104] The same verification method was used for the three individuals in the second case. The maximum peak overpressure experienced by the first two injured individuals was 831.0 kPa and 579.3 kPa, respectively (e.g., Figure 8 As shown in the figure, by solving the second-order nonlinear differential equation, vcwp were obtained as 10.77 m / s and 7.86 m / s, respectively. Based on the fact that the fatality rate of personnel 1 in the actual case was higher than 50% under the condition of 1.6 kg TNT and a detonation distance of 1 m, it can be determined that the third person in this case faced a higher risk of death under the condition of 2.5 kg TNT and a detonation distance of 1 m. Comparison with Table 2 shows that the injury levels of the three personnel are consistent with the actual situation.

[0105] The casualties of the five individuals mentioned above demonstrate that the human body model established in this invention can be used to simulate the response process of a simulated explosion shock wave.

[0106] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A computationally efficient and simplified method for modeling the impact response of the human chest, characterized in that, Includes the following steps: Step 1, Geometric simplification of the human chest: Simplify the ribs into cylinders of equal height and the lungs into solid spheres; Step 2, Hybrid Mesh Generation: a) For external muscle tissue, an adaptive mesh is used for subdivision, and the Zienkiewicz-Zhu error estimator is used to locate high-error regions: In the formula, η K σ*: Error estimate of each element; σh: Smooth recovery value of nodal stress; L: Stress at each part in the simplified model; 2 (K): L on unit K 2 Norm; Anisotropic size fields are generated based on the eigenvalues ​​of the Hessian matrix: H(u h )=QΛQ T ; In the formula, h i new : The size of the new mesh in direction i; h i current λi: The current mesh size in direction i; λi: The curvature in direction i; Q : Eigenvector matrix; τ tol Error tolerance; If |λi|>τtol, shrink hinew and refine the mesh; if |λi|<τtol, keep hinew and keep the mesh unchanged. b. For bones and lungs, a hexahedral structured mesh is used, and the hexahedral structured mesh is generated using a parametric mapping method: In the formula, x: coordinates in physical space; (ξ,η,ζ): normalized coordinates of the parameters, the natural coordinate system of the hexahedral element, with values ​​ranging from [-1,1]; N i : Trilinear function; x i : The physical coordinates of the i-th node of the hexahedral element; Using trilinear shape functions to ensure that the unit Jacobian determinant > 0: In the formula, J: Jacobian determinant, which detects whether the cell is distorted. If J < 0, the mesh needs to be re-divided.

2. The computationally efficient and simplified method for modeling the impact response of the human chest, as described in claim 1, is characterized in that... It also includes the following steps: Step 3, Material Setup: Muscle tissue with a density of 1 g / cm³ 3 A linear elastic model with a Young's modulus of 1.28 GPa and a Poisson's ratio of 0.4 was used, with the skeleton having a density of 1.561 g / cm³. 3 A linear elastic model with Young's modulus of 7.92 GPa and Poisson's ratio of 0.379 was used, and the lung density was 0.2 g / cm³. 3 A linear elastic model with Young's modulus of 0.001 GPa and Poisson's ratio of 0.3.

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