# Computational Biomechanics: In-Silico Tools for the Investigation of Surgical Procedures and Devices

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## Abstract

**:**

## 1. Introduction

## 2. Overview on Computational Modeling of Biological Structures

## 3. Geometrical Characterization of Biological Structures

## 4. Constitutive Analysis of Biological Tissues’ Mechanics

**E**and

**S**are appropriate and corresponding strain and stress tensors, respectively. Material straining entails energy consumption, as the work of the stress. Such energy is partially stored into the material in a recoverable form; the material will give it back in the form of mechanical work. The remaining part of the work of the stress is dissipated. From a microstructural point of view, material straining leads to modifications of the microstructural configuration and energy adsorption. Such microstructural modifications may be reversible, leading to recoverable energy storage; otherwise, irreversible modifications entail energy dissipation. Furthermore, friction phenomena may occur during microstructural modifications, providing further energy dissipation in heat. The development of dissipative phenomena, as the irreversible microstructural modifications and the friction effects, is described by means of internal variables

**A**

^{i}. From a phenomenological point of view, the internal variables may describe plastic strains, damage effects, non-equilibrated viscous stress components, etc. The Helmholtz free energy ψ evaluates the recoverable stored energy, which is a function of the material strain history in terms of the strain

**E**and the internal variables

**A**

^{i}. Differentiation operations lead to the stress–strain relationship (Equation (1)). Algebraic differential equations

**G**specify the evolution of internal variables with the strain history (Equation (2)). The Helmholtz free energy ψ and the evolution equations

**G**must satisfy the dissipation rule (Equation (3)), which states that the energy dissipated during the strain history, as d

_{int}, has to increase or remain constant.

_{int}remains constant). All the work of the stress is recoverably stored and is evaluated by the Helmholtz free energy. According to the dissipation rule (Equation (3)) and to the stress–strain relationship (Equation (1)), the Helmholtz free energy ψ and the stress

**S**are functions of the strain

**E**only; the Helmholtz free energy ψ is a potential of the strain

**E**and is denominated strain energy function W. Otherwise, the dissipated energy d

_{int}increases during the strain history, and the material is said to have inelastic or dissipative behavior.

**p**is related to the properties of the specific material (i.e., elastic constants, viscous parameters, yielding and hardening constants, damage limits, etc.). Once the model mathematical formulation has been defined, the next step of the constitutive analysis pertains to the identification of such parameters. The action is usually based on an inverse analysis by assuming a stress–strain history given by experimental data, designed at the purpose, and estimating the parameters that yield the best fit with analytical or computational results based on the assumed constitutive model. Aiming at the univocal identification of parameters, the required experimental situations must be accurately defined [3].

#### 4.1. Hard Tissues

**σ**is the true stress tensor (Cauchy stress tensor),

**ε**the small strain tensor and

**D**the fourth rank elasticity tensor. The microstructural configuration of bone tissues entails the anisotropic behavior, because of the orientation of osteons and trabeculae along major stiffness directions. Transversally isotropic and orthotropic formulations of the elasticity tensor

**D**have been frequently adopted, which require the identification of five or nine independent elastic parameters, respectively. Parameters’ identification requires results from mechanical tests, which can be performed on tissue specimens from different bony structures, considering both animal models and human samples [16], or processing CT data by means of relationships between Hounsfield Unit values and elastic parameters [17].

**Q**

^{i}specify non-relaxed stress components, while γ

^{i}and τ

^{i}are associated viscoelastic parameters (the relative stiffness and the relaxation time, respectively). Differential equations (Equation (6)) provide the evolution criterion of viscoelastic internal variables [15].

**ε**

_{pl}specifies permanent strain components, while the yielding criterion φ (Equation (8)) governs the development of permanent strains [18].

#### 4.2. Soft Tissues

**P**and

**F**are the nominal stress tensor (first Piola-Kirchhoff stress tensor) and the deformation gradient, respectively. The composite configuration of soft biological tissues suggests splitting the strain energy function into an isotropic contribution of the ground matrix W

^{m}and anisotropic contributions of fiber families W

_{j}

^{f}(Equation (10)). The fabric tensor

**M**

_{j}is associated with the local orientation of the j

^{th}fiber family [8]. Many different hyperelastic models of soft biological tissues have been proposed in the scientific literature [20].

**Q**

^{i}specify non-relaxed stress components, while γ

^{i}and τ

^{i}are associated viscoelastic parameters [15].

#### 4.3. Identification of Constitutive Parameters

_{mod}, the applied strain conditions E and the assumed constitutive parameters

**p**:

_{mod}are compared with experimentally measured values, such as S

_{exp}, in consideration of all the q strain conditions investigated E

_{z}. A cost function evaluates the overall discrepancy between experimental data and predictive model results:

## 5. Computational Analysis Techniques

## 6. In Silico Analysis of Surgical Procedures: Case Studies

#### 6.1. Biomechanical Tools for the Reliability Assessment of Multi-Implant Systems in Dental Implantology

#### 6.2. Biomechanical Models for the Computational Evaluation of Gastrointestinal Bariatric Procedures

^{2})) ranging between 30 and 35 kg/m

^{2}, while values between 35 and 40 kg/m

^{2}identify severe obesity and morbid obesity shows values higher than 40 kg/m

^{2}.

## 7. Future Perspectives and Concluding Remark

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Biomechanical models and computational techniques for surgery: coupled experimental and computational activities for model’s definition, identification and validation; model exploitation for designing, optimizing and certifying surgical procedures, instrumentations and devices.

**Figure 2.**Virtual solid models: the mandible, the implant and the implants’ insertion within the premolar region (

**a**). The identification of the distribution of orthotropic elastic constants and direction from CT data (contours of the Young modulus along the distal–mesial direction, ranging between 0 and 21 GPa) (

**b**). The finite elements model of the system and misfit condition (

**c**). The results from computational analyses (plots and contours of compressive stress, ranging between 0 and 90 MPa, and plastic flow, ranging between 0% and 3%) (

**d**).

**Figure 3.**Constitutive model definition and identification of parameters by means of the inverse analysis of tensile tests on stomach tissue specimens (experimental data, as empty dots, and constitutive model results, as continuous lines) (

**a**). A reliability assessment of the constitutive model and parameters by means of membrane flexural tests (statistical distribution of experimental data, as a discontinuous line and gray band, and the computational model results, as a continuous line) (

**b**). The reliability assessment of the stomach finite element model by means of insufflation tests (statistical distribution of experimental data, as a discontinuous line and blue band, and the computational model results, as a continuous line) (

**c**). A computational analysis of stomach functionality in pre- and post-surgical configurations: pressure-volume behavior (

**d**) and maximum principal strain contours (ranging between 0% and 150%) (

**e**).

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**MDPI and ACS Style**

Carniel, E.L.; Toniolo, I.; Fontanella, C.G. Computational Biomechanics: In-Silico Tools for the Investigation of Surgical Procedures and Devices. *Bioengineering* **2020**, *7*, 48.
https://doi.org/10.3390/bioengineering7020048

**AMA Style**

Carniel EL, Toniolo I, Fontanella CG. Computational Biomechanics: In-Silico Tools for the Investigation of Surgical Procedures and Devices. *Bioengineering*. 2020; 7(2):48.
https://doi.org/10.3390/bioengineering7020048

**Chicago/Turabian Style**

Carniel, Emanuele Luigi, Ilaria Toniolo, and Chiara Giulia Fontanella. 2020. "Computational Biomechanics: In-Silico Tools for the Investigation of Surgical Procedures and Devices" *Bioengineering* 7, no. 2: 48.
https://doi.org/10.3390/bioengineering7020048