Computationally modelling cell proliferation: A review.

Cell proliferation is vital for the development and homeostasis of the human body. For such to occur, cells go through the cell cycle during which they replicate their genetic material and ultimately complete cellular division, when one cell divides into two new cells with equal genetic material. However, if there are some errors or abnormalities during the cell cycle that disrupt the balance between cell death and proliferation, severe problems can occur, such as tumour development, which is currently one of the leading causes of death in the world. Nowadays, mathematical and computational models are used to understand and study several biological mechanisms and processes, namely cellular proliferation. Over the last forty-five years, several models have attempted to describe cell proliferation and its regulation. Due to the complexity of the process, numerous assumptions and simplifications have been considered. This work presents a review of some of these models, focusing mainly on mammalian or generic eukaryotic models. Previously published continuum, discrete and hybrid approaches are presented and compared, in order to understand and highlight the relevance and capabilities of these models, their shortcomings and future challenges.

Computational simulation of cellular proliferation using a meshless method.

BACKGROUND AND OBJECTIVE: During cell proliferation, cells grow and divide in order to obtain two new genetically identical cells. Understanding this process is crucial to comprehend other biological processes. Computational models and algorithms have emerged to study this process and several examples can be found in the literature. The objective of this work was to develop a new computational model capable of simulating cell proliferation. This model was developed using the Radial Point Interpolation Method, a meshless method that, to the knowledge of the authors, was never used to solve this type of problem. Since the efficiency of the model strongly depends on the efficiency of the meshless method itself, the optimal numbers of integration points per integration cell and of nodes for each influence-domain were investigated. Irregular nodal meshes were also used to study their influence on the algorithm. METHODS: For the first time, an iterative discrete model solved by the Radial Point Interpolation Method based on the Galerkin weak form was used to establish the system of equations from the reaction-diffusion integro-differential equations, following a new phenomenological law proposed by the authors that describes the growth of a cell over time while dependant on oxygen and glucose availability. The discretization flexibility of the meshless method allows to explicitly follow the geometric changes of the cell until the division phase. RESULTS: It was found that an integration scheme of 6 × 6 per integration cell and influence-domains with only seven nodes allows to predict the cellular growth and division with the best balance between the relative error and the computing cost. Also, it was observed that using irregular meshes do not influence the solution. CONCLUSIONS: Even in a preliminary phase, the obtained results are promising, indicating that the algorithm might be a potential tool to study cell proliferation since it can predict cellular growth and division. Moreover, the Radial Point Interpolation Method seems to be a suitable method to study this type of process, even when irregular meshes are used. However, to optimize the algorithm, the integration scheme and the number of nodes inside the influence-domains must be considered.

Mechanical Effects of a Maylard Scar During a Vaginal Birth After a Previous Caesarean.

Caesarean section is one of the most common surgeries worldwide, even though there is no evidence supporting maternal and perinatal long-term benefits. Furthermore, the mechanical behavior of a caesarean scar during a vaginal birth after caesarean (VBAC) is not well understood since there are several questions regarding the uterine wound healing process. The aim of this study is to investigate the biomechanical Maylard fiber reorientation and stiffness influence during a VBAC through computational methods. A biomechanical model comprising a fetus and a uterus was developed, and a chemical-mechanical constitutive model that triggers uterine contractions was used, where some of the parameters were adjusted to account for the matrix and fiber stiffness increase in the caesarean scar. Several mechanical simulations were performed to analyze different scar fibers arrangements, considering different values for the respective matrix and fibers stiffness. The results revealed that a random fiber arrangement in the Maylard scar has a much higher impact on its mechanical behavior during a VBAC than the common fibers arrangement present in the uninjured uterine tissue. An increase of the matrix scar stiffness exhibits a lower impact, while an increase of the fiber's stiffness has no significant influence.

Application of an enhanced homogenization technique to the structural multiscale analysis of a femur bone.

Bone is a complex hierarchical material that can be characterized from the microscale to macroscale. This work demonstrates the application of an enhanced homogenization methodology to the multiscale structural analysis of a femoral bone. The use of this homogenization technique allows to remove subjectivity and reduce the computational cost associated with the iterative process of creating a heterogeneous mesh. Thus, it allows to create simpler homogenized meshes with its mechanical properties defined using information directly from the mesh source: the medical images. Therefore, this methodology is capable to accurately predict bone mechanical behavior in a fraction of the time required by classical approaches. The results show that using the homogenization technique, despite the differences between the used homogeneous and heterogeneous meshes, its mechanical behavior is similar. The proposed homogenization technique is useful for a multiscale modelling and it is computationally efficient.

A general framework for the numerical implementation of anisotropic hyperelastic material models including non-local damage.

The highly nonlinear mechanical behaviour of soft tissues solicited within the physiological range usually involves degradation of the material properties. Mechanically, having these biostructures undergoing such stretch patterns may bring about pathological conditions related to the steady deterioration of both collagen fibres and material's ground substance. Tissue and subject variability observed in the phenomenological mechanical characterisation of soft tissues often hinder the choice of the computational constitutive model. Therefore, this contribution brings forth a detailed overview of the constitutive implementation in a computational framework of anisotropic hyperelastic materials with damage. Surmounting the challenge posed by the mesh dependency pathology requires the incorporation of an integral-type non-local averaging, which seeks to include the effects of the microstructure in order to limit the localisation phenomena of the damage variables. By adopting this approach, one can make use of multiple developed material models available in the literature, a combination of those, or even propose new models within the same numerical framework. The numerical examples of three-dimensional displacement and force-driven boundary value problems highlight the possibility of using multiple material models within the same numerical framework. Particularities concerning the considered material models and the damage effect implications to represent the Mullins effect, induced anisotropy, hysteresis, and mesh dependency are discussed.

A Global Numerical analysis of the "central incisor/local maxillary bone" system using a meshless method.

In this work the maxillary central incisor is numerically analysed with an advance discretization technique--Natural Neighbour Radial Point Interpolation Method (NNRPIM). The NNRPIM permits to organically determine the nodal connectivity, which is essential to construct the interpolation functions. The NNRPIM procedure, based uniquely in the computational nodal mesh discretizing the problem domain, allows to obtain autonomously the required integration mesh, permitting to numerically integrate the differential equations ruling the studied physical phenomenon. A numerical analysis of a tooth structure using a meshless method is presented for the first time. A two-dimensional model of the maxillary central incisor, based on the clinical literature, is established and two distinct analyses are performed. First, a complete elasto-static analysis of the incisor/maxillary structure using the NNRPIM is evaluated and then a non-linear iterative bone tissue remodelling analysis of the maxillary bone, surrounding the central incisive, is performed. The obtained NNRPIM solutions are compared with other numerical methods solutions available in the literature and with clinical cases. The results show that the NNRPIM is a suitable numerical method to analyse numerically dental biomechanics problems.

Mechanical characterization and constitutive modelling of the damage process in rectus sheath.

The aim of this study is to characterize and model the damage process in the anterior rectus abdominal aponeurosis (anterior rectus sheath) undergoing finite deformations. The resistance of the anterolateral abdominal aponeuroses is important when planning the surgical repair of incisional hernias, among other medical procedures. Previous experiments in prolapsed vaginal tissue revealed that a softening process occurs before tissue rupture. This nonlinear damage behaviour requires a continuum damage theory commonly used to describe the softening behaviour of soft tissues under large deformations. The structural model presented here was built within the framework of non-linear continuum mechanics. Tissue damage was simulated considering different damage behaviours for the matrix and the collagen fibres. The model parameters were fit to the experimental data obtained from anterior rectus sheath samples undergoing finite deformations in uniaxial tension tests. The tests were carried out with samples cut along the direction of the collagen fibres, and transversal to the fibres. Longitudinal and transverse mechanical properties of human anterior rectus sheath are significantly different. The damage model was able to predict the stress-strain behaviour and the damage process accurately. The error estimations pointed to an excellent agreement between experimental results and model fittings. For all the fitted data, the normalized RMS error ε presented very low values and the coefficient of determination R(2) was close to 1. The present work constitutes the first attempt (as far as the authors know) to present a damage model for the human rectus sheath.

The influence of an occipito-posterior malposition on the biomechanical behavior of the pelvic floor.

OBJECTIVES: Contribute to the clarification of the mechanisms behind pelvic floor disorders related to a vaginal delivery. Verify the effect of an occipito-posterior malposition of the fetus during delivery on the stretch values when compared to the normal occipito-anterior position. STUDY DESIGN: A numerical simulation based on the Finite Element Method was carried out. The Finite Element Model intends to represent the effects that the passage of a fetal head can induce on the muscles of the pelvic floor, from a mechanical point of view. The model used for the simulation represents the pelvic bones, with the attached pelvic floor muscles and the fetus. In this work the movements of the fetus during birth, in vertex position, with the fetus presenting in an occipito-posterior malposition were simulated. The results obtained were compared with a simulation in which the fetus presents in the normal occipito-anterior position. RESULTS: A maximum stretch value of 1.73 was obtained in the numerical simulation conducted on this work, where the occipito-posterior malposition was simulated. CONCLUSION: During a vaginal delivery, the levator ani muscle and the pubococcygeus muscle are the muscles that are subjected to the largest values of stretch and strain. These muscles are the ones at greater risk for a stretch related injury. When compared to the normal occipito-anterior position, the occipito-posterior malposition produces substantially higher stretch vales for the pelvic floor muscles, increasing the risk for a stretch related injury.

Deformation of the pelvic floor muscles during a vaginal delivery.

Pelvic floor dysfunction is a hidden problem with a magnitude unknown to many. Statistics show that one in every ten women will have pelvic floor dysfunction so severe that it will require surgery. Several studies have shown that pelvic floor injuries during a vaginal delivery can be considered a significant factor in the development of urinary incontinence, fecal incontinence, and pelvic organ prolapse. The objective of the present work is to contribute to the clarification of the mechanisms behind pelvic floor disorders related to a vaginal delivery. For this purpose, a numerical simulation based on the finite element method was carried out. The finite element model intends to represent the effects that the passage of a fetal head can induce on the muscles of the pelvic floor, from a mechanical point of view. The model used for the simulation represents the pelvic bones, with the attached pelvic floor muscles and the fetus. In this work, we simulated the movements of the fetus during birth, in vertex position. We simulated the engagement, descent, flexion, internal rotation, and extension of the fetal head. Results for the pelvic floor stretch values obtained during the passage of the fetus head are presented; the deformation field is also shown. The results were obtained using the finite element method and a three-dimensional computer model of the pelvic floor and fetus. The maximum deformation obtained was 0.66 for a vertical displacement of the fetal head of approximately 60 mm.

Finite element studies of the deformation of the pelvic floor.

This article describes research involving finite element simulations of women's pelvic floor, undertaken in the engineering schools of Lisbon and Oporto, in collaboration with the medical school of Oporto. These studies are motivated by the pelvic floor dysfunctions that lead namely to urinary incontinence and pelvic organ prolapse. This research ultimately aims at: (i) contributing to clarify the primary mechanism behind such disorders; (ii) providing tools to simulate the pelvic floor function and the effects of its dysfunctions; (iii) contributing to planning and performing surgeries in a more controlled and reliable way. The finite element meshes of the levator ani are based on a publicly available geometric data set, and use triangular thin shell or special brick elements. Muscle and soft tissues are assumed as (quasi-)incompressible hyperelastic materials. Skeletal muscles are transversely isotropic with a single fiber direction, embedded in an isotropic matrix. The fibers considered in this work may be purely passive, or active with input of neuronal excitation and consideration of the muscle activation process. The first assumption may be adequate to simulate passive deformations of the pelvic muscles and tissues (namely, under the extreme loading conditions of childbirth). The latter may be adequate to model faster contractions that occur in time intervals of the same order as those of muscle activation and deactivation (as in preventing urinary incontinence in coughing or sneezing). Numerical simulations are presented for the active deformation of the levator ani muscle under constant pressure and neural excitation, and for the deformation induced by a vaginal childbirth.

A shell finite element model of the pelvic floor muscles.

The pelvic floor gives support to the organs in the abdominal cavity. Using the dataset made public in (Janda et al. J. Biomech. (2003) 36(6), pp. 749-757), we have reconstructed the geometry of one of the most important parts of the pelvic floor, the levator ani, using NURB surfaces. Once the surface is triangulated, the corresponding mesh is used in a finite element analysis with shell elements. Based on the 3D behavior of the muscle we have constructed a shell that takes into account the direction of the muscle fibers and the incompressibility of the tissue. The constitutive model for the isotropic strain energy and the passive strain energy stored in the fibers is adapted from Humphrey's model for cardiac muscles. To this the active behavior of the skeletal muscle is added. We present preliminary results of a simulation of the levator ani muscle under pressure and with active contraction. This research aims at helping simulate the damages to the pelvic floor that can occur after childbirth.