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Multiphysics models have become a key tool in understanding the way different phenomenon are related in bone remodeling and various approaches have been proposed, yet, to the best of the author's knowledge there is no model able to link a cell population model with a mechanical stimulus model using a discrete approach, which allows for an easy implementation. This article couples two classical models, the cell population model from Komarova and the Nackenhorst model in a 2D domain, where correlations between the mechanical loading and the cell population dynamics can be established, furthermore the effect of different paracrine and autocrine regulators is seen on the overall density of a portion of trabecular bone. A discretization is performed using frame 1D finite elements, representing the trabecular structure. The Nackenhorst model is implemented by using the finite element method to calculate the strain energy as the main mechanical stimulus that determines the bone mass density evolution in time. This density is normalized to be added to the bone mass percentage proposed by the Komarova model, where coupling terms have been added as well that guarantee a stable response. In the simulations, the equations were solved employing the finite element method with a user subroutine implemented in ABAQUS (2017) and by applying a direct formulation. The methodology presented can model the cell dynamics occurring in bone remodelling in accordance with the asynchronous nature of this process, yet allowing to differentiate zones with higher density, the main trabecular groups are obtained for the proximal femur. Finally, the model is tested in pathological cases, such as osteoporosis and osteopetrosis, yielding results similar to the pathology behavior. Furthermore, the discrete modelling technique is shown to be of use in this particular application.

The umbilical cord suspends the fetus within the amniotic cavity, where fetal dynamics is one of its many functions. Hence, the umbilical cord is a viable index in determining fetal activity. Fetal movements result in mechanical loads that are fundamental for fetal growth. At present, mechanical environment during early human fetal development is still largely unknown. To determine early fetal movement dynamics at given physiological (0.060 m) and pathological umbilical cord lengths (0.030 m, 0.020 m, 0.017 m and 0.014 m) a 2D computational model was created to simulate dynamic movement conditions. Main findings of this computational model revealed the shortest umbilical cord length (0.014 m) with a 6 ( 10 - 6 ) N , twitch force amplitude had a two-fold increase on linear velocity ( 0.12 m / s ) in comparison with other lengths ( 0.05 m / s ) . Moreover, umbilical cord length effect presented an increasing exponential tension on the fetus body wall from longest to shortest, from 0 N in the control length to 0.05 N for the shortest umbilical cord. Last, tension was always present over a period of time for the shortest cord (0.03 N to 0.08 N). Collectively, for all variables evaluated the shortest umbilical cord (0.014 m) presented remarkable differences with other lengths in particular with the second shortest umbilical cord (0.017 m), suggesting a 0.003 m difference represents a greater biomechanical effect. In conclusion, this computational model brings new insights required by clinicians, where the magnitude of these loads could be associated with different pathologies found in the clinic.

Metastasis is the rapid proliferation of cancer cells (secondary tumour) at a specific place, generally leading to death. This occurs at anatomical parts providing the necessary environment for vascularity, oxygen and food to hide their actions and trigger the rapid growth of cancer. Prostate and breast cancers, for example, use bone marrow for their proliferation. Bone-supporting cancer cells thus adapt to the environment, mimicking the behaviour of genetic and molecular bone cells. Evidence of this has been given in Cecchini et al. (2005, EAU Update Ser. 3:214-226), providing arguments such as how cancer cell growth is so active during bone reabsorption. This paper simulates metastasis activation in bone marrow. A mathematical model has been developed involving the activation of molecules from bone tissue cells, which are necessary for cancer to proliferate. Here, we simulate two forms of secondary tumour growth depending on the type of metastasis: osteosclerosis and osteolysis.

Previous studies have concluded that the chemical feedback (loop) between two reagent molecular factors through a reaction-diffusion mechanism could explain the stable spatial pattern found in the origin of the secondary ossification centres (SOCs). Furthermore, the emergence of the SOC may depend on the size and shape of the head of the bone, as observed in different animals. In this paper, we develop new computer simulations that study the effect of the size of the epiphysis on the emergence of the SOC. This study determines two or more SOCs, that may appear in the head of long bones, depending on the size of the epiphysis.

This paper introduces a 'hypothesis about the growth pattern of the secondary ossification centre (SOC)', whereby two phases are assumed. First, the formation of cartilage canals as an event essential for the development of the SOC. Second, once the canals are merged in the central zone of the epiphysis, molecular factors are released (primarily Runx2 and MMP9) spreading and causing hypertrophy of adjacent cells. In addition, there are two important molecular factors in the epiphysis: PTHrP and Ihh. The first one inhibits chondrocyte hypertrophy and the second helps the cell proliferation. Between these factors, there is negative feedback, which generates a highly localised and stable pattern over time. From a mathematical point of view, this pattern is similar to the patterns of Turing. The spread of Runx2 hypertrophies the cells from the centre to the periphery of the epiphysis until found with high levels of PTHrP to inhibit hypertrophy. This mechanism produces the epiphyseal bone-plate. Moreover, the hypertrophy is inhibited when the cells sense low shear stress and high pressure levels that maintain the articular cartilage structure. To test this hypothesis, we solve a system of coupled partial differential equations using the finite element method and we have obtained spatio-temporal patterns of the growth process of the SOC. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model can be used as a methodological basis to present a complete mathematical model of the whole epiphyseal development.