A mathematical model for bone tissue regeneration inside a specific type of scaffold

Bone tissue regeneration using scaffolds is receiving an increasing interest in orthopedic surgery and tissue engineering applications. In this study, we present the geometrical characterization of a specific family of scaffolds based on a face cubic centered (FCC) arrangement of empty pores leading to analytical formulae of porosity and specific surface. The effective behavior of those scaffolds, in terms of mechanical properties and permeability, is evaluated through the asymptotic homogenization theory applied to a representative volume element identified with the unit cell FCC. Bone growth into the scaffold is estimated by means of a phenomenological model that considers a macroscopic effective stress as the mechanical stimulus that regulates bone formation. Cell migration within the scaffold is modeled as a diffusion process based on Fick's law which allows us to estimate the cell invasion into the scaffold microstructure. The proposed model considers that bone growth velocity is proportional to the concentration of cells and regulated by the mechanical stimulus. This model allows us to explore what happens within the scaffold, the surrounding bone and their interaction. The mathematical model has been numerically implemented and qualitatively compared with previous experimental results found in the literature for a scaffold implanted in the femoral condyle of a rabbit. Specifically, the model predicts around 19 and 23% of bone regeneration for non-grafted and grafted scaffolds, respectively, both with an initial porosity of 76%.     

A multiscale computational fluid dynamics approach to simulate the micro-fluidic environment within a tissue engineering scaffold with highly irregular pore geometry

Mechanical stimulation can regulate cellular behavior, e.g., differentiation, proliferation, matrix production and mineralization. To apply fluid-induced wall shear stress (WSS) on cells, perfusion bioreactors have been commonly used in tissue engineering experiments. The WSS on cells depends on the nature of the micro-fluidic environment within scaffolds under medium perfusion. Simulating the fluidic environment within scaffolds will be important for gaining a better insight into the actual mechanical stimulation on cells in a tissue engineering experiment. However, biomaterial scaffolds used in tissue engineering experiments typically have highly irregular pore geometries. This complexity in scaffold geometry implies high computational costs for simulating the precise fluidic environment within the scaffolds. In this study, we propose a low-computational cost and feasible technique for quantifying the micro-fluidic environment within the scaffolds, which have highly irregular pore geometries. This technique is based on a multiscale computational fluid dynamics approach. It is demonstrated that this approach can capture the WSS distribution in most regions within the scaffold. Importantly, the central process unit time needed to run the model is considerably low.

     

A 3D computational model of perfusion seeding for investigating cell transport and adhesion within a porous scaffold

Biomechanics and Modeling in Mechanobiology - The process of cell seeding within a porous scaffold is an essential first step in the development of tissue-engineered bone grafts. Understanding the...     

Distribution of bone-marrow stromal cells in a 3D scaffold depending on the seeding method and the scaffold inside a surface modification

The distribution of bone-marrow stromal cells (BMSC) was studied in 3D polylactide scaffolds. Seeding of cells into the scaffold by the dynamic method (with the aid of a peristaltic pump) has been shown to provide distribution of cells throughout the entire scaffold volume, unlike the static method of seeding, in which the cell suspension is applied onto the scaffold surface. Unlike the cells seeded into the scaffold by the dynamic method, the cells seeded by the static method practically completely migrate from the scaffold on the dish for the first several days. It is revealed that BMSCs cultivated in 3D polylactide scaffolds modified by fibrin form colonies, whereas BMSCs cultivated inside scaffolds modified by collagen type 1 distribute all over the scaffold volume in the form of individual cells.     

A biomechanical model to simulate the effect of a high vertical loading on trunk flexural stiffness

A human trunk model was developed to simulate the effect of a high vertical loading on trunk flexural stiffness. A force–length relationship is attributed to each muscle of the multi-body model. Trunk stiffness and muscle forces were evaluated experimentally and numerically for various applied loads. Experimental evaluation of trunk stiffness was carried out by measuring changes in reaction force following a sudden horizontal displacement at the T10 level prior to paraspinal reflexes induction. Results showed that the trunk stiffness increases under small applied loads, peaks when the loads were further increased and decreases when higher loads are applied. A sensitivity analysis to muscle force–length relationship is provided to determine the model's limitations. This model pointed out the importance of taking into account the changes in muscle length to evaluate the effect of spinal loads beyond the safe limit that cannot be evaluated experimentally and to predict the trunk instability under vertical load.     

Bone formation induced by a novel form of mechanical loading on joint tissue.

Because of insufficient mechanical loading, exposure to weightlessness in space flight reduces bone mass. In order to maintain bone mass in a weightless condition, we investigated a novel form of mechanical loading--joint loading. Since some part of gravity-induced loading to our skeletal system is absorbed by viscoelastic deformation of joint tissues, we hypothesized that deformation of joint tissues would generate fluid flow in bone and stimulate bone formation in diaphyseal cortical bone. In order to test the hypothesis, we applied directly oscillatory loading to an elbow joint of mice and conducted bone histomorphometry on the diaphysis of ulnae. Using murine femurs ex vivo, streaming potentials were measured to evaluate a fluid flow induced by joint loading. Bone histomorphometry revealed that compared to no loading control, elbow loading increased mineralizing surface, mineral apposition rate, and bone formation rate 3.2-fold, 3.0-fold, and 7.9-fold, respectively. We demonstrated that joint loading generated a streaming potential in a medullar cavity of femurs. The results support a novel mechanism, in which joint loading stimulates effectively bone formation possibly by generating fluid flow, and suggest that a supportive attachment to joints, driven passively or actively, would be useful to maintain bone mass of astronauts during an exposure to weightlessness.     

A general-purpose framework to simulate musculoskeletal system of human body: using a motion tracking approach*

Computation of muscle force patterns that produce specified movements of muscle-actuated dynamic models is an important and challenging problem. This problem is an undetermined one, and then a proper optimization is required to calculate muscle forces. The purpose of this paper is to develop a general model for calculating all muscle activation and force patterns in an arbitrary human body movement. For this aim, the equations of a multibody system forward dynamics, which is considered for skeletal system of the human body model, is derived using Lagrange–Euler formulation. Next, muscle contraction dynamics is added to this model and forward dynamics of an arbitrary musculoskeletal system is obtained. For optimization purpose, the obtained model is used in computed muscle control algorithm, and a closed-loop system for tracking desired motions is derived. Finally, a popular sport exercise, biceps curl, is simulated by using this algorithm and the validity of the obtained results is evaluated via EMG signals.     

Image-guided simulation of tissue deformation using a mechanical model on a surgical application

This paper presents a method for localizing the position of a liver and a tumor within the tissue during a minimally invasive liver operation. From pre-operative CT scans, the liver volume and its internal structures are segmented using image-processing techniques. Based on these segmentations, a three-dimensional mechanical model is built to compute the liver volume and internal structure displacement under boundary conditions such as external forces from the surgical instrument. This can help the surgeon understand the motion of internal structures when manipulating the liver. To validate our method, an experiment on a porcine liver explant was performed to assess the difference between actual tissue motion and the mechanical model.     

Effect of mechanical loading on expiratory and inspiratory muscle activity during NREM sleep

We investigated the effect of acute and sustained inspiratory resistive loading (IRL) on the activity of expiratory abdominal muscles (EMGab) and the diaphragm (EMGdi) and on ventilation during wakefulness and non-rapid-eye-movement (NREM) sleep in healthy subjects. EMGdi and EMGab were measured with esophageal and transcutaneous electrodes, respectively. During wakefulness, EMGdi increased in response to acute loading (18 cmH2O.l-1.s) (+23%); this was accompanied by preservation of tidal volume (VT) and minute ventilation (VE). During NREM sleep, no augmentation was noted in EMGdi or EMGab. Inspiratory time (TI) was prolonged (+5%), but this was not sufficient to prevent a decrease in both VT and VE (-21 and -20%, respectively). During sustained loading (12 cmH2O.l-1 s) in NREM sleep, control breaths (C) were compared with the steady-state loaded breaths (SS) defined by breaths 41-50. Steady-state IRL was associated with augmentation of EMGdi (12%) and EMGab (50%). VT returned to control levels, expiratory time shortened, and breathing frequency increased. The net result was the increase in VE above control levels (+5%, P less than 0.01). No change was noted in end-tidal CO2 or O2. We concluded that 1) wakefulness is a prerequisite for immediate load compensation (in its absence, TI prolongation is the only compensatory response) and 2) during sustained IRL, the augmentation of EMGdi and EMGab can lead to complete ventilatory recovery without measurable changes in chemical stimuli.     

Muscle cell outside and inside: the nascent approach of a clinician.

Understanding muscle cell in disease and health is an unfinished process. Following the lead of Jaime Alvarez, I have had the opportunity of working on two complementary approaches to this field. One is the study of muscle cell surface molecules. Both synaptic muscle molecules, such as the asymmetric form of acetylcholinesterase, and extrasynaptic molecules, such as the extracellular matrix proteoglycans, are regulated by the motor nerve activity. This illustrates one of Jaime's teachings: cell phenotypes are a dynamic process that reflects the influence of other cells (Alvarez, 2001). Proteoglycans have many functions, including growth factor receptors. Studying them in muscular dystrophy will contribute to the comprehension of the muscle regeneration failure, characteristic of this disease. Muscle cells are highly dependent upon energy production, and the mitochondriae produce most of it. These organelles are unique in having their own genome. Mutations in these genes have recently been recognized as the cause of human disease and originally in muscle pathology. The physiopathology of these diseases is summarized here.     

Influence of viscosity on myocardium mechanical activity: a mathematical model

We have previously proposed and validated a mathematical model of myocardium contraction-relaxation cycle based on current knowledge of regulatory role of Ca2+ and cross-bridge kinetics in cardiac cell. That model did not include viscous elements. Here we propose a modification of the model, in which two viscous elements are added, one in parallel to the contractile element, and one more in parallel to the series elastic element. The modified model allowed us to simulate and explain some subtle experimental data on relaxation velocity in isotonic twitches and on a mismatch between the time course of sarcomere shortening/lengthening and the time course of active force generation in isometric twitches. Model results were compared with experimental data obtained from 28 rat LV papillary muscles contracting and relaxing against various loads. Additional model analysis suggested contribution of viscosity to main inotropic and lusitropic characteristics of myocardium performance.     

A dynamical study of the mechanical stimuli and tissue differentiation within a CaP scaffold based on micro-CT finite element models

The control of the mechanical stimuli transmitted to the cells is critical for the design of functional scaffolds for tissue engineering. The objective of this study was to investigate the dynamics of the mechanical stimuli transmitted to the cells during tissue differentiation in an irregular morphology scaffold under compressive load and perfusion flow. A calcium phosphate-based glass porous scaffold was used. The solid phase and the fluid flow within the pores were modeled as linear elastic solid material and Newtonian fluid, respectively. In the fluid model, different levels of viscosity were used to simulate tissue differentiation. Compressive strain of 0.5% and fluid flow with constant inlet velocity of 10 μm/s or constant inlet pressure of 3 Pa were applied. Octahedral shear strain and fluid shear stress were used as mechano-regulatory stimuli. For constant inlet velocity, stimuli equivalent to bone were predicted in 80% of pore volume for the case of low tissue viscosity. For the cases of high viscosity, fluctuations between stimuli equivalent to tissue formation and cell death were predicted due to the increase in the fluid shear stress when tissue started to fill pores. When constant pressure was applied, stimuli equivalent to bone were predicted in 62% of pore volume when low tissue viscosity was used and 42% when high tissue viscosity was used. This study predicted critical variations of fluid shear stress when cells differentiated. If these variations are not controlled in vitro, they can impede the formation of new matured tissue.     

Effects of wind and thermal conditions on timberline formation in central Japan: a lattice model

The upper distribution limit of tall tree species Abies mariesii is the timberline in central Japan, and dwarf pine Pinus pumila dominates above the timberline to near the summit. My previous studies suggested that the main cause of the timberline formation is the increase in mortality due to strong wind in winter rather than low growth due to low summer temperature. This study evaluated how wind velocity affects timberline formation and if the altitude of timberline moves upward due to high thermal conditions, by using a lattice model. Increase in wind velocity throughout the altitude lowered the altitudes of upper distribution limits of the two species. On the contrary, prolonged growth period due to high thermal conditions increased the upper distribution limit of P. pumila, and the upper distribution limit of A. mariesii was hardly affected by the change of growth period. However, the upward shift of the upper distribution limit of P. pumila due to the prolonged growth period in the model would not be realistic because P. pumila had already distributed up to near the summit. This study concludes that A. mariesii is a superior competitor to P. pumila at low altitudes with low wind velocity, but dwarf pine P. pumila can dominate at higher altitudes because A. mariesii suffers severe mechanical damage due to strong wind in winter, and that the altitude of the timberline does not move upward even under high thermal conditions due to global warming.     

Development of a cell culture system loading cyclic mechanical strain to chondrogenic cells

Mechanical stimulation is considered to be one of the major epigenetic factors regulating the metabolism, proliferation, survival and differentiation of cells in the skeletal tissues. It is generally accepted that the cytoskeleton can undergo remodeling in response to mechanical stimuli such as tensile strain or fluid flow. Mechanically induced cell deformation is one of the possible mechanotransduction pathways by which chondrocytes sense and respond to changes in their mechanical environment. Mechanical strain has a variety of effects on the structure and function of their cells in the skeletal tissues, such as chondrocytes, osteoblasts and fibroblasts. However, little is known about the effect of the quality and quantity of mechanical strain and the timing of mechanical loading on the differentiation of these cells. The present study was designed to investigate the effect of the deformation of chondrogenic cells, and cyclic compression using a newly developed culture device, by analyzing mechanobiological response to the differentiating chondrocytes. Cyclic compression between 0 and 22% strains, at 23 microHz was loaded on chondrogenic cell line ATDC5 by seeding in a mass mode on PDMS membrane, assuming direct transfer of cyclic deformation from the membrane to the cells at the same frequency. The compressive strain, induced within the membrane, was characterized based on the analysis of the finite element modeling (FEM). The results showed that the tensile strain inhibits the chondrogenic differentiation of ATDC5 cells, whereas the compressive strain enhances the chondrogenic differentiation, suggesting that the differentiation of the chondrogenic cells could be controlled by the amount and the mode of strain. In conclusion, we have developed a unique strain loading culture system to analyze the effect of various types of mechanical stimulation on various cellular activities.     

The effect of perfusion on soft tissue mechanical properties: a computational model

Some simple finite element models were constructed to investigate the magnitude and character of changes in mechanical properties of very soft tissues due to the loss of perfusion. Changes in the apparent stress-strain curve were used to characterise the effect of simulated blood perfusion pressure on the engineering stress-strain curve. The results indicated that the blood to tissue volume ratio and the perfusion pressure have the strongest effect on the effective stress-strain response of a representative tissue cell. Tissue viscoelasticity increased the sensitivity of the system to perfusion pressure changes.     

A cross-scale model coupling approach to simulate the risk-reduction effect of natural adaptation on soybean production under climate change

This study establishes a procedure to couple Decision Support System for Agrotechnology Transfer (DSSAT) and China Agroecological Zone model (AEZ-China). This procedure enables us to quantify the effects of two natural adaptation measures on soybean production in China, concern on which has been growing owing to the rapidly rising demand for soybean and the foreseen global climate change. The parameters calibration and mode verification are based on the observation records of soybean growth at 13 agro-meteorological observation stations in Northeast China and Huang-Huai-Hai Plain over 1981–2011. The calibration of eco-physiological parameters is based on the algorithms of DSSAT that simulate the dynamic bio-physiological processes of crop growth in daily time-step. The effects of shifts in planting day and changes in the length of growth cycle (LGC) are evaluated by the speedy algorithms of AEZ. Results indicate that without adaptation, climate change from the baseline 1961–1990 to the climate of 2050s as specified in the Providing Regional Climate for Impacts Studies-A1B would decrease the potential yield of soybean. By contrast, simulations of DSSAT using AEZ-recommended cultivars with adaptive LGC and also the corresponding adaptive planting dates show that the risk of yield loss could be fully or partially mitigated across majority of grid cells in the major soybean-growing areas.