A mathematical model of tissue-engineered cartilage development under cyclic compressive loading

In this work a coupled model of solute transport and uptake, cell proliferation, extracellular matrix synthesis and remodeling of mechanical properties accounting for the impact of mechanical loading is presented as an advancement of a previously validated coupled model for free-swelling tissue-engineered cartilage cultures. Tissue-engineering constructs were modeled as biphasic with a linear elastic solid, and relevant intrinsic mechanical stimuli in the constructs were determined by numerical simulation for use as inputs of the coupled model. The mechanical dependent formulations were derived from a calibration and parametrization dataset and validated by comparison of normalized ratios of cell counts, total glycosaminoglycans and collagen after 24-h continuous cyclic unconfined compression from another dataset. The model successfully fit the calibration dataset and predicted the results from the validation dataset with good agreement, with average relative errors up to 3.1 and 4.3 %, respectively. Temporal and spatial patterns determined for other model outputs were consistent with reported studies. The results suggest that the model describes the interaction between the simultaneous factors involved in in vitro tissue-engineered cartilage culture under dynamic loading. This approach could also be attractive for optimization of culture protocols, namely through the application to longer culture times and other types of mechanical stimuli.     

Adhesion of tissue-engineered cartilate to native cartilage

Reconstruction of cartilaginous defects to correct both craniofacial deformities and joint surface irregularities remains a challenging and controversial clinical problem. It has been shown that tissue-engineered cartilage can be produced in a nude mouse model. Before tissue-engineered cartilage is used clinically to fill in joint defects or to reconstruct auricular or nasal cartilaginous defects, it is important to determine whether it will integrate with or adhere to the adjacent native cartilage at the recipient site. The purpose of this study was to determine whether tissue-engineered cartilage would adhere to adjacent cartilage in vivo. Tissue-engineered cartilage was produced using a fibrin glue polymer (80 mg/cc purified porcine fibrinogen polymerized with 50 U/cc bovine thrombin) mixed with fresh swine articular chondrocytes. The polymer/chondrocyte mixture was sandwiched between two 6-mm-diameter discs of fresh articular cartilage. These constructs were surgically inserted into a subcutaneous pocket on the backs of nude mice (n = 15). The constructs were harvested 6 weeks later and assessed histologically, biomechanically, and by electron microscopy. Control samples consisted of cartilage discs held together by fibrin glue alone (no chondrocytes) (n = 10). Histologic evaluation of the experimental constructs revealed a layer of neocartilage between the two native cartilage discs. The neocartilage appeared to fill all irregularities along the surface of the cartilage discs. Safranin-O and toluidine blue staining indicated the presence of glycosaminoglycans and collagen, respectively. Control samples showed no evidence of neocartilage formation. Electron microscopy of the neocartilage revealed the formation of collagen fibers similar in appearance to the normal cartilage matrix in the adjacent native cartilage discs. The interface between the neocartilage and the native cartilage demonstrated neocartilage matrix directly adjacent to the normal cartilage matrix without any gaps or intervening capsule. The mechanical properties of the experimental constructs, as calculated from stress-strain curves, differed significantly from those of the control samples. The mean modulus for the experimental group was 0.74 +/- 0.22 MPa, which was 3.5 times greater than that of the control group (p < 0.0002). The mean tensile strength of the experimental group was 0.064 +/- 0.024 MPa, which was 62.6 times greater than that of the control group (p < 0.0002). The mean failure strain of the experimental group was 0.16 +/- 0.061 percent, which was 4.3 times greater than that of the control group (p < 0.0002). Finally, the mean fracture energy of the experimental group was 0.00049 +/- 0.00032 J, which was 15.6 times greater than that of the control group. Failure occurred in all cases at the interface between neocartilage and native cartilage. This study demonstrated that tissue-engineered cartilage produced using a fibrin-based polymer does adhere to adjacent native cartilage and can be used to join two separate pieces of cartilage in the nude mouse model. Cartilage pieces joined in this way can withstand forces significantly greater than those tolerated by cartilage samplesjoined only by fibrin glue.     

Effects of mixing intensity on tissue-engineered cartilage

Mechanical forces regulate the structure and function of many tissues in vivo; recent results indicate that the mechanical environment can decisively influence the development of engineered tissues cultured in vitro. To investigate the effects of the hydrodynamic environment on tissue-engineered cartilage, primary bovine calf chondrocytes were seeded on fibrous polyglycolic acid meshes and cultured in spinner flasks either statically or at one of nine different turbulent mixing intensities. In medium from unmixed flasks, CO(2) accumulated and O(2) was depleted, whereas in medium from mixed flasks the concentrations of both gases approached their equilibrium values. Relative to constructs exposed to nonmixed conditions, constructs exposed to mixing contained higher fractions of collagen, synthesized and released more GAG, but contained lower fractions of GAG. Across the wide range of mixing intensities investigated, the presence or absence of mixing, but not the intensity of the mixing, was the primary determinant of the GAG and collagen content in the constructs. The all-or-none nature of these responses may provide insight into the mechanism(s) by which engineered cartilage perceives changes in its hydrodynamic environment and responds by modifying extracellular matrix production and release. 2001 John Wiley & Sons, Inc.     

A novel rotating-shaft bioreactor for two-phase cultivation of tissue-engineered cartilage

A novel rotating-shaft bioreactor (RSB) was developed for two-phase cultivation of tissue-engineered cartilage. The reactor consisted of a rotating shaft on which the chondrocyte/scaffold constructs (7.5 mm diameter x 3.5 mm thickness) were fixed and a reactor vessel half-filled with medium. The horizontal rotation of the shaft resulted in alternating exposure of the constructs to gas and liquid phases, thus leading to efficient oxygen and nutrient transfer, as well as periodically changing, mild shear stress exerting on the construct surfaces (0-0.32 dyn/cm2 at 10 rpm), as revealed by computer simulation. Strategic operation of the RSB (maintaining rotating speed at 10 rpm for 3 weeks and lowering the speed to 2 rpm in week 4) in combination with higher seeding density (6 x 10(6) chondrocytes/scaffold) and medium perfusion resulted in uniform cell distribution and increased glycosaminoglycan (3.1 mg/scaffold) and collagen (7.0 mg/scaffold) deposition. The 4-week constructs resembled native cartilages in terms of not only gross appearance and cell morphology but also distributions of glycosaminoglycan, total collagen, and type II collagen, confirming the maintenance of chondrocyte phenotype and formation of cartilage-like constructs in the RSB cultures. In summary, the novel RSB may be implicated for in vitro study of chondrogenesis and de novo cartilage development under periodic mechanical loading. With proper optimization of the culture conditions, a RSB may be employed for the production of cartilage-like constructs.     

A mathematical model of epiphyseal development: hypothesis on the cartilage canals growth

The role of cartilage canals is to transport nutrients and biological factors that cause the appearance of the secondary ossification centre (SOC). The SOC appears in the centre of the epiphysis of long bones. The canal development is a complex interaction between mechanical and biological factors that guide its expansion into the centre of the epiphysis. This article introduces the ‘Hypothesis on the growth of cartilage canals’. Here, we have considered that the development of these canals is an essential event for the appearance of SOC. Moreover, it is also considered to be important for the transport of molecular factors (RUNX2 and MMP9) at the ends of such canals. Once the canals are merged in the centre of the epiphysis, these factors are released causing hypertrophy of adjacent cells. This RUNX2 and MMP9 release occurs due to the action of mechanical loads that supports the epiphysis. In order to test this hypothesis, we use a hybrid approach using the finite element method to simulate the mechanical stresses present in the epiphysis and the cellular automata to simulate the expansion of the canals and the hypertrophy factors pathway. By using this hybrid approach, we have obtained as a result the spatial–temporal patterns for the growth of cartilage canals and hypertrophy factors within the epiphysis. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model may be used as a methodological basis to present a complete mathematical model of the processes involved in epiphyseal development.     

An analytical model to predict interstitial lubrication of cartilage in migrating contact areas

For nearly a century, articular cartilage has been known for its exceptional tribological properties. For nearly as long, there have been research efforts to elucidate the responsible mechanisms for application toward biomimetic bearing applications. It is now widely accepted that interstitial fluid pressurization is the primary mechanism responsible for the unusual lubrication and load bearing properties of cartilage. Although the biomechanics community has developed elegant mathematical theories describing the coupling of solid and fluid (biphasic) mechanics and its role in interstitial lubrication, quantitative gaps in our understanding of cartilage tribology have inhibited our ability to predict how tribological conditions and material properties impact tissue function. This paper presents an analytical model of the interstitial lubrication of biphasic materials under migrating contact conditions. Although finite element and other numerical models of cartilage mechanics exist, they typically neglect the important role of the collagen network and are limited to a specific set of input conditions, which limits general applicability. The simplified approach taken in this work aims to capture the broader underlying physics as a starting point for further model development. In agreement with existing literature, the model indicates that a large Peclet number, Pe, is necessary for effective interstitial lubrication. It also predicts that the tensile modulus must be large relative to the compressive modulus. This explains why hydrogels and other biphasic materials do not provide significant interstitial pressure under high Pe conditions. The model quantitatively agrees with in-situ measurements of interstitial load support and the results have interesting implications for tissue engineering and osteoarthritis problems. This paper suggests that a low tensile modulus (from chondromalacia or local collagen rupture after impact, for example) may disrupt interstitial pressurization, increase shear stresses, and activate a condition of progressive surface damage as a potential precursor of osteoarthritis.     

A reaction-diffusion system of a predator-prey-mutualist model

Mutualism is part of many significant processes in nature. Mutualistic benefits arising from modification of predator-prey interactions involve interactions of at least three species. In this paper we investigate the Homogeneous Neumann problem and Dirichlet problem for a reaction-diffusion system of three species—a predator, a mutualist-prey, and a mutualist. The existence, uniqueness, and boundedness of the solution are established by means of the comparison principle and the monotonicity method. For the Neumann problem, we analyze the constant equilibrium solutions and their stability. For the Dirichlet problem, we prove the global asymptotic stability of the trivial equilibrium solution. Specifically, we study the existence and the asymptotic behavior of two nonconstant equilibrium solutions. The main method used in studying of the stability is the spectral analysis to the linearized operators. The O.D.E. problem for the same model was proposed and studied in [13]. Through our results, we can see the influences of the diffusion mechanism and the different boundary value conditions upon the asymptotic behavior of the populations.     

碱性成纤维细胞生长因子与胶原对组织工程软骨体外构建的影响

目的：以三维成团培养为培养系统,探讨bFGF与胶原对组织工程软骨体外构建的影响。方法：成团培养兔生长板软骨细胞,设bFGF、胶原及联合作用组。HE染色观察新生组织形态;免疫组化检测Ⅰ、Ⅱ型胶原表达以观察细胞表型;Hoechst 33258法检测细胞DNA含量;羟脯氨酸法与阿新蓝法测定基质中胶原与蛋白多糖的合成。结果：新生软骨的组织学形态近似自然软骨;各实验组软骨细胞DNA含量明显上升;胶原可以显著促进基质的合成;各实验组Ⅰ型胶原的表达少于对照组,Ⅱ型胶原的表达则高于对照组;联合作用组效果更加明显。结论：三维的成团培养可以促进基质合成,有效维持软骨细胞表型;bFGF与胶原有利于工程化软骨构建,其效果具有协同效应,两者联合应用可进一步促进软骨再生。     

A reaction-diffusion model to capture disparity selectivity in primary visual cortex

Decades of experimental studies are available on disparity selective cells in visual cortex of macaque and cat. Recently, local disparity map for iso-orientation sites for near-vertical edge preference is reported in area 18 of cat visual cortex. No experiment is yet reported on complete disparity map in V1. Disparity map for layer IV in V1 can provide insight into how disparity selective complex cell receptive field is organized from simple cell subunits. Though substantial amounts of experimental data on disparity selective cells is available, no model on receptive field development of such cells or disparity map development exists in literature. We model disparity selectivity in layer IV of cat V1 using a reaction-diffusion two-eye paradigm. In this model, the wiring between LGN and cortical layer IV is determined by resource an LGN cell has for supporting connections to cortical cells and competition for target space in layer IV. While competing for target space, the same type of LGN cells, irrespective of whether it belongs to left-eye-specific or right-eye-specific LGN layer, cooperate with each other while trying to push off the other type. Our model captures realistic 2D disparity selective simple cell receptive fields, their response properties and disparity map along with orientation and ocular dominance maps. There is lack of correlation between ocular dominance and disparity selectivity at the cell population level. At the map level, disparity selectivity topography is not random but weakly clustered for similar preferred disparities. This is similar to the experimental result reported for macaque. The details of weakly clustered disparity selectivity map in V1 indicate two types of complex cell receptive field organization.     

A tissue-engineered model of fetal distal lung tissue

In extending our previous studies toward development of an engineered distal lung tissue construct (M. J. Mondrinos, S. Koutzaki, E. Jiwanmall, M. Li, J. P. Dechadarevian, P. I. Lelkes, and C. M. Finck. Tissue Eng 12: 717-728, 2006), we studied the effects of exogenous fibroblast growth factors FGF10, FGF7, and FGF2 on mixed populations of embryonic day 17.5 murine fetal pulmonary cells cultured in three-dimensional collagen gels. The morphogenic effects of the FGFs alone and in various combinations were assessed by whole mount immunohistochemistry and confocal microscopy. FGF10/7 significantly increased epithelial budding and proliferation; however, only FGF10 alone induced widespread budding. FGF7 alone induced dilation of epithelial structures but not widespread budding. FGF2 alone had a similar dilation, but not budding, effect in epithelial structures, and, in addition, significantly enhanced endothelial tubular morphogenesis and network formation, as well as mesenchymal proliferation. The combination of FGF10/7/2 induced robust budding of epithelial structures and the formation of uniform endothelial networks in parallel. These data suggest that appropriate combinations of exogenous FGFs chosen to target specific FGF receptor isoforms will allow for control of lung epithelial and mesenchymal cell behavior in the context of an engineered system. We propose that tissue-engineered fetal distal lung constructs could provide a potential source of tissue or cells for lung augmentation in pediatric pulmonary pathologies, such as pulmonary hypoplasia and bronchopulmonary dysplasia. In addition, engineered systems will provide alternative in vitro venues for the study of lung developmental biology and pathobiology.     

Construction of tissue-engineered cartilage using human placenta-derived stem cells

Human placenta-derived stem cells (hPDSCs) were isolated by trypsinization and further induced into cartilage cells in vitro. The engineered cartilage was constructed by combining hPDSCs with collagen sponge and the cartilage formation was observed by implantation into nude mice. Results showed that hPDSCs featured mesenchymal stem cells and maintained proliferation in vitro for over 30 passages while remaining undifferentiated. All results indicated that hPDSCs have the potential to differentiate into functional cartilage cells in vitro when combined with collagen sponge, which provided experimental evidence for prospective clinical application.     

Pattern formation and morphogenesis: A reaction-diffusion model

The problem of cellular differentiation and consequent pattern generation during embryonic development has been mathematically investigated with the help of a reaction-diffusion model. It is by now a well-recognized fact that diffusion of micromolecules (through intercellular gap junctions), which is dependent on the spatial parameter (r), serve the purpose of ‘positional information’ for differentiation. Based on this principle the present model has been constructed by coupling the Goodwin-type equations for RNA and protein synthesis with the diffusion process. The homogeneous Goodwin system can exhibit stable periodic solution if the value of the cooperativity as measured by the Hill coefficient (ρ) is greater than 8, which is not biologically realistic. In the present work it has been observed that inclusion of a negative cross-diffusion can drive the system into local instability for any value of ρ and thus a time-periodic spatial solution is possible around the unstable local equilibrium, eventually leading to a definite pattern formation. Inclusion of a negative cross-diffusion thus makes the system biologically realistic. The cross-diffusion can also give rise to a stationary wave-like dissipative structure.     

A delay reaction-diffusion model of the dynamics of botulinum in fish

A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.     

The influence of repair tissue maturation on the response to oscillatory compression in a cartilage defect repair model

This study examined the effects of mechanical compression on engineered cartilage in a novel hybrid culture system. Cylindrical holes were cut in discs of bovine articular cartilage and filled with agarose gels containing chondrocytes. These constructs were compressed in radiolabeled medium under static or oscillatory unconfined compression. Oscillatory compression at 1 Hz significantly stimulated synthesis above static control levels. Control experiments indicate that oscillatory compression does not stimulate freshly cast gels (without annuli), but does so after several weeks. This may be because physiologic fluid flow levels do not occur until sufficient extracellular matrix has accumulated. Finite element models predict minimal fluid flow in the gel core, and minimal differences in flow patterns between free and constrained gels. However, the models predict fluid pressures in constrained gels to be substantially higher than those in free gels. Our results suggest that pressure variations may influence synthesis of engineered cartilage matrices, with implications for construct development and post-implantation survival.     

Controlling incision-induced distortion of nasal septal cartilage: a model to predict the effect of scoring of rabbit septa

Cartilage can be shaped by scoring. In an exploratory study in living adult animals, this phenomenon was demonstrated in cartilage of the nasal septum. Bending was observed immediately after superficial scoring of the cartilage surface, and the cartilage always warped in the direction away from the scored side. The scored piece of cartilage still showed its initially distorted shape 10 weeks after primary surgery. In ex vivo experiments, a clear relation between incision depth and bending of septal cartilage was observed. Under these controlled conditions, the variation between different septa was small. Deformation of the septal specimens was increased by introducing single superficial incisions deepening to half the thickness of the cartilage. A positive correlation between incision depth and bending was demonstrated. A model was used to accurately predict the degree of bending of the cartilage after making an incision of a particular depth. Hence, the effect of cartilage scoring can be predicted. Because the results of this controlled study showed excellent reproducibility for different septa, it is expected that this model can be extrapolated to human nasal septum cartilage. This would enable the surgeon to better predict the result of cartilage scoring, either preoperatively or perioperatively.     

A theoretical model to predict the behavior of glycosylated hemoglobin levels

The measurement of glycosylated hemoglobin as a percentage of total hemoglobin is rapidly becoming the standard method of monitoring the average blood sugar level in diabetics for research purposes and may soon become the standard for clinical care and diagnosis. Much speculation exists in the literature about the nature of the glycosylation reaction. Most experimenters expect a linear relationship between the plasma glucose level and percent glycosylated hemoglobin in whole blood; however, a curve of decreasing slope with increasing glucose concentration is found.Here, a reaction model including simple first order kinetics between glucose and hemoglobin and a finite erythrocyte life of 120 days is considered. By carrying out the integration for each erythrocyte cohort followed by an integration combining all cohorts, a curve corresponding to the experimental result is found. In addition, results on expected glycosylated hemoglobin percent as a function of erythrocyte age and plasma glucose concentration are presented as well as a plot of glucose concentration versus glycosylated hemoglobin percent for the 40-day erythrocyte life in mice. All of the results correlate with experimental values in the literature if a rate constant of k = 1·0 × 10^?5dlmg^?1 day is used.The evaluation of a radioactive iron-transferrin experiment in the literature reveals the possibility that the glycosylation reaction begins during erythropoiesis.Finally, a curve is displayed which shows the expected 120-day decay during normoglycemia, of an elevated glycosylated hemoglobin level resulting from a preceding period of constant hyperglycemia.     

A reaction-diffusion model for interference in meiotic crossing over

One crossover point between a pair of homologous chromosomes in meiosis appears to interfere with occurrence of another in the neighborhood. It has been revealed that Drosophila and Neurospora, in spite of their large difference in the frequency of crossover points, show very similar plots of coincidence-a measure of the interference-against the genetic distance of the interval, defined as one-half the average number of crossover points within the interval. We here propose a simple reaction-diffusion model, where a "randomly walking" precursor becomes immobilized and matures into a crossover point. The interference is caused by pair-annihilation of the random walkers due to their collision and by annihilation of a random walker due to its collision with an immobilized point. This model has two parameters-the initial density of the random walkers and the rate of its processing into a crossover point. We show numerically that, as the former increases and/or the latter decreases, plotted curves of the coincidence vs. the genetic distance converge on a unique curve. Thus, our model explains the similarity between Drosophila and Neurospora without parameter values adjusted finely, although it is not a "genetic model" but is a "physical model," specifying explicitly what happens physically.