A Novel Simulation Model for Engineered Cartilage Growth in Static Systems

Structured population balances to support microalgae-based processes: Review of the state-of-art and perspectives analysis

Design and optimization of microalgae processes have traditionally relied on the application of unsegregated mathematical models, thus neglecting the impact of cell-to-cell heterogeneity. However, there is experimental evidence that the latter one, including but not limited to variation in mass/size, internal composition and cell cycle phase, can play a crucial role in both cultivation and downstream processes. Population balance equations (PBEs) represent a powerful approach to develop mathematical models describing the effect of cell-to-cell heterogeneity. In this work, the potential of PBEs for the analysis and design of microalgae processes are discussed. A detailed review of PBE applications to microalgae cultivation, harvesting and disruption is reported. The review is largely focused on the application of the univariate size/mass structured PBE, where the size/mass is the only internal variable used to identify the cell state. Nonetheless, the need, addressed by few studies, for additional or alternative internal variables to identify the cell cycle phase and/or provide information about the internal composition is discussed. Through the review, the limitations of previous studies are described, and areas are identified where the development of more reliable PBE models, driven by the increasing availability of single-cell experimental data, could support the understanding and purposeful exploitation of the mechanisms determining cell-to-cell heterogeneity.

Articular cartilage regeneration and tissue engineering models: a systematic review

INTRODUCTION

Cartilage regeneration and restoration is a major topic in orthopedic research as cartilaginous degeneration and damage is associated with osteoarthritis and joint destruction. This systematic review aims to summarize current research strategies in cartilage regeneration research.

MATERIALS AND METHODS

A Pubmed search for models investigating single-site cartilage defects as well as chondrogenesis was conducted and articles were evaluated for content by title and abstract. Finally, only manuscripts were included, which report new models or approaches of cartilage regeneration.

RESULTS

The search resulted in 2217 studies, 200 of which were eligible for inclusion in this review. The identified manuscripts consisted of a large spectrum of research approaches spanning from cell culture to tissue engineering and transplantation as well as sophisticated computational modeling.

CONCLUSIONS

In the past three decades, knowledge about articular cartilage and its defects has multiplied in clinical and experimental settings and the respective body of research literature has grown significantly. However, current strategies for articular cartilage repair have not yet succeeded to replicate the structure and function of innate articular cartilage, which makes it even more important to understand the current strategies and their impact. Therefore, the purpose of this review was to globally summarize experimental strategies investigating cartilage regeneration in vitro as well as in vivo. This will allow for better referencing when designing new models or strategies and potentially improve research translation from bench to bedside.

Comparison of patient-specific computational models vs. clinical follow-up, for adjacent segment disc degeneration and bone remodelling after spinal fusion

Spinal fusion is a standard surgical treatment for patients suffering from low back pain attributed to disc degeneration. However, results are somewhat variable and unpredictable. With fusion the kinematic behaviour of the spine is altered. Fusion and/or stabilizing implants carrying considerable load and prevent rotation of the fused segments. Associated with these changes, a risk for accelerated disc degeneration at the adjacent levels to fusion has been demonstrated. However, there is yet no method to predict the effect of fusion surgery on the adjacent tissue levels, i.e. bone and disc. The aim of this study was to develop a coupled and patient-specific mechanoregulated model to predict disc generation and changes in bone density after spinal fusion and to validate the results relative to patient follow-up data. To do so, a multiscale disc mechanoregulation adaptation framework was developed and coupled with a previously developed bone remodelling algorithm. This made it possible to determine extra cellular matrix changes in the intervertebral disc and bone density changes simultaneously based on changes in loading due to fusion surgery. It was shown that for 10 cases the predicted change in bone density and degeneration grade conforms reasonable well to clinical follow-up data. This approach helps us to understand the effect of surgical intervention on the adjacent tissue remodelling. Thereby, providing the first insight for a spine surgeon as to which patient could potentially be treated successfully by spinal fusion and in which patient has a high risk for adjacent tissue changes.

Extracellular Matrix Deposition in Engineered Micromass Cartilage Pellet Cultures: Measurements and Modelling

This article explores possible mechanisms governing extracellular matrix deposition in engineered cartilaginous cell pellets. A theoretical investigation is carried out alongside an experimental study measuring proteoglycan and collagen volume fractions within murine chondrogenic (ATDC-5) cell pellets. The simple mathematical model, which adopts a nutrient-dependent proteoglycan production rate, successfully reproduces the periphery-dominated proteoglycan deposition, characteristic of the growth pattern observed experimentally within pellets after 21 days of culture. The results suggest that this inhomogeneous proteoglycan production is due to nutrient deficiencies at the pellet centre. Our model analysis further indicates that a spatially uniform distribution of proteoglycan matrix could be maintained by initiating the culture process with a smaller-sized pellet. Finally, possible extensions are put forward with an aim to improve the model predictions for the early behaviour, where different mechanisms appear to dominate the matrix production within the pellets.

A novel quantitative model of cell cycle progression based on cyclin-dependent kinases activity and population balances

Cell cycle regulates proliferative cell capacity under normal or pathologic conditions, and in general it governs all in vivo/in vitro cell growth and proliferation processes. Mathematical simulation by means of reliable and predictive models represents an important tool to interpret experiment results, to facilitate the definition of the optimal operating conditions for in vitro cultivation, or to predict the effect of a specific drug in normal/pathologic mammalian cells. Along these lines, a novel model of cell cycle progression is proposed in this work. Specifically, it is based on a population balance (PB) approach that allows one to quantitatively describe cell cycle progression through the different phases experienced by each cell of the entire population during its own life. The transition between two consecutive cell cycle phases is simulated by taking advantage of the biochemical kinetic model developed by Gérard and Goldbeter (2009) which involves cyclin-dependent kinases (CDKs) whose regulation is achieved through a variety of mechanisms that include association with cyclins and protein inhibitors, phosphorylation-dephosphorylation, and cyclin synthesis or degradation. This biochemical model properly describes the entire cell cycle of mammalian cells by maintaining a sufficient level of detail useful to identify check point for transition and to estimate phase duration required by PB. Specific examples are discussed to illustrate the ability of the proposed model to simulate the effect of drugs for in vitro trials of interest in oncology, regenerative medicine and tissue engineering.

Experimental analysis and modelling of in vitro HUVECs proliferation in the presence of various types of drugs

OBJECTIVES

This study focuses on experimental analysis and corresponding mathematical simulation of in vitro HUVECs (human umbilical vein endothelial cells) proliferation in the presence of various types of drugs.

MATERIALS AND METHODS

HUVECs, once seeded in Petri dishes, were expanded to confluence. Temporal profiles of total count obtained by classic haemocytometry and cell size distribution measured using an electronic Coulter counter, are quantitatively simulated by a suitable model based on the population balance approach. Influence of drugs on cell proliferation is also properly simulated by accounting for suitable kinetic equations.

RESULTS AND DISCUSSION

The models' parameters have been determined by comparison with experimental data related to cell population expansion and cell size distribution in the absence of drugs. Inhibition constant for each type of drug has been estimated by comparing the experimental data with model results concerning temporal profiles of total cell count. The reliability of the model and its predictive capability have been tested by simulating cell size distribution for experiments performed in the presence of drugs. The proposed model will be useful in interpreting effects of selected drugs on expansion of readily available human cells.

In vitro ovine articular chondrocyte proliferation: experiments and modelling

This study focuses on analysis of in vitro cultures of chondrocytes from ovine articular cartilage. Isolated cells were seeded in Petri dishes, then expanded to confluence and phenotypically characterized by flow cytometry. The sigmoidal temporal profile of total counts was obtained by classic haemocytometry and corresponding cell size distributions were measured electronically using a Coulter Counter. A mathematical model recently proposed (1) was adopted for quantitative interpretation of these experimental data. The model is based on a 1-D (that is, mass-structured), single-staged population balance approach capable of taking into account contact inhibition at confluence. The model's parameters were determined by fitting measured total cell counts and size distributions. Model reliability was verified by predicting cell proliferation counts and corresponding size distributions at culture times longer than those used when tuning the model's parameters. It was found that adoption of cell mass as the intrinsic characteristic of a growing chondrocyte population enables sigmoidal temporal profiles of total counts in the Petri dish, as well as cell size distributions at 'balanced growth', to be adequately predicted.

A validated model of GAG deposition, cell distribution, and growth of tissue engineered cartilage cultured in a rotating bioreactor

In this work a new phenomenological model of growth of cartilage tissue cultured in a rotating bioreactor is developed. It represents an advancement of a previously derived model of deposition of glycosaminoglycan (GAG) in engineered cartilage by (i) introduction of physiological mechanisms of proteoglycan accumulation in the extracellular matrix (ECM) as well as by correlating (ii) local cell densities and (iii) tissue growth to the ECM composition. In particular, previously established predictions and correlations of local oxygen concentrations and GAG synthesis rates are extended to distinguish cell secreted proteoglycan monomers free to diffuse in cell surroundings and outside from the engineered construct, from large aggrecan molecules, which are constrained within the ECM and practically immovable. The model includes kinetics of aggregation, that is, transformation of mobile GAG species into immobile aggregates as well as maintenance of the normal ECM composition after the physiological GAG concentration is reached by incorporation of a product inhibition term. The model also includes mechanisms of the temporal evolution of cell density distributions and tissue growth under in vitro conditions. After a short initial proliferation phase the total cell number in the construct remains constant, but the local cell distribution is leveled out by GAG accumulation and repulsion due to negative molecular charges. Furthermore, strong repulsive forces result in expansion of the local tissue elements observed macroscopically as tissue growth (i.e., construct enlargement). The model is validated by comparison with experimental data of (i) GAG distribution and leakage, (ii) spatial-temporal distributions of cells, and (iii) tissue growth reported in previous works. Validation of the model predictive capability--against a selection of measured data that were not used to construct the model--suggests that the model successfully describes the interplay of several simultaneous processes carried out during in vitro cartilage tissue regeneration and indicates that this approach could also be attractive for application in other tissue engineering systems.

A 3D hybrid model for tissue growth: the interplay between cell population and mass transport dynamics

To provide theoretical guidance for the design and in vitro cultivation of bioartificial tissues, we have developed a multiscale computational model that can describe the complex interplay between cell population and mass transport dynamics that governs the growth of tissues in three-dimensional scaffolds. The model has three components: a transient partial differential equation for the simultaneous diffusion and consumption of a limiting nutrient; a cellular automaton describing cell migration, proliferation, and collision; and equations that quantify how the varying nutrient concentration modulates cell division and migration. The hybrid discrete-continuous model was parallelized and solved on a distributed-memory multicomputer to study how transport limitations affect tissue regeneration rates under conditions encountered in typical bioreactors. Simulation results show that the severity of transport limitations can be estimated by the magnitude of two dimensionless groups: the Thiele modulus and the Biot number. Key parameters including the initial seeding mode, cell migration speed, and the hydrodynamic conditions in the bioreactor are shown to affect not only the overall rate, but also the pattern of tissue growth. This study lays the groundwork for more comprehensive models that can handle mixed cell cultures, multiple nutrients and growth factors, and other cellular processes, such as cell death.

Experimental analysis and modelling of in vitro proliferation of mesenchymal stem cells

OBJECTIVES

Stem cell therapies based on differentiation of adult or embryonic stem cells into specialized ones appear to be effective for treating several human diseases. This work addresses the mathematical simulation of proliferation kinetics of stem cells.

MATERIALS AND METHODS

Sheep bone marrow mesenchymal stem cells (phenotype characterized by flow cytometry analysis) seeded at different initial concentrations in Petri dishes were expanded to confluence. Sigmoid temporal profiles of total counts obtained through classic haemocytometry were quantitatively interpreted by both a phenomenological logistic equation and a novel model based on a one-dimensional, single-staged population balance approach capable of taking into account contact inhibition at confluence. The models' parameters were determined by comparison with experimental data on population expansion starting from single seeding concentration. Reliability of the models was tested by predicting cell proliferation carried out starting from different seeding concentrations.

RESULTS AND DISCUSSION

It was found that the proposed population balance modelling approach was successful in predicting the experimental data over the whole range of initial cell numbers investigated, while prediction capability of phenomenological logistic equation was more limited.

A simulation model for stem cells differentiation into specialized cells of non-connective tissues

A novel mathematical model to simulate stem cells differentiation into specialized cells of non-connective tissues is proposed. The model is based upon material balances for growth factors coupled with a mass-structured population balance describing cell growth, proliferation and differentiation. The proposed model is written in a general form and it may be used to simulate a generic cell differentiation pathway during in vitro cultivation when specific growth factors are used. Literature experimental data concerning the differentiation of central nervous stem cells into astrocytes are successfully compared with model results, thus demonstrating the validity of the proposed model as well as its predictive capability. Finally, sensitivity analysis of model parameters is also performed in order to clarify what mechanisms most strongly influence differentiation and cell types distribution.

A novel simulation model for stem cells differentiation

A novel mathematical model to simulate mesenchymal stem cells differentiation into specialized cells is proposed. The model is based upon material balances for extracellular matrix compounds, growth factors and nutrients coupled with a mass-structured population balance describing cell growth, proliferation and differentiation. The proposed model is written in a general form and it may be used to simulate a generic cell differentiation pathway occurring in vivo or during in vitro cultivation when specific growth factors are used. Literature experimental data concerning the differentiation of mesenchymal stem cells into chondrocytes in terms of total DNA and glycosaminoglycan content are successfully compared with model results, thus demonstrating the validity of the proposed model as well as its predictive capability. A further test of the model capability is performed for the case of in vivo fracture healing during which mesenchymal stem cells differentiate into chondrocytes and osteoblasts. Considerations about the extension of the proposed model to different pathologies beside fracture healing are reported. Finally, sensitivity analysis of model parameters is also performed in order to clarify what mechanisms most strongly influence differentiation and the distribution of cell types.