Mathematical Modeling of Three-Dimensional Cell Cultures in Perfusion Bioreactors

Osteogenic Activity on NaOH-Etched Three-Dimensional-Printed Poly-ɛ-Caprolactone Scaffolds in Perfusion or Spinner Flask Bioreactor

Bioreactor systems, for example, spinner flask and perfusion bioreactors, and cell-seeded three-dimensional (3D)-printed scaffolds are used in bone tissue engineering strategies to stimulate cells and produce bone tissue suitable for implantation into the patient. The construction of functional and clinically relevant bone graft using cell-seeded 3D-printed scaffolds within bioreactor systems is still a challenge. Bioreactor parameters, for example, fluid shear stress and nutrient transport, will crucially affect cell function on 3D-printed scaffolds. Therefore, fluid shear stress induced by spinner flask and perfusion bioreactors might differentially affect osteogenic responsiveness of pre-osteoblasts inside 3D-printed scaffolds. We designed and fabricated surface-modified 3D-printed poly-ɛ-caprolactone (PCL) scaffolds, as well as static, spinner flask, and perfusion bioreactors to determine fluid shear stress and osteogenic responsiveness of MC3T3-E1 pre-osteoblasts seeded on the scaffolds in the bioreactors using finite element (FE)-modeling and experiments. FE-modeling was used to quantify wall shear stress (WSS) distribution and magnitude inside 3D-printed PCL scaffolds within spinner flask and perfusion bioreactors. MC3T3-E1 pre-osteoblasts were seeded on NaOH surface-modified 3D-printed PCL scaffolds, and cultured in customized static, spinner flask, and perfusion bioreactors up to 7 days. The scaffolds' physicochemical properties and pre-osteoblast function were assessed experimentally. FE-modeling showed that spinner flask and perfusion bioreactors locally affected WSS distribution and magnitude inside the scaffolds. The WSS distribution was more homogeneous inside scaffolds in perfusion than in spinner flask bioreactors. The average WSS on scaffold-strand surfaces ranged from 0 to 6.5 mPa for spinner flask bioreactors, and from 0 to 4.1 mPa for perfusion bioreactors. Surface modification of scaffolds by NaOH resulted in a surface with a honeycomb-like pattern and increased surface roughness (1.6-fold), but decreased water contact angle (0.3-fold). Both spinner flask and perfusion bioreactors increased cell spreading, proliferation, and distribution throughout the scaffolds. Perfusion, but not spinner flask bioreactors more strongly enhanced collagen (2.2-fold) and calcium deposition (2.1-fold) throughout the scaffolds after 7 days compared with static bioreactors, likely due to uniform WSS-induced mechanical stimulation of the cells revealed by FE-modeling. In conclusion, our findings indicate the importance of using accurate FE models to estimate WSS and determine experimental conditions for designing cell-seeded 3D-printed scaffolds in bioreactor systems. Impact Statement The success of cell-seeded three-dimensional (3D)-printed scaffolds depends on cell stimulation by biomechanical/biochemical factors to produce bone tissue suitable for implantation into the patient. We designed and fabricated surface-modified 3D-printed poly-ɛ-caprolactone (PCL) scaffolds, as well as static, spinner flask, and perfusion bioreactors to determine wall shear stress (WSS) and osteogenic responsiveness of pre-osteoblasts seeded on the scaffolds using finite element (FE)-modeling and experiments. We found that cell-seeded 3D-printed PCL scaffolds within perfusion bioreactors more strongly enhanced osteogenic activity than within spinner flask bioreactors. Our results indicate the importance of using accurate FE-models to estimate WSS and determine experimental conditions for designing cell-seeded 3D-printed scaffolds in bioreactor systems.

Bioprinting a thick and cell-laden partially oxidized alginate-gelatin scaffold with embedded micro-channels as future soft tissue platform

Despite all the advancements in tissue engineering, one of the unsolved challenges is the mass transfer limitation. Therefore, the subject of pre-vascularization in the engineered tissues gets more attention to avoid necrotic core formation. In this study, we considered a design for interconnected channels with a muscle tissue-like structure, in silico and in vitro. A sequence of simple steps make it possible for us to use the same material, gelatin, as both a sacrificial material and one of the main components of the scaffold simultaneously. We defined a new approach to quantify the repeatability of a new combination of hydrogels (Partially Oxidized Alginate + Gelatin) for extrusion-based bioprinting. Additionally, the mechanical properties, hydrogel porosity, degradation time, and swelling ratio were also evaluated. Based on all these test results, the scaffold with the optimum properties was chosen for the bioprinting of adipose derived mesenchymal stem cells (ADMSCs) in the scaffolds with and without the channels. This bioprinted scaffold with microchannels showed promising mimicry of the microenvironment, leading to higher survival and proliferation rates of the cells by up to 250%. Based on these results, it has the potential to serve as a platform for further research in vascularization, healthy/disease modelling, and stem cell differentiation.

Engineering modeling frameworks for microbial food safety at various scales

The landscape of mathematical model-based understanding of microbial food safety is wide and deep, covering interdisciplinary fields of food science, microbiology, physics, and engineering. With rapidly growing interest in such model-based approaches that increasingly include more fundamental mechanisms of microbial processes, there is a need to build a general framework that steers this evolutionary process by synthesizing literature spread over many disciplines. The framework proposed here shows four interconnected, complementary levels of microbial food processes covering sub-cellular scale, microbial population scale, food scale, and human population scale (risk). A continuum of completely mechanistic to completely empirical models, widely-used and emerging, are integrated into the framework; well-known predictive microbiology modeling being a part of this spectrum. The framework emphasizes fundamentals-based approaches that should get enriched over time, such as the basic building blocks of microbial population scale processes (attachment, migration, growth, death/inactivation and communication) and of food processes (e.g., heat and moisture transfer). A spectrum of models are included, for example, microbial population modeling covers traditional predictive microbiology models to individual-based models and cellular automata. The models are shown in sufficient quantitative detail to make obvious their coupling, or their integration over various levels. Guidelines to combine sub-processes over various spatial and time scales into a complete interdisciplinary and multiphysics model (i.e., a system) are provided, covering microbial growth/inactivation/transport and physical processes such as fluid flow and heat transfer. As food safety becomes increasingly predictive at various scales, this synthesis should provide its roadmap. This big picture and framework should be futuristic in driving novel research and educational approaches.

Numerical modelling of osteocyte growth on different bone tissue scaffolds

The most common solution for the regeneration or replacement of damaged bones is the implantation of prostheses comprising ceramic or metallic materials. However, these implants are known to cause problems such as post-operative infections, collapse of the prosthesis, and lack of osseointegration. Consequently, bone tissue engineering was established because of the limitations of such implants. Osteogenic implants offer promising solutions for bone regeneration; however, three-dimensional scaffolds should be used as supportive structures. It is challenging to correctly design these structures and their compositions or properties to provide a microenvironment that promotes tissue regeneration and expedites bone formation. Computational fluid dynamics can be used to model the main phenomena that occur in bioreactors, such as cell metabolism, nutrient transport, and cell culture growth, or to model the influence of several key mechanisms related to the fluid medium, in particular, the wall shear stress. In this work, a new numerical bone cell growth model was developed, which considered the oxygen and nutrient consumption as well as the wall shear stress effect on cell proliferation. The model was implemented using 35 three-dimensional scaffolds of different porosities, and the effect of the main geometrical parameters involved in each scaffold type was analysed. The porosity plays an important role, however, a similar porosity did not guarantee similar shear stress or cell growth among the scaffolds. Randomised trabecular scaffolds, that more closely resembled trabecular bone, showed the highest cell growth values, so these are the best candidates for cell growth in a bioreactor.

Digital Twins for Tissue Culture Techniques—Concepts, Expectations, and State of the Art

Techniques to provide in vitro tissue culture have undergone significant changes during the last decades, and current applications involve interactions of cells and organoids, three-dimensional cell co-cultures, and organ/body-on-chip tools. Efficient computer-aided and mathematical model-based methods are required for efficient and knowledge-driven characterization, optimization, and routine manufacturing of tissue culture systems. As an alternative to purely experimental-driven research, the usage of comprehensive mathematical models as a virtual in silico representation of the tissue culture, namely a digital twin, can be advantageous. Digital twins include the mechanistic of the biological system in the form of diverse mathematical models, which describe the interaction between tissue culture techniques and cell growth, metabolism, and the quality of the tissue. In this review, current concepts, expectations, and the state of the art of digital twins for tissue culture concepts will be highlighted. In general, DT’s can be applied along the full process chain and along the product life cycle. Due to the complexity, the focus of this review will be especially on the design, characterization, and operation of the tissue culture techniques.

Optimizing Bifurcated Channels within an Anisotropic Scaffold for Engineering Vascularized Oriented Tissues

Despite progress in engineering both vascularized tissues and oriented tissues, the fabrication of 3D vascularized oriented tissues remains a challenge due to an inability to successfully integrate vascular and anisotropic structures that can support mass transfer and guide cell alignment, respectively. More importantly, there is a lack of an effective approach to guiding the scaffold design bearing both structural features. Here, an approach is presented to optimize the bifurcated channels within an anisotropic scaffold based on oxygen transport simulation and biological experiments. The oxygen transport simulation is performed using the experimentally measured effective oxygen diffusion coefficient and hydraulic permeability of the anisotropic scaffolds, which are also seeded with muscle precursor cells and cultured in a custom-made perfusion bioreactor. Symmetric bifurcation model is used as fractal unit to design the channel network based on biomimetic principles. The bifurcation level of channel network is further optimized based on the oxygen transport simulation, which is then validated by DNA quantification assay and pimonidazole immunostaining. This study provides a practical guide to optimizing bifurcated channels in anisotropic scaffolds for oriented tissue engineering.

Inlet flow rate of perfusion bioreactors affects fluid flow dynamics, but not oxygen concentration in 3D-printed scaffolds for bone tissue engineering: Computational analysis and experimental validation

Fluid flow dynamics and oxygen-concentration in 3D-printed scaffolds within perfusion bioreactors are sensitive to controllable bioreactor parameters such as inlet flow rate. Here we aimed to determine fluid flow dynamics, oxygen-concentration, and cell proliferation and distribution in 3D-printed scaffolds as a result of different inlet flow rates of perfusion bioreactors using experiments and finite element modeling. Pre-osteoblasts were treated with 1 h pulsating fluid flow with low (0.8 Pa; PFF^low) or high peak shear stress (6.5 Pa; PFF^high), and nitric oxide (NO) production was measured to validate shear stress sensitivity. Computational analysis was performed to determine fluid flow between 3D-scaffold-strands at three inlet flow rates (0.02, 0.1, 0.5 ml/min) during 5 days. MC3T3-E1 pre-osteoblast proliferation, matrix production, and oxygen-consumption in response to fluid flow in 3D-printed scaffolds inside a perfusion bioreactor were experimentally assessed. PFF^high more strongly stimulated NO production by pre-osteoblasts than PFF^low. 3D-simulation demonstrated that dependent on inlet flow rate, fluid velocity reached a maximum (50-1200 μm/s) between scaffold-strands, and fluid shear stress (0.5-4 mPa) and wall shear stress (0.5-20 mPa) on scaffold-strands surfaces. At all inlet flow rates, gauge fluid pressure and oxygen-concentration were similar. The simulated cell proliferation and distribution, and oxygen-concentration data were in good agreement with the experimental results. In conclusion, varying a perfusion bioreactor's inlet flow rate locally affects fluid velocity, fluid shear stress, and wall shear stress inside 3D-printed scaffolds, but not gauge fluid pressure, and oxygen-concentration, which seems crucial for optimized bone tissue engineering strategies using bioreactors, scaffolds, and cells.

Lattice and continuum modelling of a bioactive porous tissue scaffold

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

Modeling, simulations, and optimization of smooth muscle cell tissue engineering for the production of vascular grafts

The paper presents a transient, continuum, two-phase model of the tissue engineering in fibrous scaffolds, including transport equations for the flowing culture medium, nutrient and cell concentration with transverse and in-plane diffusion and cell migration, a novel feature of local in-plane transport across a phenomenological pore and innovative layer-by-layer cell filling approach. The model is successfully validated for the smooth muscle cell tissue engineering of a vascular graft using crosslinked, electrospun gelatin fiber scaffolds for both static and dynamic cell culture, the latter in a dynamic bioreactor with a rotating shaft on which the tubular scaffold is attached. Parametric studies evaluate the impact of the scaffold microstructure, cell dynamics, oxygen transport, and static or dynamic conditions on the rate and extent of cell proliferation and depth of oxygen accessibility. An optimized scaffold of 75% dry porosity is proposed that can be tissue engineered into a viable and still fully oxygenated graft of the tunica media of the coronary artery within 2 days in the dynamic bioreactor. Such scaffold also matches the mechanical properties of the tunica media of the human coronary artery and the suture retention strength of a saphenous vein, often used as a coronary artery graft.

Curvature- and fluid-stress-driven tissue growth in a tissue-engineering scaffold pore

Cell proliferation within a fluid-filled porous tissue-engineering scaffold depends on a sensitive choice of pore geometry and flow rates: regions of high curvature encourage cell proliferation, while a critical flow rate is required to promote growth for certain cell types. When the flow rate is too slow, the nutrient supply is limited; when it is too fast, cells may be damaged by the high fluid shear stress. As a result, determining appropriate tissue-engineering-construct geometries and operating regimes poses a significant challenge that cannot be addressed by experimentation alone. In this paper, we present a mathematical theory for the fluid flow within a pore of a tissue-engineering scaffold, which is coupled to the growth of cells on the pore walls. We exploit the slenderness of a pore that is typical in such a scenario, to derive a reduced model that enables a comprehensive analysis of the system to be performed. We derive analytical solutions in a particular case of a nearly piecewise constant growth law and compare these with numerical solutions of the reduced model. Qualitative comparisons of tissue morphologies predicted by our model, with those observed experimentally, are also made. We demonstrate how the simplified system may be used to make predictions on the design of a tissue-engineering scaffold and the appropriate operating regime that ensures a desired level of tissue growth.

A computer simulation of short-term adaptations of cardiovascular hemodynamics in microgravity

Astronauts in the microgravity environment experience significant changes in their cardiovascular hemodynamics. In this study, a system-level numerical model has been utilized to simulate the short-term adaptations of hemodynamic parameters due to the gravitational removal in space. The effect of lower body negative pressure (LBNP) as a countermeasure has also been simulated. The numerical model was built upon a lumped-parameter Windkessel model by incorporating gravity-induced hydrostatic pressure and transcapillary fluid exchange modules. The short-term (in the time scale of seconds and minutes) adaptations of the cardiac functions, blood pressure, and fluid volumes have been analyzed and compared with physiological data. The simulation results suggest microgravity induces a decrease in aortic pressure, heart rate, lower body capillary pressure and volume, and an increase in stroke volume, upper body capillary pressure and volume. The activation of LBNP causes an immediate increase in lower body blood volume and a gradual decrease in upper body blood volume. As a result, the fluid shift due to microgravity could be reversed by the LBNP application. LBNP also counters the impacts of microgravity on the cardiac functions, including heart rate and stroke volume. The simulation results have been validated using available physiological data obtained from spaceflight and parabolic flight experiments.

Mesenchymal stem cell cultivation in electrospun scaffolds: mechanistic modeling for tissue engineering

Tissue engineering is a multidisciplinary field of research in which the cells, biomaterials, and processes can be optimized to develop a tissue substitute. Three-dimensional (3D) architectural features from electrospun scaffolds, such as porosity, tortuosity, fiber diameter, pore size, and interconnectivity have a great impact on cell behavior. Regarding tissue development in vitro, culture conditions such as pH, osmolality, temperature, nutrient, and metabolite concentrations dictate cell viability inside the constructs. The effect of different electrospun scaffold properties, bioreactor designs, mesenchymal stem cell culture parameters, and seeding techniques on cell behavior can be studied individually or combined with phenomenological modeling techniques. This work reviews the main culture and scaffold factors that affect tissue development in vitro regarding the culture of cells inside 3D matrices. The mathematical modeling of the relationship between these factors and cell behavior inside 3D constructs has also been critically reviewed, focusing on mesenchymal stem cell culture in electrospun scaffolds.

Modelling mesenchymal stromal cell growth in a packed bed bioreactor with a gas permeable wall

A mathematical model was developed for mesenchymal stromal cell (MSC) growth in a packed bed bioreactor that improves oxygen availability by allowing oxygen diffusion through a gas-permeable wall. The governing equations for oxygen, glucose and lactate, the inhibitory waste product, were developed assuming Michaelis-Menten kinetics, together with an equation for the medium flow based on Darcy's Law. The conservation law for the cells includes the effects of inhibition as the cells reach confluence, nutrient and waste product concentrations, and the assumption that the cells can migrate on the scaffold. The equations were solved using the finite element package, COMSOL. Previous experimental results collected using a packed bed bioreactor with gas permeable walls to expand MSCs produced a lower cell yield than was obtained using a traditional cell culture flask. This mathematical model suggests that the main contributors to the observed low cell yield were a non-uniform initial cell seeding profile and a potential lag phase as cells recovered from the initial seeding procedure. Lactate build-up was predicted to have only a small effect at lower flow rates. Thus, the most important parameters to optimise cell expansion in the proliferation of MSCs in a bioreactor with gas permeable wall are the initial cell seeding protocol and the handling of the cells during the seeding process. The mathematical model was then used to identify and characterise potential enhancements to the bioreactor design, including incorporating a central gas permeable capillary to further enhance oxygen availability to the cells. Finally, to evaluate the issues and limitations that might be encountered scale-up of the bioreactor, the mathematical model was used to investigate modifications to the bioreactor design geometry and packing density.

Numerical optimization of cell colonization modelling inside scaffold for perfusion bioreactor: A multiscale model

Part of clinically applicable bone graft substitutes are developed by using mechanical stimulation of flow-perfusion into cell-seeded scaffolds. The role of fluid flow is crucial in driving the nutrient to seeded cells and in stimulating cell colonization. A common numerical approach is to use a multiscale model to link some physical quantities (wall shear stress and inlet flow rate) that act at different scales. In this study, a multiscale model is developed in order to determine the optimal inlet flow rate to cultivate osteoblast-like cells seeded in a controlled macroporous biomaterial inside a perfusion bioreactor system. We focus particularly on the influence of Wall Shear Stress on cell colonization to predict cell colonization at the macroscale. Results obtained at the microscale are interpolated at the macroscale to determine the optimal flow rate. For a macroporous scaffold made of interconnected pores with pore diameters of above 350 μm and interconnection diameters of 150 μm, the model predicts a cell colonization of 325% after a 7-day-cell culture with a constant inlet flow rate of 0.69 mL·min^-1. Furthermore, the strength of this protocol is the possibility to adapt it to most porous biomaterials and dynamic cell culture systems.

Recent advances in bioreactors for cell-based therapies

Bioreactors have become indispensable tools in the cell-based therapy industry. Various forms of bioreactors are used to maintain well-controlled microenvironments to regulate cell growth, differentiation, and tissue development. They are essential for providing standardized, reproducible cell-based products for regenerative medicine applications or to establish physiologically relevant in vitro models for testing of pharmacologic agents. In this review, we discuss three main classes of bioreactors: cell expansion bioreactors, tissue engineering bioreactors, and lab-on-a-chip systems. We briefly examine the factors driving concerted research endeavors in each of these areas and describe the major advancements that have been reported in the last three years. Emerging issues that impact the commercialization and clinical use of bioreactors include (i) the need to scale up to greater cell quantities and larger graft sizes, (ii) simplification of in vivo systems to function without exogenous stem cells or growth factors or both, and (iii) increased control in the manufacture and monitoring of miniaturized systems to better capture complex tissue and organ physiology.

Human dental pulp stem cell adhesion and detachment in polycaprolactone electrospun scaffolds under direct perfusion

Cell adhesion in three-dimensional scaffolds plays a key role in tissue development. However, stem cell behavior in electrospun scaffolds under perfusion is not fully understood. Thus, an investigation was made on the effect of flow rate and shear stress, adhesion time, and seeding density under direct perfusion in polycaprolactone electrospun scaffolds on human dental pulp stem cell detachment. Polycaprolactone scaffolds were electrospun using a solvent mixture of chloroform and methanol. The viable cell number was determined at each tested condition. Cell morphology was analyzed by confocal microscopy after various incubation times for static cell adhesion with a high seeding density. Scanning electron microscopy images were obtained before and after perfusion for the highest flow rate tested. The wall pore shear stress was calculated for all tested flow rates (0.005–3 mL/min). An inversely proportional relationship between adhesion time with cell detachment under perfusion was observed. Lower flow rates and lower seeding densities reduced the drag of cells by shear stress. However, there was an operational limit for the lowest flow rate that can be used without compromising cell viability, indicating that a flow rate of 0.05 mL/min might be more suitable for the tested cell culture in electrospun scaffolds under direct perfusion.

Simulation of the effects of oxygen carriers and scaffold geometry on oxygen distribution and cell growth in a channeled scaffold for engineering myocardium

This study proposes a mathematical model to evaluate the impact of oxygen carriers and scaffold geometry on oxygen distribution and cell growth in a 3D cardiac construct using computational fluid dynamics (CFD). Flow equations, oxygen balance equation and cell balance equation were solved using special initial and boundary conditions. The modeling results revealed that 55% increase in cardiac cell density occurred by using 6.4% perfluorocarbon oxygen carrier (PFC) compared to pure culture medium without PFC supplementation. Moreover, the effects of the scaffold geometry on cell density were examined by changing the channel numbers and the construct length. A 30% increase in the average cells density was observed by increasing the channel numbers from 37 to 145. Furthermore, the average cell density was increased 23% by decreasing the scaffold length from 0.5 to 0.2 cm length. Overall, the average cell density of cardiac cells can be increased 2-fold by using PFC oxygen carrier and optimizing the scaffold's geometry, simultaneously.

Mathematical modeling of cell growth in a 3D scaffold and validation of static and dynamic cultures

Tissue engineering, an immensely important field in contemporary clinical practices, aims at the repair or replacement of damaged tissues. The mathematical model proposed herein shows the distribution and growth of cells in their characteristic time in a 3D scaffold model. This study contributes to the progress of simulation techniques in static and dynamic cultures of bone tissue. Brinkman, nutrient transport, and cell growth equations are brought together to quantify the growth behavior of cells. However, when a static culture is being studied, the Brinkman equation is eliminated. The model was validated by experimental cell culture using 3‐(4,5‐dimethylthiazol‐2‐yl)‐2,5‐diphenyltetrazolium bromide assay and scanning electron microscopy. Then, static and dynamic cultures were compared to assess the cell density and cell distribution in the scaffold. Cell counting after 21 days of cell culture showed that the number of cells increased 42‐fold in static and 53.5‐fold in dynamic cultures, which was in good agreement with our model estimations (37‐fold increase in the number of cells in static and 49‐fold increase in dynamic cultures). In conclusion, our mathematical model could predict cell distribution and growth in the scaffold.

Cellular Nutrition in Complex Three-Dimensional Scaffolds: A Comparison between Experiments and Computer Simulations

Studies on bone cell ingrowth into synthetic, porous three-dimensional (3D) implants showed difficulties arising from impaired cellular proliferation and differentiation in the core region of these scaffolds with increasing scaffold volume in vitro. Therefore, we developed an in vitro perfusion cell culture module, which allows the analysis of cells in the interior of scaffolds under different medium flow rates. For each flow rate the cell viability was measured and compared with results from computer simulations that predict the local oxygen supply and shear stress inside the scaffold based on the finite element method. We found that the local cell viability correlates with the local oxygen concentration and the local shear stress. On the one hand the oxygen supply of the cells in the core becomes optimal with a higher perfusion flow. On the other hand shear stress caused by high flow rates impedes cell vitality, especially at the surface of the scaffold. Our results demonstrate that both parameters must be considered to derive an optimal nutrient flow rate.

Influence of the scaffold geometry on the spatial and temporal evolution of the mechanical properties of tissue-engineered cartilage: insights from a mathematical model

The production of tissue-engineered cartilage in vitro with inhomogeneous mechanical properties is a problem yet to be solved. Different geometries have been studied to overcome this caveat; however, the reported measurements are limited to average values of some properties and qualitative measures of spatial distributions. We will apply a coupled model to extend knowledge about the introduction of a macrochannel in a scaffold by calculating spatiotemporal patterns for several interest variables related to the remodeling of the mechanical properties. Model parameters were estimated based on experimental data on the temporal patterns of glycosaminoglycans, collagen and compressive Young's modulus for channel-free constructs. The model reproduced the experimental data trends in both geometries, with experimental-numerical correlations between 0.84 and 0.97. The channel had a higher impact on the reduction in spatial heterogeneities and delay of saturation of core properties than in the improvement of average properties. Despite the possible improvement of cell densities for longer periods than 56 days, it is estimated that it will not cause further significant improvements of the mechanical properties. The degrees of spatial heterogeneity of the Young's modulus and permeability in the channeled geometry are 23 and 27 % of the channel-free values. While the average Young's modulus values are in the range of native cartilage, the permeabilities are one to three degrees of magnitude higher than the native cartilage, suggesting that limiting factors such as scaffold porosity and initial permeability are more relevant than scaffold geometry to effectively decrease the tissue permeability.

Pushing the envelope in tissue engineering: ex vivo production of thick vascularized cardiac extracellular matrix constructs

Functional vascularization is a prerequisite for cardiac tissue engineering of constructs with physiological thicknesses. We previously reported the successful preservation of main vascular conduits in isolated thick acellular porcine cardiac ventricular ECM (pcECM). We now unveil this scaffold's potential in supporting human cardiomyocytes and promoting new blood vessel development ex vivo, providing long-term cell support in the construct bulk. A custom-designed perfusion bioreactor was developed to remodel such vascularization ex vivo, demonstrating, for the first time, functional angiogenesis in vitro with various stages of vessel maturation supporting up to 1.7 mm thick constructs. A robust methodology was developed to assess the pcECM maximal cell capacity, which resembled the human heart cell density. Taken together these results demonstrate feasibility of producing physiological-like constructs such as the thick pcECM suggested here as a prospective treatment for end-stage heart failure. Methodologies reported herein may also benefit other tissues, offering a valuable in vitro setting for “thick-tissue” engineering strategies toward large animal in vivo studies.

Modeling of time dependent localized flow shear stress and its impact on cellular growth within additive manufactured titanium implants

Bone augmentation implants are porous to allow cellular growth, bone formation and fixation. However, the design of the pores is currently based on simple empirical rules, such as minimum pore and interconnects sizes. We present a three-dimensional (3D) transient model of cellular growth based on the Navier-Stokes equations that simulates the body fluid flow and stimulation of bone precursor cellular growth, attachment, and proliferation as a function of local flow shear stress. The model's effectiveness is demonstrated for two additive manufactured (AM) titanium scaffold architectures. The results demonstrate that there is a complex interaction of flow rate and strut architecture, resulting in partially randomized structures having a preferential impact on stimulating cell migration in 3D porous structures for higher flow rates. This novel result demonstrates the potential new insights that can be gained via the modeling tool developed, and how the model can be used to perform what-if simulations to design AM structures to specific functional requirements.

Applying shear stress to endothelial cells in a new perfusion chamber: hydrodynamic analysis

Perfusion bioreactors have been proved to be an impartible part of vascular tissue engineering due to its broad range of applications as a means to distribute nutrients within porous scaffold along with providing appropriate physical and mechanical stimuli. To better understand the mechanical phenomena inside a bioreactor, computational fluid dynamics (CFD) was adopted followed by a validation technique. The fluid dynamics of the media inside the bioreactor was modeled using the Navier-Stokes equation for incompressible fluids while convection through the scaffold was described by Brinkman's extension of Darcy's law for porous media. Flow within the reactor determined the orientation of endothelial cells on the scaffold. To validate flow patterns, streamlines and shear stresses, colorimetry technique was used following attained results from CFD. Our bioreactor was modeled to simulate the optimum condition and flow patterns over scaffold to culture ECs for in vitro experimentation. In such experiments, cells were attached firmly without significant detachment and more noticeably elongation process was triggered even shortly after start up.

In silico prediction of the cell proliferation in porous scaffold using model of effective pore

The mathematical prediction of cell proliferation in porous scaffold still remains a challenge. The analysis of existing models and experimental data confirms a need for a new solution, which takes into account cells" development on the scaffold pore walls as well as some additional parameters such as the pore size, cell density in cellular layers, the thickness of the growing cell layer and others. The simulations, presented below, are based on three main approaches. The first approach takes into account multilayer cell growth on the pore walls of the scaffold. The second approach is a simulation of cell proliferation in a discrete process as a continuous one. The third one is the representation of scaffold structure as a system of cylindrical channels. Oxygen (nutrient) mass transfer is realized inside these channels. The model, described below, proposes the new solution to time dependent description of cell proliferation in porous scaffold and optimized trophical conditions for tissue development.

Engineering parameters in bioreactor's design: a critical aspect in tissue engineering

Bioreactors are important inevitable part of any tissue engineering (TE) strategy as they aid the construction of three-dimensional functional tissues. Since the ultimate aim of a bioreactor is to create a biological product, the engineering parameters, for example, internal and external mass transfer, fluid velocity, shear stress, electrical current distribution, and so forth, are worth to be thoroughly investigated. The effects of such engineering parameters on biological cultures have been addressed in only a few preceding studies. Furthermore, it would be highly inefficient to determine the optimal engineering parameters by trial and error method. A solution is provided by emerging modeling and computational tools and by analyzing oxygen, carbon dioxide, and nutrient and metabolism waste material transports, which can simulate and predict the experimental results. Discovering the optimal engineering parameters is crucial not only to reduce the cost and time of experiments, but also to enhance efficacy and functionality of the tissue construct. This review intends to provide an inclusive package of the engineering parameters together with their calculation procedure in addition to the modeling techniques in TE bioreactors.

Sequential application of steady and pulsatile medium perfusion enhanced the formation of engineered bone

In native bone, cells experience fluctuating shear forces that are induced by pulsatile interstitial flow associated with habitual loading. We hypothesized that the formation of engineered bone can be augmented by replicating such physiologic stimuli to osteogenic cells cultured in porous scaffolds using bioreactors with medium perfusion. To test this hypothesis, we investigated the effect of fluid flow regime on in vitro bone-like tissue development by human adipose stem cells (hASC) cultivated on porous three-dimensional silk fibroin scaffolds. To this end, we varied the sequential relative durations of steady flow (SF) and pulsatile flow (PF) of culture medium applied over a period of 5 weeks, and evaluated their effect on early stages of bone formation. Porous silk fibroin scaffolds (400-600 μm pore size) were seeded with hASC (30×10(6) cells/mL) and cultured in osteogenic medium under four distinct fluid flow regimes: (1) PF for 5 weeks; (2) SF for 1 week, PF for 4 weeks; (3) SF for 2 weeks, PF for 3 weeks; (4) SF for 5 weeks. The PF was applied in 12 h intervals, with the interstitial velocity fluctuating between 400 and 1200 μm/s at a 0.5 Hz frequency for 2 h, followed by 10 h of SF. In all groups, SF was applied at 400 μm/s. The best osteogenic outcomes were achieved for the sequence of 2 weeks of SF and 3 weeks of PF, as evidenced by gene expression (including the PGE2 mechanotransduction marker), construct compositions, histomorphologies, and biomechanical properties. We thus propose that osteogenesis in hASC and the subsequent early stage bone development involve a mechanism, which detects and responds to the level and duration of hydrodynamic shear forces.

Mathematical model of growth factor driven haptotaxis and proliferation in a tissue engineering scaffold

Motivated by experimental work (Miller et al. in Biomaterials 27(10):2213-2221, 2006, 32(11):2775-2785, 2011) we investigate the effect of growth factor driven haptotaxis and proliferation in a perfusion tissue engineering bioreactor, in which nutrient-rich culture medium is perfused through a 2D porous scaffold impregnated with growth factor and seeded with cells. We model these processes on the timescale of cell proliferation, which typically is of the order of days. While a quantitative representation of these phenomena requires more experimental data than is yet available, qualitative agreement with preliminary experimental studies (Miller et al. in Biomaterials 27(10):2213-2221, 2006) is obtained, and appears promising. The ultimate goal of such modeling is to ascertain initial conditions (growth factor distribution, initial cell seeding, etc.) that will lead to a final desired outcome.

2-D coupled computational model of biological cell proliferation and nutrient delivery in a perfusion bioreactor

Tissue engineering aims to regenerate, repair or replace organs or tissues which have become defective due to trauma, disease or age related degeneration. This engineering may take place within the patient's body or tissue can be regenerated in a bioreactor for later implantation into the patient. Regeneration of soft tissue is one of the most demanding applications of tissue engineering. Producing proper nutrient supply, uniform cell distribution and high cell density are the important challenges. Many experimental models exist for tissue growth in a bioreactor. It is important to put experiments into a theoretical framework. Mathematical modelling in terms of physical and biochemical mechanisms is the best tool to understand experimental results. In this work a mathematical model of convective and diffusive transport of nutrients and cell evolution in a perfusion bioreactor is developed. A cell-seeded porous scaffold is placed in a perfusion bioreactor and fluid delivers the nutrients to the cells for their growth. The model describes the key features of the tissue engineering processes which includes the interaction between the cell growth, variation of material permeability due to cell proliferation, flow of fluid through the material and delivery of nutrients to the cells. The fluid flow through the porous scaffold is modelled by Darcy's law, and the delivery of nutrients to the cells is modelled by the advection-diffusion equation. A non-linear reaction diffusion system is used to model the cell growth. The growth of cells is modelled by logistic growth. COMSOL (a commercial finite element solver) is used to numerically solve the model. The results show that the distribution of cells and total cell number in the scaffold does not depend on the initial cell density but depend on the material permeability.

Modeling of cell cultures in perfusion bioreactors

Cultivating cells and tissues in bioreactors is a critical step in forming artificial tissues or organs prior to transplantation. Among various bioreactors, the perfusion bioreactor is known for its enhanced convection through the cell-scaffold constructs. Knowledge of mass transfer is essential for controlling the cell culture process; however, obtaining this information remains a challenging task. In this research, a novel mathematical model is developed to represent the nutrient transport and cell growth in a 3-D scaffold cultivated in a perfusion bioreactor. Numerical methods are employed to solve the equations involved, with a focus on identifying the effect of factors such as porosity, culturing time, and flow rate, which are controllable in the scaffold fabrication and culturing process, on cell cultures. To validate the new model, the results from the model simulations were compared to the experimental results extracted from the literature. With the validated model, further simulations were carried out to investigate the glucose and oxygen distribution and the cell growth within the cell-scaffold construct in a perfusion bioreactor, thus providing insight into the cell culture process.

Optimizing the medium perfusion rate in bone tissue engineering bioreactors

There is a critical need to increase the size of bone grafts that can be cultured in vitro for use in regenerative medicine. Perfusion bioreactors have been used to improve the nutrient and gas transfer capabilities and reduce the size limitations inherent to static culture, as well as to modulate cellular responses by hydrodynamic shear. Our aim was to understand the effects of medium flow velocity on cellular phenotype and the formation of bone-like tissues in three-dimensional engineered constructs. We utilized custom-designed perfusion bioreactors to culture bone constructs for 5 weeks using a wide range of superficial flow velocities (80, 400, 800, 1,200, and 1,800 µm/s), corresponding to estimated initial shear stresses ranging from 0.6 to 20 mPa. Increasing the flow velocity significantly affected cell morphology, cell-cell interactions, matrix production and composition, and the expression of osteogenic genes. Within the range studied, the flow velocities ranging from 400 to 800 µm/s yielded the best overall osteogenic responses. Using mathematical models, we determined that even at the lowest flow velocity (80 µm/s) the oxygen provided was sufficient to maintain viability of the cells within the construct. Yet it was clear that this flow velocity did not adequately support the development of bone-like tissue. The complexity of the cellular responses found at different flow velocities underscores the need to use a range of evaluation parameters to determine the quality of engineered bone.

Comparison of the effects of possible mechanical stimuli on the rate of biochemical reactions

The aim of this work is to address the question of what constitutes a mechanical stimulation of biochemical reactions in general and further to compare the importance of the two possible mechanical stimulations: shear rate and the rate of volume variation. Using linear nonequilibrium thermodynamics, the Curie principle (the relation for coupling phenomena) is retrieved for a phenomenological relation for a scalar flux in an isotropic system. From these phenomenological relations for the rate of chemical reaction, it is established that the only scalar quantity related to the rate of deformation tensor D that cannot be neglected is the rate of volume variation D((1)). This leads us to the conclusion that, although tissues are exposed to all variety of mechanical factors: straining, shear, pressure, and even dynamic electric fields, the volume variation rate D((1)) is the most important mechanical stimulus driving the processes in them.

Analysis of collagen and glucose modulated cell growth within tissue engineered scaffolds

The strategy of tissue engineering includes seeding cells onto porous scaffolds. The cellular construct is cultured in vitro for a period of time before transplantation for the patient. Because of the intrinsic complexity of biological systems, it is valuable to have models of simulation that can assess the culture conditions and optimize experiments. This work presents a mathematical model to account for the effects of glucose and type II collagen on chondrocyte growth under static culture conditions. Dependence of cell growth on collagen was assumed as a biphasic function of collagen quantity, whereby the cell growth rate increases and then decreases with increasing collagen content. Results from simulation were compared with experimental data in literature. The model was then applied to investigate the effects of cell seeding area, demonstrating the spatiotemporal evolution of cell distribution in scaffolds. Results show that the conventional uniform seeding method may not be a good way of gaining uniform and large cell number densities at the final stage of cultivation. A seeding mode that has cells reside initially in the middle area of scaffold was shown to be able to not only reduce the diffusion limitation of nutrients but also weaken the inhibiting impact of aggregated collagen on cell growth. Therefore the middle seeding mode may result in better cell amounts and uniformities for developing tissue engineered constructs.

Engineering anatomically shaped human bone grafts

The ability to engineer anatomically correct pieces of viable and functional human bone would have tremendous potential for bone reconstructions after congenital defects, cancer resections, and trauma. We report that clinically sized, anatomically shaped, viable human bone grafts can be engineered by using human mesenchymal stem cells (hMSCs) and a "biomimetic" scaffold-bioreactor system. We selected the temporomandibular joint (TMJ) condylar bone as our tissue model, because of its clinical importance and the challenges associated with its complex shape. Anatomically shaped scaffolds were generated from fully decellularized trabecular bone by using digitized clinical images, seeded with hMSCs, and cultured with interstitial flow of culture medium. A bioreactor with a chamber in the exact shape of a human TMJ was designed for controllable perfusion throughout the engineered construct. By 5 weeks of cultivation, tissue growth was evidenced by the formation of confluent layers of lamellar bone (by scanning electron microscopy), markedly increased volume of mineralized matrix (by quantitative microcomputer tomography), and the formation of osteoids (histologically). Within bone grafts of this size and complexity cells were fully viable at a physiologic density, likely an important factor of graft function. Moreover, the density and architecture of bone matrix correlated with the intensity and pattern of the interstitial flow, as determined in experimental and modeling studies. This approach has potential to overcome a critical hurdle-in vitro cultivation of viable bone grafts of complex geometries-to provide patient-specific bone grafts for craniofacial and orthopedic reconstructions.