Design of Polymer Scaffolds for Tissue Engineering Applications

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Design of Polymer Scaffolds for Tissue Engineering Applications Hamidreza Mehdizadeh, Sami Somo, Elif Seyma Bayrak, Eric M Brey, and Ali Cinar Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie503133e • Publication Date (Web): 06 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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Industrial & Engineering Chemistry Research

Design of Polymer Scaffolds for Tissue Engineering Applications Hamidreza Mehdizadeh, Sami I. Somo, Elif S. Bayrak, Eric M. Brey, Ali Cinar* Illinois Institute of Technology, 3300 S Federal St, Chicago, IL 60616 ABSTRACT Agent-based models (ABM) provide a flexible multi-layer platform to incorporate various modeling techniques into a single hybrid model for designing optimal biomaterial scaffolds for angiogenesis in tissue engineering applications. Scaffold geometrical variables are considered as design variables. The growth factor concentration profile is the only process variable considered in the study. The product variables used to illustrate the combined effects of scaffold design variables and process variables on the outcome of angiogenesis include the density and depth of capillary invasion within the scaffold. The scaffold design process and the ABM developed to simulate angiogenesis are described in this paper. The performance of the ABM and vascularization of polymer scaffolds are evaluated by simulation studies. The effects of pore size, pore size distribution, and interconnectivity on total blood vessel length, invasion depth, and the total number of sprouts formed during the vascularization process are reported. The integration of the simulation of angiogenesis with ABMs and scaffold design techniques provide an iterative process for designing optimal scaffold structures. This facilitates faster design of

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optimized scaffolds with significantly less cost and enables better understanding of the mechanisms of angiogenesis of polymer scaffolds for tissue engineering applications. KEY WORDS: Agent-based Modeling, Angiogenesis, Tissue Engineering, Scaffold Design 1. Introduction Growth of tissue in three-dimensional (3D) systems necessitates the presence of vasculature that supports stem cell differentiation and tissue growth and survival. In a typical application, a biocompatible scaffold is used to promote vascularization and tissue growth, with the properties of the scaffold having a strong influence on both processes.

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The use of product design

concepts can provide a valuable approach in the design of polymer scaffolds. Tissue engineering aims to produce functional replacement tissue and organs using a combination of cells, biomaterials, and engineering methods often using biochemical, bioelectrical or biomechanical signals.

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The biomaterial scaffolds serve as the substrate upon which other factors are

introduced and provide an environment for cell interactions. 4-5 These interactions are complex in nature, and include multitude of interrelated mechanisms and pathways, many of which are still unknown. Characteristics of the scaffolds impact cell behavior and the fate of the engineered tissues.

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Consequently, it is possible to influence the physical, chemical, biological, and

mechanical factors affecting cellular behavior by manipulating biomaterial scaffold properties. 78

Various fabrication techniques and biomaterials are used for tissue engineering scaffolds.

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Regardless of the type of the scaffold and its application, scaffold vascularization is essential for promoting tissue growth and production of large-volume replacement tissues.

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This is due to

the fact that cells need to be in a close vicinity of blood vessels to receive sufficient nutrients and


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oxygen required for survival and function. Hence, formation of new blood vessels is essential for the clinical translation of engineered tissues. 12 However, capillary ingrowth occurs slowly and is often limited to insufficient depths, resulting in scaffold vascularization being a major clinical barrier in the development of functional 3D tissue engineering constructs.

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Sprouting

angiogenesis, which is the development of new capillaries from pre-existing blood vessels, is the main mechanism for vascularization of biomaterials.

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Numerous research groups have used

experimental studies of angiogenesis to understand and enhance the neovascularization process. 15

In addition, various in vivo and in vitro techniques such as microfluidics strategies for

promoting formation of perfused vessels and stem cell-based approaches for vascular bone formation have been investigated. 16-20 Several groups of variables need to be considered when optimizing angiogenesis within scaffolds. The first group of variables is the “design” variables of the scaffold. The values of these variables are commonly selected prior to manufacturing the scaffold and cannot be modified after the scaffold is implanted in the body and the process of tissue regeneration is initiated.

The second group of variables is the “process” variables, which includes the

concentrations of various soluble materials in the substrate as well as the environmental variables to which the scaffold is subject. The third group of variables is the “product” variables, which defines the properties of the resulting vasculature and the regenerated tissue. Examples of design variables include porosity, pore size, and pore size distribution, while process variables include scaffold degradation behavior, growth factor concentrations and their release and diffusion rate. Typical product variables are capillary density and depths of invasion. In a tissue engineering scaffold, 3D angiogenesis is a process that takes several weeks to months to complete. Consequently, both in vivo and in vitro experimental studies are expensive


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and time consuming.

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Computational studies based on mathematical models provide an

attractive alternative. Models of angiogenesis are developed to shed light on complex interactions between numerous angiogenic factors, cells, and the surrounding microenvironment. These models enable the computation of variables and parameters that are difficult or impossible to determine experimentally, verify with experiments.

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or to test different hypothesis that are challenging to test and

In addition, models can allow high-throughput evaluation of design

strategies in a more time and cost efficient process. These simulations can be utilized to guide the design of novel strategies that can promote rapid and stable angiogenesis. Agent-based models (ABM) have been used successfully in recent years to model biological systems. ABMs are based on individual heterogeneous building blocks, referred to as agents, which are capable of performing various actions proactively and independently, based on rules that originate from knowledge of the system under study. This knowledge may be based on existing data and quantitative relations or hypotheses that need to be tested for validity. In the former case, the aim of modeling is usually to better understand the system under study and investigate the effect of changing various parameters of the system on system level behavior, while in the latter case the goal is often to test the validity of the proposed hypothesis. Biomedical ABMs are gaining acceptance among researchers and being used for developing new models in angiogenesis, tissue engineering and tumor growth or for extending existing models. One appeal of ABMs is their ability to integrate first principles (fundamental) equationbased models in their original continuous form or in discretized form, as tools for empowering the agents and enabling them in making accurate decisions. ABMs provide great flexibility to model developers, enabling them to represent the constituents of a natural system, such as cells, as discrete entities. The advantages of discrete representation and agent structure include the


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ability to incorporate the intrinsic variation and heterogeneity among living systems with the use of stochastic representations based on random variations in some parameters, rapid prototyping, and easy expansion and maintenance of the model. In this paper we focus on the application of ABM techniques in development of a flexible multi-layer platform to incorporate various modeling techniques into a single hybrid model, with the final goal of designing optimal biomaterial scaffolds for angiogenesis in tissue engineering applications. Scaffold geometrical variables that have been considered as design variables in this study include average pore size, pore size distribution (representing the standard deviation of the pore size distribution), interconnectivity, and overall scaffold porosity. To exclude the effects of random pore arrangement within the scaffold structure on angiogenesis, many scaffold structures are created with the same set of design parameters and simulations are performed using all of them and averaging the results. The growth factor (GF) concentration profile is used as the process variable for illustrating the proposed methodology and the performance of the agentbased model. The methodology can also be used to study the effects of other design variables such as scaffold degradation and diffusion rate, and their interactions.

Product variables illustrate the combined effect of scaffold design variables and process variables on the outcomes of angiogenesis such as the density and depth of capillary invasion within the scaffold. These include total blood vessel length (TBVL), blood vessel length density (BVLD), invasion depth (ID), maximum invasion depth (MID), and total number of sprouts (TNS). TBVL is the cumulative length of all blood vessels formed during the vascularization process. BVLD is equal to TBVL per unit pore volume. ID is the average ingrowth of all sprouts and shows the ability of invading sprouts to penetrate the scaffold, while MID shows the deepest


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distance inside the scaffold the sprouts have been able to invade. TNS is the total number of sprouts formed during the vascularization process. Simulation results are analyzed statistically to calculate the mean, standard deviation, and minimum and maximum values of product variables describing the vascularization process. The rest of this paper is structured as follows. Section 2 provides a brief description of the angiogenesis process, and outlines 2D and 3D angiogenesis modeling techniques. The agentbased modeling concept and applications are introduced in Section 3, followed by a description of the ABM developed in this study for designing scaffolds to optimize angiogenesis. Section 4 describes the scaffold design process, and the use of designed scaffolds as inputs to the ABM simulation for angiogenesis. Simulation case studies are defined in Section 5, outlining the input and product quality variables. Section 6 includes simulation results, and a discussion of the results, followed by conclusions in Section 7. 2. Angiogenesis in tissue engineering Angiogenesis is an important biological process that occurs during organ growth, wound healing, and reproduction in mammalian bodies. Angiogenesis is induced by the release of proangiogenic factors in the tissue in response to different causes, ranging from ischemia, hypoxia, tissue inflammation and wounds to cancer and other pathological effects. Angiogenic factors are mainly soluble proteins. Among these vascular endothelial growth factor (VEGF) is well established as one of the key regulators of the angiogenesis process and has been studied in detail by many researchers.

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Angiogenesis is the process through which scaffold

vascularization is achieved in many tissue engineering applications. 27 Growth factors are either secreted from the cells seeded within the scaffold or released from the biomaterial scaffolds preloaded with these soluble factors in strategies that result in prolonged release.

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These


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chemical signals diffuse within the scaffold and reach the host vasculature, resulting in activation of endothelial cells (EC), the cells lining the walls of blood vessels. 29 In response to these gradients of soluble GFs, activated ECs are selected through a complex series of intra- and intercellular events 30 to change their phenotype to a specialized spindle-like cell type referred to as tip cell which acts as the new sprout tip. Tip cells start migrating in the scaffold environment, following the GF gradients. The cells behind the tip cell, referred to as stalk cells, begin elongating and proliferating to maintain the connection between the tip cell and the host blood vessel, leading to formation and extension of new capillaries. These sprouts grow as a network in the scaffold towards the chemical and molecular angiogenic factors. When these sprouts grow sufficiently, they anastamose with other vessels, fusing together to form loops that can circulate blood flow. Finally, newly formed blood vessels are stabilized by cells like smooth muscle cells and pericytes that provide structural support. Blood flow through these new capillaries begins after this final step of stabilization. 31 A large number of theoretical models of angiogenesis have been developed during the last three decades, to facilitate understanding the complex behaviors of endothelial cells and their interactions with the environment. The first theoretical models of angiogenesis were simple mathematical models using one-dimensional (1D) differential equations to represent the chemotactic behavior of endothelial cells.

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Advances in modeling resulted in 2D models

including a number of partial differential equations to describe angiogenesis on a 2D surface. These models were able to describe the spatio-temporal development of the newly formed capillaries in tissue in response to gradients of diffusing soluble growth factors and insoluble ligands. 35-39 One important improvement was the addition of stochastic cell behavior for analysis of random motility and chemotaxis of ECs. Simulations were developed in which the path of


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newly formed capillaries were determined by the trajectories of the migrating tip cells, illustrating the behavior of migrating ECs as the determining factor in capillary growth rate and vasculature structure. 40 While there has been a multitude of modeling efforts aimed at understanding various aspects of the angiogenesis process, fewer models have been developed to specifically consider angiogenesis within tissue engineering scaffolds.

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Examples of such models include the

equation-based 2D model developed by Jabbarzadeh and Abrams which was developed speciﬁcally to understand and improve capillary formation within porous tissue engineering scaffolds by considering growth factor release and transport within porous scaffolds.

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Their

model enabled investigating the effect of growth factor concentration proﬁles on EC chemotaxis and blood vessel formation. Checa et al. developed a lattice-based model to consider the effect of cell seeding and mechanical stimuli in porous scaffolds on angiogenesis and tissue growth, combining angiogenesis and bone tissue formation in regular and irregular calcium phosphate scaffolds.

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Lemon et al. developed a model of angiogenesis within porous biomaterials in

which a number of rules were used to control EC migration and capillary branching and anastomosis within an implanted scaffold pore. 45 Seeded fibroblasts were considered to secrete a growth factor which was diffusing within the pore, being consumed by ECs, and simultaneously degrading with a constant rate. This model was used to study various cell seeding strategies that would enhance biomaterial vascularization. Discrete models in which individual cells are represented by agents, the so-called agent based models (ABM), have been used in recent years to model angiogenesis.

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These models can

represent cells as single independent entities that can act in their environment and interact with their neighboring cells and describe emergence of blood vessels.

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Using ABMs, researchers


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have been able to explicitly investigate the effects of EC migration, growth, proliferation, and apoptosis on angiogenesis and to show the relationship among growth factor gradients, cellular behaviors, and sprout extension. 47 ABMs have enabled researchers to develop multiscale models spanning several spatial levels.

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These models are capable of integrating fundamental models

already developed that describe dynamic changes in process variables such as GF concentrations or blood flow within the capillaries, producing more realistic outcomes as a result of the combination of the cellular level actions, variations in cell environment conditions, and interactions among them.

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ABMs have also been used to investigate the detailed behavior of

fewer numbers of cells and intracellular phenomena. In these models cell parts are represented by individual agents, resulting in higher resolution and representation of cell level behavior as the outcome of intracellular phenomena. For example, the model developed in

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simulates the

behavior of few adjacent ECs during the process of tip cell selection, based on a recently proposed feedback loop that relates tip cell induction by VEGF to Dll4/notch-mediated lateral inhibition, resulting in appearance of patterns previously observed in experiments. 30 3. Agent-based modeling Agent-based models (ABM) use a number of rules that govern the behavior of the constituents of a system to predict the system level behavior. ABMs can be used to describe, hypothesize, and predict the behavior of any system comprised of a large number of constituents. Agent-based modeling is a relatively new method that was first introduced and used in social sciences. During the past decade, software agents and multi-agent systems (MAS) have attracted significant attention in academic and industrial research. Agent-oriented design has become an active and appealing field that has found applications in diverse fields of science ranging from social sciences and economics to supply chain management, traffic control and biology, by providing a


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novel modeling paradigm for dealing with the challenges in designing computational models of interactive and complex systems. 53-56 An agent is an autonomous software entity that performs the tasks one expects from an active constituent of a large system through artificial intelligence embedded within its rule base and utilizing its ability to interact with other agents and its environment. Software agents utilize their interaction and collaboration abilities as guided by the intelligence encoded within their rule base to progress towards their goals in a context that can change in predictable or unpredictable manners. By identifying the simple, lower level actions of each of the constituents in a framework that represents the environment of these agents, emergent system level behavior are generated. Using agents for modeling becomes appealing when the system under study becomes increasingly complex, bringing about design issues that conventional equation-based methodologies fail to tackle. In the design of models for large, complex, real-life systems, agent abstractions allow the design of different components of the system separately, describe the rules for interactions with other agents and the environment, capture their stochastic variations, and enable the interactions during the execution of a simulation. Some common characteristics of agents are: 57 Autonomy: Autonomy is the most distinctive property of an agent, bringing about one agent’s capability to survive in a changing environment, while another agent may fail to do so. Communication: An agent can communicate with other agents and with its environment. Cooperation: An agent can collaborate and cooperate with other agents in order to achieve a common goal or to solve a problem pertaining to those agents.


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Mobility: An agent can have the ability to move in its environment at its own discretion. However, an agent can decide based on its logic to be mobile or to become static (or fixed). Biological systems are naturally complex systems, involving many different parameters (state variables) and functions of cells and biochemical components, and spanning different scales from molecular to cellular and tissue levels.

This complexity puts them beyond the realm of

traditional mathematical models. ABMs are suitable for representing such intricate biological systems. Each agent is represented explicitly, and the model includes context- and statedependent rules for agents’ possible actions and the emergent behaviors that result from them. Biological agents can be developed to represent organs, tissues, cells, genes, and molecules. The goal of ABMs is to describe or forecast the dynamics of system components interacting with their local environments and other agents based on the current state of other agents and the environment in the vicinity of a specific agent. Higher-scale properties can also emerge from the rules that govern the lower scale system components. Using ABM, scientists can relate the observed patterns of a system and its inherent characteristics to attributes of its constituents and rules that govern their behavior. They can experiment with assumptions about key variables, properties of constituents, rules, and processes to identify conditions that lead to desirable observable phenomena. The number of biomedical research studies that use ABMs to investigate multicellular and tissue level phenomena has been increasing. Recent publications and reviews of ABM applications of in tissue growth and angiogenesis illustrate the benefits if this framework. 14-15, 5859

ABMs are ideal for investigation of morphogenesis of tissues during development.

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They

have also been useful in understanding and explaining the underlying facts in growth and expansion of cancerous tumors. 61 A 3D multiscale ABM was developed to simulate the cellular


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decision process in the context of a virtual brain tumor.

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In bone tissue engineering, ABMs

have been used to simulate real-time signaling induced by mechanical stimuli in osteocytic networks.

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Various types of ABMs have been reported to describe several aspects of

angiogenesis. 15, 64 We developed a multi-layer ABM to simulate the process of sprouting angiogenesis within 3D porous scaffolds. Different layers in the model include a scaffold layer that is used to represent the scaffold structure and a blood vessel layer that controls the behavior of EC agents as building blocks of the blood vessels. EC agents in blood vessel layer have variables that determine the internal state of the agent and methods that define the functions that the agent can perform. A rule base contains the logic that governs the behavior of EC agents and the actions they perform during angiogenesis, including elongation, proliferation, tip cell migration, and anastomosis. Tipcell migration within the pore domain of the scaffold results in extension of the capillary branch, while it has the ability to connect to other EC agents if they are in a close proximity.

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The

leading stalk cell is following the tip cell, elongating as the tip cell migrates. Proliferation of the stalk cell occurs once it reaches a “Maximum EC Length”. In abstraction of the reality, the tip cell migration and leading stalk cell elongation were represented merely by movement of the front node of the leading stalk EC agent in the 3D space within the porous scaffold. The new branches extend stochastically in the direction of highest growth factor (GF) gradient at the location of the branched tip cell. A tip cell may connect to another EC agent only if it is from a different branch, leading to anastomosis. Figure 1 shows a set of rules derived from information in the literature that govern the behavior of individual agents, leading to tip cell migration and branching, and stalk cell elongation and proliferation. The cell agent searches its local environment to identify and avoid


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scaffold locations. The grid location resulting in highest GF concentration is selected for migration. Hence, the migration vector is a deterministic vector connecting the current location of the EC agent to the location with the highest GF gradient. After completing migration, the tip cell searches its environment and if another EC agent is found in its close neighborhood, anastomosis takes place. Tip cell migration is followed by elongation of the leading stalk cell behind the tip cell, hence maintaining the connectivity between the tip cell and stalk ECs.

Figure 1. Flowchart depicting the main rules governing the EC behavior in ABM. EC agents perform chemotactic migration, elongation and proliferation, random branching and anastomosis. Initial host blood vessels were defined and placed at the scaffold-tissue interface, representing the host vasculature. Cell adhesion ligands were distributed randomly across the scaffold. Angiogenic GF concentrations (CGF) were defined within the scaffold according to Equation 1, in which Ymax is the scaffold dimension in y-direction. GF was assumed to be an angiogenic protein that results in sprouting and chemotaxis (e.g. VEGF). CGF was a function of y-direction only, with the maximum in the top surface of the scaffold, along the x- and z-axes. To ensure


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stochastic behavior, a random change of less than or equal to ±0.05% of CGF was added to the values of CGF.

= − − +

4. Scaffold Design Designing the porous architecture of the scaffold involves definition of the variables that describe the geometrical properties of the scaffold and are expected to influence vascularization based on both experimental and theoretical studies.

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Based on these definitions and the ranges

of variable values, porous scaffold models with different architectural characteristics were designed and then used as an input to the simulations, enabling the study of the effect of scaffold architecture on vascularization. Two different general arrangements of the pores within the scaffold structure resulted in two different categories of scaffolds, termed homogeneous and heterogeneous scaffolds. Homogeneous scaffolds were designed with a constant pore size and uniform placement of the pores within the scaffold, and were characterized by specifying the pore diameter and interconnectivity. Each spherical pore is connected to six neighboring pores, and interconnectivity represents the interconnection between adjacent pores. Interconnectivity is controlled in the scaffold structure by the pore throat diameter, which is defined as the diameter of the biggest sphere that can traverse between two neighboring pores

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. In each scaffold, pore

throat diameter is normalized by dividing its value by the mean pore size, resulting in a value known as the normalized pore connectivity (NPC).

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Hence, pore size and normalized pore

connectivity were the design variables in homogeneous scaffolds, while porosity was a dependent variable that increased with the pore throat diameter and caused an increase in


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interconnectivity. Figures 2 and 3 illustrate the effect of different pore size and interconnectivity on homogenous scaffolds.

Figure 2. Three dimensional renderings of homogenous scaffolds with different pore sizes of A) 150 µm, B) 275 µm and C) 400 µm and constant interconnectivity of 140 µm. Reproduced with permission from reference 46. Copyright 2013 Elsevier.

Figure 3. Three dimensional renderings of homogenous scaffolds with constant pore size of 275 µm, and varying interconnectivities of A) 70 µm, B) 140 µm, and C) 190 µm. Heterogeneous scaffolds were designed by randomly placing spherical pores of different size within the scaffold. Pore sizes are selected from a normal distribution specified with the mean pore diameter and standard deviation values. Figure 4, illustrates the architecture of different


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scaffolds created with a constant porosity and standard deviation value. Pores were added to the scaffold at random locations until the desired scaffold porosity was achieved, as shown in Figure 5.

Figure 4. Three dimensional renderings of heterogeneous scaffolds with different pore sizes of A) 150 µm, B) 275 µm, and C) 400 µm, with constant porosity (60%) and zero standard deviation of pore size.


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Figure 5. Three dimensional renderings of heterogeneous scaffolds with constant pore size of 275 µm and zero standard deviation of pore size, and varying porosities of A) 40%, B) 60%, and C) 80%. Reproduced with permission from reference 46. Copyright 2013 Elsevier.


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Figure 6 shows the effect of different pore size distribution standard deviations on architecture of model scaffolds. In these scaffolds the design variables were the mean pore size, the standard deviation of the pore size distribution (simply referred to as pore size distribution), and porosity. To control for overlapping of the pores, the distance between the centers of two intersecting pores was defined to be greater than the sum of the two radii, plus an additional ten percent of the radius of the smaller pore.

Figure 6. Scaffolds with the same constant mean pore size (275 µm) and porosity (60%) and varying standard deviation from the mean of A) zero, B) 25 µm, C) 50 µm, and D) 75 µm.


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In heterogeneous scaffolds the random placement of pores resulted in good qualitative agreement with the structure of real biomaterials. Also, as a result of random placement of the pores, different scaffolds with unique architectures could be generated for each specific set of design variables. Randomness was introduced by producing different scaffolds with the same characteristic parameters. To average the effect of randomness out, ten scaffolds were generated for each set of design variables and simulations were run for all scaffolds and the results were averaged. Three-dimensional renderings of model heterogeneous and homogeneous scaffolds, with similar porosities and pore sizes, are compared in Figure 7 to illustrate the effect of design parameters on scaffold structure. Once the architecture of the 3D scaffold is designed and generated, a stack of 2D images is created from the 3D scaffold model and is used to input the scaffold model to ABM simulation framework. Scaffolds studied in our simulations were assumed to be implanted in vivo, in contact with host tissue.


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Figure 7. Three dimensional renderings of A) Homogenous and B) Heterogeneous scaffolds. Both types of scaffolds have similar interconnectivity (~160 µm) and porosity (80%) with zero pore size distribution. 5. Case Studies The scaffold design activity includes several stages. First, the scaffold design variables are defined. Then, 3D scaffold structure is designed and images of the scaffold with the selected variable values are generated. These images are used as an input to the agent-based simulator for modeling angiogenesis. Next, process variables are assigned at the beginning of the simulation and mathematical relations are utilized to represent their variation during the angiogenesis process. The results of the simulation are used to compute the product variables that represent the characteristics of the blood vessel network created. The average values of these variables are computed for many simulations conducted with stochastic variations for each case and they are compared with experimental results for consistency. The consumer variables can be measured only during in vivo use of the vascularized tissue and will not be considered in this communication. In order to systematically investigate the effects of different combinations of scaffold design variables and process variables on product quality variables, a number of case studies were defined and simulations were performed for each set of variables. Scaffold design variables and process variables were the input variables of the ABM. A number of stochastic variations were included in runs corresponding to each set of fixed input variables in order to represent the randomness and uncertainty associated with biological systems. Table 1 lists the values of the scaffold design variables for homogeneous and heterogeneous scaffolds. For each variable in


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Table 1, the intermediate value was considered as baseline value, and variations from the baseline were investigated to elicit the effect of that variable on product quality variables. In these studies, the pore architecture of the scaffolds was assumed to remain constant during simulation runs. Table 1. Design variables of homogeneous and heterogeneous scaffolds. Homogeneous Scaffolds Pore Size (µm)

150

275

400

Normalized Pore Connectivity 0.20-0.45 0.45-0.55 0.55-0.70 Heterogeneous Scaffolds Mean Pore Size (µm)

150

275

400

Porosity (%)

40

60

80

Pore Size Distribution (µm)

0

25

50,75

The main model parameters are summarized in Table 2. Randomness was included in some model parameters to introduce stochastic variation into the model, leading to stochastic behavior and variance among cells. For random parameters mean and the magnitude of randomness (standard deviation/± range) are reported. Parameter estimation was performed for a number of model parameters (e.g. unit elongation and sprouting probabilities) to fine tune the model and to ensure that the simulation results are in agreement with experimental data. A detailed description of model parameters and rules governing the agent behavior can be found in. 46 Table 2. Model dimensions and parameters. For random parameters, mean and the magnitude of randomness (standard deviation/± range) are reported. Parameter

Value

Explanation


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Scaffold width, height, 2000, 2000, 1000 The scaffold model represents a functional unit of an engineered tissue depth (µm) No Initial Vessels

20

The number of host blood vessels in the surrounding tissue, based on a 100 µm distance between the blood vessels

Maximum EC Length

40 (± 25%) µm

Max possible EC length for elongation. Cells that reach this length divide

Blood Vessel Diameter

10 µm

Constant diameter of blood vessels

Unit Elongation

3 µm

EC length increase at each elongation action

Speed Interval

5 (SD = 1.0) time between occurrences of migration actions), assigned randomly using a normal distribution with this mean steps

Used in Eq. 1 to calculate migration interval (interval

value

Sensing Distance

20 (± 5) µm

Maximum distance EC agents can sense [GF] in their neighborhood

Persistency Time

2 time steps

Time length a migrating tip cell does not change its direction, in accordance with persistence time of 1-5 hours reported for ECs 68-69

Branch Delay Time

8 time steps

Time length during which a new sprout does not sprout again

Anastomosis Radius

32 µm

Maximum distance a tip cell searches to connect to EC agents in the vicinity

Simulations were run for 600 time steps corresponding to six weeks in real time and outcomes were reported at 100 time step intervals. For each case, 200 simulations were run. The control case was defined as vascularization in the absence of scaffold, to represent an ideal case in which the whole scaffold volume was available to blood vessels, hence providing a basis for comparing the different case studies with. The results of various runs were statistically analyzed to calculate the mean, standard deviation, and minimum and maximum values of product quality variables of all the runs performed for each set of design variables. The quality variables investigated included total blood vessel length (TBVL), blood vessel length density (BVLD), invasion depth


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(ID), maximum invasion depth (MID), and total number of sprouts (TNS). TBVL was defined as the cumulative length of all blood vessels formed during the vascularization process. BVLD represented the ratio of TBVL to scaffold pore volume. ID represented the average ingrowth of all sprouts and shows the ability of all invading sprouts to penetrate the scaffold, while MID is the maximum depth of scaffold penetrated by any single invading capillary. TNS represents total number of sprouts formed, being an indication of how branched the newly formed vasculature is. 6. Discussion of results In simulation runs, the activated tip cells sprout from the host blood vessels in response to GF gradients. The tip cells migrated and led capillary ingrowth into the pore domain of the scaffold, resulting in scaffold perfusion. 3D images depicting scaffold vascularization for sample homogeneous (Fig. 8) and heterogeneous (Fig. 9) scaffolds illustrate the invasion of blood vessels into the scaffolds and the effects of scaffold design variables on product quality variables qualitatively. Figure 8 qualitatively demonstrates the positive effect of increasing pore size on angiogenesis in homogeneous scaffolds, as observed from the increased depth of capillary invasion and vascularized pores at the end of simulation run. Figure 9 depicts a comparison of the effect of pore size in heterogeneous scaffolds at the end of simulation runs (mean pore size 150, 275, and 400 µm, PSD = 75, and porosity of 80%). Capillary invasion is shown to lead to vascularization of the pores close to the interface during the early stages of the simulation. TBVL increases in the pores close to the interface while invasion of blood vessels to the deeper regions of the scaffold occurs with difficulty. Consequently, the majority of the pores located far from the interface are not vascularized even at the end of simulation runs (corresponding to 6 weeks). Steric hindrances result in smaller number of capillaries that continue invading the depths of the scaffold particularly in the smaller mean pore size.


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Figure 8. 3D illustration of homogeneous scaffold vascularization for different pore sizes of (A) 150, (B) 275, and (C) 400 µm after 600 time steps. In all depicted cases, the overall scaffold porosity is 80% and NPC is 0.60.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

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Figure 9. 3D renderings of heterogeneous scaffold vascularization after 6 weeks for different pore sizes of (A) 150, (B) 275, and (C) 400 µm after 600 time steps. Overall scaffold porosity is 80% in all pore sizes, and pore size distribution is 75 µm. Figure 10 illustrates the effect of heterogeneous scaffold design parameters on TBVL, ID, and TNS at the end of simulation time. Similar trends are observed for homogeneous scaffolds, and hence the results are not presented here. Increasing the scaffold pore size continuously improves angiogenesis within the scaffold, leading to considerable difference in TBVL between pore size 275 and 400 µm. Depth of invasion (ID) and number of sprouts (TNS) show the same trend, consistently increasing with pore size and porosity. Pore size distribution plays an important role. The maximum ID achieved after 6 weeks is approximately 25% of the whole scaffold depth, indicating the difficulty in vascularizing the deeper parts of the scaffold.


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Figure 10. TBVL (A,B, and C), ID (D, E, and F) , and TNS (G, H, and I) in heterogeneous scaffolds after 6 weeks as a function of porosity for different pore sizes of 150 (A, D and G), 275 (B, E and H), and 400 (C, F and I) µm, for various pore size distributions of 0-75 µm.

Sensitivity analysis was performed to compare the effect of different design parameters on improving scaffold vascularization. Baseline values were defined as the intermediate values


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(pore size = 275 µm, pore size distribution = 5, normalized pore connectivity = 0.6, and porosity = 60%). Figures 11 and 12 illustrate the effect of homogeneous and heterogeneous scaffold design variables on TBVL, ID, and TNS throughout the simulation time, respectively. All scaffold design variables considered in this study impact scaffold vascularization, hence providing flexibility in scaffold design. As an example, one practical conclusion is the possibility to improve scaffold vascularization and increase the number and length of the formed blood vessels as well as the depth of scaffold invasion by increasing the pore size distribution and keeping porosity constant at an acceptable value when there are limitations on the upper limit of scaffold porosity due to mechanical strength issue. It should be noted that in heterogeneous scaffolds, TBVL showed less dependence on pore size distribution compared to ID and TNS (Fig. 12).


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Figure 11. Effect of pore size (A, C and E) and interconnectivity (B, D and F) on TBVL (A and B), ID (C and D), and TNS (E and F) over time in homogeneous scaffolds.

Figure 12. Effect of pore size (A, D and G), pore size distribution (B, E and H), and porosity (C, F and I) on TBVL (A, B, and C), ID (D, E and F) and TNS (G, H and I) over time in heterogeneous scaffolds.


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Simulation runs using tools such as ABM presented in this study facilitate designing optimal scaffold structures. In addition, simulations can be used to evaluate new scaffold designs for specific applications (e.g. bone tissue engineering), highlighting the value of ABM as a novel tool in product design. 70 Appropriate experimental techniques should be in place to fabricate the scaffolds with well-defined design parameters. Porous scaffolds with well-designed architecture and controlled porosity and interconnectivity are manufactured using a variety of fabrication methods.

1, 71

Novel scaffold fabrication techniques using a bottom up approach have been

developed to produce microscale building blocks of tissue constructs and combining these subunits to fabricate larger tissues.

9, 72

It is expected that combination of experimental and

computational techniques with novel scaffold fabrication approaches would lead to generation of enhanced tissue engineering scaffolds for specific applications. 7. Conclusions Simulations of angiogenesis with ABMs can be coupled with scaffold design techniques to select optimal values for design variables and process variables that influence the product variables, namely the characteristics of the vascularization process. Scaffold design and simulation of angiogenesis within the designed scaffolds can be used as an iterative process for designing optimal scaffold structures. This technique not only facilitates designing optimized scaffolds with significant reductions in cost and in time, but also provides a better understanding of the mechanisms of the system under study. Corresponding Author * Email: [email protected], Phone: +1-312-567-3042 ACKNOWLEDGMENT


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This material is based upon work supported in part by the National Science Foundation (CBET0731201 and IIS-1125412) and the Veterans Administration.


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Flowchart depicting the main rules governing the EC behavior in ABM. EC agents perform chemotactic migration, elongation and proliferation, random branching and anastomosis. 82x80mm (299 x 299 DPI)


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Three dimensional renderings of homogenous scaffolds with different pore sizes of A) 150 µm, B) 275 µm and C) 400 µm and constant interconnectivity of 140 µm. 257x64mm (96 x 96 DPI)



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Three dimensional renderings of homogenous scaffolds with constant pore size of 275 µm, and varying interconnectivities of A) 70 µm, B) 140 µm, and C) 190 µm. 164x61mm (300 x 300 DPI)


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Three dimensional renderings of heterogeneous scaffolds with different pore sizes of A) 150 µm, B) 275 µm, and C) 400 µm, with constant porosity (60%) and zero standard deviation of pore size. 151x46mm (300 x 300 DPI)



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Three dimensional renderings of homogenous scaffolds with constant pore size of 275 µm and zero standard deviation of pore size, and varying porosities of A) 40%, B) 60%, and C) 80%. 91x178mm (72 x 72 DPI)


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Scaffolds with constant mean pore size (275 µm) and A) zero, B) 25 µm, C) 50 µm, and D) 75 µm standard deviation from the mean. 352x225mm (72 x 72 DPI)



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Three dimensional renderings of A) Homogenous and B) Heterogeneous scaffolds. Both types of scaffolds have similar interconnectivity (~160 µm) and porosity (80%) with zero pore size distribution. 352x162mm (72 x 72 DPI)


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3D illustration of homogeneous scaffold vascularization for different pore sizes of (A) 150, (B) 275, and (C) 400 µm after 600 time steps. In all depicted cases, the overall scaffold porosity is 80% and NPC is 0.60. 187x415mm (300 x 300 DPI)



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3D renderings of heterogeneous scaffold vascularization after 6 weeks for different pore sizes of (A) 150, (B) 275, and (C) 400 µm after 600 time steps. Overall scaffold porosity is 80% in all pore sizes, and pore size distribution is 75 µm. 189x425mm (300 x 300 DPI)


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TBVL (A,B, and C), ID (D, E, and F) , and TNS (G, H, and I) in heterogeneous scaffolds after 6 weeks as a function of porosity for different pore sizes of 150 (A, D and G), 275 (B, E and H), and 400 (C, F and I) µm, for various pore size distributions of 0-75 µm. 1087x1312mm (96 x 96 DPI)



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Effect of pore size (A, C and E) and interconnectivity (B, D and F) on TBVL (A and B), ID (C and D), and TNS (E and F) over time in homogeneous scaffolds. 103x102mm (300 x 300 DPI)


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Effect of pore size (A, D and G), pore size distribution (B, E and H), and porosity (C, F and I) on TBVL (A, B, and C), ID (D, E and F) and TNS (G, H and I) over time in heterogeneous scaffolds. 209x216mm (300 x 300 DPI)


Design of Polymer Scaffolds for Tissue Engineering Applications

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