Mathematical model of growth and neuronal differentiation of human induced pluripotent stem cells seeded on melt electrospun biomaterial scaffolds

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by

Meghan Hall

B.Sc., University of Victoria, 2013

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in the Department of Mathematics and Statistics

c

Meghan Hall, 2016 University of Victoria

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Mathematical Model of Growth and Neuronal Differentiation of Human Induced Pluripotent Stem Cells Seeded on Melt Electrospun Biomaterial Scaffolds

by

Meghan Hall

B.Sc., University of Victoria, 2013

Supervisory Committee

Dr. Roderick Edwards, Co-supervisor

(Department of Mathematics and Statistics)

Dr. Stephanie Willerth, Co-supervisor (Department of Mechanical Engineering)

Dr. Mohsen Akbari, Outside Member (Department of Mechanical Engineering)

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Supervisory Committee

Dr. Roderick Edwards, Co-supervisor

(Department of Mathematics and Statistics)

Dr. Stephanie Willerth, Co-supervisor (Department of Mechanical Engineering)

Dr. Mohsen Akbari, Outside Member (Department of Mechanical Engineering)

ABSTRACT

Human induced pluripotent stem cells (hiPSCs) have two main properties: pluripo-tency and self-renewal. Physical cues presented by biomaterial scaffolds can stimulate differentiation of hiPSCs to neurons. In this work, we develop and analyze a math-ematical model of aggregate growth and neural differentiation on melt electrospun biomaterial scaffolds. An ordinary differential equation model of population size of each cell state (stem, progenitor, differentiated) was developed based on experimen-tal results and previous literature. Analysis and numerical simulations of the model successfully capture many of the dynamics observed experimentally. Analysis of the model gives optimal parameter sets, that correspond to experimental procedures, to maximize particular populations. The model indicates that a physiologic oxygen level (∼_{5%) increases population sizes compared to atmospheric oxygen levels (}∼_21%).

Model analysis also indicates that the optimal scaffold porosity for maximizing ag-gregate size is approximately 63%. This model allows for the use of mathematical analysis and numerical simulations to determine the key factors controlling cell be-havior when seeded on melt electrospun scaffolds.

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List of Tables

Table 2.1 Comparison of data of three experiments for two scaffold porosi-ties 12 days after seeding. . . 11 Table 2.2 Proliferation data for two neural aggregates seeded on loop mesh

200 and loop mesh 500. . . 12 Table 3.1 Components of functional effects on parameters and extremal

val-ues. . . 32 Table 3.2 Experimental and compound parameter values. The functional

effect is the product of feedbacks for each parameter, e.g. fα =

f1f7. The compound value is the experimental value multiplied

by the functional effect, e.g. αfα. ∗ No measurements available.

Closest related measurements were taken. For α, the upper bound for β was used. For r, the upper bound for d1 was used. In both

cases, it is taken that 0.000001 is the lower bound. Note that C ∈ [1, 10]. . . 40 Table 4.1 Summary of optimal parameter values for maximizing cell

popu-lations. . . 63 Table 4.2 Optimized fixed points with O-independent parameters. . . 63 Table 4.3 Optimized fixed points with dependent parameters. Note that

W∗ = 5 in all cases. Coupled indicates dependence of the pa-rameters on O, W and C. Uncoupled indicates the independent optimal parameter value was used. For the optimal T∗ popula-tions, the populations are given for the minimum and maximium values of γ, γmin and γmax, as these can vary depending on the

value of γ taken, noting that the value of T∗ does not change. Also note that these fixed points are stable as the condition for stability, 2p2 > β + d2, is met in each case. . . 68

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Table 5.1 Summary of optimization results for the SP D model. Note that the parameter sets for max S∗ and max T∗ are the same. † indi-cates a condition on the optimal value. . . 104 Table 5.2 Summary of SP DOW optimized populations. Note that the

in-dep P∗ and indep D∗ are unstable, with all other being equilibria stable. Numeric values are denoted ˆS∗_{, ˆ}_P∗_{, and ˆ}_D∗_{, while}

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List of Figures

Figure 1.1 Reprogramming and differentiation process of hESCs and hiP-SCs into neuronal cell types. . . 3 Figure 1.2 Overview of experimental protocol for neural aggregate

forma-tion and seeding. Isolated hiPSCs (left) are cultured in ag-grewells with NIM (center) to form aggregates of neural progeni-tor cells. The neural aggregates are then seeded onto biomaterial scaffolds in NIM (right) to produce neurons. . . 5 Figure 2.1 Scanning electron microscope images of loop mesh 200 and loop

mesh 500 scaffolds. . . 10 Figure 2.2 Top and side views of a simplified experimental setup. . . 10 Figure 2.3 Fluorescence images of neuronal marker Tuj1 expressed in neural

aggregates 12 days after seeding on loop mesh scaffolds. (A) Scaffold with higher porosity. (B) Scaffold with lower porosity. Scale bar is 400 µm. . . 11 Figure 2.4 Cell body cluster area of two neural aggregates on loop mesh 200

and loop mesh 500 over 12 days. . . 12 Figure 2.5 Bright field images of neural aggregates on loop mesh 200 (top)

and loop mesh 500 (bottom) scaffolds. . . 13 Figure 3.1 Schematic diagram of three cell states with cellular feedback.

Black arrows indicate transitions between states. Red arrows indicate negative feedback. . . 15 Figure 3.2 Function for the effect of oxygen on proliferation, f5, and

exper-imental data points used for fitting. . . 20 Figure 3.3 Function for the effect of waste on proliferation, f6, and

experi-mental data points used for fitting. . . 21 Figure 3.4 Function for the effect of contact on proliferation, f10 and

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Figure 3.5 Function for the effects of oxygen on death, f1, f2 and f3, and

experimental data points used for fitting. . . 24

Figure 3.6 Function for the effect of waste on death, f7, and experimental data points used for fitting. . . 26

Figure 3.7 Function for the effect of oxygen on differentiation, f4, and ex-perimental data points used for fitting. . . 27

Figure 3.8 Function for the effect of contact on differentiation, f9. . . 28

Figure 3.9 Function for the effect of oxygen on reversion, f8. . . 29

Figure 3.10Function for the effect of oxygen on oxygen consumption and waste production, f13. The range of O shown is 0 to 2 as the function values above this range are nearly constant. . . 30

Figure 3.11Functions for the effect of contact on oxygen and waste diffusion, f11 and f12. . . 31

Figure 3.12Functional effects of O. . . 33

Figure 3.13Functional effects of W . . . 33

Figure 3.14Functional effects of C. . . 34

Figure 3.15Side view of experimental setup including distances used in dif-fusion terms. . . 41

Figure 4.1 Schematic diagram of two cell states with cellular feedback. Black arrows indicate transitions between states. Red arrows indicate negative feedback. . . 55

Figure 4.2 Functional effects of W with optimal points for maximizing S∗, P∗ and T∗. . . 64

Figure 4.3 Functional effects of C with optimal points for maximizing S∗, P∗ and T∗. . . 65

Figure 4.4 Functional effects of W with optimal points for maximizing D∗. 65 Figure 4.5 Functional effects of C with optimal points for maximizing D∗. 66 Figure 4.6 Functional effects of O with optimal points for maximizing P∗ and T∗. . . 67 Figure 4.7 Functional effects of O with optimal points for maximizing D∗. 67 Figure 5.1 Example of hyperbola intersections for N1 and N2. Note that

there is only one positive intersection. The parameter set used is for optimal S∗ with dependent parameters (see Section 5.4). 82

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Figure 5.2 Example of hyperbola and asymptote intersection for N1 and

N2. Note that the intersections of the asymptotes is near the

intersection of the hyperbolas at large P . . . 83 Figure 5.3 Example of effects of altering dependence of parameter on

pop-ulations. Independent parameters (dot); d1, d2, r independent

(solid); d1, d2 independent (dash); dependent (dash-dot).

Param-eter set used is for maximizing S∗, with C = 3.75 and O∗, W∗, Oair, and Wair values from Table 5.2. Experimental parameters:

α = 0.000001, β = 0.000016, γ = 0.000026, p1 = 0.00119, p2 =

0.0016, d1 = 0.0001, d2 = 0.0000729, r = 0.00017. Independent

parameters: d1 = 0.0000005, d2 = 0.0000003645, r = 0.00051. . . 107

Figure 5.4 Population dynamics with initial population of 5000 progenitor cells for C = 7.7 (solid) and C = 6 (dash) with O = 21 and W = 5. Experimental parameters: α = 0.0000001, β = 0.0000016, γ =

0.0000016, p1 = 0.00119, p2 = 0.0016, d1 = 0.000135, d2 = 0.00007745, r =

0.000001. . . 108 Figure 5.5 Population dynamics with initial population of 5000 stem cells

and C = 10 for O = 21 (solid) and O = 5 (dash). Same experi-mental parameters used as in Figure 5.4. . . 109 Figure 5.6 Population dynamics after switching to parameters for

maximiz-ing D∗ from the initial point of S∗, with independent d1, d2, and

r. See Figure 5.3 for parameters used in maximizing S∗, and Figure 5.4 for D∗ maximizing parameters. . . 110

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Introduction

1.1 Tissue Engineering

Throughout the history of medicine, many ways to treat illnesses by external means or by inducing a natural response have been employed. However, there are certain injuries and diseases that cannot be currently resolved as the body’s mechanisms are unable to repair the damage, even with medical intervention. Among these illnesses are organ failure and damage, which can sometimes be resolved using transplantation. However, transplantation is reliant on a limited donor supply and the donor-recipient combination requires many factors in order to avoid complications such as rejection of the donor organ. To this end, tissue engineering is a thriving field that aims to provide replacement tissues and organs that can replace diseased or damaged tissue, circumventing the need to fix the current tissues. The process of tissue engineering is the engineering of combinations of materials and cells to create a functional structure that can be used in place of the original tissue [30, 58]. Engineered tissues would also eliminate the limitations of traditionally transplanted tissues as the supply could be made to order and there is the possibility of using the patient’s own cells to create the tissue, eliminating issues of rejection [30].

1.2 Stem Cells

Stem cells are unique in that they can differentiate into multiple cell types [30, 58, 38]. A specific type of stem cells are called pluripotent stem cells, referring to their pluripo-tency, the ability to differentiate into any cell type, not just a particular subset of

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cell types [30, 58, 38]. Tissue engineering relies on cells that can divide and per-form the necessary functions to simulate the functions of natural tissues. Terminally differentiated cells have limited to no capacity for division or adaptive capacity, so when introduced into a new environment, cell viability becomes an issue. Another problem is that tissues have heterogeneous cell populations. Determining the nature of and being able to reproduce these populations are not simple, or reasonable, goals. The discovery of embryonic stem cells (ESCs) avoids these limitations as they have the capacity for nearly unlimited replication and can differentiate into any cell type needed, including heterogeneous populations [32]. ESCs also have their own draw-backs as they have tumorigenic potential [6] and would not be derived from the future recipient. The lack of host source cells caused problems such as rejection [44]. Also, the source of embryonic stem cells became a highly contentious topic and research was highly restricted in some countries [47]. In 2006, however, it was determined that terminally differentiated cells could be transformed into stem cells using transcrip-tion factors, termed Yamanaka factors [51]. Human induced pluripotent stem cells (hiPSCs) removed the controversy of using embryonically derived stem cells and can also be sourced from the recipient’s own cells [50, 59], limiting rejection. In theory, these cells could be combined with the right materials to produce the structure and cell populations needed to generate tissue.

The differentiation of stem cells into the appropriate structure and functions is not straightforward. The differentiation process is not fully understood and con-trolling the process in vitro is a difficult task. The molecular pathways concon-trolling differentiation are partially known, but the entire network is yet to be determined [25, 26, 43, 49, 29]. Factors that control differentiation are both physical and chemical [25, 26, 43, 49, 29], with speculation that mechanisms involved are controlled, chaotic, random, or some combination thereof [31, 21]. The process of reprogramming somatic cells to hiPSCs and subsequent differentiation into neural cell types is illustrated in Figure 1.1 [57].

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Figure 1.1: Reprogramming and differentiation process of hESCs and hiPSCs into neuronal cell types.

1.3 Stem Cell Behavior

Stem cells are undifferentiated cells that can divide indefinitely and differentiate into multiple cell types, where pluripotent stem cells can give rise to any cell type [30, 29, 58]. Stem cells divide and differentiate, which can produce structures such as organs and tissues. The differentiation process is a continuum where the stem cell’s gene expression is altered until the terminally differentiated state is reached [29]. These changes result in altered behaviors including the development of new functions and the loss of indefinite division. Although a continuum, it has been shown that the differentiation process has somewhat distinct transition states where the cell is neither stem nor differentiated [31, 57]. This transitory state is commonly referred to as the progenitor cell state. Certain markers can be used to identify which state a cell is in. For example, the stem cell state for hiPSCs can be identified by the expression of the markers SSEA-3 and SSEA-4 [57]. The progenitor cell state can be defined by the expression of nestin [34, 49]. Finally, when expression of β-III-tubulin (Tuj1) occurs, the cell is considered to be in the differentiated cell state because this is a neuron-specific protein [34]. Other markers for mature neuronal lineages include neuronal nuclear antigen (NeuN), microtubule-associated protein 2 (MAP2), synap-tophysin, and tau [24].

Until recently, differentiation was thought to be immutable once initiated. How-ever, it has been shown that differentiated cells can undergo a process of reversion

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from the terminally differentiated state back to the stem state, both in nature and in the lab [29, 51]. Reversion from the differentiated state is highly uncommon, but it has been indicated that reversion from the progenitor state to the stem state may be a relatively more common process [29, 39].

There are functional differences between ESCs and hiPSCs. ESCs tend to differ-entiate more efficiently than hiPSC’s [38, 4]. One explanation for this difference is the epigenetic changes that occur during differentiation and that may remain once differentiation is reversed [38, 4]. Epigenetic refers to the modification of gene ac-tivity without altering the DNA directly, where these modifications are passed on to daughter cells [55]. Termed ”epigenetic memory”, epigentic remenants could lead to certain characteristics of the differentiated cell type to remain in the hiPSCs [38, 4]. These epigenetic alterations would be present in the hiPSCs and not in the ESC’s, causing this difference in functionality [38, 4].

The experiment that we are modelling is the growth and differentiation of the neural aggregate once seeded on a biomaterial scaffold. The experimental procedure that preceeds the aggregate seeding is outlined in Figure 1.2. Full details can be found in [34] and [42]. The hiPSCs are cultured in aggrewells to produce aggregates. The stem cells differentiate to the progenitor cell state in response to culturing in aggrewells with Neural Induction Medium (NIM), producing aggregates of neural progenitor cells, called neural aggregates. These aggregates are then placed onto a scaffold (see Section 1.5) and cultured in NIM to complete differentation of the progenitor cells to the terminally differentiated state. The cells modelled here require contact with a substrate to initiate differentation [34]. Experimental observations indicate that the aggregates can be held in solution without progressing from the progenitor to the terminally differentiated state, but once placed on a substrate, begin differentiation [34].

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Figure 1.2: Overview of experimental protocol for neural aggregate formation and seeding. Isolated hiPSCs (left) are cultured in aggrewells with NIM (center) to form aggregates of neural progenitor cells. The neural aggregates are then seeded onto biomaterial scaffolds in NIM (right) to produce neurons.

1.4 Cell Signalling

It is well known that cells interact with each other using various mechanisms, includ-ing release of soluble factors and bindinclud-ing of membrane proteins [2]. The mechanisms involved in a particular signalling pathway affects the speed and effective distance of the signal [2]. These signalling mechanisms can affect only cells in immediate contact with the signalling cell (juxtacrine), cells in a particular vicinity of the signalling cell (paracrine), or cells distant from the signalling cell (endocrine) [2].

Each of these signalling mechanisms is associated with different molecular com-ponents. Juxtacrine signalling involves membrane-bound receptors and signalling molecules, whereas paracrine and endocrine signalling involves soluble factors that leave the signalling cell and bind to receptors on or in the target cell [2].

The various forms of signalling could have distinct implications on the dynamics of the individual processes as each would require a different length of time to take effect and have different regions of influence [2]. For example, the signal could be immediate or delayed, depending on the signalling mechanism. Dynamically, combi-nations of these mechanisms could cause major differences in the overall performance of the signal.

In theory, these signalling methods could lead to different outcomes of prolifera-tion and differentiaprolifera-tion of a three dimensional aggregate of cells. For example, it has

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previously been shown that contact with a substrate initiates differentiation of cells within a neural aggregate [34]. However, the signalling mechanisms at play in this differentiation process are unknown. Is it the case that the outer cells that contact the substrate signal the rest of the aggregate via contact-dependent mechanisms, i.e. juxtacrine signalling, or does the contact elicit a soluble factor to be released by the cell in contact with the substrate which then affects cells further away in the aggre-gate, i.e. paracrine or endocrine signalling. The time-course and spatial dispersion of these two signalling options would be very different, resulting in distinct evolutions of differentiation within the aggregate. Therefore, it is important to consider the mechanisms of signalling in modelling a three-dimensional cellular system.

In the model considered in this thesis, we have included many processes that undoubtedly have different signalling methods. For example, the responses to contact and oxygen concentrations would likely have very different mechanisms, as one is a mechanical process and the other chemical. However, when we consider the size of the experimental setup we are modelling, the maximal distance between cells within the neural aggregate is taken to be small enough to allow for the assumption that any signalling has negligible differences in spatial dispersion and delay. Thus, the various types of signalling are assumed to act with no delay and to reach the entire population of cells in the aggregate. The same assumptions are made regarding the inclusion of soluble factors in the medium and their effects on the aggregate.

1.5 Biomaterial Scaffolds

Biomaterial scaffolds are substrates on which cells can be seeded [8, 34, 7, 27, 22, 11]. Scaffolds provide an appropriate substrate and environment for cells [34, 7, 8]. In tis-sue engineering, the scaffold provides a suitable environment to develop tistis-sue from an initial population of seeded cells [30, 8]. Scaffolds can also initiate certain behav-iors. An example of this is inducing differentiation by providing mechanical input [25, 27, 13, 34]. Another benefit of scaffolds is that they provide a vehicle for cell transplantation [30].

Biomaterial scaffolds can be two-dimensional (2D) or three-dimensional (3D) and be made of many different materials, organic and inorganic [8, 34, 22]. Depending on the intended use, the properties can vary. Relevant properties include stiffness,

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porosity, topology, degradability and bioactivity [8, 33, 13]. Stiffness has been shown to affect the type of cell resulting from differentiation with cells differentiating into neural, mesenchymal, and bone, as scaffold stiffness increases [13]. Moreover, sub-strate stiffness has been shown to modulate the resulting neural subtypes of neurons, astrocytes, and oligodendrocytes [25]. Porosity affects differentiation and prolifera-tion as contact is required to induce these processes, with excessive contact having an inhibitive effect on aggregate growth. The topology is also a relevant factor as it affects the growth and differentiation of the aggregate[34]. For 2D manufactured fibrous scaffolds, it has been shown that the fibre diameter of the scaffold can affect aggregate growth as fibres that are too small or large inhibit proliferation of the cells within the aggregate [11]. The similarity of the topology of the scaffold to the natural environment of the desired cell type can influence the overall outcome of cell seeding and differentiation [8]. This is illustrated by the seeding of stem cells on decellularized organs that then grow and differentiate to become fully functioning organs [8]. The degradability of scaffolds can also be a major factor. Some scaffolds are designed to degrade once implanted into a patient so that the host will completely integrate the implanted tissue as the scaffold degrades [8]. This is an important factor as the factors previously mentioned all change as the scaffold degrades as the stiffness will decrease, porosity will increase and the topology will obviously be affected. Bioactivity can also play a role once the scaffold in implanted. A bioactive scaffold induces a reaction from the host to facilitate processes such as cellular invasion and regeneration [8].

1.6 Previous Work on Modelling Stem Cells

There has been significant research into stem cell proliferation and differention, both in the experimental sense and with mathematical models [21, 25, 26, 31, 35, 43, 54, 58, 57, 60]. Our current work stands out from previous research by including mulit-ple factors intrinsic and extrinsic to the cell population. The intrinsic characteristics include cell-cell signalling, differential responses to extrinsic effects, and state-specific metabolic properties. The extrinsic detail includes scaffold effects, oxygen and waste effects, depth of medium (see Section 3.5), and control of differentiation via growth factors. By approximating these processes, we are able to include more experimen-tal properties. Moreover, by including these properties in a way that can be related back to experimental procedure, the model can indicate how this procedure could be altered to generate optimal results.

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In [31] ordinary differential equations (ODEs) were used to model neurogenesis in the mouse olfactory epithelium (OE), a layer of cells that includes neurons which lines approximately half of the nasal cavities [41]. In the OE, a lineage of cell states forms terminally differentiated olfactory neurons [31]. In the OE, there are self-renewing stem cells, neuronal precursors, and a terminally differentiated state of olfactory receptor neurons (ORNs) [31, 41]. Although not studying the same cell type as we consider here, [31] considered the three maturity-based states of the neuronal lineage in order to determine the likely modes of feedback that form differentiated tissue. Due to the similarity of the systems and cell type being studied, a minimal version of the feedback explored in [31] was employed here. It is clear that this is only one possible mechanism for feedback in our system. However, it makes for a good starting point that could be expanded upon and explored further.

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Chapter 2 Experimental Procedure and

Results

2.1 Methods

A custom-made melt electrospinning setup was used to fabricate

Poly-(-caprolactone) (PCL, number average molar mass (Mn)∼45,000, Sigma Aldrich Chemical Co) biomaterial scaffolds [34]. Scaffolds were fabricated by melt electro-spinning using 200 µm and 500 µm nozzle sizes and are referred to as loop mesh 200 and loop mesh 500, respectively [34]. Note that increasing nozzle size increases fiber diameter and decreases porosity [34]. The resulting porosities were 23% for loop mesh 200 and 40% for the loop mesh 500. Figure 2.1 shows two examples of the final scaffolds. hiPSCs were cultured on a Vitronectin XFTM matrix in the presence of TeSRTM_-E8TM _{medium [7], then in STEMdiff}TM _{Neural Induction Medium (NIM)}

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(a) Loop 200 (b) Loop 500

Figure 2.1: Scanning electron microscope images of loop mesh 200 and loop mesh 500 scaffolds.

Aggregates containing neural progenitor cells were then seeded onto scaffolds and cultured in NIM for 12 days to induce differentiation to the terminally differentiated cell state. A simplified diagram of the experimental setup is shown in Figure 2.2.

Figure 2.2: Top and side views of a simplified experimental setup.

To measure the growth of the aggregates, bright field images were taken daily of three neural aggregates for each set of scaffolds with IncuCyteTM _{Software measuring}

the cell body cluster area over 12 days. Viability of the neural aggregates on the PCL scaffolds was analyzed after 12 days using a LIVE/DEAD R _{Viability/Cytotoxicity}

Kit (Invitrogen) [36, 37]. Terminal neuronal differentiation of hiPSCs was assessed by immunocytochemistry targeting the neuron-specific protein Tuj1 [34, 37].

2.2 Results

Neural aggregates derived from hiPSCs were seeded on two different scaffolds for 12 days. Figure 2.3 shows that both scaffold topographies support cell adhesion and cell

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migration.

Figure 2.3: Fluorescence images of neuronal marker Tuj1 expressed in neural aggre-gates 12 days after seeding on loop mesh scaffolds. (A) Scaffold with higher porosity. (B) Scaffold with lower porosity. Scale bar is 400 µm.

As shown in Figure 2.4, cell body cluster area of neural aggregates cultured on more porous scaffolds was consistently larger than that of neural progenitors seeded on less porous scaffolds. The average results of 3 experiments 12 days after seeding are summarized in Table 2.2.

Table 2.1: Comparison of data of three experiments for two scaffold porosities 12 days after seeding.

Scaffold Type Loop Mesh 200 Loop Mesh 500

Porosity (%) 40 23

Fluorescence (%) 71.5 ± 1 58.4 ± 3

Cell Body Cluster Area (mm2₎ _{2.04 ± 0.1} _{0.87 ± 0.27}

For two of these samples, aggregate growth data was collected at 2 day intervals. This data is shown in Table 2.2 and plotted in Figure 2.4.

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Day Cell Body Cluster Area on Loop Mesh 200 (mm2₎

Cell Body Cluster Area on Loop Mesh 500 (mm2₎ 0 0.85 0.75 2 0.92 0.78 4 1.11 0.80 6 1.44 0.88 8 1.67 0.91 10 1.82 1.00 12 2.24 1.1

Table 2.2: Proliferation data for two neural aggregates seeded on loop mesh 200 and loop mesh 500. 0 2 4 6 8 10 12 0.5 1 1.5 2 2.5 Time (days)

Cell Body Cluster Area (mm

2) Best Fit Line_{Best Fit Line}

Loop Mesh 200 Loop Mesh 500

Figure 2.4: Cell body cluster area of two neural aggregates on loop mesh 200 and loop mesh 500 over 12 days.

Bright field images show the growth of a neural aggregate on the scaffold in figures 2.5.

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(a) Day 0. (b) Day 6. (c) Day 12.

Figure 2.5: Bright field images of neural aggregates on loop mesh 200 (top) and loop mesh 500 (bottom) scaffolds.

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Chapter 3 Model Development

For the derivation of the model, the aggregate cell population is divided into three subpopulations: stem, progenitor and differentiated cells, denoted respectively by S, P , and D, measured in number of cells. The total number of cells in the aggregate is then the sum of the three subpopulations, denoted by T . The feasible region for these variable is S, P, D ∈ [0, ∞). The rates of the cellular processes are non-negative and are given the following notation:

α : death rate of stem cells β : death rate of progenitor cells γ : death rate of differentiated cells p1 : proliferation rate of stem cells

p2 : proliferation rate of progenitor cells

d1 : differentiation rate of stem cells to progenitor cells

d2 : differentiation rate of progenitor cells to differentiated cells

r : reversion rate of progenitor cells to stem cells.

The units for the above rates are proportion per minute.

All three subpopulations undergo death, with only the stem and progenitor pop-ulations proliferating as the differentiated state is terminal, i.e. do not proliferate. The stem and progenitor cell states differentiate with rates d1 and d2, respectively.

The progenitor cell population can also revert to the stem cell state by reversion with rate r. With these processes included, the differential equations for S, P and D are

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given by dS dt = − αS − d1S + 2p1S + rP (3.1) dP dt = − βP − d2P − rP + 2p2P + d1S dD dt = − γD + d2P

Once feedback between the compartments as previously discussed is included, this model becomes dS dt = − αS − d1S + 2p1S 1 + S + P + D + rP (3.2) dP dt = − βP − d2P − rP + 2p2P 1 + P + D + d1S dD dt = − γD + d2P

A schematic for this system is shown in Figure 3.1. The form of feedback included in this model was based on the feedback used in [31]. This particular form of feedback was used as it provides a simple starting point that would not overly complicate the analysis.

S

P

D

Death Death Death

p1 p2 d1 α d2 r β γ

Figure 3.1: Schematic diagram of three cell states with cellular feedback. Black arrows indicate transitions between states. Red arrows indicate negative feedback.

This model is further expanded by including the oxygen and waste concentrations experienced by the aggregate, denoted O and W , respectively. The units of O and W are in percentage, with feasible regions O, W ∈ [0, 100]. The concentrations of O and W in the air surrounding the culture are denoted Oairand Wair, with the same feasible

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produce waste. Also included in the model, is diffusion from the air-medium interface. The resulting model is

dS dt = − αS − d1S + 2p1S 1 + S + P + D + rP (3.3) dP dt = − βP − d2P − rP + 2p2P 1 + P + D + d1S dD dt = − γD + d2P dO dt = − u1S − u2P − u3D + Of lux(S, P, D, O) dW dt =w1S + w2P + w3D + Wf lux(S, P, D, W ) ,

where ui are the oxygen consumption rates, wi are the waste production rates, and

Of lux and Wf lux are the rates of diffusion of O and W between the neural aggregate

and air above the medium.

Note that the units used throughout the model are minutes for time, centimeters for distance, and percentage for gas concentration within the medium.

3.1 General Structure of the Model

The general structure is a compartmental population model. The variables are the populations of stem, progenitor and differentiated cells, as well as the concentrations of oxygen and waste. This choice of model was based on the following factors. First, these cell states can be distinguished in the lab and cells can be held at each state. Second, each state has unique properties, some of which have been determined exper-imentally. Finally, the scale of the cellular level is coarse enough that there is useful data from experimental work and literature that can be included in the model. The cellular scale is also fine enough that the results of the model can be interepreted and transferred to the lab protocol. Each of the cell populations undergoes the ap-propriate cellular processes for it’s state. The stem cells undergo three processes: proliferation via division, differentiation to progenitor cells and cell death. The pro-genitor cells undergo four processes: proliferation via division, differentiation to ter-minally differentiated cells, reversion to stem cells, and cell death. The only process of differentiated cells is death as the cells are fully committed to their differentiated

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state and do not proliferate. There is also signalling between and within the cell states. Division of earlier states is inhibited by the presence of cells in the same and and later states. This signalling scheme is based on a model developed by [31]. This scheme was chosen because the cell types and model structure used by [31] are similar.

An obvious consideration when using ODEs to model populations is that the cell poplulations are considered to be continuous, rather than composed of discrete indi-viduals. In this work, the cell populations being considered and analyzed are generally in the hundreds to thousands of cells. The large cell number limits any inconsistentcy that may arise out of the use of ODEs in this fashion, as well as the effects of noise within the system.

In addition to the three cell states, local oxygen and waste concentrations are also variables. These were included because oxygen is an important factor for proliferation and differentiation. It has also been shown in other systems that different levels of oxygen affect cell types differently, and so could be an integral component when modelling three distinct cell populations. In addition, it is feasible to include oxygen and waste as there is data on both O2 and CO2 in the neural stem cell differentiation

pathway in previous literature and these are factors that could be changed in the lab by altering protocols [12, 14, 17, 35, 49, 54]. Thus, the inclusion of these variables could be useful for determining how to optimize current experimental protocols. It should be noted that the model uses CO2 as a proxy for waste as it is a cellular waste

product that has been studied and measured experimentally, while data is lacking for other waste products.

3.2 Scaffold Modelling

The scaffold is included in the model by a cell-scaffold contact rate, C, which takes values from 1 to 10, with 1 being a 90% porous scaffold and 10 being a solid scaffold, i.e. 0% porosity. Note that C does not go below 1 as it is taken that the cells would not have a suitable substrate to attach to and would not go through any of the processes included in the model. Therefore, we exclude the range from 0 to 1 for C from the model. This rate increases with decreasing porosity. It should be noted that due to the spherical nature of the neural aggregate, 100% contact does not mean that all the cells are in contact with the scaffold, but rather that the maximum that

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could be in contact is in contact. This maximum possible contact is estimated on the basis of heuristics to be 10%. The relationship between porosity and C is

C =(100% − Porosity) · (0.1) ,

where the porosity has units of percent and the factor of 0.1 represents the maximum possible contact.

In reality, the scaffold porosity is altered by changing the fiber diameter of the scaffold rather than density of the fibers of the same density. Altering porosity by changing fiber diameter is not optimal, however, due to experimental limitations, this was the course taken. This change in fiber diameter, which would affect proliferation of the cells in the aggregate as previously mentioned, is not explicitly included in the model. However, the data used to fit some of the effects of scaffold porosity come from scaffolds with different fiber diameters, so the effects are implicitly included in the model. Another factor related to scaffolds is the topology. The scaffolds in the experiments used for data are called loop mesh scaffolds. These are 2D scaffolds formed by randomly aligned layers of loop fibers. This topology is ignored in the model as this was previously determined to be less critical in that it could be opti-mized experimentally [34] and would have added a significant level of complexity.

The effect of scaffold porosity on differentiation and proliferation is not linear. If the scaffold is too porous, first, the aggregates cannot adhere and simply fall through the gaps, and second, the scaffold does not act on the cells enough to signal for proliferation and differentiation. On the opposite end of the scale, a scaffold that is too solid inhibits proliferation because contact inhibition comes into play and limits proliferation. A non-porous scaffold does not affect differentiation to the same degree, as it is likely the effect of contact on differentiation plateaus at some rate of contact. These effects are modelled as functional terms in proliferation and differentiation rates of the stem and progenitor cell variables.

3.3 Determination of Functional Effects

The functional effects multiply the experimental rates for each of the processes. This is a reasonable method of incorporating each effect as they are all independent

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pro-cesses, with data taken from experiments that were only testing one alteration. The qualitative and quantitative data are taken from literature using similar cells and un-der similar culture conditions in orun-der to minimize differences stemming from factors that were not of interest. For example, if data were only available from human or mouse cells, the human data were used.

In general, each effect was first determined qualitatively, then fit to quantitative data. Due to the limited data available, the functions were fitted manually. All of the functions for the effects are greater than 0, with values above 1 increasing the rate and values below 1 decreasing the rate. It should be noted that O ∈ [0, 100] and W ∈ [0, 100] as they are both concentrations with units of percent. The contact parameter range is C ∈ [1, 10], as described in Section 1.5.

For the concentrations of oxygen and waste, in the conversion of the literature measurements of mmHg to our units of percent we use the fact that 760mmHg = 1atm = 100% [16]. When given a measurement in mmHg, it was converted to percent by multiplying the value by 100%/760mmHg. For example, to convert 160 mmHg

160mmHg =(160mmHg) 100% 760mmHg =21%

3.3.1 Effect of O

2

on Proliferation

The qualitative effect of oxygen on proliferation occur in three phases. First, at very low oxygen, proilferation is slowed as oxygen is required for metabolic processes, with no proliferation occurring at 0% oxygen. As oxygen levels increase to physiologic oxygen levels (∼_{1-10%), the proliferation peaks as this is the most hospitable}

envi-ronment for the stem and neural progenitor cells. Lastly, as oxygen levels increase towards 21%, the effect decreases and plateaus at 1. A general functional form that fits this effect is

(aO + 1)(aO − 1)e−bO+ c .

Studer et al. (2000) found that there was a relative increase in the percentage of proliferating cells of between 0.17 and 0.6 in 3 % oxygen versus 20 % oxygen. Using quantitative data from [49], this function fits the data when a = 0.6, b = 0.5 and

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c = 1.01, giving the final equation

(0.6O + 1)(0.6O − 1)e−0.5O+ 1.01 ,

which is denoted by f5. For S and P , the experimental data gives points at (3, 1.6) and

(20, 1). The function has points at (3, 1.51) and (20, 1.02), and thus closely matches the experimental data. Figure 3.2 shows the function f5 and the experimental data

points. 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 O (%) Function Value f 5 Experimental Data

Figure 3.2: Function for the effect of oxygen on proliferation, f5, and experimental

data points used for fitting.

3.3.2 Effect of Waste on Proliferation

The qualitative effect of waste levels on proliferation is that as waste increases, the proliferation of the cells decreases because CO2 becomes toxic. In response to high

CO2 levels, the cells halt proliferation as the environment becomes inhospitable. A

general functional form for this effect is a sigmoid of the form

c 1 − 1 1 + e(−a(W −b)) .

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Using quantitative data from [40], the function is fitted to this data when a = 0.1, b = 25, and c = 10/9. The final function, denoted by f6, is

10 9 1 − 1 1 + e(−0.1(W −25)) .

With this function, we get the points (0, 1.03), (5, 0.98), (21, 0.67), (25, 0.56), (100, 0.00). Pattison et al. (2000) found that cell proliferation decreases by 40% as pCO2increases

from 53 to 165 mmHg (∼_{5 to 21% CO}₂_{). Taking the value at 5% CO}₂ _{as the}

stan-dard, this data would indicate a value of 0.6 at 21% CO2. Thus, with the functional

values of 0.98 at W = 5 and 0.67 at W = 21, the function is reasonably close to the experimental data. Figure 3.3 shows the function f6 and the experimental data

points of (5, 1) and (21, 0.6). 0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 W (%) Function Value f 6 Experimental Data

Figure 3.3: Function for the effect of waste on proliferation, f6, and experimental

3.3.3 Effect of Contact on Proliferation

The effect of contact on proliferation is derived from work done by our partners in the Willerth lab. Experiments done with varying porosities of scaffolds show a significant impact on the proliferation during a 12 day culture period. Qualitatively, the effect of scaffold porosity has four components.

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First, with too porous a scaffold, the cells cannot attach and simply fall through. This effect is not modelled here as we make the assumption that the aggregate does attach.

The next step is a scaffold that is dense enough to allow for aggregate attachment, but is too porous to encourage proliferation. This failure to proliferate is caused by a minimal contact rate that does not signal the aggregate to expand because the substrate does not support contact-dependent proliferation of the cells.

As contact is increased within a certain range, the cells switch to proliferation. This is indicated by the experimental observations [34], although the reason is not fully understood. One possibility is that this range of porosity is the most similar to the natural environment of the cells.

Decreasing porosity, i.e. increasing contact, further also decreases the prolifera-tion. This effect is attributed to the “contact inhibition” experienced by cells. With typical tissues and cells, a certain amount of contact signals the cells that the appro-priate density has been reached and proliferation should only maintain the current population. A functional form that matches these dynamics is

aC3e−bC−c.

The function fits the data when a = 4, b = 0.8, and c = 0.5, resulting in the function 4C3e−0.8C−0.5,

denoted f10. To fit this equation to the experimental results, we first take a 0%

porosity scaffold (i.e. a solid surface) as a standard and set the function value at 0 to 1. The experimental data showed 1.8 times increase in population size with 23% porosity (i.e. C∼_{8), with a 4.3 times increase of proliferation at 40% porosity (i.e.}

C∼_{6). Along with the standard with a solid scaffold, these give the data points (6,4.3),}

(8,1.8), (10,1). This gives the points (10,0.81), (8,2.06), (6,4.31). These values do not match up as closely as for other effects, but as the data is limited, this is taken as a sufficient approximation. Figure 3.4 shows the function f10 and the experimental

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1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 C Function Value f 10 Experimental Data

Figure 3.4: Function for the effect of contact on proliferation, f10 and experimental

3.3.4 Effect of O

2

on Death

The qualitative effects of oxygen levels on cell death are in three parts, with the effects described in relation to 21% oxygen. First, at a very low oxygen levels the lack of oxygen is toxic, increasing cell death. Second, for a physiologic oxygen level (∼_{1 to}

10%) the death rate decreases as these are the typical environmental oxygen levels for neural cells. Third, the death increases somewhat as oxygen levels increase to 21%. A general functional form for this effect is

1

1 + aO − bOe

−cO_{+ 1 .}

Using quantitative data from Studer et al. (2000), the function is fitted to this data for S when a = 10, b = 0.3 and c = 0.3, for P when a = 5, b = 0.3 and c = 0.3, and for D when a = 2, b = 0.5 and c = 0.3. The final functions for S, P and D, denoted by f1, f2, and f3 are 1 1 + 10O − 0.3Oe −0.3O + 1 , 1 1 + 5O − 0.3Oe −0.3O + 1 ,

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1

1 + 2O − 0.5Oe

−0.3O

+ 1 ,

respectively. The value of a differs between the equations for S, P , and D as each of these cell types have a different ability to adapt to anoxia. Stem cells are highly adaptive and thus can continue to consume oxygen under lower conditions than the progenitor and differentiated cells. This adaptability decreases as the cells mature. Thus, the value of a decreases with progressing differentiation, resulting in a more gradual increase in death by anoxia (i.e. O going to 0) as differentiation progresses. For stem and progenitor cells, the experimental results show a decrease from 51% cell death at 20% O2 to 34% cell death at 3% O2. For differentiated cells, the

experimen-tal results show a decrease from 45% cell death at 20% O2 to 21% cell death at 3% O2.

For S and P , this experimental data gives points of (3, 0.67), (20, 1). The functions give points at (3.5, 0.66), (20, 0.99). For D the experimental has points at (3.5, 0.47), (20, 1). The function gives points at (3.5, 0.51), (20, 1.00). Thus, the functional effects closely match the experimental data. Figure 3.5 shows the functions f1, f2, and f3,

as well as the experimental data points.

0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 O (%) Function Value f 1 f2 f3 Experimental Data

Figure 3.5: Function for the effects of oxygen on death, f1, f2and f3, and experimental

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3.3.5 Effect of Waste on Death

The qualitative effect of waste levels on cell death is that as waste increases, the death rate of the cells increases as CO2 becomes toxic. A general functional form for this

effect is a sigmoid of the form

1 + 1

1 + e(−a(W −b)) .

Using quantitative data from [17], the function is fitted to this data when a = 0.5 and b = 16. The final function, denoted f7, is

1 + 1

1 + e(−0.5(W −16)) .

With this function, we get the points (0, 1), (5, 1.00), (15, 1.34), (21, 1.92), (100, 2). Gray et al. (1996) found that percent viability decreases by 7 and 21% as pCO2

increases from 35 to 103 and 148mmHg (∼_{5 to 15 and 21% CO}₂_{). Taking the value}

at 5% CO2 as the standard, this data would indicate a value of 1.33 at 15% CO2 and

2 at 21% CO2. This experimenta data corresponds to the points (5, 1), (12, 1.33),

and (21, 2). Thus, with the functional values of 1.00 at W = 5, 1.34 at W = 15, and 1.92 at W = 21, the function is close to the experimental data. Figure 3.6 shows the functions f7 and the experimental data points.

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0 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 2 2.5 W (%) Function Value f 7 Experimental Data

Figure 3.6: Function for the effect of waste on death, f7, and experimental data points

used for fitting.

3.3.6 Effect of O

2

on Differentiation

The qualitative effect of oxygen on differentiation occurs in three phases. First, at very low oxygen, differntiation is halted as very low oxygen promotes stemness, with no differentiation occurring at 0% oxygen. As oxygen levels increase to physiologic oxygen levels (∼_{1-10%), the differentiation peaks as this is the most hospitable}

envi-ronment for the stem and neural progenitor cells. Lastly, as oxygen levels increase towards 21%, the effect decreases and plateaus at 1. A general functional form for this effect is

(aO + 1)(aO − 1)e−bO+ c .

Studer et al. (2000) found that there was an increase in the percentage of differenti-ated cells from 63 to 74% in 3% oxygen versus 20% oxygen. Using quantitative data from [49], this function fits the data when a = 0.4, b = 0.5 and c = 1.01. Denoted f4,

the resulting function is

(0.4O + 1)(0.4O − 1)e−0.5O+ 1.01 .

For S and P , the experimental data gives points at (3, 1.13) and (20, 1). The function has points at (3, 1.03) and (20, 1.01), and thus closely matches the experimental data.

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Figure 3.7 shows the function f4 and the experimental data points. 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 O (%) Function Value f 4 Experimental Data

Figure 3.7: Function for the effect of oxygen on differentiation, f4, and experimental

3.3.7 Effect of Contact on Differentiation

Experiments have shown that contact initiates differentiation. There are no available quantitative data for this effect, however, there are experimental qualitative obser-vations. We have observed that when simply placed in medium, no differentiation is observed. When a scaffold is provided and the cells adhere, differentiation is initiated. It is also assumed that there is a saturation of the effect of contact on differentation. This effect is modelled by the function

C 1 + C .

This function is 0 at C = 0, corresponding to the situation with no scaffold. As C increases the effect increases, rising to a value of 0.91 at C = 10. Figure 3.8 shows the function f9.

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1 2 3 4 5 6 7 8 9 10 0.5 0.6 0.7 0.8 0.9 1 C Function Value f 9

Figure 3.8: Function for the effect of contact on differentiation, f9.

3.3.8 Effects of O

2

on Reversion

The qualitative effects of oxygen on reversion are less clear. It is known that very low oxygen increases stemness [49]. It is assumed here that reversion contributes to this increase in stemness. Thus, the qualitative effect is that lower oxygen increases reversion. It is also taken that the level of reversion decreases as oxygen increases.

A general functional form for this effect is a O + c.

Experimental measurements were not available for this effect, thus the function is based purely on qualitative observations. The values chosen to fit this function into reasonable values are a = 15 and b = 5, giving the final function

15 O + 5,

denoted f8. This is a monotone decreasing function with the points (0, 3.00), (10, 1.00),

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0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 O (%) Function Value f 8

Figure 3.9: Function for the effect of oxygen on reversion, f8.

3.3.9 Effect of O

2

on O

2

Consumption and Waste Production

The effect of oxygen on oxygen consumption only becomes a factor when oxygen levels become very low. As oxygen becomes scarce, cells limit their consumption [52]. Thus, this function must decrease as oxygen levels approach 0, while having no effect at higher oxygen levels. This effect is given by the function

1

1 + e−10(O−0.25),

which is denoted f13. By multiplying the oxygen consumption rates with this function,

the consumption remains normal above 0.5% oxygen and sharply decreases to near 0 (actual point is (0, 0.076)) as the oxygen approaches 0%. As CO2 production is

related/proportional to oxygen consumption, this function is also applied to CO2

production rates. Thus, as oxygen goes to 0, oxygen use and CO2 production decrease

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 O (%) Function Value f 13

Figure 3.10: Function for the effect of oxygen on oxygen consumption and waste production, f13. The range of O shown is 0 to 2 as the function values above this

range are nearly constant.

3.3.10 Effect of Contact on Diffusion of O

2

and Waste

The porosity of the scaffold affects the diffusion of oxygen and waste around the cells. With no scaffold, diffusion is unimpeded. As porosity decreases, the diffusion also decreases. This decrease is small as it is only the diffusion around and below the aggregate that is affected, not the diffusion from above. A functional form that fits these qualitative effects is

a a + C ,

where a = 100 and a = 200 for the functional effects on diffusion of O and W , respectively. The final functions, denoted f11 and f12, are

100

100 + C and

200 200 + C.

These terms will be applied to the flux terms in the differential equations for O and W . Figure 3.11 shows the functions f11 and f12.

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1 2 3 4 5 6 7 8 9 10 0.9 0.92 0.94 0.96 0.98 1 C Function Value f 11 f12

Figure 3.11: Functions for the effect of contact on oxygen and waste diffusion, f11

and f12.

3.3.11 Summary of Functional Effects

With the functional effects determined in the previous sections, it is useful for later analysis to find the maximum and minimum values for each. Table 3.1 gives the indi-vidual functional effects, as well as the ranges for each over the appropriate domains of O, W and C.

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Function Name Function Minimum Point Maximum Point Function Range

f1 _1+10O1 − 0.3Oe−0.3O+ 1 (3.57, 0.66) (0, 2) [0.66, 2]

f2 _1+5O1 − 0.3Oe−0.3O+ 1 (3.77, 0.69) (0, 2) [0.69, 2]

f3 _1+2O1 − 0.5Oe−0.3O+ 1 (3.89, 0.51) (0, 2) [0.51, 2]

f4 (0.4O − 1)(0.4O + 1)e−0.5O+ 1.01 (0, 0.01) (5.2, 1.26) [0.01, 1.257]

f5 (0.6O − 1)(0.6O + 1)e−0.5O+ 1.01 (0, 0.01) (4.6, 1.67) [0.01, 1.67]

f6 10₉(1 −_1+e−0.1(W −25)1 ) (100, 0) (0, 1.03) [0, 1.03] f7 1 + _1+e−0.5(W −16)1 (0, 1) (100, 2) [1, 2] f8 _O+515 (100, 0.14) (0, 3) [0.14, 3] f9 _C+1C (1, 0.5) (100, 0.99) [0.5, 0.99] f10 4C3e−0.5−0.8C (100, 0) (3.75, 6.37) [0, 6.37] f11 _200+C200 (10, 0.95) (1, 0.99) [0.95, 0.99] f12 _100+C100 (10, 0.91) (1, 0.99) [0.91, 0.99] f13 _1+e−10(O−0.25)1 (0, 0.076) (100, 1.00) [0.076, 1.00]

Table 3.1: Components of functional effects on parameters and extremal values.

To give a clear picture of the functional effects, the plots of each component for O, W and C are shown in figures 3.12, 3.13, and 3.14, respectively.

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0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 O Function Value f1 f2 f3 f4 f5 f8

Figure 3.12: Functional effects of O.

0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 W Function Value f6 f7

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1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 C Function Value f9 f10 f11 f12

Figure 3.14: Functional effects of C.

The complete parameter coefficients of the model are the product of the above functional components and the experimental parameter values. With the ranges of the functional effects determined, the remaining unknown components of the parameter coefficients of the model are the experimental parameter values. In the following section, the experimental parameters are determined.

3.4 Determination of Experimental Rates

The experimental rates were determined from multiple sources, including data from our collaborators and data from literature. When sourcing data from literature, care was taken to use sources whose experimental setups and procedures were as close to our collaborators’ as possible. Considerations include cell type and culture condi-tions (O2 concentration of 21%, CO2concentration of 5%, temperature of 37◦ C, etc.).

3.4.1 Death Rates: α, β, γ

The death rates are derived from experimental data from our collaborators. At the end of the experiment, a stain for viable cells was performed. This stain can be used to deterimine the ratio of dead cells present, however no distinction can be

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made between the cell states. Also, the values determined come from two different scaffold porosities, but it is unlikely that cell death would be significantly influenced by scaffold porosity. The range for the percentage of viable cells at the end of 12 days is 55.4% to 72.5%, giving a range of dead cells of 27.5% to 44.6%. Thus, the minimum death rate is calculated by

0.275

(12d)(24h)(60min) = 0.000016/min and the maximum by

0.446

(12d)(24h)(60min) = 0.000026/min . This gives a range for α, β, and γ of [0.000016, 0.000026].

3.4.2 Proliferation Rates: p

1

, p

2

The proportion of cells dividing per minute is taking the ratio of the proportion of cells dividing in the generation time and the generation time in minutes.

The proliferation rates are derived from the generation times for each of the cell states. We assume all the cells divide in the generation time. Thus, taking the generation time and converting it to the proportion of cells dividing per minute is achieved by taking the ratio of the proportion of cells dividing in the generation time and the generation time in minutes. The generation times in the literature are given in hours, so they are rescaled to minutes. For example, if the given generation time is 24 hours, the calculation for the proliferation rate p is

p = 1

(24h)(60min) = 0.00069/min .

This gives a proportion of cells proliferating per minute. The cell generation times for S and P found in [43], [5], and [18] range from 14 to 24.2 hours and 10.3 to 37 hours, respectively. These values give ranges for p1 and p2 of [0.00069, 0.0012] and

[0.00045, 0.0016]. Recall that the differentiated cells do not proliferate. Therefore, there is no proliferation rate for the D compartment.

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3.4.3 Differentiation and Reversion Rates: d

1

, d

2

, r

The differentiation rates are based on data from [49] that quantified the proportion of progenitor and differentiated cells. Studer et al. (2000) determined that after 4 to 6 days, at least 90% of stem cells differentiated into progenitor cells. These values are used to calculate s maximum and minimum for d1, the differentiation rate from S

to P . The minimum is determined by taking the longest time span and the minimal differentiation, giving

min d1 =

0.9

(6d)(24h)(60min) = 0.00010/min .

The maximum is determined by taking the shortest time span and the maximal differentiation, giving

max d1 =

1

(4d)(24h)(60min) = 0.00017/min . This gives a range for d1 of [0.00010, 0.00017].

Studer et al. (2000) also determined the level of differentation to terminally dif-ferentiated cells from progenitor cells to be 63% 6 days after their measurements of the proportion of progenitor cells. If it is assumed that all the cells were progenitors at the beginning of these 6 days, then the calculation is

0.63

(6d)(24h)(60min) = 0.000073/min .

If there were 90% progenitor cells and 10% stem cells, then part of this 6 days must be the completion of differentiation of all cells to P . This time is given by

0.1

0.0001/min = 1000min .

Thus, the differentiation from P to D must occur in the remaining time which is 6d − 1000min

(60min)(24hr) = 5.306d .

Although this method of calculation assumes that all the stem cells progress to pro-genitor cells before cells already at the propro-genitor state transition to the differentiated

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state, this still gives a reasonable approximation in our model. Using this time span for the differentiation to 63% terminally differentiated cells gives

0.63

(5.306d)(24h)(60min) = 0.000082/min . Thus, we have a range for d2 of [0.000073, 0.000082].

There is no available data for the reversion rate. As there is no quantitative evidence for reversion in our system, the lower bound of r is taken as 0.000001. For the upper bound, the upper bound for the differentiation rates of S to P is taken as it is extremelt unlikely that reversion would overtake this value. Therefore, the bounds for r are [0, 0.00017].

3.4.4 Oxygen Consumption and Waste Production Rates

The oxygen consumption rates of cells have been well studied, but there is little data for the three cellular compartments as defined here. Birket et al. (2011) studied the oxygen conumption rates (OCRs) of hESCs and hESC-derived neural stem cells (NSCs), which are similar to the cells used by our collaborators. The quantitative data obtained from [5] has the units pmol O2/min/10µg cell protein. The amount

of cell protein is not included in our units and must be removed by the appropriate conversion to number of cells. Thus we must find a conversion factor for the amount of protein in a cell. From [2], the average amount of cellular protein in a given cell is 7 × 10−10g. Thus, the units from [5] can be converted using the factor

1cell 7 × 10−10_g =

1cell 7 × 10−4_µg

For example, for hESCs, a close approximation to hiPSCs (i.e. S), in [5], the OCR was determined to be 163.4 pmol/cell/10µ g cell protein.

163.4 × 10−12mol O2/min/10µg cell protein

=(163.4 × 10−12mol O2/min/10µg)

7 × 10−4_{µg cell protein}

1 cell

=1.14 × 10−14mol O2 /min/cell

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this conversion, we need the amount of oxygen in the region around our aggregate. For this conversion, we use the ideal gas law. This law is given by

p ˆV = nR ˆT ,

where p = gas pressure, ˆV = gas volume, n = amount of gas, R = ideal gas constant, and ˆT = gas temperature [15]. The temperature used here is the standard cell culture temperature of 310.15◦ K, or 37◦C. To get the units in percent, we first convert mmHg to mol/L. After this is determined, a conversion factor is used to convert to units of percent. The ideal gas law states

p ˆV =nR ˆT =⇒ n/ ˆV =p/R ˆT .

Under our conditions, p = 1mmHg = ₇₆₀1 atm, R = 0.082, and ˆT = 310.15◦K. This gives n/ ˆV = 1mmHg (0.082)(310.15◦_K) = 1atm (760)(0.082)(310.15◦_K) =0.000051737mol/L .

With 21% oxygen equal to 160mmHg and estimating the volume interacting with the aggregate as 0.25mL, this gives

160mmHg =(160)(0.000051737mol/L)(1L/1000mL)(1/4mL) =2.069 × 10−6mol

as the amount of oxygen present locally. It is of note that in numerical simulations, the estimate of 0.25mL was not considerably different compared to simulations with other values. Thus [O]21,1/4mL = 2.069 × 10−6mol.

This value is used to convert the oxygen consumption rate from mol to percent, giving 1.14 × 10−14_mol/cell/min

2.069 × 10−6_mol

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Similar calculations were carried out for the OCRs of P and D. The conversions for P and D were 83.6×10−12mol O2/min/10µg cell protein [5] to 2.83×10−7%O2/min/cell

and 95 × 10−12mol O2/min/10µg cell protein [5] to 3.21 × 10−7 %O2/min/cell,

respec-tively.

It is assumed that the (total) amount of O2 consumed is equal to the amount of

CO2 produced, but as the total amounts of the gases are not equal (i.e. 21% O2 and

5% CO2), a different conversion factor is used to change the units from mol to percent.

Given that 5% oxygen is equal to 38mmHg and taking the volume interacting with the aggregate as 0.25mL, this gives

38mmHg =(38)(0.000051737mol/L)(1L/1000mL)(1/4mL) =4.92 × 10−7mol

as the amount of CO2 present. The values determined for oxygen consumption in

mols are then divided by this value and multiplied by 100% as was done for oxy-gen consumption, giving the percentage CO2 production. For examples, the CO2

production rate for S was calculated as 1.14 × 10−14_mol/cell/min

4.92 × 10−7_mol

(100%) = 2.32 × 10−6%/cell/min .

Similar calculations for the P and D compartments give CO2 production rates of

1.19 × 10−6%/cell/min and 1.35 × 10−6%/cell/min, respectively.

3.4.5 Summary of Experimental Rates

For each parameter, a combination of the functions in Table 3.1 multiplies the ex-perimental parameter value to produce the compound parameter value as detailed in the following table.

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Par Experimental Value Range Ref Functional Effect Range Compound Value Range α [0.000001, 0.000026]∗ [34] [0.66, 4] [0.00000066, 0.00010] β [0.000016, 0.000026] [34] [0.685, 4] [0.000011, 0.000104] γ [0.000016, 0.000026] [34] [0.51, 4] [0.00000816, 0.000104] p1 [0.00069, 0.0012] [5, 43, 18] [0, 11.81] [0, 0.014] p2 [0.00045, 0.0016] [5, 43, 18] [0, 11.81] [0, 0.019] d1 [0.00010, 0.00017] [49] [0.0050, 1.24] [0.00000050, 0.00021] d2 [0.000073, 0.000082] [49] [0.0050, 1.24] [0.00000036, 0.000102] r [0.000001, 0.00017]∗ [49] [0.14, 3] [0.00000014, 0.00051]

Table 3.2: Experimental and compound parameter values. The functional effect is the product of feedbacks for each parameter, e.g. fα = f1f7. The compound value is

the experimental value multiplied by the functional effect, e.g. αfα.

∗ _{No measurements available. Closest related measurements were taken. For α, the}

upper bound for β was used. For r, the upper bound for d1 was used. In both cases,

it is taken that 0.000001 is the lower bound. Note that C ∈ [1, 10].

Throughout the model analysis, when referring to a parameter as simply α, for example, the value being referenced is the compound parameter value, rather than the experimental parameter value.

3.5 Diffusion

As previously noted, oxygen and waste can significantly affect many cellular pro-cesses. Because of the importance of these effects, oxygen and waste are included in the model as the variables O and W , where CO2 is used as a proxy for cellular waste.

The levels of O and W that affect the neural aggregate can be controlled by altering the oxygen and CO2 present in the air above the medium during culturing, i.e. Oair

and Wair. Control of these levels can be achieved by culturing in a chamber that

Mathematical model of growth and neuronal differentiation of human induced pluripotent stem cells seeded on melt electrospun biomaterial scaffolds

Contents

List of Tables

List of Figures

Introduction

1.1

Tissue Engineering

1.2

Stem Cells

1.3

Stem Cell Behavior

1.4

Cell Signalling

1.5

Biomaterial Scaffolds

1.6

Previous Work on Modelling Stem Cells

Chapter 2

Experimental Procedure and

Results

2.1

Methods

2.2

Results

Chapter 3

Model Development

S

P

D

3.1

General Structure of the Model

3.2

Scaffold Modelling

3.3

Determination of Functional Effects

3.3.1

Effect of O

on Proliferation

3.3.2

Effect of Waste on Proliferation

3.3.3

Effect of Contact on Proliferation

3.3.4

Effect of O

on Death

3.3.5

Effect of Waste on Death

3.3.6

Effect of O

on Differentiation

3.3.7

Effect of Contact on Differentiation

3.3.8

Effects of O

on Reversion

3.3.9

Effect of O

on O

Consumption and Waste Production

3.3.10

Effect of Contact on Diffusion of O

and Waste

3.3.11

Summary of Functional Effects

3.4

Determination of Experimental Rates

3.4.1

Death Rates: α, β, γ

3.4.2

Proliferation Rates: p

, p

3.4.3

Differentiation and Reversion Rates: d

, d

, r

3.4.4

Oxygen Consumption and Waste Production Rates

3.4.5

Summary of Experimental Rates

3.5