experiments and mathematical model - Springer Link

Biomech Model Mechanobiol (2011) 10:539–558 DOI 10.1007/s10237-010-0254-2

ORIGINAL PAPER

Growth of the chorioallantoic membrane into a rapid-prototyped model pore system: experiments and mathematical model Greg Lemon · Daniel Howard · Hongyi Yang · Svetan M. Ratchev · Joel I. Segal · Felicity R. A. J. Rose · Oliver E. Jensen · Sarah L. Waters · John R. King

Received: 9 March 2010 / Accepted: 28 August 2010 / Published online: 5 October 2010 © Springer-Verlag 2010

Abstract This paper presents a mathematical model to describe the growth of tissue into a rapid-prototyped porous scaffold when it is implanted onto the chorioallantoic membrane (CAM). The scaffold was designed to study the effects of the size and shape of pores on tissue growth into conventional tissue engineering scaffolds, and consists of an array of pores each having a pre-specified shape. The experimental observations revealed that the CAM grows through each pore as an intact layer of tissue, provided the width of the pore exceeds a threshold value. Based on these results a mathematical model is described to simulate the growth of the membrane, assuming that the growth is a function of the local isotropic membrane tension. The model predictions are compared against measurements of the extent of membrane growth through the pores as a function of time for pores with different dimensions.

G. Lemon (B) · O. E. Jensen · J. R. King School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK e-mail: [email protected] G. Lemon · D. Howard · F. R. A. J. Rose Wolfson Centre for Stem Cells, Tissue Engineering and Modelling (STEM), Centre for Biomolecular Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK H. Yang · S. M. Ratchev · J. I. Segal Manufacturing Research Division, Faculty of Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK S. L. Waters Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 24–29 St Giles’, Oxford OX1 3LB, UK

Keywords Tissue engineering · Mathematical model · Biomechanics · Partial differential equations · Angiogenesis assay · Chorioallantoic membrane

1 Introduction Rapid-prototyping technologies have an important role to play in tissue engineering, being a means of producing biomimetic scaffolds to optimise tissue growth and ensure biocompatibility when implanted in vivo (Cui et al. 2009; Mattioli-Belmonte et al. 2008; Yeong et al. 2004). In the present study a novel type of rapid-prototyped tissue engineering scaffold was devised, hereafter named the porearray scaffold, which consists of an array of defined pores (Fig. 1a–c) specifically having a shape chosen to represent two half-pores joined by a narrow constriction (Fig. 1d). Such a topographical feature is typical of that encountered by cells migrating into non-rapid prototyped tissue engineering scaffolds, produced for example by the method of supercritical fluids (Barry et al. 2004) or salt leaching (Murphy et al. 2002). The aim of this work was to investigate the effect of pore size and shape on the growth of tissue into such scaffolds when they are implanted into living tissue. In this study pore-array scaffolds were manufactured using a rapid-prototyping technique based on photopolymerisation. The biocompatibility of the arrays were assessed by implanting them onto the chorioallantoic membrane (CAM) of a fertilised chicken egg (Figs. 2a, b and 3a). Interestingly the experimental results showed that a portion of the CAM grew through the pore as an intact membrane (Fig. 4). Motivated by the observed phenomenon, it was the aim of this work to develop a mathematical model to explain the observed growth of the intact membrane through a single pore

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Fig. 1 Diagrams showing the design of the pore-array scaffold including a a schematic view from above (dimensions in mm), b a schematic view from the side (dimensions in mm), d a pictorial representation of the pore-array, and c a close-up of an individual pore within the array

Fig. 2 Images showing a a schematic diagram of some features of a fertilised chicken egg with a pore-array scaffold implanted onto the chorioallantoic membrane (CAM), and b histology of the excised CAM (scale bar is 100 µm), and c schematic diagram of the pore-array scaffold on the CAM. The weight of the scaffold causes it to sink into the allantoic vesicle to a depth z h below the phreatic surface of the allantoic fluid labelled H . The plane of the top of the scaffold is labelled T

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indicating the shape parameters w, d and h (dimensions in µm). The scaffold consists of 4 sets of 4 pores, each set having constriction diameters of w = 0.1, 0.15, 0.2, 0.4 mm, with all pores having an exterior diameter of d = 0.4 mm and length of h = 1 mm

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Fig. 3 Images showing a a pore-array scaffold on the CAM viewed through the egg-shell window from above (scale bar is 5 mm), b a µCT cross section of a pore-array scaffold with fragments of the CAM remaining inside it after excision (scale bar is 1 mm), and c an histological cross section of a scaffold showing intrusion of the membrane into the lower half of a pore (scale bar is 200 µm)

Fig. 4 Images showing membrane growth through a defined pore. Part a shows the CAM separating into two membranes and the breaching of the lower membrane resulting in hydrostatic pressure, with magnitude P, from within the allantoic vesicle acting on the upper part of the membrane. Part b shows how growth of the membrane and the hydrostatic pressure gives rise to directed growth of the membrane into the pore. Here the quantity N is the pressure exerted by the pore surface on the membrane in reaction to the pressure exerted by the membrane. Part c shows a PCNA antibody stained image of the excised CAM (scale bar is 300 µm) which confirms that cell proliferation occurs across the whole surface of the membrane (individual nuclei appear as light dots)

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in the pore-array scaffold, and to compare experimental measurements of the extent of growth to the theoretical results. Similar phenomena to that observed in the experiments, involving the constrained deformation of expanding membranes due to boundary constraints, arise in a diverse range of areas of biology and engineering. Consequently there are a variety of different modelling approaches that are relevant for describing the growth of the CAM into the pore-array scaffold. Studies from the engineering literature includes the blow moulding of elastic membranes during plastics manufacture (Khayat and Derdouri 1994), the movement of bubbles of oil or water through porous media (Ajaev and Homsy 2006), and the analysis of inflated structures (Bruyneel et al. 2006). Particularly relevant are works analysing an initially flat cylindrical rubber membrane undergoing unconstrained (Yang and Feng 1970) or constrained (Charrier et al. 1987) hyperelastic inflation, however it is unlikely that the CAM encounters the same degree of stretch as in these problems. Relevant studies from medicine and biology include the buckling of cells under a compressive load (Preston et al. 2008) or during micro-pipette aspiration (Baaijens et al. 2005), and the modelling of balloon angioplasty (Gasser and Holzapfel 2007). A novel and challenging aspect of modelling the growth of the CAM, which is not addressed in the aforementioned models, is to include in the model the growth of the membrane, as well as the effect that the physical forces acting on the membrane may have on this growth. A possible way of incorporating growth into the mechanics of the membrane is given by Goriely and Tabor (2003), who treated the growth of an actinomycete filament as a sequence of steps each consisting of an incremental inflation of a linearly-elastic membrane followed by a remodelling step. In this approach the stretched configuration after the inflation becomes the new unstretched initial configuration following the remodelling. Although this approach is natural and intuitively appealing, there are complications arising from the need to combine elastic deformations with growth (Goriely and Amar 2007; Rodriguez et al. 1994). Hence in the present paper a simpler approach was adopted to model the growth of the CAM whereby the membrane was treated as being instantaneously inextensible (but allowing for slow remodelling) and in-plane tensions were assumed isotropic. The structure of this paper is as follows. Firstly, in Sect. 2 some details of the experimental procedures (Sect. 2.1) and results (Sect. 2.2) of applying the CAM assay to the porearray scaffold are given. Then, in Sect. 3, the theory of the growth of the membrane through the scaffold is described, beginning with the formulation of the mathematical model (Sect. 3.1), an explanation for the choice of some of the parameter values in the model (Sect. 3.2), and then a detailed exposition of the solutions of the model equations and comparison with the experimental results (Sect. 3.3). Discussion of the results and conclusions are drawn in Sect. 4. The

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Appendices include further details of the experimental methods used (Appendix A), and details of a method for deriving approximate solutions to the model equations (Appendix B).

2 Experiments 2.1 Experimental procedure The design of the pore-array scaffolds used for the experiments is shown in Fig. 1, each scaffold being an array of pores with the characteristics of the single pore feature shown in Fig. 1d. The design of the pore is the shape of a sinusoidal curve rotated about a vertical axis (the formula given by Eq. (1) in Sect. 3.1), where d is the diameter of the pore entrance, w is the diameter of the constriction half way along the pore, and h is the length of the pore. This shape is intended to represent two spherically-shaped pores joined together by a narrow constriction, which would be a typical feature encountered by cells as they migrate through a tissue engineering scaffold. There are 4 sets of 4 such pores in the pore-array, with each set having the same specified values of the constriction widths, w = 0.1, 0.15, 0.2 and 0.4 mm. All pores have the same specified external diameter, d = 0.4 mm, and pore length of h = 1 mm. The value of d used is typical of the diameter of pores inside a scaffold made using super-critical fluids (Tai et al. 2007), and the range of values of w reflects that the connection sizes between the pores in such scaffolds takes values ranging from being nearly equal to the diameter of the pores down to where the pores are nearly closed off (Barry et al. 2004). To stop water and egg albumen from falling into the pores through the top of the scaffold, and thereby impeding the progress of tissue growing upwards through the pores, the design includes a wall that forms a perimeter around the top of the scaffold, having height 1 mm and thickness 0.4 mm (Fig. 1b). According to the design, pore-arrays were manufactured from a photocurable acrylic resin using a commerciallyavailable rapid-prototyping machine (Perfactory® type III mini system, EnvisionTec, Germany), which allowed the precise mathematical form of the shape of the pore to be specified (see Appendix A for further details). To determine any possible cytotoxic effects of the resin from which the pore-arrays was made, 3T3 dermal fibroblasts were cultured on a sample of the resin, and a Live/Dead® stain of the cells was made and imaged using fluorescence microscopy. It was found that the cells adhered and proliferated well on the resin and that there were relatively low levels of cell necrosis. To test the compatibility of the scaffolds with living tissue the chorioallantoic membrane (CAM) assay was used (Ribatti et al. 2003; Valdes et al. 2002; Zwaldo et al. 2001). The chorioallantoic membrane surrounds the developing


chick (Fig. 2a), the function of which is to provide ion transport and physically support respiratory capillaries (Fig. 2b). Starting with a batch of fertilised chicken eggs, the experimental procedure consisted of creating a small opening in the top of each egg at day 4 after fertilisation, placing a single pore-array onto the CAM (Figs. 2a and 3a), and then resealing the egg. After the required number of days of incubating the eggs, some of the eggs were sacrificed and the scaffolds were excised with the membrane still attached (Fig. 3b). For these scaffolds the extent of tissue growth through each of the pores was assessed using micro-computed tomography (µCT) and histological analysis (see Appendix A for details). Due to animal license restrictions the incubation could be carried only up to day 10 after fertilisation i.e. day 6 after implantation. 2.2 Experimental results When a pore-array was placed on the CAM, it sank a short distance into the membrane and the liquid underneath, while the top of the array remained relatively dry. Thus it appeared that the weight of the pore-array was being supported by both the CAM and by the hydrostatic pressure within the allantoic fluid (Fig. 2b), much as if it were placed on a water bed. Soon after implantation, the CAM underneath each pore apparently split into two separate membranes (Fig. 4a). The lower membrane, seemingly corresponding to the layer of stromal tissue in Fig. 2b, was breached by fluid from inside the allantoic vesicle, whereas the upper membrane, corresponding to the epithelial layer in Fig. 2b, remained intact and grew upwards through the pore (Fig. 4b). Histological analysis showed the upward-growing membrane to be less than 20 µm thick, presumably comprising a single layer of epithelial cells attached to basement membrane. It was hypothesised that hydrostatic pressure due to fluid from within the allantoic vesicle forces the intact part of the membrane upwards into the pore (Fig. 4a). Part of this intact membrane is forced against the inside surface of the pore, and the other part remains free and billows out inside the pore under pressure from the allantoic fluid (Fig. 4b). This results in the top of the membrane having a rounded appearance when imaged in situ or after removal from the pore (Fig. 3c). Figure 4c shows a light-microscopy image depicting proliferating cell nuclear antigen (PCNA) staining of the excised membrane. The PCNA analysis highlights cell nuclei where DNA is being synthesised, thus Fig. 4c suggests that the membrane can continue to grow while inside the pore. The expansion of the membrane due to this growth, together with the pressure acting behind it from the allantoic fluid directs the growth upwards through the pore (Fig. 4a, b). The growth of the membrane and the influence that mechanical forces acting on the membrane has on this growth is a very complex process. Growth and expansion of the CAM requires that individ-

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ual cells and blood vessels migrate within the extra-cellular matrix (ECM) of the membrane (Fig. 2b), and the ECM is itself being constantly remodelled by stromal cells embedded in the matrix. Scaffolds were excised from eggs at days 2, 4 and 6 after implantation for analysis. On day 2 there was insufficient adhesion between the membrane and the scaffolds for imaging of the pore-array with the membrane still attached to be possible. On day 4 and day 6, in the cases where the membrane did remain attached, the membrane-scaffold construct was treated with an X-ray dense stain (osmium tetroxide) and imaged using µCT equipment. From the resulting three-dimensional images, the distance between the top of the membrane and the lower surface of the scaffold, z m (indicated in Fig. 5), was measured together with the actual measured diameter of the constriction, w. It was found necessary to measure w because there was some deviation from the value specified in the design, although the ‘hour glass’ shape of the pore was generally well preserved (Fig. 3b). The data are shown in Fig. 10a for the pores of 3 arrays excised at day 4, and in Fig. 10b for 12 arrays excised at day 6 after implantation. Each data point in that figure represents a measurement taken from a single pore, where the height of the membrane, z m , is plotted against the measured width of the constriction, w. It can be seen that some of the experimental data points in Fig. 10a, b show no incursion of the membrane into the pores at all i.e. z m = 0, suggesting that in those cases the membrane failed to attach to the scaffold at the edges of the pore. The proportion of pores where the membrane was unattached was higher at day 4 (≈18%) than at day 6 (≈1%). In order to discern more clearly the underlying trends in the data and to be able to compare z m values from pores with similar values of w, the data were first sorted into order of increasing values of w and then combined into bins each containing 25 pairs of w and z m measurements. The means and standard deviations of the w and z m values in each bin were then plotted in Fig. 10a, b as horizontal and vertical error bars. The data are characterised by a large degree of variability in the measured values of z m and w. However, two trends can be discerned in Fig. 10b. The first is that there is a pinching-off effect whereby the membrane cannot grow through the pore if the size of the constriction is too small. The other is that even when the width of the constriction is large enough not to impede its progress, the membrane only rises to a level which is close to the top of the scaffold, with no further significant increase in z m after day 6 of incubation. This latter observation was also ascertained by visual inspection through the egg shell window of the membrane growing through the pore-array in situ. The observed saturation of the membrane displacement was hypothesised to be due to the approach of the membrane towards the phreatic surface of the liquid in the allantoic vesicle i.e. the level at which the hydrostatic pressure inside the vesicle falls to

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(a)

Table 1 Definitions of the variables used in the model Variable

Definition

t

Time

td

Time for completion of initial phase of growth

r

Radial displacement

rc

Radial displacement of contact line

z

Axial displacement

zc

Axial displacement of contact line

zm

Axial displacement of top of membrane

θ

Angular displacement

θc

Angular displacement of contact line

s

Arc length displacement

sc

Arc length displacement of contact line

sm

Arc length displacement of top of membrane

κs , κ ψ

Principal curvatures of membrane

A

Membrane area

T

Membrane tension

Tc

Membrane tension at contact line

N

Pressure exerted by pore wall on membrane

P

Pressure exerted by allantoic fluid on membrane

p

Pressure difference across membrane

v

Sliding speed of membrane

Table 2 Definitions of the symbols used to denote parts of the scaffold and membrane in the model

(b)

Fig. 5 Schematic diagrams showing a the labelled parts of the poremembrane system used in the derivation of the mathematical model in Sect. 3.1, and b the different stages of penetration of the membrane into the pore. Definitions of the variables and symbols are given in Tables 1 and 2 and in the text

Symbol

Definition

H

Position of phreatic surface

T

Position of uppermost face of the construct

P

Position of point of interest on the membrane

Bc

Free part of membrane

Lc

Part of membrane against the pore surface

observations, namely the blocking of the membrane for small constriction diameters and the saturation of the membrane displacement at a level near the top of the scaffold. It is shown how these two phenomena can arise from a tension-dependent growth mechanism in the membrane.

3 Mathematical model 3.1 Model formulation

atmospheric levels (Fig. 2c). As the membrane approaches this level the net force acting on the membrane forcing it to grow up through the pore falls, and so its displacement begins to saturate. In the following section a mathematical model is presented that reproduces the key features of the experimental

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Here the derivation is given of the mathematical model to predict the growth of the membrane through a single pore. For convenience all variables used in the model and the symbols used to explain the derivation are defined in Tables 1 and 2.


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The function r (z) used to specify the radially symmetric profile of the pore (Fig. 1d) is (1) r (z) = 21 d 1 + 21 (1 − w/d) [cos (2π z/ h) − 1] , where d > 0 is the diameter of the pore where it opens to the exterior, w is the diameter of the constriction half way along the pore, with 0 ≤ w ≤ d, andh is the length of the pore. Hence r (0) = r (h) = 21 d and r 21 h = 21 w. A schematic diagram of the membrane growing upwards into a pore is shown in Fig. 5a. The membrane configuration is taken to be axi-symmetric, and is described using cylindrical coordinates. Hence the coordinates of a point, P, on the membrane surface are (r, θ, z), where r and z are its radial and axial displacement respectively, and θ is the angle between the z−axis and the normal to the membrane. These variables are parametrised using the arc length, s, measured from z = 0. The assumption of radial symmetry implies that the free surface osculates with the pore surface along a circular line of contact. There are assumed to be two phases of the growth of the membrane into the pore. The first phase begins at t = 0 when the scaffold is first placed onto the CAM and the membrane across the pore opening has the shape of a disc. As the membrane expands, the hydrostatic pressure from inside the allantoic vesicle forces the membrane to grow into the pore. During this time the line of contact is assumed to remain fixed at z = 0. However when the growing membrane becomes tangential to the pore i.e. when θc = π/2 at time t = td , the line of contact can begin to move up into the pore, with axial displacement z c (Fig. 5b). The membrane behind the line of contact, where 0 ≤ z < z c is forced against the wall of the pore by pressure exerted from within the allantoic vesicle. In the model the membrane is assumed to have negligible thickness, to be inextensible and to be free of bending stiffness. Also the rate of growth of the membrane is assumed to be sufficiently slow compared to the rate of deformation that it can be considered to be in equilibrium under the forces acting on it. The configuration of the membrane at a particular time t is described using the standard equations for the equilibrium of an axisymmetric inflated membrane (see for example Evans and Skalak 1980) which are κs Ts + κψ Tψ = p, ∂ ∂r (r Ts ) − Tψ = −r σm , ∂s ∂s

(2) (3)

where Ts and Tψ are respectively the meridional and zonal components of the tension in the membrane, ∂θ sin θ and κψ = κs = − ∂s r

(4)

are the corresponding principal curvatures, p is the pressure difference across the membrane, and σm is the tangential component of the shear stress due to contact with the pore.

The tension is assumed to be isotropic everywhere in the membrane i.e. Ts = Tψ = T , thus obviating the need to specify a constitutive assumption to relate Ts and Tψ in the model. This simplifying assumption means that the membrane exhibits a fluid-like behaviour analogous to a soap film. Swelling of the allantoic vesicle (Fig. 2a) with water from the egg albumen is driven by osmosis (Needham 1931) and the resulting hydrostatic pressure inside the part of the vesicle within the pore causes a pressure difference across the membrane equal to p = P − N , where P is the pressure inside the allantoic vesicle and N is the pressure exerted by the pore wall on the membrane. Sedimentation of the construct into the allantoic vesicle occurs due to the polymer from which the pore-array is made having a density of ρs = 1.2 g cm−3 and the allantoic fluid having a slightly lower density in the range ρa =1.00–1.02 g cm −3 (Romanoff and Hayward 1943). For simplicity it is assumed that this sedimentation results in the phreatic surface of the allantoic fluid (marked by the symbol H in Fig. 2c and Fig. 5a) being located at a fixed height z = z h above the lower surface of the scaffold, and hence P = ρa g(z h − z) is the hydrostatic pressure for z ≤ z h . To account for the effect of sliding friction between the membrane and the scaffold surface, the shear stress acting on the membrane is related to the normal force using σm = −μN f (v) where μ is the coefficient of sliding friction and v is the speed with which the membrane slides across the pore at some fixed location on the pore surface behind the contact line. This description of the friction force is similar to the Coulomb static friction model (see for example Olsson et al. 1998) but includes a dependence on the sliding speed which is taken as v 2 , (5) f (v) = tan−1 π vf where v f is a shape parameter. This dependence has been included to facilitate the calculation of the numerical solution, with a value of v f 1 chosen to produce a piece-wise constant behaviour in f (v) as explained further in Sect. 3.3. There is ample evidence that tensile stresses within biological tissues can stimulate cell proliferation (Pietramaggiori et al. 2007; McAdams et al. 2006; Tanner et al. 1995). To allow for this possibility it is assumed that the rate of growth of the area of the membrane is γ k(T ) where T β 2 −1 , (6) k(T ) = tan π Tp for constants γ , β, T p ≥ 0. The function k(Tc ) is plotted in Fig. 7b for T p = 0.01 and β = 100, where Tc is the tension in the membrane specified at the contact line between the free part of the membrane and the pore surface. The curve tends to unity for large values of the tension and tends to zero as the tension tends to zero. The parameter T p can be thought of as

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the cut-off level of tension below which membrane growth is inhibited, indeed k(T p ) = 21 , and the parameter β specifies the steepness of the curve in the vicinity of T = T p . The membrane is assumed to be pinned at z = 0 i.e. no membrane is inserted into the pore from underneath the array. By integrating γ k(T ) over the whole membrane, the rate of change of the total membrane area can be shown to be sm T (˜s ) β dA −1 r (˜s ) d s˜ . (7) = 4γ tan dt Tp 0

The equation for the sliding speed (8) written in terms of the variable z, z T β 2 1 −1 γ tan rq d z˜ , (14) v= r π Tp 0

The sliding speed, v, can be calculated by considering the motion of a point on the membrane having arc length coordinate 0 ≤ s ≤ sc at time t. In a small amount of time δt the area of the membrane behind this point will expand by an amount s δ A = 0 2π γ k(T (˜s ))r (˜s )d s˜ δt, but this corresponds to an additional amount of arc length δs where δ A = 2πr (s)δs. Equating these two expressions for δ A and taking the limit δt → 0 gives s T (˜s ) β 2 1 −1 γ tan v(s) = r (˜s ) d s˜ . (8) r (s) π Tp 0

In the free part of the membrane (Bc in Fig. 5a) N = 0 so σm = 0 and Eq. (3) becomes ∂ T /∂s = 0 thus T = Tc for constant Tc specified at the contact line. Following Goriely and Tabor (2003), the equations for the membrane are written in terms of the arc length variable s as follows ∂r = − cos θ, (9) ∂s ∂θ sin θ ρa g (z h − z) + =− , (10) ∂s Tc r ∂z = sin θ, (11) ∂s with the boundary conditions at the contact line being r = rc , θ = θc , z = z c at s = sc and the boundary conditions at the top of the membrane being r = 0, θ = 0, z = z m at s = sm . Since the membrane and the pore surface have a common tangent at the line of contact, the values of rc and θc are prescribed for a given value of z c . On the part of the membrane that is in contact with the pore (Lc in Fig. 5a) the variables r (s), θ (s), z(s) are prescribed from the shape of the pore. Thus the principal curvatures of the membrane can be calculated from Eq. (1) using κs = −r (1 + (r )2 )−3/2 and κψ = r −1 (1 + (r )2 )−1/2 (see for example Humphrey 2003, p. 140), where the prime denotes differentiation with respect to z. Using these expressions for the principal curvatures and the previous expression for the friction force, Eq. (3) gives an equation for the tension, which when written in terms of the independent variable z is v 2 ∂T −1 , (12) = μq N tan ∂z π vf

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where q = (1 + (r )2 )1/2 . Also, letting p = P − N = ρa g(z h − z) − N in Eq. (2) and rearranging gives r 1 N = ρa g(z h − z) − T − 3 + . (13) q rq

completes the specification of the equations governing the tension in the part of the membrane against the pore surface. Continuity of the tension at the contact line requires T (z c ) = Tc . The method of solution of the model equations is, for a given membrane area A at time t, to determine the quasiequilibrium configuration and tension distribution by solving Eqs. (9)–(14) subject to the constraint sm 2πr (˜s ) d s˜ ,

A=

(15)

0

then using Eq. (7) to evolve the membrane area. The given value of A uniquely determines z m and z c . The model equations are nondimensionalised using the following substitutions ⎫ r = h r, ˆ z = h zˆ , t = γ −1 tˆ, s = h sˆ , ⎬ ˆ N = ρa gh Nˆ , T = ρa gh 2 Tˆ , v = hγ v, ˆ A = h 2 A, (16) ⎭ ˆ w = h w, ˆ d = h d, and dropping the carets the equations become, for the free part of the membrane, ∂r = − cos θ, ∂s ∂θ 1 sin θ = − (z h − z) + , ∂s Tc r ∂z = sin θ, ∂s

(17) (18) (19)

and

v ∂T 2 2 1/2 −1 , = μ(1 + (r ) ) N tan ∂z π vf 1 − rr + (r )2 N = zh − z − T , r (1 + (r )2 )3/2 z 1 T β 2 v= d z˜ , r (1 + (r )2 )1/2 tan−1 r π Tp

(20) (21) (22)

0

for the part of the membrane in contact with the pore surface. The total area and rate of change of the total area of the


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membrane with respect to time are given by

3.2 Parameter value selection

sm A=

2πr d s˜ ,

(23)

0

dA = dt

sm 4 tan

−1

0

T Tp

β

r d s˜ ,

(24)

respectively and Eq. (1) becomes r (z) = 21 d 1 +

1 2

(1 − w/d) [cos (2π z) − 1] .

(25)

At the instant the scaffold is placed onto the CAM the membrane is assumed to be almost flat, thus A(0) = 14 π d 2 . It is assumed that initially the ballooning membrane grows into the pore while the line of contact between the pore and the membrane remains at z = 0. The procedure to calculate the configuration of the membrane as a function of time is implemented numerically using a shooting method. For a given numerical approximation for the area, At , at time t, trial values are presented for z m , sm and Tc . Equations (17)– (19) are integrated from s = sm to s = 0, and the trial values iterated to make r = 21 d and z = 0 at s = 0, and sm At = 0 2πr d s˜ simultaneously. The tension and membrane configuration so determined are used to update the membrane area using Eq. (24). This process is repeated until time t = td , when the membrane becomes tangential to the pore at z = 0 i.e. θ = 21 π at s = 0 (Fig. 5b). For t > td the line of contact between the pore and the membrane begins to move inwards as the membrane continues to grow. During this next phase of the growth Eqs. (17)–(19) are integrated from s = sm to s = sc with trial values of sc , Tc and z m iterated to make z r = rc and θ = θc at s = sc , and s At = scm 2πr d s˜ + 0 c 2πr (1+(r )2 )1/2 d z˜ simultaneously. Once the position of the contact line has been determined, the tension in the part of the membrane behind the contact line is found by integrating Eqs. (20)–(22) from z = 0 to z = z c . Initial conditions T = T0 and v = 0 at z = 0 are used, and T0 is iterated to make T = Tc at z = z c . The resulting tension distribution and membrane configuration is then used to update the membrane area using Eq. (24). All numerical calculations and graphical presentation of the results were performed using the MATLAB computer package. Specifically, integration of the ODEs was carried out using the function ode45, and iteration of the variables with the non-linear equation solver fsolve. The definite integrals in Eqs. (22)–(24) were computed by numerically integrating the corresponding z ODEs. For example, writing Eq. (22) as v = (1/r ) 0 I (ˆz ) d zˆ and differentiating with respect to z it can be shown that v satisfies dv/dz = I /r − (v/r ) dr/dz.

The parameter values used in the model are listed in Table 3. Some details and discussion are now given for how these values were chosen. Explanation for the choices of parameter values listed as ‘Fit to experimental data’ in Table 3 is given in the context of the results of the model solutions described in Sect. 3.3. To nondimensionalise the model equations, the characteristic length scale in Eq. (16) was chosen to be h = 1 mm, which is the length of the pore in the original scaffold design (Fig. 1). The value d = 0.4 mm for the exterior diameter of a pore in the model was also taken from the construct design, however because there was considerable deviation of the measured constriction diameter from that in the design specification (Fig. 10a, b), constriction diameters of w = 0.05, 0.1 and 0.4 mm were chosen for calculating the solutions of the model equations rather than the design values. The values of the shape parameters w and d were chosen to avoid the possibility of the free part of the membrane, Bc , determined from the model solutions having multiple points of intersection with the pore surface. When the axial displacement of the point of contact, z c , is fixed and the pore-shape parameter values are varied, the single contact line bifurcates into two when the principal curvature, κs , of Bc and the pore have the same magnitude and sign. For the choice of r (z) defined by Eq. (1) this can only occur for z c within the intervals 0 ≤ z c < 41 and 34 < z c ≤ 1 because the curvatures have opposite sign where 14 ≤ z c ≤ 43 . Neglecting the effects of the gradient of the hydrostatic pressure inside the pore, it is shown in Appendix B that Bc has the shape of a spherical segment. The radius of curvature of Bc is thus R = r (1 + (r )2 )1/2 , where the prime denotes differentiation with respect to z. Thus multiple points of intersection arise where r (1 + (r )2 )−3/2 = R −1 i.e. r r = 1 + (r )2 . Substituting Eq. (1) into this expression yields the quadratic equation 2π 2 (d − w)2 u 2 + π 2 (d 2 − w 2 )u − (4h 2 + π 2 (d − w)2 ) = 0, where u = cos(2π z). The absence of real solutions for all z c requires u > 1 which implies that

2 1 (26) d< π 1 − w/d must hold and hence there will be no multiple intersections of √ the free part of the membrane with the pore provided d < 2/π ≈ 0.45. The values of w and d shown in Table 3 were chosen to satisfy Eq. (26). Experiments by Byerly (1932) showed that the weight of the allantois (the membrane that forms the surface of the allantoic vesicle: see Fig. 2a) grows rapidly between about day 4 and day 8 of incubation and saturates around day 10. Assuming that its thickness remains constant, uniform

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548 Table 3 Definitions and values of the parameters used in the model

Units are not given for dimensionless parameters

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Parameter

Definition

Value(s)

Source

d w

Exterior diameter of pore

0.4 mm

Design parameter used in this study

Diameter of pore constriction

0.05,0.1,0.4 mm


h

Length of pore

1 mm


ρs

Density of scaffold polymer

1.2 g cm −3

Data from manufacturer

ρa

Density of allantoic fluid

1.01 g cm −3

Romanoff and Hayward (1943)

g

Acceleration due to gravity

9.8 m s−2

γ

Membrane growth rate

1.1 day−1

Fit to the data from Byerly (1932)

zh

Phreatic surface displacement

1.05 mm

Fit to experimental data

μ

Friction coefficient

0,0.2,1.0,100


vf

Friction shape parameter

0.01


Tp

Inhibition tension

0.01


β

Tension shape parameter

100


expansion of the allantois at a constant rate, γ , would result in the total surface area and mass growing exponentially with time. Performing a least-squares fit of the function Ceγ (t−4) to the data in Fig. 3 of Byerly (1932), for the weight of the allantois in grams for standard eggs between day 4 and day 8 of incubation, gave C = 0.012 and γ = 1.1 day−1 . 3.3 Model solutions In this section the solutions of the mathematical model are described for different choices of the parameter values. Specifically an investigation is made of how the rate of growth of the membrane into the pore-array is affected by the shape of the pore and by the amount of friction between the membrane and the pore surface. To facilitate comparison with the experimental data, values of displacements and time are given in physical units, which from Eq. (16) are obtained by multiplying them by 1 mm and 1/γ ≈ 0.833 days respectively. Figure 6a shows curves of the axial displacement of the top of the free part of the membrane, z m , as a function of time, for an exterior diameter of d = 0.4 mm and for constriction diameters of w = 0.05, 0.1 and 0.4 mm. The value used for the coefficient of friction parameter is μ = 1. The corresponding results for the axial position of the line of contact between the membrane and the pore, z c , is given in Fig. 6b. The top of the pore-array is located at z = 1 mm, which is indicated with a dotted line and the symbol T . The axial displacement of the phreatic surface is z h = 1.05 mm, indicated by a dotted line and the symbol H (see also Fig. 5a). Shortly after implantation as the membrane first starts to penetrate into the pore, z m increases while z c remains zero. At t = td ≈ 0.5751 the membrane becomes tangential to the pore surface, and for t > td a portion of the membrane begins to grow flat against the pore surface while the free part of the

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membrane grows upwards into the pore, thus the z c curves leave the z c = 0 axis at t = td . Note that the smaller the diameter of the constriction the longer it takes for the membrane to grow through the pore. This is because the smaller the radius of the constriction, the smaller the force exerted across the membrane when it is in the vicinity of the constriction, and hence the smaller the tension in the membrane. Therefore by Eq. (6) the lower the rate of growth of the membrane and hence the longer it will take the membrane to grow through the constriction. In the case of w = 0.4 it can be seen in Fig. 6a that the growth of the membrane appears to slow as it approaches the top surface of the scaffold, because z m and z c appear to approach constant values after about t = 3.5 days. This is because the drop in hydrostatic pressure as the membrane approaches the phreatic surface causes a drop in membrane tension. Again by Eq. (6), this lowering in membrane tension lowers the rate of growth of the membrane. These tension-dependent growth effects are illustrated in more detail in Fig. 7. Part (a) of this figure shows how the constant tension in the free part of the membrane, Tc , varies with the height of the membrane, z m , for the three different values of the constriction diameter, w = 0.05, 0.1 and 0.4 mm. The reason for the drop in tension as the membrane passes through the constriction can be understood by noting that Eqs. (2) and (3) for a free membrane, σm = 0, and with constant pressure difference p, and isotropic tension T = Ts = Tψ , has a solution where the membrane is spherical with constant radius R, provided T = 21 p R. Thus the narrower the constriction, the smaller the value of R and therefore the smaller the tension. From the equation T = 21 p R it can be seen that the tension also falls as the membrane approaches the phreatic surface because the value of p drops. If the tension falls below the cut-off tension for growth, T p in Eq. (6), the rate of growth of the membrane falls


(a)

(b)

Fig. 6 Graphs showing the axial displacement of a the top of the free part of the membrane, z m , and b the contact line between the free part of the membrane and the pore surface, z c , as functions of time for different constriction diameters, w = 0.05, 0.1 and 0.4 ( mm). Results are given for a friction coefficient of μ = 1

markedly due to the step-like functional dependence of the rate of growth of the membrane on tension shown in Fig. 7b. This causes the slowing of the growth of the membrane as it passes through the constriction and as it approaches the top surface of the scaffold, consistent with the experimental observations. The values T p = 0.01 and β = 100 were chosen so as to produce the blocking effect for w = 0.05 mm, but to leave the membrane unimpeded for larger values of w, consistent with the experimental data shown in Fig. 10b. The vertical displacement of the phreatic surface, z h = 1.05 mm, was chosen so that the height of the membrane is slightly above the top surface of the pore-array after 6 days of incubation, consistent with the experimental data shown in Fig. 10b. For comparison with the experimentally-derived value of the sedimentation depth, z h , an estimate can be obtained by

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(a)

(b)

Fig. 7 Graphs showing a the tension in the free part of the membrane, Tc , as a function of the axial displacement of the top of the membrane, z m , for different values of the constriction diameters, w = 0.05, 0.1 and 0.4, for a friction coefficient of μ = 1, and b the dependence on the rate of growth of the free part of the membrane as a function of the free-membrane tension, defined by Eq. (6) i.e. k(Tc ) = 2/π tan−1 ((Tc /T p )β ), where T p = 0.01 and β = 100

ignoring the forces exerted by the membrane on the scaffold and equating the weight of the scaffold with the force exerted on its lower face. The pore-array (Fig. 1a, b) is a rectangular prism of height h = 1 mm with horizontal faces having area a = 4.7 mm × 4.7 mm. For simplicity all the pores are taken to be cylindrical with diameter 0.4 mm therefore the total volume of the pores is, V p = (16 × π(0.2)2 × 1) mm3 . Each of the four sides of the wall are approximately 1 mm high with a thickness of 0.4 mm giving a total wall volume Vw = (4 × 4.3 × 0.4) mm3 . Summing the forces on the lower surface of the scaffold gives ρs g(ah − V p + Vw ) = ρa gaz h , which implies z h ≈ 1.45 mm. Although this estimate is larger than the experimentally-derived value, a more detailed calculation would take into account that the tension

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in the membrane partially supports the scaffold’s weight (see Press 1978) and this would result in a smaller estimate for the value of z h . More detailed modelling could also take into consideration that the tension and the hydrostatic forces acting on the construct change as the membrane grows into the pores. Because a definitive value for the friction coefficient μ could not be determined directly from experiments or biophysical data, a range of values for μ were tried in the simulations. In Fig. 8a the displacement of the top of the free part of the membrane, z m , is plotted as a function of time for four different values of the friction coefficient: μ = 0 (no friction), 0.2, 1 and 100. An exterior diameter of d = 0.4 and constriction diameter of w = 0.4 i.e. a cylindrically-shaped pore is considered. In Fig. 8b the corresponding results are shown for the displacement of the contact line between the free part of the membrane and the pore wall, z c . Increasing μ reduces the total rate of growth of the membrane and hence the rate of its penetration into the pore by reducing the tension in the membrane behind the line of contact; the effect is most pronounced for the case of μ = 100. For this value of the friction coefficient the membrane is bound tightly to the pore wall and tissue growth is limited mainly to the free part of the membrane. Hence for μ = 100 the friction between the membrane and pore can be likened to static friction. Again, evident in Fig. 8 is the slowing of the membrane as it approaches the top surface of the scaffold. Although z m and z c appear to have reached equilibrium values well within the 7 day simulated incubation period, this is due to the membrane growth rate, k, falling to almost negligible but non-zero values. In fact z m and z c will continue to increase with respect to time but over a very long time scale. The limiting values of both z m and z c will be equal to the displacement of the phreatic surface, z h , because only there will the tension and the growth rate of the membrane become zero. However this configuration can never be reached because the solutions of the model equations can only proceed until such time as z c = 1, where the contact point reaches the top surface of the scaffold. As discussed above in the context of Fig. 6, the free part of the membrane can be considered to be spherical in shape if the hydrostatic pressure gradient through the pore is neglected, however accounting for this pressure gradient it is shown in Appendix B that the shape of the membrane is a slightly oblate spheroid. Also in Appendix B a method is presented for obtaining an approximation to the curve for z m as a function of time during the initial phase of entry of the membrane into the pore. This approximation is defined by Eqs. (B.18) and (B.20), and is plotted in Fig. 8a for t < td (line with triangles). Note that z c = 0 during this initial phase of entry of the membrane. An approximate expression for td , the time at which the membrane becomes tangential to the pore at z = 0, is given by Eq. (B.19). This expression gives a numer-

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(a)

(b)

Fig. 8 Graphs showing the axial displacement of a the top of the free part of the membrane, z m , and b the contact line between the free part of the membrane and the pore surface, z c , as functions of time for different values of the friction coefficient i.e. μ = 0, 0.2, 1 and 100. Also shown are graphs of the approximate solutions for the limit μ → ∞ derived in Appendix B (lines with triangles). Results are given for an exterior diameter of d = 0.4 mm and constriction diameter of w = 0.4 mm i.e a cylindrically-shaped pore

ical value of td ≈ 0.5751 days which is close to the value determined from the simulations. Also in Appendix B a derivation is given for an approximate solution to the model equations for t ≥ td in the limit of infinite friction i.e. μ → ∞. The result is Eqs. (B.22) and (B.30), and these solutions are plotted in Fig. 8 for t ≥ td (lines with triangles). In the limit μ → ∞, growth is confined only to the free part of the membrane since the tension falls to zero in a vanishingly small region behind the contact line. Therefore, since the area of the free part of the membrane is approximately uniform as it traverses the pore, the amount of new membrane added per unit time is constant and so the rate of penetration of the membrane into the pore is constant. The speed of the membrane in the limit μ → ∞ is smaller


than when μ is finite, although Fig. 8 shows that the case of μ = 100 is a good approximation to the infinite limit. Considering the case of a cylindrically-shaped pore for t ≥ td in the absence of friction, μ = 0, and ignoring the perturbing effect of the hydrostatic pressure gradient, the membrane can be thought of as comprising a cylinder and a hemisphere, therefore A(t) = 21 π d 2 + π dz c (t). Assuming the tension in the membrane to be much greater than T p allows the effects of tension-dependent growth to be ignored, thus Eqs. (23) and (24) can be combined to give d A/dt = A. Solving this with the condition that at t = td the membrane forms a hemisphere i.e. A(td ) = 21 π d 2 , gives A(t) = 21 π d 2 exp (t − td ). Equating the two expressions for A(t) it follows that z m = z c + 21 d = 21 d exp (t − td ). Thus the membrane displacement varies exponentially in time in the absence of friction rather than linearly as in the case of infinite friction. The exponential shape of the curve of z m as a function of time for the case μ = 0 is evident in Fig. 8a. In Fig. 9 the membrane tension, T , and the sliding speed, v, as functions of z are plotted for the four different values of the friction coefficient i.e. μ = 0, 0.2, 1 and 100. Again, a cylindrically-shaped pore with d = 0.4 mm and w = 0.4 mm is considered. The curves are plotted at the instant in time when z c = 0.5, although the value of t when this occurs depends on the particular value of μ used. Figure 9a shows that when μ = 0 the tension in the membrane is uniform everywhere, but for μ > 0 the tension is uniform only in the free part of the membrane, and decreases with distance behind the line of contact. This effect is analogous to the loss of tension that occurs in ropes being drawn over pulleys (Dufva et al. 2007). As μ is increased the growth becomes confined to the free part of the membrane and to an eversmaller region in the part of the membrane lying against the pore surface behind the line of contact. Figure 9b shows that the sliding speed, v, tends to zero as μ becomes large. Shown in Fig. 9 are solutions derived using an approximate method for solving the model equations described in Appendix B, for the case μ = 1 (lines with crosses). The approximate solutions for T and v for z ≤ z c are given by Eq. (B.25) and (B.26), and their graphs show good agreement with the curves of the numerical solutions. The parameter value v f = 0.01 used in the Eq. (20) to relate the sliding speed to the friction force, was chosen to be sufficiently small so that in the simulations the function f (v), defined by Eq. (5), was either approximately equal to unity or zero over most of the membrane surface. That this results from choosing v f to be small can be understood by considering the case where friction has a negligible effect on the tension and growth, e.g. where μ 1. Differentiating Eq. (22) it can be shown that ∂v/∂z = q − vr /r , where q = (1 + (r )2 )1/2 , so ∂v/∂z → q as v → 0. Thus if v f is taken to be small, f (v) saturates over small axial distances, thereby exhibiting piecewise constant behaviour. Thus from

551

(a)

(b)

Fig. 9 Graphs showing a the membrane tension T and b the sliding speed v, as functions of position, z, along the pore for different values of the friction coefficient μ = 0, 0.2, 1 and 100. Results are given for an exterior diameter and constriction diameter of d = 0.4 mm and w = 0.4 mm, respectively i.e. a cylindrically-shaped pore. Also shown are the solutions for μ = 1 calculated using the approximate method described in Appendix B (curves with crosses). The solutions are shown at the instant in time when z c = 0.5 mm

Eqs. (20) and (22) both T and v are piecewise linear functions of z, as seen in Fig 9a, b, respectively. Using Eq. (16) the maximum value of the nondimensional tension from Fig. 9a, i.e. Tc ≈ 0.043, corresponds to a physical tension of T ≈ 4.3 × 10−4 N.m −1 . Assuming the thickness of the membrane to be 20 µm, this would correspond to a tensile stress of approximately 20 Pa, well below the breaking stress of soft tissues (typically > 10 KPa, see Lepetit et al. 2004). Figure 10c shows the final displacement of the top of the membrane, z m , predicted by the model after 2, 4 and 6 days of growth on the CAM. The curves are characterised by incomplete growth of the membrane through the pore for small

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(a)

(b)

(c)

Fig. 10 Graphs showing experimental data of the axial displacement of the top of the membrane, z m , as functions of the constriction width, w, after a 4 days and b 6 days of growth of the CAM into the pore-array, and c results of simulations after 2, 4 and 6 days. In a and b the crosses

indicate measurements taken from each pore, and the error bars indicate the means and standard deviations of the measured values of w and z m , which were distributed into bins of 25 pairs taken in order of increasing value of w

values of the constriction diameter, w, and a saturation of the vertical displacement of the membrane just above the top surface of the scaffold for large values of w. There is good general agreement between the shape of the theoretical curves and the trends exhibited by the experimental data points shown in Fig. 10a, b, despite the large variability in the measured values of z m and w. This variability made numerical fitting of the model to the data problematic so only a qualitative fit using manually selected parameter values was attempted. Variations in the w values arise from lack of resolution in the rapid-prototyping technique (see Sect. 4 for more details). Variations in z m for similar values of w may have arisen from spatial inhomogeneities in the growth rate across the membrane surface resulting in the membrane growing into the pores at differing rates, or as a result of perturbations to the integrity of the attachment between the membrane and the pore due to movement of the developing chick. As mentioned in Sect. 2 some of the data points in Fig. 10a, b indicate that the membrane failed to attach to the pore-array at the edges of some of the pores (z m = 0). However, attachment of the membrane may have occurred a few days after implantation, which may explain why in some cases the membrane only grew part of the way through the pore after the 6 days

of incubation. This scenario would also explain why there were a higher percentage of pores where the membrane was unattached at day 4 than at day 6. Alternatively, attachment and growth of the membrane into the pore may have occurred directly after implantation but growth may have slowed due to adaptation of the membrane to the tension, resulting in the membrane passing only part of the way through the pore by day 6.

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4 Discussion This paper describes experiments of the implantation onto the chorioallantoic membrane (CAM) of a tissue engineering scaffold consisting of an array of pores each having a predefined shape. The aim was to study how the size and shape of pores affects how cells and tissues grow into a porous scaffold when it is implanted in vivo. The experimental results revealed that part of the CAM grew as an intact membrane through the pores of the scaffold, and measurements of the displacement of the membrane relative to the lower surface of the scaffold were taken over the time course of the incubation. It was found that growth of the membrane was impeded


if the diameter of the constriction in the centre of the pore was too small and that the growth of the membrane stopped when it reached the top of the pore array. A simple mathematical model was formulated for how the membrane grows through the pore-array scaffold, which reproduces the general trends seen in the experiments. While the manner of entry of the CAM into the porearray is a striking phenomenon, it is not fully appropriate as a paradigm for how tissue grows into a porous scaffold when implanted into a human patient or mammalian laboratory animal. In those cases tissue growth would more likely occur in the form of a wavefront of cells that moves through the pores and fills the voids of the pores as it goes, rather than in the present case where the cells are confined to a membrane that slides over the surfaces of the pores. Nevertheless the experimental observations described in this paper are of particular interest to the study of the growth and mechanics of biomembranes. The pore-array represents a potentially valuable tool for tissue engineering since it facilitates the imaging of cells and tissues as they interact with defined pore features. This paper has focused on how the size of the openings between spherical pores affects the way tissue grows into porous scaffolds, produced for example by super-critical fluids (Barry et al. 2004) or salt-leaching (Murphy et al. 2002) using spherically-shaped porogens. The design of the pore-array could be easily modified to study the effects of using square pores which are more typical of rapid-prototyped scaffolds (Seitz et al. 2005) and salt-leached scaffolds made using cubic porogens. Other studies using the pore-array could investigate the effects of different surface coatings inside the pores, or the application of growth factors through the top of the array during the incubation on the CAM. This work has highlighted the importance of the use of rapid-prototyping technologies in regenerative medicine. This research has used the technique of projection microstereolithography for the rapid manufacturing of the pore-array, which has a planar resolution of 5 µm and a depth direction resolution of 15 µm. The ability to produce the scaffolds with pore dimensions close to the given specifications, using the Perfactory® system, represents a considerable achievement considering the very small dimensions involved. Other commercially available rapid prototyping technologies such as selective laser sintering (SLS), fused deposition modelling (FDM) and 3D printing (3DP) can only produce a manufacturing resolution in the range of 100 to 300 µm (Yeong et al. 2004). In recent years dual photon stereolithography technology has been proposed for rapid prototyping, and it is claimed to have a manufacturing precision of less than 100 nm (Lee et al. 2000) which is higher than the projection microstereolithography adopted in the current study. However, the dual photon stereolithography technology is still in the experimental stage and no mature commercial systems are

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currently available. Also dual photon stereolithography has a much slower fabrication speed than projection microstereolithography. Thus projection microstereolithography was chosen for this research, and it represents the most advanced concurrent RP technique available for the manufacture of the pore-array. For the purposes of investigating the effects of pore size on tissue ingrowth, the projection microstereolithography technique has produced satisfactory results. The constriction diameter in the manufactured pore-array differed from that in the design by less than 10 µm. Although there is potential to further reduce this tolerance level, currently there are no established operating routines to address and minimise the influence of the multiple factors that affect the tolerance, including the physical properties of the photocurable resin, the curing time, light intensity and the microstructure profile. As a result, the tolerance of the manufacturing process to a great extent depends on the expertise of the operator. Currently efforts are being directed towards optimising the microstereolithography manufacturing process using numerical simulations and experimentation. In the solutions of the model equations, the passage of the membrane through the constriction is impeded by a drop in membrane tension, resulting in a decreased rate of growth which is postulated to be tension dependent. In reality the neglect of bending stiffness of the membrane (Fig. 2b), as well as the lowering of the tensile stress acting due to the narrowing of the free part of the membrane, may cause growth-induced compressive stress to be induced in the membrane as it approaches the constriction. This scenario cannot be captured with the present model because of the neglect of bending effects in the membrane. Nevertheless, making the rate of growth a decreasing function of tension in the model goes some way towards describing the reduction in the rate of cell proliferation that is likely to result from spatial confinement of the membrane in the region of the constriction. This tension-stimulated growth mechanism may play an important role in the growth and development of the fertilised egg. The surface area of the CAM must increase rapidly after fertilisation but its growth has to be constrained by physical barriers such as the egg shell, the embryo, and various extraembryonic membranes (Fig. 2a). Without a reduction in the growth rate of the membrane when it comes into contact with such obstacles the CAM would buckle and fold, possibly to the detriment of the proper functioning of the allantoic vesicle. However, with the growth being tension dependent the membrane stops growing once it has fully covered the surface of any obstruction, much as it does when it grows into the pore-array but stops when it encounters a narrow constriction (Fig. 3c). Thus tension-dependent growth could be the means by which an allantoic vesicle is produced with an optimum surface area to volume ratio while at the same time

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accommodating the tissues surrounding it inside the egg. It is interesting to speculate as to whether this growth regulation mechanism also plays a role in the growth of membranous tissues in other species during development. In this paper a simplified approach has been taken to incorporate growth of the membrane into the model, whereby the elastic deformation is treated as being negligible, and so the growth merely results in an increase in the area of an unstretched membrane. A natural development of the present approach would be to adapt the techniques of Goriely and Tabor (2003) so as to combine the tension-dependent growth of the membrane with a realistic constitutive law for the deformation, such as that of a Mooney-Rivlin solid (Humphrey 2002) or (as a special case) the neo-Hookean solid. To include in such a model the friction force and the confinement of the growth of the membrane could be pursued as future work. Acknowledgments The authors wish to acknowledge the financial support of the Biotechnology and Biological Sciences Research Council (BBSRC). The authors would also like to thank C M Cuesta for the drawing in Fig. 2a.

Appendices A Experimental methods Fabrication of pore-array scaffolds using rapid prototyping The pore-array scaffolds were fabricated with a newly developed mask-projection based photopolymerisation system, the Perfactory® type III mini system (EnvisionTec, Germany). This machine has a 2,800 × 2,100 pixel Digital Light Processing (DLP) projector allowing a resolution of 5 µm. The 3D solid model of the pore-array scaffold designed with CAD software (Solidworks) was first saved as an STL file and then converted to a job file using Perfactory RP version 2.1 proprietary software, which sliced the 3D solid model into a series of 2D layers of uniform thickness. A visible light beam with λ ≈ 475 nm which passes through the DLP projector, is modulated into the layer pattern, and focused onto the resin surface for the polymerisation of the exposed areas. Photo curable acrylic resin, R5 from EnvisionTec, was used. The fabricated scaffolds were washed with isopropanol in an ultrasonic bath several times to remove any uncured resin, then sterilised by soaking in 70% ethanol for 1 h then rinsing in warmed phosphate buffered saline prior to implantation. Chick chorioallantoic membrane culture of scaffolds Fertilised eggs were incubated until day 4 for the vascular bed of the chick to become large enough to support the scaf-

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fold. An air cavity was formed to allow cutting of a 2 cm2 window in the top of the egg without damaging the chorioallantoic membrane (CAM). The scaffolds were then placed on the membrane and the eggs sealed and incubated at 38◦ C in a humidified atmosphere for up to 6 days. At day 2, 4 and 6 pore-arrays were carefully removed together with the underlying membrane. Arrays and membrane were fixed in neutral buffered formalin for 24 h then heavy metal stained using a 1% solution of osmium tetroxide. Arrays were then rinsed extensively in distilled water for 2 days to remove any salt, soaked in 10% glycerol for a further day (to prevent excessive contraction of the drying tissue) before rinsing for 1 h and drying prior to X-ray µCT analysis. Scaffold histology Once the scaffolds were dry they were scanned for X-ray µCT 3D reconstruction and analysis (see below). Samples were then vacuum infiltrated with a stabilising CY212 resin before being sectioned to 10 µm transparent slices on a microtome for microscopy to verify the information seen using the µCT (Fig. 3c). Micro-computed tomographic (μCT) imaging of scaffolds Constructs were imaged using a high-resolution µCT system (Skyscan 1,174). Samples were scanned to produce 10 µm voxel resolution 3D Feldkamp algorithm reconstructed images for analysis. Scaffold and osmium tetroxide stained X-ray dense tissue locations within the arrays were analysed from the scan information using the CTAn software program. Tissue immuno-histochemistry Scaffold samples were fixed in formyl saline for 1 h before embedding in wax using standard histology procedures. 10 µm sections were taken to water and incubated with blocking solution before addition of a primary PCNA antibody. Sections were extensively washed of primary antibody and detected with a secondary antibody labelled with Fluorescein isothiocyanate (FITC). Nuclei were further highlighted with Hoescht nucleic acid staining solution. B Approximate solution methods In this section a method for deriving approximate solutions to the model equations is presented, the main simplifying assumption being that changes in the z coordinate over the free part of the membrane are small compared to the distance from the membrane to the phreatic surface, or equivalently, to a first order approximation the hydrostatic pressure acting behind the free part of the membrane is treated as being constant. Also the effects of tension on growth are neglected by


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taking T p in Eqs. (22) and (24) to be sufficiently small so that the growth rate of the membrane can be taken to be constant and equal to γ . A change of variables is made by substituting r = R(θ ) sin θ , where R is the radius of curvature, into Eqs. (17)– (19). Using the Chain Rule to eliminate the variable s between those equations gives

tan θ (Tc − R(z h − z)) R − (z h − z)R 2 + 2Tc R = 0, (B.1)

z + sin θ (R + tan θ R ) = 0,

(B.2)

where ( ) denotes differentiation with respect to θ . Equations (B.1) and (B.2) are to be solved subject to the boundary conditions at the contact line, which are that R = Rc = rc / sin θc and z = z c at θ = θc . The problem is to find Tc so that when (B.1) and (B.2) are integrated, finite values of R and z are obtained when θ = 0. The assumption that the variations in the hydrostatic pressure are small over the free membrane surface requires that rc z h − z c , 1. Thus the solutions are expressed as series in small powers of ε = rc , i.e. R = ε R1 + ε 2 R2 + · · · ,

(B.3)

z = z c + εz 1 + · · · ,

(B.4)

Tc = εT1 + ε2 T2 + · · · ,

(B.5)

which are substituted into Eq. (B.1) and (B.2). Expanding and collecting terms to leading order in ε gives tan θ (T1 − (z h − z c )R1 ) R1 − (z h − z c )R12 + 2T1 R1 = 0, (B.6)

z 1 + sin θ (R1 + tan θ R1 ) = 0,

Integrating this equation and enforcing the boundary condition R2 = 0 at θ = θc implies R2 =

2(cos3 θc −cos3 θ)+3(2T2 sin2 θc −cos θc )(cos2 θc −cos2 θ ) 3(z h − z c ) sin2 θc sin2 θ

(B.11) Because the denominator in (B.11) is zero at θ = 0, for limθ→0 R2 to be defined a necessary condition is that the numerator of Eq. (B.11) also vanishes at θ = 0. This requires T2 = − 16 (2 + cos θc )/(1 + cos θc )2 , thus Eq. (B.11) becomes R2 =

2(1−cos θ )[cos θ +cos θc (1+cos θ )](cos θ −cos θc ) 3(z h −z c ) sin2 θc (1+cos θc ) sin2 θ (B.12)

By l’Hôpital’s rule limθ→0 R2 = 13 (1 + 2 cos θc )/((z h − z c ) (1 + cos θc )2 ) which is finite as required. Substituting Eq. (B.12) into Eq. (B.9), integrating, and determining the constant of integration to enforce z 2 = 0 at θ = θc , it can be shown that 1 + cos θc 2 z 2 = 2 (1 + cos θc ) cos θ − ln 1 + cos θ 2 3(z h +(1 + cos θc + cos θc ) cos θ + cos θc −z c )(cos θc + 1) sin2 θc ,

(T1 −(z h −z c )R1 ) (tan θ R2 +2R2 )+2T2 R1 +z 1 R12 = 0, (B.8) z 2 + sin θ (R2 + tan θ R2 ) = 0.

(B.9)

z 2 (0) 2 cos θc (1 − cos θc ) + 2(1 + cos θc ) ln 21 (1 + cos θc ) = . 3(z h − z c )(1 + cos θc ) sin2 θc (B.14) The area of the free part of the membrane is θc B= 0

∂s dθ = 2π 2πr ∂θ

(z h − z c ) sin2 θc (sin2 θ R2 ) = 2(cos θc − 2T2 sin2 θc − cos θ ) cos θ sin θ.

(B.10)

θc 0

Tc R 2 sin θ dθ, R(z h − z) − Tc

(B.15)

where Eq. (18) has been used to substitute for ∂s/∂θ . Substituting the expansions given by Eqs. (B.3)– (B.5) into Eq. (B.15) and collecting terms in like powers of ε gives θc B =ε

2 0

+ε

After substituting the results for R1 , z 1 and T1 , Eq. (B.8) can be recast as

(B.13)

and therefore

(B.7)

which has the solution R1 = 1/ sin θc and z 1 = (cos θ − cos θc )/ sin θc , provided T1 = 21 (z h − z c )/ sin θc . To the next highest order in ε, the expansions of Eqs. (B.1) and (B.2) yield

.

2π T1 R12 sin θ R1 (z h − z c ) − T1

2π(2T2 R12 + 2T1 R1 R2 + R13 z 1 − R12 R2 (z h − z c )) sin θ T1

+···

dθ.

(B.16)

Substituting into this equation the previous results for R1 , T1 , R2 and T2 as functions of θ and evaluating the

123

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G. Lemon et al.

resulting definite integral gives 2π 2π(cos θc − 1)3 + ε3 1 + cos θc 3(z h − z c ) sin3 θc (cos θc + 1) +··· . (B.17)

B = ε2

During the initial phase of penetration of the membrane into the pore, where z c = 0 and there is no friction force acting, the membrane is expanding at a constant rate so B = 1 2 t 4 π d e . Equating this expression for B with that given by Eq. (B.17), where ε = rc = 21 d, it follows that (cos θc − 1)3 d 2 . + t ≈ ln 1 + cos θc 3 (z h − z c ) sin3 θc (cos θc + 1) (B.18)

This initial phase of penetration commences when θc = 0, at which time the right-hand side of Eq. (B.18) is zero. It terminates when θ = 21 π at t = td , at which time Eq. (B.18) gives d , (B.19) td ≈ γ −1 ln 2 − 3z h when expressed in terms of dimensionalised parameters. For the case of the cylindrical pore with w = d = 0.4 mm, described in Sect. 3.3, the approximate curve in Fig. 8 (curve with triangles) was calculated by evaluating Eq. (B.18) on a uniform grid of values of θc between 0 and 21 π then using z m ≈ εz 1 (0) + ε2 z 2 (0) i.e. zm ≈ d

1 − cos θc 2 sin θc

+d 2

cos θc (1 − cos θc ) + (1 + cos θc ) ln 21 (1 + cos θc ) 6(z h − z c )(1 + cos θc ) sin2 θc

,

(B.20) to calculate the axial displacement of the top of the membrane. Considering still the case of a cylindrical pore, once t ≥ td the membrane begins to move up into the pore, but the contact angle between the pore and the membrane remains constant at θc = 21 π . Substituting θc = 21 π into Eqs. (B.5) and (B.20) the expansions of Tc and z m become Tc ≈ 41 d(z h − z c ) − z m ≈ z c + 21 d −

1 6

1 2 12 d ,

ln 2

d2 zh − zc

(B.21) .

(B.22)

Next the motion of the membrane behind the contact line for t ≥ td for the case of the cylindrical pore is investigated. To make progress analytically the friction is treated as being constant by taking the limit δ f → 0 in Eq. (20). In this limit

123

Eqs. (20)–(22) reduce to 2 ∂T = μ zh − z − T , ∂z d ∂v = 1. ∂z

(B.23) (B.24)

Integrating Eqs. (B.23) and (B.24), and applying continuity of the tension at the contact point i.e. T = Tc at z = z c , gives the solution d d d zh − z + + Tc − zh − zc T = 2 2μ 2 d 2μ + (B.25) exp − (z − z c ) , 2μ d v = z, (B.26) for z p ≤ z ≤ z c , where z p is the value of z where T given by (B.25) becomes zero. Calculating Tc using Eq. (B.21), the approximations for T and v given by Eqs. (B.25) and (B.26) respectively, are plotted in Fig. 9 for the case of μ = 1 (lines with crosses). Note that T is constant and equal to Tc in the free part of the membrane for z c ≤ z ≤ z m , the value of z m being calculated using Eq. (B.22). An approximation for z p for large values of the friction coefficient, μ, is obtained by assuming z p − z c ∼ 1/μ for μ 1, which by expanding Eq. (B.25) in powers of 1/μ with z = z p implies to leading order 21 d(z h − z c ) + Tc − 21 d(z h − z c ) exp 2μ(z c − z p )/d = 0 i.e. 2Tc d ln 1 − . (B.27) z p ≈ zc + 2μ d(z h − z c ) Thus z p → z c as μ → ∞, and because the tension drops to zero in a vanishingly small distance behind the contact point, the growth of the membrane becomes confined to the free part. In the limit of large friction, μ → ∞, an equation of motion can be derived for the axial displacement of the contact line, z c , for a pore of any shape, as follows. In a small time δt the area of the free part of the membrane, B(z), with contact line at z = z c at time t, increases by an amount B(z c + δz c ) − B(z c ) ≈ B (z c )δz c , where the prime denotes differentiation with respect to z c . In the same amount of time, the area of the membrane behind the contact line increases by an amount 2πr (z c )δz c , while cell proliferation makes the total membrane area increase by an amount B(z c )δt. Conservation of membrane area thus requires B (z c )δz c + 2πr (z c ) δz c ≈ B(z c )δt. Dividing this equation by δ and taking the limit δt → 0 implies B(z c ) dz c = . dt B (z c ) + 2πrc (z c )

(B.28)

Considering again the case of the cylindrical pore, θc = is substituted into (B.17) to obtain a series approximation

1 2π


557

where 0 ≤ θc ≤ π/2. For t ≥ td , z c and z m are calculated using t ≈ td +

2z c d

(B.33)

and z m ≈ z c + 21 d.

Fig. 11 Graph showing a comparison of the full numerical solution with the approximate analytical solutions of the equations for the growth of the membrane into the pore. The axial displacements of the top of the free part of the membrane, z m (dotted line), and the contact line, z c (dashed line), are plotted as functions of time for the case of a cylindrical pore, w = d = 0.4 mm and a friction coefficient of μ = 100. Also shown are the curves of the approximate solutions in the limit μ → ∞, derived in Appendix B. The curve labelled ‘First approximation’ is a plot of the solution where only the leading order terms in the series solutions is used, whereas the curve labelled ‘Refined approximation’ includes in the solution the next highest order terms in the series

for B(z c ) in the small parameter ε = the result into (B.28) gives 1 1 dt + ··· = + dz c ε 3(z h − z c )

1 2 d,

and substituting

(B.29)

Solving this equation with the initial condition z c = 0 at t = td gives 1 zh − zc 2z c . (B.30) − ln t ≈ td + d 3 zh This solution is plotted in Fig. 8 for t > td (lines with triangles), where td is approximated using (B.19) and z m is calculated from z c using Eq. (B.22). To compare the approximate solutions with the numerical solution in more detail, Fig. 11 shows the numerical solutions of z m (dotted curve) and z c (dashed curve), for t ≤ 2, for the case of the cylindrical pore, i.e. w = d = 0.4 mm, and a friction coefficient of μ = 100. The curve labelled ‘First approximation’ is a plot of the approximate solution where only the leading order terms in the series are used to construct the solution. Thus for t < td , where z c = 0, Eq. (B.19) reduces to td ≈ γ −1 ln 2, and z m is calculated using 2 (B.31) t ≈ ln 1 + cos θc and zm ≈ d

1 − cos θc , 2 sin θc

(B.32)

(B.34)

The curve labelled ‘Refined approximation’ in Fig. 11 includes the next higher order terms in the series solution. Thus for t < td where td is given by Eq. (B.19), z c = 0, and z m is calculated using Eqs. (B.18) and (B.20). For t ≥ td , z c and z m are calculated using Eqs. (B.30) and (B.22). Comparing the two approximations in Fig. 11 shows that the refined approximation better approximates the numerical solution of z m , and also gives a better approximation to the numerical value of td . Although for t > td the refined approximation starts out more closely following the numerical solution, the curve of the approximate solution and that of the numerical solution eventually diverge (see also Fig. 9) because the calculation of the approximate solution is based on the assumption of there being infinite friction, but in the numerical solution the amount of friction is finite albeit large (μ = 100).

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