Modelling oxygen diffusion and cell growth in a porous, vascularising scaffold for soft tissue engineering applications

Chemical Engineering Science 60 (2005) 4924 – 4934 www.elsevier.com/locate/ces

Modelling oxygen diffusion and cell growth in a porous, vascularising scaffold for soft tissue engineering applications Tristan I. Crolla , Silke Gentzb , Kilian Muellerb , Malcolm Davidsona , Andrea J. O’Connora , Geoffrey W. Stevensa , Justin J. Cooper-Whitec,∗ a Department of Chemical and Biomolecular Engineering, The University of Melbourne, Victoria 3010, Australia b Lehrstuhl für Fluidverfahrenstechnik, Technische Universität München, Germany c Division of Chemical Engineering, The University of Queensland, St. Lucia QLD 4072, Australia

Received 15 July 2004; received in revised form 13 January 2005; accepted 18 March 2005

Abstract Soft tissue engineering presents significant challenges compared to other tissue engineering disciplines such as bone, cartilage or skin engineering. The very high cell density in most soft tissues, often combined with large implant dimensions, means that the supply of oxygen is a critical factor in the success or failure of a soft tissue scaffold. A model is presented for oxygen diffusion in a 15–60 mm diameter dome-shaped scaffold fed by a blood vessel loop at its base. This model incorporates simple models for vascular growth, cell migration and the effect of cell density on the effective oxygen diffusivity. The model shows that the dynamic, homogeneous cell seeding method often employed in small-scale applications is not applicable in the case of larger scale scaffolds such as these. Instead, we propose the implantation of a small biopsy of tissue close to a blood supply within the scaffold as a technique more likely to be successful. Crown Copyright 䉷 2005 Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion; Numerical model; Oxygen concentration; Oxygen uptake rate; Cell characteristics

1. Introduction Tissue engineering is the process of synthesising natural tissue for the repair or replacement of damaged or failed organs and tissues, for example cartilage (Freed et al., 1994), bone (Agrawal and Ray, 2001; Buma et al., 2004; Burdick and Anseth, 2002; Karp et al., 2003), skin (Groth et al., 2002), liver (Hasirci et al., 2001), muscle (Agrawal and Ray, 2001) or breast (Patrick, 2000, 2001). One of the most promising approaches to tissue engineering to date is the formation of a porous biodegradable, biocompatible polymer scaffold to provide a substrate for cell adhesion and proliferation, and to control the final shape of the regenerated tissue (Agrawal and Ray, 2001; Ganta et al., 2003; Gümüsderelioglu and Türkoglu, 2002; ∗ Corresponding author. Tel.: +61 7 3346 8715.

E-mail address: [email protected] (J.J. Cooper-White).

Hollister et al., 2002; Vyakarnam et al., 2002; Cao et al., 2004). Over time, the scaffold material, for example, poly(lactic-co-glycolic acid) (PLGA), degrades away, leaving nothing but native tissue. Regeneration of soft tissue is one of the most demanding applications of tissue engineering. In contrast to other tissue engineering applications such as bone, skin and cartilage, targets for soft tissue engineering such as liver, kidney, muscle or adipose tissue for cosmetic applications often have large volumes, and require a high degree of vascularisation. Even in the case of cortical bone and cartilage (both relatively acellular tissues), to date it has only been possible to grow functional tissue in vitro with a thickness less than a few hundred micrometres, due to the constraints of oxygen transport (Kellner et al., 2002). In a liver tissue engineering study, 5 mm thick PLGA foam scaffolds seeded with rat hepatocytes at a very modest initial density of 106 /mL and cultured under static conditions were found to

0009-2509/$ - see front matter Crown Copyright 䉷 2005 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.03.051

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

Polycarbonate chamber

Dome-shaped scaffold Arterio-venous loop Scaffold base Polycarbonate base Fig. 1. Patented in vivo tissue generation model of Cassell et al. (2001), consisting of a dome-shaped PLGA scaffold sandwiching a blood vessel loop, enclosed within a polycarbonate chamber.

lose 50% of their DNA content after 12 days (Hasirci et al., 2001). The lack of an adequate in vivo model to study such constraints has limited systematic investigations into scale-up of tissue engineering scaffolds. Recently an appropriate in vivo model has been developed to investigate angiogenesis and soft tissue generation (Fig. 1) (Cassell et al., 2001). An arteriovenous (A-V) loop is formed by dissecting a section of the femoral vein from the groin of a rat, and inserting this as a shunt between the femoral artery and vein in the opposite groin (Fig. 2). This is sandwiched between two pieces of a dome-shaped PLGA scaffold, and enclosed within a polycarbonate chamber to provide a degree of isolation from the surrounding tissue. Over time, capillaries and tissue grow from the A-V loop to fill the chamber. In tissue engineering applications such as bone or cartilage where the scaffold thickness is generally quite small, scaffolds are often pre-seeded with high densities of cells extracted from the patient (Freed et al., 1994; Burdick and Anseth, 2002). However, as the smallest dimension of a scaffold increases past a few millimetres, this approach becomes questionable, due to the constraints of oxygen diffusion. A method of predicting the effect of geometric and

4925

physiological parameters on growth rates is necessary for scaling up this process. A possibly more useful option is to do away with, or reduce to very low levels, homogeneous seeding, and instead implant a small biopsy of tissue adjacent to the blood vessel loop in a model such as the A-V loop model described above. Given a suitable environment, over time cells migrate from this tissue into the surrounding scaffold. While in vivo models are extremely important in proving the utility of tissue engineering approaches, it is very difficult to obtain real-time information on spatially dependent parameters such as oxygen concentration or the health of the cells. Thus, it would appear that a mathematical model is the best option to investigate the detailed microprocesses occurring in such scaffolds in vivo. It is widely accepted that the cell growth-limiting nutrient in most tissue engineering applications is oxygen (Kellner et al., 2002; Annesini et al., 2000; Balis et al., 1999; Chow et al., 2001; Hay et al., 2000). Thus, a mathematical model for oxygen distribution throughout an implanted scaffold is an important tool for the informed design of scaffold structures and implantation techniques. Cell migration and angiogenesis are complex, interrelated phenomena critical to the success or failure of a soft tissue scaffold. During the wound-healing process, soluble factors released by hypoxic cells are known to stimulate chemotaxis and capillary growth (Pettet et al., 1996). A simplified model of this behaviour has been previously formulated (Pettet et al., 1996),makingtheassumptionthatlocaloxygen concentration correlates to the distance to the nearest blood vessel. In this paper, we develop a numerical model for oxygen diffusion and tissue growth in the in vivo A-V loop chamber model for a number of physical situations, taking into account cellular oxygen usage and growth kinetics, simple models for cell migration and capillary growth, and the effects of cell density on the effective oxygen diffusivity in the scaffold. Furthermore, we present a brief review of the available data in the literature for growth and oxygen usage kinetics. Finally, we present a sensitivity analysis for the model parameters within physiologically realistic ranges. This allows us to draw some conclusions about the effectiveness of different seeding strategies.

Fig. 2. Implantation of a typical dome-shaped scaffold into the groin of a rat (reproduced with permission from the Bernard O’Brien Institute for Microsurgery).

4926

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

2. Model development

centration throughout. Thus,

2.1. Geometry

c(r, t) = c0 ,

Modelling the uniform outward growth of capillaries from the blood vessel loop in the system described in Fig. 1 is a very difficult task, due to the somewhat complex geometry of the loop. However, if we make the simplifying assumption that the capillaries grow outwards in a hemispherical fashion (Fig. 3), the oxygen concentration between the capillary “front” and the wall can be calculated by a relatively simple one-dimensional spherical diffusion calculation:
 jc 1 j 2 jc = 2 DO2 r − S, r0 (t) < r < r1 ; t > 0, jt r jr jr c(r, 0) = cin , r > r0 , (1) where r is the radial dimension, t is the time from implantation, c(r, t) is the concentration of oxygen within the scaffold, DO2 (r, t) is the diffusivity of oxygen within the medium, and S(r, t) is the rate of uptake of oxygen by the cells. For the sake of simplicity, it is assumed that the oxygen concentration throughout the scaffold is initially at a constant value cin . This assumption, coupled with the choice of initial value of r0 , is most valid in the direction perpendicular to the scaffold base. The inner radius, r0 (t) is defined as the capillary front, i.e., the radius to which capillaries have grown at time t (the inner boundary shown in Fig. 3). While it is known that oxygen concentration in blood varies between approximately 115 nmol/mL for arterial blood and 50 nmol/mL for venous blood, in representing the A-V loop configuration it has been assumed for the sake of simplicity that the blood oxygen concentration has a constant value c0 . Furthermore, it is assumed that the region within the capillary network, i.e., behind the capillary front, has this same oxygen con-

r0 (0)
r
r0 (t),

t > 0.

The growth of capillaries is a complex process, believed to be mainly driven by growth factors released by hypoxic cells within the avascular area (Pettet et al., 1996). However, quantitative data describing this behaviour is extremely difficult to determine. As a first approximation, it may be assumed that capillaries grow at a constant rate, leading to: r0 (t) = r0 (0) +
t,

r1, c1

Capillary front

r0, c0

dc (r1 , t) = 0. dr

(4)

The scaffolds used in the in vivo model have very highly interconnected pores, with a porosity of ca. 96% (Cao et al., 2004). Thus, it is reasonable to assume that the scaffold structure has little effect on the diffusion rate, and the scaffold can be treated as a two-phase system consisting of cells suspended in fluid. Wood et al. (2002) recently developed a detailed theoretical description of the effective diffusivity in cellular media. However, very little experimental data exist to support this model. Since the cell density within the avascular portion of the scaffolds is quite low during the period of interest, it can be assumed that bulk transport processes dominate compared to transmembrane transport, and so the effective diffusivity Deff (r, t) can be described by Maxwell’s solution for porous media (Wood et al., 2002): (5)

where ε , the volume fraction of cells within the scaffold, can be related to the cell diameter dcell and the cell density (r, t) by 1 3 (r, t). ε = dcell 6

(6)

Two similar models for oxygen consumption of cells are commonly used in modelling cell cultures under oxygenlimited conditions: the Michaelis–Menten (Hay et al., 2000, 2001; Fassnacht and Portner, 1999) and the Monod (Picioreanu et al., 1998) models, described by Vmax c(r, t) (r, t), (KS + c(r, t)) 
max c(r, t) + ms (r, t) . S(r, t) = YXS KS + c(r, t)

S(r, t) = Fig. 3. Simplified geometry used for modelling purposes. Capillaries are assumed to grow uniformly outwards from an initial hemisphere of radius equal to that of the blood vessel loop.

(3)

where
is the capillary growth rate. This may be the case, for example, in a sustained-release situation where the gradient in growth factor concentration is held relatively constant. At the outer boundary r1 , the wall of the chamber provides a physical barrier to oxygen diffusion. Thus, the outer boundary condition is

DO2 ,eff (r, t) 2(1 − ε (r, t)) = , DO2 2 + ε (r, t)

Chamber wall

(2)

(7) (8)

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

4927

Table 1 Characteristics of different cell lines. References are shown in brackets Cell line

max (s−1 )

Chondrocytes Baby hamster kidney Porcine epithelial Porcine hepatocytes Hepatocytes Rat hepatocytes

3.7 × 10−6 3.4 × 10−6 3.0 × 10−6 4.5 × 10−6 — —

Human umbilical vein endothelial Chinese hamster ovary Granulocytes Lymphocytes Erythrocytes Megakaryocytes Adipocytes Monocytes

— — — — — — — —

(Freed et al., 1994) (Jorjani and Ozturk, 1999) (Kussow et al., 1995) (Balis et al., 1999)

Here Vmax is the maximum oxygen consumption rate, KS is the saturation, or half-rate constant, defined as the oxygen concentration at which the consumption rate drops to half of its maximum value, max is the maximal cell specific growth rate, YXS is the yield of cells per unit oxygen, and ms is the maintenance coefficient, the minimum oxygen consumption required to keep the cells alive. While both models are of essentially the same form, the Monod model has the advantage that the oxygen consumption rate can be directly coupled to the cell growth rate as
 S(r, t) (9) (r, t) = YXS − mS . (r, t) Note that this equation has no inhibition term related to cell density, and is hence not self-limiting to a physically realistic maximum cell density. However, when combined with Eq. (5) the cell growth rate at high cell density becomes restricted by very low oxygen diffusivity, and hence the density is limited to a physically realistic level. Cell migration and capillary growth are known to be quite complex and interdependent (Pettet et al., 1996; Calabrese, 2001; Franz et al., 2002). As mentioned previously, however, while models exist for this behaviour, very little experimental data are available. In the absence of any chemoattractants or haptotactic agents, it is reasonable to assume that cell migration is a random process, and hence may be described by the equation
 j j 1 j (10) = 2 Dcell r 2 + , r0 (0) < r < r1 , jt r jr jr

(r, 0) = f (r), (r0 (0), t) = max , j (r1 , t) = 0, jr

Vmax (nmol/s) (106 cells)−1

Cell diameter (
m)

— — — 0.25–0.91 (Balis et al., 1999) 0.21–0.23 (Annesini et al., 2000) 0.28–0.56 (Wu et al., 1996) 0.15–0.43 (Foy et al., 1994) 0.33–0.35 (Balis et al., 1999) 0.03–0.05 (Steinlechner-Maran et al., 2000) 0.071–0.079 (Ducommun et al., 2000) 0.006–0.18 (Chow et al., 2001) 0.0002–0.014 (Chow et al., 2001) 0.0005–0.002 (Chow et al., 2001) 0.4 (Chow et al., 2001) 0.09 (Chow et al., 2001) —

— — — — — 20 (Wu et al., 1996)

— — 10–25 (Chow et al., 2001) 9–20 (Chow et al., 2001) 6.7–8 (Chow et al., 2001) 30–160 (Chow et al., 2001) 50–300 (Chow et al., 2001) 12–50 (Chow et al., 2001)

where Dcell is an effective “diffusion coefficient” for cells and f (r) is a function describing the cell seeding profile. While in the case of oxygen diffusion, the concentration is constant behind the capillary front, this is not the case for cell migration, and hence  must be calculated over the entire region r0 (0) < r < r1 . 2.2. Parameter values 2.2.1. Maximal cell growth rate max Values for the maximal specific cell growth rate, max , were estimated for various cell lines from data presented in the literature. Values were taken from the exponential growth phase. These are summarised in Table 1. 2.2.2. Oxygen uptake rate Data for cellular specific oxygen uptake rate in the literature are generally reported as the value Vmax from the Michaelis–Menten equation, Eq. (7). This can be adapted to the Monod mechanism via Eq. (11) if max and the maintenance coefficient, ms , are known or estimated. Data for various cell types are summarised in Table 1. YXS =

max . Vmax − ms

(11)

The half-rate constant KS in Eqs. (7) and (8) ranges between 0.17 and 5.9 mmHg (0.2 and 7.0 nmol/mL) for human cells (Chow et al., 2001). Values for rat (Foy et al., 1994) and porcine (Balis et al., 1999) hepatocytes have been reported as 4.2 and 6.0 nmol/mL, respectively. The maintenance coefficient, ms in Eqs. (8) and (9), is defined as the minimum amount of oxygen per unit time the cells need simply to survive. For mammalian cells, it can generally be assumed that this is very low or negligible, since under hypoxic conditions mammalian cells are able to shift to an anaerobic metabolic pathway, while slowing their

4928

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

metabolism by as much as 50% (Boutilier and St-Pierre, 2000). However, it must be kept in mind that under these conditions production of lactic acid increases significantly, with a concomitant increase in the rate of glucose consumption. This is further discussed in the results and discussion section. 2.2.3. Cell volume An important parameter in the calculation of the effective diffusivity, Deff , through ε as described in Eq. (5), is the cell volume, which may be estimated via the cell diameter dcell . Cell diameters range from as low as 7
m for red blood cells (Chow et al., 2001) to as high as 300
m for adipocytes (Chow et al., 2001). “Average” cells are considered to be 10–20
m in diameter. Data from the literature for various cell lines are listed in Table 1. 2.2.4. Cell migration Tjia and Moghe (2002) reported a “population averaged cell migration rate” for human keratinocytes on PLGA films, with values ranging between 2 and 50
m2 /min, or 3.3 × 10−10 and 8.3 × 10−9 cm2 /s. As a first estimate, a cell “diffusion coefficient”, Dcell , of 5.0 × 10−9 cm2 /s was used. 2.2.5. Capillary growth rate Based on early experimental results (Cao et al., in preparation) in which a scaffold of 15 mm diameter was fully infiltrated with a low density vascularised tissue within 5–10 days, the one-dimensional capillary growth rate has been estimated at approximately 0.5–1 mm/day (0.6–1.2 × 10−3
m/s). The choice of the starting capillary radius r0 (0) is somewhat arbitrary. In a real situation, the diameter of the blood vessel loop and thickness of the blood vessels themselves varies from animal to animal. Typical blood vessels may vary from approximately 0.05 cm (rat) up to a few millimetres (pig or human) in diameter. For the purposes of this analysis, r0 (0) has been fixed at a value of 0.25 cm, and only the outer radius r1 varied.

3. Computational methods The oxygen diffusion through the scaffold is a typical moving boundary problem, and hence the spatial coordinate r was transformed to a new dimensionless coordinate, y, in order to simplify the numerical solution: r − r0 y= . r1 − r 0 Thus, Eq. (1) becomes 

jc jc jc 1−y jr0 1 j = 2 2 Deff r 2 −S+ , jt r h jy jy jy r1 −r0 jt

(12)

(13)

where h = r1 − r0 and 0
y
1. Thus the step size in y is invariant with respect to t, whereas the step size in r changes as t increases. Since the cell distribution for mobile cells was calculated over the entire range r0 (0) < r < r1 , Eq. (10) is unchanged, and is solved directly for r and t. For the purposes of numerical simulation, Eq. (13) can be approximated at time n and space i via finite difference methods by 
2 cin+1 − cin 1 n n n D 1 r 1 ci+1 = − cin 2 n 2 n 2 i+ i+ t (h ) (ri ) y 2 2 
2
2  × Dn 1 r n 1 + Dn 1 r n 1 i+ 2

+ Dn

rn

1 i− 2

+

i+ 2




2

1 i− 2



i− 2

i− 2

n − Sin ci−1

n − cn )
(1 − yi ) (ci+1 i−1 , hn 2y

(14)

where D refers to Deff,O2 , the superscripts denote the time points and the subscripts denote the spatial points. The subscripts i + 21 or i − 21 indicate the value at a point midway between spatial points. Sin was calculated for each spatial point i from Eq. (8). The central differencing scheme used for the advection term has second-order accuracy and is stable as long as the diffusion term is large. Rearrangement yields an explicit solution for the next time point:  cin+1



1

= cin +t

(hn )2 (rin )2 y 2 
2 n n × D 1 r 1 + Dn i+ 2

2


× rn

1 i− 2

n ci−1







rn 1 i+ 2

n ci+1 − cin

+ Dn

1 i− 2


(1 − yi ) −Sin + n h

2

2 

rn 1 i− 2

1 i− 2

i+ 2



Dn 1 i+ 2

n −cn ) (ci+1 i−1

2y

 .

(15) Eq. (10) can be approximated via a directly analogous method to Eq. (13), however, in this case the diffusion coefficient Dcell is assumed to be constant and remains as a single term: 

n+1 i

= ni +t

Dcell




(rin )2 r 2 2 

n + rn + r 1 i− 2

rn 1 i+ 2

1 i− 2

2

2

ni+1 −ni






ni−1

 + ni

rn 1 i+ 2

.

2

(16)

The value of  at each spatial point for the calculation of c and the value of c at each spatial point for the calculation of  were calculated using cubic interpolation of the respective profiles.

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

The spatial domain 0
y
1 was subdivided into 50 intervals in the case of homogeneous seeding, and 100 intervals in the case of heterogeneous seeding. Numerical stability was ensured by choosing the time steps such that

t = 0.25 MIN

2

hy 1 h2 y . ,
2 DN+1

(17)

Note that in all cases simulated, the oxygen diffusivity at the outer radius, DN+1 , had the highest value, since this point always had the lowest cell density. The described model was implemented using Fortran 77. Numerical accuracy of the solution was tested by carrying out a limited number of simulations using 100, 200 or 300 spatial steps. No significant differences in the solutions were noted in these cases.

4. Results and discussion The most important factor in the success or failure of a scaffold is arguably the length of time that cells remain under hypoxic conditions. Under hypoxic conditions, cells shift to an anaerobic metabolic pathway, converting glucose into lactic acid (Boutilier and St-Pierre, 2000). This process yields only 2 ATP molecules (the basic energy unit in a cell) per glucose molecule, whereas under aerobic conditions one glucose molecule can theoretically yield up to 36 molecules of ATP (Stephanopoulos et al., 1998). Thus, under hypoxic conditions glucose utilisation quickly increases, causing a drop in serum glucose concentration, coupled with an increase in lactate concentration and subsequent decrease in pH. If the lactate concentration becomes too high or the glucose concentration drops too low, cellular ATP is depleted to the point where transmembrane ion gradients can no longer be maintained, and necrosis quickly occurs (Boutilier and St-Pierre, 2000). In the complete absence of oxygen (anoxia), this can occur within minutes, however, under more mild hypoxic conditions, cells may survive for up to a few hours (Boutilier and St-Pierre, 2000). For example, it has been found that under hypoxic conditions more than 80% of pre-adipocytes become non-viable in in vitro culture within 3 hours (Patrick, 2000). Lactate has been shown to decrease the growth rate of baby hamster kidney (BHK) cell cultures, leading to negative growth rates at concentrations above approximately 35 mM (Cruz et al., 2000). While tolerance of hypoxia differs widely between cell types, fibroblasts and pre-adipocytes are particularly susceptible, becoming necrotic after only a few hours under hypoxic conditions. Thus, while short periods of hypoxia are acceptable and even desirable in that they stimulate the release of vasculogenic factors (Pettet et al., 1996), longer periods of more than a few hours are indicative of large-scale necrosis.

4929

Table 2 Parameters used for sensitivity analysis of the homogeneously seeded model. Each parameter was tested with all other parameters held fixed at the middle values. r0 (0) was held constant at 0.25 cm

initial (cells/mL)

c0 (nmol/mL) max (s−1 )
(cm/day) YXS (cells/nmol) KS (nmol/mL) r1 (cm) dcell (cm) ∗r

Low value

Middle value

High value

105 47.6 10−6 0.01 10 0.2023 0.75* 10−3

5 × 105 80 5 × 10−6 0.05 20 3.6176 1.5 2 × 10−3

2 × 106 113.05 10−5 0.1 100 7.021 3 10−2

1 was fixed at 0.75 cm for testing of other parameters.

4.1. General observations For all model parameters used, infiltration of the scaffolds showed two distinct phases. During the first phase, the capillary network grows out to fill the scaffold, but the cell density in the outer region remains very low compared to native tissue. Depending on the model parameters, this phase lasted for a few days up to approximately two weeks. In the second (consolidation) phase the scaffold is fully vascularised, and hence cells proliferate without oxygen limitation, eventually forming fully consolidated tissue. Only the beginning of this phase was modelled. In vivo trials using 15 mm diameter dome-shaped scaffolds (Cao et al., in preparation) in rats showed that after one week a loose network of capillaries had penetrated throughout the scaffold, indicating that the vascularisation phase was complete. The second, consolidation phase in these trials appeared complete after approximately 4–6 weeks. 4.2. Homogeneously seeded scaffolds It is often assumed in the literature that homogeneous seeding of cells within scaffolds, for instance via the use of a stirred bioreactor, is the best option. This may indeed be the case for scaffolds of small dimensions and low vascularity (e.g. articular cartilage or bone). However, its applicability to scaffolds larger than a few millimetres in their smallest dimension is somewhat more questionable. The oxygen concentration and cell density at three radial points within the scaffold, calculated by the model as a function of time using the parameters in the middle column of Table 2, are shown in Fig. 4. While cells close to the blood vessel loop (r = 0.35 cm) have ample oxygen available and hence proliferate without limit, cells at any greater distance (r = 0.55 cm or 0.75 cm) remain under hypoxic conditions until the capillary front comes within approximately 1–2 mm of their position, a period of at least a few days. This is a highly undesirable situation, indicating that necrosis throughout the majority of the scaffold is the likely outcome of this seeding method.

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934 1.00E+07

90 80 70 60 50

1.00E+06

40 30 20 10

cell density (cells/ml)

O2 concentration (nmol/ml)

4930

1.00E+05

0 0

1

2

3

4

5

6

time (days)

Fig. 4. Model output using middle values of parameters from Table 2. Solid symbols represent oxygen concentration, while open symbols represent cell density. Radius = 0.35 cm(), 0.55 cm() and 0.75 cm(). Note the extended hypoxic period at the outer periphery.

In order to calculate the effective diffusivity as a function of cell density, Eq. (5), treats cells as non-permeable objects, the most pessimistic approach available. However, even under these conditions, the effect of the cell density on the effective diffusivity of the scaffold only becomes apparent for very high cell loadings. For small cells of 10 or 20
m diameter, the effect of the diffusivity model on the outcome is generally negligible even for quite high numerical cell densities of 106 − 107 cells/mL. Therefore, it can be concluded that in most cases the effect of cell density on the effective diffusion coefficient can be neglected, since cell densities only become high enough to have an effect during the consolidation phase, when the scaffold is fully vascularised. A sensitivity analysis was carried out on the major model parameters within physiologically realistic ranges (Table 2). Each parameter was tested with all other parameters except r1 held constant at the middle values. A value of 0.75 cm (the radius of the scaffolds used in in vivo trials (Cao et al., in preparation)) for r1 was used for testing the other parameters. We defined the hypoxic period as the period for which the serum oxygen concentration remained below 5 nmol/mL, the approximate half-rate constant for most mammalian cell types. Fig. 5 shows the hypoxic period for each of these combinations. As can be seen, there was no physiologically realistic combination of parameters which gave a hypoxic period at a radius of 7.5 mm (5 mm from the blood vessel loop) of less than 3 days. At a radius of 5 mm only the use of a very low cell seeding density of 105 cells/mL or the use of a slow-growing or efficient (high YXS ) cell line gave no hypoxic period. This is in reasonably good agreement with experimental results for non-vascularised bovine cartilage tissue and polymer scaffolds found in the literature (Kellner et al., 2002), where cells more than approximately 2 mm from oxygenated medium were found to be hypoxic. 4.3. Heterogeneous seeding Heterogeneous seeding is a fundamentally different approach to scaffold implantation. Rather than soaking the

scaffold in a suspension of cultured or extracted cells, a small biopsy of the desired tissue is taken from elsewhere in the body and placed adjacent to the blood vessel loop during implantation. Given a suitable environment, cells migrate from this tissue to invade the surrounding scaffold, followed closely by capillaries. The seeding situation described above was modelled as a step change in cell density from 90% of the theoretical maximum cell density (native tissue) to a minimal density of 1 cell/mL at a distance of 0.3 mm from the blood vessel loop. This minimal, but non-zero, density was necessary to avoid divide-by-zero errors during numerical simulation. Fig. 6 shows the calculated oxygen concentration and cell density profiles after two days for a cell diffusivity of 5 × 10−9 cm2 /s and a range of capillary growth rates, with all parameters except seeding density held fixed at the values shown in the middle column of Table 2. The oxygen profile was very strongly dependent on capillary growth rate; a fairly modest increase from 0.05 to 0.07 cm/day had a profound impact on the oxygen concentration in the outer regions of the scaffold. However, the cell density profile showed relatively little dependence on
, indicating that cell migration, rather than proliferation, is the dominating factor at this early stage of infiltration. The cell distribution as a function of radius and time under the same conditions is shown in Fig. 7, and the oxygen profile in Fig. 8 . The capillary growth rate again had very little effect on the cell density profile (not shown), indicating that over the first few days of infiltration cell migration, rather than proliferation, is by far the dominant controlling factor. After 4 days, infiltration of cells reaches the periphery of the scaffold, albeit to a very low density. Consolidation of the tissue occurs over the next few weeks. This correlates quite well with histological examination of scaffolds tested in vivo in rats (Cao et al., in preparation) as discussed above. In contrast to the cell density profile, the effect of capillary growth rate on oxygen concentration was once again profound. At a capillary growth rate of 0.05 cm/day (Fig. 8(a)), cells at the periphery of the scaffold remain hypoxic for extended periods of time, an undesirable physiological situation. However, given a slightly higher capillary growth rate of 0.06–0.07 cm/day (Fig. 8(b),(c)) it can be seen that, after a relatively short period of hypoxia while the blood vessels re-infiltrate the implanted tissue, the oxygen levels in the outer area of the scaffold increase sharply, even with quite high cell densities of > 106 cells/mL at the capillary front. Even this relatively short induction period may be too long for some cell types, however. In addition, in larger scaffolds (for example scaffolds for breast reconstruction) it is desirable to seed with a larger volume of tissue, to minimise the number of cell generations required to completely fill the scaffold. Under these conditions, it is no longer possible to ensure that all cells are at an adequately small distance from the blood supply. An ideal approach here would be to transplant a piece of tissue with an intact blood vessel

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

4931

7

Hypoxic period (days)

6 5 High value

4

Middle value 3

Low value

2 1 0

(a)

co

dcell

Ks




max

r1

initial

Yxs

7

Hypoxic period (days)

6 5 High value

4

Middle value 3

Low value

2 1 0

(b)

co

dcell

Ks




max

r1

initial

Yxs

Fig. 5. Effect of different parameters on the period for which oxygen concentration remained below 5 nmol/mL at (a) 5 mm or (b) 7.5 mm from the centre of a homogeneously seeded 7.5 mm radius scaffold. Simulations were run for a 7-day period.

network, and attach this to the blood supply at the implant site, ensuring that all cells have adequate oxygen from the start. While this is a simple procedure for some tissue types such as muscle, it is somewhat more difficult for less structured tissues such as fat (unpublished data). The behaviour shown here, in which the capillary front slowly overtakes the invading cells, is not physiologically realistic in a wound-healing situation (Pettet et al., 1996). Under these conditions, a complex interrelation is known to exist in which the early invading cells (mostly macrophages) become hypoxic and release growth factors to encourage ingrowth of capillaries. However, recent evidence (Cronin et al., 2004) shows that, given a suitable environment and stimulation by growth factors, capillaries can be encouraged to grow even in the absence of any other cell type. The results shown here indicate that vascularisation rate is by far the most important factor in the success or failure of a large tissue engineered scaffold, and hence that any such scaffold should ideally actively encourage the growth of capillary endothelial cells. Of course, tissue engineering scaffolds used in any real situation would often have a geometry quite different from the idealised model described here, and would most likely not be sealed at the outer surface. Human-scale scaffolds (for example, for breast reconstruction) may have linear dimensions more than ten times greater than the 7.5 mm radius

scaffolds considered throughout most of this analysis. The oxygen diffusion limitation and importance of rapid vascularisation under these conditions will hence be even more profound than described here. Cell migration also is not completely random, but may be influenced by gradients in oxygen, nutrients, extracellular matrix components, and growth factors (Calabrese, 2001; Franz et al., 2002). Random migration, however, appears to be a reasonable assumption during the early infiltration of the scaffold. It is interesting to note the apparent agreement between estimates for capillary growth rate (Cao et al., in preparation) and cell migration rate (Tjia and Moghe, 2002) obtained from two independent sources in a model considering only random migration.

5. Conclusions Due to the extreme difficulty in collecting reliable data relating to cell growth and infiltration in an in vivo situation, mathematical modelling appears at present to be the only available option to probe the relative importance of various parameters. We have developed a model for cell growth and oxygen diffusion during the early stages of implantation of a soft tissue scaffold which, while relatively simple in design, appears to re-create the behaviour observed from

4932

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934 80

λ=0.1 c (nmol/mL)

70

c (nmol/mL)

60 50 λ=0.07

40 30 20 10

λ=0.05 cm/day 0.35

0.45

0.55

0.65

0.75

r (cm) c (nmol/mL)

1e9 1e8 1e7 1e6 1e5

λ=0.1

1e4 1e3

0.5

1

0.35

0.45

(b)

0.55

0.65

0.75

r (cm)

Fig. 6. Effect of capillary growth rate on (a) oxygen profile and (b) cell density after 2 days simulated growth in the migration model. Dcell = 10−8 cm2 /s all other parameters at their average values.

c (nmol/mL)

1e0 0.25

90 80 70 60 50 40 30 20 10 0

r=0.3

90 80 70 60 50 40 30 20 10 0

r=0.3 0.35

2

0.5

0.35

2.5

1

0.4

1.5 2 Time (day)

0.4

3

3.5

0.45

0.5+

2.5

3

3.5

2.5

3

3.5

0.45+

0.35 r=0.3 0

(c)

1.5

Time (day)

(b)

1e1

0.4

0.5+

0

λ=0.05 cm/day

1e2

1e9 1e8 1e7 1e6 1e5 1e4 1e3 1e2 1e1 1e0

0.35

0.45

0

(a)

 (cells/mL)

r=0.3

(a)

0 0.25

 (cells/mL)

90 80 70 60 50 40 30 20 10 0

0.5

1

1.5

2

Time (day)

Fig. 8. Effect of capillary growth rate
on oxygen distribution. (a)
= 0.05 cm/day, (b)
= 0.06 cm/day, (c)
= 0.07 cm/day. Dcell = 5 × 10−9 cm2 /s, dcell = 20
m, all other parameters at average values.

0.4 0.45 0.5 0.55 0.6 0.65

0

0.5

1

1.5

2

2.5

0.7 3

0.75 3.5

Time (day)

Fig. 7. Cell density as a function of time and radius calculated using the migration model. Dcell = 5 × 10−9 cm2 /s, dcell = 20
m,
= 0.6 mm/day, all other parameters at average values.

histological evaluation of explanted scaffolds (Cao et al., in preparation). Hence, this model appears to be a powerful tool for the prediction of optimal strategies for implantation of soft tissue scaffolds. The model incorporates established relationships for cellular oxygen usage and growth, as well as more recent developments such as the dependence of the effective diffusivity on the cell density. However, the model indicates that under the conditions found during the early stages of infiltration, the effect of the cell density on the diffusion coefficient can be ignored. In addition, we have incorporated a simple model for cell migration based on Fick’s law. The range of values for the diffusion coefficient used to

describe cell migration was obtained from Tjia and Moghe (2002) and appears to correlate strongly to the approximate capillary growth and cell migration rate observed from in vivo results (Cao et al., in preparation). In the past the standard approach to the problem of introducing tissue into scaffolds has been to pre-seed them homogeneously using a cell suspension. While this appears to be a viable approach for thin scaffolds, such as cartilage, dermis and in some cases bone, this model demonstrates that, even at relatively low seeding densities, large scaffolds homogeneously pre-seeded with cells will invariably fail. Cells more than a few millimetres from the nearest blood vessel experience hypoxic conditions for at least 3 days, which may be expected to lead to high levels of necrosis. Implantation of a large scaffold with a small amount of native tissue adjacent to the blood vessel(s) appears to be the most feasible option for large-scale soft tissue engineering. Furthermore, the results shown here lend weight to the view that early and fast vascularisation is the most important

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

factor in the success or failure of a large-scale tissue engineering scaffold. Thus, efforts to tailor the scaffold environment to improve tissue growth should include components aimed at stimulating the growth of vascular endothelial cells as a matter of course. Once this early vascularisation has occurred, the migration of cells into the scaffold and consolidation into functional tissue can occur with little or no oxygen limitation.

6. Notation c c0 dcell Dcell DAB Deff h KS ms r r0 r1 S t Vmax y YXS

oxygen concentration, nmol/mL oxygen concentration in blood, nmol/mL diameter of a cell,
m cell diffusion coefficient, cm2 /s diffusivity of oxygen in plasma, cm2 /s effective diffusivity of oxygen in scaffold, cm2 /s r1 − r0 , cm Michaelis–Menten or Monod half-rate constant, nmol/mL maintenance coefficient, nmol/cell/mL/s radial distance from centre of chamber base, cm inner radius, cm outer radius, cm oxygen uptake rate, nmol/mL/s time, s maximum specific oxygen uptake rate, nmol/s/cell dimensionless spatial coordinate, dimensionless yield of cell biomass from oxygen, cell/nmol

Indices C i i+1 i + 21 i−1 i − 21 n n+1 n−1 S

cell value of parameter at current spatial step value of parameter at next spatial step value of parameter between current and next spatial step value of parameter at previous spatial step value of parameter between previous and current spatial step value of parameter at current time step value of parameter at next time step value of parameter at previous time step substrate, i.e., oxygen (original Monod equation)

Greek letters ε




volume fraction of cells in scaffold, dimensionless capillary growth rate, cm/day

 max 

4933

cell specific growth rate, s−1 maximum cell specific growth rate, s−1 cell density, cells/mL

Acknowledgements The input and insight of Professor Wayne Morrison (Director) and other members of the Bernard O’Brien Institute for Microsurgery, St. Vincent’s Hospital, Melbourne, and their permission to use Fig. 2 is gratefully acknowledged. Funding support from the Australian Research Council, without which this work could not have been completed, is also received with thanks. References Agrawal, C.M., Ray, R.B., 2001. Biodegradable polymeric scaffolds for musculoskeletal tissue engineering. Journal of Biomedical Materials Research 55 (2), 141–150. Annesini, M.C., Castelli, G., Conti, F., De virgiliis, L.C., Marrelli, L., Miccheli, A., Satori, E., 2000. Transport and consumption rate of O-2 in alginate gel beads entrapping hepatocytes. Biotechnology Letters. 22 (10), 865–870. Balis, U.J., Behnia, K., Dwarakanath, B., Bhatia, S.N., Sullivan, S.J., Yarmush, M.L., Toner, M., 1999. Oxygen consumption characteristics of porcine hepatocytes. Metabolic Engineering 1 (1), 49–62. Boutilier, R.G., St-Pierre, J., 2000. Surviving hypoxia without really dying. Comparative Biochemistry and Physiology a-Molecular and Integrative Physiology 126 (4), 481–490. Buma, P., Schreurs, W., Verdonschot, N., 2004. Skeletal tissue engineering—from in vitro studies to large animal models. Biomaterials 25, 1487–1495. Burdick, J.A., Anseth, K.S., 2002. Photoencapsulation of osteoblasts in injectable RGD-modified PEG hydrogels for bone tissue engineering. Biomaterials 23 (22), 4315–4323. Calabrese, E.J., 2001. Cell migration/chemotaxis: biphasic dose responses. Critical Reviews in Toxicology 31 (4–5), 615–624. Cao, Y., Davidson, M.R., O’Connor, A.J., Stevens, G.W., CooperWhite, J.J., 2004. Architecture control of three-dimensional polymeric scaffolds for soft tissue engineering I. establishment and validation of numerical models. Journal of Biomedical Materials Research A 71 (1), 81–89. Cao, Y., Mitchell, G., Messina, A., Penington, A.J., Morrison, W., O’Connor, A.J., Stevens, G.W., Cooper-White, J.J., The influence of architecture on degradation and vascularisation of three dimensional poly(lactic-co-glycolic acid) scaffolds in vitro and in vivo, in preparation. Cassell, O.C.S., Morrison, W.A., Messina, A., Penington, A.J., Thompson, E.W., Stevens, G.W., Perera, J.M., Kleinman, H.D., Hurley, J.V., Romeo, R., Knight, K.R., 2001. The influence of extracellular matrix on the generation of vascularized, engineered, transplantable tissue. Annals of the New York Academy of Sciences 944, 429–442. Chow, D.C., Wenning, L.A., Miller, W.M., Papoutsakis, E.T., 2001. Modeling pO2 distributions in the bone marrow hematopoietic compartment. I. Krogh’s model. Biophysical Journal 81 (2), 675–684. Cronin, K.J., Messina, A., Knight, K.R., Cooper-White, J.J., Stevens, G.W., Penington, A.J., Morrison, W.A., 2004. New murine model of spontaneous autologous tissue engineering, combining an arteriovenous pedicle with matrix materials. Plastic and reconstructive surgery 113 (1), 260–269. Cruz, H.J., Freitas, C.M., Alves, P.M., Moreira, J.L., Carrondo, M.J.T., 2000. Effects of ammonia and lactate on growth, metabolism, and

4934

Tristan I. Croll et al. / Chemical Engineering Science 60 (2005) 4924 – 4934

productivity of BHK cells. Enzyme and Microbial Technology 27 (1–2), 43–52. Ducommun, P., Ruffieux, P.A., Furter, M.P., Marison, I., von Stockar, U., 2000. A new method for on-line measurement of the volumetric oxygen uptake rate in membrane aerated animal cell cultures. Journal of Biotechnology 78 (2), 139–147. Fassnacht, D., Portner, R., 1999. Experimental and theoretical considerations on oxygen supply for animal cell growth in fixed-bed reactors. Journal of Biotechnology 72 (3), 169–184. Foy, B.D., Rotem, A., Toner, M., Tompkins, R.G., Yarmush, M.L., 1994. A device to measure the oxygen-uptake rate of attached cells—importance in bioartificial organ design. Cell Transplantation 3 (6), 515–527. Franz, C.M., Jones, G.E., Ridley, A.J., 2002. Cell migration in development and disease. Developmental Cell 2 (2), 153–158. Freed, L.E., Vunjaknovakovic, G., Marquis, J.C., Langer, R., 1994. Kinetics of chondrocyte growth in cell-polymer implants. Biotechnology and Bioengineering 43 (7), 597–604. Ganta, S.R., Piesco, N.P., Long, P., Gassner, R., Motta, L.F., Papworth, G.D., Stolz, D.B., Watkins, S.C., Agarwal, S., 2003. Vascularization and tissue infiltration of a biodegradable polyurethane matrix. Journal of Biomedical Materials Research 64A, 242–248. Groth, T., Seifert, B., Malsch, G., Albrecht, W., Paul, D., Kostadinova, A., Krasteva, N., Altankov, G., 2002. Interaction of human skin fibroblasts with moderate wettable polyacrylonitrile-copolymer membranes. Journal of Biomedical Materials Research 61 (2), 290–300. Gümüsderelioglu, M., Türkoglu, H., 2002. Biomodification of non-woven polyester fabrics by insulin and RGD for use in serum-free cultivation of tissue cells. Biomaterials 23, 3927–3935. Hasirci, V., Berthiaume, F., Bondre, S.P., Gresser, J.D., Trantolo, D.J., Toner, M., Wise, D.L., 2001. Expression of liver-specific functions by rat hepatocytes seeded in treated poly(lactic-co-glycolic) acid biodegradable foams. Tissue Engineering 7 (4), 385–394. Hay, P.D., Veitch, A.R., Smith, M.D., Cousins, R.B., Gaylor, J.D.S., 2000. Oxygen transfer in a diffusion-limited hollow fiber bioartificial liver. Artificial Organs 24 (4), 278–288. Hay, P.D., Veitch, A.R., Gaylor, J.D.S., 2001. Oxygen transfer in a convection-enhanced hollow fiber bioartificial liver. Artificial Organs 25 (2), 119–130. Hollister, S.J., Maddox, R.D., Taboas, J.M., 2002. Optimal design and fabrication of scaffolds to mimic tissue properties and satisfy biological constraints. Biomaterials 23, 4095–4103. Jorjani, P., Ozturk, S.S., 1999. Effects of cell density and temperature on oxygen consumption rate for different mammalian cell lines. Biotechnology and Bioengineering 64 (3), 349–356.

Karp, J.M., Schoichet, M.S., Davies, J.E., 2003. Bone formation on twodimensional poly (DL-lactide-co-glycolide) (PLGA) films and threedimensional PLGA tissue engineering scaffolds in vitro. Journal of Biomedical Materials Research 64A, 388–396. Kellner, K., Liebsch, G., Klimant, I., Wolfbeis, O.S., Blunk, T., Schulz, M.B., Göpferich, A., 2002. Determination of oxygen gradients in engineered tissue using a fluorescent sensor. Biotechnology and Bioengineering 80, 73–83. Kussow, C.M., Zhou, W.C., Gryte, D.M., Hu, W.S., 1995. Monitoring of mammalian-cell growth and virus production process using online oxygen-uptake rate measurement. Enzyme and Microbial Technology 17 (9), 779–783. Patrick, C.W., 2000. Adipose tissue engineering: the future of breast and soft tissue reconstruction following tumor resection. Seminars in Surgical Oncology 19 (3), 302–311. Patrick, C.W., 2001. Tissue engineering strategies for adipose tissue repair. Anatomical Record 263 (4), 361–366. Pettet, G.J., Byrne, H.M., Mcelwain, D.L.S., Norbury, J., 1996. A model of wound-healing angiogenesis in soft tissue. Mathematical Biosciences 136 (1), 35–63. Picioreanu, C., van Loosdrecht, M.C.M., Heijnen, J.J., 1998. A new combined differential-discrete cellular automaton approach for biofilm modeling: application for growth in gel beads. Biotechnology and Bioengineering 57 (6), 718–731. Steinlechner-Maran, R., Eberl, T., Kunc, M., Margreiter, R., Gnaiger, E., 2000. Oxygen dependence of respiration in coupled and uncoupled endothelial cells. American Journal of Physiology-Cell Physiology 40 (6), C2053–C2061. Stephanopoulos, G.N., Aristidou, A.A., Nielsen, J., 1998. Metabolic Engineering Principles and Methodologies. Academic Press, San Diego. Tjia, J.S., Moghe, P.V., 2002. Regulation of cell motility on polymer substrates via “dynamic,” cell internalizable, ligand microinterfaces. Tissue Engineering 8 (2), 247–261. Vyakarnam, M.N., Zimmerman, M.C., Scopelianos, A.G., Roller, M.B., Gorky, D.V., 2002. Porous tissue scaffoldings for the repair or regeneration of tissue. United States Patent. Ethicon Inc, US. Wood, B.D., Quintard, M., Whitaker, S., 2002. Calculation of effective diffusivities for biofilms and tissues. Biotechnology and Bioengineering 77 (5), 495–516. Wu, F.J., Friend, J.R., Hsiao, C.C., Zilliox, M.J., Ko, W.J., Cerra, F.B., Hu, W.S., 1996. Efficient assembly of rat hepatocyte spheroids for tissue engineering applications. Biotechnology and Bioengineering 50 (4), 404–415.