A mathematical model for the design of fibrin microcapsules with skin cells

ORIGINAL PAPER

A mathematical model for the design of fibrin microcapsuleswith skin cells

Cristian A. Acevedo Æ Caroline Weinstein-Oppenheimer Æ Donald I. Brown ÆHolger Huebner Æ Rainer Buchholz Æ Manuel E. Young

Received: 1 May 2008 / Accepted: 22 July 2008 / Published online: 13 August 2008

� Springer-Verlag 2008

Abstract The use of fibrin in tissue engineering has

greatly increased over the last 10 years. The aim of this

research was to develop a mathematical model to relate the

microcapsule-size and cell-load to growth and oxygen

depletion. Keratinocytes were isolated from rat skins and

microencapsulated dropping fibrinogen and thrombin

solutions. The cell growth was measured with MTT-assay

and confirmed using histochemical technique. The oxygen

was evaluated using a Clark sensor. It was found that Fick–

Monod model explained the cell growth for the first 48 h,

but overestimated the same thereafter. It was necessary to

add a logistic equation to reach valid results. In relation to

the preferred implant alternative, when considering large

initial cell loads, the possibility to implant small loads of

fast-growing cells arises from the simulations. In relation to

the microcapsule size, it was found that a critical diameter

could be established from which cell growth velocity is

about the same.

Keywords Fibrin � Microencapsulation � Keratinocytes �Logistic equation � Oxygen diffusion

List of symbols

C oxygen concentration

Deff effective diffusivity

K Monod constant

k logistic parameter

KLa oxygen transfer coefficient

PR proliferation rate

q respiration rate

r radius of cell cluster

R radius of microcapsule

t time

V volume of cell cluster

X cell concentration

l specific growth rate

Subscripts

AP apparent

EQ equilibrium

M maximum

O initial

SAT saturation

Introduction

The pioneering work of James Rheinwald and Howard

Green allowed for the successful culture of keratinocytes

utilizing a substrate of irradiated 3T3 fibroblast feeder layers

[1]. This became the basis for constructing therapeutic sys-

tems to be applied in the treatment of burned patients. This

3T3 feeder layer culture takes too long to cultivate enough

keratinocytes to built implants large enough to treat large

areas of damaged skin, during which period, the condition of

C. A. Acevedo (&) � M. E. Young

Biotechnology Center, Universidad Tecnica Federico Santa

Marıa, Av. Espana 1680, Valparaıso, Chile

e-mail: [email protected]

C. Weinstein-Oppenheimer

Departamento de Bioquımica, Facultad de Farmacia,

Universidad de Valparaıso, Av. Gran Bretana 1093, Valparaıso,

Chile

D. I. Brown

Departamento de Biologıa y Ciencias Ambientales, Facultad de

Ciencias, Universidad de Valparaıso, Av. Gran Bretana 1111,

Valparaıso, Chile

H. Huebner � R. Buchholz

Department of Bioprocess Engineering,

Friedrich-Alexander University of Erlangen-Nuremberg,

Paul Gordan Str. 3, 91052 Erlangen, Germany

123

Bioprocess Biosyst Eng (2009) 32:341–351

DOI 10.1007/s00449-008-0253-1

the burned patient often deteriorated beyond recovery [2].

However, the repair of damaged human skin can be

improved greatly by organizing the skin cells [3], such as

using the cells when they are immobilized in the matrix, to

construct the scaffolds. The most frequent matrices are those

made out of alginate [4], even though chitosan, glycosami-

noglycans, polyacrylates and poly (vinyl alcohol) are also

used [5]. Fibrin has also being included as a possible

material for cell support, being a natural polymer with a

well-known function related to cell differentiation. Fibrin

microcapsules may be easily obtained by thrombin cata-

lyzed polymerization of fibrinogen.

The application of a commercial fibrin gel containing

cultured keratinocytes was first reported by Hunyadi et al.

[6]. Fibrin gels have since been used as a dermal matrix

containing keratinocytes [7]. The use of commercial Fibrin

in tissue engineering practices has increased over the last

10 years [9–11]. Fibrin has been proposed as a delivery

matrix in seeding urothelial cells for urethral reconstruction

[12], and in the form of microcapsules as effective carriers

to support the bone and cartilage-forming potential of

different types of cells [13]. It has also been tested as a

vehicle to deliver viable cells to infarcted myocardium,

which increases cell viability [10, 11].

Its use as a polymeric support for the transplant of

human skin cells has also been reported [2]. The first in

vitro studies were done in rat models. The results were

successful, showing an improvement in the proliferation,

migration and differentiation of the cells compared to

keratinocytes cultured in traditional cell culture flasks [8].

It was later reported that keratinocyte culture in fibrin,

maintains the cells in a proliferative state and improves the

grafts containing these cells [14].

To successfully grow skin cells, it is important to pre-

vent uncontrolled cell growth and overcrowding of cells

within the immobilization device, a situation which would

lead to nutrient and oxygen depletion resulting in cell death

[15]. Oxygen and nutrient supply to cells inside a micro-

capsule mainly depend on mass diffusion. In large

microcapsules, diffusion gradients are formed so that the

cells in the center are not liable to get sufficient nutrients to

support growth. Conversely, the rate of waste removal from

the center is insufficient and thus the cells eventually die

[16]. Oxygen transportation is normally considered as the

main limiting factor for nutrient exchange in cells, such as

hepatocytes [16], chondrocytes [17], hybridoma [18],

fibroblasts [19], mesenchymal stem cells [20] and many

other cells. In implying that fact, hypoxia is being reported

as an important mechanism for regulation of gene expres-

sion in eukaryotic cells [21–23]. In particular for

keratinocytes, anoxia conditions alter cell morphology and

seem to delay senescence, allowing for more passages in

surface growth [24].

All of these conditions, which are related to the kinetic

growth of keratinocytes during in vitro cultures, indicate a

strong dependence on oxygen concentration. The cell

growth in flasks is characterized by a thin liquid layer with

nutrients designed to contain all excess requirements, and

the oxygen present in the environment is sometimes reg-

ulated to contain a lower oxygen concentration from that in

the air, thereby indirectly controlling the maximum load of

oxygen that a solution might have. In addition, the cells

attached to the flask in monolayers have easy access to

dissolved oxygen. When keratinocytes are encapsulated a

new barrier to oxygen availability is created, because the

liquid phase is separated from the cells generating an

oxygen profile inside the microcapsule. This becomes

critical for larger microcapsule diameters.

There are many studies available which are trying to

predict nutrient behavior in solid matrices by applying

mathematical modeling [16–18, 20, 25]. All theoretical

approaches consider oxygen diffusion explained by Fick’s

law, and oxygen consumption kinetics linked to the Monod’s

equation through a yield parameter. In our work, we

hypothesized that the proliferation rate in the microencap-

sulated cells cannot be fully explained by the Monod model

in its simplest version which requires additional complexity.

We proposed to apply a modified form of the model by

including an additional parameter given by a logistic

equation. The aim of this research was to develop and

validate such a mathematical model for keratinocyte

growth in encapsulated fibrin beads. The developed model

will provide guidance in designing the system intended for

tissue engineering purposes.

Materials and methods

Cell isolation and culture

The primary cell culture was obtained from rats (strain cpr

100). They were anesthetized with ketamine/xilacine (5 mg

and 2 mg per 100 g of body weight, respectively), shaved,

and disinfected with a povidone-iodine solution. A biopsy

of 1 cm2 was taken from the dorsal area of the animal.

Animals were treated according to the Guide for the Care

and Use of Laboratory Animals (National Academy of

Sciences, USA).

The technique utilized for isolation and culture of skin

cells was adapted from previously published reports [26].

Briefly, the biopsy was washed three times with phosphate

buffer saline, 0.1 M, pH 7.4 (PBS) containing penicillin

(100 U/mL), streptomycin (100 lg/mL) and amphotericin

B (25 lg/mL). The visible fat was mechanically removed.

The clean biopsy was incubated overnight at 48C in trypsin-

EDTA (0.05%–0.53 mM).

342 Bioprocess Biosyst Eng (2009) 32:341–351

123

The epidermis layer was removed, sliced into 1-mm

pieces and incubated for 1 h in trypsin-EDTA (0.05%–

0.53 mM). The recovered cells were filtered through sterile

gauze and cultured in Defined Keratinocyte Media. All

reagents were purchased from Gibco-Invitrogen.

Cells microencapsulated in fibrin beads

Cells were microencapsulated using a coaxial microen-

capsulation apparatus [18], with fibrinogen and thrombin

solutions. Fibrinogen solution (80 mg-lyophilized/mL;

lyophilized: 78% protein, 87% clottable protein; Sigma-

Aldrich) and thrombin solution (130 units-NIH/mL,

80 mM CaCl2; MP ICN-Biomedicals) carrying the cells

[12], were coagulated forming a drop on oil phase (tri-

glyceride mixture of capric/caprylic acid) [27].

Evaluation of oxygen consumption by keratinocytes

Free cells (4 9 106) were introduced in a batch micro-

bioreactor with 5 mL of media at 37 8C. The concentration

of dissolved oxygen was measured using a Clark type

polarographic sensor (WTW Multiline P4) for 2 h. The

correction, due to sensor response time delay was taken

into account as described by Jorjani and Ozturk [28]. The

resulting data was modeled, taking into consideration, the

fact that the change in oxygen concentration increases by

mass transport and is consumed by the cells, as follows:

oC

ot¼ KLaðCSAT � CÞ � X qM

C

K þ C

� �

This equation was solved with numerical solutions and

fit with experimental values using minimum square

criterion in Frontline Solver Software. The parameter KLa

was determined under steady-state conditions characterized

by no time change in oxygen concentration.

Viable cell count

The number of viable cells was evaluated by MTT-assay,

which measures the ability of mitochondrial dehydroge-

nase enzymes to convert the soluble yellow MTT salt (3-

(4,5-dimethyl-2-thiazolyl)-2,5-diphenyl-2H-tetrazolium

bromide) into insoluble purple formazan salt [29, 30].

Microcapsules were incubated with MTT solution

and Defined Keratinocyte Medium (Gibco-Invitrogen) in

the proportion of 1:4, for 4 h at 37 8C. MTT solution

(Sigma–Aldrich) was prepared at 0.5% in PBS. Then, the

microcapsules were disaggregated with trypsin 0.5%

(Gibco-Invitrogen) at 37 8C. Lysis buffer (SDS 3%w/v and

HCl 40 mM, in isopropanol) with ultrasounds (15 min)

were used to solubilize formazan.

The factor used to estimate viable biomass was the total

formazan production after reaching a plateau level (end

point). To be certain that absorbance proportional to cell

number is measured instead of enzyme activity, previous

experiments were performed with known number of

immobilized keratinocytes confirming that before 4 h the

plateau is reached indicating cessation of enzyme activity.

For each experiment, a calibration curve was produced

from known microencapsulated viable cell previously

counted in a Neubauer chamber and assayed by MTT. Cells

before microencapsulation were stained with trypan-blue

solution (Gibco-Invitrogen) in order to recognize viable

and unviable cells.

Histochemistry and immuno-histochemistry

The cells microencapsulated were fixed in Bouin’s solution

for 24 h, dehydrated in alcohol solutions (ethanol 70%,

ethanol 95% and butanol), and embedded in Paraplast Plus

(Sigma–Aldrich). Serial sections of 5 lm thickness were

cut in a microtome, mounted on microscope slides,

hydrated and stained or immunostained.

To perform the histochemistry analysis, the Arteta tri-

chrome stain was applied, using Harris–hematoxylin

solution for 75 s (the nuclei became dark blue). Afterward,

erythrosine-B/orange-G (0.5/0.5%) was used for 30 min (to

stain the cytoplasm red) and finally aniline-blue (1%) for

75 s (to stain the extracellular matrix or collagen light-

blue). All dyes were obtained from Sigma–Aldrich.

Keratinocytes were identified by immunostaining using

an AE1/AE3 monoclonal antibody for cytokeratin (Dako-

Cytomation). To recognize possible fibroblast cross-

contamination, another set of slides was incubated with the

polyclonal antibody for vimentin (c-20): sc-7557 (Santa

Cruz Biotechnology). The amplification was developed

with the immuno-peroxidase technique using a commercial

kit of avidin-biotin-peroxidase complex (ABC) (Vectastain-

ABC, Vector) and DAB (Sigma–Aldrich) as chromogen

which produces a brown color in positive detection.

Results

Histochemistry and inmunohistochemistry

The histological sections from microcapsules with cells

stained with Arteta thrichrome show that the fibrin is

present in a heterogeneous matrix that is mainly stained

with the aniline blue and red with erythrosine in the free

borders (Fig. 1a–d). In this matrix there are spaces where

the cells are located (Fig. 1a–e). In the first 3 days,

individual cells observed (Fig. 1a–c), among those mitotic

Bioprocess Biosyst Eng (2009) 32:341–351 343

123

figures are frequent (Fig. 1b; metaphase in polar view). In

Fig. 1d a pair of cells is observed, with their characteristic

slightly ovoid nuclei with a very marked nucleole. Over

longer periods, cells joined in clusters appear in the

spaces inside the fibrin (Fig. 1e, f). In these groups typical

mitotic figures are common (Fig. 1f; metaphase in polar

view).

In the serial sections of microcapsules, in this case with

clusters of cells (Fig. 1g), a clear positive reaction with the

antibody to anti-cytokeratin is present (Fig. 1h). On the

contrary, there is not evident reaction with the anti-

vimentin antibody (Fig. 1i). Therefore, the presence of

keratinocytes without fibroblasts contamination is

indicated.

Simple model: diffusion and Monod

From the kinetics experiments with microencapsulated

cells, increasing cell mass was observed with the cells

arranging in aggregates inside the microcapsule. To

establish a relationship between the observed viable cells

increase and oxygen availability, a diffusion model based

on Fick’s behavior for oxygen was used which explains

oxygen transportation, only in terms of the concentration

Fig. 1 Histology and immuno-

histochemistry of

microencapsulated cells.

Histology of cells

microencapsulated in fibrin

stained with Arteta thrichrome

stain. a One cell in a zone of

homogeneous fibrin stained

with aniline blue (1 h after

microencapsulation). b One cell

in metaphase (polar view) in a

space near fibrin stained with

erythrosin (48 h after

microencapsulation). c Three

cells in a fibrin fold (24 h after

microencapsulation).

d A couple of cells between

fibrin matrix (48 h after

microencapsulation). e Clusters

of cells in spaces (arrowheadsand arrow) inside the fibrin

(96 h after microencapsulation).

f A high magnification view of a

cluster of cells from e (arrow),

where there is a cell in

metaphase (polar view).

Immuno-histochemistry of a

cluster of cells growing in fibrin.

g Control section stained with

Arteta thrichrome stain.

h Consecutive sections with

strong positive immuno-

localization for cytokeratin in

the cytoplasm of cells; it is

evident that nuclei are negative

(white round spaces).

i Consecutive section showing

negative reaction for vimentin

immunolocalization


123

gradient inside the microcapsule, assuming initial homo-

geneous distribution of cells [25]:

oC

ot¼ Deff

o2C

oR2þ 2

R

oC

oR

� �� qM

C

K þ C

� �X

The parameters K and qM were evaluated by using

respirometric experiments with free cells—without diffusion

limitations—as previously described in methods, where

4 9 106 cells were introduced in a micro-reactor containing

5 mL of cell culture medium at 37 8C and oxygen

consumption monitored over a short period of time with a

polarographic Clark sensor. Figure 2 shows the experimental

data together with the adjusted values. A good agreement was

found using the non-linear methods correlation. The specific

rate of oxygen consumption obtained for the rat

keratinocyctes was similar to that reported for other animal

cells as seen in Table 1, with maximum oxygen demand

qM of 9.7 9 10-17 mol/cell s; this result is in the order

of magnitude of other cells like fibroblasts (4–7 9 10-17),

stem cells (1–3 9 10-17), hybridomas (6 9 10-17) and

monocytes (1–18 9 10-17) [31].

The adjusted ‘‘K’’ value was found to be 0.2 mg/L; this

value is very close to the 0.19 mg/L reported for chon-

drocytes [32].

To include the effect of cell growth in the microen-

capsulated experiment (qX/qt = lX), Monod’s equation

was applied to the specific growth as follows:

l ¼ lM

C

K þ C

The short time experiments performed to determine the

duplication time using the MTT proliferation assay (as

described in the experimental section) in triplicate, resulted

in a duplication time of 28 h, close to 1.2 days. Therefore,

in lM equal to 0.6 day-1 or 6.88 9 10-6 s-1 (l = 0.693/

duplication-time). For cell growth in skin, the duplication

time of keratinocytes is 300 h, but it is being reported that

for psoriasis: it is approximately 30 h [33]. In cell cultures

of human keratinocytes, a duplication time of 20 h has

been reported [24].

With all these parameters having already been evalu-

ated, oxygen concentration and cell growth in the

microcapsule can be predicted using the following

boundary conditions: symmetry in the center of the sphere

(microcapsule) and mass transfer resistance at the micro-

capsule-bulk interface evaluated considering Sherwood

Number equal 2, for static sphere [34]. The oxygen diffu-

sion coefficient was estimated as a value in water

(2.4 9 10-5 cm/s2 at 25 8C) [34], corrected for tempera-

ture and a hydrogel environment [25]:

Deff ¼ 2:4 � 10�5ðcm=s2Þ 310�K

298�K

� �ðe�4�0:04Þ

¼ 2:1� 10�5ðcm=s2Þ

For microcapsules 5 mm in diameter with initial cell

concentration of 9 9 105 cells/mL, Fig. 3 shows that the

model explains the cell number data for the first 48 h very

well, overestimating thereafter. A possible explanation for

the decrease in growth [16] is the development of local

extreme hypoxia inside the observed clusters, which appear

as the cells grow in the beads (Fig. 1e, f). This situation

was not accounted for by the mathematical model where

cells were assumed to be well dispersed individuals inside

the microcapsule. Therefore, a modification is necessary if

the histological observations are to be taken into account,

which leads toward the application of a logistic equation.

Logistic equation to predict cell growth

From histological observations, a sample of which are

presented in Fig. 1a, c show that at the beginning of the

process, the microcapsules contain individual cells with

well-delimited nucleus and cytoplasm. At day 2, cells in

division are observed (Fig. 1b, d), and from day 3 the

dividing cells form clusters of many individuals.

Fig. 2 Respirometric experiment. The cells (4 9 106) were intro-

duced in a batch micro-reactor with 5 mL of cell culture medium at

37 8C. Oxygen consumption was monitored with a polarographic

Clark sensor. The experimental data were fitted to Monod’s equation

using minimum square criterion. The blank-experiment without cells

did not show a significant oxidation into the medium (p [ 0.05)

Table 1 Values of oxygen uptake rate in mammalian cells

Cell type Oxygen uptake

rate (mol/cell s)

Rat hepatocytes [42] 2.0 9 10-16

Rat keratinocytes (our data) 9.7 9 10-17

Hybridoma [18] 6.0 9 10-17

Chinese hamster ovary [42] 8.3 9 10-18

Human embryonic lung fibroblasts [42] 2.5 9 10-18

Bovine chondrocytes [17] 1.9 9 10-18


123

The cells aggregated in spheroids exhibit extensive cell–

cell contacts and tight junctions [35]. Thus, cell aggrega-

tion is assumed to be an important factor in tissue culture

processes. From the perspective of mass transfer, however,

cell aggregation in culture generates an additional resis-

tance effect on oxygen transportation implying local

extreme hypoxia at the center, thereby affecting cell via-

bility and function [16]. In our experiments, these effects

appear after 48 h when cluster formation is evident.

To include this additional resistance to oxygen diffusion

transport in the equations, the use of a logistic equation is

proposed which takes cell growth in aggregates into

account. Following Glicklis et al. [16], the logistic equation

assumes that initially, cluster radius increases following a

first order kinetics which is balanced by a second order

term thereby reaching a maximum radius, rM where growth

stops.

or

ot¼ kr 1� r

rM

� �same

o

ot

r

rM

� �¼ k

r

rM

1� r

rM

� �

Furthermore, assuming that cluster radius is related to

cell mass: X, proportional to r3, the kinetic equation

becomes:

oX

ot¼ 3k 1� X

XM

� �1=3 !

X same

oPR

ot¼ 3k 1� PR

PRM

� �1=3 !

PR

Rearranging in terms can be made by including the

proliferation rate PR concept (PR = X/X0), which reaches

a maximum PRM = XM/X0.

Experimental data were fitted to the modified form of

the logistic equation assuming that initially all cells are

individual, and each individual cell produces one cluster,

all well distributed inside the microcapsule. In Fig. 4, it is

shown that the growth data can be explained by the logistic

model where the k logistic parameter is 1.28 9 10-5 (s-1),

with a maximum proliferation rate PRM of 2.9.

Modified model: diffusion, Monod and logistic

Coupling Monod’s equation with the logistic model, we

produce a new specific growth rate expression taking into

account both, cell growth in cluster and oxygen limitation

in the fibrin microcapsules:

l ¼ lM;AP

C

K þ C

where the apparent specific rate growth lM,ap includes the

parameters of the logistic equation:

lM;AP ¼ 3k 1� PR

PRM

� �1=3 !

same

lM;AP ¼ 3k 1� X=Xo

PRM

� �1=3 !

Figure 5 shows the oxygen concentration in the fibrin

matrix and the cell concentration in time.

Additional kinetic data for sphere-microcapsules 3 and

5 mm in diameter, and initial concentration between

8 9 104 and 8 9 105 (cells/mL) were simulated using the

same adjusted parameters presented in Table 2. Figure 6

shows simulated and real data. It is observed that the

simulations adjust well to the experimental values. When

the bisector statistical criterion was used, a unit slope was

Fig. 3 Simple model. Experimental data and simulated values of cell

growth were obtained with fibrin microcapsules of 5 mm diameter

and initial cell concentration of 9 9 105 cells/mL. For the first 48 h,

the error between experimental and simulated data with the simple

model was not significant (p [ 0.05). The oxygen concentrations on

the surface and center of microcapsule were simulated with the simple

model (oxygen concentration in the center of the microcapsules is

asymptotic to zero)

Fig. 4 Logistic equation. Experimental data of cell growth were

obtained microencapsulated keratinocytes in fibrin at initial concen-

tration of 1 9 105cells/mL. The logistic equation was used to fit

experimental data (the mathematical correlation was significant,

p \ 0.05)


123

obtained (p [ 0.05) with zero offset (p [ 0.05), thereby

pointing toward the validity of the proposed model.

In Table 2, the simple and modified models are pre-

sented in a comprehensive manner, showing the common

characteristics and highlighting the differences.

Design of microcapsules

With this modified model (diffusion, Monod and logistic), it

is now possible to analyze the effect of diameter and initial

concentration of cells on cell growth over time. Figure 7

shows the simulated kinetic until steady state is reached in

3 days. It is clear that only upto 105 cells/mL initial con-

centration the size of a microcapsule is not an important

parameter, thereby indicating that all nutrients diffuse

freely and all limitation resides in the aggregates. As the

initial cell concentration increases above106 cells/mL, the

maximum attainable concentration is lower for larger mi-

crocapsules suggesting nutrient limitation by diffusion. This

effect in terms of proliferation rate (PR) can be seen in

Fig. 8; microcapsules including 105 cells/mL reach the

maximum PR (2.9) for all microcapsule diameters, but

those including 107 cells/mL approach 2.9 PR value near to

1 mm with lower PR for larger microcapsules.

Fig. 5 Simple and modified model. Experimental data and simulated

values of cell concentration were obtained with fibrin microcapsules

of 5 mm diameter and initial cell concentration of 9 9 105 cells/mL.

The prediction error of simple model is not significant only in the first

48 h (p [ 0.05). The prediction error of modified model is not

significant in whole kinetic (p [ 0.05)

Table 2 Mathematics models

Fig. 6 Bisector criterion to validate the modified model. Four

experimental kinetics (four sampling-times per experiments) were

used to develop the curves (regression and confidence interval). The

size and initial cell concentration in the microcapsule were: First

experiment, 3 mm and 8 9 104 cell/mL; Second experiment, 3 mm

and 3 9 105 cell/mL; Third experiment, 5 mm and 7 9 105 cell/mL;

and Fourth experiment, 5 mm and 8 9 105 cell/mL


123

Discussion

Microencapsulated keratinocytes behavior

The growth of keratinocytes microencapsulated in fibrin

matrix of different sizes was studied in relation to oxygen

consumption taken as the limiting substrate. Keratinocytes

were obtained from epidermis, mechanically separated from

other skin layers after trypsin incubation of a rat skin biopsy.

The pre-confluent cells in culture were recovered and

encapsulated in fibrin, following the procedure described

earlier. Histological sections showed healthy cells lodged in

sites inside the heterogeneous fibrin matrix. The cells with

their characteristic round morphology are in active prolif-

eration, as revealed by the presence of mitotic figures, until

they organize in three-dimensional compact clusters. At this

stage they were immuno-stained with anti-Cytokeratin, clon

AE1/AE3, which reacts with most cytokeratins [36, 37]

allowing positive identification of keratinocytes inside the

microcapsule. Moreover, when histological sections were

immuno-stained with polyclonal anti-Vimentin, looking for

the presence of fibroblasts, scarce reaction was verified,

thereby indicating a predominant population of keratino-

cytes with meaningless presence of fibroblasts.

During the initial 2 days, sustained cell growth was

verified by MTT assay and histochemical observations.

After this, the MTT analysis shows a decrease in cell

activity which could be attributed to a reduction in the cell

growth rate. This reduction in cell activity concurs with

evidence of growth in three-dimensional clusters observed

inside the microcapsule, which may explain the change in

kinetic behavior related to local nutrient depletion inside

the clusters. At this stage, the observed keratinocytes

exhibit a morphology which is different from the one

usually observed when growth occurs in flasks. This

change in morphology was also reported by Weiss et al.

[38] for keratinocytes cultured in plastic flasks with and

without fibrin coating where the usual polyedric shape

observed in monolayers and developed in plastic flasks

turned into a rounded shape which duplicates in three

dimensions forming cord-like structures. In our case, the

round-shaped cells predominated, arranging in spherical

three-dimensional compact clusters.

Cells growing inside the microcapsule obtain oxygen

from the growth medium by a diffusion mechanism

through the fibrin matrix. Depending on the location with

respect to the microcapsule, they are exposed to different

degrees of hypoxia. Improved proliferation of skin cells

under hypoxia has been reported [19, 24], which may

explain the initial sustained growth during the first 2 days.

The observed compact clusters of cells, leaving no apparent

interstitial space, suggest that oxygen and most probable

nutrient starvation for the cells inside the cluster is taking

place, with the adverse effect for the cell proliferation

observed after day 2. An association could be made with

quiescent cell growth when reaching confluence in con-

ventional cell culture flasks. Even though the cells inside

the microcapsule were far from reaching confluence, they

behave in a similar fashion with regard to oxygen.

Commencing day 4, during the course of the experi-

ment, the microcapsule showed the first signs of

fibrynolysis starting to disintegrate. We chose not to use

that data, because cell growth could be affected by other

phenomenon besides oxygen diffusion. This phenomenon

has been discussed by other authors: for commercial hard

fibrins, where degradation times of 15 days for culture of

keratinocytes have been reported [39], and degradation

times less to 21 days in rat grafts [40]. Other authors, using

the same protocol, reported a different fibrynolysis time

Fig. 7 Simulated growth kinetics with the modified model. Three

initial cell concentrations in fibrin microcapsules were used: 105, 106

and 107 (cell/mL), and three sizes of microcapsule were used: 1, 3

and 5 mm diameter

Fig. 8 Simulated values using the modified model for microcapsule

design. Three initial cell concentrations in fibrin microcapsules were

used: 105, 106 and 107 (cell/mL). Proliferation rate (PR) was defined

as the rate between viable cell number after 3 days and initial viable

cell number


123

depending on the use of primary keratinocytes or estab-

lished cell lines [27].

Model and design of microcapsules

It was found that the simple Fick’s diffusion model toge-

ther with a Monod’s growth kinetics well represents the

data of cell growth associated with predicted values for

oxygen concentration in the microcapsule over the first

2 days. For longer periods, this simple model fails in the

interpretation of data since growth starts to decrease. In

Fig. 3, it is predicted that during the first 24 h, oxygen

concentration in the center of the microcapsule decreases to

about 0.4 mg/L, the same as the concentration in a solution

in equilibrium with air at 1% oxygen concentration, and

there is evidence in literature that in this condition, kerat-

inocytes are viable and proliferate [24] (our data shows

cells to be growing fast). At 48 h, Fig. 3 shows oxygen

depletion in the center of the microcapsule but in the

largest sections of it, concentration is well above 0.4 mg/L.

Nevertheless, by this time, growth starts declining. In 72 h

the oxygen concentration in the microcapsule surface

reaches 0.4 mg/L and cell growth is observed to decline

fast; however, it is also observed that clusters of cell col-

onies appear, including a situation of additional oxygen

limitation for the cells inside the cluster creating micro-

environments not taken into account by the model, where a

major assumption was the homogeneous distribution of

single cells inside the microcapsule. The effect of cluster

formation on cell growth was included into the model

making use of a logistic type equation relating mass cell

increase with increase in the radius of the cluster.

The logistic equation (also known as the Verhulst model

or logistic growth curve) is a model of population growth

first published by P.F. Verhulst in the nineteenth century

[41], who suggested that the rate at which any population

increases may reach a limit. Logistic model therefore plays

two biological processes in opposition: reproduction and

restriction. Both processes depend on population numbers

(or density). The rate of both processes is explained by the

mass-action law with coefficients: ‘‘k’’ for the first order

kinetics (kr), modeling reproduction and ‘‘k/rM’’ for the

second order kinetics (kr2/rM) modeling restriction.

Our data show that proliferation dominates over the first

2 days when the first order kinetics is the larger term due to

small radius size. Thereafter, competition for space,

nutrients and oxygen causing restriction hinder the prolif-

eration process, when the second order term becomes the

same order of magnitude as the first order kinetics due to a

larger radius with cluster formation, reaching steady state

when the population has increased by approximately three

times and both terms are equal. The presence of a first

order kinetics is clear; nevertheless the occurrence of the

second order kinetics could be derived from several factors.

Competition for space is associated to increase in volume

and the rate of change in volume of a cluster assumed

spherical with respect to the radius (qV/qr) is a second

order function of the radius (qV/qr = 4pr2), which implies

that if oxygen availability affects consumption due to the

size of the colony, then the relationship should be

quadratic.

On the other hand, from the simulation of growth lim-

itation by oxygen concentration, it is possible to conclude

that at cell loads of 105 cell/mL, the effect of the micro-

capsule size is not significant, and that during the first day,

all cultures show exponential cell growth. In addition, the

microcapsule diameter of 1 mm always shows exponential

cell growth. The cell load of 105 cell/mL and size of 1 mm

are data of special consideration, because they could be

critical parameters to design microcapsules without diffu-

sion limitation (see Figs. 7, 8). For instance, it may be

predicted that low cell loads immobilized in small micro-

capsules for 1 day would be the optimum for implants

because the cells are in the highest proliferative stage under

these conditions. It is important to point out that this cri-

terion is not universal, but the methodology to establish the

best condition for cell growth on a particular scaffold may

be applied to other tissue engineering systems.

Conclusions

In the field of tissue engineering, it is important to choose

the optimum time for performing the implant. For that, it is

necessary to estimate the viable cell density and predict the

proper time for the implant-surgery. If a traditional model

is used as a predictor—based solely on a Diffusion-Monod

approach—the cell growth can be overestimated because

the cells grow forming clusters and the assumption of

disperse growth is invalid. We concluded that there is a

critical time when it is possible to use a simple model. In

our case, 48 h is the limit where fast growth occurs and an

arrest due to oxygen limitation takes place.

Another important issue that the proposed model revealed

is related to initial cell concentration, demonstrating that for

low cell loads, the microcapsule size is not relevant. This is

not the case for high cell loads, where significant differences

in growth were detected indicating the existence of a critical

cell load. In our case the critical load is 105 cells/mL as the

cells grew in all microcapsule sizes. This information could

be used to develop a useful criterion for tissue engineering, as

the preferred strategy is large seed populations used to insure

a successful implant. Our simulations show that low cell

loads have a period of fast growth not observed when large

populations are applied, making allowance for an alternative

approach in implant design.


123

On the other hand, when the microcapsule size is very

small, the cells grow fast at low and high initial concen-

trations, indicating the existence of a critical diameter. The

critical load and critical diameter are data of special con-

sideration, because they could be important parameters to

design skin microcapsules without diffusion limitations. In

our case, these parameters are 105 cell/mL and of 1 mm.

This information could be used to develop a useful crite-

rion to design cell growth in a three-dimensional matrix.

Acknowledgment The authors thank CONICYT from Chile by:

FONDEF Grant (# DO2I1009) and Doctoral Scholarship for Cristian

Acevedo (# D-21050588).

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A mathematical model for the design of fibrin microcapsules with skin cells

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