A simulation model for stem cells differentiation into specialized cells of non-connective tissues

Computational Biology and Chemistry 32 (2008) 338–344

Contents lists available at ScienceDirect

Computational Biology and Chemistry journal homepage: www.elsevier.com/locate/compbiolchem

A simulation model for stem cells differentiation into specialized cells of non-connective tissues Massimo Pisu a , Alessandro Concas a , Sarah Fadda b , Alberto Cincotti b , Giacomo Cao a,b,∗ a b

CRS4 (Center for Advanced Studies, Research and Development in Sardinia), Località Piscinamanna, Edificio 1, 09010 Pula, Cagliari, Italy Dipartimento di Ingegneria Chimica e Materiali, Università degli Studi di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy

a r t i c l e

i n f o

Article history: Received 28 September 2007 Accepted 16 June 2008 Keywords: Simulation model Population balance Stem cell differentiation Non-connective tissue Growth factor

a b s t r a c t A novel mathematical model to simulate stem cells differentiation into specialized cells of non-connective tissues is proposed. The model is based upon material balances for growth factors coupled with a massstructured population balance describing cell growth, proliferation and differentiation. The proposed model is written in a general form and it may be used to simulate a generic cell differentiation pathway during in vitro cultivation when specific growth factors are used. Literature experimental data concerning the differentiation of central nervous stem cells into astrocytes are successfully compared with model results, thus demonstrating the validity of the proposed model as well as its predictive capability. Finally, sensitivity analysis of model parameters is also performed in order to clarify what mechanisms most strongly influence differentiation and cell types distribution. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Stem cell therapies based on the differentiation of adult or embryonic cells into specialized ones appear to be an effective method in the treatment of human diseases (Henningson et al., 2003; Baksh et al., 2004). Stem cells, the progenitors of all body tissues, have the remarkable and critical abilities to exist in vivo in a quiescent, undifferentiated state and to propagate. They are characterized initially as being totipotent, pluripotent, or multipotent (Henningson et al., 2003). These cells may differentiate into functional cells of various tissues (Baksh et al., 2004; Khademhosseini and Zandstra, 2004; Henningson et al., 2003). Since the proliferative capacity of many adult tissue-specific cells is very limited, thus making difficult their in vitro expansion, current research is focused on the use of stem cells instead of tissue-specific cells (Kuo and Tuan, 2003). Today, adult stem cell therapies are very promising in medicine, even if the diseases they are used to treat are limited to very specific types of disorders. Autologous stem cells may be obtained from tissues (skin, muscle, retina, neural, liver, intestine, mammary glands and others) of individual patients so that reimplantation of in vitro cultivated cells/tissues would avoid rejection problems. A wide range of diseases may be potentially treated

∗ Corresponding author at: Dipartimento di Ingegneria Chimica e Materiali, Università degli Studi di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy. Tel.: +39 070 675 5058; fax: +39 070 675 5057. E-mail address: [email protected] (G. Cao). 1476-9271/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiolchem.2008.06.001

by using totipotent embryonic stem cells which are object of ethic debates and discussions (Macklin, 2000; Moore et al., 2006). A crucial role for cell differentiation is played by growth factors (GFs), which generally are proteins that bind to receptors on the cell surface, with the primary result of activating cellular proliferation and/or differentiation. Several growth factors are quite versatile, stimulating cellular division in numerous different cell types, while others are specific to a particular cell type. In this context the well-known TGF-ˇ superfamily plays an important role in the development of cells. This family comprises a variety of growth factors that feature different functions in stem cells biology. These factors regulate both “stemness” and various cell differentiation pathways (Pucéat, 2007). Despite intensive research work, the role of this family of growth factors in stem cells differentiation is still unclear (Pucéat, 2007). Further studies are required since the effect of this type of growth factors are in some cases contradictory and mechanisms concerning cell proliferation/differentiation as well as the interaction with growth factors need to be elucidated. An important contribution along these lines may be provided by predictive models which should facilitate the experiments, thus helping researchers to find optimal operating conditions and at the same time contributing to the understanding of biological mechanisms and stem cell behavior. For this reason several papers on modeling stem cell proliferation/differentiation are available in the literature starting with the stochastic model proposed by Till et al. (1964). A remarkable attempt to simulate cell differentiation for ex vivo hematopoiesis in the presence of TGF-ˇ1 was done by Nielsen et al. (1998). These authors developed a mathematical

M. Pisu et al. / Computational Biology and Chemistry 32 (2008) 338–344

Nomenclature a b CGF CO2 Cm d f(m) m NC NGF Nm n nt p q t V

parameter appearing in Eqs. (13) and (14) (1/h) parameter appearing in Eqs. (13) and (14) (ng/mm3 ) concentration of growth factor (ng/mm3 ) concentration of O2 at saturation condition (mmol/mm3 ) oxygen concentration at half-maximal consumption (mmol/mm3 ) mass density (ng/mm3 ) division probability density function (1/ng) single cell mass (ng) number of cell type number of growth factors number of grid points for the mass domain cell number total cell number partitioning function coefficient appearing in symmetric beta function cultivation time (h) total cultivation volume (mm3 )

Greek letters ˇ(q, q) symmetric beta function  (q) gamma function F division rate function (1/h) differentiation rate function (1/h) T 0 average mass of dividing cells (ng) c catabolic rate (1/h)  maximum rate of cell growth (ng/(mm2 h))  time rate of change of cell mass m (ng/h)  standard deviation of the Gaussian distribution (ng)  yield appearing in Eq. (14) (ng of GF/ng of cells) cell distribution function (cells/(ng mm3 )) Superscripts 0 initial conditions  mother cell Subscripts c cells GF growth factor i ith cell type j jth growth factor k kth cell type O2 oxygen

model based on population balance which simulates hematopoiesis starting from a colony of hematopoietic stem cell. Nielsen et al. (1998) use a tank and tubular reactor metaphor to describe the (pseudo)-stochastic and deterministic elements of hematopoiesis. More recently, Bailon-Plaza and van der Meulen (2001) proposed a two-dimensional mathematical model to describe the effect of growth factor on fracture healing. The model simulates the differentiation of mesenchymal stem cells into chondrocytes and osteoblasts during a fracture healing by accounting for material balance of the involved cells (mesenchymal stem cells, chondrocyte and osteoblast cells) coupled with a material balance on the extracellular matrix (ECM) components and growth factors. Hentschel et al. (2004) proposed a mathematical model which simulates the dynamic mechanisms for skeletal pattern formation in the vertebrate limb when fibroblast growth factor, FGF, and TGF-ˇs are present. Beside the works considered above, other authors have

339

Table 1 Models concerning the simulation of cell differentiation processes Model

Reference

Stochastic model of stem cell proliferation for spleen colony-forming cells Model of differentiation processes in the thymus Stochastic model of brain cell differentiation Pseudo-stochastic and deterministic model for ex vivo hematopoiesis Stochastic model of proliferation and differentiation of O2 -A progenitor cells Deterministic two-dimensional model of fracture healing Dynamic mechanisms for skeletal pattern formation in the vertebrate limb when the growth factors are present Dynamic model for cellular differentiation and co-expression properties of switch networks Mathematical model for the interaction of transcription factors

Till et al. (1964) Mehr et al. (1995) Yu et al. (1998) Nielsen et al. (1998) Zorin et al. (2000) Bailon-Plaza and van der Meulen (2001) Hentschel et al. (2004)

Cinquin and Demongeot (2005)

Roeder and Glauche (2006)

addressed the simulation of cell differentiation. For the sake of brevity, a list of the most interesting models available in the literature on this subject is summarized in Table 1. The main limitation of these models is represented by the absence of the description of the cell size distribution and its influence on cellular metabolism and differentiation. With the aim to improve these models a novel mathematic approach to simulate in vivo or in vitro stem cell differentiation into specialized cells of connective tissues where cells may grow, differentiate and synthesize components of the extracellular matrix (ECM) has been developed by Pisu et al. (2007). The model, which may also describe cell mass distribution, was validated by comparison with available literature experimental data concerning the differentiation of mesenchymal stem cells into chondrocytes. In the present work we propose a mathematical model to simulate stem cells differentiation into specialized cells of non-connective tissues. The model, along the lines of our previous contributions (Pisu et al., 2003, 2004, 2006, 2007), is based upon material balances for growth factors coupled with a mass-structured population balance to simulate cell growth, differentiation and proliferation in vivo or during in vitro cultivation. The proposed model is written in a general form and may be used to describe a generic cell differentiation pathway occurring during in vitro cultivation of non-connective tissues. Literature experimental data concerning the differentiation of murine central nervous stem cells into astrocytes are successfully compared with model results, thus demonstrating the validity of the proposed model as well as its predictive capability. Finally, sensitivity analysis of model parameters is also performed in order to clarify what mechanisms most strongly influence differentiation and the distribution of cell types. This approach may be followed to guide the investigation on cell cultivation, growth and differentiation for stem cell therapies or to describe different pathologies involving cell growth and differentiation (i.e., tumors development, infections, etc.). 2. Mathematical modeling The mathematical model proposed in the present work accounts for the cell generic differentiation pathway schematically shown in Fig. 1. Stem cells (e.g. central nervous system stem cells) may differentiate into specialized cells of type 1 (e.g. astrocytes, i = 2) under the influence of one or more specific growth factors, GF1 and GF2 . Stem cells may also differentiate into specialized cells of

340

M. Pisu et al. / Computational Biology and Chemistry 32 (2008) 338–344

cell, i , and the cell division rate, iF , appearing in Eq. (1), are expressed as reported below:

 i (mi , CO2 ) = iF (mi , CO2 ) =

Fig. 1. Schematic representation of the mathematical model for cell differentiation into specialized cell of non-connective tissues.

type 2 (e.g. neurons, i = 3) by means of a different class of growth factors, GF3 . This scheme may involve more cell types and different pathways of cell specialization starting from a stem cell population (i = 1). Specifically, the mathematical model proposed in the present work describes cell growth, proliferation and differentiation into intermediate or specialized cells of non-connective tissues during cell cultivation in the presence of culture medium and specific growth factors. The mathematical model is written in a general form and may be used to simulate the differentiation of stem cells into intermediate and specialized cells of various non-connective tissues. To describe cell growth, proliferation and differentiation into intermediate or specialized cells during cell cultivation in the presence of culture medium and specific growth factors, the following mass-structured population balance for the generic cell of ith type may be written: ∂

i (mi , t)

∂t



+

∞

=2 mi

∂[vi

i (mi , t)]

−

F i (mi , t)i (mi , CO2 ) +

−

T i (mi , t)i (mi , CGF,j )

i = 1, . . . , NC and k = / i

i CO2

Cm + CO2

− c,i mi

 mi 0

(4)

(5)

f (mi ) dmi

2i2

exp

−(mi − 0,i )2

(6)

2i2

The partitioning function, p(mi , mi ), appearing in Eq. (1), specifies how cellular material is partitioned among the two daughter cells when mother cell divides (Hatzis et al., 1995). According to Ramkrishna (2000) the partitioning function must satisfy the symmetry condition: p(mi , mi ) = p(mi − mi , mi )

(7)

and the normalization condition:



m

i

0

p(mi , mi ) dmi = 1.

(8)

Moreover conservation of mass requires that: m

i

mi p(mi , mi ) dmi =

mi

(9)

2

It is convenient to represent the partitioning function as a symmetric beta distribution:

T k (mk , t)k (mk , CGF,j )

for

2/3

(4)1/3 mi

i (mi , CO2 )f (mi ) 1−

1



f (mi ) =

0

 F    i (mi , t)i (mi , CO2 )p(mi , mi ) dmi

2/3

where symbol’s significance is reported in the nomenclature section. The time rate of change of cell mass, i , is derived from the postulate of von Bertalanffy where the anabolic term for a single cell is proportional to its surface area, while the catabolic term to the cell mass (Himmelblau and Bischoff, 1968; Pisu et al., 2004). The cell division rate iF (mi ), is derived from Koch and Schaecter (1962) where cell mass may be used as a predictor of cell division. If it is assumed that cell division occurs only when cell reaches a critical mass, the probability density function of a cell with mass mi to divide, f(mi ), may be expressed as a Dirac delta function (deterministic division). In order to account for heterogeneity in cell cycle distribution the function f(mi ) may be assumed as a distribution of dividing mass around a mean (Eakman et al., 1966; Mantzaris et al., 1999):



∂mi

3 dc,i

(1)

p(mi , mi ) =

1 1 ˇ(qi , qi ) mi



mi mi

qi −1 

1−

mi mi

qi −1

(10)

where ˇ(qi , qi ) is the symmetric beta function:

along with: 0 (mi , 0) i

i (mi , t)

=

i (mi , t)

= 0 for

for t = 0 and ∀mi

t > 0 and mi = 0

(2) (3)

where symbol’s significance is reported in the nomenclature section. In Eq. (1) the first two terms of the right-hand side represent the birth (i.e., a mother cell may be divided in two daughter cells) and the death of the cell of ith type due to mitosis. The third term of the right-hand side of Eq. (1) accounts for the gain of the cell of ith type due to differentiation from cells of kth type (with k = / i) as stimulated by one or more growth factors. Finally, the fourth term is represented by the loss of the cell of ith type due to differentiation. It is worth mentioning that the cellular death by apoptosis is neglected since it may be relevant only in the case of apoptotic/necrotic tissues. Following the approach described in detail in previous works (Pisu et al., 2004, 2006, 2007), the time rate of change of cell mass

2

ˇ(qi , qi ) =

( (qi ))  (2qi )

(11)

and  (qi ) is the gamma function:



+∞

 (qi ) =

t qi −1 e−t dt.

(12)

0

By taking into account all growth factors j that stimulate the differentiation of the cell of ith type and assuming the dependence upon each specific growth factor concentration up to a maximum constant saturation value (Bailon-Plaza and van der Meulen, 2001), the differentiation rate iT , appearing in Eq. (1), may be expressed as follows: iT =

 aij CGFj j

bij + CGFj

(13)

M. Pisu et al. / Computational Biology and Chemistry 32 (2008) 338–344

Finally, the consumption of growth factor may be calculated by following equation: ∂CGFj ∂t



=−



mi

i (mi , t) dmi

0

j

×



∞

ij

aij CGFj bij + CGFj

for

j = 1, . . . , NGF

(14)

0 CGFj = CGF at t = 0 j

(15)

Also in this case symbols’ significance is reported in the nomenclature section. It should be noted that Eq. (14) accounts for the yield ij which relates the differentiation rate of the generic ith cell to the consumption of the jth growth factor. The system of ordinary differential Eqs. (14) along with initial conditions (15) requires the knowledge of the cell distribution function, t of ith type of cells. The latter one is obtained from the solution of the population balance expressed by Eq. (1) which represents a partial differential equation in the variables t and m. This is solved by discretizing the derivative whose independent variable is the cell mass m, in order to obtain a system of ordinary differential equations. The latter one, coupled with Eq. (14) along with initial conditions (15), represents a larger system of ordinary differential equations which is solved as an initial value problem with the Gear method by taking advantage of standard numerical libraries. The number of cells of ith type, ni (i = 1 for stem cells; i = 2 for astrocytes) and the total cell number used to illustrate the capability of the proposed model, are calculated as follows:



∞

ni = V

i (mi , t) dmi

(16)

0

nt =



ni for

i = 1, . . . , NC

Table 2 Model parameters used in the simulation of the differentiation of murine central nervous system stem cells into astrocytes adding 10 ng/ml of LIF in the cultivation system Parameter

Value

Unit

Reference

n0 (i = 1) i n0 (i = 2) i 0 (j = 1) CGF

2.0 × 105 0 10 × 10−3

cells cells ng/mm3

Satoh et al. (2000) Satoh et al. (2000) Satoh et al. (2000)

dc,i (i = 1, 2) C O2 Cm o,i (i = 1, 2)  i (i = 1, 2) qi (i = 1, 2) C,i (i = 1, 2)  (i = 1, 2) i ij (i = 1; j = 1) aij (i = 1; j = 1) bij (i = 1; j = 1)

1.14 × 106 0.124 × 10−6 0.006 × 10−6 0.4 0.125 40 1.0 × 10−3 18.8 3.80 × 10−4 1.05 × 10−2 0.07 × 10−3

ng/mm3 mmol/mm3 mmol/mm3 ng ng – 1/h ng/(mm2 h) ng of GF/ng of cells 1/h ng/mm3

Jakob et al. (2003) Obradovic et al. (2000) Obradovic et al. (2000) Kupsco (2001) Mantzaris et al. (1999) Mantzaris et al. (1999) Munteanu et al. (2002) This work This work This work Pisu et al. (2007)

j

along with the initial conditions:

(17)

i

where V represent the total cultivation volume. It should be noted that in the simulations we typically use a number of grid points in the mass domain equal to Nm = 30, since finer grids do not provide significant changes in the numerical solution. 3. Results and discussion The mathematical model proposed in this work is compared with literature experimental data concerning the differentiation of central nervous system stem cells (CNSSCs) into astrocytes cells (ACs) stimulated by a specific growth factor (leukemia inhibitor factor, LIF, and Activin A of the TGF-ˇ superfamily). Under these conditions, the differentiation of stem cells into neurons, described by the pathway schematically reported in Fig. 1, is neglected. In particular, we compare model results with data reported by Satoh et al. (2000) who investigated the LIF-induced differentiation of multipotent neural stem cells of murine central nervous system into astrocytes enhanced by Activin A. Our attention was focused on the experiments concerning the differentiation of cells of MEB5 line which is a multipotent stem cell line that can differentiate (Satoh et al., 2000) into neurons and glial cells (astrocytes and oligodendrocytes). In these experiments murine stem cells were seeded onto poly-l-lysine, fibronectin, laminin-coated glass coverslips and were cultivated with 10 ng/ml of LIF. The effect of Activin A on the stem cell differentiation of MEB5 line into astrocytes was investigated by adding 1 or 10 or 100 ng/ml of this growth factor. Model parameters used in simulation run, whose values are reported in Table 2, are taken from the literature related to either

341

i = 1 for stem cells, i = 2 for astrocytes; j = 1 for LIF. Experimental data from Satoh et al. (2000).

the original work from which the experimental data have been generated or to specific sources where the corresponding values are available. It should be noted that parameters qi , 0i ,  i and  i are the same for each type of cells since we do not have specific information for murine stem cells and astrocytes. Fig. 2a and b shows a good agreement between model results and experimental data in terms of total cells and astrocytes number, respectively, as a function of cultivation time when 10 ng/ml of LIF are used during the cell cultivation. It is worth noting that the unknown model parameters (i.e., a11 , 11 and i ) are properly tuned to simulate the experimental data. Next, we compare model results with experimental data obtained when cultivation was carried out by using 10 ng/ml of LIF with an addition of 100 ng/ml of Activin A. Model parameters used in this simulation run are reported in Table 3. Also in this case parameters qi , 0i ,  i and i are the same for each type of cells (i = 1 and i = 2). As shown in Fig. 3a and b, the agreement between model results and experimental data in terms of total cells and astrocytes number as a function of cultivation time, is satisfactory. It should be noted that, in this simulation, the only adjusted parameter is a12 (related to Activin A growth factor), while no change has been made in parameters a11 , 11 (related to LIF) obtained from the previous calculation. Furthermore, it is assumed that b12 = b11 and 12 = 11 . Table 3 Model parameters used in the simulation of the differentiation of murine central nervous system stem cells into astrocytes adding 10 ng/ml of LIF and 100 ng/ml of Activin A in the cultivation system Parameter

Value

Unit

Reference

n0 (i = 1) i n0 (i = 2) i 0 (j = 1) CGF j 0 (j = 2) CGF j

2.0 × 105 0 10 × 10−3

cells cells ng/mm3

Satoh et al. (2000) Satoh et al. (2000) Satoh et al. (2000)

100 × 10−3

ng/mm3

Satoh et al. (2000)

1.14 × 106

ng/mm3 mmol/mm3 mmol/mm3 ng ng – h−1 ng/(mm2 h) ng of GF/ng of cells h−1 h−1 ng/mm3

Jakob et al. (2003) Obradovic et al. (2000) Obradovic et al. (2000) Kupsco (2001) Mantzaris et al. (1999) Mantzaris et al. (1999) Munteanu et al. (2002) This work This work This work This work Pisu et al. (2007)

dc,i (i = 1, 2) C O2 Cm o,i (i = 1, 2)  i (i = 1, 2) qi (i = 1, 2) C,i (i = 1, 2)  (i = 1, 2) i ij (i = 1; j = 1; j = 2) aij (i = 1; j = 1) aij (i = 1; j = 2) bij (i = 1; j = 1; j = 2)

0.124 × 10−6 0.006 × 10−6 0.4 0.125 40 1.0 × 10−3 18.8 3.80 × 10−4 1.05 × 10−2 0.39 × 10−2 0.07 × 10−3

i = 1 for stem cells, i = 2 for astrocytes; j = 1 for LIF, j = 2 for Activin A. Experimental data from Satoh et al. (2000).

342

M. Pisu et al. / Computational Biology and Chemistry 32 (2008) 338–344

Fig. 2. Comparison between model results and experimental data (Satoh et al., 2000) in terms of total cells (a) and astrocytes number (b) as a function of cultivation time when 10 ng/ml of LIF are used during cell cultivation.

To test the predictive model capability we then simulated the experimental data performed when different dosages of Activin A growth factor were used during the cell cultivation. By using the same parameters of Table 3, i.e., without any fitting procedure, we compare model predictions and experimental data by Satoh et al. (2000) in terms of astrocytes percentage as a function of added Activin A in the concentration range 0–100 ng/ml after 3 days of cell cultivation. The good agreement between model results and experimental data shown in Fig. 4 demonstrates the predictive capability of the model. It is possible to observe that the augmentation of Activin A increases the differentiation of stem cells into astrocytes but this effect is not evident when the concentration of this growth factor is equal or greater to about 20 ng/ml. This behavior is well predicted by the model which therefore could be an useful tool to optimize the growth factor dosage during cell cultivation and differentiation. With the aim of identifying the mechanisms and interactions that most strongly influence differentiation, a sensitivity analysis on the main model parameters is performed. In Fig. 5 it is shown the effect of population balance parameters qi , 0i ,  i and i (i = 1 and i = 2) on total cell (a) and astrocytes number (b), in the case of murine central nervous system stem cells differentiation after 4 days of cultivation with 10 ng/ml of LIF and 100 ng/ml of Activin

Fig. 3. Comparison between model results and experimental data (Satoh et al., 2000) in terms of total cells (a) and astrocytes number (b) as a function of cultivation time when 10 ng/ml of LIF and 100 ng/ml of Activin A are used during cell cultivation.

Fig. 4. Comparison between model prediction and experimental data (Satoh et al., 2000) in terms of astrocytes content (%) as a function of Activin A added during cell cultivation.

M. Pisu et al. / Computational Biology and Chemistry 32 (2008) 338–344

343

Fig. 5. Sensitivity analyses on total cell number (a) and astrocyte number (b) by varying population balance parameters (i.e., qi , 0i ,  i and  ) after 4 days of cultivation i with 10 ng/ml of LIF and 100 ng/ml of Activin A.

Fig. 6. Sensitivity analyses on the astrocyte number by varying differentiation parameter aij , bij and ij for LIF (a) and Activin A (b) after 4 days of cultivation with 10 ng/ml of LIF and 100 ng/ml of Activin A.

A. Specifically, modeling results are reported in terms of percentage variation of each model output with respect to the values obtained for the base case, when the considered parameters values are modified by ±50%. It may be seen that the most sensitive model parameters are the rate of cell growth, i , and the average mass of dividing cells, 0i . It is then apparent that the latter ones represent the model parameters which most strongly influence differentiation. This result may be ascribed to the effect of such parameters on cell growth function and mitosis rate which may significantly modify cell content and its distribution during cell differentiation. It should be noted that, this effect could not be predicted by using a more simple averaged description. The sensitivity analysis has been also performed by varying the differentiation parameters aij , bij and ij (appearing in Eqs. (8) and (9)) related to LIF (j = 1) and Activin A (j = 2) growth factors, respectively. Results of these simulations are reported in Fig. 6 where it is possible to observe the effect of parameter aij , bij and ij for LIF (Fig. 6a) and Activin A (Fig. 6b) on astrocytes number variation. It is apparent that the main effect could be ascribed to the parameter aij while no significant effect may be ascribed to the parameter bij and ij . A very small influence of the latter one on the astrocytes number may be observed

only in Fig. 6b when parameters related to Activin A are varied. It is worth noting that no significant effect of parameters aij , bij and ij on the total cell number has been revealed by the sensitivity analysis. This fact is fully expected since the parameter aij , bij and ij should not affect the cell division and growth but only the differentiation process. A predictive model of the type proposed in this work may be useful for the identification of optimal experimental conditions as for example the growth factor dosage to investigate the stem cell differentiation into specialized cells of non-connective tissues. Such a model capability arises from the detailed level of description which includes metabolic kinetics and its relationship with cell growth, proliferation and differentiation. Another important feature of the proposed model is its potential ability to simulate a variety of differentiation processes and transformation pathways. Nevertheless, it should be noted that the use of the model to investigate different specific growth factors would require a deeper knowledge and description of cell biology including the metabolic paths which are missing at this stage. A possible future direction of our work may concern the introduction of spatial coordinates in the model together with the description of mass transfer pro-

344

M. Pisu et al. / Computational Biology and Chemistry 32 (2008) 338–344

cesses and migration which may involve cells, growth factors and nutrients. Acknowledgements Ministero dell’Università e della Ricerca (MUR), Italy, Regione Autonoma della Sardegna, Italy, Fondazione Banco di Sardegna and Aria Srl, Italy, are gratefully acknowledged for the financial support of projects CYBERSAR, CARDIOSTAM, and “Crescita, caratterizzazione e simulazione modellistica del processo di crescita di tessuti cartilaginei ingegnerizzati e di cellule staminali del midollo osseo”, respectively. This work has been also carried out with the financial contribution of the Sardinian Regional Authorities. One of us (S.F.) acknowledges the PhD School of Industrial and Chemical Engineering of Politecnico di Milano, Italy. References Bailon-Plaza, A., van der Meulen, M.C.H., 2001. A mathematical framework to study the effects of growth factor influences on fracture healing. Journal of Theoretical Biology 212, 191–209. Baksh, D., Song, L., Tuan, R.S., 2004. Adult mesenchymal stem cells: characterization, differentiation, and application in cell and gene therapy. Journal of Cellular Molecular Medicine 8, 301–316. Cinquin, O., Demongeot, J., 2005. High-dimensional switches and the modelling of cellular differentiation. Journal of Theoretical Biology 233, 391–411. Eakman, J.M., Fredrickson, A.G., Tsuchiya, H.M., 1966. Statistics and dynamics of microbial cell populations. Chemical Engineering Progress Symposium Series 62, 37–49. Hatzis, C., Srienc, F., Fredrickson, A.G., 1995. Multistaged corpuscolar models of microbial growth: Monte Carlo simulations. Biosystems 36, 19–35. Henningson Jr., C.T., Stanislaus, M.A., Gewirtz, A.M., 2003. Embryonic and adult stem cell therapy. Journal of Allergy and Clinical Immunology 111, 745–753. Hentschel, H.G.E., Glimm, T., Glazier, J.A., Newman, S.A., 2004. Dynamical mechanisms for skeletal pattern formation in the vertebrate limb. Proceedings of the Royal Society of London. Series B 271, 1713–1722. Himmelblau, D.M., Bischoff, K.B., 1968. Process Analysis and Simulation: Deterministic Systems. Wiley, New York. Jakob, M., Demarteau, O., Schafer, D., Stumm, M., Heberer, M., Martin, I., 2003. Enzymatic digestion of adult human articular cartilage yields a small fraction of the total available cells. Connective Tissue Research 44, 173–180. Koch, A., Schaecter, M., 1962. A model for statistics of the cell division process. Journal of General Microbiology 29, 435–444. Khademhosseini, A., Zandstra, P.W., 2004. Engineering the in vitro cellular microenvironment for the control and manipulation of adult stem cell responses. In: Turksen, K. (Ed.), Adult Stem Cells. The Humana Press Inc., Totowa, NJ, pp. 289–314.

Kuo, C.K., Tuan, R.S., 2003. Tissue engineering with mesenchymal stem cells. IEEE Engineering in Medicine and Biology Magazine 22, 51–56. Kupsco, J., 2001. Master’s Thesis. University of Richmond. Available at http://oncampus.richmond.edu/%7Egradice/Kupsco thesis.html. Macklin, R., 2000. Ethics, politics, and human embryo stem cell research. Women’s Health Issues 10, 111–115. Mantzaris, N.V., Liou, J.J., Daoutidis, P., Srienc, F., 1999. Numerical solution of a mass structured cell population balance in an environment of changing substrate concentration. Journal of Biotechnology 71, 157–174. Mehr, R., Globerson, A., Perelson, A.S., 1995. Modeling positive and negative selection of differentiation processes in the thymus. Journal of Theoretical Biology 175, 103–126. Moore, K.E., Mills, J.F., Thornton, M.M., 2006. Alternative sources of adult stem cells: a possible solution to the embryonic stem cell debate. Gender Medicine 3, 161–168. Munteanu, S.E., Ilic, M.Z., Handley, C.J., 2002. Highly sulfated glycosaminoglycans inhibit aggrecanase degradation of aggrecan by bovine articular cartilage explant culture. Matrix Biology 21, 429–440. Nielsen, L.K., Papoutskis, L.K., Miller, W.M., 1998. Modeling ex-vivo hematopoiesis using chemical engineering metaphors. Chemical Engineering Science 53, 1913–1925. Obradovic, B., Meldon, J.H., Freed, L.E., Vunjak-Novakovic, G., 2000. Glycosaminoglycan deposition in engineered cartilage: experiments and mathematical model. AIChE Journal 46, 1860–1871. Pisu, M., Lai, N., Cincotti, A., Delogu, F., Cao, G., 2003. A simulation model for the growth of tissue engineered cartilage on polymeric scaffolds. International Journal of Chemical Reactor Engineering, Available at http://www.bepress.com/ijcre/vol1/A38. Pisu, M., Lai, N., Cincotti, A., Concas, A., Cao, G., 2004. Model of engineered cartilage growth in rotating bioreactors. Chemical Engineering Science 59, 5035–5040. Pisu, M.A., Concas, A., Lai, N., Cao, G., 2006. A novel simulation model for engineered cartilage growth in static systems. Tissue Engineering 12, 2311–2320. Pisu, M., Concas, A., Cao, G., 2007. A novel simulation model for stem cells differentiation. Journal of Biotechnology 130, 171–182. Pucéat, M., 2007. TGF-ˇ in the differentiation of embryonic stem cells. Cardiovascular Research 74, 256–261. Ramkrishna, D., 2000. Population Balances—Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego. Roeder, I., Glauche, I., 2006. Towards an understanding of lineage specification in hematopoietic stem cells: a mathematical model for the interaction of transcription factors GATA-1 and PU-1. Journal of Mathematical Biology 241, 852–865. Satoh, M., Sugino, H., Yoshida, T., 2000. Activin promotes astrocytic differentiation of a multipotent neural stem cell line and an astrocyte progenitor cell line from murine central nervous system. Neuroscience Letters 284, 143–146. Till, J.E., McCulloch, E.A., Siminovitch, L., 1964. A stochastic model of stem cell proliferation, based on growth of spleen colony-forming cells. Proceedings of the National Academy of Sciences United States of America 51, 29–36. Yu, A., Mayer-Proschel, M., Noble, M., 1998. A stochastic model of brain cell differentiation in tissue culture. Journal of Mathematical Biology 37, 49–60. Zorin, A., Mayer-Proschel, M., Noble, M., Yakovlev, A.Y., 2000. Estimation problem associated with stochastic modeling of proliferation and differentiation of O2-A progenitor cells in vitro. Mathematical Biosciences 167, 109–121.