Sarah Fadda

A novel population balance model to investigate the kinetics of in vitro cell proliferation: Part I. model development

In biotechnology and biomedicine reliable models of cell proliferation kinetics need to capture the relevant phenomena taking place during the mitotic cycle. To this aim, a novel mathematical model helpful to investigate the intrinsic kinetics of in vitro culture of adherent cells up to confluence is proposed in this work. Specifically, the attention is focused on the simulation of proliferation (increase of cell number) and maturation (increase of cell size and DNA content) till contact inhibition eventually takes place inside a Petri dish. Accordingly, the proposed model is based on a population balance (PB) approach that allows one to quantitatively describe cell cycle progression through the different phases the cells of the entire population experienced during their own life. In particular, the proposed model has been developed as a 2D, multi‐staged, and unstructured PB, by considering a different sub‐population of cells for any single phase of the cell cycle. These sub‐populations are discriminated through cellular volume and DNA content, that both increase during the mitotic cycle. The adopted mathematical expressions of the transition rates between two subsequent phases and the temporal increase of cell volume and DNA content are thoroughly analyzed and discussed with respect to those ones available in the literature. Specifically, the corresponding uncertainties and pitfalls are pointed out, by also taking into account the difficulties and the limitations involved in the quantitative measurements currently practicable for these biological systems. A novel mathematical expression for contact inhibition in line with the PB model developed is also formulated, along with a proper comparison between modeled and measurable DNA distributions. The strategy for a reliable, independent tuning of the adjustable parameters involved in the proposed model along with its numerical solution is outlined in Part II of this work, where it is also shown how it can be profitably used to gain a deeper insight into the phenomena involved during cell cultivation under microgravity conditions. Biotechnol. Bioeng. 2012; 109:772–781.

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

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 mass-structured 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.

A novel population balance model to investigate the kinetics of in vitro cell proliferation: Part II. numerical solution, parameters' determination, and model outcomes

Based on the general theoretical model developed in Part I of this work, a series of numerical simulations related to the in vitro proliferation kinetics of adherent cells is here presented. First the complex task of assigning a specific value to all the parameters of the proposed population balance (PB) model is addressed, by also highlighting the difficulties arising when performing proper comparisons with experimental data. Then, a parametric sensitivity analysis is performed, thus identifying the more relevant parameters from a kinetics perspective. The proposed PB model can be adapted to describe cell growth under various conditions, by properly changing the value of the adjustable parameters. For this reason, model parameters able to mimic cell culture behavior under microgravity conditions are identified by means of a suitable parametric sensitivity analysis. Specifically, it is found that, as the volume growth parameter is reduced, proliferation slows down while cells arrest in G0/G1 or G2/M depending on the initial distribution of cell population. On the basis of this result, model capabilities have been tested by means of a proper comparison with literature experimental data related to the behavior of synchronized and not‐synchronized cells under micro‐ and standard gravity levels. Biotechnol. Bioeng. 2012; 109:782–796.

The effect of EIF dynamics on the cryopreservation process of a size distributed cell population.

Typical mathematical modeling of cryopreservation of cell suspensions assumes a thermodynamic equilibrium between the ice and liquid water in the extracellular solution. This work investigates the validity of this assumption by introducing a population balance approach for dynamic extracellular ice formation (EIF) in the absence of any cryo-protectant agent (CPA). The population balance model reflects nucleation and diffusion-limited growth in the suspending solution whose driving forces are evaluated in the relevant phase diagram. This population balance description of the extracellular compartment has been coupled to a model recently proposed in the literature [Fadda et al., AIChE Journal, 56, 2173-2185, (2010)], which is capable of quantitatively describing and predicting internal ice formation (IIF) inside the cells. The cells are characterized by a size distribution (i.e. through another population balance), thus overcoming the classic view of a population of identically sized cells. From the comparison of the system behavior in terms of the dynamics of the cell size distribution it can be concluded that the assumption of a thermodynamic equilibrium in the extracellular compartment is not always justified. Depending on the cooling rate, the dynamics of EIF needs to be considered.

Modeling of Point Defects Annihilation in Multilayered Cu/Nb Composites under Irradiation

This work focuses on a mathematical modeling of the response to irradiation of a multilayer composite material. Nonstationary balance equations are utilized to account for production, recombination, transport, and annihilation, or removal, of vacancies and interstitials at interfaces. Although the model developed has general validity, Cu/Nb multilayers are used as case study. Layer thickness, temperature, radiation intensity, and surface recombination coefficients were varied systematically to investigate their effect on point defect annihilation processes at interfaces. It is shown that point defect annihilation at interfaces mostly depends on point defect diffusion. The ability of interfaces to remove point defects can be described by a simple map constructed using only two dimensionless parameters, which provides a general tool to estimate the efficiency of vacancy and interstitial removal in multilayer composite materials.

Role of Interface in Multilayered Composites under Irradiation: A Mathematical Investigation

A continuum model of point-defects evolution during irradiation of a multilayer composite material is presented in this work. Nonstationary balance equations are used to describe production, recombination, transport, and annihilation, or removal, of vacancies and interstitials in a β-α-β three-layer system (α = Cu and β = Nb, V, or Ni). In addition, transport and trapping of point-defects at interfaces are taken into account. Numerical investigation on similarities and differences between Cu/Nb, Cu/V, and Cu/Ni systems is also performed. A general comparison of model results reveals that average vacancy concentration is typically higher than SIA one in both layers for all the systems investigated. This is a consequence of the higher diffusion rate of SIAs with respect to vacancies. Stationary state is reached without saturating interface point-defect traps by all systems but Cu/Ni for the case of SIAs. It can be also seen that Cu/Nb and Cu/V systems have a very similar behavior regarding point-defect temporal evolution in copper (layer α), while higher SIA concentration at steady state is shown therein by the Cu/Ni structure. Moreover, Cu/V system displays the lower stationary vacancy concentration in layer β.

Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population

Cryopreservation of biological material is a crucial step of tissue engineering, but biological material can be damaged by the cryopreservation process itself. Depending on some bio-physical properties that change from cell to cell lineages, an optimum cryopreservation protocol needs to be identified for any cell type to maximise post-thaw cell viability. Since a prohibitively large set of operating conditions has to be determined to avoid the principal origins of cell damage (i.e., ice formation and solution injuries), mathematical modelling represents a valuable alternative to experimental optimisation. The theoretical analysis traditionally adopted for the cryopreservation of a cell suspension addresses only a single, average cell size and ascribes the experimental evidence of different ice formation temperatures to statistical variations. In this chapter our efforts to develop a novel mathematical model based on the population balance approach that comprehensively takes into account the size distribution of a cell population are reviewed. According to this novel approach, a sound explanation for the experimental evidence of different ice formation temperatures may now be given by adopting a fully deterministic criterion based on the size distribution of the cell population. In this regard, the proposed model represents a clear novelty for the cryopreservation field and provides an original perspective to interpret system behaviour as experimentally measured so far. First our efforts to successfully validate the proposed model by comparison with suitable experimental data taken from the literature are reported. Then, in absence of suitable experimental data, the model is used to theoretically investigate system behaviour at various operating conditions. This is done both in absence or presence of a cryo-protectant agent, as well as when the extra-cellular ice is assumed to form under thermodynamic equilibrium or its dynamics is taken into account consistently by means of an additional population balance. More specifically, the effect of the cell size distribution on system behaviour when varying cooling rate and cryo-protectant content within practicable values for a standard cryopreservation protocol is investigated. It is demonstrated that, cell survival due to intra-cellular ice formation depends on the initial cell size distribution and its osmotic parameters. At practicable operating conditions in terms of cooling rate and cryo-protectant concentration, intra-cellular ice formation may be lethal for the fraction of larger size classes of the cell population whilst it may not reach a dangerous level for the intermediate size class cells and it will not even take place for the smaller ones.

Osmotic behaviour of human mesenchymal stem cells: implications for cryopreservation

Aimed at providing a contribution to the optimization of cryopreservation processes, the present work focuses on the osmotic behavior of human mesenchymal stem cells (hMSCs). Once isolated from the umbilical cord blood (UCB) of three different donors, hMSCs were characterized in terms of size distribution and their osmotic properties suitably evaluated through the exposure to hypertonic and isotonic aqueous solutions at three different temperatures. More specifically, inactive cell volume and cell permeability to water and di-methyl sulfoxide (DMSO) were measured, being cell size determined using impedance measurements under both equilibrium and dynamic conditions. Experimental findings indicate that positive cell volume excursions are limited by the apparent increase of inactive volume, which occurs during both the shrink-swell process following DMSO addition and the subsequent restoration of isotonic conditions in the presence of hypertonic solutions of impermeant or permeant solutes. Based on this evidence, hMSCs must be regarded as imperfect osmometers, and their osmotic behavior described within a scenario no longer compatible with the simple two-parameter model usually utilized in the literature. In this respect, the activation of mechano-sensitive ion-channels seemingly represents a reasonable hypothesis for rationalizing the observed osmotic behavior of hMSCs from UCB.

A population balance model to investigate the effect of microgravity on the kinetics of in vitro cell proliferation

In this work a mathematical model helpful to investigate the effects of microgravity on the kinetics of in vitro proliferation of adherent cells is proposed. The model is based on a Population Balance (PB) approach that allows to describe cell cycle progression through the different phases experienced by any cell of the entire population during its own life. Specifically, the proposed model has been developed as a multi-staged 2-D PB, by considering a different sub-population of cells for any single phase of the cell cycle. These sub-populations composing the entire population of cells of the cultivation system are discriminated through cellular volume and DNA content, that both increase during the mitotic cycle. A series of numerical simulations related to the in vitro proliferation kinetics of adherent cells is here reported for illustrating model capabilities. It is found that, the change of one adjustable parameter related to cell volume growth rate in response to a change of gravity is able to mimic cell culture behaviour under microgravity conditions, as reported in the technical literature.

Experimental analysis and modeling of in vitro mesenchymal stem cells proliferation

Stem cell therapies based on the differentiation of adult or embryonic cells into specialized ones appear to be effective for treating several human diseases. This work addresses the mathematical simulation of the proliferation kinetics of stem cells. Sheep bone marrow mesenchymal cells seeded at different initial concentrations in Petri dishes are expanded up to confluence. The sigmoidal temporal profiles of total counts obtained through classic hemocytometry are quantitatively interpreted by a novel model based on a 1-D, single-staged population balance approach capable to take into account the contact inhibition at confluence. The models’ parameters have been determined by comparison with experimental data on population expansion starting from a single seeding concentration. The reliability of the model is tested by predicting the cell proliferation carried out starting from different seeding concentrations.