Page 361 - IJB-9-4

International Journal of Bioprinting A computational model of cell viability and proliferation of 3D-bioprinted constructs

works, a set of limiting assumptions were again considered, the generated tissue. It also acts as a powerful tool for the
namely the 1D simplification and the neglection of cellular design of tissues embedded with proper vascular networks
proliferation. to guarantee viability for all cells and proper tissue
formation. To the best of our knowledge, no other work on
With regards to cell proliferation and death, many
models have been developed and exploited to explain PDE-based models and finite element simulation of these
phenomena present in the literature is specifically applied
various cell growth curves under different conditions. Jin to bioprinting. In the present work, some important details
and Lei developed a mathematical model based on the in the model with respect to the already proposed ones
[14]
logistic growth model to study the role of autophagy in were added. First, the dynamic state of the 3D-bioprinted
yeast cell population dynamics in response to starvation. constructs was modeled, considering both temporal and
The logistic model is a basic paradigm in population spatial variations. Besides, glucose and oxygen diffusion
ecology, first established by Verhulst in 1838 . The logistic were analyzed at the same time to study their effect on
[15]
model is a reasonable approximation of growth behavior in cellular proliferation. The influence of the substance
many situations and is qualitatively correct, as it captures concentration on the consumption rate was considered,
the phenomenon of exponential growth at low population and cell proliferation and death were introduced. To
levels and saturation in the case of high population consider the competitive and cooperative behaviors of cells
levels . Several works are found in the literature about in the different phases of proliferation, we combined the
[14]
mathematical modeling of nutrient diffusion and cell typical Monod expression with the Lotka–Volterra model
proliferation and death [16-19] . They take place within the of population growth. This combination accounts for the
context of avascular tumor models and cell spheroids, transition from the substrate-limiting phase, in which the
where cells occupy the entire volume of the spheroid. Cell limiting growth factor is the substance shortage, to the self-
proliferation and death is modeled through a change in limiting phase, in which cells decrease their proliferation
the spheroid size, which is not applicable to 3D-printed rate. To account for both substrate‐limiting and self‐
constructs where cells are embedded in a hydrogel. In inhibiting factors and to describe the transition between
2009, Higuera et al. proposed a mathematical model the two phases, the Monod equation was simply multiplied
[20]
to assess the proliferation of human mesenchymal stem with the self-inhibiting factor.
cells (hMSCs) in 2D culture over glucose and glutamine.
Oxygen, on the other hand, was neglected in the model All the input parameters required by the model were
since it was deemed to be sufficient for cell proliferation. derived from the existing literature. A second step of
The growth phenomenon was modeled using the Monod model validation was carried out with an experiment
equation, an empirical model very similar to the Michaelis– measuring cell viability observed over time on extrusion-
Menten law in its formulation, where the consumption rate based 3D-bioprinted samples. This validation step allowed
is replaced by the growth rate. In the model, the possible a clear understanding of the most relevant parameters
cell death was considered through a constant death rate in affecting the prediction capability of the simulation model.
the mass balance. Yet, because of the short time of culture In fact, by comparing the cell viability predicted by the
considered, it was then neglected. Besides, the model model with the one empirically assessed, the first step of
would predict a forever lasting growth, because no terms model calibration was performed by tuning all the model
to stop cellular proliferation are considered. In 2019, Xu parameters in order to minimize the discrepancy between
et al. proposed an analytical solution for a hybrid the predicted and the real cell viability observed over time.
[21]
Logistic–Monod equation. In this work, the authors A further sensitivity analysis on input parameters allowed
developed a realistic PDE model based on diffusion and us to identify the most crucial parameters influencing the
consumption of nutrients and cell growth and applied it prediction capability of the simulation model, namely the
to 3D-printed constructs. Yet, the model did not consider growth rate and the maximum cell density. Given the high
the three-dimensionality of the phenomena and the model number of input parameters and the difficulty in measuring
was solved analytically. them, which results in a broad range of values available in
the literature, this sensitivity analysis represents a relevant
The aim of our work is to develop a versatile
computational model combining the diffusion and byproduct of this work, as it allows the interested reader
to clearly identify critical parameters that need to be
consumption of nutrients and the consequent cell accurately estimated.
proliferation and death in 3D-bioprinted construct.
The differential model, a system of PDEs approximated Eventually, the last part of this work is meant to show
by means of the finite element method, enhances the how the simulation model can guide 3D-printed construct
understanding of how these phenomena influence cell design. In fact, the model is eventually applied to 3D-printed
distribution in the construct and the consequent success of constructs including internal vascularization channels to

Volume 9 Issue 4 (2023) 353 https://doi.org/10.18063/ijb.741