In silico Tissue Engineering and Cancer Treatment Using Cellular

Automata and Hybrid Cellular Automata-Finite Element Models

Andrés Díaz Lantada

, Miguel Urosa Sánchez and David Fernández Fernández

Department of Mechanical Engineering, ETSI Industriales, Universidad Politécnica de Madrid,

c/ José Gutiérrez Abascal 2, 28006 Madrid, Spain

Keywords: Software as Medical Device, in silico Tissue Engineering, Cell Simulation, Cellular Automata, Finite Element

Modelling, Multi-Scale Modelling.

Abstract: An innovative approach for in silico tissue engineering and cancer treatment is presented in this study. It is

based on the employment of cellular automata (CA) and cellular automata hybridised with finite-element

models (FEM) for simulating cells within tissue engineering scaffolds. Thanks to the presented strategy, it

has been possible to model cells colonising scaffolds, the interactions among different populations of cells

and between the cells and the scaffolds as extracellular matrices, and the effects of external stimuli, like

temperature, for treating disease. Among the advances incorporated to conventional models based on cellular

automata it is important to mention: the establishment of a direct connection between CAD models and the

simulation workspace, the incorporation of a wall factor for considering the affinity of cells for the

extracellular matrix, the coupling of FEM simulations to the cellular automata for rendering them more

versatile, and the modelling of interactions among different types of cells. Results, limitations, and potentials

of these simulation approaches are presented and discussed, in connection with current trends in software as

a medical device (SaMD).

1 INTRODUCTION

Software as a medical device (SaMD) is gaining

momentum and transforming healthcare. For decades

active medical devices have been smartly driven by

embedded software, but nowadays medical apps and

different kinds of standalone software are emerging

for a wide set of prevention, diagnosis and monitoring

purposes and must be considered medical devices in

themselves (Ludvigsen, 2022). These SaMDs not

only support medical practice but may render it much

more efficient and sustainable, from the different

economic, environmental, and social perspectives. In

large part they can also contribute to the 3R principles

(Replacement, Reduction, Refinement) for ethical

biomedical research (Aske, 2017), as the use of

simulations can be an excellent alternative to other in

vivo studies with animals or in vitro studies with cells

and tissues, along the development lifecycle of

innovative medical devices and drugs or in parallel to

medical practice.

https://orcid.org/0000-0002-0358-9186

Contact: andres.diaz@upm.es

To cite some examples, minimising the number of

animals required for validating innovative therapies

or reducing the use of cells and tissues employed for

studying disease, by means of in silico strategies -

based on software and simulations-, can have highly

positive ethical, economical, and procedural impacts

in remarkable fields such as tissue engineering,

biofabrication and cancer therapies. Indeed, in silico

tissue engineering (Geris, 2018, Keshavarzian, 2019)

and in silico cancer research (Edelman, 2010, Jean-

Quartier, 2018) constitute important trends aimed at

speeding up the (R & D & I) Research – Development

– Innovation cycle, while reducing associated costs

and minimising negative social impacts without

compromising safety. Regarding the simulations of

cells, several computational approaches, both

continuum and discrete, enable the modelling of their

collective behaviours, their interactions with

extracellular matrices, the progress of disease and the

eventual success or failure of a healing or

regenerative strategy (Spencer, 2013, Geris, 2013,

2016).

Díaz Lantada, A., Urosa Sánchez, M. and Fernández Fernández, D.

In silico Tissue Engineering and Cancer Treatment Using Cellular Automata and Hybrid Cellular Automata-Finite Element Models.

DOI: 10.5220/0011742300003414

In Proceedings of the 16th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2023) - Volume 1: BIODEVICES, pages 56-63

ISBN: 978-989-758-631-6; ISSN: 2184-4305

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

In vitro, the invention of tissue engineering

scaffolds, 3D or 4D porous structures that mimic the

extracellular environment providing cells in culture

with biomimetic cell niches, has been fundamental

for setting the foundations of tissue engineering and

biofabrication and for providing more physiological

environments for studying disease and therapies

(Khademhosseini, 2016).

In many ways, scaffolds can be employed to study

the healing and regeneration of tissue and to test the

development of innovative therapies for cancer, but

the cell sources, growth factors, reagents and cell

culture processes involved are often challenging,

expensive and require highly trained professionals. If

such processes could be simulated, fields like tissue

engineering and cancer research could advance even

more efficiently and sustainably. Therefore, it is

necessary to further progress in the simulation of cells

within scaffolds, in the modelling of cell-cell and cell-

material interactions, and in the coupling of these

simulations with the effects from environmental cues

and stimuli. Linking computer-aided designs of

scaffolds and biomaterials with the workspaces

employed for agent-based models, for considering

both the individual and collective behaviours of cells,

and with the input from FEM simulations, for

considering the effects of different physical/chemical

fields on cells’ behaviour and fate, is hence required.

Some recent inspiring studies can be cited, which

evolve from the foundational works with cellular

automata (CA) (Von Neumann, 1966). In short, CA

were developed as collections of elements or cells

defined upon grids that evolve through time steps or

iterations following certain rules. Along the time

steps, the state (i.e. colour or value, typically “0” or

“1”) of the cells within the grid changes according to

the rules and to the previous states of the neighbour

cells. Since the beginning, these models were

conceived as possible simulators for biological

systems with remarkable examples, such as

Conway’s game of life (Gardner, 1970), in which the

cells upon a 2D grid have two possible states, dead or

alive, and in which cells survive, reproduce, migrate,

or die, depending on the 8 neighbouring cells or the

previous state. Further studies led to verifying that

extremely complex systems could be modelled with

CA (Wolfram, 1984).

In connection with biodevices, these models have

also proven useful for studying the biodegradation of

tissue engineering scaffolds (Erkizia, 2010), for

studying scaffolds’ colonization processes (Garijo,

2012, Vivas, 2015), and, by our team, for simulating

and optimising biomimetic cell culture systems

(Ballesteros Hernando, 2019).

In this study, we intend to advance to the next step

by linking three-dimensional CAD models of tissue

engineering scaffolds with the grids of CA and by

hybridising CA and FEM simulations for obtaining

multi-scale and multi-physical/chemical simulators

with more versatile functionalities, as described in the

following sections. Applications in tissue engineering

and cancer research are foreseen and discussed.

2 MATERIALS AND METHODS

2.1 Software Resources

Siemens NX 12.0 and Autodesk Inventor 2020 are

employed as main CAD software resources. The

FEM capabilities of NX 12.0 are used for the thermal

simulations performed. Regarding programming,

Matlab r2020a is used as main resource for creating

the codes for cellular automata. Ultimaker Cura, a 3D

printing slicer is utilised for slicing the CAD models

and obtaining images used as input for generating the

workspaces for the CA models, as described below.

2.2 Fundamentals of Models Used

2.2.1 From CAD Models to Cellular

Automata

Once the usual lattice-like or porous structures of

tissue engineering scaffolds are designed, it is

possible to generate the workspaces or grids for CA

models by slicing their geometries, performing digital

tomographs, and processing the images obtained. The

process is based on previous studies by our team with

some minor modifications that allow us to work with

voxels, instead of pixels (Ballesteros Hernando,

2019), and is schematically illustrated in figure 1.

Figure 1: From CAD models of tissue engineering scaffolds

to the working grids for cellular automata. Left: CAD

models are sliced to generate grayscale images with

allowed and forbidden regions. Right: model space in

Matlab with allowable (red) and restricted (blue) voxels -or

volumetric pixels-.

In silico Tissue Engineering and Cancer Treatment Using Cellular Automata and Hybrid Cellular Automata-Finite Element Models

2.2.2 Modelling the Colonization of Tissue

Engineering Scaffolds

Once the working space is obtained, cell proliferation

can be modelled following different proliferation

rules and illustrated along the temporal iterations by

means of colour changes to the voxels, as shown in

figure 2. Depending on the resolution of the images

obtained through the slicing process and on the

distance between slices, voxel size can be adjusted to

represent single cells (i.e., voxels of c.a. 10 x 10 x 10

μm

) or cell populations or clusters. The size of the

scaffold employed as extracellular matrix and its

porosity, which defines the allowed space for cell

proliferation, together with the resolution or number

of voxels employed per volume unit, define the

computational cost of these simulations.

By means of example, figure 2a presents three

iterations of a cell or cell cluster proliferating

following a rule, by which all voxels normally

connected to a voxel filled with a seed cell or cluster

become populated by cells or clusters in the following

step. However, figure 2b presents four iterations of a

cell or cell cluster proliferating following an irregular

pattern, giving options for asymmetric growth

patterns and even for steps without any proliferation,

based on the incorporation of random functions.

Apart from the proliferation, the possibility of cell

death is taken into account by adding a probability of

death in each iteration. This is represented in figure

2c, in which green voxels represent living cells, while

dead cells are represented in red and, in general,

occupy that space until the end of the simulation.

In order to consider the affinity of cells for the

scaffold’s trusses and the effects of adhesion, we have

also decided to study the incorporation of a “wall

factor”, which modifies the proliferation rules or

probabilities, by employing different probability

proliferation values for cells surrounded by cells and

for cells in contact with trusses. To our knowledge,

this is reported for the first time and leads to results

that better mimic what happens with cells cultured

within real scaffolds, as additionally analysed in

section 3.1.

2.2.3 Modelling Interactions Among Cells

Current tissue engineering strategies face the great

challenge of reconstructing large defects involving

different types of cells and tissues. In cancer research,

the progression of tumours affecting the various kinds

of cells and tissues within organs is also pivotal. In

consequence, simulating interactions among different

cell populations is essential.

The presented CA models can be also applied to

simulate the interactions among different types of

cells. From a visual point of view the voxels (e.g.,

green and blue in figure 2d). From a modelling

perspective, different kinds of cells are employed as

proliferation seeds, by initially selecting one or more

voxels in distinct regions of the allowable space of the

grid. The invasiveness of one cell type can be

modelled by establishing a simple colour change rule

whenever one invasive cell reaches the boundaries of

a normal cell, respectively illustrated in blue and

green in figure 2d. In this way, tumoral processes can

be simulated as further discussed in section 3.2.

Figure 2: Examples of cell proliferation and cell-cell

interactions along different iterations using CA models. a)

Three iterations of a symmetric growth pattern. b) Four

iterations of an asymmetric growth pattern with a

proliferation probability lower than 1, due to which dead

cells (red) appear after some time steps. c) Four iterations

showing proliferation after including the probability of

death. d) Cell-cell iterations showing an invasive cell (blue)

attacking, invading, or cannibalising (Fais, 2018) a healthy

cell (green), subsequently proliferating, or dying (pink).

BIODEVICES 2023 - 16th International Conference on Biomedical Electronics and Devices

2.2.4 Coupling Finite Element Models and

Cellular Automata

More complex behaviours of cells within scaffolds

should take account of existing physical/chemical

fields, microenvironmental cues and external stimuli

that may affect processes like gene expression, cell

differentiation and final cell fate. FEM prove

excellent for numerically solving partial differential

equations upon complex geometries and domains,

Figure 3: Temperature fields obtained by heating the central

upper (a) and upper corner (b) regions of tissue engineering

scaffolds and their mapping upon CA grids as ranges

associated to death probabilities.

hence being fundamental in modern engineering for

mechanical, electromagnetic, fluidic, and thermal

problems. All these physical domains can be used for

modulating cellular behaviour. Thus, the connection

of FEM simulations to agent-based models can prove

extremely useful, as we aim to demonstrate.

Accordingly, results from FEM simulations

stored in matrices have been mapped upon the three-

dimensional grids of CA models. So as to modulate

cellular responses, the values mapped can be

employed to modify the proliferation and survival or

death probabilities, depending on the actual fields

calculated with FEM simulations. To illustrate this

possibility, thermal simulations have been performed,

in connection with the possible cancer treatment

employing high temperatures (hyperthermia), and the

survival probabilities modified. Figure 3 presents the

temperature fields obtained by heating two tissue

engineering scaffolds and their mapping upon CA

grids, as ranges associated to death probabilities. In

these examples we consider temperatures around

37ºC as adequate, temperatures in the 38-43ºC as

risky, temperatures in the 43-50ºC as critical and

temperatures above 50ºC as necessarily deadly.

3 RESULTS

3.1 Cells Colonising Scaffolds

Figures 4 and 5 provide examples of cells colonizing

scaffolds measuring 10 mm in height and 10 in

diameter, which correspond to 100 x 100 x 100 voxels

with the slicing and resolution employed. The voxels

corresponding to scaffolds’ trusses are represented in

blue, while those voxels corresponding to living/dead

cells are respectively drawn in green/red. Figure 4

shows different colonization patterns emulating

colonization of scaffolds starting from distinct

regions. Proliferation and death probabilities of 0.9

and 0.05% are used. Four selected iterations (100,

150, 200, 250) of a simulation performed along 400

steps. Iterations represent time steps, which should be

adjusted by means of in vitro experiments monitoring

cell growth within real tissue engineering scaffolds,

as we reported previously for lab-on-a-chip devices

(Ballesteros Hernando, 2019). The challenges linked

to these experimental validations are discussed in

section 4. Figure 5 presents the influence of using a

wall factor for adjusting the simulations to the fact

that cells tend to colonize scaffolds by growing

preferentially along the trusses and finally filling the

voids.

In silico Tissue Engineering and Cancer Treatment Using Cellular Automata and Hybrid Cellular Automata-Finite Element Models

a) b)

Figure 4: Cells (green) colonizing scaffolds (blue). a) Four

iterations showing colonization from above. b) Four

iterations starting from a lower side.

a) b)

Figure 5: Influence of wall factor on colonization patterns.

a) Without wall factor. b) With increased proliferation

probability for cells touching scaffold’s trusses as

compared to cells far from the trusses (0.9 vs 0.5% as

proliferation probabilities for this model).

3.2 Cellular Interactions

To illustrate the possibility of modelling these

interactions, Figure 6 provides two examples of cells

interacting within a tissue engineering scaffold. For

visualization purposes the trusses of the scaffold are

not drawn, although they are considered as forbidden

regions for the cells, and the different cell types are

shown in green/red and in blue/pink, respectively for

living/dead healthy and invasive cells. Both examples

show features from patterns typically obtained when

diseases progress within in vitro, which have been

previously reported as extension along the scaffold,

cell clumps on the borders, and large cell clusters

(Zhang, 2013). In these examples no wall factor is

used and, for each iteration, proliferation probabilities

of 0.5 and 0.9 and death probabilities of 0.05% and

0.5% respectively for healthy and cancerous or

invasive cell types are employed.

a) b)

Figure 6: Examples of cell-cell interactions within a tissue

engineering scaffold. a) Scaffold being colonized by

healthy and invasive (cancerous) cells. b) Tumour growing

within an already colonized scaffold. Colour code: green =

healthy cells, blue = invasive cells, red = dead healthy cells,

pink = dead invasive cells. Four selected iterations for each

case.

BIODEVICES 2023 - 16th International Conference on Biomedical Electronics and Devices

3.3 Hyperthermia Therapy

Hyperthermia, as a therapy, refers to the controlled

heating of a region of the human body for a medical

purpose, usually cancer treatment. It has shown high

potential for cancer therapy, either in conjunction

with immunotherapy, chemotherapy, radiotherapy,

and surgery (Yagawa, 2017), or as standalone

technique, although the heating affects surrounding

healthy tissues, which is still concerning. Different

approaches are being studied for minimizing its

invasiveness and reaching remote regions, based on

magnetic and optical systems (Casanova-Carvajal,

2021, Zeinoun, 2021). The use of simulations is

expected to support the 3Rs in this field.

a) b)

Figure 7: CA simulation coupled to thermal FEM

representing cancer treatment by hyperthermia. Misaligned

intervention with large HAZ.

a) Rapid intervention stopping the tumour.

b) Delayed and unsuccessful intervention.

Figures 7 and 8 present two examples of CA

simulations coupled to thermal FEM representing

cancer treatments by hyperthermia. First the scaffolds

are colonized by healthy cells, as shown in the first

rows of images corresponding. Just before iteration

150 a tumoral seed is added by modifying some

voxels in the upper regions of the scaffolds. Rapid

interventions (figs. 7a and 8a) show the killing effect

of the thermal hyperthermia applied in iteration 150,

while delayed interventions (figs. 7b and 8b) apply

the thermal field in iteration 200, when the tumour

progress cannot be halted anymore. Figure 8a shows

the result of a thermal field more aligned with the

tumour, which helps to minimise the heat-affected

zone (HAZ) without compromising effectivity.

a) b)

Figure 8: CA simulation coupled to thermal FEM

representing cancer treatment by hyperthermia. Focused

intervention with reduced HAZ.

a) Rapid intervention stopping the tumour.

b) Delayed and unsuccessful intervention.

In silico Tissue Engineering and Cancer Treatment Using Cellular Automata and Hybrid Cellular Automata-Finite Element Models

4 LIMITATIONS AND FUTURE

RESEARCH PROPOSALS

Main limitation of the study is linked to the still

pending experimental validation with real cells. In the

previous study from our team, in which CA were

employed for modelling cells within lab-on-a-chip

devices (Ballesteros Hernando, 2019), we

demonstrated the possibility of adjusting cell growth

patterns using proliferation rates obtained directly

from cultures in Petri dishes. In those systems, simple

microscopy upon the lab-on-a-chip platforms is

adequate for monitoring growth. However, within

tissue engineering scaffolds the visualization is much

more challenging and monitoring the colonization

and the cell-cell and cell-material interactions in their

core requires alternatives to microscopy. Currently

we are exploring the applicability of magnetic

resonance microscopy, as a non-invasive technology

capable of exploring the inside of scaffolds with cells,

following the example of pioneering research

(Führer, 2017).

Considering future research, together with the

experimental validation employing in vitro cultures

for adjusting the CA models, it is important to

consider the following: First, the coupling of FEM

simulations with CA models has proven useful for

mapping temperature fields and simulating cancer

treatment using hyperthermia. Apart from that,

several phenomena can be modelled following this

hybrid CA-FEM approach, including: the effects of

fluid flow and shear stresses on scaffolds’

colonization, if computational fluid dynamics

simulations are used; the impact of nutrients and

drugs’ diffusion on cell viability or disease

progression; or, even more challenging, potential

mechanobiological effects (vibrations, cyclic

compressions / tractions, pulsatile stimulation) on cell

differentiation during tissue repair processes.

Second, the geometries of scaffolds employed

and the control volume and working space for the

performed simulations remain fixed during the

calculations, that is, a 3D grid with a fixed number of

elements with fixed sizes is always employed. The

size of the grid and elements depends only on the

actual scaffold’s size and slicing employed, which

can be adjusted to the dimensions of individual cells

or cells’ clusters. From a computational point of view

this is not yet optimal; it would be interesting to

explore models, in which the grid’s size and the

number of elements varies along the simulation, as

has proven useful in advanced FEM simulations, in

which elements can be switched off and on during a

simulation.

In addition, size changes are often involved in

gene expression and differentiation processes, so

counting with agents or cells within the automata

capable of modifying their size would be an

interesting incorporation.

Finally, towards societal impact, once validated,

these models should undergo a certification process

under the appropriate regulation, the Medical Device

Regulation 2017/745 in the case of the European

Union. Thus, they could become commercially

available solutions that healthcare professionals and

engineers devoted to medical technologies could

employ, in parallel to their research on tissue

engineering and cancer therapies, for supporting both

the design of medical devices like scaffolds and the

development of drugs for cancer treatment. Ideally,

for increased reach out, these SaMDs may be shared

as open-source solutions accessible to all.

5 CONCLUSIONS

This study has presented an innovative approach for

modelling cells colonising scaffolds, the interactions

among different populations of cells and between the

cells and the scaffolds as extracellular matrices, and

the effects of external stimuli, like temperature, for

treating disease. To this end, different advances have

been incorporated to conventional models based on

cellular automata. First, a direct connection between

CAD models and the simulation workspace has been

provided, by using digital tomographs, employing

additive manufacturing slicers upon the CAD files, to

create the grid. Furthermore, the option of

incorporating a sort of wall factor, capable of

considering the affinity of cells for the extracellular

matrix, has been discussed. In addition, FEM

simulation upon the scaffolds’ environment have

been coupled to the cellular automata for making

them more versatile. Finally, interactions among

different types of cells have been simulated.

Despite the pending in vitro experiments,

simulations presented show the possibility of

modelling complex interactions and phenomena

directly working with the CAD models of

biomaterials and scaffolds. Besides, a connection

between designed geometries and the grids used for

agent-based simulations has been established, and the

utility of hybridising CA with FEM for studying cells,

tissues, scaffolds, and cancer therapies has been

illustrated and discussed. The fact that wisely

implemented cellular automata can perform as

universal Turing machines (Wolfram, 1984) points

out the relevance and potentials of this approach.

BIODEVICES 2023 - 16th International Conference on Biomedical Electronics and Devices

ACKNOWLEDGEMENTS

This research was performed with the support of the

following programmes and projects: Programa estatal

de generación de conocimiento y fortalecimiento

científico y tecnológico del sistema de I+D+i,

subprograma estatal de generación de conocimiento

del Ministerio de Ciencia e Innovación y Agencia

Estatal de Investigación, ref. PGC2018-097531-B-

I00 (Talenano: Estudio de la eficacia de tecnologías

alternativas de liberación de energía térmica y

mecánica mediante nanoestructuras de óxido de

hierro y de oro con aplicación en terapias), as regards

the modelling of cancer hyperthermia; European

Union’s Horizon 2020 Research and Innovation

Programme under grant agreement n. 953134

(INKplant project: Ink-based hybrid multi-material

fabrication of next generation implants), as regards

the modelling of cell-scaffold interactions; and

Programa de estancias de movilidad de profesores e

investigadores en centros extranjeros de enseñanza

superior e investigación del Ministerio de

Universidades, ref. PRX21/00460 (Ingeniería de

tejidos in silico facilitada mediante microscopía de

resonancia magnética), as regards validation strategy.

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