Tommaso Lorenzi

Journal articles

62 R.M. Crossley, P.K. Maini, T. Lorenzi, R.E. Baker, Travelling waves in a coarse-grained model of volume-filling cell invasion: Simulations and comparisons, submitted

Many reaction-diffusion models produce travelling wave solutions that can be interpreted as waves of invasion in biological scenarios
such as wound healing or tumour growth. These partial differential equation models have since been adapted to describe the
interactions between cells and extracellular matrix (ECM), using a variety of different underlying assumptions.
In this work, we derive a system of reaction-diffusion equations, with cross-species density-dependent diffusion,
by coarse-graining an agent-based, volume-filling model of cell invasion into ECM. We study the resulting travelling
wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic
comparison between the behaviours observed in this model and those predicted by simpler models in the literature which do
not take into account volume-filling effects in the same way. Our study justifies the use of some of these simpler,
more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes,
but it also reveals some interesting properties arising from the introduction of cell and ECM volume-filling effects,
where standard model simplifications might not be appropriate.

61 L. Almeida, J. Denis, N. Ferrand, T. Lorenzi, M. Sabbah, C. Villa, Evolutionary dynamics of glucose-deprived cancer cells: insights from experimentally-informed mathematical modelling, submitted

Glucose is a primary energy source for cancer cells. Several lines of evidence support the idea that monocarboxylate transporters, such as MCT1,
elicit metabolic reprogramming of cancer cells in glucose-poor environments, allowing them to reuse lactate, a byproduct of glucose metabolism,
as an alternative energy source with serious consequences for disease progression. We employ a synergistic experimental and mathematical
modelling approach to explore the evolutionary processes at the root of cancer cell adaptation to glucose deprivation, with particular focus
on the mechanisms underlying the increase in MCT1 expression observed in glucose-deprived aggressive cancer cells.
Data from in vitro experiments on breast cancer cells are used to inform and calibrate a mathematical model that comprises a
partial integro-differential equation for the dynamics of a population of cancer cells structured by the level of MCT1 expression.
Analytical and numerical results of this model indicate that environment-induced changes in MCT1 expression mediated by
lactate-associated signalling pathways enable a prompt adaptive response of glucose-deprived cancer cells,
whilst spontaneous changes due to non-genetic instability create the substrate for environmental selection to act upon,
speeding up the selective sweep underlying cancer cell adaptation to glucose deprivation, and may constitute a long-term bet-hedging mechanism.

60 A. Lasri Doukkali, T. Lorenzi, B.J. Parcell, J.L. Rohn, R. Bowness, A hybrid individual-based mathematical model to study bladder infections, Front. Appl. Math. Stat., 9, doi: 10.3389/fams.2023.1090334, 2023

Bladder infections are common, affecting millions each year, and are often recurrent problems.
We have developed a spatial mathematical framework consisting of a hybrid individual-based model to simulate these
infections in order to understand more about the bacterial mechanisms and immune dynamics. We integrate a varying bacterial replication rate and model bacterial
shedding as an immune mechanism. We investigate the effect that varying the initial bacterial load
has on infection outcome, where we find that higher bacterial burden leads to poorer outcomes,
but also find that only a single bacterium is needed to establish infection in some cases. We
also simulate an immunocompromised environment, confirming the intuitive result that bacterial
spread typically progresses at a higher rate. With future model developments, this framework is capable of providing new clinical insight into bladder infections.

59 L. Almeida, C. Audebert, E. Leschiera, T. Lorenzi, A hybrid discrete-continuum modelling approach to explore the impact of T-cell infiltration on anti-tumour immune response, Bull. Math. Biol., 84:141, 2022

We present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells,
which play a pivotal role in the immune response against tumours. In this framework, tumour cells and T cells are modelled as individual agents
while chemokines that drive the chemotactic movement of T cells towards the tumour are modelled as a continuum. We formally derive the continuum
counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells,
a partial differential equation for the density of T cells, and a partial differential equation for the concentration of chemokines. We report
on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions
of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of infiltration
of T cells into the tumour and elucidate how T-cell infiltration shapes anti-tumour immune response. Moreover, to present a proof of concept for
the idea that, exploiting the computational efficiency of the continuum model, extensive numerical simulations could be carried out,
we investigate the impact of T-cell infiltration on the response of tumour cells to different types of anti-cancer immunotherapy.

58 F.R. Macfarlane, T. Lorenzi, K.J. Painter, The impact of phenotypic heterogeneity on chemotactic self-organisation, Bull. Math. Biol., 84:143, 2022

The capacity to aggregate through chemosensitive movement forms a paradigm of self-organisation, with examples
spanning cellular and animal systems. A basic mechanism assumes a phenotypically homogeneous population that
secretes its own attractant, with the well known system introduced more than five decades ago by Keller and
Segel proving resolutely popular in modelling studies. The typical assumption of population phenotypic
homogeneity, however, often lies at odds with the heterogeneity of natural systems, where populations
may comprise distinct phenotypes that vary according to their chemotactic ability, attractant secretion, etc.
To initiate an understanding into how this diversity can impact on autoaggregation, we propose a simple extension
to the classical Keller and Segel model, in which the population is divided into two distinct phenotypes:
those performing chemotaxis and those producing attractant. Using a combination of linear stability analysis
and numerical simulations, we demonstrate that switching between these phenotypic states alters the capacity
of a population to self-aggregate. Further, we show that switching based on the local environment (population density or chemoattractant level)
leads to diverse patterning and provides a route through which a population can effectively curb the size and density of an aggregate.
We discuss the results in the context of real world examples of chemotactic aggregation, as well as theoretical aspects
of the model such as global existence and blow-up of solutions.

57 F.R. Macfarlane, X. Ruan, T. Lorenzi, Individual-based and continuum models of phenotypically heterogeneous growing cell populations, AIMS Bioeng., 9, 68-92, 2022

Existing comparative studies between individual-based models of growing cell populations and their continuum counterparts
have mainly been focused on homogeneous populations, in which all cells have the same phenotypic characteristics.
However, significant intercellular phenotypic variability is commonly observed in cellular systems. In light of these considerations,
we develop here an individual-based model for the growth of phenotypically heterogeneous cell populations. In this model, the phenotypic
state of each cell is described by a structuring variable that captures intercellular variability in cell proliferation and migration rates.
The model tracks the spatial evolutionary dynamics of single cells, which undergo pressure-dependent proliferation, heritable phenotypic changes
and directional movement in response to pressure differentials. We formally show that the continuum limit of this model comprises a non-local
partial differential equation for the cell population density function, which generalises earlier models of growing cell populations.
We report on the results of numerical simulations of the individual-based model which illustrate how proliferation-migration tradeoffs
shaping the evolutionary dynamics of single cells can lead to the formation, at the population level, of travelling waves whereby
highly-mobile cells locally dominate at the invasive front, while more-proliferative cells are found at the rear. Moreover,
we demonstrate that there is an excellent quantitative agreement between these results and the results of numerical simulations
and formal travelling-wave analysis of the continuum model, when sufficiently large cell numbers are considered.
We also provide numerical evidence of scenarios in which the predictions of the two models may differ due to demographic
stochasticity, which cannot be captured by the continuum model. This indicates the importance of integrating individual-based
and continuum approaches when modelling the growth of phenotypically heterogeneous cell populations.

56 T. Lorenzi, Cancer modelling as fertile ground for new mathematical challenges, Phys. Life Rev., 40, 3-5, 2022

55 L. Almeida, C. Audebert, E. Leschiera, T. Lorenzi, Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells, Math. Med. Biol., dqac017, 2023

We present an individual-based model for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes (CTLs) and tumour cells.
In this model, every cell is viewed as an individual agent whose phenotypic state is modelled by a discrete variable. For tumour cells
this variable represents a parameterisation of the antigen expression profiles, while for CTLs it represents a parameterisation of the
target antigens of T-cell receptors (TCRs). We formally derive the deterministic continuum limit of this individual-based model,
which comprises a non-local partial differential equation for the phenotype distribution of tumour cells coupled with an integro-differential
equation for the phenotype distribution of CTLs. The biologically relevant homogeneous steady-state solutions of the continuum model equations
are found. The linear-stability analysis of these steady-state solutions is then carried out in order to identify possible conditions
on the model parameters that may lead to different outcomes of immune competition and to the emergence of patterns of phenotypic
coevolution between tumour cells and CTLs. We report on computational results of the individual-based model, and show that there
is a good agreement between them and analytical and numerical results of the continuum model. These results shed light on the way
in which different parameters affect the coevolutionary dynamics between tumour cells and CTLs. Moreover, they support the idea that
TCR-tumour antigen binding affinity may be a good intervention target for immunotherapy and offer a theoretical basis for the development
of anti-cancer therapy aiming at engineering TCRs so as to shape their affinity for cancer targets.

54 T. Lorenzi, K.J. Painter, Trade-offs between chemotaxis and proliferation shape the phenotypic structuring of invading waves, Int. J. Non Linear Mech., 139:103885, 2022

Chemotaxis-driven invasions have been proposed across a broad spectrum of biological processes, from cancer to ecology.
The influential system of equations introduced by Keller and Segel has proven a popular choice in the modelling of such phenomena,
but in its original form restricts to a homogeneous population. To account for the possibility of phenotypic heterogeneity,
we extend to the case of a population continuously structured across space, time and phenotype, where the latter determines variation in chemotactic
responsiveness, proliferation rate, and the level of chemical environment modulation. The extended model considered here comprises a non-local partial
differential equation for the local phenotype distribution of cells which is coupled, through an integral term, with a differential equation for the
concentration of an attractant, which is sensed and degraded by the cells. In the framework of this model, we concentrate on a chemotaxis/proliferation
trade-off scenario, where the cell phenotypes span a spectrum of states from highly-chemotactic but minimally-proliferative to minimally-chemotactic but
highly-proliferative. Using a combination of numerical simulation and formal asymptotic analysis, we explore the properties of travelling-wave solutions.
The results of our study demonstrate how incorporating phenotypic heterogeneity may lead to a highly-structured wave profile, where cells in different
phenotypic states dominate different spatial positions across the invading wave, and clarify how the phenotypic structuring of the wave can be shaped by
trade-offs between chemotaxis and proliferation.

53 C. Colson, F. Sánchez-Garduño, H.M. Byrne, P.K. Maini, T. Lorenzi, Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion, Proc. Roy. Soc. A, 477:20210593, 2021

In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion.
We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue.
These types differ according to whether the density of ECM far ahead of the wave front is maximal or not.
In the former case, we use a shooting argument to prove that there exists a unique travelling wave solution for any positive propagation speed.
In the latter case, we further develop this argument to prove that there exists a unique travelling wave solution for any propagation speed greater
than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that
the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.

52 E. Leschiera, T. Lorenzi, S. Shen, L. Almeida, C. Audebert,
A mathematical model to study the impact of intra-tumour heterogeneity
on anti-tumour CD8^{+} T cell immune response, J. Theor. Biol., 538:111028, 2022

Intra-tumour heterogeneity (ITH) has a strong impact on the efficacy of the immune response against solid
tumours. The number of sub-populations of cancer cells expressing different antigens and the percentage
of immunogenic cells (i.e. tumour cells that are effectively targeted by immune cells) in a tumour are both
expressions of ITH. Here, we present a spatially explicit stochastic individual-based model of the interaction
dynamics between tumour cells and CD8+ T cells, which makes it possible to dissect out the specific impact
of these two expressions of ITH on anti-tumour immune response. The set-up of numerical simulations of
the model is defined so as to mimic scenarios considered in previous experimental studies. Moreover, the
ability of the model to qualitatively reproduce experimental observations of successful and unsuccessful
immune surveillance is demonstrated. First, the results of numerical simulations of this model indicate
that the presence of a larger number of sub-populations of tumour cells that express different antigens
is associated with a reduced ability of CD8^{+} T cells to mount an effective anti-tumour immune response.
Secondly, the presence of a larger percentage of tumour cells that are not effectively targeted by CD8^{+} T cells
may reduce the effectiveness of anti-tumour immunity. Ultimately, the mathematical model presented in
this paper may provide a framework to help biologists and clinicians to better understand the mechanisms
that are responsible for the emergence of different outcomes of immunotherapy.

51 C. Giverso, T. Lorenzi, L. Preziosi, Effective interface conditions for continuum mechanical models describing the invasion of multiple cell populations through thin membranes, Appl. Math. Letters, 125:107708, 2022

We consider a continuum mechanical model for the migration of multiple cell populations through parts of tissue separated by thin membranes.
In this model, cells belonging to different populations may be characterised by different proliferative abilities and mobility,
which may vary from part to part of the tissue, as well as by different invasion potentials within the membranes.
The original transmission problem, consisting of a set of mass balance equations for the volume fraction of cells of every population
complemented with continuity of stresses and mass flux across the surfaces of the membranes, is then reduced to a limiting transmission
problem whereby each thin membrane is replaced by an effective interface. In order to close the limiting problem, a set of
biophysically-consistent transmission conditions is derived through a formal asymptotic method. Models based on such a limiting
transmission problem may find fruitful application in a variety of research areas in the biological and medical sciences,
including developmental biology, immunology and cancer growth and invasion.

50 T. Lorenzi, A. Pugliese, M. Sensi, A. Zardini, Evolutionary dynamics in an SI epidemic model with phenotype-structured susceptible compartment, J. Math. Biol., 83:72, 2021

We present an SI epidemic model whereby a continuous structuring variable captures variability in proliferative potential and resistance
to infection among susceptible individuals. The occurrence of heritable, spontaneous changes in these phenotypic characteristics and the
presence of a fitness trade-off between resistance to infection and proliferative potential are explicitly incorporated into the model.
The model comprises an ordinary differential equation for the number of infected individuals that is coupled with a partial integrodifferential
equation for the population density function of susceptible individuals through an integral term. The expression for the basic reproduction number
R0 is derived, the disease-free equilibrium and endemic equilibrium of the model are characterised and a threshold theorem involving R0 is proved.
Analytical results are integrated with the results of numerical simulations of a calibrated version of the model based on the results of artificial
selection experiments in a host-parasite system. The results of our mathematical study disentangle the impact of different evolutionary parameters
on the spread of infectious diseases and the consequent phenotypic adaption of susceptible individuals. In particular, these results provide a theoretical
basis for the observation that infectious diseases exerting stronger selective pressures on susceptible individuals and being characterised by higher
infection rates are more likely to spread. Moreover, our results indicate that heritable, spontaneous phenotypic changes in proliferative potential
and resistance to infection can either promote or prevent the spread of infectious diseases depending on the strength of selection acting on susceptible
individuals prior to infection. Finally, we demonstrate that, when an endemic equilibrium is established, higher levels of resistance to infection and
lower degrees of phenotypic heterogeneity among susceptible individuals are to be expected in the presence of infections which are characterised by lower
rates of death and exert stronger selective pressures.

49 C. Villa, M.A.J. Chaplain, A. Gerisch, T. Lorenzi, Mechanical models of pattern and form in biological tissues: the role of stress-strain constitutive equations, Bull. Math. Biol., 83:80, 2021

Mechanical and mechanochemical models of pattern formation in biological tissues
have been used to study a variety of biomedical systems, particularly in developmental
biology, and describe the physical interactions between cells and their local
surroundings. These models in their original form consist of a balance equation for the
cell density, a balance equation for the density of the extracellular matrix (ECM), and a
force-balance equation describing the mechanical equilibrium of the cell-ECM system.
Under the assumption that the cell-ECM system can be regarded as an isotropic linear
viscoelastic material, the force-balance equation is often defined using the Kelvin-Voigt
model of linear viscoelasticity to represent the stress-strain relation of the ECM.
However, due to the multifaceted bio-physical nature of the ECM constituents, there
are rheological aspects that cannot be effectively captured by this model and,
therefore, depending on the pattern formation process and the type of biological tissue
considered, other constitutive models of linear viscoelasticity may be better suited. In
this paper, we systematically assess the pattern formation potential of different stress-strain
constitutive equations for the ECM within a mechanical model of pattern
formation in biological tissues. The results obtained through linear stability analysis and
the dispersion relations derived therefrom support the idea that fluid-like constitutive
models, such as the Maxwell model and the Jeffrey model, have a pattern formation
potential much higher than solid-like models, such as the Kelvin-Voigt model and the
standard linear solid model. This is confirmed by the results of numerical simulations,
which demonstrate that, all else being equal, spatial patterns emerge in the case
where the Maxwell model is used to represent the stress-strain relation of the ECM,
while no patterns are observed when the Kelvin-Voigt model is employed. Our findings
suggest that further empirical work is required to acquire detailed quantitative
information on the mechanical properties of components of the ECM in different
biological tissues in order to furnish mechanical and mechanochemical models of
pattern formation with stress-strain constitutive equations for the ECM that provide a
more faithful representation of the underlying tissue rheology.

48 G. Fiandaca, M. Delitala, T. Lorenzi, A mathematical study of the influence of hypoxia and acidity on the evolutionary dynamics of cancer, Bull. Math. Biol., 83:83, 2021

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype.
As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic
and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to
study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours.
The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of
cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism:
oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate
the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments.
Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients
which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour
phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia
and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of
cancer cells.

47 G. Estrada-Rodriguez, T. Lorenzi, Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions, Eur. J. Appl. Math., 34, 1-27, 2023

Experimental results on the immune response to cancer indicate that activation of cytotoxic T lymphocytes (CTLs) through interactions with dendritic cells (DCs)
can trigger a change in CTL migration patterns. In particular, while CTLs in the pre-activation state move in a non-local search pattern,
the search pattern of activated CTLs is more localised. In this paper, we develop a kinetic model for such a switch in CTL migration modes.
The model is formulated as a coupled system of balance equations for the one-particle distribution functions of CTLs in the pre-activation state,
activated CTLs and DCs. CTL activation is modelled via binary interactions between CTLs in the pre-activation state and DCs.
Moreover, cell motion is represented as a velocity-jump process, with the running time of CTLs in the pre-activation state following a long-tailed distribution,
which is consistent with a Lévy walk, and the running time of activated CTLs following a Poisson distribution, which corresponds to Brownian motion.
We formally show that the macroscopic limit of the model comprises a coupled system of balance equations for the cell densities whereby activated CTL movement
is described via a classical diffusion term, whilst a fractional diffusion term describes the movement of CTLs in the pre-activation state.
The modelling approach presented here and its possible generalisations are expected to find applications in the study of the immune response
to cancer and in other biological contexts in which switch from non-local to localised migration patterns occurs.

46 T. Lorenzi, B. Perthame, X. Ruan, Invasion fronts and adaptive dynamics in a model for the growth of cell populations with heterogeneous mobility, Eur. J. Appl. Math., 33, 766-783, 2022

We consider a model for the dynamics of growing cell populations with heterogeneous mobility and proliferation rate.
The cell phenotypic state is described by a continuous structuring variable and the evolution of the local cell population density
(*i.e.* the cell phenotypic distribution at each spatial position) is governed by a non-local advection-reaction-diffusion equation.
We report on the results of numerical simulations showing that, in the case where the cell mobility is bounded, compactly supported travelling fronts emerge.
More mobile phenotypic variants occupy the front edge, whereas more proliferative phenotypic variants are selected at the back of the front.
In order to explain such numerical results, we carry out formal asymptotic analysis of the model equation using a Hamilton-Jacobi approach.
In particular, we show that the locally dominant phenotypic trait (*i.e.* the maximum point of the local cell population density along
the phenotypic dimension) satisfies a generalised Burgers' equation with source term, we construct travelling-front solutions of such transport
equation and characterise the corresponding minimal speed. Moreover, we show that, when the cell mobility is unbounded, front edge acceleration
and formation of stretching fronts may occur. We briefly discuss the implications of our results in the context of glioma growth.

45 F.R. Macfarlane, M.A.J. Chaplain, T. Lorenzi, A hybrid discrete-continuum approach to model Turing pattern formation, Math. Biosci. Eng., 17, 7442-7479, 2020

Since its introduction in 1952, Turing's (pre-)pattern theory (“the chemical basis of morphogenesis”) has been widely applied to a
number of areas in developmental biology. The related pattern formation models normally comprise a system of reaction-diffusion equations
for interacting chemical species (“morphogens”), whose heterogeneous distribution in some spatial domain acts as a template for
cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern.
Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism.
In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system
for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are
governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with morphogens in their local
area through either of two forms of chemically-dependent cell action: chemotaxis and chemically-controlled proliferation.
We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains.
In both cases, we formally derive the corresponding deterministic continuum limit and show that there is an excellent
quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart,
when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework,
with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future.

44 A. Ardaševa, R.A. Gatenby, A.R.A. Anderson, H.M. Byrne, P.K. Maini, T. Lorenzi, A comparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environments, Phys. Rev. E, 102:042404, 2020

Deterministic continuum models formulated as non-local partial differential equations for the evolutionary dynamics of populations structured
by phenotypic traits have been used recently to address open questions concerning the adaptation of asexual species to periodically fluctuating
environmental conditions. These models are usually defined on the basis of population-scale phenomenological assumptions and cannot capture
adaptive phenomena that are driven by stochastic variability in the evolutionary paths of single individuals. In light of these considerations,
in this paper we develop a stochastic individual-based model for the coevolution between two competing phenotype-structured cell populations
that are exposed to time-varying nutrient levels and undergo spontaneous, heritable phenotypic variations with different probabilities.
Here, the evolution of every cell is described by a set of rules that result in a discrete-time branching random walk on the space of phenotypic states,
and nutrient levels are governed by a difference equation in which a sink term models nutrient consumption by the cells. We formally show that
the deterministic continuum counterpart of this model comprises a system of non-local partial differential equations for the cell population density
functions coupled with an ordinary differential equation for the nutrient concentration. We compare the individual-based model and its continuum analogue,
focussing on scenarios whereby the predictions of the two models differ. The results obtained clarify the conditions under which significant differences
between the two models can emerge due to stochastic effects associated with small population levels. In particular, these differences arise in the presence
of low probabilities of phenotypic variation, and become more apparent when the two populations are characterised by less fit initial mean phenotypes and
smaller initial levels of phenotypic heterogeneity. The agreement between the two modelling approaches is also dependent on the initial proportions of the
two populations. As an illustrative example, we demonstrate the implications of these results in the mathematical modelling of metastatic colonisation
of distant organs.

43 F. Bubba, T. Lorenzi, F.R. Macfarlane, From a discrete model of chemotaxis with volume-filling to a generalised Patlak-Keller-Segel model, Proc. Roy. Soc. A, 476:20190871, 2020

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is
described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at
high cell densities (*i.e.* volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density.
We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum
limit of this discrete model. The family of steady-state solutions of such a generalised PKS model are characterised and the conditions for the emergence
of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of
the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that
there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the
two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.

42 C. Villa, M.A.J. Chaplain, T. Lorenzi, Evolutionary dynamics in vascularised tumours under chemotherapy: Mathematical modelling, asymptotic analysis and numerical simulations, Vietnam J. Math., 49, 143–167, 2021

We consider a mathematical model for the evolutionary dynamics of tumour cells in vascularised tumours under chemotherapy.
The model comprises a system of coupled partial integro-differential equations for the phenotypic distribution of tumour cells,
the concentration of oxygen and the concentration of a chemotherapeutic agent. In order to disentangle the impact of different evolutionary parameters
on the emergence of intra-tumour phenotypic heterogeneity and the development of resistance to chemotherapy, we construct explicit solutions
to the equation for the phenotypic distribution of tumour cells and provide a detailed quantitative characterisation of the long-time asymptotic behaviour
of such solutions. Analytical results are integrated with numerical simulations of a calibrated version of the model based on biologically consistent
parameter values. The results obtained provide a theoretical explanation for the observation that the phenotypic properties of tumour cells
in vascularised tumours vary with the distance from the blood vessels. Moreover, we demonstrate that lower oxygen levels may correlate with higher
levels of phenotypic variability, which suggests that the presence of hypoxic regions supports intra-tumour phenotypic heterogeneity.
Finally, the results of our analysis put on a rigorous mathematical basis the idea, previously suggested by formal asymptotic results
and numerical simulations, that hypoxia favours the selection for chemoresistant phenotypic variants prior to treatment. Consequently, this facilitates
the development of resistance following chemotherapy.

41 A. Ardaševa, R.A. Gatenby, A.R.A. Anderson, H.M. Byrne, P.K. Maini, T. Lorenzi, A mathematical dissection of the adaptation of cell populations to fluctuating oxygen levels, Bull. Math. Biol., 82:81, 2020

The highly disordered network of blood vessels that arises from tumour angiogenesis results in variations in blood flow characterised
by cycles of perfusion, cessation of flow, and then re-perfusion, which generate fluctuations in the delivery of oxygen into the tumour tissue.
This brings about regions of chronic hypoxia (*i.e.* sustained low oxygen levels) and cycling hypoxia (*i.e.* alternating phases of
low and relatively higher oxygen levels) within vascularised tumours, and makes it necessary for cancer cells to adapt to fluctuating environmental
conditions. Here we use a phenotype-structured model to dissect the evolutionary dynamics of cell populations exposed to fluctuating oxygen levels.
In this mathematical model, the phenotypic state of every cell is described by a continuous variable that provides a simple representation of its
metabolic phenotype, ranging from fully oxidative to fully glycolytic, and cells are grouped into two competing populations that undergo heritable,
spontaneous phenotypic variations at different rates. Model simulations indicate that, depending on the rate at which oxygen is consumed by the cells,
nonlinear dynamic interactions between cells and oxygen can stimulate chronic hypoxia and cycling hypoxia. Moreover, the model supports the idea that
under chronic-hypoxic conditions lower rates of phenotypic variation lead to a competitive advantage, whereas higher rates of phenotypic variation can
confer a competitive advantage under cycling-hypoxic conditions. In the latter case, the numerical results obtained show that bet-hedging evolutionary
strategies, whereby cells switch between oxidative and glycolytic metabolic phenotypes, can spontaneously emerge. Finally, the model suggests that,
under cycling hypoxia, higher rates of oxygen consumption by the cells and higher fitness costs associated with glycolytic metabolism can promote the
transient coexistence of competing cell populations that undergo heritable, spontaneous phenotypic variations at different rates. We explain how the
results of our theoretical study can shed light on the evolutionary process that may underpin the emergence of phenotypic heterogeneity in vascularised tumours.

40 C. Villa, M.A.J. Chaplain, T. Lorenzi, Modelling the emergence of phenotypic heterogeneity in vascularised tumours, SIAM J. Appl. Math., 81, 434–453, 2021

We present a mathematical study of the emergence of phenotypic heterogeneity in vascularised tumours.
Our study is based on formal asymptotic analysis and numerical simulations of a system of non-local parabolic equations that describes
the phenotypic evolution of tumour cells and their nonlinear dynamic interactions with oxygen, which is released from the intratumoural
vascular network. Numerical simulations are carried out both in the case of arbitrary distributions of intratumour blood vessels and
in the case where the intratumoural vascular network is reconstructed from clinical images obtained using dynamic optical coherence tomography.
The results obtained support a more in-depth theoretical understanding of the eco-evolutionary process which underpins the emergence of phenotypic
heterogeneity in vascularised tumours. In particular, our results offer a theoretical basis for empirical evidence indicating that the phenotypic properties
of cancer cells in vascularised tumours vary with the distance from the blood vessels, and establish a relation between the degree of tumour tissue
vascularisation and the level of intratumour phenotypic heterogeneity.

39 A. Ardaševa, R.A. Gatenby, A.R.A. Anderson, H.M. Byrne, P.K. Maini, T. Lorenzi, Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments, J. Math. Biol., 80, 775–807, 2020

Living species, ranging from bacteria to animals, exist in environmental conditions that exhibit spatial and temporal heterogeneity
which requires them to adapt. Risk-spreading through spontaneous phenotypic variations is a known concept in ecology, which is used to
explain how species may survive when faced with the evolutionary risks associated with temporally varying environments. In order to support a deeper
understanding of the adaptive role of spontaneous phenotypic variations in fluctuating environments, we consider a system of non-local partial differential
equations modelling the evolutionary dynamics of two competing phenotype-structured populations in the presence of periodically oscillating nutrient levels.
The two populations undergo spontaneous phenotypic variations at different rates. The phenotypic state of each individual is represented by a continuous variable,
and the phenotypic landscape of the populations evolves in time due to variations in the nutrient level. Exploiting the analytical tractability of our model,
we study the long-time behaviour of the solutions to obtain a detailed mathematical depiction of evolutionary dynamics. The results suggest that when nutrient
levels undergo small and slow oscillations, it is evolutionarily more convenient to rarely undergo spontaneous phenotypic variations. Conversely, under relatively
large and fast periodic oscillations in the nutrient levels, which bring about alternating cycles of starvation and nutrient abundance, higher rates of spontaneous
phenotypic variations confer a competitive advantage. We discuss the implications of our results in the context of cancer metabolism.

38 T. Lorenzi, C. Pouchol, Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks, Nonlinearity, 33:5791, 2020

We study the long-time behaviour of phenotype-structured models describing the evolutionary
dynamics of asexual populations whose phenotypic fitness landscape is characterised by multiple peaks.
First we consider the case where phenotypic changes do not occur, and then we include the effect of heritable phenotypic changes.
In the former case the model is formulated as an integrodifferential equation for the phenotype distribution of the individuals in
the population, whereas in the latter case the evolution of the phenotype distribution is governed by a non-local parabolic equation whereby
a linear diffusion operator captures the presence of phenotypic changes. We prove that the long-time limit of the solution to the
integrodifferential equation is unique and given by a measure consisting of a weighted sum of Dirac masses centred at the peaks of the
phenotypic fitness landscape. We also derive an explicit formula to compute the weights in front of the Dirac masses.
Moreover, we demonstrate that the long-time solution of the non-local parabolic equation exhibits a qualitatively similar
behaviour in the asymptotic regime where the diffusion coefficient modelling the rate of phenotypic change tends to zero.
However, we show that the limit measure of the non-local parabolic equation may consist of less Dirac masses, and we provide
a sufficient criterion to identify the positions of their centres. Finally, we carry out a detailed characterisation of the speed
of convergence of the integral of the solution (*i.e.* the population size) to its long-time limit for both models.
Taken together, our results support a more in-depth theoretical understanding of the conditions leading to the emergence
of stable phenotypic polymorphism in asexual populations.

37 F.R. Macfarlane, M.A.J. Chaplain, T. Lorenzi, A stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumours, J. Theor. Biol., 480, 43–55, 2019

Spatial interactions between cancer and immune cells, as well as the recognition of tumour antigens by cells of the immune system,
play a key role in the immune response against solid tumours. The existing mathematical models generally focus only on one of these key aspects.
We present here a spatial stochastic individual-based model that explicitly captures antigen expression and recognition. In our model, each cancer
cell is characterised by an antigen profile which can change over time due to either epimutations or mutations. The immune response against the cancer
cells is initiated by the dendritic cells that recognise the tumour antigens and present them to the cytotoxic T cells. Consequently, T cells become
activated against the tumour cells expressing such antigens. Moreover, the differences in movement between inactive and active immune cells are explicitly
taken into account by the model. Computational simulations of our model clarify the conditions for the emergence of tumour clearance, dormancy or escape,
and allow us to assess the impact of antigenic heterogeneity of cancer cells on the efficacy of immune action. Ultimately, our results highlight the complex
interplay between spatial interactions and adaptive mechanisms that underpins the immune response against solid tumours, and suggest how this may be exploited
to further develop cancer immunotherapies.

36 T. Lorenzi, P.J. Murray, M. Ptashnyk, From individual-based mechanical models of multicellular systems to free-boundary problems, Interface Free Bound., 22, 205-244, 2020

In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell
populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered
whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described
via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart
of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the
parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove a local
existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the
case where the cellular interaction forces are described by the celebrated Johnson-Kendall-Roberts model of elastic contact,
which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the
simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis.

35 R.E.A Stace, T. Stiehl, M.A.J. Chaplain, A. Marciniak-Czochra, T. Lorenzi, Discrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapy, Math. Mod. Nat. Phen., 15:14, 2020

We present a stochastic individual-based model for the phenotypic evolution of cancer
cell populations under chemotherapy. In particular, we consider the case of
combination cancer therapy whereby a chemotherapeutic agent is administered as the
primary treatment and an epigenetic drug is used as an adjuvant treatment. The cell
population is structured by the expression level of a gene that controls cell proliferation
and chemoresistance. In order to obtain an analytical description of evolutionary
dynamics, we formally derive a deterministic continuum counterpart of this discrete
model, which is given by a nonlocal parabolic equation for the cell population density
function. Integrating computational simulations of the individual-based model with
analysis of the corresponding continuum model, we perform a complete exploration of
the model parameter space. We show that harsher environmental conditions and
higher probabilities of spontaneous epimutation can lead to more effective
chemotherapy, and we demonstrate the existence of an inverse relationship between
the efficacy of the epigenetic drug and the probability of spontaneous epimutation.
Taken together, the outcomes of our model provide theoretical ground for the
development of anticancer protocols that use lower concentrations of
chemotherapeutic agents in combination with epigenetic drugs capable of promoting
the re-expression of epigenetically regulated genes.

34 M.A.J. Chaplain, C. Giverso, T. Lorenzi, L. Preziosi, Derivation and application of effective interface conditions for continuum mechanical models of cell invasion through thin membranes, SIAM J. Appl. Math., 79, 2011–2031, 2019

We consider a continuum mechanical model of cell invasion through thin membranes.
The model consists of a transmission problem for cell volume fraction complemented with continuity of
stresses and mass flux across the surfaces of the membranes. We reduce the original problem to a limiting transmission
problem whereby each thin membrane is replaced by an effective interface, and we develop a formal asymptotic method that enables
the derivation of a set of biophysically consistent transmission conditions to close the limiting problem. The formal results obtained
are validated via numerical simulations showing that the relative error between the solutions to the original transmission problem and
the solutions to the limiting problem vanishes when the thickness of the membranes tends to zero. In order to show potential applications
of our effective interface conditions, we employ the limiting transmission problem to model cancer cell invasion through the basement
membrane and the metastatic spread of ovarian carcinoma.

33 M.A.J. Chaplain, T. Lorenzi, F.R. Macfarlane, Bridging the gap between individual-based and continuum models of growing cell populations, J. Math. Biol., 80, 343-371, 2020

Continuum models for the spatial dynamics of growing cell populations have been widely used to
investigate the mechanisms underpinning tissue development and tumour invasion.
These models consist of nonlinear partial differential equations that describe the evolution of cellular densities
in response to pressure gradients generated by population growth.
Little prior work has explored the relation between such continuum models and related single-cell-based models.
We present here a simple stochastic individual-based model for the spatial dynamics of multicellular systems whereby
cells undergo pressure-driven movement and pressure-dependent proliferation. We show that nonlinear partial differential
equations commonly used to model the spatial dynamics of growing cell populations can be formally derived from the branching
random walk that underlies our discrete model. Moreover, we carry out a systematic comparison between the individual-based model
and its continuum counterparts, both in the case of one single cell population and in the case of multiple cell populations
with different biophysical properties. The outcomes of our comparative study demonstrate that the results of computational
simulations of the individual-based model faithfully mirror the qualitative and quantitative properties of the solutions
to the corresponding nonlinear partial differential equations. Ultimately, these results illustrate how the simple
rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial
patterns of population growth observed in continuum models.

32 L.C. Franssen, T. Lorenzi, A. Burgess, M.A.J. Chaplain, A mathematical framework for modelling the metastatic spread of cancer, Bull. Math. Biol., 81, 1965-2010, 2019

Cancer is a complex disease that starts with mutations of key genes in one cell or a small
group of cells at a primary site in the body. If these cancer cells continue to grow successfully and, at some
later stage, invade the surrounding tissue and acquire a vascular network (tumour-induced angiogenesis),
they can spread to distant secondary sites in the body. This process, known as metastatic spread, is
responsible for around 90% of deaths from cancer and is one of the so-called hallmarks of cancer.
To shed light on the metastatic process, we present a mathematical modelling framework that captures
for the first time the interconnected processes of invasion and metastatic spread of individual cancer
cells in a spatially explicit manner — a multi-grid, hybrid, individual-based approach. This framework
accounts for the spatio-temporal evolution of mesenchymal- and epithelial-like cancer cells,
membrane type-1 matrix metalloproteinase (MT1-MMP) and the diffusible matrix metalloproteinase-2 (MMP-2), and for their interactions with the extracellular matrix.
Using computational simulations, we demonstrate that our model captures all the key steps of
the invasion-metastasis cascade, *i.e.* invasion by both heterogeneous cancer cell clusters and by single
mesenchymal-like cancer cells; intravasation of these clusters and single cells both via active mechanisms
mediated by matrix-degrading enzymes and via passive shedding; circulation of cancer cell clusters and
single cancer cells in the vasculature with the associated risk of cell death and disaggregation of clusters;
extravasation of clusters and single cells; and metastatic growth at distant secondary sites in the body.
By faithfully reproducing experimental results, our simulations support the evidence-based hypothesis
that the membrane-bound MT1-MMP is the main driver of invasive spread rather than diffusible
matrix-degrading enzymes like MMP-2.

31 T. Lorenzi, A. Marciniak-Czochra, T. Stiehl, A structured population model of clonal selection in acute leukemias with multiple maturation stages, J. Math. Biol., 79, 1587-1621, 2019

Recent progress in genetic techniques has shed light on the complex co-evolution of malignant cell clones in leukemias.
However, several aspects of clonal selection still remain unclear. In this paper, we present a multi-compartmental continuously structured
population model of selection dynamics in acute leukemias, which consists of a system of coupled integro-differential equations. Our model
can be analysed in a more efficient way than classical models formulated in terms of ordinary differential equations. Exploiting the analytical
tractability of this model, we investigate how clonal selection is shaped by the self-renewal fraction and the proliferation rate of leukemic cells at
different maturation stages. We integrate analytical results with numerical solutions of a calibrated version of the model based on real patient data.
In summary, our mathematical results formalise the biological notion that clonal selection is driven by the self-renewal fraction of leukemic stem cells
and the clones that possess the highest value of this parameter are ultimately selected. Moreover, we demonstrate that the self-renewal fraction and
the proliferation rate of non-stem cells do not have a substantial impact on clonal selection. Taken together, our results indicate that interclonal
variability in the self-renewal fraction of leukemic stem cells provides the necessary substrate for clonal selection to act upon.

30 L. Almeida, P. Bagnerini, G. Fabrini, B.D. Hughes, T. Lorenzi, Evolution of cancer cell populations under cytotoxic therapy and treatment optimisation: insight from a phenotype-structured model, ESAIM Math. Model. Numer. Anal., 53, 1157-1190, 2019

We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to
the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the
cell population density function. We develop a novel method for constructing exact solutions to the model equation,
which allows for a systematic investigation of the way in which the size and the phenotypic composition of the cell
population change in response to variations of the drug dose and other evolutionary parameters. Moreover, we address
numerical optimal control for a calibrated version of the model based on biological data from the existing literature,
in order to identify the drug delivery schedule that makes it possible to minimise either the population size at the end
of the treatment or the average population size during the course of treatment.
The results obtained challenge the notion that traditional high-dose therapy represents a
'one-fits-all solution' in anticancer therapy by showing that the continuous administration
of a relatively low dose of the cytotoxic drug performs more closely to the optimal dosing
regimen to minimise the average size of the cancer cell population during the course of treatment.

29 T. Lorenzi, C. Venkataraman, A. Lorz, M.A.J. Chaplain, The role of spatial variations of abiotic factors in mediating intratumour phenotypic heterogeneity, J. Theor. Biol., 451, 101-110, 2018

We present here a space- and phenotype-structured model of selection dynamics between
cancer cells within a solid tumour. In the framework of this model, we combine
formal analyses with numerical simulations to investigate *in silico* the
role played by the spatial distribution of abiotic components of the tumour
microenvironment in mediating phenotypic selection of cancer cells. Numerical
simulations are performed both on the 3D geometry of an *in silico*
multicellular tumour spheroid and on the 3D geometry of an *in vivo* human
hepatic tumour, which was imaged using computerised tomography.
The results obtained show that inhomogeneities in the spatial distribution
of oxygen, currently observed in solid tumours, can promote the creation of
distinct local niches and lead to the selection of different phenotypic variants
within the same tumour. This process fosters the emergence of stable phenotypic
heterogeneity and supports the presence of hypoxic cells resistant to
cytotoxic therapy prior to treatment. Our theoretical results demonstrate
the importance of integrating spatial data with ecological principles when
evaluating the therapeutic response of solid tumours to cytotoxic therapy.

28 F.R. Macfarlane, T. Lorenzi, M.A.J. Chaplain, Modelling the immune response to cancer: an individual-based approach accounting for the difference in movement between inactive and activated T cells, Bull. Math. Biol., 80, 1539-1562, 2018

A growing body of experimental evidence indicates that immune cells
move in an unrestricted search pattern if they are in the pre-activated state, while
they tend to stay within a more restricted area upon activation induced by the
presence of tumour antigens. This change in movement is not often considered in
the existing mathematical models of the interactions between immune cells and
cancer cells. With the aim to fill such a gap in the existing literature, in this
work we present a spatially structured individual-based model of tumour-immune
competition that takes explicitly into account the difference in movement between
inactive and activated immune cells. In our model, a Lévy walk is used to capture
the movement of inactive immune cells, whereas Brownian motion is used to describe
the movement of antigen-activated immune cells. The effects of activation
of immune cells, the proliferation of cancer cells and the immune destruction of
cancer cells are also modelled. We illustrate the ability of our model to reproduce
qualitatively the spatial trajectories of immune cells observed in experimental data
of single cell tracking. Computational simulations of our model further clarify the
conditions for the onset of a successful immune action against cancer cells and
suggest possible targets to improve the efficacy of cancer immunotherapy. Overall,
our theoretical work highlights the importance of taking into account spatial
interactions when modelling the immune response to cancer cells.

27 T. Lorenzi, A. Lorz, B. Perthame, On interfaces between cell populations with different mobilities, Kinet. Relat. Models, 10, 299-311, 2017

Partial differential equations describing the dynamics of cell population densities from a fluid
mechanical perspective can model the growth of avascular tumours.
In this framework, we consider a system of equations that
describes the interaction between a population of dividing cells and a population of non-dividing cells.
The two cell populations are characterised by different mobilities.
We present the results of numerical simulations displaying two-dimensional spherical waves
with sharp interfaces between dividing and non-dividing cells.
Furthermore, we numerically observe how different ratios between the mobilities
change the morphology of the interfaces, and lead to the emergence of nger-like patterns of invasion above a threshold.
Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions.

26 A.E.F. Burgess, T. Lorenzi, P.G. Schofield, S.F. Hubbard, M.A.J. Chaplain, Examining the role of individual movement in promoting coexistence in a spatially explicit prisoner's dilemma, J. Theor. Biol., 419, 323-332, 2017

The emergence of cooperation is a major conundrum of evolutionary biology. To unravel this evolutionary riddle,
several models have been developed within the theoretical framework of spatial game theory, focussing on the interactions between
two general classes of player, "cooperators" and "defectors". Generally, explicit movement in the spatial domain
is not considered in these models, with strategies moving via imitation or through colonisation of neighbouring sites.
We present here a spatially explicit stochastic individual-based model in which pure cooperators and defectors undergo
persistent spontaneous motion via diffusion and also directed movement in response to gradients of chemoattractants. Individual movement rules are derived from an
underlying system of reaction-diffusion-taxis partial differential equations which describes the dynamics
of the local number of individuals and the concentration of semiochemical. Local interactions are governed
by the payoff matrix of the classical prisoner's dilemma, and accumulated payoffs are translated into offspring.
We investigate the cases of both synchronous and non-synchronous generations. Focussing on an ecological scenario
where defectors are parasitic on cooperators, we find that spontaneous motion and semiochemical sensing bring about
self-generated patterns in which resident cooperators and parasitic defectors can coexist in proportions that fluctuate
about non-zero values. Remarkably, coexistence emerges as a genuine consequence of the natural tendency of cooperators to aggregate into clusters, without the need for them to find physical shelter or outrun the parasitic defectors.
This provides further evidence that spatial clustering enhances the benefits of mutual cooperation and plays a crucial role in preserving cooperative behaviours.

25 M. Delitala, T. Lorenzi, Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells, Math. Biosci. Engin., 14, 79-93, 2017

Accumulating evidence indicates that the interaction between epithelial
and mesenchymal cells plays a pivotal role in cancer development and
metastasis formation. Here we propose an integro-dierential model for the
co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model
takes into account the effects of chemotaxis, adhesive interactions between
epithelial-like cells, proliferation and competition for nutrients. We present a
sample of numerical results which display the emergence of spots, stripes and
honeycomb patterns, depending on parameters and initial data. These simulations
also suggest that epithelial-like and mesenchymal-like cells can segregate
when there is little competition for nutrients. Furthermore, our computational
results provide a possible explanation for how the concerted action between
epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation
of ring-like structures, which resemble the fibrous capsules frequently
observed in hepatic tumours.

24 A.E.F. Burgess, P.G. Schofield, S.F. Hubbard, M.A.J. Chaplain, T. Lorenzi, Dynamical patterns of coexisting strategies in a hybrid discrete-continuum spatial evolutionary game model, Math. Mod. Nat. Phen., 11, 49-64, 2016

We present a novel hybrid modelling framework that takes into account two aspects which have been
largely neglected in previous models of spatial evolutionary games: persistent spontaneous motion and chemotaxis.
A stochastic individual-based model is used to describe the player dynamics, whereas the evolution of the chemoattractant
is governed by a reaction-diffusion equation. The two models are coupled by deriving individual movement rules via the
discretisation of a taxis-diffusion equation which describes the evolution of the local number of players.
In this framework, individuals occupying the same position can engage in a two-player game, and are awarded a payoff,
in terms of reproductive fitness, according to their strategy. As an example, we let individuals play the Hawk-Dove game.
Numerical simulations illustrate how persistent spontaneous motion and chemotactic response can bring about self-generated
dynamical patterns that create favourable conditions for the coexistence of hawks and doves in situations in which the two
strategies cannot coexist otherwise. In this sense, our work offers a new perspective of research on spatial evolutionary games,
and provides a general formalism to study the dynamics of spatially-structured populations in biological and social contexts where
individual motion is likely to affect natural selection of behavioural traits.

23 T. Lorenzi, R.H. Chisholm, J. Clairambault, Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations, Biol. Direct, 11, 1-17, 2016

A thorough understanding of the ecological and evolutionary mechanisms that drive the
phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer
research community. In this respect, mathematical modelling can complement experimental
cancer research by offering alternative means of understanding the results of
*in vitro* and *in vivo* experiments, and by allowing for a quick and easy exploration
of a variety of biological scenarios through *in silico* studies.
To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here
a phenotype-structured model of evolutionary dynamics in a cancer cell population
which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows
us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size
of the cell population are shaped by the strength of natural selection, the rate of random epimutations,
the intensity of the competition for limited resources between cells, and the drug dose in use.
Our analytical results clarify the conditions for the successful adaptation of cancer cells faced
with environmental changes. Furthermore, the results of our analyses demonstrate that the same
cell population exposed to different concentrations of the same cytotoxic drug can take different
evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by
different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents
is more complex than a simple binary outcome, *i.e.*, extinction of sensitive cells and selection of
highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents
at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones.
Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols
that go beyond the 'maximum-tolerated-dose paradigm', as they may be more effective than traditional protocols
at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.

22 R.H. Chisholm, T. Lorenzi, J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: biological and mathematical assessment, theoretical treatment optimisation, Biochim. Biophys. Acta, Gen. Subj., 1860, 2627-2645, 2016

BACKGROUND: Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms
at the individual cell or cell population scale, relying on stochastically or
epigenetically varying expression of phenotypes at the single cell level,
and on the adaptability of tumours at the cell population level.

SCOPE OF REVIEW: We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies.

MAJOR CONCLUSIONS: Plasticity in cancer cells,*i.e.*, partial reversal to a stem-like status in individual cells and resulting
adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant
to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity
through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for
the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies.

GENERAL SIGNIFICANCE: Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics.

SCOPE OF REVIEW: We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies.

MAJOR CONCLUSIONS: Plasticity in cancer cells,

GENERAL SIGNIFICANCE: Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics.

21 R.H. Chisholm, T. Lorenzi, L. Desvillettes, B.D. Hughes, Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences, Z. angew. Math. Phys., 67, 1-34, 2016

Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species
that face environmental changes. In particular, empirical work has provided important insights
into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a
number of verbal explanations have emerged that are suited to logical testing by proof-of-concept
mathematical models. Here, we present a stochastic agent-based model and a related deterministic
integro-differential equation model for the evolution of a phenotype-structured population composed
of asexually-reproducing and competing organisms which are exposed to novel environmental conditions.
This setting has relevance to the study of biological systems where colonising asexual
populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion
and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene
expression can change the reproductive capacity of individuals within the population in the new
environment. Simulations and analyses of our models clarify the conditions under which certain
evolutionary paths are possible, and illustrate that whilst epigenetic mechanisms may facilitate
adaptation in asexual species faced with environmental change, they can also lead to a type of
"epigenetic load" and contribute to extinction. Moreover, our results offer a formal basis for the
claim that constant environments favour individuals with low rates of stochastic phenotypic variation.
Finally, our model provides a "proof of concept" of the verbal hypothesis that phenotypic
stability is a key driver in rescuing the adaptive potential of an asexual lineage, and supports the
notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic
instability and biased epimutations, and steer an asexual population back from the brink of
an evolutionary dead end.

20 R.H. Chisholm, T. Lorenzi, A. Lorz, Effects of an advection term in nonlocal Lotka-Volterra equations, Commun. Math. Sci., 14, 1181-1188, 2016

Nonlocal Lotka-Volterra equations have the property that solutions concentrate as Dirac masses in the limit of small diffusion. In this paper, we show how
the presence of an advection term changes the location of the concentration points in the limit of small diffusion and slow drift. The mathematical interest lies in the formalism of constrained Hamilton-Jacobi equations.
Our motivations come from previous models of evolutionary dynamics in phenotype-structured populations [R.H. Chisholm, T. Lorenzi, A. Lorz, et al., Cancer Res., 75, 930-939, 2015], where the diffusion operator models
the effects of heritable variations in gene expression, while the advection term models the effect of stress-induced adaptation.

19 T. Lorenzi, R.H. Chisholm, L. Desvillettes, B.D. Hughes, Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environments, J. Theor. Biol., 386, 166-176, 2015

An enduring puzzle in evolutionary biology is to understand how individuals and populations adapt to fluctuating environments.
Here we present an integro-differential model of adaptive dynamics in a phenotype-structured population whose fitness landscape
evolves in time due to periodic environmental oscillations. The analytical tractability of our model allows for a systematic
investigation of the relative contributions of heritable variations in gene expression, environmental changes and natural selection as drivers of phenotypic adaptation.
We show that environmental fluctuations can induce the population to enter an unstable and fluctuation-driven epigenetic state.
We demonstrate that this can trigger the emergence of oscillations in the size of the population,
and we establish a full characterisation of such oscillations. Moreover, the results of our analyses provide a formal basis for
the claim that higher rates of epimutations can bring about higher levels of intrapopulation heterogeneity, whilst intense selection pressures
can deplete variation in the phenotypic pool of asexual populations. Finally, our work illustrates how the dynamics of the population size is led
by a strong synergism between the rate of phenotypic variation and the frequency of environmental oscillations, and identifies possible ecological
conditions that promote the maximisation of the population size in fluctuating environments.

18 T. Lorenzi, R.H. Chisholm, M. Melensi, A. Lorz, M. Delitala, Mathematical model reveals how regulating the three phases of T-cell response could counteract immune evasion, Immunology, 46, 271-280, 2015

T cells are key players in immune action against the invasion of target cells expressing non-self antigens.
During an immune response, antigen-specific T cells dynamically sculpt the antigenic distribution of target cells,
and target cells concurrently shape the host's repertoire of antigen-specific T cells.
The succession of these reciprocal selective sweeps can result in "chase-and-escape" dynamics and lead to immune evasion.
It has been proposed that immune evasion can be countered by immunotherapy strategies aimed at regulating the three phases of the immune
response orchestrated by antigen-specific T cells: expansion, contraction and memory. Here, we test this hypothesis with a
mathematical model that considers the immune response as a selection contest between T cells and target cells.
The outcomes of our model suggest that shortening the duration of the contraction phase and stabilising as many T cells as possible
inside the long-lived memory reservoir, using dual immunotherapies based on the cytokines IL-7 and/or IL-15 in combination with molecular
factors that can keep the immunomodulatory action of these interleukins under control, should be an important focus of future immunotherapy research.

17 R.H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L. Neves de Almeida, A. Escargueil, J. Clairambault, Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, non-genetic instability and stress-induced adaptation, Cancer Res., 75, 930-939, 2015

In recent experiments on isogenetic cancer cell lines, it was observed that exposure to high doses of anti-cancer drugs can induce the emergence of a subpopulation of weakly-proliferative and drug-tolerant cells, that display markers associated with cancer stem cells.
After a period of time, some of the surviving cells were observed to change their phenotype to resume normal proliferation, and eventually repopulate the sample.
Furthermore, the drug-tolerant cells could be drug resensitized following drug washout. Here we propose a theoretical mechanism for the transient emergence of such drug tolerance.
In this framework, we formulate an individual-based model and an integro-differential equation model of reversible phenotypic evolution in a cell population exposed to cytotoxic drugs.
The outcomes of both models suggest that selection, non-genetic instability, stress-induced adaptation and the interplay between these mechanisms can push an actively proliferating cell population to transition into a weakly-proliferative and drug-tolerant state.
Hence, the cell population experiences much less stress in the presence of the drugs and, in the long run, reacquires a proliferative phenotype, due to selection pressure and phenotypic fluctuations.
These mechanisms can also reverse epigenetic drug tolerance following drug washout. Our study highlights how the transient appearance of the weakly-proliferative and drug-tolerant cells is related to the use of high-dose therapy.
Furthermore, we show how stem-like characteristics can act to stabilize the transient, weakly-proliferative and drug-tolerant subpopulation for a longer time window.
Finally, using our models as *in silico* laboratories, we propose new testable hypotheses that could help uncover general principles underlying the emergence of cancer drug tolerance.

16 C.J. Torney, T. Lorenzi, I.D. Couzin, S.A. Levin, Social information use and the evolution of unresponsiveness in collective systems, J. R. Soc. Interface, 12(103):20140893, 2015

Animal groups in nature often display an enhanced collective information-processing capacity. It has been speculated that natural selection will tune this response to be optimal,
ensuring that the group is reactive while also being robust to noise. Here, we show that this is unlikely to be the case. By using a simple model of decision-making in a dynamic environment,
we find that when individuals behave rationally and are subject to selection based on their accuracy, optimality of collective decision-making is not
attained. Instead, individuals overly rely on social information and evolve to be too readily influenced by their neighbours. This is due to a classic evolutionary
conflict between individual and collective interest. The result is a sub-optimal system that is poised on the cusp of total unresponsiveness.
Individuals in the evolved group exhibit delayed reactions to changes in the environment, before responding with rapid, socially reinforced transitions,
reminiscent of familiar human and animal social systems (markets, stampedes, fashions, etc.). Our results demonstrate that behaviour of this type may not be pathological,
but instead could represent an evolutionary attractor for such collective systems.

Herd mentality: are we programmed to make bad decisions? — ScienceDaily

Group-think: decision-making is best done alone, relying too much on friends and family isn't a good thing — Medical Daily

The power of social influence — The Times of Malta

15 A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame, Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77, 1-22, 2015

Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to
cytotoxic drugs can be considered as a process of selection in tumor cell populations.
In this framework, can we explain intra-tumor heterogeneity in terms of cell adaptation to local conditions?
Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies?
Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects
of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables
standing for space position and the expression level of a phenotype of resistance to cytotoxic drugs. The model takes
explicitly into account the dynamics of resources and anti-cancer drugs as well as their interactions with the cell
population under treatment. We analyze the effects of space structure and combination therapies on phenotypic
heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols
based on constant infusion and/or bang-bang delivery of cytotoxic and cytostatic drugs.

14 E. Faggiano, T. Lorenzi, A. Quarteroni, Metal artifact reduction in computed tomography images by a fourth-order total variation flow, CMBBE: Imaging & Visualization, DOI: 10.1080/21681163.2014.940629, 2014

Permanent metallic implants, such as dental fillings and cardiac devices, generate streaks-like artifacts in computed tomography images. In this paper, we propose a strategy to perform metal artifact reduction that relies on the TV-H^{-1} inpainting, a variational approach based on a fourth-order total variation flow. This approach has never been used to perform metal artifact reduction, although it has been profitably
employed in other branches of image processing. A systematic evaluation of the performance is carried out. Comparisons are made with the results obtained using classical linear interpolation and two other PDE-based approaches relying, respectively, on the Fourier's heat equation and on a second order total variation flow. Visual inspection of both synthetic and real computed tomography images, as well as
computation of similarity indexes, suggest that our strategy for metal artifact reduction outperforms the others considered here, as it provides best image restoration, highest similarity indexes and for being the only one able to recover hidden structures, a task of primary importance in the medical field.

13 G. Dimitriu, T. Lorenzi, R. Stefanescu, Evolutionary dynamics of cancer cell populations under immune selection pressure and optimal control of chemotherapy, Math. Mod. Nat. Phen., 9, 88-104, 2014

Increasing experimental evidence suggests that epigenetic and microenvironmental factors play a key role in cancer progression.
In this respect, it is now generally recognized that the immune system can act as an additional selective pressure,
which modulates tumor development and leads, through cancer immunoediting, to the selection for resistance to immune effector mechanisms.
This may have serious implications for the design of effective anti-cancer protocols. Motivated by these considerations, we present
a mathematical model for the dynamics of cancer and immune cells under the effects of chemotherapy and immunity-boosters.
Tumor cells are modeled as a population structured by a continuous phenotypic trait, that is related to the level of resistance
to receptor-induced cell death triggered by effector lymphocytes.
The level of resistance can vary over time due to the effects of epigenetic modifications. In the asymptotic regime of small
epimutations, we highlight the ability of the model to reproduce cancer immunoediting.
In an optimal control framework, we tackle the problem of designing effective anti-cancer protocols. The results obtained suggest
that chemotherapeutic drugs characterized by high cytotoxic effects can be useful for treating tumors of large size.
On the other hand, less cytotoxic chemotherapy in combination with immunity-boosters can be effective against tumors of smaller size.
Taken together, these results support the development of therapeutic protocols relying on combinations of less cytotoxic agents
and immune-boosters to fight cancer in the early stages.

12 M. Delitala, T. Lorenzi, A mathematical model for value estimation with public information and herding, Kinet. Relat. Models, 7, 29-44, 2014

This paper deals with a class of integro-differential equations modeling the dynamics of a market where
agents estimate the value of a given traded good. Two basic mechanisms are assumed to concur in value estimation:
interactions between agents and sources of public information and herding phenomena. A general well-posedness result
is established for the initial value problem linked to the model and the asymptotic behavior in time of the related
solution is characterized for some general parameter settings, which mimic different economic scenarios. Analytical
results are illustrated by means of numerical simulations and lead us to conclude that, in spite of its oversimplified nature,
this model is able to reproduce some emerging behaviors proper of the system under consideration.
In particular, consistently with experimental evidence, the results obtained suggest that if agents are highly
confident in the product, imitative and scarcely rational behaviors may lead to an over-exponential rise of the
value estimated by the market, paving the way to the formation of economic bubbles.

11 M. Delitala, T. Lorenzi, Evolutionary branching patterns in predator-prey structured populations, Disc. Cont. Dyn. Syst. B, 18, 2267-2282, 2013

Predator-prey ecosystems represent, among others, a natural context where evolutionary branching patterns may arise.
Moving from this observation, the paper deals with a class of integro-differential equations modeling the dynamics of
two populations structured by a continuous phenotypic trait and related by predation. Predators and prey proliferate
through asexual reproduction, compete for resources and undergo phenotypic changes. The asymptotic behavior of the
solution of the mathematical problem linked to the model is studied in the limit of small phenotypic changes. Analytical
results are illustrated and extended by means of numerical simulations with the aim of showing how the present class of
equations can mimic the formation of evolutionary branching patterns. All simulations highlight a chase-escape dynamics,
where the prey try to evade predation while predators mimic, with a certain delay, the phenotypic profile of the prey.

10 T. Lorenzi, A. Lorz, G. Restori, Asymptotic dynamics in populations structured by sensitivity to global warming and habitat shrinking, Acta Appl. Math., 131, 49-67, 2013

How to recast effects of habitat shrinking and global warming on evolutionary dynamics into continuous mutation/selection models?
Bearing this question in mind, we consider differential equations for structured populations, which include mutation,
proliferation and competition for resources. Since mutations are assumed to be small, a parameter ε is introduced
to model the average size of phenotypic changes. A well-posedness result is proposed and the asymptotic behavior of the
density of individuals is studied in the limit ε → 0. In particular, we prove the weak convergence of the density
to a sum of Dirac masses and characterize the related concentration points. Moreover, we provide numerical simulations
illustrating the theorems and showing an interesting sample of solutions depending on parameters and initial data.

9 M. Delitala, U. Dianzani, T. Lorenzi, M. Melensi, A mathematical model for immune and autoimmune response mediated by T-cells, Comp. Math. Appl., 66, 1010-1023, 2013

How to recast the effects of molecular mimicry and genetic alterations affecting the T-cell response against
self and non-self antigens into a mathematical model for the development of autoimmune disorders? Bearing
this question in mind, we propose a model describing the evolution of a sample composed of immune cells
and cells expressing self and non-self antigens. The model is stated in terms of integro-differential equations
for structured populations and ordinary differential equations for unstructured populations. A global existence
result is established and computational analysis are performed to verify the consistency with experimental data,
making particular reference to the Autoimmune LymphoProliferative Syndrome (ALPS) as model-disease. Using our model
as a virtual laboratory, we test different hypothetical scenarios and come to the conclusion that, besides molecular
mimicry, genetic alterations leading to an over-proliferation of the T-cells and a less effective action against
non-self antigens can be the driving forces of autoimmunity.

8 M. Delitala, T. Lorenzi, Drift-diffusion limit of a model for the dynamics of epithelial and mesenchymal cell mononalyers, Appl. Math. Letters, 26, 826-830, 2013

This paper is devoted to carry out formally the drift-diffusion limit for a kinetic-like model describing the
dynamics of a monolayer sample of epithelial and mesenchymal cells, which move via chemotaxis on a flat surface,
proliferate and interact among themselves. The aim is to verify if the macroscopic equations resulting from the
underlying model are able to mimic a biologically consistent scenario, where epithelial cells tend to adhere to
one another while mesenchymal cells diffuse through the sample.

7 D. Borra, T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12, 1487-1499, 2013

This paper deals with the asymptotic behavior of mathematical models for opinion dynamics under bounded
confidence of Deffuant-Weisbuch type. Focusing on the Cauchy Problem related to compromise models with
homogeneous bound of confidence, a general well-posedness result is provided and a systematic study
of the asymptotic behavior in time of the solution is developed. More in detail, we prove a theorem
that establishes the weak convergence of the solution to a sum of Dirac masses and characterizes the
concentration points for different values of the model parameters. Analytical results are illustrated by means of numerical simulations.

6 M. Delitala, T. Lorenzi, Recognition and learning in a mathematical model for immune response against cancer, Disc. Cont. Dyn. Syst. B, 18, 891-914, 2013

This paper presents a mathematical model for immune response against cancer aimed at reproducing
emerging phenomena arising from the interactions between tumor and immune cells. The model is
stated in terms of integro-differential equations and describes the dynamics of tumor cells,
characterized by heterogeneous antigenic expressions, antigen-presenting cells and T-cells.
Asymptotic analysis and simulations, developed with an exploratory aim, are addressed to verify
the consistency of the model outputs as well as to provide biological insights into the mechanisms
that rule tumor-immune interactions. In particular, the present model seems able to mimic the recognition,
learning and memory aspects of immune response and highlights how the immune system might act as
an additional selective pressure leading, eventually, to the selection for the most resistant cancer clones.

5 A. Lorz, T. Lorenzi, M.E. Hochberg, J. Clairambault, B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM Math. Model. Numer. Anal., 47, 377-399, 2013

Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern.
Among the many mechanisms at work in cancers, one of the most important is the selection of tumor
cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection
in adaptive evolution, we propose a model based on a continuous variable that represents the expression
level of a resistance gene (or genes, yielding a phenotype) influencing in healthy
and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations.
We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic
and cytostatic treatments may induce different levels of resistance. The mathematical interest of our
study is in the formalism of constrained Hamilton-Jacobi equations in the framework of viscosity solutions.
We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations.
In the context of adaptive cancer management, we also analyze whether an optimal drug level is better than the maximal tolerated dose.

4 D. Borra, T. Lorenzi, A hybrid model for opinion formation, Z. angew. Math. Phys., 64, 419-437, 2013

This paper presents a hybrid model for opinion formation in a large group of agents exposed
to the persuasive action of a small number of strong opinion leaders. The model is defined by
coupling a finite difference equation for the dynamics of leaders opinion with a continuous integro-differential
equation for the dynamics of the others. Such a definition stems from the idea that the leaders are few and
tend to retain original opinions, so that their dynamics occur on a longer time scale with respect
to the one of the other agents. A general well-posedness result is established for the initial
value problem linked to the model. The asymptotic behavior in time of the related solution is characterized
for some general parameter settings, which mimic distinct social scenarios, where different emerging
behaviors can be observed. Analytical results are illustrated and extended through numerical simulations.

3 M. Delitala, T. Lorenzi, Asymptotic dynamics in continuous structured populations with mutations, competition and mutualism, J. Math. Anal. Appl., 389, 439-451, 2012

This paper deals with a class of integro-differential equations arising in evolutionary biology to model the dynamics of specialist
and generalist species related by mutualistic interactions. The effects of mutation events, proliferative phenomena and competition
are taken into account. Specialist population is assumed to be structured by a continuous phenotypical trait related to the ability of
individuals to ingest specific resources and a parameter εis introduced to model the average size of mutations. A well-posedness result
is proposed here and the asymptotic behavior of the density of specialist individuals in the space of the phenotypical traits is studied in the limit ε → 0.
In particular, under a suitable time rescaling, we prove the weak convergence of such a density to a sum of Dirac masses. A characterization of the set of
concentration points is provided.

2 M. Delitala, T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theor. Biol., 297, 88-102, 2012

This paper deals with the development of a mathematical model for the *in vitro* dynamics of malignant
hepatocytes exposed to anti-cancer therapies. The model consists of a set of integro-differential
equations describing the dynamics of tumor cells under the effects of mutation and competition phenomena,
interactions with cytokines regulating cell proliferation as well as the action of cytotoxic drugs and
targeted therapeutic agents. Asymptotic analysis and simulations, developed with an exploratory aim,
are addressed to enlighten the role played by the biological phenomena under consideration in the dynamics
of hepatocellular carcinoma, with particular reference to the intra-tumor heterogeneity and the response to therapies.
The results obtained suggest that cancer progression selects for highly proliferative clones.
Moreover, it seems that intra-tumor heterogeneity makes targeted therapeutic agents to be less effective
than cytotoxic drugs and a joint action of these two classes of agents may mutually increase their efficacy.
Finally, it is highlighted how targeted therapeutic agents might act as an additional selective pressure
leading to the selection for the fittest, and then most resistant, cancer clones.

1 M. Delitala, T. Lorenzi, A mathematical model for progression and heterogeneity in colorectal cancer dynamics, Theor. Popul. Biol., 79, 130-138, 2011

This paper deals with the development of a mathematical model that
describes cancer dynamics at the cellular scale. The selected case study
concerns colon and rectum cancer, which originates in colorectal crypts.
Cells inside the crypts are assumed to be organized according to a compartmental-like
arrangement and to be homogeneously mixing. A mathematical model for cancer progression
is proposed here. This model describes the generation of multiple clonal
sub-populations of cells at different progression stages in a single crypt.
Asymptotic analysis and simulations are developed with an exploratory aim.
The results obtained offer some insights into the role played by mutation, proliferation
and differentiation phenomena on cancer dynamics. In particular, the acquisition of
an additional growing power and a reduction for cellular differentiation seem more
likely to be the driving force behind carcinogenesis rather than an increase in the mutation rate.
The mutation rate instead seems to affect progression dynamics and intra-tumor heterogeneity.
The role played by cells, at different differentiation stages, in the onset and progression
of colorectal cancer is highlighted. The results support the fact that stem cells play a
key role in cancer development and the idea that transit-amplifying cells could also take on an active role in carcinogenesis.

Book chapters

6 T. Lorenzi, F.R. Macfarlane, C. Villa, Discrete and continuum models for the evolutionary and spatial dynamics of cancer: a very short introduction through two case studies, (pp. 359-380) in Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment, Ed. R. Mondaini, Springer, Cham, 2019

We give a very short introduction to discrete and continuum models for the evolutionary and spatial
dynamics of cancer through two case studies: a model for the evolutionary dynamics of cancer cells under cytotoxic
therapy and a model for the mechanical interaction between healthy and cancer cells during tumour growth.
First we develop the discrete models, whereby the dynamics of single cells are described through a set of rules that result in branching random walks.
Then we present the corresponding continuum models, which are formulated in terms of non-local and nonlinear partial differential equations,
and we summarise the key properties of their solutions. Finally, we carry out numerical simulations of the discrete models and we construct
numerical solutions of the corresponding continuum models. The biological implications of the results obtained are briefly discussed.

5 M.A.J. Chaplain, T. Lorenzi, A. Lorz, C. Venkataraman, Mathematical modelling of phenotypic selection within solid tumours (pp. 237-245) in Numerical Mathematics and Advanced Applications ENUMATH 2017, Eds. F.A. Radu, K. Kumar, I. Berre, J.M. Nordbotten, I.S. Pop, Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham, 2019

We present a space- and phenotype-structured model of selection dynamics between cancer cells
within a solid tumour. In the framework of this model, we combine formal analyses with numerical
simulations to investigate *in silico* the role played by the spatial distribution of oxygen and therapeutic agents
in mediating phenotypic selection of cancer cells. Numerical simulations are performed on the 3D geometry of an *in vivo* human hepatic
tumour, which was imaged using computerised tomography. Our modelling extends our previous work in the area through the inclusion of
multiple therapeutic agents, one that is cytostatic, whilst the other is cytotoxic. In agreement with our previous work, the results
show that spatial inhomogeneities in oxygen and therapeutic agent concentrations, which emerge spontaneously in solid tumours, can promote
the creation of distinct local niches and lead to the selection of different phenotypic variants within the same tumour. A novel conclusion
we infer from the simulations and analysis is that, for the same total dose, therapeutic protocols based on a combination of cytotoxic and
cytostatic agents can be more effective than therapeutic protocols relying solely on cytotoxic agents in reducing the number of viable cancer cells.

4 L. Almeida, R.H. Chisholm, J. Clairambault, T. Lorenzi, A. Lorz, C. Pouchol, E. Trélat, Why is evolution important in cancer and what mathematics should be used to treat cancer? Focus on drug resistance (pp. 107-120) in Trends in Biomathematics: Modeling, Optimization and Computational Problems, Ed. R. Mondaini, Springer, Cham, 2018

The clinical question of drug resistance in cancer, our initial motivation to study continuous models of adaptive cell population dynamics,
leads naturally and more generally to consider the cancer disease itself from an evolutionary biology viewpoint, a consideration without
which even the best targeted therapies will likely most often eventually fail. Among the challenging questions to mathematicians who
tackle the task of understanding this disease and optimising its treatment are the representation of phenotypic heterogeneity of cancer
cell populations and of their plasticity in response to anticancer drug insults. Such representation can be obtained using phenotype-structured
models of healthy and cancer cell populations, and optimal control methods to optimise drug effects, with the perspective to implement them
in the therapeutics of cancer, aiming at both avoiding the emergence of drug resistance in tumours and taking into account a constraint of
limiting unwanted adverse effects to healthy tissues.

3 M. Delitala, T. Lorenzi, M. Melensi, A structured population model of competition between cancer cells and T-cells under immunotherapy (pp. 47-58) in Mathematical Models of Tumor-Immune System Dynamics, Eds. A. Eladdadi, P. Kim, D. Mallet, Springer Proceedings in Mathematics & Statistics, Vol. 107, 2014

How does immunotherapy affect the evolutionary dynamics of cancer cells? Can we enhance the anti-cancer
efficacy of T cells by using different types of immune boosters in combination? Bearing these questions in mind,
we present a mathematical model of cancer-immune competition under immunotherapy.
The model consists of a system of structured equations for the dynamics of cancer cells and activated T cells.
Simulations highlight the ability of the model to reproduce the emergence of cancer immunoediting, that is,
the well-documented process by which the immune system guides the somatic evolution of tumors by eliminating
highly immunogenic cancer cells. Furthermore, numerical results suggest that more effective immunotherapy
protocols can be designed by using therapeutic agents that boost T cell proliferation in combination with boosters of immune memory.

2 M. Delitala, T. Lorenzi, Mathematical modeling of cancer cells evolution under targeted chemotherapies (pp. 81-89) in Managing Complexity, Reducing Perplexity, Eds. M. Delitala, G. Ajmone Marsan, Springer Proceedings in Mathematics & Statistics, Vol. 67, 2014

This chapter focuses on selection and resistance to drugs in an integro-differential model describing
the dynamics of a cancer cell population exposed to targeted chemotherapies. Mutations, proliferation
and competition for resources are assumed to occur under the cytotoxic action of targeted therapeutic agents.
The results obtained support the idea that cancer progression selects for highly proliferative clones.
Moreover, it is highlighted how targeted chemotherapies might act as an additional selective
pressure leading to the selection for the fittest, and thus eventually most resistant, cancer clones.

1 M. Delitala, T. Lorenzi, Formations of evolutionary patterns in cancer dynamics (pp. 179-190) in Pattern Formation in Morphogenesis. Problems and mathematical issues, Eds. V. Capasso, M. Gromov, A. Harel-Bellan, N. Morozova and L.L. Pritchard, Springer Proceedings in Mathematics, Vol. 15, 2013

This chapter originated from the idea that carcinogenesis can be considered as a multiscale
morphogenetic process. A specific mathematical model for cancer evolution in epithelial cells is defined,
which describes the morphogenesis of multiple sub-populations of cancer cells at different malignancy stages.
Simulations are developed with an exploratory aim. The results obtained offer insights into the role played by
mutation, proliferation and differentiation phenomena on the morphogenesis of sub-populations of cancer cells.

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