RESOLVE ProjectActivity Report - WP4 - POLITO

Nicola Bellomo, Elena De Angelis, and Damian Knopoffnicola.bellomo@polito.it

Department Mathematical Sciences, Politecnico di Torino,Corso Duca degli Abruzzi 24,

I-10129 Torino, Italy

30 September 2013

Index .

1. Motivations, Role of WP4, and Plan of the Report

2. Summary of the Activities up to the Mid Term Meeting (in Vienna) ofNovember 2010

3. Activities from Mid Term Meeting to March 2012

4. Activities from March 2012 to September 2013

5. Recruitment-Training and Future Steps

6. Appendix I - Mathematical Tools

7. Appendix II - RESOLVE Papers and Books

Thanks: The authors of this report need forwarding a special thank to Professors Lutz-Henning Bloch and Rolf Ziesche for having designed and promoted this important and chal-lenging project. Nicola Bellomo from his position of President of the Italian MathematicalSociety of Applied and Industrial Mathematics I wish also to join to the thanks the Italianand European Community of Mathematicians. Moreover, he feels personally indebted withBrigitte Rohner. Her constant and effective monitoring of the activities of the project hasdefinitely contributed to create a highly stimulating research environment.

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1 Motivations and Plan of the Report

The scientific community has achieved the awareness that this century will see an effectiveinterplay between mathematical and biological sciences, which will drive the traditional heuris-tic approach of biology towards rigorous methods. Biology, similarly to physics in the pastcenturies, will link to conjectures and experiments, also formal structures, where mathemat-ics provides tools for the interpretation of biological phenomena and, possibly, prediction ofevents. Definitely, medicine can take advantage of this interplay as it is already taking ben-efit of advanced technology. The final target should be the development of bio-mathematicaltheories.

An important hint towards this challenging, may be ambitious objective, is given by theNobel laureate Hartwell [1], who first illustrates the conceptual differences between the livingand the inert matter, and subsequently invites mathematicians and physicists to compete todevelop a new system approach theory. This seminal paper [1], also explains why traditionalmathematical approaches failed, and suggests some guidelines to be followed for a successfulattempt. Basically, understanding the complexity of living systems and looking for new toolsto capture this essential feature.

System biology has a recent story [2] and aims at understanding biological systems at sys-tem level, where the behavior of a system cannot be explained by its components alone but it isnecessary to examine the cellular dynamics, the organism functions and mechanisms processes.Thus, the approach needs to study the interactions between components and analyze in whichway these interactions give rise the behavior of the whole system. The conceptual difficultiesto pursue this objective, namely lack of conservation rules and background paradigms, areproperly put in evidence by May [3] and Reed [4], while hints towards a new system biologyapproach are presented by Woese [5], who identifies the multiscale and network structure ofbiological systems. Perspective ideas, still to be developed, are presented in [6] and [7] focusingon the interplay between pathologic agents and immune system.

In general focusing on the objectives of the Project RESOLVE and of the activities sched-uled in WP4 the aforesaid system biology approach should be focused on the following mainobjectives:

1. Developing a new system biology approach;

2. Understanding if there is a way, and how, for mathematical prediction;

3. Developing the study of a number of case studies planned within the activitiesof RESOLVE

4. Looking ahead to forward steps

Bearing all above in mind, this final report is presented through five additional sections.In details, Section 2 focuses on the activities from the start of the project up to the MidTerm meeting held in Vienna on November 2010, namely on the development of the aforesaidnew system biology approach with some preliminary applications suitable to test its validity;

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Section 3 is devoted to the activities from Mid Term meeting to March 2012 specificallydeveloped on the case studies scheduled at the Mid Term meeting; Section 4 presents thecontinuation of the activities in the period from April 2012 to September 2013; Section 5 isdevoted to the development of future steps also related to the training of young scientists ableto transfer the knowhow achieved within the project to Society and Research Institutions.

Indeed, it is worth stressing that the contribution of WP4 has needed sophisticated math-ematical tools. In fact, the complexity of the various topics under consideration cannot beachieved by standard methods, while it soon appeared, out of the first meetings, that a newmathematical theory needed to be developed. The presentation of this report will simplycontain concepts, while formal mathematical tools are reported in Appendix I. The practicalimplementation of the said tools has needed appropriate softwares which have been posted inthe WEB site of RESOLVE introduced by a Notebook for users. Finally, Section 5 reports theessential mathematical tools, even a mathematical theory, that has been used to contributeto the Project. The reader who is not familiar with mathematical tools can skip over thissection as all essential concepts are presented in the preceding sections. The bibliography, atthe end of the report, presents all papers and books published during the period covered bythe Project.

Bibliography - Section 1

[1] H.L. Hartwell, J.J. Hopfield, S. Leibner, and A.W. Murray, From molecular tomodular cell biology, Nature, 402, (1999), c47-c52.[2] H. Kitano, Foundations of Systems Biology, The MIT Press, Cambridge, 2001.[3] R.M. May, Uses and abuses of mathematics in biology, Science, 303, (2004) 790–793.[4] R. Reed, Why is mathematical biology so hard?, Notices American Mathematical Society,51, (2004) 338–342.[5] C.R. Woese, A new biology for a new century, Microbiology Molecular Biology Reviews,68, (2004) 173–186.[6] N. Bellomo and G. Forni, Complex multicellular systems and immune competition:New paradigms looking for a mathematical theory, Current Topics in Developmental Biology,81, (2008) 485–502.[7] N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic gametheory to modelling mutations, onset, progression and immune competition of cancer cells,Physics of Life Reviews, 5 (2008) 183–206.

2 Summary of the Activities from the Start-up to

the Mid Term Meeting

The activity within the period indicated in the title was essentially devoted to the developmentof a new system biology approach and to its preliminary application to a case study chosen totest such theory. Namely the activity was finalized to design all effective tools to pursue the

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objectives of the project related to WP4. The Mid Term Meeting November 2010 appearedthe right period to plan and schedule the activities to be properly developed in collaborationwith the Partners. Accordingly, the main objective of this period was the development of anew system biology approach.

The system biology approach is based on the idea that the interplay between mathe-matics and biology should be focused on the complexity features of living systems. Namely,mathematics should be able to retain, as far as possible, this crucial aspect. Moreover, theconceptual difficulties of the lack of first principles makes us understand that a great deal ofresearch activity and human energy is needed.

Biological systems are constituted by living entities, which have the ability to developa specific strategy depending on the state of the surrounding environment on the search ofindividuals for their best well-being. Living systems receive inputs from the environment andhave the ability to learn from past experience, in order to adapt themselves to the changing-in-time external conditions [7]. These strategies are not the same for all entities, namelyheterogeneity characterizes a great part of living systems. Interacting entities can appear andbehave differently to many extent, even though they share the same molecular structure, forinstance due to different phenotype expression generated by the same genotype.

Let us start, according to the above reasonings, by an assessment of the main complexityfeatures of biological (living) systems. The list given below does not claim to be exhaustive,but hopefully already provides the main hallmarks.

1. Ability to express a strategy: Living entities are capable to develop specific strategiesgenerated by their organization ability depending on the state and on entities in theirsurrounding environment. These ones can be expressed without the application of anyprinciple imposed by the outer environment. In general, they typically operate out-of-equilibrium. For example, a constant struggle with the environment is developed toremain in a particular state, namely stay alive.

2. Heterogeneity: The said ability is heterogeneously distributed, namely the character-istics of interacting entities can differ from an entity to another belonging to the samefunctional subsystem. For instance, this is due to different phenotype expressions gen-erated by the same genotype. In some cases, these expressions induce genetic diseases[8].

3. Learning ability: Living systems receive inputs from their environments and have theability to learn from past experience. Therefore, the strategy expressed by living systemsevolves in time. Therefore, also the dynamics of interactions evolves in time. In fact,living systems have the ability to learn from past experience. Learning dynamics appearsin the immune defence [9], when immune cells transfer the so called innate immunityinto the acquired immunity.

4. Nonlinear interactions: Interactions are nonlinearly additive and involve immediateneighbors, but in some cases also distant particles. This is what happens at level of cells,which have the ability to communicate by signaling and can choose different observationpaths within networks that evolve in time. Living entities play a game at each interaction

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with an output that is technically related to their strategy often associated to survivingand adaptation ability.

5. Darwinian selection and time as a key variable: All living systems are evolution-ary [7,10]. Birth processes can generate individuals that fit better the outer environment,which in turn generate new ones fitting better and better. In each birth process gen-eration, mutations bring new genetic variants into populations. Natural selection thenscreens them: the rigors of the environment reduce the frequency of “bad” (relativelyunfit) variants and increase the frequency of “good” (relatively fit) ones. There existsa time scale of observation and modeling of the system itself long enough to observeevolutionary events. In biology such a time scale can be very short for cellular systemsand very long for vertebrates. In some cases, such as the generation of daughters frommother cells, new cell phenotypes can originate from random mistakes during replication.

These features are represented, with the aid of a scientific-artistic fantasy by the represen-tation offered Escher of a village on the Amalfitan coast in the Mediterranean sea. The pictureshows nonlinear interactions, along a Darwinian selection, where time is a key variable.

The system biology approach needs to capture these features by designing mathematicalstructures that have the ability to retain them. This is a conceptual method to overcome thelack of first principles that characterizes living systems. This is only a preliminary requirementto develop models with a predictive ability. These can be obtained by inserting into the saidstructure the specific behaviors of each system object of modeling. The derivation of modelsneeds tackling various technical. Among others:i) Large number of components: Complexity in living systems is induced by a largenumber of variables, which are needed to describe their overall state. Therefore, the numberof equations needed for the modeling approach may be too large to be practically treated.Reduction of complexity is the first step of the modeling approach.ii) Multiscale aspects: The study of complex living systems always needs a multiscaleapproach. For instance, the functions expressed by a cell are determined by the dynamics at

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the molecular (genetic) level. Moreover, the structure of macroscopic tissues depends on sucha dynamics.iii) Time varying role of the environment: The environment surrounding a living systemchanges in time, in several cases also due to the interaction with the inner system. Thereforethe output of this interaction and the number of components of a living system can evolve intime.

From a mathematical point of view, the complexity is translated in a large number ofvariables, and hence in a large, may be excessively, number of equations able to describe theoverall system. On the other hand, this implies a high computational effort. Consequently,the first step consists in reducing this specific complexity. Technically the approach providesan effective answer to the following 3 key questions:

1. Which are the most important complexity features of living systems?

2. Which is the interplay between complexity and Darwinian evolution?

3. How a new system approach can be designed and which are the appropriate mathemat-ical tools?

The reply to the above queries is given by the hallmarks of the new approach to systembiology, which can be briefly summarized as follows:

• The overall system is subdivided into functional subsystems constituted by entities,called active particles, whose individual state is called activity;

• The state of each functional subsystem is defined by a suitable, time dependent, prob-ability distribution over the microscopic state, which includes position, velocity, andactivity variables;

• Interactions are modeled by games, more precisely stochastic games, where the state ofthe interacting particles and the output of the interactions are known in probability;

• Modeling the interactions among the various components of the overall system, takinginto account the networks and the multiscale aspects of their connection.

• The evolution, in time and space, of the probability distribution is obtained by a balanceof particles within elementary volume of the space of the microscopic states, where thedynamics of inflow and outflow of particles is related to interactions at the microscopicscale.

The system biology approach needs mathematical tools, therefore the mathematical kinetictheory for active particles is proposed as appropriate to deal with them. The method refers toa large system of interacting entities. The essential feature of the method are the following:

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1. The interacting entities, which compose each functional subsystem, are called activeparticles. Their physical state is described by a set of variables, in particular a vari-able called activity representing the individual ability to express a specific strategy.The activity variable is heterogeneously distributed, while the state in each functionalsubsystem is defined by a probability distribution over the microscopic state.

2. Interactions involve not only immediate neighbors (short range interactions) but alsothe distant ones (long range interactions). Indeed, living systems communicate eachother directly or through media. Consequently, each entity interacts with all the othersin a domain whose elements are able to communicate. In some cases, such a domain isidentified with the visibility zone, in other cases by a communication network.

3. Interactions are complex, namely the overall output of the game that an active particleplays with the ones lying in its interaction domain is not the linear superposition of itsseparated interactions with all of them, but a complex combination whose form dependson the strategy that all particles can develop.

4. The output of the game modifies the activity of interacting particles and may alsogenerate, in the proliferative process, particles with a different structure (for instance,entities with a different phenotype).

5. Derivation of the evolution equation by a balance of particles within elementary volumeof the space of the microscopic states, where the dynamics of inflow and outflow ofparticles is related to interactions at the micro-scale.

Technical mathematical details of the theory are reported in Section 6 (Appendix II)starting from papers [11,12,13].

Finally, it is worth mentioning that the guidelines concerning the interactive collabora-tions were given at Mid Term meeting in Vienna. More precisely a number of case studieswere identified to develop a scientific collaboration involving clinicians, biologists and math-ematicians to bring to the third level of mathematical modeling the fundamental studies onthe ground of clinics and biology. More precisely the following activities were scheduled:

• Development of mathematical tools for a system biology approach to a new systemapproach consistent with the aims of RESOLVE and testing these tools in a number ofcases of interest for the project;

• Optimization of surgery processes for removal of keloids;

• Analysis of expression profiling to compare human epithelial 3p hepatocynes with mes-enchymal 3ps cells in the liver fibrosis;

• Simulation of lung dynamics in view of tissue corruption and genetic modifications;

• Development of software codes for simulation of biological phenomena;

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• Planning of the activities in the fourth and last year of the project.

All activities mentioned above were technically scheduled after meetings with the partnersinvolved in them, and were approved at the Mid Term meeting.

Bibliography - Section 2

[7] E. Mayr, What Evolution Is, Basic Books, New York, (2001).[8] R.A. Weinberg, The Biology of Cancer, Garland Sciences - Taylor and Francis, NewYork, (2007).[9] E.L. Cooper, Evolution of immune system from self/not self to danger to artificial immunesystem, Physics of Life Reviews, 7, (2010) 55–78.[10] F. Mynard and G.J. Seal, Phenotype spaces, J. Mathematical Biology, 60, (2010)247–266.[11] C. Bianca and N. Bellomo, Towards a Mathematical Theory of Multiscale ComplexBiological Systems, Series in Mathematical Biology and Medicine, Vol. 11, World Scientific,London, Singapore (2010).[12] N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic gametheory to modelling mutations, onset, progression and immune competition of cancer cells,Physics of Life Reviews, 5 (2008) 183–206.[13] N. Bellomo, Modelling Complex Living Systems - A Kinetic Theory andStochastic Game Approach (Birkhauser, Boston, 2008).[14] C. Bianca, and L. Fermo, Bifurcation diagrams for the moments of a kinetic typemodel of keloid-immune system competition, Computers and Mathematics with Applications,61, (2011) 277–288.

3 From the Mid Term Meeting to March 2012

The activities planned in the third year where motivated by the Mid Term Meeting inVienna 2011, where the guidelines concerning the interactive collaborations, as alreadymentioned, were given. This activity was developed after two meetings in Vienna involv-ing POLITO and some of the Partners, namely W. Mikulits, B.D. Lumenta. The study wasdeveloped according to the system biology approach presented in the preceding sections.

This section presents the activity developed by Team Polito-WP4 in collaboration withthe Partners of the Resolve Project. Moreover, the organization of the Notebook used for thescientific computing is also briefly presented in the last section.

3.1 Optimization of the surgery action for keloids

We investigated, in collaboration with Professor David B. Lumenta (WP1) of the MedicalUniversity of Vienna, on the modeling of the surgery action to remove keloids. More precisely,

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the study has been reduced to an optimization problem related to the choice of the shape ofsurgery removal.

In this research program, we focused on the surgical excision as primary form of its treat-ment. Surgical excision can be performed extralesionally or intralesionally. Intralesional exci-sion denotes the central excision of a keloid, which ignores its peripheral borders and leavesthis portion of the keloid in the wound. Extralesional excision describes the entire removal ofa keloid inclusive of its borders, resulting in a larger scar. Currently, both forms of surgicalexcision of keloid scars are readily performed by surgeons. In vitro studies have demonstratedthat the central subpopulation of fibroblasts in keloid scars have lower rates of apoptosis ascompared to the superficial and deep borders, supporting the practice of intralesional exci-sion. Additionally, it is hypothesized, that the intralesional excision with lateral underminingof the wound provides a rim of the keloid scar that serves as a splint in order to reduce thetensile stress, possibly reducing the stimulus for increased collagen synthesis. In contrast tothat a pathological study reviewing keloid specimens and correlating them with 6-12 monthspatient follow-up revealed that incomplete surgical excision was associated with higher recur-rence rates, and therefore favored an extralesional approach. No clinical studies comparingthe effectiveness of intralesional versus extralesional excision are currently available.

From a mathematical point of view, the problem, that has been outlined above, can beregarded as an optimization problem. It is worth stressing, that the problem is dealt with usingsimple geometries for the shape of the keloid, while more sophisticated approaches need to beemployed for the nontrivial mathematical calculations. The overall activity is documented inpaper [15].

3.2 Analysis of expression profiling to compare human epithe-lial 3p hepatocynes with mesenchymal 3ps cells in liverfibrosis

In collaboration with Professor Wolfgang Mikulits (WM, WP7) of the Institute of Cancer Re-search, we constructed a mathematical model that gives an answer to the following questions:What is the driving force and its underlying molecular mechanisms for the fibro-proliferationof mesenchymal 3sp cells, which includes regulatory circuits required for proliferation, sur-vival, invasion, transmigration and re-modulation of extracellular matrix (degradation anddeposition)?

The overall activity focused on cell monolayers composed of cancer hepatocytes expressingepithelial and mesenchymal phenotypes, which move via chemotaxis, proliferate and interactamong themselves. In particular, taking advantage of experimental data from the group leadedby Professor Mikulits, we are aimed at defining a mathematical model able to mimic the col-lective behaviors expressed by these cells in co-culture. In fact, since evidence is accumulatingthat the trans-differentiation of cancer epithelial cells of the liver into motile and invasivemesenchymal derivatives can play a crucial role in metastatic processes, such a model mayallow a deeper comprehension of some mechanisms underling the progression of hepatocellularcarcinoma.

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This activity fills part of the PhD dissertation by Tommaso Lorenzi defendedin March 2013.

3.3 Development and evolution of lung fibrosis First steps

A scientific interaction was developed with the team of Professor Marco Chilosi (WP3), Uni-versity of Verona, with the aim of deriving a mathematical model able to identify the zones ofthe lung which are at most risk and how the disease develops.

More precisely, aim of this activity consists in developing a mathematical-mechanicalmodel suitable to describe the dynamics of the lung in order to investigate on the area moresusceptible to loss of elasticity. The lung is viewed as a two-dimensional surface compactedinto the thoracic space. Really, the lung is a three dimensional organ but this simple approachis sufficient to investigate on the aforesaid problem. The formulation of the model involves twokey issues, namely the analysis of the geometric structure of the parenchima and the study ofthe mechanical dynamics of the lung. The parenchima is assumed to be a viscoelastic tissueand its microscopic images show that alveoli are arranged in a hexagonal-like structure. Hence,we have modeled it as a two-dimensional network of purely viscous damper and purely elasticspring connected in parallel and arranged in a hexagonal array. Simulations have been devel-oped based on a model derived according to the aforesaid assumptions. These have confirmedthat the highest concentration of stress is in the extreme end of the lung and in the near zoneover the diaphragm.

The activity in the period was limited to a first approach and to the development of theappropriate software to simulate a large number of equations as reported in the next subsection.This preliminary activity is documented in the report [17], to be followed by additional activity.

3.4 Software for Simulation

This activity, scheduled in the milestones, was devoted to organize the scientific programsimplemented to simulate the biological problems treated within the RESOLVE project. Forpractical purposes it is divided into three folders:

1. Keloid,

2. Surgery,

3. Lung,

Each of these case studies present a different type of computing, namely the first oneconcerns progression and mutation at the cellular scale. The second one an optimizationproblem, while the third one a problem at the level of mechanics of biological tissues. Indeed,these were the case studies selected at the Mid Term meeting. Therefore their ensembleconstituted a significant panorama of the computing activity within the consortium.

All folders contain a text file having the instructions to run the programs which are writtenin Matlab.

The folder Keloid contains nine files as follows:

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1. Bifurcation

2. Initial Conditions

3. Model

4. Model Bif

5. NodWei

6. Par alpha

7. RK4sist

8. Shift Node

9. System

The main routines are:

Bifurcation that gives the bifurcation diagrams of the local number density and the localmean activity when a specific parameter varies in a fixed range. To run this program, it issufficient to write the name of the file on the command window of Matlab. At this point,Matlab asks to write the values of all the parameters appearing in the model. Then therequired graphs are automatically generated.

Par alpha that graphically represents the number density of the cells involved in the model(see [12] for details). To run this program it needs to write Par alpha on the command windowof Matlab. Then it is necessary to write the values of all parameters involved in the modeland the graphs are automatically given.

These two programs use all the other sub-routines: Initial Conditions containing theinitial conditions; NodWei that defines the zeros and the weights; RK4sist having the instruc-tion to apply the Runge Kutta method; System containing the system of equations to besolved; Model Bif that is used by the program Bifurcations; Model that is used by the pro-gram Par alpha; Shift Node that is a technical program aiming to avoid numerical problems.

All programs and a Notebook for their use have been uploaded on the WEB page of theproject.

Bibliography - Section 3

[13] N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systemsfocusing on developmental biology and evolution: a review and perspectives, Phys. Life Rev.,8, (2011) 1–18.[14] V. Coscia, L. Fermo, N. Bellomo, On the mathematical theory of living systems II:The interplay between mathematics and system biology, Computers Mathematics Application,62(10):3902-3911, 2011.

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[15] L. Fermo, N. Bellomo, and D. Lumenta, Assessment of surgical strategies for ad-dressing keloids: An optimization problem, Computers Mathematics Application, 62(6), (2011),2417-2423.[16] T. Lorenzi, PhD Dissertation, Politecnico of Torino, supervisor Prof. M. Delitala.[17] M. Chilosi, A. Carloni, V. Poletti, N. Bellomo, and L. Fermo, Modeling andsimulation of lung fibrosis, Internal Report (2012).

4 From April 2012 to September 2013

The activity in this period is strongly related to that of the preceding period. In fact thepreliminary studies documented in [17] have shown that the biological conjecture offered byProfessor Chilosi and his team was confirmed by a mechanical-biological model derived fol-lowing the guidelines of the system biology approach presented in Section 2. Therefore wedecided to continue our activity in the field. Moreover, the dissertation [16], based on theactivity of Professor Mikulits, motivated a deeper analysis of the immune competition relatedto Darwinian phenomena at the cellular scale. Finally, the team WP4 decided to work at anoverall revisiting of the system biology approach focusing on the role of nonlinear interactionsat the cellular scale.

4.1 Development and evolution of lung fibrosis

A scientific interaction was developed with the team of Professor Marco Chilosi (WP3) ofUniversity of Verona, with the aim of deriving a mathematical model able to identify thezones of the lung which are at most risk and how the disease develops.

Idiopathic pulmonary fibrosis (IPF) is the most common and severe form of idiopathicinterstitial pneumonia, and its median survival is 3-4 years. New concepts have been recentlyproposed regarding the biology and pathogenesis of this devastating disease, suggesting thatthe abnormal remodeling of lung parenchyma in IPF is not related to inflammatory mecha-nisms as previously considered, but is rather the effect of a series of pathologic events includingepithelial alveolar cell injury, and deranged activation of lung reparative processes. Accumu-lating evidence is available that in IPF the abnormal re-epithelialization after injury can berelated to a progressive stem-cell exhaustion due to intrinsic cellular defects either related toa predisposing genetic background (familial IPF is a well recognized entity), or to the aging-related accumulation of metabolic alterations (e.g. due to toxic effects of smoking, pollution,metabolic abnormalities, or other causes). The remodeling process in this scheme is likely re-lated to the abnormal triggering at sites of disease progression of different molecular pathwayscrucial to lung tissue development and regeneration, including the wnt-beta-catenin pathway,TGF-beta, NOTCH, BMP, and others.

A relevant missing point in this pathogenic scenario is related to the peculiar localizationof IPF lesions, that typically start at the posterior bases of the lower lobes, progressivelyextending in a caudal-cranial mode. Several lines of evidence suggest that these anatomical

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parts of the lung are in fact sites where mechanical forces can be particularly concentrated,thus triggering the formation of microscopic tears in the alveolar structure, which may resultin repetitive small scarring events (fibroblast foci), and eventual honeycomb changes. The mi-croscopic damages due to chronic mechanical stress located in these areas can in fact cooperatein a sequence of pathogenic events, including:

1. localized accelerated pneumocyte turnover, with eventual increase of replicative senes-cence;

2. occurrence of a senescence-induced secretive phenotype (SASP) within these parts ofthe lung parenchyma;

3. abnormal activation of reparative molecular pathways (e.g. wnt-pathway), leading toabnormal remodeling of the lung parenchyma.

Evidence of a close correspondence existing between the early IPF lesions and the anatom-ical distribution of mechanical stress in human lung is still incomplete. In this study we haveproposed a mathematical-mechanical model to cover the aforesaid gap by comparing the distri-bution of IPF lesions, determined on HRTC images, with the hypothetical distribution of maxi-mal mechanical stress obtained by a simplified mathematical model. Mechanical-mathematicalmodel and simulations are reported in paper [18]. The following figure shows the concentrationof stress from the initial status to the stressed one denoted by red color.

4.2 Derivation of tissue models related to ageing

Part of the final activity related to RESOLVE Project was devoted to the derivation of macro-scale models of biological tissues from the underlying description at the cellular scale. It iswell known that the properties of tissues are determined by the dynamics at the lower cellularscale that include, among others, those induced by the biological functions they express. Forinstance mutations, evolution, proliferation and ability to communicate at distance inducingnon local, and in some cases, nonlinearly additive interactions.

Specific cases that have been treated refer to chemotaxis and angiogenesis phenomenagenerated by the ability that inflamed cells have to attract capillary sprouts from a pre-existing

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vasculature. Namely, cells acquire the ability of producing angiogenic signals to be feeded byvessels. This phenomenon is a crucial component of many mammalian growth processes. Forexample, it occurs in adult mammals during tissue-repair and wound healing. In particular,an uncontrolled or excessive blood-vessel formation is essential for tumourigenesis [19]. Morein details on cancer phenomenon, the first event of tumor-induced angiogenesis involves thecancerous cells of a solid tumor secreting a number of chemicals, collectively known as tumorangiogenic factors into the surrounding tissue. These factors (cytokines) diffuse through thetissue (extracellular matrix), thereby creating chemical gradients in the tissue surrounding thetumor. Once the cytokines reach any neighboring blood vessels, endothelial cells lining thesevessels are stimulated into a sequence of events affecting their migration and proliferation,which culminates in the tumor being penetrated by a capillary network of blood vessels.

Derivation of models at the macro-scale is a key issue toward the modeling of therapeuticalactions as in the various study cases presented in this report. It is worth stressing that the saidderivation needs highly sophisticated mathematical tools. The obvious advantage is that thismultiscale micro-meso-macro approach can lead not only to the rigorous derivation of modelsof biological tissues, but also to the revisiting of classical models to improve the quality oftheir descriptive ability based on a deep understanding of the dynamics at the scale of cells.Models are related to real biological parameters, which can be specifically referred also to theage and overall pathological state of patients.

As already mentioned, two case studies have been developed, namely the derivation ofchemotaxis models [20] and formation of angiogenesis capillary sprouts [21]. Additional bibli-ography is reported in Section 7 by the final lists of papers published within the activities ofthe RESOLVE Project.

4.3 Darwinian modeling of the immune competition related toageing

The activity on the topic in the title has been focused on the modeling of the competitionbetween cancer and immune cells based on the idea, proposed in [22], that mutations in theimmune cells can follow, by natural evolution or induced by medical actions, those of cancercells. The activity in the field is documented in paper [23] just recently published.

The system approach was developed according to the following assessment of the com-plexity features of cancer and immune cells:

• Cancer is a genetic disease: Its evolution is related, since the very early stage, tomutations that give acquired abilities to cells;

• Critical changes in cell physiology: characterize malignant cancer growth;

• Hallmarks: self-sufficiency in growth signals, insensitivity to anti-growth signals, evad-ing apoptosis, limitless replicative potential, sustained angiogenesis, evading immunesystem attack, and tissue invasion and metastasis - incorporate some aspects of geneticmutation, gene expression, and evolutionary selection, leading to malignant progression;

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• The role of the immune system: immune cells have a strategy to learn the presenceof carriers of a pathology and attempt to deplete them, the dynamics involves hallmarksof immune cells; It is a complex process, where immune cells, starting from the innateimmunity, improve their action by learning the so-called acquired immunity.

• Both types of cells (cancer and immune) interact with normal epithelial cells,which can undergo a first stage mutation toward cancer progression. Cells within eachpopulation can modify the level of their specific biological expression.

• The expression of a strategy is heterogeneously distributed among cells even whenthey share the same molecular structure, for instance due to different phenotype expres-sion generated by the same genotype;

• Mutations and selections are generated by net destructive and/or proliferative events.Indeed, all living systems are evolutionary: birth processes can generate individuals thatfit better the outer environment, which in turn generate new ones better and betterfitted.

• Interactions are nonlinear and involve immediate neighbors, but in some cases alsodistant entities. For instance, cells have the ability to communicate by signalling andcan choose different observation paths within networks that evolve in time.

Focusing on the identification of the functional subsystems, tumor cells have beendistinguished according to their progressive hallmarks, while the immune cells are characterizedby the capability to recognize specific hallmarks.

1. i = 1 labels epithelial cells, whose selected function is the ability, supposed uniform forall cells, to feed proliferative phenomena. Proliferative events can generate cells with thesame phenotype, but also cells with different phenotype toward the onset of cancer cells.It is supposed that the organism is a source of epithelial cells, so that their quantity canbe regarded as constant in time;

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2. i = 2 labels cells, generated by the first functional subsystem, that have the ability tothrive in a chronically inflamed micro-environment;

3. i = 3 denotes the functional subsystem of cells, generated by the previous subsystem,that have the ability to evade the immune recognition;

4. i = 4 refers to cells that have acquired the ability of suppressing the immune reaction;

5. i = 5 labels cells of the innate immune system which have the ability to acquire, by alearning process, the capacity of contrasting the development of cancer cells;

6. i = 6 labels cells generated by the innate immune system, which have acquired theability of contrasting the development of cancer cells labelled by i = 2, i.e. cancer cellsfrom the first hallmark;

7. i = 7 labels cells of the immune system generated from the previous two subsystems,which have acquired the ability of contrasting the development of cancer cells labeledby i = 3, i.e. cancer cells from the second hallmark;

8. i = 8 labels cells of the immune system generated from the previous three subsystems,which have acquired the ability of contrasting the development of cancer cells labeledby i = 4, i.e. cancer cells from the third hallmark.

The games modeling interactions at the cellular scale are reported in paper [23]. Simula-tions have put in evidence the role of the parameters that characterize the model on the overalldynamics. All expected emerging behaviors have been foreseen. Moreover, considering thatall parameters have a well defined biological meaning, simulations offer to biologists a detailedpicture of all possible evolution patterns. This contributes to focus the most effective ther-apeutical actions. The following figures show some simulations, where the immune reactioncontrasts the growth of cancer cells at the last stage of mutation. An asymptotic equilibriumstate is reached with a value reported in the second figure. The parameter that rules thisdynamics can be related to the age of the patients [23].

0

0.6

1

1.2

t

n 2

ε = 0.1ε = 0.5ε = 0.9

16

0 0.1 0.2

0.03

0.06

ε2

Asym

ptotic

value

for n

2

Bibliography - Section 4

[18] A. Carloni, V. Poletti, N. Bellomo, L. Fermo and M. Chilosi, Heterogeneous dis-tribution of mechanical stress in human lung: a mathematical approach to evaluate abnormalremodeling in IPF, Journal Theoretical Biology, 332, (2013) 136-140.[19] R.A. Weinberg, The Biology of Cancer, Garland Sciences - Taylor and Francis, NewYork, (2007).[20] N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, On the Asymptotic Theory fromMicroscopic to Macroscopic Tissue Models: an Overview with Perspectives, Math. ModelsMethods Sci., 22 (2012), Paper No. 1130001 (37 pages).[21] N. Bellomo and A. Bellouquid, On the Derivation of Angiogenesis Models: From theMicro- to the Macro-Scale, Mathematics and mechanics of Solids, Submitted, (2013).[22] F. Cavallo, C. De Giovanni, P. Nanni, G. Forni, and P-L. Lollini, 2011: Theimmune hallmarks of cancer, Cancer Immunology Immunotherapy, 60, (2011) 319–326.[23] A. Bellouquid, E. De Angelis, and D. Knopoff, From the modeling of the hallmarksof cancer to a black swan in biology, Mathematical Models and Methods in Applied Sciences,23, (2013) 949–978.

5 Recruitment-Training and Future Steps

Some conclusive remarks can be proposed out of the overall activity reviewed in the precedingsections. The reasonings presented in the following look ahead to conceivable future steps.

1. On the interdisciplinary activity: The author of this report feels comfortable tostate that the interactions of applied mathematicians with biologists and clinicians has beenfruitful as documented in various papers that have faced important problems related to theobjectives of the project. The development of dedicated softwares for the simulations allowa use and future development of the results of the research activity by future generations ofresearchers.

It is worth mentioning that the interplay has not only brought to specific progresses such asthe optimization of surgical action, the modeling of keloid formation, and the study of stress

17

localization in lungs, namely the three specific topics selected as case studies, but also tofundamental problems. In fact, paper [24] proposes a new mathematical-biological approach,within the general framework of developmental biology. Hopefully, this paper represents a firststep toward the ambitious aim of a mathematical theory of biological systems.

As an applied mathematician, I wish to mention that the activity has also contributedto the development of mathematical sciences. In fact, a new mathematical theory has beendeveloped, namely the the kinetic theory of active particles with nonlinearly additive inter-actions. This theory will engage a future generation of applied mathematicians and will takethem towards applications. This is an important achievement in our society, where the interestof young generation to fundamental disciplines is witnessing a remarkable decay.

2. Recruitment and training: The development of the project has allowed the trainingof young researchers in the field of mathematics towards interdisciplinary research linkingmathematical tools and biology. In details the following researchers have been involved andtrained in the project:Carlo Bianca, 24 months devoted to the design of a new system biology approach and modelingthe formation and evolution of keloids;Luisa Fermo, 18 months devoted to optimization problems in the surgery of keloids as well ason the modeling of the mechanics of lungs;Damian Knopoff 18 months devoted to modeling cancer-immune system competition and inrevisiting the system biology approach.

3. Future steps: Future steps are planned based on the idea that the Project RESOLVEis a unique experiment that has brought progress not only to Biological and Medical Sciences,but also on the complex interplay between biology and mathematics, namely between twodisciplines that have a very different research tradition. The first one based on heuristic ap-proaches related to the design of experiment and collection, plus interpretation, of empiricaldata, while the second one on rigorous methods typical of hard sciences. The natural ques-tion: can biological systems be constrained into the framework of mathematicalequations? has a positive answer from this present project. Therefore, we can state that thevarious results achieved within the project already opens new paths to a rigorous approach tobiological sciences by tools from hard sciences such as Mathematics and Physics.

Therefore, future steps should care about additional training of young researchers anddiffusion of the results. In detail:

1. Luisa Fermo is now researcher in the University of Cagliari. She will bring to hernew university her expertise for future activities in the field. Damian Knopoff willreturn, after the experience in Torino, to his origin University of Cordoba (Argentina)and organize there future activity in the field. Indeed, this activity is already beingorganized as a Post Doc from Cordoba (Argentina), namely Juan Pablo Agnelli will bevisiting in Torino for a six months period with the support of an European Programdevoted to scientific interactions between European countries and Argentina.

18

2. This present report will be inserted in the WEB page of Nicola Bellomo to be availableto scientists interested to the interplay between biology and hard sciences. This page isfrequently visited and will contribute to dissemination.

3. Paper [22] is planned to be developed into a book to be offered to the internationalscientific community, where the acknowledgement to RESOLVE Project will be properlygiven.

4. Diffusion by International Conferences: The results of the activity developedwithin the project have been presented in important scientific conferences generally as keynotelectures. The following conferences can be listed in the period IV of the project: “KineticDescription of Social Dynamics” University of Maryland, 5-9-November, 2012.

• Advances in Computational Mechanics, February, 21-27, 2013, San Diego, USA.

• Evolution and Cooperation in Social Sciences and Biomedicine, June 17-21, Granada,Spain.

• BIOMAT 2013 - ”Luis Santalo” School: Mathematics of Planet Earth - Scientific Chal-lenges in a Sustainable Planet - July 15 - 19, 2013, Santander, Spain.

• Mean Fields Games and Related Topics , September 4-6, Padova, Italy.

• ESF OPTPDE Workshop InterDyn - Modeling and Control of Large Interacting Dy-namical Systems, September 10 - 12, 2013, Paris, France.

Bibliography - Section 5

[24] N. Bellomo, D. Knopoff, and J. Soler, On the Difficult Interplay Between Life,“Complexity”, and Mathematical Sciences, Math. Models Methods Appl. Sci., 23 (2013),1861–1913.

6 Appendix I - Mathematical Tools

This final section, proposed as a technical appendix, presents the mathematical tools that havebeen used to chase the objectives of WP4. Clinicians and experimental biologists might possi-bly skip over this presentation that transfers into equations the various concepts described inthe preceding sections. As a matter of fact, the difficult interplay between biology and math-ematical sciences should end up with a unified conceptual framework, while the formalizationis left to mathematical sciences. The related research activity is necessarily interdisciplinary,where biologist should also design experiments to validate models.

19

Let us consider a system of active particles, whose micro-state includes geometrical andmechanical variables, for instance position and velocity, as well as an activity variable suitableto model their strategy. These particles interact within a closed domain Ω, whose largestdimension is denoted by `. In some cases, the dynamics occurs in unbounded domains. Then,a reference length ell is still needed by reasonings on the physics and geometry of the systemunder consideration. Moreover, let us consider, for simplicity of notations, particles modeled aspoints. In this specific case, their microscopic state is identified by the following dimensionlessvariables:

• x = (x, y, z) is the position, referred to `, of each particle;

• v is the velocity, whose modulus is referred to the maximal velocity vM , which eachparticle can attain for their own physical limit;

• u ∈ Du is the activity variable, which models the strategy developed by particles;

• the microscopic state w = (x,v, u) is the set of all the variables, and it defines the stateof each individual.

These particles are subdivided into functional subsystems which correspond to groups ofparticles that express the same activity. In some cases, although the activity is the same, adifferent way to express it can characterize different subsystems.

The overall state of an homogeneous system is described by the probability distribution

functions over the micro-state:

fi = fi(t,x,v, u) : [0, T ]× Ω×Dv ×Du → IR+ , i = 1, . . . , n, (6.1)

where i denotes the functional subsystem. The distribution functions are positive defined, andare referred to the total number of particles N0 at t = 0. Under suitable local integrabilityassumptions, it provides via fi(t, ·) dx dv du the number of active particles that are, at time t, inthe elementary volume of the space of the microscopic states [x,x+dx]×[v,v+dv]×[u, u+du],for each functional subsystem.

If the distribution functions fi are known, macroscopic quantities can be computed asweighted moments by standard calculations [23] that are not repeated here. For instance,zero-th order moments correspond to number density, while first and second order moments

20

to quantities that can be interpreted as somehow equivalent to linear momentum and energy.However, it is generally useful looking at marginal densities, which correspond either to theactivity variable, after integration over the mechanical variables, or to mechanical quantitiesafter averaging over the activity variables. The representation strategy by moments dependson the specific object of the modeling approach.

Living entities, at each interaction, play a game with an output that technically dependson their strategy often related to surviving and adaptation ability, namely to an individualor collective search for fitness [25]. The output of the game generally is not deterministiceven when a causality principle is identified. This dynamics is also related to the fact thatagents receive a feedback from their environments, which modifies the strategy they expressadapting it to the mutated environmental conditions [26]. The interaction modifies the outerenvironment, hence the dynamics of interactions evolves in time.

The following active particles are involved, for each functional subsystem, in the interac-tions:• Test particles from the i-th functional subsystem with microscopic state, at the time t,defined by the variable (x,v, u) := w, whose distribution function is fi = fi(t,x,v, u) =fi(t,w). The test particle is assumed to be representative of the whole system.• Field particles from the k-th functional subsystem with microscopic state, at the time t,defined by the variable (x∗,v∗, u∗) := w∗, whose distribution function is fk = fk(t,x∗,v∗, u∗) =fk(t,w∗).• Candidate particles from the h-th functional subsystem with microscopic state, at the timet, defined by the variable (x∗,v∗, u∗) := w∗, whose distribution function is fh = fh(t,x∗,v∗, u∗) =fh(t,w∗).

Let us now consider short range interactions, when particles interact within a domainΩS ⊂ Ω, generally small with respect to Ω. Interactions at the micro-scale can be describedby the following quantities:Interaction rates, denoted by ηhk(w∗,w∗) and µhk(w∗,w∗), which model the frequency ofthe interactions between a candidate particle from functional subsystem h with state w∗ anda field particle from functional subsystem k with state w∗. Different rates η and µ are usedcorresponding to conservative and proliferative/destructive interactions, respectively.Transition probability density Ci

hk(w∗ → w;w∗), which denotes the probability densitythat a candidate particle ends up into the state of the test particle of the i-th functionalsubsystem after interaction (with rate ηhk(w∗,w∗)) with a field particle, while test particlesinteract with field particles and lose their state;Proliferative term P i

hk(w∗ → w;w∗), which models the proliferative events for a candidateparticle from h into the i-th functional subsystem after interaction (with rate µhk(w∗,w∗))with a field particle from k;Destructive term Dik(w;w∗), which models the rate of destruction for a candidate particlefrom i in its own functional subsystem after an interaction (with rate µik(w,w∗)) with a fieldparticle from k;

21

Pictorial illustration of (a) competitive,(b) cooperative, (c) hiding-chasing and (d) learninggame dynamics between two active particles. Black and grey bullets denote, respectively, thepre- and post-interaction states of the particles.

where dependence on the distribution function is formally denoted by square brackets over fto denote that the various terms can be viewed as operators over f .

The modeling of these terms can take advantage of suitable elaboration of the conceptof distance between particles for the encounter rate, and of game theory for the transitionprobability density, while proliferative and destructive terms occur with the said encounterrate with an intensity depending on the properties, namely state and functional subsystem, ofthe interacting active particles. Technical and analytic details on the aforesaid quantities willbe given in the next subsection.

Bearing all above in mind, let us introduce some ideas borrowed from game theory [27,28]by selecting a certain typology of games. Their modeling provides information of the dynam-ics at the micro-scale that can be introduced, as we shall see in the next section, into thegeneral mathematical structure that can provide the overall dynamics of the systems underconsideration.

In details, let us consider the following types of games:

1. Competitive (dissent): When one of the interacting particle increases its status by takingadvantage of the other, obliging the latter to decrease it. Therefore the competitionbrings advantage to one of the two. This type of interaction has the effect of increasingthe difference in the states of interacting particles, due to a kind of driving back effect.

2. Cooperative (consensus): When the interacting particles exchange their status, oneby increasing it and the other one by decreasing it. Therefore, the interacting activeparticles show a trend to share their micro-state. Such type of interaction leads toa decrease of the difference between the interacting particles’ states, due to a sort ofdragging effect.

3. Learning: One of the two modifies, independently from the other, the micro-state, inthe sense that it learns by reducing the distance the distance between them.

22

4. Hiding-chasing: One of the two attempts to increase the overall distance from the other,which attempts to reduce it.

Bearing all above in mind, let us focus on a large system of interacting active particlessubdivided into n functional subsystems labeled by the subscript i. As already mentioned, ageneral mathematical structure suitable to describe the dynamics of the distribution functionsfi is obtained by a balance of number particles in the elementary space of the micro-states asfollows:

Variation rate of the number of active particles

= Inlet flux rate caused by conservative interactions

−Outlet flux rate caused by conservative interactions

+Inlet flux rate caused by proliferative interactions

−Outlet flux rate caused by destructive interactions,

where the inlet flux includes the dynamics of mutations. This flowchart corresponds to thefollowing structure:

(∂t + v · ∂x

)fi(t,x,v, u) =

(JC

i − JLi + JP

i − JDi

)[f ](t,x,v, u)

=n∑

h,k=1

∫

Ω[f ]×D2u×D2

v

ηhk[f ](x,x∗,v∗,v∗, u∗, u∗)

×Cihk[f ](v∗ → v, u∗ → u|v∗,v∗, u∗, u∗)

×fh(t,x,v∗, u∗)fk(t,x∗,v∗u∗) dx∗ dv∗ dv∗ du∗ du∗

−n∑

k=1

fi(t,x,v)∫

Ω[f ]×Du×Dv

ηik[f ](x,x∗,v,v∗, u∗, u∗)

×fk(t,x∗,v∗, u∗) dx∗ dv∗ du∗

+n∑

h,k=1

∫

Ω[f ]×D2u×Dv

µhk[f ](x,x∗,v∗,v∗, u∗, u∗)P ihk[f ](u∗, u

∗)

×fh(t,x,v, u∗)fk(t,x∗,v∗, u∗) dx∗ dv∗ du∗ du∗,

−n∑

k=1

fi(t,x,v)∫

Ω[f ]×Du×Dv

µij [f ](x,x∗,v∗,v∗, u∗, u∗) Dij [f ](u∗, u∗)

×fk(t,x∗,v∗, u∗) dx∗ dv∗ du∗, (6.2)

where the various terms Ji can be formally expressed, consistently with the definition of η, µ,C, P , and D given in the preceding section; while when space and velocity variables are not

23

significant, this framework simplifies as follows:

∂tfi(t, u) =(JC

i − JLi + JP

i − JDi

)[f ](t, u)

=n∑

h,k=1

∫

Du[f ]×Du[f ]ηhk[f ](u∗, u∗)Ci

hk[f ] (u∗ → u|u∗, u∗) fh(t, u∗)fk(t, u∗) du∗ du∗

−fi(t, u)n∑

k=1

∫

Du[f ]ηik[f ](u∗, u∗) fk(t, u∗) du∗

+n∑

h,k=1

∫

Du[f ]

∫

Du[f ]µhk[f ] P i

hk[f ](u∗, u∗)fh(t, u∗)fk(t, u∗) du∗ du∗

−fi(t, u)n∑

k=1

∫

Du[f ]µik[f ](u∗, u∗) Dik[f ] fk(t, u∗) du∗, (6.3)

Bibliography - Section 6

[25] N. Bellomo, Modelling Complex Living Systems. A Kinetic Theory and Stochastic GameApproach, Birkhauser-Springer, Boston, (2008).[26] E. Mayr, 80 Years of watching the evolutionary senary, Science, 305, 46–47 (2004).[27] H. Gintis, Game Theory Evolving: A Problem-Centered Introduction to Modeling Strate-gic Interaction, Princeton University Press, Princeton, (2009).[28] M.A. Nowak, Evolutionary Dynamics - Exploring the Equations of Life, Princeton Univ.Press, (2006).

7 Appendix II - RESOLVE Papers and Books

[1] N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic gametheory to modelling mutations, onset, progression and immune competition of cancer cells,Physics of Life Reviews, 5, (2008) 183–206.[2] N. Bellomo and A. Bellouquid, On the derivation of macroscopic hyperbolic equationsfor binary multicellular growing mixtures, Computers and Mathematics with Applications, 57,(2009) 744–756.[3] N. Bellomo and M. Delitala, On the coupling of higher and lower scales using themathematical kinetic theory of active particles, Applied Mathematics Letters, 22, (2009) 646–650.[4] C. Bianca and N. Bellomo, Towards a Mathematical Theory of Multiscale ComplexBiological Systems, Series in Mathematical Biology and Medicine, Vol. 11, World Scientific,London, Singapore (2010).

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[5] N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, Multiscale biological tissue modelsand flux-limited chemotaxis for multicellular growing systems, Mathematical Models MethodsApplied Sciences, 20, (2010) 1179–1207.[6] N. Bellomo, Modeling the hiding-learning dynamics in large living system, Applied Math-ematics Letters, 23, (2010) 907–911.[7] N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, Complexity and mathematicaltools towards the modelling multicellular growing systems in Biology, Mathematical ComputerModelling, 51, (2010) 441–451.[8] N. Bellomo, A. Bellouquid and E. De Angelis, On the Derivation of Biological TissueModels From Kinetic Models of Multicellular Growing Systems, in B. Albers, Ed., ContinuousMedia with Microstructure, pp. 131-145, Springer Berlin Heidelberg, 2010.[9] S. De Lillo and N. Bellomo, On the modeling of collective learning dynamics, AppliedMathematics Letters, 24, (2011) 1861-1866.[10] C. Bianca, and L. Fermo, Bifurcation diagrams for the moments of a kinetic typemodel of keloid-immune system competition, Computers Mathematics Application, 61, (2011)277–288.[11] V. Coscia, L. Fermo, N. Bellomo, On the mathematical theory of living systems II:The interplay between mathematics and system biology, Computers Mathematics Application,62, (2011) 3902–3911.[12] L. Fermo, N. Bellomo, and D. Lumenta, Assessment of surgical strategies for ad-dressing keloids: An optimization problem, Computers Mathematics Application, 62(6), (2011)2417–2423.[13] N. Bellomo and B. Carbonaro, Toward a mathematical theory of living systemsfocusing on developmental biology and evolution: A review and perspectives, Physics of LifeReviews, 8(1) (2011), 1–18.[14] N. Bellomo and J. Soler On the mathematical theory of the dynamics of swarmsviewed as a complex system, Mathematical Models Methods Applied Sciences, 22 (Supp01)(2012), Paper No. 1140006, 29 pages.[15] N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, On the Asymptotic Theoryfrom Microscopic to Macroscopic Tissue Models: an Overview with Perspectives, MathematicalModels Methods Applied Sciences, 22 (2012), Paper No. 1130001 (37 pages).[16] L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz, and N. Bellomo, On a Classof Integro-Differential Equations Modeling Complex Systems with Nonlinear Interactions, Ap-plied Mathematics Letters, 25 (2012), 490-495.[17] A. Bellouquid, E. De Angelis, and D. Knopoff, From the modeling of the hallmarksof cancer to a black swan in biology, Mathematical Models and Methods in Applied Sciences,23, (2013) 949–978.[17] A. Carloni, V. Poletti, N. Bellomo, L. Fermo and M. Chilosi, Heterogeneous dis-tribution of mechanical stress in human lung: a mathematical approach to evaluate abnormalremodeling in IPF, Journal Theoretical Biology, 332, (2013) 136-140.

25

[18] N. Bellomo and A. Bellouquid, On the Derivation of Angiogenesis Models: From theMicro- to the Macro-Scale, Mathematics and Mechanics of Solids, Submitted, (2013).[19] A. Bellouquid, E. De Angelis, and D. Knopoff On a 4x3 Model of DarwinianDynamics and Competition between Tumor and Immune Cells submitted to: Computationaland Mathematical Methods in Medicine, (2013).[20] D. Knopoff, D. Fernandez, G. Torres, and C. Turner A mathematical method forparameter estimation in a tumor growth model, submitted to: Computational and Mathemat-ical Methods in Medicine, (2013).[21] N. Bellomo, Commentary to the paper “Morphogenetic action through flux-limitedspreading”, Physics of Life Reviews, (2013), to appear.

26