recently , a great interest has been devoted to consensus problems ( see , e.g. , , and the literature cited within ) .consensus refers to a scenario where a number of agents with interdependent dynamics converge to a common value .when this happens we say that the agents reach an agreement or _consensus_. if state is the resource owned by agent , consensus means that strong agents , those with large resources , transfer part of these resources to weak agents .interdependency has a `` local '' flavor , in the sense that each agent can exchange resources only with a subset ( say it _ neighborhood _ ) of adjacent agents . in its simplest formthe consensus problem can be modeled as an autonomous cooperative linear system with a laplacian dynamic matrix .  in this work ,we study a discrete event system ( ,  ) that behaves in a complementary manner with respect to the above system .we model a competitive scenario where strong agents , gain more and more resources at the expenses of weak ones .this unstable dynamics can be captured by assuming that between two consecutive events , the system evolves as a discrete time autonomous linear system as before , but now the dynamic matrix is the opposite of a time dependent laplacian matrix .deviation between neighbors states / resources increases and for this reason the system is named _ competitive _ system .  with this premise , it makes sense to assume that when an agent looses all its resources , its state goes to zero , then it `` dies '' .the dead agent is then removed from the system together with all its connections ._ death _ introduces a discrete event into the system dynamics .  on the contrary ,when an agent gains a critical amount of resources , then it _duplicates_. the agent , say it _ parent _ , divides in two new agents , say it _ childrens_. both the childrens initial states and neighborhoods are functions of the parent state and neighborhood respectively ._ duplication _ introduces furtherly a second discrete event into the system dynamics .such a modeling framework which we formalize in this paper for the first time , goes under the name of _ dissensus_. the law ruling the discrete time evolution will be called _dissensus protocol_.  dissensus is motivated by applications where multiple agents compete for scarce resources . in marketing , as an example , agents represent firms or companies and resources are market shares or number of customers .competition often results in firms with a large customer base attracting more and more customers at the expenses of the smaller firms  .death describes a firm abandoning the market for lack of customers .duplication captures the situation where a firm expands and then divides into smaller firms .duplication may be forced by different causes .for example , the intervention of an anti - trust authority ; the presence of physical constraints in large infrastructures such as airports ; the departure of apprentices / young partners from the mother firm to open a new one , in case of service businesses or handicraft industry .other applications are in biological networks , e.g. , adjacent cells compete for nutrients .the state of an agent describes its size , that in turn is ( approximately ) proportional to the amount of nutrients the cell subtracts to its adjacent cells .finally , as a theoretical example , let us introduce what we call the _ dynamic pie diagram_. in a dynamic pie diagram , the amplitude of each slice evolves over time .the larger slices become ampler and ampler at the expenses of their adjacent slices .whenever the amplitude of a slice becomes zero , the slice disappears ( dies ) . on the other hand , whenever the amplitude of a slice reaches a given angle the slice is partitioned ( duplicates ) .resources flows are quantized and involve at each time a single pair of agents selected according to a gossip rule .dissensus in continuous time and continuous / unquantized flows have been introduced by the same author in  .quantization in consensus problems can be found in  .gossip algorithms in consensus problems have been introduced in  .the paper is organized as follows . in section[ sec : cp ] , we present the dissensus protocol . in section [ sec : basicproperties ] , we discuss invariancy of the states and network connectivity . in section [ sec : division ] , we comment invariancy of topologies under specific death and duplication rules .some final comments are drawn in section [ sec : conclusions ] .the system evolves according to a quantized discrete time dynamics interrupted at _ critical times _ by some discrete events which we will call _ critical events_. between each two consecutive critical times  and  , with  , the system is made of a set of agents whose cardinality  is a function of the time instants  .the agents  are characterized by states  that assume only discrete values in  .the same agents are organized in a single component _ connection network _ whose edgeset  includes all the non oriented pairs  of agents that bilaterally exchange information about their respective states . at the occurrence of critical event boththe number of agents and the topology of the connection network are modified .we describe how the system state evolves in the following subsection , whereas we describe how the system structure is modified at the critical events in subsection [ subsec : critev ] .  hereafter ,for each  , we call the set _ neighborhood _ of agent  .in addition , for the easy of notation , we omit the dependence on  when there is no risk of ambiguity . then , e.g. , we write  instead of  .the evolution of each agent state depends on local information meaning that it depends only on the state of the agent neighbors .it is governed by a _ quantized gossip algorithm _ ( qga ) in the spirit of what described for cooperative systems .  at each time , for any , one edge  is selected , see , e.g. , in fig .[ fig : gossiponetwo ] , and the states  and  of agents  and  are updated as follows :    for some integer  such that .dynamics ( [ eq : goss1 ] ) makes the deviation between neighbor agents _ diverge _ meaning that if , e.g. , , the greater state tends to increase , and lower state tends to decrease .also , dynamics ( [ eq : goss1 ] ) is _ zero - sum _ meaning that a same quantity is gained by and lost by , thus resulting in the local invariance property .( left ) and ( right ) . ]differently from , we additionally require that edges are selected in a deterministic way according to the next assumption .[ as : t ] there exists a big enough integer value  such that , within each time interval  , either each edge  is selected at least once or a critical event occurs .had we assumed , in line with , that each edge in  is selected randomly and with a positive probability , some of the results in this paper would have held in probability .upper and lower thresholds are given : the upper threshold is the positive integer value  , the lower threshold is the value zero .critical event _ occurs at time if an agent  reaches a threshold at time  , i.e , when either or .more formally , hereafter , when it is not stated otherwise , we assume without loss of generality that , within each interval , the agents are renumbered so that the agent reaching a threshold value is always agent  .  on the occurrence of a critical event ,the number of agents and the connection network are modified . in this context , we denote as  ( respectively  ) the set of agents ( respectively edges ) involved in a critical event and still present in the network at the end of the event . at the system structure modifies as follows .  *if , we say that agent  _ dies _ and is consequently removed from the system .the agents in the neighborhood of  inherit the connections of  as depicted in fig .[ fig : networkdeath ] .+  + in particular , if , for each agent , we denote with , the set of new neighbors that inherits from the dead agent , we can write : + then , in the occurrence of a death , the set of involved agents is and the set of inherited links is . + as it will be clearer in section [ sec : basicproperties ] , in order to preserve the connectivity of the network , we require that i ) all neighbors of the dead agent are linked to at least one other neighbor , ii ) the set of neighbors of the dead agent , together with the inherited links , form a connected subnetwork.formally , regarding the connection network  , the dead agent  is removed from the set of agents  , all links of the dead agent are removed from the set of edges  , finally the new links in  are added to  : * if , the parent agent  _ _ divides _ _ producing two children agents  and  .the two children agents inherit the parent connections and state ( see , e.g. , fig .[ fig : death1dupli2 ] ) .+  + this means that , if we denote with and  the new neighbors of the two children agents  and  , we can write : then , in the occurrence of a duplication , the set of involved agents is and the set of the links to the children agents is .+ in order to preserve the connectivity of the network , we impose that one of the following two rules hold + furthermore , in order to maintain the sum of the states invariant , we impose also that the values  and  are integers such that and regarding the connection network  , a new agent  , replacing the old one , and an additional agent  are added to the set of agents  , all the links to the father agent  are removed from the set of edges  and the set  of new links to the children agents are added to  :   figure [ fig : piediagram ] shows the possible evolution of a dynamic pie diagram .an agent / slice divides at time  , having reached an amplitude of at the previous time instant .differently , an agent / slice dies at time .    in the next session ,we discuss invariancy and connectivity associated to death and duplication rules .in this section , we introduce some bounds on the number  of agents . to this end, we initially discuss the connectivity and invariancy properties that are exploited in the bounds determination .  preserving the agents connectivitymotivates the choice for death ( [ conn1 ] ) and duplication rules ( [ connectivity1 ] ) . before discussing them in details , we need the following definition .( _ local connectivity _ ) is _ locally connected _ if the subnetwork is connected .note that we can impose the local connectivity of  exploiting the knowledge of  , i.e. , the information available to the agent  just before its death or duplication . conditions for the local connectivity to imply the connectivity of  are established in the next theorem .if  is connected , then both the death ( [ conn1 ] ) and duplication rules ( [ connectivity1 ] ) are sufficient conditions for the connectivity of  .these rules become even necessary to guarantee the connectivity of  on the basis of the only knowledge of neighborhood  available to agent and disregarding any information on other agents neighborhoods  for  .  the proof is based on the following two facts :  * differs from  only in the subnetwork induced by the agents in  , * both the death ( [ conn1 ] ) and duplication rules ( [ connectivity1 ] ) guarantee the local connectivity of  .this latter fact hold true by definition of rule ( [ connectivity2 ] ) in case of a death . in the case of a duplication, we note that  and that both rules ( [ connectivity1 ] ) imply that each  is connected with either  or  .in addition , if the first condition of ( [ connectivity1 ] ) holds , then and , hence , includes  .if the second condition of ( [ connectivity2 ] ) holds , then there exists  such that  includes  and  . in both cases, we can conclude that each  is connected to  through the edges in  . by transitivity is connected .( sufficiency ) as  has a single component , we know that , in  , each  is connected to at least an agent through a path that does not include agent  and any other agent in  different from  .these paths connecting  with  remain identical in  .then , as  , all the agents are connected in  with at least an agent in  .as  is connected , all the agents in  , and hence the agents in , are connected to each others in  . finally , as the set of the agents in  is , we can conclude that  is connected .( necessity ) if we have no knowledge of neighborhoods  for , we must conservatively assume that all the links between  and the agents  are necessary for the connectivity of  , since  may be , e.g. , a tree .consequently , to guarantee the connectivity of  , we must guarantee that the edges in keep the agent in  connected to each others and to the new agents eventually added in .in other words , we require that  is connected .figure [ fig : examples ] left shows that if ( [ connectivity1 ] ) does not hold  may turn to be disconnected .however , in general , rules ( [ conn1 ] ) and ( [ connectivity1 ] ) are not necessary to the connectivity of  .as an example , in case of a duplication , the connectivity of  holds even if for all , if all are adjacent to a common agent ( see , e.g. , fig .[ fig : examples ] right ) .  )are necessary ( left networks ) and not necessary to the connectivity of  ( right networks ) . ]  ) are necessary ( left networks ) and not necessary to the connectivity of  ( right networks ) . ]  as regards invariancy , we observe that dynamics described in ( [ eq : goss1 ] ) , ( [ eq : death ] ) and ( [ eq : division ] ) imply that the sum of the agents states is a constant value over time , i.e. ,   invariancy is a necessary condition to avoid that the number of agents neither diverges to infinity nor converges to one .assume that in consequence of a duplication at time the sum of the states decreases over time , i.e , or that an agent may die having a state .it is immediate to see , that under these hypotheses either the system reaches consensus or .differently , if the sum of the states increases over time , which corresponds to saying , then either the system reaches consensus or .invariancy is also used in the next result to establish that the number of agents never goes above neither below certain bounds .[ th : bounds ] let be the critical time , possibly infinite , at which the first duplication occurs .then the following bounds holds on the number of agents :   the lower bound is a straightforward consequence of the invariancy ( [ eq : stateinvariance ] ) .as regards to the upper bounds , we can have in the following trivial situation . at the initial time the agents states have value for all .differently , from on , the maximum number of agents can be reached only when a duplication occurs . in this case, the agents state invariancy imposes    in ( [ eq : lb1q ] ) , the inequality holds as , after a duplication event , at least one of the system agents has state equal to , a second agent has state equal to , and the remaining agents have state .finally , the bound in ( [ eq : nlowerbound ] ) trivially follows from ( [ eq : lb1q ] ) .note that the fact that , and hence the cardinality of , is bounded from the above for all guarantees the possibility that the value  required by assumption [ as : t ] exists .invariancy and connectivity play a critical role when proving that under a qga the intervals between consecutive critical events is finite .indeed , this fact holds true unless the agents reach _ consensus_. we say that the agents reach consensus at time if all states are equal , i.e. , for all .the next lemma is a prtelude to show that the interval between consecutive critical events is finite .[ lem : cv ] consensus can be reached only at times , for .  for each time , let be the agent with minimum state , be the agent with maximum state .by definition of qga we have that the values of , respectively , may vary but can not increase , respectively decrease , for . as a consequence, the difference between and can not reduce to zero for .[ th : critev ] assume that the agents do not reach consensus in  , then a critical event will occur in a finite time  .we prove the statement by contradiction .in particular , we show that if no critical event occurs in a finite time the system state diverges . to this end , we assume that the agents have not reached consensus at time  and that no critical event occurs from  on .let , be the agents selected by the qga at the generic time and consider difference in view of assumption [ as : t ] , we have that for all  , as , if i ) no critical events occurs after  , ii )  is connected and iii ) the agents have not reached consensus in  , within every  instants at least a couple of agents with different states is selected . then, as , we have that .the value of the latter limit is in contradiction with the fact that must hold for any .hence , either the agents have reached consensus in  or a critical event occurs after  .an immediate consequence of theorems [ th : bounds ] and [ th : critev ] is that a first duplication event occurs at time after at maximum death events .in addition , the is finite unless the agents reach consensus at time or at onethe above death events .  in the rest of this section ,we analyze possible scenarios where the agents reach consensus .consider systems evolving according a protocol in which the duplication rule ( [ eq : division ] ) requires that and that both the two children inherit all their parent connections and connect to each other ( see , e.g. , rule ( [ eq : likefather ] ) discussed in the next section ) .such systems may reach consensus .note that in this case each duplication generates twin agents .consider , as an example , the evolution of a system where and and . by theorem[ th : bounds ] , we have for all . we can easily define a qga such that at time , we have and . at time , the agent  dies ( we have not renumbered the agents for the easy of exposition ) and the remaining two twin agents reach an equilibrium corresponding to a state value equal to .  the next theorem establishes some necessary conditions for the agents to reach consensus .[ th : cv ] consensus can be reached provided that for all .furthermore , where is the critical time of the first duplication event .the first part of the theorem is a trivial consequence of the definition of consensus and of the invariancy property .the first lower bound is true as , for , we have and , obviously , the invariancy property holds . in proving the second lowerbound , we observe that , for , we have .actually , any qga prevents from decreasing between two consecutive critical duplication events .a direct consequence of theorem [ th : cv ] is that there exist initial states  such that the agents will never reach a consensus as stated in the following corollary .let be the critical time of the first duplication event .no consensus can be attained for , if is not divisible for some integer less than or equal to .no consensus can be attained for , if is not divisible for some integer greater than or equal to .the corollary thesis is an immediate consequence of of theorem [ th : cv ] and the fact that both the number of agents that reach consensus and their states must assume integer values .we conclude this section by observing that trajectory may become periodic in the long run when the agent do not reach a consensus . as an example, this situation occurs if the system evolution evolves according to death / duplication rules that depend on the agents states and on the connection network topology and no stochasticity or time dependency is allowed.indeed , we can observe that our systems evolve in a space defined by the number of agents , their states and the topologies of their possible connection networks.as both the number of agents and their states may assume only integer values bounded from above , then the number of topologies of the possible connection networks is also finite , hence the thesis follows .let us now investigate conditions on death and duplication rules that result in the invariancy of specific network topologies like complete , hole or chain networks .as we study the asymptotic evolution of a system , let us initially observe that theorem [ th : bounds ] implies that there always exists a critical time  an two integer values and such that , for each satisfying we have an infinite sequence of critical time instants such that . in this context, we say that the _ density of the connection network _ of a system does not increase asymptotically if there exists a value  such that for each  satisfying condition ( [ eq : nbar ] ) , the value does not increase over the critical times in  and greater than or equal to  .we start by considering the duplication rule which equally divides the parent s connections between the two children .formally , where function returns a random subset of elements of .with such a rule , if is a hole network , respectively a chain network ( see fig . [fig : holechain ] ) , then are hole network , respectively chain network , for every , whatever death rule is implemented .again , this corresponds to saying that hole and chain networks are invariant as established in the next theorem . before giving the theorem , we recall that a network is a hole , respectively a chain network , if it is connected and all the agents have degree two , respectively all the agents have degree two a part from two agents at the extreme of the chain whose degree is one ( see , e.g. , fig . [fig : holechain ] ) .hole and chain networks are invariant under duplication rule ( [ eq : part ] ) .the idea of the proof is that critical events do not change , in general , the degree of the agents .this is always true unless the dying or duplicating agent is an extreme of the chain . actually , when an extreme agent dies , its neighbor , whose degree changes from two to one , becomes the new extreme .similarly , when an extreme agent duplicates , one children is the new extreme with degree one , while the other children has degree two as all other agents .consider now the death rule ( [ eq : death ] ) that assign all the connections of a dead agent to just one of its neighbors .this can be formally described by a death rule that , for each , satisfies : where is arbitrarily picked .  from a practical point of view , these choices for are the simplest ones to implement that guarantee the connectivity of after the removal of the dying agent .[ th : lowden ] under duplication rule ( [ eq : part ] ) and death rule ( [ eq : death1 ] ) , both the density and the number cycles of a connection network can not increase asymptotically .the thesis is trivially true if the agents reach consensus .otherwise , it is immediately to see that , if at  a duplication occurs , ; if at  a death occurs , as at least the connection  is not substituted by a new connection . now , observe that , for each  satisfying condition ( [ eq : nbar ] ) , an equal number of death and duplication events must occur between each two critical times and , with , hence . then , we can conclude that the density of the connection network can not increase over time . finally , the observation that the considered duplication and death rules forbid the creation of new cycles in the connection network concludes the proof .authors computational experiments show that the density of the connection network usually indeed decreases until presents a single or no cycle at all if the assumptions of theorem [ th : lowden ] hold and the agents do not reach consensus .although no formal proof confirms such observations , it appears reasonable to expect that the the connection network becomes sparser and sparser .indeed , inequality ( [ eq : ineq ] ) in the proof of theorem [ th : lowden ] holds strictly in the occurrence of a death event whenever , situation quite common if the network is not sparse .let us now consider the duplication rule which makes both children inherit all the parent connections and connect to each other :   observe that the above rule preserves complete connectivity of the subnetwork induced by the critical set after a duplication event .this also induces the fact that if is a complete network then are complete networks for every , whatever death rule is implemented .this corresponds to saying that the complete network is _ invariant _ under the duplication rule ( [ eq : likefather ] ) as established in the next theorem .the complete network is _ invariant _ under the duplication rule ( [ eq : likefather ] ) .assume is a complete network .if a death occurs at time , each agent is already adjacent to all the other agents . if a duplication occurs at time , because of rule ( [ eq : likefather ] ), the two new agents are adjacent to each other and to all the other agents .the authors computational experiments show that , for a generic systems with not complete , the density of the connection network usually increases until becomes a complete network if the rule ( [ eq : likefather ] ) is applied and the agents do not reach consensus , see .in this paper we have introduced systems of competitive agents exchanging quantized flows according to a dissensus protocol .we have discussed some invariancy properties of the systems and observed that the number of agents is bounded from below and above over time . as a future direction of research we are oriented toward the definition of more general rules for the inheritance of the connections .indeed , the main limit of the current version of the dissensus protocol seems the fact that the inheritance rule of the connections applies only when an agent dies . in many real competitive systems the larger agents may acquire the connections of the smaller agents when the latter ones are still alive .consider the following two examples .the number of the cells adjacent to ( that is , in the neighborhood of ) a cell  of a tissue is a function of the size of the cell  and not only of the possible occurrence of a death of an adjacent cell .similarly , the rules used for the dynamic pie diagram in fig . [ fig : piediagram ] can not trivially be extended to a general dynamic tessellation .d. bauso , l. giarr , r. pesenti . quantized dissensus in switching networks with nodes death and duplication2 , _ proc . of the 1st ifac workshop on estimation and control of networked systems _ ,venice , i , sept .24 - 26 , pp . 1 - 6 , 2009 .    p. frasca , r. carli , f. fagnani , s. zampieri , `` average consensus by gossip algorithms with quantized communication '' _ proc .47th ieee conference on decision and control _, cancun , mex , pp .4831 - 4836 , 2008 .g. karev , y. wolf , a. rzhetsky , f. berezovskaya , e. koonin , `` birth and death of protein domains : a simple model of evolution explains power law behavior '' , _ bmc evol .1 , pp . 1 - 8 , 2002 .a. nedi , a. olshevsky , a. ozdaglar , j.n .tsitsiklis , `` on distributed averaging algorithms and quantization effects ? '' , _ proc .47th ieee conference on decision and control _, cancun , mex , pp .4825 - 4830 , 2008 .w. ren , r. beard , e.m .atkins , `` information consensus in multivehicle cooperative control : collective group behavior through local interaction '' , _ ieee cont .syst . mag .2 , pp . 71 - 82 , 2005

Abstract: _ dissensus _ is a modeling framework for networks of dynamic agents in competition for scarce resources . originally inspired by biological cells behaviors , it fits also marketing , finance and many other application areas . competition is often unstable in the sense that strong agents , those having access to large resources , gain more and more resources at the expense of weak agents . thus , strong agents duplicate when reaching a critical amount of resources , whereas weak agents die when loosing all their resources . to capture all these phenomena we introduce systems with a discrete time gossip and unstable state dynamics interrupted by discrete events affecting the network topology . invariancy of states and topologies and network connectivity are explored .  * keywords * : consensus protocols , quantized control , dynamic programming , network based marketing , dynamic pie diagram .


there are many exciting challenges facing mathematical modelling in biology . in this articlewe review some of these challenges in the context of the healthy and diseased dynamics of a specific biological system , the epithelial tissues that line internal and external body surfaces .much of the interest in epithelial tissues arises because almost all solid cancers are epithelial in origin ; however , many of the modelling issues we discuss are relevant to a wider range of problems in mathematical biology .  the challenges we will highlight are : ( i ) _ scale and resolution of models_. the desire for a model to address a specific biological question often motivates model simplicity ; however , it is now well - known that biological function arises from the integration of coupled processes acting across a range of spatial and temporal scales . often these processes can not be easily separated and this creates technical challenges for mathematical and computational modelling .( ii ) _ model - model comparison_. currently , many different models have been developed to address the same problem , leading us to ask under what conditions do the different models generate the same predictions ? further , when models disagree , which approach is better ? questions of this nature motivate the development of new methods of model - model comparison .( iii ) _ model - data comparison_. validation of multiscale models is typically performed in a coarse - grained manner . with the advent of improved imaging techniques, there is now the potential to compare such models with high - resolution experimental data .modern statistical inference methods , made available with increased computing power , have yet to be fully exploited , however , and offer exciting opportunities for the future .the remainder of the paper is structured as follows .section [ sec : epithelial_biology ] contains a brief biological introduction to epithelial tissues .we take the view that many questions regarding epithelial pathology such as tumorigenesis can be profitably understood in the context of failure of normal homeostasis and repair mechanisms .thus we consider a range of phenomena in epithelial tissue dynamics . in sections [ sec : compartment_models ] and [ sec : individual_based_models ] , respectively , we review compartmental and individual - based approaches that have been used to model epithelial cell populations , emphasising how the scale and resolution of the models ( challenge i ) have yielded insights into epithelial dynamics , and how some organising concepts have emerged within these model classes . at the same time , the absence of a common structure linking the different compartmental and cell - based models makes their comparison ( challenge ii ) difficult .this motivates our discussion in section [ sec : continuum_models ] of continuum limits of discrete , cell - based models , considered in the context of mechanically - interacting epithelial cells . in the discussion in the final section , section [ sec : discussion ] , we focus on the opportunities that new data raise for measurement and parameter estimation ( challenge iii ) , referencing examples from tumour dynamics and epithelial development and repair , while emphasising the intertwined nature of the three challenges i - iii .we also speculate on possible approaches for developing unified frameworks for using modelling , measurement and inference to better understand epithelial tissue dynamics .the epithelial tissues that line the internal and external surfaces of the body are a primary source of physical protection and act as barriers to unwarranted substances  .maintenance of normal physiological function requires that many epithelia simultaneously carry out directed transport of substances such as water and electrolytes between bathing solutions on either side of the epithelial tissue ; this is crucial for tasks such as regulating body fluid composition and absorbing nutrients  . as a result , many epithelial tissues , such as the skin and intestinal epithelia , have rapid rates of self - renewal and distinctive structural and functional organisation ( see below ) .importantly , dysregulation of the renewal process is thought to be associated with the large number ( up to 90% ) of cancers originating in epithelia  .epithelial tissues thus serve as important biological model systems for studying the dynamics of multicellular populations ; further , understanding them under both healthy and diseased conditions is important for understanding the transition from one state to the other .epithelial tissues comprise sheets of polarised cells , as well as glandular ingrowths ( or crypts ) and finger - like protrusions ( or villi ) formed from these sheets ; the arrangement of these layers is maintained by specialised cell - cell and cell - matrix junctions  ( figure [ fig : epi]a ) .these junctions , especially the tight junctions , are crucial not only for structural integrity of the epithelial layers , but also for maintaining cell polarity and , hence , the ability of epithelial tissues to perform directed transport of solutes and water .individual epithelial cells are subject to strong mutual mechanical interactions mediated through these proteins .  the self - renewal of epithelial tissues is driven by stem cells which generate , in a tightly - controlled manner , a sequence of increasingly specialised descendants known collectively as a lineage structure ( figure [ fig : epi]b ) .this sequence must produce a sufficient number of descendants to maintain those parts of the epithelium exposed to the greatest levels of damage and so the spatial arrangement of the cell lineages frequently corresponds to functional requirements .an archetypal example is provided by intestinal crypts ( figure [ fig : epi]c ) , finger - like invaginations of the intestinal mucosa . the intestinal crypt is structured spatially into regions corresponding to a stem cell niche at the base , an intermediate region of rapidly proliferating transit - amplifying cells , and a region of differentiated , functional cells nearest to the crypt orifice , where most transport and damage occur  .this lineage structure maximises the number of functional descendants of stem cells while protecting the stem cells from damage .in addition to extensive experimental study , the intestinal epithelium has been the focus of numerous mathematical models , especially in the context of colorectal carcinogenesis  . in the sections that follow we will review some of these models .compartmental models provide a flexible framework for studying cell population dynamics . as a result, they have been widely used to study epithelial tissue homeostasis , growth , regeneration and carcinogenesis , and the interplay between these dynamics .these models account for processes such as cell proliferation , differentiation and death in generic terms , without specifying details of the mechanisms ( e.g. signalling pathways ) by which they are regulated .the resolution of these models is coarse ; compartments typically correspond to either spatial domains or distinct cell populations , with movement or other exchange between compartments .using too many compartments is not typically a good idea - it can lead to parameters that are redundant or poorly constrained . in this caseone might seek to reduce the number of compartments or , alternatively , use partial differential equation ( pde ) or delay - differential equation ( dde ) models , which replace the discrete parameters with some constrained ( e.g. smooth ) function or ` lumping ' of parameters . in what followswe restrict attention to deterministic compartment models comprising systems of ordinary differential equations ( odes ) .further details on more complex compartment models , including those that account for age - structure and stochastic effects , may be found in  and references therein .later sections address alternative models for introducing finer - grained spatial resolution .the general form of an ode - based compartment model for a sequence of cell lineages ( see figure [ fig : epi]b ) , which incorporates the processes of cell renewal , differentiation and death , is given by here denotes the number of cells in compartment at time , while the parameters , and denote the renewal , differentiation and death rates of cells in compartment .these parameters may be constants ( for a given biological scenario ) or may be prescribed functions of the different cell populations . in the latter case ,the functional forms encode key hypotheses about how signalling and other control mechanisms regulate cell kinetics ; we consider some examples of this next .johnston et al . construct a model of the form  to investigate homeostasis and the transition to tumorigenesis in the intestinal crypt .they consider three compartments , corresponding to stem , transit amplifying and differentiated cells respectively .they note that models in which the kinetic parameters , and are constants are _ structurally unstable _ : slightly different parameter values can lead to qualitatively different population dynamics . in their model ,the total number of crypt cells may exhibit exponential growth , exponential decay or converge to a finite equilibrium , according to whether the net growth rate in the stem cell compartment , , is greater than , less than or equal to zero , respectively .such behaviour is physically unrealistic , not least because the model parameters are idealisations , which are unlikely to remain constant during tissue homeostasis .  to obtain a model in which homeostasis is preserved as system parameters vary , johnston et al . incorporate local feedbacks within the stem and transit amplifying compartments , such that the proportion of cells differentiating in a given compartment varies with the compartment size .they find that a feedback which increases linearly with compartment population size , taking the form for constants and , leads to a model for which homeostasis is a stable steady state . while the total number of crypt cells at equilibrium changes as parameter values vary , representing the effect of pre - cancerous mutations on cell kinetics ,the system does not exhibit the unbounded growth that is characteristic of tumorigenesis .improving on this , the authors find that including a saturating local feedback of the form , for constants , and , leads to a model that , as parameter values are varied to represent the effects of mutations , exhibits both changes to equilibrium cell numbers as well as eventual unbounded growth after sufficient mutations accumulate .the authors note that this behaviour recapitulates the natural pattern of tumorigenesis , where periods of rapid expansion alternate with long periods of equilibrium in a stepwise manner .  in other work ,lander et al . ( previously analysed in more generality in ) develop an ode - based compartment model to study the renewal of mammalian olfactory neural epithelium .in contrast to the model of johnston et al . , this model allows for feedback mechanisms whereby cells that are further along the lineage modulate the behaviour of cells earlier in the lineage , this feedback being mediated by chemicals sometimes given the generic name ` chalones ' .lander et al . consider the cases of two , three and an arbitrary number of compartments , and assume that cell death is restricted to the most differentiated compartment only : in all other compartments , dividing cells either renew or differentiate .lander et al . denote by the renewal probability and by the rate of cell division , and so we may identify the link between our notation and theirs through .lander et al . use an engineering approach to analyse their model .they consider a variety of control ` objectives ' such as control of size , growth rate and cell - type proportions . in agreement with johnston et al . , they observe that without feedback the system is sensitive to variations in parameter values .they consider several feedback mechanisms by which chalones may achieve the control objectives .these include feedback on renewal probabilities , division rates , and the compartments between which the feedbacks operate .feedback on renewal probabilities , rather than division rates , is found to give the most robust form of control , with reduced parameter sensitivity , faster regeneration in response to injury and less sensitive dependence on initial conditions .  the work of lander et al . illustrates how even simple theoretical analyses can generate new biological insight when carefully combined with experiments .the authors progress from a model prediction , which was unexpected without formal analysis , to experimental validation by considering the protein growth differentiation factor 11 ( gdf11 ) , a member of the transforming growth factor beta superfamily . while already known as an inhibitor of the mammalian olfactory neural epithelium , gdf11 s primary target was believed to be the rate of cell division rather than the renewal probability .lander et al .show experimentally that gdf11 appears negatively to regulate both rates , and conclude that many experimental observations are consistent with the theoretical expectations of ` best ' control .the generic nature of compartment models is both a strength and a weakness .since they are formulated in phenomenological terms , they can be used to determine what types of feedback mechanisms will produce robust dynamic behaviour when model parameters fluctuate , and how these might fail .however , it can be difficult to relate these general features to specific measurable quantities and/or to rule out alternative mechanisms .furthermore , while capable of representing coarse spatial structure , such as cell lineages in the intestinal crypt , compartmental models lack the spatial resolution needed to track cell movement or to account for mechanical effects that regulate such movement .this can have important implications for understanding both healthy and diseased state dynamics .we next consider a modelling approach that can address such issues .in order to account for phenomena such as cell sorting , rearrangements , mechanical interactions and pattern formation , mathematical models with spatial resolution are often required .individual - based models ( ibms ) offer a natural framework for describing the behaviour of a finite number of individual cells or cellular components .these models are typically termed either _ ` on lattice ' _ or _ ` off lattice ' _ , depending on how individual cells are represented ( figure [ fig : ibm ] ) . in the following we focus on general features of ibms that have been used to study the dynamics of cell sorting in embryonic epithelia and the influence of mechanical forces on the dynamics of intestinal crypt cells : for more extensive reviews , see  and  ..5   .5   .5     on - lattice models usually discretise space using a fixed , regular , lattice .for example , in cellular automata ( ca ) , each cell occupies a single lattice site , which may change over time  .the system dynamics are encoded in a simple , rule - based manner , with changes in cell configuration occurring at discrete time steps , using synchronous or asynchronous updating or an event - driven approach based on the gillespie algorithm  .ca have no _ a priori _ requirement to satisfy physical principles and hence may exhibit unphysical dynamics .for example , when a cell divides in a ca , its daughter cells may ` shove ' their neighbours to make space , resulting in instantaneous displacements over many cell diameters .model simulations may also suffer from artificial anisotropies induced by symmetries of the underlying lattice . despite their relative simplicity , camodels have been successfully used to predict aspects of epithelial carcinogenesis .for example , gatenby et al . combined ca simulations with histological studies of tumour biopsies to demonstrate the key role that adaptation to hypoxia and acidosis may play in the transition from a precancerous lesion ( such as ductal carcinoma in situ in the breast ) to invasive cancer .the cellular potts model ( cpm ) is another popular , on - lattice - model , based on simple models of magnetism from statistical mechanics , and adapted by graner and grazier to study cell sorting in embryonic epithelia . in the cpm , cells may span several , connected lattice sites , rather than being in one - to - one correspondence as for the ca models .if lattice sites are indexed by a single integer , then cells correspond to collections of lattice sites which share the same value of an integer label . using to denote the set of all pairs of neighbouring lattice sites , the possible interactions between neighbouring cells are encoded via a hamiltonian ( interaction energy ) , such as  where the additional label defines cell type and is the kronecker delta .the first summation in equation ( [ eqn - cpm ] ) represents an adhesion interaction energy and the second term an energy associated with cell stretching . in ( [ eqn - cpm ] ) ,different adhesion energies may be assigned to interactions involving different types of cells , but individual cells do not have adhesive interactions with themselves .the area of a cell ( assuming we are modelling cells in two dimensions ) is denoted by , is its target area and is an tunable phenomenological parameter weighting the contribution of the target area energy to the adhesive energy .additional interaction or cell property terms may be added as desired .the model dynamics are implemented via a markov chain monte carlo ( mcmc ) updating method ( the metropolis algorithm ) rather than an explicit equation of motion .we note that the original statistical mechanical models that the cpm is based on were developed to study equilibrium systems and there is some difficulty in interpreting the dynamic features of the cpm ( see e.g. the discussion in ) .as mentioned , the cpm was introduced by glazier and graner to explain experiments performed on embryonic epithelial cell populations .after artificial mixing , the different types of cells rearrange until they recover their pre - mixed configurations .steinberg  introduced the differential adhesion hypothesis ( dah ) to explain this phenomenon , the general principle being that intermixed cell populations rearrange so as to minimise their collective adhesive free energy .the cpm , as implemented by graner and glazier , successfully , albeit qualitatively , reproduces observations of cell sorting  based on the dah .this concept has subsequently been applied to processes in development , wound repair , and tumorigenesis ; for example , used a cpm to investigate the influence of changes in cell - cell and cell - substrate adhesion on cancer invasion .their model notably incorporates a description of cell proliferation in which the mitotic rate depends on the relative magnitudes of homotypic and heterotypic adhesion . despite difficulties in precisely relating some aspects of this model framework to biological processes ,the cpm remains an informative and popular framework  .off - lattice ibms describe the dynamics of discrete , mechanically - interacting cell entities . at their simplestthese entities are typically either cell - centres or cell vertices , while more complex models allow for additional resolution of the cell interior in terms of further ` subcellular'-scale entities .the basic governing principles are the same in all cases , however . as for the cpm, these models can be formulated in terms of potential functions encoding the interaction energy between cells or cellular elements .in contrast to the cpm , here an explicit equation of motion is specified for the cellular entities .  for vertex models we consider cells indexed by , as in the cpm , but now with vertices , rather than grid points , indexed by and with each vertex having an associated position vector . herecell vertices represent the tight junction proteins connecting cells .the free energy of each cell is then defined as a sum of contributions , such as adhesive energy , target area ( assuming for convenience that we are in two - dimensions ) energy and target perimeter energy , giving  here is the set of lengths of the lines , or ` bonds ' , connecting the vertices of cell ; the first term , summing over these lengths , then represents the total vertex bond energy of cell . and are the area and perimeter of cell , respectively , and and are its target area and perimeter , respectively . finally , , and , respectively , are phenomenological parameters weighting the contributions of the above energies .as indicated , each cell has an associated set of vertices with position vectors and each of the above cell properties are then defined by functions of these position vectors .this allows an equation of motion to be written for the coordinate vectors of the vertices , here an overdamped newtonian equation .this is driven by ( generalised ) forces obtained by differentiating the free energy function considered as a function of vertex coordinates .this gives where is the viscous drag coefficient of vertex and the gradient operator with respect to .while cell - centre models may be formulated in energetic terms , they are typically ( but equivalently ) formulated by applying force balances . thus the equation governing the motion of , where now indexes cell centres rather than the cell vertices , is similar in form to ( [ eq : motion - vertex ] ) : where is the assumed equal and opposite force that cell centre exerts on cell centre , an example being the linear spring law for which , where is the target length of the spring connecting cell centres and . in ( [ eq : motion - centre ] ) ,the sum is over all cell centres in a well - defined neighbourhood of the cell with centre ( e.g. by a delaunay triangulation ) .the ` subcellular element ' approach was introduced by newman as a general framework for investigating the emergent behaviour of cells and tissues , including tumorigenesis ; the mechanics of an epithelial sheet was chosen as an initial example but many other systems have been studied ( see below ) . in this model ,cells are defined by a finer set of vertices than just cell boundary or centre positions . in this case indexes a set of subcellular - scale vertices called ` subcellular elements ' , which have a phenomenological correspondence to internal cellular architecture .the total potential is split into a set of interaction potentials which here are further divided into intracellular and intercellular components depending only on the distance between element positions .a noise term is often added ( this may similarly be added to the above vertex or cell - centre models ) , and is typically chosen to be gaussian with zero mean . again , however , the principles are the same . here the equation of motion for the position vector of a subcellular element , with associated cell , takes the form    where the first sum of accounts for interactions between element of cell and all other intracellular elements of the same cell , while the second sum governs interactions of element with each element of each other cell .though the above models are all similar in form , each comes with its own conceptual framework for specifying the particular interaction potentials ( i.e constitutive modelling ) ; thus detailed comparisons of all three approaches under different circumstances would be helpful .this will be greatly aided by the development and sharing of open source software ; an example is chaste , which currently implements vertex and cell - centre models . fornow we consider a range of different examples , where one particular approach was typically chosen .intestinal crypt mechanics have been widely studied with ibm models .meineke et al . developed the first lattice - free ibm model of cell motion in the intestinal crypt ; the crypt is modelled as a cylindrical surface and cell centres interact via linear spring forces .the model was developed to overcome limitations of earlier , lattice - based ca models of the intestinal crypt such as .in particular , the model allows for continuous time and space steps ( to computational resolution ) , avoids unrealistic shifting of columns of cells , and cell motion is not based on rules involving age or knowledge of neighbouring cell types , in contrast to .furthermore , the model can be adapted to determine how the cell dynamics change when the idealised cylindrical geometry of the crypt is replaced by a more realistic test - tube shaped geometry .more recently , other authors have included more biological detail into ibm models of the intestinal crypt  .for example , van leeuwen et al . construct a multiscale model of the intestinal crypt which couples dynamic , subcellular models of the cell - cycle and wnt signalling pathway with a cell - scale mechanical model based on meineke et al.s .their model simulations reveal that relating cell proliferation explicitly to the cells biochemical environment or ` niche ' more naturally explains the experimental data than the idea that cells are intrinsically ` stem ' or ` non - stem ' cells independent of their environment . building on the van leeuwen et al .model , mirams et al . studied how monoclonal conversion in the colonic crypt ( i.e. when all cells in a crypt have emerged from the same stem cell ) is affected by mutations causing changes in the proliferative and adhesive properties of cells , as well as the spatial location at which these mutations occur .a particularly robust finding of theirs was that mutations must occur in the bottom few locations of the crypt in order to have any chance of their descendants taking over the crypt .thus a better understanding of spatial dynamics can lead to a better understanding of the transitions from healthy to diseased states .  a notable early application of the subcellular element method was to individual cell rheology ; here the authors found evidence that they need to incorporate active cytoskeletal rearrangement to account for long - time / large - strain behaviour .the incorporation of active processes was subsequently investigated by the same authors in both a cellular and multicellular context .they give favourable comparisons to experiments of a range of different types - cell stretching under large - strain conditions , tissue fluidity in embryos and streaming patterns in collective cell motion ( see also ) .other applications of this method to epithelia include those by nie and co - workers , who applied the subcellular element method , giving particular attention to its implementation on graphics processing units ( gpus ) , to investigate the epidermal epithelium , finding for example that a polarized distribution of adhesion proteins , which inhibits cell detachment , in combination with asymmetric cell divisions provides a robust mechanism for the formation of the stratified structure of skin layers .the subcellular element method is one of the most detailed frameworks available for modelling epithelial dynamics and will likely continue to be applied to a range of systems .biological tissues such as epithelia have been modelled as continua for almost a century .macroscale representations of growth have been of particular interest , stimulated initially by darcy thompson s  simple geometric theories .recent reviews of continuum theories for growth and remodelling , and on continuum and discrete representations of growth , can be found in  and  , respectively .simple continuum models , based on conservation principles and constitutive assumptions ( e.g. darcy s law for cell movement through a fluid ) , have been developed to describe epithelial tissues including the intestinal crypt .such models are reviewed and compared to cell - based models in  , in the context of cancer in the colorectal crypt epithelium .simple convection - diffusion - reaction models have recently been used to study the effects of chemical signalling on crypt formation in the intestine , tissue stratification in epithelia and for formulating an inverse problem for proliferation rates in the crypt .it is often difficult , however , to relate mechanisms included in the continuum models to mesoscopic processes such as those involved in individual cellular rearrangements . in this sectionwe primarily study epithelial relaxation and rearrangement by taking continuum limits of some of the ibms discussed above , before returning briefly to recent developments in continuum modelling directly incorporating more mesoscopic information .murray et al . argue that nonlinear diffusion equations may represent a conceptual continuum framework for comparing different discrete models of epithelial cell movement , and also for ` inverting ' discrete to continuum derivations in order to formulate discrete models that are compatible with a particular nonlinear diffusion equation .in particular , they note that continuum limits of ibm models frequently result in equations of the form , here in one - dimension ,    where is the cellular density at position and time and ) is a nonlinear diffusion coefficient . herewe have neglected growth , and just consider the relaxation of tissues towards equilibrium ; we return to this issue in the discussion .a benefit of deriving such a coarse - grained description is the availability of mathematical analysis tools for the study of such models . before we discuss this further , we consider some examples .murray et al . consider a discrete model of one - dimensional cell motion , in the overdamped limit for which viscous forces are dominant and similar in form to the cell - centre model in . by converting the discrete cell index into a continuous index and assuming that cell numbers are large, they obtain an equation of the form ( [ eq : diff ] ) , with    where , as above , represents the cell density , is the spring constant associated with cell - cell interactions and is a viscous drag coefficient characterising resistance to cell motion .murray et al . demonstrate good agreement between simulations of the discrete and continuum models and note that models of this sort have appeared in the context of ` fast ' diffusion phenomena .using an alternative approach , lushnikov et al . consider continuum limits for a ( lattice - based ) cellular potts model of collective cell movement .as discussed above , interactions are incorporated via an energetic ( hamiltonian ) approach ; for example , cell - cell adherence , interaction with the extracellular matrix and an excluded volume constraint ( ensuring cells do not overlap ) are all included in the ` effective energy ' .they take the surrounding medium to be viscous and neglect inertial effects .they first derive a chemical master equation before taking a continuum limit . despite their different approach and model type ,this leads to a nonlinear diffusion equation of the same form as ( [ eq : diff ] ) , this time with where is the cell density , is a constant and is the equilibrium cell density , equal to one over the local cell rest length ( see also ) .  as noted above ,murray et al . explain how different discrete models can be compared by analysing the functional form of the nonlinear diffusion coefficient that is generated when a continuum limit of the cell - based equations is taken . as further support for this approach , they found that they could derive an equivalent newtonian force law at the cell level which recovers the same continuum description of the cpm as lushknikov et al .this illustrates how information at the continuum level can be used to infer information about possible behaviour at the cell scale and to compare alternative cell - based descriptions .furthermore continuum models are generally much easier to analyse than cell - based models ; they can be used either independently or in combination with cell - level simulations .for example , murray et al . found , by adding a cell proliferation term to equation ( [ eq : diff ] ) with diffusion coefficient ( [ eq : murray ] ) , that a bifurcation in model behaviour occurs - if the diffusion is too slow relative to cell division rate , the system does not reach a steady state but instead ` blows up ' . since this is biologically unrealistic under normal conditions , this aids in understanding the important balances of different biological processes , as well as providing guidance for parameterisation of the ibm model .there are several directions that require further work , however .it is not clear , in particular , how growth , topological changes , subcellular signalling and energy transduction should be rigorously incorporated .extensions of these derivations to higher dimensions also remain to be established . while fozard et al . also investigate continuum limits of a discrete mechanical model of cell motion , systematically considering both slowly varying and heterogeneous cell parameter limits , they also limit their analysis to one dimension . in addition , the approaches of murray et al . and fozard et al . are based on limits of deterministic models ; it is likely valuable to retain more explicit considerations of stochastic effects , as in lushknikov et al.s approach .middleton et al . , building on the work of newman and grima , present a similar approach for a stochastically - forced point - mass cell - centre model ( i.e. a langevin equation model ) of cell migration in the presence of adhesive effects , taking careful account of the correlations induced by strong mechanical interactions between cells . in principlethis approach is applicable in three - dimensions but only one - dimensional comparisons are carried out .contrasting with the above ` bottom - up ' approach , we briefly mention the more ` top - down ' generalised continuum theories , which may provide one alternative or , more appropriately , a complement to ` bottom - up ' modelling . the sophistication of continuum models has also increased significantly in parallel with discrete models , with the introduction of additional mesoscopic state variables , multiple phases and extensions of continuum kinematics .these generalised continuum theories are an important approach to representing phenomena such as growth , rearrangements , plastic deformations and other dissipative phenomena , as well as coupling to processes such as chemical signalling and mechanotransduction . herewe can only briefly review this large topic and refer the reader to the books for excellent introductions to such generalised continuum theories ( here ultilising subtle concepts such as ` internal state variables ' and ` configurational forces ' ) , and the articles reviewing , among other topics , multiphase theories , continuum thermodynamics and the so - called multiplicative decomposition of the continuum deformation gradient in the context of growth and remodelling of biological materials .this approach can help bypass the technical difficulties of homogenisation of underlying discrete models ; however it is probably best combined with partial homogenisation , statistical mechanics and simulation targeted at modelling particular state variables such as the ` growth tensor ' often used in continuum biomechanics .this filling in of mesoscopic details , providing appropriate interpretations and determining the validity of a continuous description is the central difficulty involved in applying these generalised continuum models .an interesting practical example of tackling this challenge is ( see also ) where the kinematic tools of continuum field theory were used , along with statistical fitting methods , to quantify deformation rates at both the tissue and mesoscopic cellular level ( cell shape changes , sliding and rearrangements ) in epithelial tissues to assess their impact on development .the rearrangement / intercalation deformation tensor they define provides an example of a mesoscopic ` microstructural ' variable that fits well with the approach of generalised continuum theories .similarly , bonnet et al . investigated the applicability of continuum theory to epithelia through the use of laser ablation experiments , live imaging and statistical fitting methods , finding that the continuum description was well justified ( see also their related earlier work on connecting statistical processing of foam structures to continuum theories ) .as detailed in this review , epithelial tissue provides an important experimental model system for investigating a broad range of intertwined biological phenomena , including cell lineage dynamics , tissue repair , tissue homeostasis , tissue development and tumorigenesis .many features of these processes are still not fully understood , however , and in particular we face three key challenges , raised in the introduction : i ) choosing which of the diverse biological processes , spanning multiple spatial and temporal scales , to include in our mathematical models ( _ scale and resolution of models _ ) ; ii ) how to best compare models of different types ( _ model - model comparison _ ) and iii ) how to best connect our models to data ( _ model - data comparison _ ) .we first summarise the features of the various modelling approaches considered in this review , and then discuss some possible approaches for tackling the above challenges more systematically .compartmental models provide a robust and high - level approach suitable for investigating feedbacks and signalling , while having limitations on the resolution of spatial and mechanical factors .individual - based models allow for higher - resolution descriptions of cellular motion and mechanics .as such they offer many new insights but can be difficult to analyse and are still subject at present to computational limitations for large simulations .continuum models provide an approach with a spatial and mechanistic resolution lying between broad - brush compartment models and detailed discrete models , and standard formulations typically lack the detail to permit comparisons with important cellular - level data .while each approach has its strengths , and in many cases each can be used independently of the other , there is much to be gained from methods allowing the comparison of different models .continuum limits of discrete models are one approach to tackling this ; alternatively , modern continuum models tend to include much more ` microstructural ' information than they previously did .this apparent ` meeting in the middle ' offers many opportunities to incorporate and combine the strengths of different approaches , but requires practitioners to first be aware of the range of alternatives - one aim of this review is to help further this goal .addressing questions about how much detail to include , which model class(es ) to use and how to relate models to each other and to real biological phenomena remains an art rather than science .these questions have typically been tackled in an _ ad - hoc _ manner in mathematical biology ; on the other hand , advances in computation , formalisation of inference methods and new imaging methods should enable some aspects of these questions to be addressed more systematically .in particular , in statistics , the rise in computational power has seen a resurgence of interest in bayesian inference methods , typically through markov chain monte carlo sampling methods ( see for example the account in ) .one of the biggest strengths of the bayesian approach is that it provides a formalised inference framework for combining models with disparate data sources and other information and for characterising the uncertainty and/or ` underdetermination ' in parameters .this provides one possible inference - based perspective for assessing models and relating them to data .the key here is to interpret probabilistically _ both _observable data and model parameters ( this latter part being particular to the bayesian inference approach ) , which allows / forces one to define a ` full probability model ' or joint distribution for observable data and model parameters . by conditioning on a sample of observed data, one can update _ prior _ estimates of model parameters ( expressed in terms of probability distributions ) and as - yet unobserved data to obtain a new ( _ posterior _ ) predictive model .one appealing aspect of this approach is that this posterior can then be used either as a prior to be updated on the basis of further data or as a predictive model , which can also be subject to model checking / tests .there are of course a number of mathematical and modelling complexities , especially when dealing with the high or even infinite - dimensional nature of parameter estimation and uncertainty quantification for truly spatio - temporal models .this represents an exciting interface between applied mathematics and statistics ; we refer the reader to for more mathematical details in the context of pde models .we illustrate the general bayesian approach in figure [ fig : bayes ] .    the recent work of oden et al . illustrates this approach in the context of tumour modelling .they combine a modelling framework of appropriate ( spatio - temporal ) resolution - the generalised phenomenological framework of continuum thermodynamics of mixtures - with the formalised methods of bayesian inference and uncertainty quantification as outlined above .their approach is still rather general and we refer the reader to this work for more information .of particular note , however , is that even a deterministic continuum model such as that developed in is now interpreted stochastically , here as a set of stochastic partial differential equations . with this in mind , we speculate that this ` inference - centric ' approach may further bridge the gap between model types , with both initially discrete or continuous and deterministic or stochastic models ultimately having common interpretations in terms of probabilistic models linking both observable data and model parameters . by translating models into this common languageone could , in principle , facilitate model comparisons and derivations of one type from another ( both ` bottom - up ' and ` top - down ' ) .mathematical biology is an area ripe for application of methods such as these - not just to the newest or most complicated models , though these are certainly of major interest , but also to return to previous models or theories that often far outstripped available data .this will should enable us to systematically test , validate and improve on these models , starting from good empirical information .in general , advances in imaging , computation and statistical methods have brought models and data closer together and this means that we can develop better ways of testing , comparing and choosing between different modelling approaches .methods are needed for organising and summarising , in an understandable way , different models , interesting data features and complex couplings of biological processes . in this discussionwe have speculated that treating specific problems of mathematical biology as problems of inference provides a promising guide to choosing and comparing appropriate models , characterising uncertainty and even deriving one model type from another .this perspective is but one promising approach to be further developed and employed .furthermore , as we have noted with earlier models , modeller creativity will often outstrip available data ; in the right balance this is not necessarily a bad thing . while we feel that some of the most exciting challenges presently involve directly relating models and data, novel models will likely continue to provide conceptual foundations extending even further into the future .o.j.m . is funded by the biotechnology and biological sciences research council through grant bb / k017578/1 .is funded by the engineering and physical sciences research council and microsoft research , cambridge through grant ep / i017909/1 .10 d. ambrosi , g.a .ateshian , e.m .arruda , s.c .cowin , j. dumais , a. goriely , g.a .holzapfel , j.d .humphrey , r. kemkemer , e. kuhl , j.e .olberding , l.a .taber , and k. garikipati .perspectives on biological growth and remodeling . , 59(4):863883 , 2011 .  a .-baker , b. cereser , s. melton , a.g .fletcher , m. rodriguez - 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Abstract: the majority of solid tumours arise in epithelia and therefore much research effort has gone into investigating the growth , renewal and regulation of these tissues . here we review different mathematical and computational approaches that have been used to model epithelia . we compare different models and describe future challenges that need to be overcome in order to fully exploit new data which present , for the first time , the real possibility for detailed model validation and comparison .   oliver j. maclaren  helen m. byrne  alexander g. fletcher and philip k. maini