Mhorag Goff
email: m.goff@edu.salford.ac.uk

Electronic Patient Records and Benefits to Clinicians: an Actor-Network Study of a Technological Innovation in the NHS

Keywords:

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Introduction

The National Programme for IT (NPfIT) was launched in 2002, as part of which a centrally defined programme of EPR development was set out for both primary and secondary care organisations across the NHS in England. Following the devolution of healthcare within the UK in 1999, England, Scotland, Wales and Northern Ireland have each taken a distinct approach to EPR development [START_REF] Greenhalgh | Introducing a nationally shared electronic patient record: Case study comparison of Scotland, England, Wales and Northern Ireland[END_REF]. Electronic patient records (EPRs) are fundamental to visions of future health care delivery in the UK [START_REF] Wanless | Securing Good Health for the Whole Population: final report[END_REF]3]. They are expected to provide safer, quicker and better quality patient care as a key element of modernised information systems within the NHS [START_REF] Burns | Information for Health: An Information Strategy for the Modern NHS 1998-2005[END_REF][START_REF]Building the information core: implementing the NHS Plan[END_REF]. Discourses from government and IT suppliers about the promise of this technology hinge on notions of modernisation and 'seamless' care driven by pursuit of clinical and management information to evidence performance, reduce errors and achieve efficiencies [START_REF]The power of information: Putting all of us in control of the health and care information we need[END_REF]. This is to be realised through an 'information revolution', which involves improving the flow of information within the NHS [START_REF] Takian | NHS information: Revolution or evolution?[END_REF].

The rationales for EPR development imply that in addition to its properties as a digital object its value derives from its networked-ness in terms of allowing wider sharing of patient information within and between organisations locally, regionally and nationally. The notion of improved information flows is based on the idea of abstracting clinical information captured at the point of care from the local context of production [START_REF] Berg | The contextual nature of medical information[END_REF]. EPRs are therefore predicated on the 'networkedness' or scope of sharing of records and the patient information within them, apparent in the UK government's ambition to develop a nationally networked longitudinal electronic record. 'Networkedness' in this respect refers to the extent to which the EPR technology is being used to exchange information about patients and their care between NHS Trusts, which represent semi-autonomous organisational units under the direction of the government Department of Health.

In this respect the National Programme for IT embodies infrastructural ambitions for EPR systems, which can be seen as enablers of a range of benefits and transformative improvements to the ways in which healthcare is delivered including, for example, virtual consultations [START_REF] Nicolini | The work to make telemedicine work: A social and articulative view[END_REF].

A key driver for the large scale development of electronic patient records is the opportunity for secondary uses of patient data to support care commissioning, administration and research [START_REF]Building the information core: implementing the NHS Plan[END_REF], [START_REF] Wanless | Securing Good Health for the Whole Population: final report[END_REF], [START_REF] Thorp | Secondary Uses Service Strategic Direction[END_REF] at Trust and national levels, including for example, analyses of patient 'casemix', which categorises patients on the basis of consumption of resources, and comparisons of patient outcomes between Trusts [START_REF]Report of the Care Record Development Board Working Group on the Secondary Uses of Patient Information[END_REF]. This relies on the development of standardised data formats and recording practices and the ability to (easily) extract and aggregate data from local EPRs for central use. However, the literature highlights the risks of prioritising a secondary uses agenda which places a burden of information provision on clinicians to capture patient data in the EPR when it adds little value to clinical work practices [START_REF] Berg | The contextual nature of medical information[END_REF][START_REF] Greenhalgh | Tensions and Paradoxes in Electronic Patient Record Research: A Systematic Literature Review Using the Meta-narrative Method[END_REF].

The National Programme for IT has run over-budget, costing to date an estimated £9.8 billion [START_REF]House of Commons Committee of Public Accounts: The dismantled National Programme for IT in the NHS[END_REF]. It has been criticised for being some years behind schedule having been due to be completed in 2010 [START_REF]The National Programme for IT in the NHS: an update on the delivery of detailed care records systems[END_REF], against a backdrop of mixed success of EPR programmes globally [START_REF] Payton | Health Care IT: Process, People, Patients and Interdisciplinary Considerations[END_REF], and a perception among some clinicians that EPRs are an administrative rather than a clinical tool [START_REF] Jones | Computers can land people on Mars, why can't they get them to work in a hospital[END_REF].

The objectives of the National Programme for IT in relation to EPRs were only partially achieved before the NPfIT was dismantled in late 2011 [START_REF] Takian | NHS information: Revolution or evolution?[END_REF], [START_REF]The National Programme for IT in the NHS: an update on the delivery of detailed care records systems[END_REF], notably in terms of the implementation of local EPR systems in hospitals, and the rollout of a nationally accessible Summary Care Record. The failure of the NPfIT to meet its original objectives, the significant costs and the scale of the ambitions with respect to EPR development justify questioning this technology. In particular it is recognised that a national EPR presents ethical issues, especially in relation to the uses of patient data [START_REF] Anderson | Do summary care records have the potential to do more harm than good[END_REF], manifested in the ongoing controversy about the privacy of patient records. Therefore the potentially profound implications for healthcare professionals as users, for patients and for the nature of healthcare make it important to critically discuss this technology.

This paper focuses on exploring clinical users' experiences of EPRs, and the benefits and disadvantages to their work practices. In relation to their position at the forefront of healthcare provision, I start from the assumption that technologies such as the EPR are intended to support the work practices of healthcare professionals and that this in turn will have ramifications for healthcare provision. I therefore aim to under-stand users' actor-networks for the EPR, as a means to discuss how the technology benefits them. Cresswell et al. [START_REF] Cresswell | Actor-Network Theory and its role in understanding the implementation of information technology developments in healthcare[END_REF] note that using Actor-Network Theory means being selective about what to include within scope and what to exclude, because of the impossibility of mapping the entire network. For this reason I explicitly include those involved with healthcare provision under the auspices of the NHS (whether within a Trust or in related bodies), while stakeholders such as patients, software providers and pharmaceutical companies, although they have an interest in EPRs, are out of scope.

In the context of a government vision of a nationally networked record the study sets out to find out what this means for the work practices of clinicians by investigating the reality of the technology as experienced in use. For this reason I have taken a meso level approach to investigating this technology using Actor-Network Theory, not within an individual organisational context, but in relation to multiple NHS organisations which will, if government strategy is realised, entail significantly interconnected EPR systems and extensive information flows.

Methodological and Theoretical Background

Data were collected through interviews with EPR users across 10 NHS Trusts in England in a number of roles in primary, secondary and ambulatory care as indicated in Figure 1 below. Participants include a number of non-clinical users, whose perspectives were originally sought as a means to better understand the secondary uses of EPRs, in addition to the use of documentary sources such as government reports.

Through an iterative methodology of identifying actors and refining understanding of the actor-network, it was found that substantial investigation of non-clinical uses would be out of scope due to the explosion of complexity in the actor-networks, a challenge acknowledged in the literature [START_REF] Bonner | Restoring balance: How history tilts the scales against privacy. An Actor-Network Theory investigation[END_REF]. Participant Trusts were selected for diversity of healthcare settings and on the basis of an existing EPR implementation. In addition to participant Trusts having a local EPR system Fig. 1 also shows the instances of inter-organisationally shared EPRs encountered in this study, including the Diabetes Record, the Summary Care Record and the Specialist Hospital EPR for which the scope of information sharing is indicated. Among broadly socio-technical systems perspectives on information systems it has long been recognised that technological artefacts inscribe certain interests, values and ideals in relation to their domains of use, which tend to reflect and reproduce a particular view of organisational reality belonging to the dominant group or groups [START_REF] Bloomfield | Information Technology, Control and Power: the Centralization and Decentralization Debate Revisited[END_REF][START_REF] Doolin | Narratives of Change: Discourse, Technology and Organization[END_REF]. This implies that a system does not necessarily support the work practices of all users [START_REF] Berg | Considerations for sociotechnical design: experiences with an electronic patient record in a clinical context[END_REF][START_REF] Hedström | The Socio-Political Construction of CareSys[END_REF], and that power may be exercised through the use of a technology. This means that technologies cannot be considered neutral tools, rather they are political in nature, entangled with and emerging from a particular set of socio-material arrangements in a given implementation context [START_REF] Law | Notes on the theory of the actor-network: ordering, strategy, and heterogeneity[END_REF].

Actor-Network Theory (ANT) holds that an entity exists as a product of a particular configuration of associations with a constellation of human (or social) and nonhuman (material) entities [START_REF] Law | Actor Network Theory and Material Semiotics[END_REF]. The hybrid nature of entities arises because the social is defined in relation to, and presupposes the material, and vice versa, such that they are inseparable [START_REF] Doolin | Narratives of Change: Discourse, Technology and Organization[END_REF]. Understanding the actor-network for a given phenomenon, such as the EPR, entails identifying the entities and relationships that constitute that actornetwork, and in doing so appreciating the influences of various actors on the outcome. Any entity may have membership of multiple actor-networks [START_REF] Star | Power, technologies and the phenomenology of standards: On being allergic to onions[END_REF] and that this also means a research object may be constituted by multiple actor-networks [START_REF] Law | Object lessons[END_REF]. My focus will be on actor-networks for the EPR as defined by its users in their particular healthcare roles and settings.

The concept of inscription, central to Actor-Network Theory, is the translation of interests into material form [START_REF] Callon | Techno-economic networks and irreversibility[END_REF] through the design, configuration and use of material artefacts. This concept is used to discuss the manifestation of interests within EPR systems in specific local settings, in doing so revealing the ways in which power and politics emerge from the actor-network. This paper therefore presents an Actor-Network Theory analysis of the implications of EPRs for clinicians' work practices through discussing their actor-networks for the technology. Through its agnosticism about scale [START_REF] Latour | On actor-network theory: a few clarifications[END_REF] and its ability to trace connections between entities at micro, meso and macro level [START_REF] Nicolini | Zooming In and Out: Studying Practices by Switching Theoretical Lenses and Trailing Connections[END_REF], ANT is able to support analysis of how this technology is manifested across multiple organisations, and therefore be sensitive to its networked aspects.

Experiences of EPRs

These findings derive from interviews with EPR users and draw out aspects of discussions that relate to benefits and disadvantages of EPR use, or non-use. EPR users interviewed indicate a consensus that electronic patient records are beneficial through being more accessible than paper records in a number of ways. All clinical users agreed that EPRs are a significant improvement on paper records in terms of legibility and because they are not lost or misplaced. They can also be accessed concurrently by multiple individuals rather than being tied up by a single user at a time. The improved accessibility of patient records entails advantages in terms of time and human resources which would previously have been spent tracking down paper case notes, and reduced delays in getting access to record information which could otherwise affect the responsiveness and timeliness of patient care. Furthermore the flexibility and convenience of 'anytime, anywhere' access to electronic records was noted by a number of clinicians, enabling them to access EPRs from different locations on site and potentially off-site.

A second area in which users see EPRs as beneficial is in terms of more and better quality clinical information about patients, with access to the integrated clinical contributions of different professional groups, including doctors, nurses and Allied Healthcare Professionals, where their records would previously have been relatively inaccessible. Easier access to patient information also supports information sharing with healthcare professionals in other organisations as clinicians can respond more quickly and accurately to enquiries.

In this respect, a specialist hospital, which by definition treats patients usually treated by other healthcare providers, allows access to their EPR for local GPs and hospitals via web portal. Nevertheless access was found to be little used in practice, challenging the idea that EPRs might fulfil a knowledge management function.

Challenges are experienced in terms of the continued need for paper documents to bridge between Trusts. This reflects the distributed-ness of care pathways for patients, especially those with chronic conditions or multiple co-morbidities, so that Trusts need the means to communicate effectively with each other about their patients. The persistence of paper makes it necessary to scan forms and letters into electronic format, creating demand for additional human resource. Moreover users complain that scanned documents are inconvenient to use because of the time taken to open individual files and the difficulty of finding detail within them, although it was noted that thumbnails and bookmarking functionality help to mitigate this.

The continued reliance on paper also indicates that EPR systems remain largely local and are not integrated between Trusts, as originally envisioned by NPfIT. Certain types of information are shareable via EPR applications between Trusts, with local EPR systems of participant Trusts incorporating functionality allowing requests for tests and treatments, such as blood tests, X Rays and MRI scans, and communication of results to be done electronically through the EPR. The NHS healthcare model requires clinicians to request services from service provider facilities, often in other Trusts; therefore this functionality within EPR systems supports the model and is reported as being heavily used by clinicians.

Clinical specialisms are a significant shaper of EPRs and have a bearing on how well EPR systems fit with and support (or fail to support) work practices. In Critical Care and surgery EPRs are perceived as being of less value than in other specialisms. For chronic conditions such as asthma and diabetes that entail ongoing demand for healthcare services treatment cuts across multiple healthcare organisations, entailing a need for greater collaboration and information sharing. EPRs are seen as essential for effective management of chronic disease because of the need to track and monitor a range of physiological data from tests and readings over time, for example when monitoring of a patient's weight or blood sugar levels. This is impossible without structured records that facilitate filtering and organising of that information, and moreover EPRs provide the ability to manipulate and visualise that data. "There's certain things that are very difficult to do without a computer, chronic disease management is one of them" (Business Change Manager, PCT A).

Diabetes is a specialism where the EPR is perceived as particularly beneficial, and for this reason a participant hospital Trust has developed a diabetes-specific area within their EPR which is shared with and contributed to by the community diabetes team and GPs in the local area. This presents a relatively rare instance of an interorganisational EPR being used to provide 'seamless' care, and while still partial in content, is the closest to the vision of whole record sharing.

The patient empowerment and self-care agendas in the NHS aim to enable patients with chronic and lifestyle-related conditions to take greater responsibility for their own care [START_REF] Darzi | NHS next stage review: interim report[END_REF], not least because such conditions are expensive to treat and potentially detrimental to long-term quality of life. A number of users reported that the ability to graph clinical data, such as blood pressure readings, for individual patients is particularly valuable in supporting effective consultations in relation to behaviour change, and the development of patients' own expertise.

Reduced duplication of recording is cited as an anticipated benefit of EPR systems [START_REF] Greenhalgh | The devil's in the detail: final report of the independent evaluation of the Summary Care Record and HealthSpace programmes[END_REF] in relation to the integration of previously disparate records systems within Trusts. For example, clinical entries made in the EPR can be reused for hospital discharge letters to GPs by cutting and pasting content, thereby reducing rework. However, the efficiency and value of EPRs in terms of capture and sharing of (close to) real time patient data is predicated on the assumption of contemporaneous recording [START_REF] Eason | Local sociotechnical system development in the NHS National Programme for Information Technology[END_REF][START_REF] Sheikh | Implementation and adoption of nationwide electronic health records in secondary care in England: final qualitative results from prospective national evaluation in "early adopter" hospitals[END_REF], and where consultations are not office-based or clinicians are unable to enter data into the EPR at the bedside they rely on recording notes on paper or using a dictaphone for input later.

Materiality

The extent to which EPRs benefit users relates not only to content and scope of information sharing but also to material aspects of EPR use including interweaving of access and use with work practices. These considerations are closely tied to users' roles and healthcare settings, and represent one way in which non-human actors are significant within the actor-network.

Computing hardware presents physical considerations for EPR use in terms of whether there are enough devices and whether they are available in the locations they are needed, and the suitability of hardware, whether desktop or mobile devices, for particular settings and clinical functions. A nurse participant from a Critical Care ward complained of the lack of physical space for the Computers on Wheels, comprising a workstation attached to a trolley, provided for consultants to access EPRs at the bedside, and intended to be moved around the ward during ward rounds. There is also insufficient hardware for staff in other work spaces, such as the nurses' stations, as well as challenges with fitting EPR use around unpredictable work practices characteristic of urgent care settings.

Users are reliant on adequate broadband connection for network access, highlighted as an issue for community services where clinicians go out on home visits. In one Trust mobile broadband had been trialled with community Occupational Therapists, who reported problems with accessing and using EPRs in some geographic areas due to poor network coverage. A specialist consultant who regularly works outside his own Trust noted that network access permissions can be a barrier to EPR use when working away from his 'home' Trust site, highlighting that assumptions on which access is based do not fit all modes of working.

Material access issues highlight the localness of EPRs, in tension with the ambition for inter-organisational sharing which demands inter-connected access and recordingrelated processes. That material aspects of use may be a constraint reflects the largely autonomous nature of NHS Trusts and the need for robust information governance to manage the sharing of patient records in ways which protect patient privacy.

Distribution of Benefits

Benefits and disadvantages are not equally distributed, with hospital consultants acknowledging that junior doctors experience greater difficulties in gaining access to computers to view records, and that they do not have secretarial support to input clinical entries into EPRs on their behalf. In this respect secretaries shield consultants from having to alter their work practices too much and in this way consultants experience the benefits of the EPR without as many of the disadvantages. Access to hardware also operates hierarchically so that if there is a shortage of computers there is an informal expectation that nurses, administrators, Allied Healthcare Professionals and junior doctors should give up a computer they are using for a consultant. Although not necessarily experienced as a burden or disadvantage by secretaries, the use of EPRs in hospital settings is reliant on their work, and it is acknowledged that anticipated efficiency gains from EPRs in terms of clinical activity must be offset against increased demands for secretarial time.

Users also reported that EPR use is associated with new information requirements and work responsibilities, reflected, for example, in an increase in recording compared with paper records. Furthermore in order to have an input into the ongoing development of EPRs users must invest time in keeping up to date with technical developments, and in training and familiarising themselves with the functionalities of the system.

Secondary Uses

An information-driven NHS entails increased demand for greater quantities of and more granular data. This is brought about in part by demand for internal and external accountability for use of resources and quality of care, and EPRs are envisaged as essential in supporting this strategy [START_REF]The Information Centre: Health and Social Care Information Centre Strategic Plan 2012 -2015 and Business Plan[END_REF]. In this respect EPRs inscribe non-clinical purposes and agendas in addition to supporting clinical practice, and these include provision of data quantifying clinical activities and numbers and types of patients treated.

"Now you have to be able to not only do things to patients, you have to prove that you've done it. That's the only way you get paid." (GP, General Practice D).

This generates a requirement for diagnostic and treatment information to be captured in coded and structured form, for example, using Read codes, an alphanumeric coding scheme for representing diagnoses and treatments within electronic patient records, or drop down menus which provide a fixed set of options for documenting aspects of care. This places a demand on clinicians to record information in structured form, which they may not otherwise capture in this way for their own clinical purposes. One consultant expressed frustration that he could not order an ultrasound scan for a patient because the given body part was not available in the options for that field within the EPR. Restricting options for recording not only facilitates standardised clinical recording, it also constrains representation of reality and can therefore be used to present organisational activity in a better light [START_REF] Ramiller | Constructing safety: System designs, system effects, and the play of heterogeneous interests in a behavioral health care setting[END_REF].

The evidence-based medicine agenda is reflected in the use of EPRs as a mechanism to support the standardisation of clinical practice. In one participant hospital Trust, the EPR enables pre-configured sets of medications for anaesthetics and pain management use with certain categories of patient, which are provided in a drop down menu. This encourages the use of the standard prescribing protocols by making it easier to prescribe in this way and also therefore discourages clinicians from selecting more freely from available medications when prescribing. The protocols derive from National Institute of Health and Care Excellence1 guidelines, thereby also making the EPR a tool to promote and legitimise the evidence-based medicine agenda. This use of EPRs to encourage standardisation is linked with clinical audit and associated targets for care quality improvement, which acts as a form of performance management in relation to clinical practice. Recording in the EPR therefore reflects the accountability agenda, including that of medico-legal accountability.

"What is does is back up what you're doing so anything to do you can be held accountable for so if anything went to coroner's you've got proof that you've been doing your job." (Nurse, Hospital B).

For certain conditions explicitly targeted by the Department of Health the EPR is used as a checklist of activities to ensure not only that a minimum standard of care, but that there is recorded evidence to demonstrate that targets have been met. Care quality targets are also tied to payments for Trusts' services and additional financial incentives, for example, through the Quality Outcomes Framework for GPs and Payment by Results for hospitals. EPRs play a critical role in providing the evidence of such activity and of the meeting of targets and other measures of performance against centrally defined criteria.

Discussion

Clinicians' experiences of EPR use are strongly mediated both by their material aspects and by the inscriptions that derive from non-clinical agendas, as outlined above. These inscriptions serve as mechanisms to enforce standards in clinical activity, not only through focussing attention on the achievement of targets and measurable indicators of good practice but also by selectively enabling and constraining particular recording practices which then feed into clinical behaviours. Unsurprisingly users who have an interest in clinical research and managementrelated uses of data are more concerned about the availability of structured data than with narrative-style recording, with structured data more easily aggregated, while for clinical purposes narrative recording, for example in clinical notes, remains important. This highlights the tension between what is clinically useful and support for broader organisational objectives.

In general users are ambivalent but accepting of EPRs as a technology, creating a sense that use is inevitable and that there is no going back to paper records. This lack of criticality is recognised in other studies that investigate the implications of health information systems e.g. [START_REF] Mensink | The role of expectations in system innovation: the Electronic Health Record, immoderate goal or achievable necessity[END_REF][START_REF] Checkland | Ticking Boxes and Changing the Social World: Data Collection and the New UK General Practice Contract[END_REF], and arises, I suggest, because EPR use is entangled with the use of those inscriptions such that non-use is all but impossible and would disadvantage users. Moreover clinicians acknowledge the need to carry out some information activities for the wider benefit of the organisation.

In so far as non-clinical managerial and other agendas are inscribed within EPRs they are impossible to resist and as such power effects are played out through a variety of controls over clinical and non-clinical behaviours, mediated by the EPR. This pervasiveness therefore has implications for the ability of users to challenge or avoid non-clinical agendas. It is argued that the insinuation of an information system into the organisational context restructures users' entire social world, such that opposition becomes lost [START_REF] Lowe | Casemix accounting systems and medical coding: Organisational actors balanced on "leaky black boxes[END_REF]. Their embeddedness within clinical activity means that EPR use tacitly acts to stabilise these inscriptions and legitimise the agendas of which they are a part. For example, their use to support performance measures such as those used in national league tables provides reciprocal legitimacy for both performance management mechanisms and for the EPR.

It is argued that an actor-network reshapes reality such that its existence becomes taken for granted [START_REF] Bowker | How things (actor-net) work: Classification, magic and the ubiquity of standards[END_REF]. In terms of users' limited criticality about EPRs it can also be argued from an Actor-Network perspective that the stabilisation of the EPR in any organisational context necessarily entails a reconfiguration of reality to accommodate the EPR such that it is hard to imagine the world without it. In this respect EPR use has become part and parcel of what clinicians do on a daily basis not only on a clinical level but also in terms of non-clinical activities around performance reporting and accountability. Once established such an entity may be taken for granted, hidden from view and closed to debate and scrutiny. This is particularly the case for infrastructures, which are by definition, intended to become 'part of the furniture' [START_REF] Star | Power, technologies and the phenomenology of standards: On being allergic to onions[END_REF].

Nevertheless, it has been recognised within the Information Systems literature on power and information systems that users are not necessarily passive subjects in this respect, and may use EPRs to 'play the system' to their advantage e.g. [START_REF] Doolin | Casemix Management in a New Zealand Hospital: Rationalisation and Resistance[END_REF], for example to produce more positive self-representations. Within this study users reported using data from EPRs as evidence to support their own performance appraisals and also to exploit the ways in which EPRs focus attention on particular measures to present their organisation in the best light in various 'figures'. One GP noted that performance targets imposed by the UK Department of Health had skewed clinical recording such that it had become a running joke among GPs that the Quality Outcomes Framework has "cured depression overnight", alluding to the observation that GPs are more reluctant to record diagnoses which entail setting in motion a set of clinical activities that perceived as laborious.

Where benefits are experienced from inter-organisational sharing of patient information, this has been partial and related to particular sub-sets of patient information, such as the diabetes element of the EPR, and communication of tests and results.

Conclusion

Using Actor-Network Theory as a methodological approach reveals a range of interests and relationships implicated in the achievement of the EPR as a technological phenomenon, incorporating both clinical and non-clinical agendas. Reporting findings about clinicians' experiences of day to day EPR use grounds the envisioned aim of a nationally networked EPR within local realities. The ideal of a networked EPR as an infrastructure and enabler of wider uses of patient information has only been partially realised. Clinicians make use of certain aspects of inter-organisational sharing via EPRs, however low uptake and use of the national Summary Care Record, and low usage of the specialist hospital's EPR indicates that these instances have as yet failed to translate clinical interests in such a way that they can stabilise. While the nationally networked EPR has not yet been realised, additional risks and responsibilities associated with EPR use are nevertheless related to the overall information strategy of which this is a part. While there are undoubtedly benefits to users, disadvantages also have to be absorbed in terms of additional responsibilities, such as increased recording, capturing information in coded form, and greater scrutiny of clinical practice which relate both to material aspects of use that mediate access within given use settings and inscriptions of non-clinical agendas.

EPRs bring benefits to clinicians in a number of fundamental ways, centred on more reliable access to patient information as compared with paper records, both for the records themselves and in terms of the availability of greater quantities and more sources of information about patients.

However, benefits and disadvantages are not experienced equally by all groups of users, such that EPRs may in some respects reinforce existing hierarchies and professional boundaries. This supports Vikkelso's [START_REF] Vikkelso | Subtle redistribution of work, attention and risks: Electronic patient records and organisational consequences[END_REF] conceptualisation of technologically enabled organisational change in terms of shifts and redistributions rather than simply in terms of improvements. In this way the study reinforces findings from the literature that suggest that EPRs may be less transformative than expected because users try to minimise the impact on their work practices [START_REF] Latour | On actor-network theory: a few clarifications[END_REF]. This also challenges the assumption that EPRs and health information systems will straightforwardly empower healthcare professionals by liberating them from certain activities and through increased and more democratic access to patient information, as acknowledged in the literature e.g. [START_REF] Vikkelso | Subtle redistribution of work, attention and risks: Electronic patient records and organisational consequences[END_REF].

This study, by taking a meso level approach, offers an analysis of the EPR across a number of local instances, recognising it as a multiple research object. The research contributes to the literature on this topic by analysing benefits to clinical practice in relation to EPRs as local applications and as national infrastructure. It suggests that to date benefits have largely been experienced in relation to local organisational EPRs, restricted in networkedness to individual Trusts, challenging expectations behind the original objective of infrastructure-like nationally networked records. The benefits of the various types of EPR and the roles they play in supporting clinician's work practices will have implications for the successful achievement of EPRs as technological innovations, and will influence the feasibility of achieving the government's longterm vision for the information revolution in healthcare.

Fig 1 .

 1 Fig 1. Participant Trusts and EPR users

http://www.nice.org.uk/


Chiara Giverso

Pasquale Ciarletta

On the morphological stability of multicellular tumour spheroids growing in porous media

Keywords: 87, 18, Hf Spatiotemporal pattern formation in cellular populations -87, 18, Gh Cell-cell communication; collective behavior of motile cells -46, 32, +x Static buckling and instability

Multicellular tumour spheroids (MCTSs) are extensively used as in-vitro system models for investigating the avascular growth phase of solid tumours. In this work, we propose a continuous growth model of heterogeneous MCTSs within a porous material, taking into account a diffusing nutrient from the surrounding material directing both the proliferation rate and the mobility of tumour cells. At the time scale of interest, the MCTS behaves as an incompressible viscous fluid expanding inside a porous medium. The cell motion and proliferation rate are modelled using a non-convective chemotactic mass flux, driving the cell expansion in the direction of the external nutrients' source. At the early stages, the growth dynamics is derived by solving the quasi-stationary problem, obtaining an initial exponential growth followed by an almost linear regime, in accordance with experimental observations. We also perform a linear-stability analysis of the quasi-static solution in order to investigate the morphological stability of the radially symmetric growth pattern. We show that mechano-biological cues, as well as geometric effects related to the size of the MCTS subdomains with respect to the diffusion length of the nutrient, can drive a morphological transition to fingered structures, thus triggering the formation of complex shapes that might promote tumour invasiveness. The results also point out the formation of a retrograde flow in the MCTS close to the regions where protrusions form, that could describe the initial dynamics of metastasis detachment from the in-vivo tumour mass. In conclusion, the results of the proposed model demonstrate that the integration of mathematical tools in biological research could be crucial for better understanding the tumour's ability to invade its host environment.

sorbs vital nutrients via diffusion from the external envi-13 ronment 1,3,5 . Thus, diffusion may become suddenly inef-14 fective in the center of the tumour mass, forming a charac-15 teristic necrotic core (see Fig. 1-a). At later stages, a solid 16 tumour is characterized by the occurrence of angiogenesis a e-mail: pasquale.ciarletta@polimi.it (i.e. the process by which the tumour induces new blood vessels formation from the nearby existing vasculature), thus switching to a vascular growth phase [START_REF] Folkman | Tumour angiogenesis[END_REF][START_REF] Muthukarruppan | Tumour-induced neovascularisation in the mouse 624 eye[END_REF] .

The analysis of the avascular growth phase in tumours has attracted a lot of interest in the mathematical and physical research communities, and a large number of in silico mathematical models has been proposed [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Adam | A simplified mathematical model of 627 tumor growth[END_REF][START_REF] Adam | A mathematical model of tumour 630 growth. ii. effects of geometry and spatial unifor-631 mity on stability[END_REF][START_REF] Adam | Diffusion regu-634 lated growth characteristics of a spherical prevascular 635 carcinoma[END_REF][START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Chaplain | Experimental and Theoretical Advances in Biological Pattern Formation, ch. The development of a spatial pattern in a model for cancer growth[END_REF][START_REF] Greenspan | Models for the growth of a solid tumour by diffusion[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Maggelakis | Mathematical model of prevascular growth of a spherical carcinoma[END_REF] . Thanks to the controllability and the reproducibility of the experimental setting, MCTS has become a widely used system model for the development of theoretical models.

The classical approach of deterministic tumour model comprised an ordinary differential equation (ODE), derived from either mass conservation or population dynamics, coupled with at least one reaction-diffusion equation, representing the spatio-temporal distribution of vital nutrients or chemical signals inside the tumour [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Adam | A mathematical model of tumour 630 growth. ii. effects of geometry and spatial unifor-631 mity on stability[END_REF][START_REF] Adam | Diffusion regu-634 lated growth characteristics of a spherical prevascular 635 carcinoma[END_REF][START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Greenspan | Models for the growth of a solid tumour by diffusion[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Mcelwain | Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth[END_REF] . Only recently, many authors have extended such models including the pivotal role of mechanics in tumour growth. In most cases, fluid-like constitutive equations have been used to model the tumour mass [START_REF] Ambrosi | On the closure of mass balance models for tumor growth[END_REF][START_REF] Araujo | A history of 668 the study of solid tumour growth: The contribution 669 of mathematical modeling[END_REF][START_REF] Lowengrub | Non-673 linear modelling of cancer: Bridging the gap between 674 cells and tumours[END_REF][START_REF] Mcelwain | Cell migration in mul-677 ticell spheroids: Swimming against the tide[END_REF][START_REF] Chen | The influence of 681 growth-induced stress from the surrounding medium 682 on the development of multicell spheroids[END_REF][START_REF] Landman | Tumour dynamics 685 and necrosis: surface tension and stability[END_REF][START_REF] Steinberg | Reconstruction of tissues by dissoci-689 ated cells. some morphogenetic tissue movements and 690 the sorting out of embryonic cells may have a common 691 explanation[END_REF][START_REF] Foty | Liquid properties of embryonic tissues: Mea-695 surement of interfacial tensions[END_REF][26] . This choice is obvi-ously only an approximation of the by far more complex behaviour of cellular aggregates, that also display solidlike properties related to the adhesive characteristics of cells 27 and to the mechanical properties of the single cell in the cluster. Thus, in some limiting cases, cell aggregates are better described as solids with linear or eventually nonlinear elasticity, in which compressive and shear loads are balanced by the solid stress in the body, depending on the strain of its material points [28][29][30][31][32][33] . A solid-like constitutive equation has been advocated for its suitability of accounting for both residual stresses 29,32,34 and the plastic behaviour of cellular aggregates [START_REF] Ambrosi | Cell adhesion mecha-738 nisms and stress relaxation in the mechanics of tu-739 mours[END_REF][START_REF] Ambrosi | The inter-742 play between stress and growth in solid tumors[END_REF][START_REF] Giverso | Growing avas-747 cular tumours as elasto-plastic bodies by thetheory of 748 evolving natural configurations[END_REF] . Even though these considerations support the idea that a cellular aggregate can behave as a solid at some extent, experimental evidences 26,[START_REF] Aigouy | Cell flow reori-752 ents the axis of planar polarity in the wing epithelium 753 of drosophila[END_REF][START_REF] Vasilica Stirbat | Multicellular aggregates: a model system for tissue rheology[END_REF] have shown that aggregates behave as elastic solids on short timescales (of the order of a few minutes) but display a fluid-like behaviour at longer times. Furthermore, it was shown that cellular aggregates behave as an elastic solids at time scales short compared to that of cell division and apoptosis, and as a fluid (with the traceless stress that relaxes to zero) for long times [START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF] . Thus the description of MCTSs as a liquid is widespread.

Even though the existing mathematical models on both solid tumours and MCTSs successfully reproduce the experimentally observed growth dynamics [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Adam | A mathematical model of tumour 630 growth. ii. effects of geometry and spatial unifor-631 mity on stability[END_REF][START_REF] Adam | Diffusion regu-634 lated growth characteristics of a spherical prevascular 635 carcinoma[END_REF][START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Greenspan | Models for the growth of a solid tumour by diffusion[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Mcelwain | Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth[END_REF][START_REF] Casciari | Mathematical modelling of microenvironment and growth in emt6/r0 multicellular tumour spheroids[END_REF][START_REF] Marusic | Analysis of growth of multicellular tumour spheroids by mathematical models[END_REF] , they poorly consider the mechanical and chemical interaction with the surrounding environment. Furthermore, most approaches assume that the initial spherical symmetry is preserved during the growth of the aggregate 28-30 , whilst only in few cases [START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF] the development of tumour irregular contours has been taken into account. Indeed, it is known that some solid tumours, e.g. carcinomas, grow almost spherically only in the first stages of their progression 1,3,5 , while they might show a less defined and even asymmetric outer boundary 43 (see Fig. 1-b). Since higher irregular contours usually indicate aggressive tumours, the capability to undergo a morphological transition might promote tumour infiltration and invasion within the surrounding tissue [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Cross | The fractal dimension may be a useful morphometric discriminant in histopathology[END_REF][START_REF] Cross | Fractal geometric analysis of colorectal polyps[END_REF][START_REF] Landini | How important is tumour shape?[END_REF] . Thus, it has been proposed that some measure of the irregularity of a tumour boundary (e.g. its fractal index measured via particular medical imaging techniques such as computerized tomography scans), may provide clinicians with useful information for its prognosis and treatment [START_REF] Cross | The fractal dimension may be a useful morphometric discriminant in histopathology[END_REF][START_REF] Cross | Fractal geometric analysis of colorectal polyps[END_REF][START_REF] Landini | How important is tumour shape?[END_REF] , being potentially useful in predicting the efficacy of drug treatment or chemotherapy [START_REF] Sutherland | Radiation response 785 of multicell spheroids: An in vitro tumour model[END_REF][START_REF] Tubiana | The kinetics of tumour cell proliferation 789 and radiotherapy[END_REF] .

In this work we go beyond the state-of-the art in the field [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Greenspan | On the growth and stability of cell 792 cultures and solid tumours[END_REF][START_REF] Byrne | Modelling the role of 795 cell-cell adhesion in the growth and development of 796 carcinomas[END_REF] by proposing a mathematical model that accounts for the presence of a surrounding porous media with a finite thickness. Thus, nutrient diffusion from the external environment creates a chemical gradient that directs both the proliferation rate and the motility of the tumour cells. MCTS is modelled as a viscous fluid with adhesive interactions at the border, expanding inside a porous material. This work is organized as follows. First, we introduce in Section 2 the mathematical model describing the expansion of an initially spherical tumour. In Section 3, we derive the radially-symmetric solution of the quasi-stationary problem. Then, we perform a linear stability on the quasi-97 static tumour growth. Finally, in Section 4, we discuss the 98 modelling results with respect to the key chemo-mechanical 99 and geometric parameters that govern the mathemati-100 cal problems, highlighting the key mechano-biology effects 101 that promote a morphological transition during tumour 102 invasion. The MCTS is modelled as a three dimensional contin-105 uum growing inside a rigid porous structure, representing 106 the surrounding environment, usually extracellular matrix 107 (ECM) or matrigel. In this respect, the proposed model 108 refers to the in vitro case in which MCTS grows inside 109 a three dimensional either natural medium (e.g. agarose 110 gel, hyaluronic acid gel) or synthetic matrices scaffolds 111 (e.g. polylactide and polyglycolide biodegradable struc-112 tures mimicking a tissue-like environment) [START_REF] Nyga | 3D tumour 804 models: novel in vitro approaches to cancer studies[END_REF] .

113

The outer boundary of the tumour is considered as a 114 freely moving material interface separating the tumour 115 cells from the surrounding medium.

In particular, we account for the presence of a central region of necrotic cells, surrounded by a layer of quiescent and proliferating cells. Thus, the whole domain Ω is divided in different regions, depending on the residing cellular population (see Fig. 2):

the necrotic cells are located in the central core of the spheroid, in a region called Ω N (t), with

Ω N (t) = {(r, θ) : r < R N (t), 0 < ϕ ≤ π, 0 < θ ≤ 2π} ,
where R N is the radius of the necrotic core, that might evolve in time;

the proliferative and quiescent tumour cells are located in the region

Ω T (t) = {(r, θ) : R N (t) < r < R T (t), 0 < ϕ ≤ π, 0 < θ ≤ 2π} ,
where R T is the radius of the spheroid, whose evolution in time represents the growth of the MCTS;

the healthy space, composed by either the in vitro scaffold or the extracellular matrix, the extracellular liquid and possibly healthy cells (in vivo),

Ω H (t) = {(r, θ) : R T (t) < r < R out , 0 < ϕ ≤ π, 0 < θ ≤ 2π} ,
being R out the outer boundary of the whole domain.

The boundary between the necrotic core and the proliferative region is called ∂Ω N (t), whereas the moving interface between the tumour region and the healthy space is denoted with ∂Ω T (t). In the following we will consider that the interior boundary between the necrotic core and the quiescent-proliferative region does not evolve in time, since we are interested only in the evolution of the MCTS boundary, which is related to tumour infiltration inside the healthy region. Furthermore, we assume that the porous material is homogeneously distributed in the whole region Ω = Ω N ∪ Ω T (t) ∪ Ω H (t) and it is neither produced/degraded (i.e. behaves as inert matter), nor deformed (i.e. structurally rigid) by the moving tumour cells.

We will consider a single nutrient species (e.g. oxygen)

with volume concentration n(x, t), diffusing from the fixed outer boundary ∂Ω through the porous material. Thus, we assume that the vascular network providing the source of nutrients is outside the modelled domain, and can be represented by a boundary term at ∂Ω. The diffusion coefficient is a constant value D n everywhere, but the nutrient is only consumed, with an uptake rate γ n , in the region occupied by the proliferative and quiescent cells. Indeed we consider that the consumption of nutrients in the healthy region is negligible. This is certainly the case of MCTS growing inside artificial/natural scaffolds, but, in a first approximation, it can be used also to model the in vivo condition [START_REF] Lowengrub | Non-673 linear modelling of cancer: Bridging the gap between 674 cells and tumours[END_REF][START_REF] Wise | Three-dimensional multispecies nonlinear tumor 809 growth-I Model and numerical method[END_REF] , since the net consumption of nutrients in the extracellular healthy space is negligible compared to the uptake by tumor cells [START_REF] Ramanathan | Pertur-812 bational profiling of a cell-line model of tumorigenesis 813[END_REF] .

Thus, the 3D homogenized concentration per unit volume of this generic chemical species, indicated with n(x, t), obeys the following reaction-diffusion equation

ṅ(x, t) =      D n ∇ 2 n(x, t) in Ω N , D n ∇ 2 n(x, t) -γ n n(x, t) in Ω T (t) , D n ∇ 2 n(x, t)
in Ω H (t) .

(

) N T H R N R T R out Figure 2. 1
Representation of the domain used for the analytical analysis. At time t = 0, the three domains ΩN , ΩT and ΩH are concentric spherical shells, with radius RN , RT and Rout, respectively. In this work, we consider that only the tumour boundary ∂ΩT evolves in time.

We remark that, in principle, the uptake rate γ n should 159 depend on the tumour cell density, although, in the follow-160 ing, it will be considered homogeneous and constant over 161 time. Even the diffusion coefficient D n can be affected by 162 the cell packing inside the tumor and by the extracellular 163 matrix alignment and distribution. However, coherently 164 with the hypothesis of an inert, rigid and homogeneous 165 extracellular matrix distributed in the whole domain, the 166 diffusion of nutrients can be assumed to be constant [START_REF] Wise | Three-dimensional multispecies nonlinear tumor 809 growth-I Model and numerical method[END_REF]55 . 167

The diffusing nutrient notably not only affects the growth 168 of single individuals in the tumour but also directs cell 169 movements, e.g. through chemotaxis 56,57 . Therefore, we 170 consider a non-convective mass flux term, m, taking into 171 account both tumour proliferation and chemotactic mo-172 tion, differently from the standard volumetric production 173 rate considered in literature [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Chaplain | Experimental and Theoretical Advances in Biological Pattern Formation, ch. The development of a spatial pattern in a model for cancer growth[END_REF][START_REF] Greenspan | Models for the growth of a solid tumour by diffusion[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF] . Accordingly, the mass 174 balance inside Ω T (t) reads

175 dρ dt + ρ∇ • v = ∇ • m in Ω T (t). ( 2
)
where ρ is the tumour cell density, which is approximately 176 the same of water. Since mass transport phenomena in 177 MCTSs are driven by the local concentration of chemicals, 178 the mass flux vector appearing in Eq. ( 2) should depend 179 on nutrient availability. A simple constitutive law for m 180 can be taken in the form of a chemotactic term 56,58 , i.e. 181 m = χρ∇n, where χ is the chemotactic coefficient, here 182 considered constant. Consequently, the mass flux m de-183 scribes the expansion of the tumour due to proliferation 184 and driven by chemotaxis towards higher concentration of 185 nutrients.

186

Assuming that the living aggregate can be macroscopically 187 modelled as a Newtonian fluid, Darcy's law describes its 188 motion inside the inert, porous surrounding medium [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Ambrosi | On the closure of mass balance models for tumor growth[END_REF] . 189 Thus, the cell velocity v is related to the pressure field

p 190 through v = -K p ∇p , (3)
where K p is related to the permeability of the medium, k, and the viscosity of the cellular material, µ, by

K p = k/µ.
Assuming the incompressibility of the cellular spheroid, which is mostly composed by water, we impose dρ/dt = 0 in Eq. ( 2), so that the relation between the pressure p and the nutrient concentration n reads

∇ 2 p = - χ K p ∇ 2 n in Ω T (t) , (4)
which has been obtained substituting the Darcy's law (3) and the constitutive relations for m in the mass balance equation ( 2). In summary, the coupling of Eq. ( 1) with

Eq.( 4), complemented by a proper set of boundary conditions (BCs), describes the macroscopic evolution of the avascular tumour inside the healthy tissue.

In particular, for the pressure we impose the Young-Laplace equation at the moving boundary ∂Ω T (t) and the null velocity of the tumour cells at the fixed boundary ∂Ω N , i.e. For what concerns the chemical species, in absence of an interfacial structure, the continuity for the nutrient concentration and flux can be assumed (both in ∂Ω T (t) and in ∂Ω N ), and the concentration at the outer boundary can be assumed constant (to model the source of nutrients from the external vascular network), so that

p = p 0 -σ b C on ∂Ω T (t) , (5) v ∂Ω N • n N = 0 → (∇p)| ∂Ω N • n N = 0 on ∂Ω N , (6)
n | ∂Ω = n out on ∂Ω , (7) n | ∂Ω T = 0 , ∇n | ∂Ω T • n = 0 on ∂Ω T , (8) n | ∂Ω N = 0 , ∇n | ∂Ω N • n N = 0 on ∂Ω N , ( 9
)
where n is the outward normal vector at the boundary ∂Ω T and (•) | ∂Ωj denotes the jump of the quantity between brackets across the boundary ∂Ω j , with j = N, T .

Finally, the compatibility condition at the free interface imposes

dx ∂Ω T dt • n = v ∂Ω T • n on ∂Ω T . (10)
In the following we will work with dimensionless equations, obtained writing the system of Eqs. ( 1)-( 4) in terms of the dimensionless chemical concentration, n = n/n c , and the dimensionless pressure, p = p/p c and referring to the geometry outlined in Fig. 2. The dimensionless quantities are obtained using the following characteristic time t c , length l c , velocity v c , pressure p c and chemical concentration n c :

t c = γ -1 n , l c = D n γ -1 n , v c = √ D n γ n , p c = D n K -1 p , n c = n out . Finally, the resulting dimensionless systems of equations reads ṅ =      ∇2 n for r < RN ∇2 n -n for RN < r < RT (t) ∇2 n for RT (t) < r < Rout (11a) ∇2 p = -β ∇2 n for RN < r < RT (t) (11b) n | RN = 0, ∇n RN • nN = 0, ( ∇p) • nN = 0 for r = RN (11c) n | RT = 0 , ∇n RT • n = 0 , p = p0 -σ C for r = RT (t) (11d) n( t, Rout ) = 1 for r = Rout (11e) dx RT d t • n = v RT • n = -∇p RT • n for r = RT (t) . (11f)
The nondimensionalization leads to the definition of five 231 dimensionless parameters, classified into two broad cate-232 gories:

233 -β := χn c /D n and σ := σ b K p γ 1/2 n D -3/2 n =σ b K p l -1/2 c D -1 n 234
define mechano-biology effect on the aggregate expan-235 sion, and are called motility parameters;

236

-RN , RT and Rout (i.e. the dimensionless radii of the 237 necrotic core, of the MCTS and the whole domain, 238 respectively) define the geometrical properties of the 239 system with respect to the diffusive length l c , and are 240 denoted as size parameters.

241

In particular, the dimensionless parameter β represents 242 the chemical effects associated to the expansion of MCTSs, 243 since it can be regarded as the ratio between the typical 244 time-scales of mass production over nutrient diffusion. On 245 the other hand, the parameter σ defines the influence of 246 mechanical cues over tumour development, representing 247 the ratio of the surface tension of the aggregate over the 248 characteristic viscous pressure of the fluid ensemble.

249

For sake of simplicity, in the following we will omit the 250 barred notation to denote dimensionless quantities, e.g. 251 R T stands for RT and so on. In this Section, we first derive the quasi-static solution of 255 the proposed model in order to mimic the early avascular 256 growth. We later perform a linear stability analysis to investigate the occurrence of a morphological instability at later growth stages.

Quasi-stationary solution

At early stages of avascular growth MCTSs maintain a spherical shape [START_REF] Folkman | Self-regulation of 600 growth in three dimensions[END_REF][START_REF] Sutherland | Growth and cellular 607 characteristics of multicell spheroids[END_REF][START_REF] Sutherland | Cell and environment interactions in 618 tumor microregions: the multicell spheroid model[END_REF] . Thus, we look for a radially symmetric quasi-stationary solution, assuming that the diffusive process is much faster than the MCTS expansion, so that it is possible to drop the time derivative in Eq. (11a).

This assumption is valid in many biological conditions, since a fast-growing tumour may expand at a rate of up to 0.5 mm/day, whereas a typical diffusion time scale is about 1 min (considering a typical length scale L ≈ 10 -2 cm and a typical diffusion coefficient D ≈ 10 -6 cm 2 s -1 ) 11 .

Thus, it is clear that the diffusion timescale of nutrients is much shorter than the growth timescale, so that the quasi-stationary assumption can be effectively formulated.

Furthermore for such long time scale the MSC can be actually treated as a viscous fluid.

Specializing our analysis to the case of a spherical tumour of radius R T , we will denote with n * = n * (r, t)

the quasi-stationary solution of Eq. 11a and with p * = p * (r, t) the quasi-stationary pressure field satisfying (11b).

Given the boundary conditions (11c)-( 11d)-(11e) and considering that n * and p * should be bounded, the quasistationary fields read

n * =              2R out e RT +RN e 2RT w + T -e 2RN w - T if r ≤ R N - e 2RN e 2(r-RN ) (R N + 1)R out + (R N -1)R out r w - T e r-RT +2RN -r w + T e r+RT if R N < r ≤ R T Rout e 2RT (RN + 1)(r -RT + 1) -e 2RN (RN -1)(r -RT -1) r e 2RT w + T -e 2RN w - T otherwise ( 12
)
p * = p 0 + σ R T + β(n * R T -n * ) , (13)
where we called n * R T = n * (R T ) the concentration of the nutrient at the boundary of the aggregate and we defined

w + T = (R N +1)(R out -R T +1) and w - T = (R N -1)(R out - R T -1)
, being w T = (R out -R T ) the width of the region occupied by the tumour. Then, using Eq. (11f), it is possible to compute the quasi-stationary velocity of the front, which is directed along the radial direction for symmetry considerations, i.e. v * = v * r e r , with

v * r (R T ) = β Rout(e 2R N (RN -1)(RT +1)-(RN +1)e 2R T (RT -1)) R 2 T (e 2R N (RN -1)(w - T )-(RN +1)e 2R T (w + T ))
.( 14)

Equation ( 14) can be integrated numerically to determine the evolution of the spheroid border over time. The result, reported in Fig. 3 time exhibits an early stage of exponential growth, cor-301 responding to spheroid volumetric growth, since nutrients 302 are available everywhere in the spheroid bulk 32,61 . Subse-303 quently, when the diameter of the spheroid becomes much 304 larger than the penetration length of the nutrient, the cel-305 lular growth becomes mainly localized on the surface of 306 the tumor, leading to a linear growth in time. In this paragraph, we investigate the stability of the steady, 309 radially-symmetric solution by applying small perturba-310 tions of the MCTS boundary.

311

Let R * T be the unperturbed position of the moving inter-312 face, we consider a small perturbation (ε 1) of the kind 313

R(θ, ϕ, t) = R * T (t) + εe λt Re [Y m (θ, ϕ)] . (15)
where

(r) + 2rn 1 (r) -( + 1) + (λ + 1 ΩT )r 2 n 1 (r) = 0 , ( 18
)
where primes denote derivatives on r and

1 Ω T = 1 if 327 R N < r ≤ R * T ,
1 Ω T = 0 otherwise. The solution of Eq. 328 (18), for λ = {0, -1} is

n 1 (r) =      C 1 i ( √ λr) if r ≤ R N B 1 i ( √ λ + 1r) + B 2 k ( √ λ + 1r) if R N < r ≤ R * T A 1 i ( √ λ + 1r) + A 2 k ( √ λ + 1r) if R * T < r ≤ R out , (19)
where i (r) and k (r) are the modified spherical Bessel

n 1 | R N = 0 , ∂n 1 ∂r R N = 0 , ( 20
)
n 1 | R * T = 0 , ∂n 1 ∂r R * T = n 0 , n 1 (R out ) = 0 . ( 21
)
The perturbed pressure field p 1 in Ω T is obtained from 336 Eq.(11b) that leads to

337 p 1 (r) = Qr + W r --1 -β B 1 i ( √ λ + 1r) + B 2 k ( √ λ + 1r) , (22)
where the constants Q and W can be determined from the it is possible to obtain the following dispersion equation

345 λ = -p * (R * T ) -p 1 (R * T ) , (24)
which has the same form of the relation found for the 346 rectilinear front on an infinite domain [START_REF] Ciarletta | Free boundary morphogenesis in liv-843 ing matter[END_REF] or an expand-347 ing circular colony [START_REF] Giverso | Branching 846 instability in expanding bacterial colonies[END_REF][START_REF] Giverso | Emerg-849 ing morphologies in round bacterial colonies: compar-850 ing volumetric versus chemotactic expansion[END_REF] . The dispersion equation ( 24) is The dispersion equation ( 24) has been solved numerically 357 in order to investigate the global stability of the solutions 358 depending on the system parameters. The corresponding 359 dispersion diagrams are reported in Fig. 4 for different 360 values of both the size and the motility parameters. As 361 in classical perturbation theory 62 , a positive real part of the growth rate λ implies global instability, thus highlight a critical spatial mode of the perturbation defined by the degree associated with the highest positive growth rate. Interestingly, Fig. 4 shows that the spheroid front is linearly unstable at small , with = 1 being always unstable. Indeed, whilst for a spheroid growing inside an infinite homogeneous domain with constant chemical concentration, one would expect to find λ = 0 for = 1, due to translational symmetry [START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Byrne | Modelling the role of 795 cell-cell adhesion in the growth and development of 796 carcinomas[END_REF] , we must remind that in our case, due to the presence of the external environment the translational symmetry is no longer preserved.

Furthermore, the dispersion diagrams in Fig. 4 also indicate the emergence of a characteristic mode different from = 1 in the cases of bigger size parameters (see Fig. 4a), as well as of small values of the motility parameters σ (see Fig. 4-c) and β (see Fig. 4-d). Interestingly, the characteristic mode is not significantly affected by varying only the dimension of the external domain, while keeping the necrotic radius R N and the initial tumour radius R T fixed (Fig. 4-b) Moreover, whether the range of unstable modes is highly influenced by the sizes parameters and by the motility parameter σ (Fig. 4-a-c), it is not deeply influenced by variations of R out and β (Fig. 4-b-d). Indeed as either the size of the domains decreases (Fig. 4-a) or σ increases, the range of unstable modes decreases, up to a range where only = 1 is unstable. The dependency on the size of the domains states that smaller diffusive lengths (i.e. smaller diffusion coefficient or higher absorption rate of the nutrients) lead to highly irregular contours during the growth of the tumour. On the other hand, the effect on the mechanical parameter σ on the dispersion diagram shows that, as expected, the surface tension σ b , along with an high permeability of the surrounding porous environment k act a stabilizing effect on the front (see Fig. 4-c), whereas the viscosity of the tumour cluster destabilize the border of the MCTS leading to more aggressive tumours. As β settles the velocity of the quasi-stationary moving front (see Eq. ( 14)), the dispersion diagram in Fig. 4-d shows that the tumour developed highly irregular contour only in the case of slowly-moving front (i.e. small chemotactic coefficient and proliferation), since for fast moving front the characteristic mode decrease, until only = 1 is unstable.

Moreover, it is interesting to consider the role played by the radius of the growing tumour in the development of instabilities, while keeping all the other parameters fixed (see Fig. 5). Fig. 5-a reports the results for a set of parameters R out , R N , β and σ for which, independently from R T , the most unstable mode is always = 1. This situation corresponds to a sort of translation of the spheroid inside the domain (see Fig. 5-a on the right). On the other hand, the characteristic mode depends on the MCTS size in a certain range of material parameters (see Fig. 5-b ). Indeed, it increases for increasing R T , so that bigger tumours show more irregularities at their border. Therefore, a growing MCTS can undergo a morphological transition 418 that may significantly affect the invasion pattern towards

Varying σ R out =50, R T =10, R N =5, β= 1 Varying β R out =50, R T =10, R N =5, σ=0.01 Varying R out R T =10, R N =5, σ=0.01, β=1
Varying sizes the typical finger-like structures observed for invasive carcinomas (see Fig. 5-b).

R T /R N =2, R out /R T =5, β= 1, σ=0.01 σ=0.01 σ=0.1 σ=1 σ=10 β=0.1 β=1 β=5 β=10 R out =25 R out =50 R out =250 R out =10 R out =25 R out =50 R out =100 R out =500 (a) (b) (c) (d)
Finally, Fig. 6 depicts the perturbed pressure and velocity fields for a linearly unstable perturbation, given by a spherical harmonic of the kind Y 6 10 (θ, ϕ). The highest variation of the pressure is located in a thin shell closer to the interface of the tumour, so that in the bulk of the tumour the velocity is almost null. In the region just at the rear of small protrusions (due to the perturbation of the boundary), the pressure field increases, so that the velocity at the border of the MCTS where a protrusion 430 form, for the unstable modes (such as the one reported 431 in Fig. 6), is higher than the velocity in the invagination 432 on the contour. Furthermore, from the perturbed field it 433 is possible to appreciate small negative radial velocities in 434 the bulk, just at the rear of the region where protrusion 435 forms. Thus, while the spheroid surface moves outward, 436 some cells inside the cluster move inward. This result con-437 firms the existence of a radial convergent flow, in addition 438 to the divergent flow that makes the aggregate expand, as 439 pointed out in [START_REF] Dorie | Migration and 855 internalization of cells and polystyrene microsphere 856 in tumor cell spheroids[END_REF][START_REF] Delarue | Mechanical control of cell flow in multi-861 cellular spheroids[END_REF] . This effect combined with the higher 440 velocity associated to the protrusion border could explain 441 the possible detachment of carcinoma cells that lead to 442 metastasis and thus the higher invasivity of tumours with 443 irregular contours.

444

Even though the onset of irregular contours and the de-445 velopment of a retrograde flow are in qualitative agree-446 ment with biological experiments [START_REF] Dorie | Migration and 855 internalization of cells and polystyrene microsphere 856 in tumor cell spheroids[END_REF][START_REF] Delarue | Mechanical control of cell flow in multi-861 cellular spheroids[END_REF][START_REF] Frieboes | An integrated computa-865 tional/experimental model of tumor invasion[END_REF] , a direct quantita-

447 R out =50, R N =1, β= 1, σ=0.01 R out =10, R N =1, β= 1, σ=0.01 =2 =4 =6 =8 =1 R T R T R T =2 =4 =6 =8 =10 =1 R T

Conclusions

In this work we have presented a continuum model for describing the avascular growth of a multicellular tumour spheroid, comprising a fixed necrotic core surrounded by a region of proliferative cells, guided by the uptake of a diffusing nutrient. The proposed model encapsulates the diffusion of a chemical species from the vasculature of the healthy region and the tumour cell response to nutrients, via their proliferation and their chemotactic migration inside the extracellular space. The proposed model differs from existing approaches [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Greenspan | On the growth and stability of cell 792 cultures and solid tumours[END_REF][START_REF] Byrne | Modelling the role of 795 cell-cell adhesion in the growth and development of 796 carcinomas[END_REF] since it considers a growth though a rigid, porous surrounding material. Moreover, the MCTS expansion is guided not only by cell proliferation as in [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Greenspan | On the growth and stability of cell 792 cultures and solid tumours[END_REF][START_REF] Byrne | Modelling the role of 795 cell-cell adhesion in the growth and development of 796 carcinomas[END_REF] , but also by the chemotactic motion of cells, through a non-convective mass flux term. Differently from [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Byrne | Modelling the role of 795 cell-cell adhesion in the growth and development of 796 carcinomas[END_REF] , that assumed a Gibbs-Thompson relation [START_REF] Langer | Instabilities and pattern formation in crys-868 tal growth[END_REF] marginally stable, i.e. λ = 0 for = 1 [START_REF] Byrne | Growth of non-necrotic tumours in the presence and absence of inhibitors[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Byrne | Modelling the role of 795 cell-cell adhesion in the growth and development of 796 carcinomas[END_REF] , the proposed 497 model is always linearly unstable, since translational sym-498 metry is broken by considering a finite dimension of the 499 surrounding media. Furthermore, differently from existing 500 works [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Adam | A simplified mathematical model of 627 tumor growth[END_REF][START_REF] Adam | A mathematical model of tumour 630 growth. ii. effects of geometry and spatial unifor-631 mity on stability[END_REF][START_REF] Greenspan | Models for the growth of a solid tumour by diffusion[END_REF][START_REF] Byrne | A weakly nonlinear analysis of a model of avascular solid tumour growth[END_REF][START_REF] Mcelwain | Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth[END_REF] , the perturbation analysis is conducted 501 here without neglecting the diffusion timescale in the un-502 stable growth rate.

503

The analysis of the perturbed field also pointed out a 504 possible mechanism that could lead to the detachment of 505 metastasis from the primary tumour mass, based on the 506 development of higher velocity at the border of the MCTS 507 and a convergent flow inside where protrusions form. This 508 mechanism could explain why the propensity for asym-509 metric invasion and the installation of irregular morphol-510 ogy characterize the growth of aggressive carcinomas in 511 vivo. Thus, this approach has the potential to foster our 512 understanding on the process of transition from the be-513 nign to the aggressive tumour stage and might provide also 514 some indications for improving therapeutic treatments. In-515 deed, more blurred and irregular contours detected in vivo 516 can be related to more malignant tumour, with respect to 517 smoother and clearer contours that can be associated to 518 benign carcinomas. 10 (θ, ϕ). Since higher changes in the velocity and pressure field occurs only at the interface, we use a logarithmic scale in the velocity plot in order to show small variations of the perturbed field inside the bulk of the tumour. The perturbed velocity field highlights the existence of negative radial velocities (i.e. radial convergent flow), as pointed out in [START_REF] Dorie | Migration and 855 internalization of cells and polystyrene microsphere 856 in tumor cell spheroids[END_REF][START_REF] Delarue | Mechanical control of cell flow in multi-861 cellular spheroids[END_REF] .

However, the present model considers a really simplified 520 geometry and adopts some simplifications in order to ob-521 tain a model that can be studied analytically. Thus, fu-522 ture improvements of the proposed mathematical model 523 should focus on the explicit description of quiescent cell re-524 gion (that in the present model corresponds to the region 525 of the spheroid in which we have an almost null velocity) 526 and on tracking the evolution of the inner necrotic core, 527 occurring, for instance, when the nutrients concentration 528 attains a specific value [START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF][START_REF] Byrne | Growth of necrotic tumours in the presence and absence of inhibitors[END_REF] (whereas in the present work 529 the fixed radius R N of the necrotic core is a parameter in 530 the sensitivity analysis). Then, the stability analysis can 531 be enriched by considering the weakly nonlinear interac-532 tions of the asymmetric modes, as well as their evolution 533 depending on the order m of the spherical harmonic per-534 turbation (as done for example in 15 in a simplified case) 535 and numerical techniques should be developed in order to 536 simulate the fully nonlinear evolution of the morphologi-537 cal transition.

538

From the modelling point of view, future studies should 539 also consider the effect of the cells populating the sur-540 rounding healthy environment on the consumption of nu-541 trients and the effect of varying local densities (both inside 542 the healthy tissue and inside the different tumor regions) 543 on the nutrient diffusion coefficient. Then, the effect of 544 solid mechanical stresses on the growth dynamics of tu-545 mours 32,61,67,70-73 and the effect of the possible deforma-546 tion, degradation and reorganization of extracellular ma-547 trix fibres [START_REF] Kopanska | Tensile Forces Originating from Can-895 cer Spheroids Facilitate Tumor Invasion[END_REF][START_REF] Provenzano | Collagen reorganization at 899 the tumor-stromal interface facilitates local invasion[END_REF][START_REF] Kaufman | Glioma Ex-903 pansion in Collagen I Matrices: Analyzing Collagen 904 Concentration-Dependent Growth and Motility Pat-905 terns[END_REF] should be included to move towards a more 548 realistic representation of the problem. Indeed, for tumor 549 growing both in vivo and in xenograft animal models, the 550 description of the system evolution is far more complex 551 than the one proposed in this work, referred to MCTS growth inside inert and rigid ECM scaffolds. In particular, it has been shown that the geometrical and mechanical properties of the ECM [START_REF] Kopanska | Tensile Forces Originating from Can-895 cer Spheroids Facilitate Tumor Invasion[END_REF][START_REF] Provenzano | Collagen reorganization at 899 the tumor-stromal interface facilitates local invasion[END_REF] play an important role for the possible formation of metastasis, since they can lead to growth arrest (i.e. spheroid compartmentalization) or, on 557 the contrary, foster the detachment of invasive cells. Along 558 with the rigidity of the matrix, its density, and the ten-559 sile forces generated in the ECM [START_REF] Kopanska | Tensile Forces Originating from Can-895 cer Spheroids Facilitate Tumor Invasion[END_REF][START_REF] Provenzano | Collagen reorganization at 899 the tumor-stromal interface facilitates local invasion[END_REF] , more recent studies 560 identify the matrix pore size as the critical property mod-561 ulating cancer cell invasion [START_REF] Wolf | Physical limits of cell migration: Con[END_REF]78 . Based on these biologi-562 cal observations, some recent mathematical models have

1 Introduction 1 A 3 6

 1136 multicellular tumour spheroid (MCTS) is an ensemble 2 of tumour cells organized in a multi-layered structure[START_REF] Folkman | Self-regulation of 600 growth in three dimensions[END_REF][START_REF] Byrne | Free boundary value 603 problems associated with the growth and development 604 of multicellular spheroids[END_REF] .In general, a MCTS consists of a central core of necrotic 4 cells, surrounded by a layer of quiescent (i.e. dormant) 5 cells and an outer rim of proliferating cells 1-4 . MCTSs are widely used in vitro to study the early stages 7 of avascular tumour growth and to assess the efficacy of 8 anti-cancer drugs and therapies, since their growth and 9 structure resemble the in vivo avascular phase of solid 10 tumour invasion. Such an early growth phase is charac-11 terized by diffusion-limited growth, since the tumour ab-

12

 12

Figure 1 .

 1 Figure 1. (a) Morphological evolution of a multicellular tumour spheroid of HeLa cells, showing the development of an undulated contour and a necrotic core (reproduced with permission from 51 ). HeLa cells were trypsinized, counted and grown as multicellular spheroids using the liquid overlay technique. The sections were counterstained with hematoxylin and eosin to visualize the cytoplasms of the cells. The multicellular spheroid section is reproduced at days 0, 4 and 12, from left to right. (b) Solid tumours extracted from mice after orthotopic implant of MCF10CA1a cell lines in the mammary fat pad of the nude mice (courtesy of T. Stylianopoulos, Cancer Biophysics Laboratory, University of Cyprus).

  being n N the normal at the fixed boundary ∂Ω N , C the local curvature of the free boundary ∂Ω T (t), p 0 the constant pressure in the outer healthy domain and σ b the surface tension at the moving interface. The surface tension σ b arises from the collective adhesive interaction among tumour cells at the MCTS boundary, primarily mediated by cadherins,59 and from the differential contractility between the cell-cell and cell-medium interfaces, mainly mediated by α-catenin 60 . Even if, in principle the surface tension σ b depends on the density of cells, the distribution of cadherins and the presence of α-catenin 60 , we will assume that it can be considered constant, for the chosen cellular population composing the aggregate.

252 3

 3 Linear stability analysis of the quasi-static 253 solution 254

Figure 3 .

 3 Figure 3. Quasi-stationary solution of the proposed model, depicting the radius of the tumour over time for different values of the motility parameter β. At early stages the growth is exponential, as a consequence of the bulk availability of nutrients. At later stages, the growth law is almost linear, reflecting the higher nutrient concentration on the outer surface of the growing spheroid.

307 3 . 2

 32 Perturbation of the quasi-stationary solution 308

330

  function of the first and second kind, respectively, evalu-331 ated in r. The coefficients A 1 , A 2 , B 1 , B 2 , C 1 appearing in 332 the expression of n 1 (r) can be determined imposing the in-333 cremental boundary conditions for the concentration field 334 (11c), (11d) and (11e), being 335

338|

  boundary conditions on the pressure field (11c) and (11d), 339 considering only the first order terms, i.e. RN = 0 . (23) perturbation theory 62 Finally, using standard procedures 341 in perturbation theory 62 , imposing the boundary condi-342 tion (10) at the perturbed interface and neglecting the 343 terms of order higher than the first in the series expansion,

  344

348 355 4

 355 an implicit function of the time-growth mode λ and the 349 spherical harmonic degree , depending on the five dimen-350 sionless parameters β i , σ, R N , R * T and R out . Interestingly, λ does not depend on the azimuthal component of the 352 model solutions Y m (φ, θ), i.e. the solutions are indepen-353 dent of the order m, as observed also in previous works 354 based on different models 15,50 . Results and Discussion 356

Figure 4 .

 4 Figure 4. Dispersion diagrams for different values of the model parameters (a) Rout keeping q = Rout/RN constant, (b) Rout keeping RN constant, (c) σ and (d) β. The solid lines in the graphs are obtained by interpolating the discrete values (see the dots on the curves) of the time growth rate of the perturbation, λ, calculated for integer values of from eq. (24). In (a) the shapes of the tumour corresponding to the characteristic mode ( = 1, 2, 3, 6) is reported.

Figure 5 .

 5 Figure 5. Evolution of the time growth rate of the perturbation λ with respect to the dimensionless tumour radius, RT , for different values of (with = 1, 2, 4, 6, 8, 10). (a) For the chosen set of parameters, = 1 is the most unstable mode, whatever the tumour radius is. The deformed shapes corresponding to RT = 2, 5, 8 are reported aside (the gray region represents the outer environment). (b) The characteristic mode changes for different values of the tumour radius. Aside the dispersion curves, the section of the tumour perturbed shapes are reported for RT = 2.5, 8, 12.5, 20, 26, 35 to which the corresponding characteristic modes are = 1, 2, 4, 6, 8, 10 respectively.

  on the moving boundary for the chemical potential, we considered a mechanical effect in term of a surface tension at the MCTS outer boundary, leading to the Young-Laplace equation at the interface. The proposed model is governed by five dimensionless pa-483 rameters: two of them, β and σ are called motility pa-484 rameters and representing the mechano-biology cues, the 485 other three are denoted size parameters and are related 486 to the typical sizes of the domains with respect to the dif-487 fusive length. The analytic results predicted the existence 488 of a quasi-stationary radially-symmetric tumour configu-489 ration that is always linearly unstable to asymmetric per-490 turbations involving spherical harmonics Y m (θ, φ), with 491 the range of the unstable modes depending on the dimen-492 sion of the domain with respect to the diffusive length 493 and on the motility parameter β, related to the chemo-494 tactic growth of the tumour. We remark that, whilst a 495 MCTS growing inside an infinite homogeneous domain is 496

Figure 6 .

 6 Figure 6. Evolution of the quasi-stationary and perturbed pressure and velocity fields for Rn = 1, R * T = 35, Rout = 50 for a perturbation of the kind e λt Y 610 (θ, ϕ). Since higher changes in the velocity and pressure field occurs only at the interface, we use a logarithmic scale in the velocity plot in order to show small variations of the perturbed field inside the bulk of the tumour. The perturbed velocity field highlights the existence of negative radial velocities (i.e. radial convergent flow), as pointed out in[START_REF] Dorie | Migration and 855 internalization of cells and polystyrene microsphere 856 in tumor cell spheroids[END_REF][START_REF] Delarue | Mechanical control of cell flow in multi-861 cellular spheroids[END_REF] .

  563 been developed to take into account, on one hand, MCTS 564 segregation by thick porous (but still rigid and homoge-565 neous) structures 79-81 and, on the other, ECM deforma-566 tion 82 . Furthermore, not only ECM fibers can accumulate 567 or being degraded at the host-MCTS interface, but they 568 strongly reorganize, aligning parallel to the tumor bor-569 der, in a first stage, and then perpendicular to the tumor 570 boundary 74 . 571 Thus, to take into account all these aspects and more 572 ealistically describe tumor growth in vivo, an anisotropic 573 poro-elasto-visco-plastic model with a threshold (based on 574 microscopic arguments) for cell motion should be devel-575 oped. However in that case tumor irregular contours will 576 likely arise for inhomogeneity and anisotropy in the ECM, 577 whereas this work demonstrates that mechano-biological 578 and (macroscopic) geometrical cues can determine the oc-579 currence of a morphological transition in growing tumours 580 that can promote invasiveness, even in an homogeneous 581 environment. The theoretical results push towards the 582 developments of further biological experiments for accu-583 rate characterization of MCTS morphology and careful 584 measures of the surface tension and the interstitial pres-585 sure within MCTSs 59,83 , as well as growth and mobility 586 properties of the tumour cells to validate the predictions of 587 the model. Indeed, the integration of mathematical tools 588 in biological research could be crucial for estimating the ters, vol. 72, no. 14, pp. 2298-2301, 1994. 697 26. G. Forgacs, R. Foty, Y. Shafrir, and M. Steinberg, 698 "Viscoelastic properties of living embryonic tissues: 699 a quantitative study," Biophysical Journal, vol. 74, 700 no. 5, pp. 2227-2234, 1998. 701 27. G. Vitale and L. Preziosi, "A multhiphase model crovascular Research, vol. 66, no. 3, pp. 204-212, 2003.

  722 32. C. Voutouri, F. Mpekris, P. Papageorgis, A. Odysseos, by using metabolic measurements," PNAS, vol. 102, 814 no. 17, pp. 992-7, 2005. 815 55. H. Byrne and L. Preziosi, "Modelling solid tumour 816 growth using the theory of mixtures," Mathematical 909 trol by ECM space and nuclear deformation and tun-910 ing by proteolysis and traction force," The Journal of 911 Cell Biology, vol. 201, pp. 1069-1084, 2013. 912 78. A. Haeger, M. Krause, K. Wolf, and F. P., "Cell jam-913 ming: collective invasion of mesenchymal tumor cells 914 imposed by tissue confinement," Biochimica et Bio-915 physica Acta, vol. 1840, no. 8, pp. 2386-95, 2014. 916 79. C. Giverso, A. Grillo, and L. Preziosi, "Influence of nu-917 clear deformability on cell entry into cylindrical struc-918 tures," Biomechanics and Modeling in Mechanobiol-919 ogy, vol. 13, pp. 481-502, 2014. 920 80. A. Arduino and L. Preziosi, "A multiphase model of 921 tumour segregation in situ by a heterogeneous extra-922 cellular matrix," International Journal of Non-Linear 923 Mechanics, vol. 75, pp. 22-30, 2015.

  924 81. C. Giverso, A. Arduino, and L. Preziosi, "How Nu-925 cleus Mechanics and ECM Microstructure Influence 926 the Invasion of Single Cells and Multicellular Aggre-927 gates," [submitted], 2016.

  928 82. G. Sciumé, R. Santagiuliana, M. Ferrari, P. Decuzzi, 929 and B. Schrefler, "A tumor growth model with de-930 formable ECM," Physical Biology, vol. 11, no. 6, p. 065004, 2014. 83. R. K. Jain, "Barriers to drug delivery in solid tumours," Scientific American, vol. 271, no. 1, pp. 58-65, 1994.

  λ ∈ R is the amplification rate (or time-growth rate) 314 of the perturbation and Y m (θ, ϕ) is the spherical harmonic 315 of degree and order m, with m ∈ N, ∈ N + and |m| ≤ .316 The spherical harmonics Y m (θ, ϕ) form a complete set 317 of orthonormal functions and thus any square-integrable 318 function can be expanded as a linear combination of spher-319 ical harmonics. For physical consistency, the variations of 320 n and p from the quasi-stationary solutions n * and p *

		321
	should be in the form	322
	n(r, θ, ϕ, t) = n

* (r, t) + εn 1 (r)e λt Re [Y m (θ, ϕ)] (16) p(r, θ, ϕ, t) = p * (r, t) + εp 1 (r)e λt Re [Y m (θ, ϕ)] , (17) Using Eq. (11a) and the relation ∇ 2 Ω Y m + ( +1)Y m = 0, 323 where we set the angular part of the Laplacian operator as 324 ∇ 2 Ω (•) = 1/ sin θ ∂/∂θ(sin θ ∂(•)/∂θ)+ 1/ sin 2 θ ∂ 2 (•)/∂φ 2 , 325 the term n 1 must obey the following ODE 326 r 2 n 1

tumour's ability to invade its host environment. The long-time dynamics of elastic growth," Journal of 729 Biological Dynamics., vol.