Botswana, Chair Paul T Nleya

Sethunya P Kgaodi

Botswana Rapporteur

Botswana, Rapporteur Letlhogonolo Moaneng

Botswana Godfrey S Kalanke

Javier Osorio

Nicholas B Mavengere

Online Learning Features and Factors in Getting the Blend Right Conference discussion group paper

Introduction and Context

This presentation discusses online learning provision using blended approach in educational settings. Online learning provision in the African setting includes both the traditional e-Learning management systems (e. g. proprietary -blackboard and webCT), and Open Source (i. e. Moodle, KEWL etc.). Online learning can feature at primary, junior and senior secondary as well as tertiary levels. However, within the African context, much emphasis is placed on the tertiary level and the scope of this paper focuses on eLearning provision within the tertiary level where it is believed that necessary resources are available. The paper firstly outlines the features of online learning, secondly outlines several factors that would determine the level and success of online provision in tertiary institutions in Africa; and lastly, discusses challenges and blended learning trends regarding online provision given the several factors in place.

Features of Online Learning Settings

There exist several features of online learning provision that make it a distinct mode of delivery. Certain prerequisite requirements would enable adequate online provision and the following have been identified as critical attributes of online learning:

• Smart classrooms/Laboratory

Factors Determining Online Learning Provision

Several general factors tend to determine the level of provision of online learning and the following have been identified:

• Strategic policy Frameworks The identified factors also tend to determine the blending of online learning as there can be no 'one size fits all solution' in online provision. Each institution will have to weigh its capability so as to strike an adequate balance regarding a blending configuration that best fits its needs. Furthermore, some key success factors regarding online learning provision have also been identified as follows:

Institutional support -addresses technological infrastructure issues, a technology plan, and professional incentives for faculty.

Getting the Blending Right

Creating a blended learning strategy is viewed as an evolutionary process. One needs to explore the capabilities of the team, the organization's infrastructure, and learners' receptiveness to the new learning formats. For many, the first stage in their blended learning program initiative is to supplement their current programs, either traditional classroom or self-paced content libraries, with live eLearning activities (coaching, virtual classrooms or workshops) to extend the learning process and better integrate it with the work environment (Lily Wong and Arthur Tatnall, 2010). There are critical factors that have some bearing on blending configurations that institutions have adopted. These range from staff/students awareness levels to the capacity of the institution in terms of available resources. Some institutions will have acquired higher readiness levels and would therefore aspire to have a blending that is biased in favor of technology while some will have less readiness levels in technology applications and will infuse/integrate less technological applications. Motivation of students and staff regarding the use of technology is also another factor and the time taken by teachers to organize materials online.

Availability of resources has some bearing on online provision. Given that institutions have varying levels of provision of resources. Furthermore, some resources are restricted as laboratories close early while other institutions have open access and unrestricted access to resources and therefore have the potential of integrating more technologies that promote eLearning at an accelerated rate and leading to some rich blending. Technical and pedagogical support is also critical as some institutions provide eLearning centres that not only train staff and students, but also assist with instructional design, graphics and other online aspects critical in blended learning. There is need for both technical and pedagogical support in eLearning infusion. This could provide the required capacity building in the eLearning initiatives of institutions of higher learning.

The pedagogical aspect in the implementation of online learning is a critical attribute. Every learning tool has its own modalities and users (students and staff) therefore require training on how to use it in the most effective way. Some areas of specialization (e. g. Math and Science) have the reputation of promoting aspects of online learning while some areas lag behind, and yet technology could be integrated in all areas. Some instructional technologies are commonly used in eLearning while others have not been used significantly for learning purposes despite their being available (e. g. mobile telephone). Both asynchronous and synchronous technologies are used in eLearning together with other convergence technologies depending on how they suit the institution. Copyright/plagiarism and intellectual property issues have become critical when planning online learning delivery. Online courses also require accreditation to make them authentic given the prevalence of fraudulent tendencies practiced by some online institutions.

The critical question for institutions of higher learning to ask is; to what extent can online learning replace classroom teaching (face-to-face)? Blended learning has become the popular mode given that online learning has not yet developed to the extent of replacing face-to-face delivery mode in recent years. Blended learning is viewed as learning that is facilitated by the effective combination of different modes of delivery, models of teaching and styles of learning, and is based on transparent communication amongst all parties involved with a course. The potential of blended learning is almost limitless and represent a naturally evolving process from traditional forms of learning to a personalised and focused development path.

Some professionals merely upload materials onto available online management systems without examining what to put online and how to use it to enhance learning. There is also need for strategic eLearning policies that would guide implementation of effective online practices. One of the most important factors in creating blended learning is to find out where it fits in the broader context of organisational learning and development. The interplay of these factors in institutions of higher learning will determine the appropriate mix of blended learning as there is no one size fits all situation.

Challenges in Online Learning

Several challenges tend to exist regarding the implementation of eLearning provision at the tertiary level within the African setting. Some of these challenges are as follows:

• Technophobia -attitude towards computers (lecturers and students)

Conclusion

The paper has firstly outlined the characteristic features of online learning in the context of tertiary institutions in developing countries with some focus on the African context. Secondly, it discusses several factors that would determine the level and success of online learning provision and lastly, the resulting blended learning trends regarding online provision within tertiary institutions in the African setting. Several challenges have also been associated with the implementation of online learning provision at the tertiary level. It can be concluded that creating a blended learning strategy in online learning provision at the tertiary level in Africa and other developing countries is an evolutionary process, and that there is no 'one size fits all solution'.

  Course Development -this category involves benchmarks for the development of courseware which is produced largely either by individual faculty (or groups of faculty members) on campus, subject experts in organizations and/or commercial enterprises. Teaching/Learning Process -this category addresses the array of activities related to pedagogy, the art of teaching. Included in this category are process benchmarks involving interactivity, collaboration, and modular learning. Course Structure -addresses those policies and procedures that support and relate to the teaching/learning process. They include course objectives, availability of library resources, types of material provided to students, response time to students, and student expectations. Student Support -this category includes the array of student services normally found on a college campus including admissions, financial aid, etc. as well as student training and assistance while using the Internet. Faculty Support -address activities that assist faculty in teaching online, including policies for faculty transition help as well as continuing assistance throughout the teaching period. Evaluation Assessment -this category relate to policies and procedures that address how, or if, the institution evaluates Internet-based distance learning.

Masoumi D. (n. d.)

  Accreditation of programmes (Stephen Asunka and Hui Soo Chae, 2009)

	•	Cultural changes
	•	Skills and expertise for technical support
	•	Cost of proprietary learning management systems
	•	Policies issues, (institutional and pedagogical)
	•	Copyright
	•	Plagiarism
	•	Intellectual property
	•	Data security
	•	Quality control and assurance issues
	•	Access to connectivity
	•	Discomfort with self-directed learning activities, and
	•
	•	Computer literacy level for users
	•	Access to resources


Jacques Bulchand-Gidumal

Santiago Melián-González

Javier Osorio- Acosta

Value Chain of Technology in Higher Education Institutions: from IT Resources to Technological Performance

Keywords: IT performance, firm performance, IT resources, IT strategic planning, Value chain, resource-based view of the firm, universities, Spain

Different studies confirm that the presence of IT in firms, together with human and other organizational resources, has a positive influence on the performance of organizations. However, the details of the process through which that influence is produced have not been clarified. This study is based on an extensive IT data base corresponding to a sample of universities and presents an IT-technological performance value chain and confirms the hypotheses about its functioning. The result is a value chain that begins with the IT planning, passes through different components related to technology in organizations and ends with the performance of the technology. We believe that this research is useful to higher education institutions managers by allowing them to have a clear path on how to improve the return of IT investments.

Introduction

Today, almost every higher education institution uses IT. Still, we know that not all of them get the same value out of IT. This is, the organizational performance is not increased in higher education institutions as the investment in IT. Some obtain more, some obtain less.

Although a lot of different explanations can be given, starting by how we measure organizational performance in the case of higher education institutions, and passing by concepts such as complementary resources [START_REF] Brynjolfsson | The IT productivity gap[END_REF], the effect of the learning curve deferring the performance and the sample size [START_REF] Hitt | Productivity, Profit and Consumer Welfare: Three Different Measures of Information Technology Value[END_REF][START_REF] Brynjolfsson | Beyond the productivity paradox[END_REF][START_REF] Schreyer | Information and communication technology and measurement of real output, final demand and productivity[END_REF][START_REF] Kholi | Measuring Information Technology Payoff: A Meta-Analysis of Structural Variables in Firm-Level Empricial Resarch[END_REF], we understand it would be very interesting for organizations to have a path, a value chain, that defines what to do with IT investments in order to improve organizational performance. Since not many works address this issue, this article aims to clarify this question, specifically in the context of higher education institutions.

To that end, we first explain our proposed model and the data base we used to test it. This is followed by the methodology used, the results obtained and, finally, the conclusions and limitations of the work.

Preparation of the Model and Work Hypotheses

According to [START_REF] Barney | Firm Resources and Sustained Competitive Advantage[END_REF], a firm's resources can be classified into resources of physical, human and organizational capital. Thus, as drivers of the process, those resources must appear in the initial phase of the process of IT value creation.

In a first design of our proposal, the three types of resources constitute the starting point of the chain through which IT begins to add value. Like [START_REF] Bharadwaj | A resource-based perspective on information technology capability and firm performance: An empirical investigation[END_REF], we consider that an organization's IT capability consists of organizing those three classes of resources in a way that is effective for their performance: IT strategic planning (ITPLAN), IT area human resources (ITHR) and infrastructure IT resources (ITINFR) and we understand that ITPLAN is the guiding structure that defines what to do with the other two categories, ITHR and ITINFR.

With these categories organizations are able to create IT applications (ITAPPS), guarantee system reliability (ITRELIAB) and define a series of norms for IT use (ITPROCS).

After these three items we understand we arrive to the part of the chain dealing with performance. Its position is in the final part, which is where the cited works also place it. Thus, in the analyses, we show that many of the data that we have refer to the output of the IT processes or tasks, in other words, they constitute what we finally called technological performance.

We believe there are really two kinds of technological performance: performance referring to the technology's contribution to the management of the organization (ITPERFORG) and performance related to the use of the technology by users who are not IT personnel (ITPERFUSER). Those two dimensions match the possibilities that [START_REF] Sambamurthy | IT Management Competency Assessment: A Tool for Creating Business Value through IT[END_REF] consider for the IT impact.

So we derive a model that is presented in Figure 1. In this article, we are not going to test the final part of the model, but, like [START_REF] Sambamurthy | IT Management Competency Assessment: A Tool for Creating Business Value through IT[END_REF] and [START_REF] Soh | How IT Creates Business Value: A Process Theory Synthesis[END_REF], we propose that organizational performance will be influenced by the technological performance

To end this section, we break down the functioning of the proposed model into hypotheses. There are many authors who indicate the need to plan the development of IT (i.e. Boar, 2001). In light of the above, our first hypothesis is the following:

H1. ITPLAN has a positive influence on the level of (a) ITHR and (b) ITINFR.

The importance of human resources to IT performance has been confirmed in various studies [START_REF] Powell | Information technology as competitive advantage: The role of human, business, and technology resources[END_REF][START_REF] Francalanci | Information Technology and worker composition: determinants of productivity in the Life Insurance Industry[END_REF]Galal, 1998, Bharadwaj. 2000). Therefore, we propose that the greater the deployment of IT human resources, the higher the performance in the three IT processes identified from our data base. Thus:

H2. ITHR has a positive influence on (a) ITAPPS, (b) ITPROCS and (c) ITRELIAB.

It is also logical that it is necessary to have a basic IT infrastructure in order to perform well in IT [START_REF] Ross | Develop long-term competitiveness through IT assets[END_REF][START_REF] Bharadwaj | A resource-based perspective on information technology capability and firm performance: An empirical investigation[END_REF][START_REF] Broadbent | The implications of information technology infrastructure for business process redesign[END_REF][START_REF] Bhatt | Types of Information Technology Capabilities and Their Role in Competitive Advantage: An Empirical Study[END_REF]. Thus, as in the case of human resources, we consider that the physical IT resources exercise a positive influence on the three established IT processes.

H3. ITINFR has a positive influence on (a) ITAPPS, (b) ITPROCS and (c) ITRELIAB.

Moving onto the final part of the model, [START_REF] Cooper | Data warehousing supports corporate strategy at first American corporation[END_REF] indicate that the presence of certain management applications (e.g., datawarehouses, work flows, content and news managers) in organizations favors organizational performance. Therefore:

H4. ITAPPS has a positive influence on (a) ITPERFORG and (b) ITPERFUSER. Secondly, [START_REF] Broadbent | Management by Maxim: How Business and IT Managers can Create IT Infrastructures[END_REF] indicate that the standardization and formalization of the IT in organizations is a positive value that helps improve the organization's technological performance. Therefore, we propose that:

H5. ITPROCS has a positive influence on (a) ITPERFORG and (b) ITPERFUSER.

Finally, authors such as [START_REF] Teo | Critical success factors in the alignment of IS plans with business plans[END_REF] indicate that the fact that the IT services are efficient and reliable constitutes one of the factors critical to the alignment of IT:

H6. ITRELIAB has a positive influence on (a) ITPERFORG and (b) ITPERFUSER.

Description of the IT Data Base

In order to test the model, we had access to a data base of IT data of Spanish universities that was created by the universities themselves trough their Rectors Conference (CRUE), specifically the group dedicated to IT.

The data base in question contains a vast amount of information related to IT in university institutions and has a total of 118 indicators. Table 1 shows examples of data for each of the six strategic axes in which that work group organized the information. It also includes some additional data about each institution in a specific area.

Management processes 12

Computerized management processes, backup system to activate the services that suffer availability problems, eadministration technologies being exploited.

Management of institutional information 21

Procedures available in the records management, departments that use the content manager to publish on the web, existence of a contingency plan in case of emergencies.

IT training and culture 21

Individuals on the management staff who have received training in IT competencies, average budget for specialized training of IT personnel, users who access the intranet through a VPN.

Organization of IT 16

Existence of a strategic plan for IT, IT personnel in relation to number of students, incidents successfully resolved by IT department in relation to number of users.

Context data 13

Administrative personnel, faculty staff, classrooms, research groups, subjects, budget, staff budget.

Source: CRUE-TIC group

In turn, each of those six axes comprises up to a total of twenty-eight objectives. Some examples of these were:

•

Methodology, Data Analysis and Results

The population of our study comprises all 73 Spanish universities, 50 of which are state universities and 23 private. Data was collected by the Rectors Conference in 2006 and 2007 through a questionnaire completed by the CIO. A total of 59 completed questionnaires were received, of which three were eliminated because they were not properly completed, giving a response rate of 76.7%. The chosen method of data analysis was structural equations using the Partial Least Squares technique (PLS), specifically, we use SmartPLS 3.0 build M3 [START_REF] Ringle | Smart-PLS Version 2.0 M3[END_REF].

Our model has eight constructs, three of which have formative indicators and the other five reflective indicators. In our case, the indicators of the constructs ITHR, ITINFR, ITAPPS, ITPROCS and ITRELIAB are reflective and, therefore, should display high levels of correlation. However, the indicators of the variables ITPLAN, ITPERFORG and ITPERFUSER do not have that property, thus this constructs are considered as formative.

To test the hypotheses, it is necessary to begin by analyzing the measurement model (in other words, the relationships between the constructs and their indicators) and then the structural model (that is, the relationships between constructs). All the test were sufficiently passed by the model: individual item reliability of reflective constructs, construct reliability (Cronbach's alpha coefficient and for composite reliability), average variance extracted (AVE) and discriminant validity. Formative constructs also passed the non-multicolinearity test.

We now analyze the structural model, summarized in Figure 2. That figure shows the explained variance of the constructs (R 2 ) and the standardized coefficients (β). Other tests such as the Q 2 test for predictive relevance and the test for the stability of the estimations were all exceeded comfortably by the model.

Fig. 2. Estimated causal relationships in the structural model

As can be seen, out of the 14 proposed hypotheses, 10 were confirmed and 4 were not (H2c, H5a, H4b and H6b). With regard to the explained variance (R 2 ) of the constructs the structural model displays adequate predictive power. The constructs obtain R 2 values that are never below 0.290, the explained variance of the variable ITPERFORG is 69.9% and that of the variable ITPERFUSER is 36.6%.

Apart from examining the R 2 , the model is evaluated by observing the predictive power Q 2 of the constructs of the model [START_REF] Geisser | A predictive approach to the random effects model[END_REF][START_REF] Stone | Cross-validatory choice and assessment of statistical predictions[END_REF]. The model passed the test of the Q 2 being greater than 0 for the seven constructs.

As figure 2 shows, the model predicts 51.5% of the variance of the variable ITHR, regarding the deployment of HR in the IT area, and 31.4% of the variance of ITINFR, referring to the deployment of IT resources.

Discussion

It can be seen that the deployment of IT human resources (ITHR) helps explain the 12.8% variation in the existence of computer solutions to improve management (ITAPPS) and 12.8% in the existence of norms and procedures for the use of IT in organizations. However, it only explains 5.8% of the variation in systems to ensure the functioning of the IT, this last prediction being non-significant. We consider that the first two results are quite logical since an adequate deployment of human resources in the IT area enables the implementation of the automation processes. In other words, to be able to progress in the automation processes, it is not enough to purchase technology, it is also necessary to have the human resources required for the technology to function and be correctly applied. It is also logical that the deployment of human resources also helps and contributes to the development of norms and procedures for the use of IT in organizations, since it is a type of job that is usually associated with the technicians of the area. It is later shown that the existence of norms and procedures improves the users' perception and use of the technologies.

However, it is somewhat surprising that adequate deployment of human resources does not have a positive influence on the guarantee of reliability (i.e., high availability, back-up copies). This may indicate that it is more a question of having sufficient economic resources (and ultimately, technological resources) than anything else. In other words, the non-confirmation of the hypothesis may point to the fact that the need for reliability in the IT area is something already present in organizations and is also a service offered by numerous suppliers.

The basic IT infrastructure (ITINFR) explains 19.0% of ITAPPS, 20.0% of ITINFR and 23.2% of ITRELIAB. Thus, an appropriate level of physical resources in the IT area influences the existence of computer solutions to improve management, the existence of norms and procedures, and the systems to ensure reliable IT functioning. It is interesting to see that all those values are higher than those of the variable ITHR. This means that, although some authors, including [START_REF] Marchand | Information orientation: people, technology and the bottom line[END_REF], have indicated that in the implementation of technologies, people must be taken more into consideration than has traditionally been the case, our study indicates that machines and finance are still relevant.

Another interesting result is the extent to which organizations progress in the existence of computer solutions to improve management (ITAPPS); in other words, whether they have applications for the most common tasks, such as publication on the web, workflow management, datawarehouse, single login, etc. We confirm that this variable explains 18.9% of the technological performance in the organization's activities (ITPERFORG) but is non-significant in the case of technological performance related to users (ITPERFUSER). This indicates that the technological output in the organization is explained by the development of automation processes but that, in the case of users, this type of issue is not so important. It should be borne in mind that a good part of the university communities comprises faculty staff and students and that such users do not normally have direct contact with management procedures. Therefore, issues related to the reliable functioning of the technology are more important to them than automation.

With regard to the existence of norms for the use of the technology (ITPROCS), we find that this variable has no significant relationship with performance referring to the technology's contribution to the management of the organization (ITPERFORG). However, it has a significant relationship with, and explains 13.2% of, the use of the technology by users that are not IT personnel (ITPERFUSER). The non-significance of the first relationship indicates that the use of technology in the organization's management area is not explained by the existence of norms. That finding is logical since the norms for the use of technology are, above all, oriented to the use of technologies on campus, especially by faculty staff (e.g., on the web, and related to respect for intellectual property) and students (e.g., in the use of classrooms, mail and related to respect for intellectual property). Therefore, the significant predictive capability of ITPROCS over ITPERFUSER is logical.

Finally, we analyze the predictive capability of ITRELIAB. We find that it is able to explain 58.4% of the variance of ITPERFORG but displays no significant relationship with ITPERFUSER. In other words, the implementation of technology in organizations must be accompanied by guarantees that it all functions correctly and that any problems are rapidly resolved. Thus, the continuity of the business is a fundamental issue in the internal use of technology. However, the non-significance of the relationship between ITRELIAB and ITPERFUSER indicates that such reliability is very important and decisive from the management point of view but much less so from the perspective of the use of technology by users.

Conclusions and Implications

The objective of this study was to expand knowledge of the process of how IT contributes to the performance of higher education institutions. We believe that this work sheds light on the space left by most works between the technological and organizational variables, on the one hand, and organizational performance, on the other. Although our model only goes as far as the phase immediately prior to organizational performance, logic and the opinion of other authors [START_REF] Sambamurthy | IT Management Competency Assessment: A Tool for Creating Business Value through IT[END_REF][START_REF] Soh | How IT Creates Business Value: A Process Theory Synthesis[END_REF] suggest that organizational performance is the step that follows technological performance. Thus, we have been able to show a path, based on the results and arguments found in previous literature, of how IT acts to increase technological performance. Moreover, the elements of that value chain are specific links that generate less ambiguity than statements such as a suitable combination of human and technological resources or organizational structures that support the IT tasks. In that respect and except for the work of [START_REF] Bhatt | Types of Information Technology Capabilities and Their Role in Competitive Advantage: An Empirical Study[END_REF], the rest of the studies neither generate nor test specific hypotheses regarding the IT-performance process. However, it would be naïve to aim for high specificity about how all the elements of the value chain are managed since that would be too specific to be able to be generalized and applied to different organizations.

A result of the above is that an overall analysis of the value chain enables university managers to draw many interesting conclusions about how to orient their IT management policies. Thus, this work has shown the importance of planning in the context of technology in higher education. This is the variable that organizes and guides all IT activity and has a direct influence on the two types of resources most studied in the IT-performance literature. Hence, the human resources and the basic IT infrastructure constitute the basis for the development of IT activities. With regard to those activities, our sample contains three that have different effects on technological performance. On the one hand, the existence of computer solutions to improve management, and systems to ensure the reliable functioning of IT (this is determined only by the basic IT resources) that influence the extent to which IT contribute to and support the organization's activities and processes and, on the other, the generation of norms and procedures for the use of technology, which influences the use of technology by non IT-personnel users.

  Fig. 1. Proposed model

  Axe 1, Objective 1 -To incorporate IT in the classrooms (computer and noncomputer) -9 indicators. • Axe 2, objective 4 -To promote the expansion of research activity by means of IT tools -4 indicators. • Axe 4, objective 2 -To be prepared to manage institutional knowledge on the basis of statistics, indicators, balanced scorecards and data analysis -2 indicators. • Axe 5, objective 2 -To ensure specific training for IT personnel -3 indicators. • Axe 6, objective 1 -To have strategic IT planning -3 indicators.

Table 1 . Structure and content of the data base

 1

	Axes	Indicators	Examples of data
	1. Teaching-learning	16	Students per computer in classrooms, classrooms with Wi-Fi coverage, institutional plan for the development of on-line teaching.
	2. Research	19	Scientific calculation capability, application for research management, research groups with web page.

Appendix I. Constructs and short description

• ITPLAN (Formative construct, 5 indicators). IT planning process.

• ITHR (Reflective, 4). Deployment of HR in the IT area.

• ITINFR (Reflective, 10). Basic IT infrastructure.

• ITAPPS (Reflective, 8). Development of IT applications to automate management.

• ITPROCS (Reflective, 4). Proceduralization of tasks related to norms and procedures. • ITRELIAB (Reflective, 2). Systems to ensure the IT functioning.

• ITPERFORG (Formative,6). Performance referring to the contribution of technology to the management of the organization. • ITPERFUSER (Formative,6). Performance referring to the use of the technology by non IT-personnel users.


Don Passey
email: d.passey@lancaster.ac.uk

The 'Educational Lot' for Young People who are NEET: Quality Management and Roles of Management Information Systems

Keywords: Learning management systems, learner-centred learning, motivation, evaluation, policy

The United Kingdom (UK) government, in common with those in many other countries, recognises the significant numbers of young people who are not in education, employment or training (NEET) from the age of 16 years. Perceptions associated with this group of young people have often been negative, yet recent research suggests that reasons for the lack of involvement of this group of young people often arises from situations where there is extreme trauma, linking to desires of individuals to seek alternative forms of educational and economic involvement from the traditionally cognitively focused opportunities accessible normally in schools. Alternatives, sometimes involving uses of technology, are being offered in a range of school and support situations. The importance of management information systems in both identifying and positively tracking these young people is recognised, yet the development of systems to cope with policy and support needs is currently at an early stage. This paper offers an overview of the current, largely UK, context, and indicates how data management information systems are likely to support important policy, practice and research needs.

Introduction

It is well recognised that many young people in the United Kingdom (UK) are not in education, employment or training (NEET) from the age of 16 years. The British Broadcasting Corporation (BBC) reported [START_REF] Eason | NEETs are an unknown quantity[END_REF] that: "At the end of 2006 there were said to be 206,000 Neets aged 16 to 18 in England in the target group for the government's new strategy". [START_REF] Raffe | Young People Not in Education, Employment or Training[END_REF] found, in a study conducted in Scotland, that: "more than three in ten (31%) young people were NEET at some time during the three years after the end of compulsory schooling". However, the exact numbers of young people who are NEET, particularly at any one point in time, are not at all clear. As the BBC article stated: "Shortcomings in the system that tracks what becomes of young people mean that officials do not know what 5% are up to -they are simply not on the databases".

Many governments wish to address what is seen as a 'NEETs issue'. In 2000, heads of state and governments of the European Union (EU) agreed the 'Lisbon Objectives in Education and Training'; these objectives included the reduction of the high number of early school leavers aged 18 to 24 years to 10% by 2010, in order to reduce the danger of them 'being on the fringes of the knowledge society' (Commission of the European Communities, 2005). The second report on progress towards the meeting of those objectives indicated that in 2004 the EU average ratio of early school leavers (referred to as 'drop outs' in EU documentation and NEETs in UK documentation), for those with only 'lower-secondary education' was the same as that in 2003, and that the 2010 target would not be met if this trend were to continue. Within that report, the UK rate was reported as 16.7% for 2004 (16.4% for females, and 17.0% for males). In a UK context, concerns about this group, and perceptions associated with them, are themselves quite diverse. The approaches that can be taken by policy makers and those who support this group can be wide, and these approaches appear to be associated with perceptions of who constitutes this group. At times, perceptions of this group of young people have been painted in rather negative ways. But, as the BBC report stated: "We imagine them to be the idle feckless -good-fornothing layabouts, content to subsist on benefits and/or family handouts. But the government's own strategy document makes it clear this is a misleading impression".

It is clear that many different individuals (and sub-groups) make up this wider NEET group. It is commonly believed that certain factors impact often at early stages, and the potential levels of impact of some factors is alarming; consider, for example, that "Some 78 million Europeans live at risk of poverty, 19 million of whom are children" (European Communities, 2008), or that "6 million young people leave school without any qualifications" (Commission of the European Communities, 2009). There have been a number of different attempts to categorise and characterise this group more exactly, in order that policy makers and support practitioners can be in a position to understand them in greater detail, and support them (or address issues associated with them) more effectively. The BBC report stated that: "Researchers working for the then Department for Education and Skills came up with various characteristics of youngsters who were likely to drop out: Angry young rebels; Quitters; Rebels without a cause; cool dudes; Hedgers; Settlers; Escapists; and Strugglers". A recent study that looked at uses of technology by this group of young people [START_REF] Passey | Assessing the potential of e-learning to support reengagement amongst young people with Not in education, employment or training (NEET) status: An independent research and evaluation study[END_REF] categorised them in ways that were concerned with a focus on potential future support. They suggested using four different categories: those young people who are employment and education ready (EERs); those who are nearly employment and training ready (NEERs); those who have chosen alternative lifestyles (ALTs); and those who have multiple long-term problems (MULPs). Understanding the characteristics of this group of young people, to consider how to associate most appropriately the alternative forms of support available, is clearly likely to be of importance. The range of characteristics of this group is wide, and some of the features of this group bring particular challenges to any implementation approach. A phenomenon of note is that of NEET 'churn'. A report by the National Audit Office (2004) found that while the majority of young people do leave NEET status, this is not necessarily a permanent exit. They found that a substantial minority continued to join and rejoin the NEET group, with numerous short periods of education or employment in between. The NEET group is not a stable group -young people are continually leaving and joining it. [START_REF] Sachdev | Regional and sub-regional variation in NEETsreasons, remedies and impact[END_REF] reported, for example, that only 1% of the group remain NEET right through from 16 to 18 years of age.

To positively support these young people, it is clear that there are potential roles for management information systems: to help identify the population more exactly; to identify characteristics that can suggest appropriate engagement; to track and record success as well as indications of weakness; and to provide research with opportunities to address key questions relating to this area. These key areas will be explored further within this paper. It should be noted in this context, however, that inappropriate or inadvertent uses of management information systems could create significant future issues. 'Escapists', for example, are a group of young people likely to be seeking ways to be (rightly) individual, related with high degrees of freedom. Data management systems used to track individuals so that those individuals feel that their freedom is being taken away from them are likely to feel negatively, they may well disengage further, rather than being encouraged to engage positively.

Understanding the Scope of the Problem

There is a clear need to understand the scope of the problem, if it is to be addressed. Currently, the lack of detailed quantitative information is in itself a weakness. Within the UK, data management systems within schools retain detailed information on individuals. While individuals are within school systems, the recording and retaining of data is widespread and usually thorough. However, when individuals move outside a school system (to another school or beyond), the transfer of data has been less well developed technologically and operationally.

When young people leave the school system, local authority (LA) Connexions services are key agents in providing support for young people who are NEET. Connexions services generally would find it useful to have access to certain types of data that are not currently necessarily easily accessible to them. For example, having end of Key Stage 2 (at 11 years of age) and end of Key Stage 3 (at 14 years of age) subject attainment data would be helpful to them. Some services in some LAs and regions have started to look at how to address this type of need. There is a need, however, for appropriate transfer of data not just from schools to the Connexions services, but also from other LA services to the Connexions service. Currently, this form of transfer is not developed in any common form, but the importance of information and management systems in this respect is stressed by the government education department (DCSF, 2008a) through its website. The website states that: "Connexions has developed excellent case management systems for effectively tracking young people's progress. These systems formed the basis of the Client Caseload Information System (CCIS) specification which enables support to young people across local authority boundaries, ensures multi-agency working and gives better targeting of services; all of which form the heart of any integrated youth support service. Local authorities will be expected to ensure that the client tracking systems set up by Connexions services are maintained and developed to inform local planning and measurement of the NEET group. ... Local authorities must have in place a robust client management system and client tracking arrangements that meet the Client Caseload Information System (CCIS) specification." Some local exemplars do currently exist where implementation has been undertaken in particular ways. A report by Citizens Online and the National Centre for Social Research ( 2008) identified a number of key exemplars of technologies in LAs supporting young people who are NEET, including the YorOK Database (information sharing in multi-agency working). The DCSF (n.d.) website provides a range of case studies, including an example where data is transferred between schools and the Connexions service, called Pan London CCIS. In the state of Victoria in Australia, a Schools Interoperability Framework (SIF) is being used to support data transfer with a focal purpose of enabling the identification and tracking of young people who are NEET [START_REF] Shoesmith | The use of data and associated issues[END_REF]. Although schools in the state record relevant data, details about young people are not held in the same software applications as data relating to attendance and to timetable. The SIF framework works by using a central Zone Integration Server, which reads data from different sources, and writes them to agreed applications. Each application across the system that contains data of value is read through the use of a SIF agent (a specific piece of software to read that agreed data). In this way, data already across the system can be read and written to other parts of the system, so that a more complete picture becomes accessible to users. This system is in use in certain local authorities and states in the UK, the United States, and Australia. Within the UK, SIF has now been agreed as the form of data transfer that will be central to national policy (see Becta, 2008). It is clear overall that, until this form of data transfer is more commonly accessible, the scope of the problem is not likely to be known exactly. However, even when this is the case, the forms of data retained, to fulfil specific purposes, will need to be considered carefully.

Tracking and Recording Success as well as Weakness

Perceptions of 'the NEET issue' could well frame the forms of data that are retained in management systems. It is clear from those working within the Connexions and support services currently that tracking and recording success is as important a need as identifying those aspects of weakness that need to be addressed. In the UK, the DCSF has set up a national dedicated website to support young people who are NEET, called Connexions Direct (CXD). As the DCSF (2008b) says, it: "offers advice, support and information for young people aged 13 to 19 on a wide range of issues relevant to their age group. CXD consists of a dedicated website -for online searching on subjects such as careers, learning, health, relationships, finance and leisure activities and a helpline where young people can access information, advice and support via the telephone, text messaging, webchat and email. The helpline operates from 8.00 am until 2.00 am, 365 days a year, offering useful out-ofhours support to complement local provision". However, although this website clearly encourages contact with young people, the ways in which those contacts relate to records created, or how they are integrated with wider uses of a virtual learning environment to support e-portfolio provision, is not indicated there, although a recent report (DCSF, 2009a) states commitment to developing innovative online services, including a growing provision for young people to let others know of their experiences, as well as a commitment to developing wider ranges of facilities to link existing online provision. Currently, and in the future, it is arguable that these forms of issue will need to be met and to be considered at a much more local level, with LAs and Connexions services considering the ways in which specific local technological facilities can be integrated. Some Connexions services have already looked at ways to support their needs, and the needs of young people in these respects. A report by Citizens Online and the National Centre for Social Research report ( 2008) identified an exemplar of use of a form of e-portfolio, called the Virtual Ruksak (allowing young people to record and retain examples of work done in a variety of situations, collected together so that employers and admissions tutors can easily view their achievements). Although these forms of technologies are likely to support these young people, they are not widely available currently. The transfer of records between such systems is also not widespread.

Identifying the Population more Exactly

Connexions services use particular categories to maintain records of young people, which have been provided from the UK government department, for policy reporting purposes. Experience has shown that placing young people within these categories is not always easy or consistent. The category of 'unemployed' can be very wide, and this group is registered as unemployed with Connexions. Some will also be registered unemployed with Job Centre Plus (when they are post 18 years of age). Other young people may be grouped in categories of 'illness' (which in one Connexions service is used for short term illness), 'unavailable on religious grounds', 'not economically active' (which is used for long-term illness over 12 months or for disability), 'unavailable for other reasons' (which is used when it is not clear that any other category is appropriate), 'young carer' (which is concerned with those caring for relatives), 'young parents', 'pregnant', and 'personal development opportunity' (which covers voluntary work that is both paid and unpaid).

These forms of category, while necessary from a government record viewpoint, appear to be of much less value to those supporting the young people. There is currently no consistent recording approach used across Connexions services in the UK, for example, to identify features that will help those who support the young people. Yet important features of these young people are being identified, and could be considered in the formation of appropriate data recording elements. For example, young people who are NEET commonly exhibit limited capabilities to support decision-making (reported by [START_REF] Passey | Assessing the potential of e-learning to support reengagement amongst young people with Not in education, employment or training (NEET) status: An independent research and evaluation study[END_REF]. In addition to this limited capability (in terms of decision making), other factors such as traumatic backgrounds and poor experiences of learning environments can mean that young people who are NEET often do not sustain employment or educational opportunities when these are offered to them. These decision-making limitations tend to be linked for many young people who are NEET to particular ways of thinking. They do not tend to think in abstract or theoretical ways, and value 'real-world' experiences. Many young people who are NEET think along practical lines or creative lines, rather than theoretical or cognitive lines that are more abstract. Young people who are NEET often engage most readily in practical and creative endeavour, and desire involvement in social environments. How these features can be related to recorded details for those who support these young people is a clear need.

Identifying Predictors of the Issues

Some schools have already begun to take preventative action with regard to 'the NEET issue'. Some schools are now involved in identifying young people who are not benefiting from the 'traditional' curriculum. They are involved in providing alternative curriculum opportunities, with the intention of supporting the needs of young people who might have moved into the NEET group. In one school federation, inclusion advocates in each school identify young people (through their behaviour patterns) who might in the future become NEET.

However, this identification of the group of 'potential young people who will become NEET' is based on particular assumptions. These assumptions are not necessarily universally held. Some managers of Connexions services, for example, feel that those young people who become NEET may actually be in a category in school associated with the quiet, low achievers, rather than those identified as disengaging and identifiable through their external behaviour patterns. There is clearly a role for management information systems in helping to explore and address this question; the data held in systems is likely already to enable a study to look at the match of these features. It is generally accepted that there is a need to identify early the characteristics which put young people at risk of becoming long-term NEET. UK government policy (DCSF, 2009b) now expects this group to be supported by alternative curriculum provision (including a range of new more practically focused diplomas accessible to young people 14 to 19 years of age); it is clear that research is needed in order to demonstrate much more precisely what the outcome of this provision will be for the young people. A long-term study of this form is currently being set up in Scotland (Longitudinal Studies Centre, 2009).

Ethical Issues, Positive Intentions and Engagement

Although this paper is not specifically focusing on an in-depth discussion of ethical or wider issues concerned with engagement of young people who are NEET, it is worth highlighting a number of points of pertinence within this context. What is clear from background literature is that young people who are NEET may well have had negative experiences in school situations, have found concentrating within classrooms to be difficult, have sought engagement in training and employment that is practically, creatively or socially based, but have been hampered by their limitations with regard to a range of positive experiences and how to handle decision-making. If numbers of these young people are to be engaged positively (helping them to explore opportunities and find unrecognised areas of interest), then evidence of outcomes so far suggests that there is a need to do more; and that that need should include more positive support, using data management and associated research positively within this context; the purpose of the use of data, and research using data, needs to positively support young people, rather than encroach on their privacy, or individualism, or to 'keep track of them'. Research in this context has a role in identifying answers that can be of benefit to the young people; this paper does not argue for widespread tracking, or for an encroachment that could lead to a generation of antagonism. In that respect there is a need to consider individuals, as well as to consider protection of data, appropriate use of data, and appropriate elimination of data when answers have been identified. While the rights of individuals need to be upheld, a further dimension to consider is the financial support that agencies and governments provide; it is arguable that those who provide funds have a right to know how those funds are used and what outcomes arise.

Research, Policy and Practice Needs

Evidence indicates that roles for management information systems in supporting 'the issue of NEETs' are well and widely recognised, but that implementation, practice and features of use are only at early stages. The current situation indicates that there are some key potential roles in the areas of research, policy, and practice.

In terms of policy, key questions with regard to data management systems relate to the ways in which data transfer will be enabled, the details recorded to enable an understanding of scope, the forms of presentation that will be of value to different users of the system, how systems will track and record successes as well as weaknesses, how the details presented will be of direct value to those supporting the young people (or to the young people themselves even more ideally), and how the data systems will provide timely data to support prevention of severe situations arising. In terms of practice, key questions with regard to data management systems relate to the ways in which data will be accessible to the user, the details recorded to enable an understanding of scope within their localities, the forms of presentation that will offer value to that particular user, how systems will provide a view of the young people's successes as well as their weaknesses, and how the data systems will provide alerts of severe situations arising. In terms of research, key questions with regard to data management systems relate to the ways in which data can provide an exploration of scope of the issues, how the data can enable a more thorough understanding of the population and how support will relate to those features that are identified, how systems will provide a view of the young people's successes as well as their weaknesses to enable measures of success that can be made more widely accessible to the profession, and how the data systems can provide opportunities to explore predictors of 'the issues of NEETs', so that young people can gain from positive, supportive intervention. Data management systems have the potential to offer substantial future value to those concerned with 'NEETs issues' in policy, practice and research areas. Although there is likely to be a range of technological issues that will need to be addressed if they are to fulfil their widest potential, even exploring uses of current data management systems to look at some of the many assumptions that exist are likely to be fruitful at this stage.

Acknowledgements

The author would like to thank Becta for their funding of key elements of the background research reported in this paper. This paper does not, however, represent any official statement or view of Becta.


Anass Bouchnita
email: anass.bouchnita@univ-lyon1.fr

Fatima-Ezzahra Belmaati
email: fati.belmaati@gmail.com

Rajae Aboulaich
email: aboulaich@emi.ac.ma

Mark J Koury
email: mark.koury@vanderbilt.edu

Vitaly Volpert
email: volpert@math.univ-lyon1.fr

A Hybrid Computation Model to Describe the Progression of Multiple Myeloma and Its Intra-Clonal Heterogeneity

Keywords: multiple myeloma, intra-clonal heterogeneity, hybrid model

Multiple myeloma (MM) is a genetically complex hematological cancer that is characterized by proliferation of malignant plasma cells in the bone marrow. MM evolves from the clonal premalignant disorder monoclonal gammopathy of unknown significance (MGUS) by sequential genetic changes involving many different genes, resulting in dysregulated growth of multiple clones of plasma cells. The migration, survival, and proliferation of these clones require the direct and indirect interactions with the non-hematopoietic cells of the bone marrow. We develop a hybrid discrete-continuous model of MM development from the MGUS stage. The discrete aspect of the model is observed at the cellular level: cells are represented as individual objects which move, interact, divide, and die by apoptosis. Each of these actions is regulated by intracellular and extracellular processes as described by continuous models. The hybrid model consists of the following submodels that have been simplified from the much more complex state of evolving MM: cell motion due to chemotaxis, intracellular regulation of plasma cells, extracellular regulation in the bone marrow, and acquisition of mutations upon cell division. By extending a previous, simpler model in which the extracellular matrix was considered to be uniformly distributed, the new hybrid model provides a more accurate description in which cytokines are produced by the marrow microenvironment and consumed by the myeloma cells. The complex multiple genetic changes in MM cells and the numerous cell-cell and cytokine-mediated interactions between myeloma cells and their marrow microenviroment are simplified in the model such that four related but evolving MM clones can be studied as they compete for dominance in the setting of intraclonal heterogeneity.

Introduction

Multiple myeloma (MM) is initiated through the acquisition of genetic changes that transform the plasma cells from normal to malignant. These changes often result in the development of selective advantage, leading to the excessive proliferation of myeloma cells. The development of MM leads to several harmful clinical conditions including anemia, renal failure, recurrent infections, hypercalcemia, and osteoporosis with bone fractures. These pathological conditions can frequently result in the death of the patient.

The evolution of myeloma from the clonal premalignant plasma cell condition termed monoclonal gammopathy of unknown significance (MGUS) through the intermediate stage of smoldering myeloma to the malignant MM stage is mediated by multiple sequential genetic changes, including chromosomal translocations, hyperdiploidy, and mutations, which permit independent growth and spread of MM in the bone marrow [START_REF] Palumbo | Multiple myeloma[END_REF][START_REF] Morgan | The genetic architecture of multiple myeloma[END_REF]. A wide variety of genetic changes involving many different genes have been documented in MM cases. In addition to these sequential genetic changes, MM cells require specific interactions with the non-hematopoietic cells of the bone marrow including the stromal cells (BMSCs), osteoblasts, osteoclasts, and cells associated with vascular supply of the marrow [START_REF] Palumbo | Multiple myeloma[END_REF][START_REF] Morgan | The genetic architecture of multiple myeloma[END_REF]. These marrow microenvironment interactions with the MM cells include direct binding of cell surface adhesion proteins and their binding partners on the other cell types as well as diffusible, soluble molecules that are secreted by one cell type and are bound and internalized by another cell type. These soluble molecules are often chemokines and cytokines produced by other marrow cells, and their receptors are expressed on the MM cells. However, specific molecules produced by MM cells indirectly affect growth of the malignancy by inducing localized resorption of bone, death of hematopoietic cells, and expansion of vessels supplying blood to the MM. The combined effects of sequential genetic changes within the evolving myeloma cells and interactions of the MM cells with the marrow microenvironment affect the migration, proliferation, and survival of the MM cells.

There are three types of mathematical models of tumor growth and intra-clonal heterogeneity. The first type is the continuous model in which partial and ordinary differential equations are used to describe the density of premalignant and malignant cells [START_REF] Adam | A simplified mathematical model of tumor growth[END_REF][START_REF] Glass | Instability and mitotic patterns in tissue growth[END_REF][START_REF] Mcelwain | Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth[END_REF]. Some continuous models study the effects of nutrients and inhibitors on tumor growth that may or may not be accompanied by necrosis [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], while others focus on the nonlinear three-dimensional spatial aspects of tumors [START_REF] Wise | Three-dimensional multispecies nonlinear tumor growth-I: Model and numerical method[END_REF]. The question of clonal competition within tumors has been studied by other continuous models [START_REF] Stiehl | Clonal selection and therapy resistance in acute leukaemias: Mathematical modelling explains different proliferation patterns at diagnosis and relapse[END_REF][START_REF] Walenda | Feedback Signals in Myelodysplastic Syndromes: Increased Self-Renewal of the Malignant Clone Suppresses Normal Hematopoiesis[END_REF], and the role of clonal heterogeneity in drug resistance has been previously explored [START_REF] Panetta | A mathematical model of drug resistance: Heterogeneous tumors[END_REF]. While these continuous models sustain mathematical analysis and provide key insights in the development of tumors, they do not describe the cell-cell interactions and intracellular regulation of single cells. The second type of mathematical models describing tumor growth is the discrete model which usually represents cells as individual objects that interact with each other. These interactions determine the fate of each cell [START_REF] Basanta | Studying the emergence of invasiveness in tumours using game theory[END_REF][START_REF] Enderling | Migration rules: Tumours are conglomerates of self-metastases[END_REF][START_REF] Piotrowska | A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth[END_REF][START_REF] Drasdo | A single-cell-based model of tumor growth in vitro: Monolayers and spheroids[END_REF][START_REF] Shirinifard | 3D Multi-Cell Simulation of Tumor Growth and Angiogenesis[END_REF][START_REF] Swat | Emergent Stratification in Solid Tumors Selects for Reduced Cohesion of Tumor Cells: A Multi-Cell, Virtual-Tissue Model of Tumor Evolution Using CompuCell3D[END_REF]. Cellular automatons provide a simple yet powerful tool to understand the mechanisms of tumor growth [START_REF] Hatzikirou | Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion[END_REF][START_REF] Aubert | A cellular automaton model for the migration of glioma cells[END_REF]. The discrete models usually describe cells in on-lattice or off-lattice grids. They focus on the interactions between cells and how they lead to the development of tumors, but discrete models do not link the different processes taking place at different scales. The third type of tumor growth model, which is known as a hybrid model, combines the features of both continuous and discrete models, thereby benefiting from the advantages of both of them [START_REF] Ramis-Conde | Modeling the influence of the E-cadherin-β-catenin pathway in cancer cell invasion: A multiscale approach[END_REF][START_REF] Ramis-Conde | Multi-scale modelling of cancer cell intravasation: The role of cadherins in metastasis[END_REF][START_REF] Zhang | Multiscale agent-based cancer modeling[END_REF]. In hybrid models, the concentrations of intracellular and extracellular proteins are described by partial and ordinary differential equations while the cells are introduced as individual objects. The heterogeneous nature of tumors has been studied by some of these models [START_REF] Anderson | A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion[END_REF][START_REF] Fang | Adaptation to hypoxia and acidosis in carcinogenesis and tumor progression[END_REF][START_REF] Vincent | An evolutionary model for initiation, promotion, and progression in carcinogenesis[END_REF][START_REF] Chisholm | Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation[END_REF]. The models describing MM development usually study the impact of MM on different biological systems such as bone remodeling [START_REF] Ayati | A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease[END_REF] and erythropoiesis [START_REF] Bouchnita | Bone marrow infiltration by multiple myeloma causes anemia by reversible disruption of erythropoiesis[END_REF].

In this work, we develop a hybrid model describing the development and intra-clonal heterogeneity in MM as described in Walker et al. [START_REF] Walker | Intraclonal heterogeneity and distinct molecular mechanisms characterize the development of t(4;14) and t(11;14) myeloma[END_REF]. In this model, malignant cells are represented by elastic spheres that can move, grow, interact, divide and die by apoptosis. Their fate is determined by intracellular and extracellular regulation networks. The growth of the cellular radii between cell divisions and the formation of two daughter cells with each division lead to the expansion of the tumor. To investigate the role of the bone marrow (BM) microenvironment in the growth of myeloma tumors, we specifically consider another population of cells, the bone marrow stromal cells (BMSCs). These cells secrete cytokines such as stromal cell-derived factor 1 (SDF-1), insulin-like growth factor 1 (IGF-1),and interleukin 6 (IL-6) which are necessary to the survival, homing and growth of malignant plasma cells. The concentration of each of these cytokines is modelled by a reaction-diffusion equation. The intracellular regulation of malignant cells is represented by a system of ordinary differential equations which depends on the concentration of extracellular cytokines. We characterize each cell by a specific genotype, and we consider that after its division, daughter cells will either inherit the same genotype or acquire a slightly different one due to random mutations. As the result, aggressive clones emerge during tumor progression in a parallel pattern. Although experiments show the possibility of clones emerging in both linear and branching evolutionary patterns, we restrict this study to the parallel evolutionary case. The MM cells consume cytokines, leading to competition between clones for the available resources. We study the dynamics of clonal competition and its role in the progression of MM.

Hybrid Model of MM Development

We developed a hybrid discrete-continuous model of MGUS progression to MM. The discrete aspect of the model is observed at the cellular level: cells are represented as individual objects which move, interact, divide, and die by apoptosis. Each of these actions is regulated by intracellular and extracellular processes described by continuous models. The hybrid model consists of the following submodels: cell motion due to chemotaxis, intracellular regulation of plasma cells, extracellular regulation in the bone marrow, and acquisition of mutations upon cell division. It is an extension of a previously developed simpler model [START_REF] Bouchnita | Mathematical modelling of intra-clonal heterogeneity in multiple myeloma[END_REF]. While the previous model considers the cytokines in the extracellular matrix to be uniformly distributed, the present study provides a more accurate description by considering that these cytokines are produced by the BMSCs and consumed by the myeloma cells. Furthermore, a more detailed description of the intracellular pathways regulating the fate of plasma cells is provided in the present model.

The model is based on the direct effects of sequential genetic changes and marrow microenvironmental chemokine and cytokine activity that influence the chemotaxis, proliferation and survival of MM, but does not include the MM effects on bone resorption, hematopoietic cell loss, or development of specialized vasculature. The complex multiple genetic changes in MM cells and the numerous cell-cell and cytokine-mediated interactions between myeloma cells and their marrow microenviroment are simplified in the model so that four related but evolving clones develop in a process termed intra-clonal heterogeneity [START_REF] Brioli | The impact of intra-clonal heterogeneity on the treatment of multiple myeloma[END_REF][START_REF] Melchor | Single-cell genetic analysis reveals the composition of initiating clones and phylogenetic patterns of branching and parallel evolution in myeloma[END_REF] (See Figure 1). Competition among these four MM clones is based on differences in cellular growth and survival rates and interactions with the marrow microenvironment. This competition results in predominance of the more fit clones and decline and ultimate extinction of the less fit ones. An early event in the MM model (Figure 1a) is a standard-risk genetic change, a t (11;14) translocation that involves the immunoglobulin heavy chain switch region on chromosome 14, inducing overexpression of the gene encoding cyclin D1, a regulator of cell cycle progression located on chromosome 11 [START_REF] Chesi | Dysregulation of cyclin D1 by translocation into an IgH gamma switch region in two multiple myeloma cell lines[END_REF]. This t(11;14) clone has deregulated cell cycle progression due to the translocation, and it undergoes two separate secondary genetic events: mutations involving the oncogenes N-RAS and K-RAS, which are common secondary events in the development of multiple myeloma [START_REF] Morgan | The genetic architecture of multiple myeloma[END_REF]. The resultant clone with t(11;14)/mutant N-RAS has significantly enhanced proliferation compared to the parent t(11;14) clone, while the t(11;14)/ mutant K-RAS clone has much less of an increase in proliferation relative to the t(11;14) parental clone. Thus, the two descendant clones have differing degrees of increased RAS activity, resulting in a proliferative advantage for the t(11;14)/mutant N-RAS clone compared to both the t(11;14) parental clone and t(11;14)/ mutant K-RAS clone. A subsequent genetic event in the t(11;14)/K-RAS clone is, however, a mutation in the gene encoding interferon regulatory factor 4 (IRF4). Mutant IRF4 is a protein that can enhance survival and proliferation of MM cells. IRF4 mutation has been associated with mutations of N-RAS or K-RAS [START_REF] Melchor | Single-cell genetic analysis reveals the composition of initiating clones and phylogenetic patterns of branching and parallel evolution in myeloma[END_REF], and it allows the t(11;14)/K-RAS mutant/IRF4 mutant clone in the model to compete more successfully with the t(11;14)/N-RAS clone than either the parent t (11;14) Among the many possible microenvironmental factors that may influence the growth of MM in the marrow, the simplified model includes the chemokine stromal cell-derived factor 1 (SDF-1). It is produced by multiple cell types but mainly by BMSCs. Other extracellular cytokines included in the model are interleukin 6 (IL-6) and insulin-like growth factor 1 (IGF-1), which are produced by other types of non-hematopoietic cells in the marrow as well as BMSCs. The marrow stromal cells are considered as sources of SDF-1, IL-6, and IGF-1 in the model (Figure 1a). SDF-1, through its receptor CXCR4, mediates the homing of circulating myeloma cells to the marrow and their migration within the marrow space [START_REF] Bouyssou | Targeting SDF-1 in multiple myeloma tumor microenvironment[END_REF][START_REF] Vanderkerken | Homing of the myeloma cell clone[END_REF]. IL-6 and IGF-1 induce multiple effects after binding to their respective specific surface receptors on the MM cells, but in the model we restrict our study to their effects on the RAS/Extracellular signal-regulated kinases (RAS/ERK) pathway that promotes proliferation and the phosphatidylinositol-3 kinase/protein kinase B/Forkhead in rabdomyosarcoma (Akt/FKHR) pathway that regulates apoptosis [START_REF] Morgan | The genetic architecture of multiple myeloma[END_REF] (Figure 1a).

Cell Motion

We represent each cell as an elastic sphere. It contains two parts: an incompressible part and a compressible one. As the cell grows, its radius increases and hence it pushes the neighboring cells. The motion of each cell is determined by Newton's second law. Let us denote the cell number by i. Then we have the following equation for the coordinate x i of the center of the ith cell:

m ẍi + mµ ẋi -∑ j =i f ij -f ch = 0, ( 1
)
where m is the mass of cell, µ is the friction coefficient, f ij is the interaction force between the cells i and j, f ch is the chemotactic force which depends on the concentration of SDF-1. We consider f ij in the following form:

f ij =      K h 0 -hij h ij -(h 0 -h 1 ) , h 0 -h i < h ij < h 0 0, h ij ≥ h 0 . ( 2
)
Here h ij is the distance between the center of the cells i and j, h 0 is the sum of their radii, K is a positive parameter and h 1 represents the incompressible part of each cell. The numerical implementaton of Newton's equation is presented in Appendix A.1. The chemotactic force represents the motion of the cell in response to SDF-1 stimulus. Let us denote the concentration of SDF-1 by S. Then the expression of f ch is given by:

f ch = κ∇S,
where κ is a positive constant and ∇S denotes the SDF-1 gradient.

Extracellular Regulation

We consider a square two-dimensional computational domain with the side equal to 1000 microns. The hybrid model contains two types of cells: myeloma cells and BMSCs. The former have a diameter of 10 microns while the latter are considered to have six-fold larger radii as observed in experimental data. Furthermore, they secrete cytokines that are necessary for the survival and proliferation of myeloma cells such as SDF-1, IL-6, and IGF-1. Let us denote the total concentration of the last two cytokines by I and the concentration of SDF-1 by S. Assuming that these cytokines have the same diffusion coefficient, production rate, consumption rate, and degradation rate, we describe their concentrations in the extracellular matrix by:

∂I ∂t = D∆I + W -∑ i λI -σI, ( 3
)
∂S ∂t = D∆S + W -∑ i λS -σS, ( 4
)
where D is the diffusion coefficient, W the production factor, λ the consumption rate by each myeloma cell, and σ the degradation rate. We set the Dirichlet boundary condition I = 0 and S = 0 at all boundaries. We have prescribed zero Dirichlet boundary condition to represent the local dynamics of tumor growth in a bone marrow site surrounded by adipose tissue. The finite difference method was used to implement the reaction-diffusion equations. The used numerical methods and implementation algorithm are presented in Appendix A.2.

Intracellular Regulation

Although myeloma cells are genetically complex [START_REF] Morgan | The genetic architecture of multiple myeloma[END_REF], we will restrict the intracellular regulation to two pathways (Figure 1a). These pathways are the RAS/ERK pathway which is responsible for cell proliferation and the Akt/FKHR pathway which regulates its survival. Other pathways such as the janus kinase/signal transducer and activator of transcription (JAK/STAT) pathway are not considered in the model. We have chosen these two pathways in order to study the role of aquired RAS mutations in the progression of MM and how it is affected by the extracellular matrix. Let us denote the concentrations of ERK, Akt, and FKHR as e, a, and f , respectively. We describe these concentrations inside each cell as follows:

de dt = α 1 (z)(κI) -β 1 e, da dt = α 2 (κI) -β 2 a, d f dt = α 3 -β 3 a f -γ 3 f . ( 5
)
where α i and β i , i = 1, 2, 3 and γ 3 are positive constants. The coefficient α 1 (z) depends on the genotype z, and this coefficient varies depending on the effect of RAS mutations. The cell will die by apoptosis if f > f * during its lifetime cycle. If the myeloma cell survives, then it will self-renew, if e > e * by the end of its life cycle. Otherwise, the cell will die by apoptosis. We consider that the proliferation threshold e * depends on the IRF4 gene expression. The values of both intracellular and extracellular regulation parameters are provided in Appendix B.

Myeloma Cells Division and Mutations

In the present study, we investigate the progression of MGUS into MM through RAS mutations [START_REF] Walker | Intraclonal heterogeneity and distinct molecular mechanisms characterize the development of t(4;14) and t(11;14) myeloma[END_REF]. We characterize each cell by an aggressiveness phenotype which represents the cumulative effects of acquired RAS mutations. We introduce an aggressiveness phenotype function f (z) resulting from the RAS genotype for each cell. After its division, the daughter myeloma cell will keep the genome of the mother cell with a probability of one-third. Otherwise, it will acquire a mutation and either increases or diminishes its aggressiveness by a positive value . We represent the genotype and resultant RAS activation that determine aggressiveness as due to describing the frequency and effects of RAS mutations in the resulting clones in Figure 2a. Furthermore, we assume that the ERK threshold for cell division is reduced upon the mutation of the IRF4 gene as shown in Figure 2b. We consider four clones of MM denoted by c 1 , c 2 , c 3 , and, c 4 (Figure 1b). The clone c 1 is the initial clone which does not harbour any RAS mutations. The clone c 2 is more aggressive than clone c 1 . It consists of MM cells that acquired the N-RAS gene mutation. The clones c 3 and c 4 belong to a lineage that is independent from the clone c 2 . They emerge when the cells of c 1 acquire the K-RAS gene mutation for clone c 3 and an additional IRF4 gene mutation for clone c 4 . The latter clone is therefore more aggressive than the former. We show the relationships between these different clones in Figure 1b.

Results

Initially, there are 47 myeloma cells in a bone marrow site containing a single BMSC, although the initial number of these cells is not essential for the results of the simulations. All of these malignant cells are considered to be tumor-initiating cells because they belong to the same initial clone. The motion of the malignant cells depends on their distance from the BMSC. Closer cells move faster than farther ones due to the increase in SDF-1 gradient near the BMSC. During their motion, myeloma cells consume the cytokines necessary for their survival such as IL-6 and IGF-1. These cytokines promote survival by activating the RAS/ERK pathway, and dowregulating the Akt/FKHR pathway. Depending upon its initial distance from the BMSC, the myeloma cell will either survive and get closer to the BMSC or it will die by apoptosis. The surviving myeloma cells will surround the BMSC and form the tumor niche. Initially, this niche consists of cells belonging to the same clone. After some time, other subclones will emerge in the course of tumor growth. We have shown the different steps of myeloma cells homing to a BMSC in Figure 3. The initial condition for the simulation is shown in Figure 3a. Due to the low number of myeloma cells in the beginning of the simulation, the BMSC secretes a high concentration of the chemokine SDF-1 which attracts the myeloma cells located close to it (Figure 3b). The surviving myeloma cells adhere to the BMSC and surround it while other cells divide and remain farther away (Figure 3c). During this process, cells whose aggressiveness phenotype belongs to the intermediate state between the clones c 1c 2 and c 1c 3 emerge (Figure 3d) and the global concentration of SDF-1 starts to reach stability due to its consumption by myeloma cells (Figure 4a). MGUS progresses to aggressive MM through the acquisition of random mutations by myeloma cells. In this respect, more aggressive subclones emerge in the course of MGUS progression due to the acquisition of RAS-associated mutations. The aggressive subclones need lower concentrations of IL-6 and IGF-1 to survive, resulting in the expansion and the persistence of the tumor (Figure 4b). Because of their rapid proliferation, they crowd out other clones and consume most of the available resources. Ultimately, these less aggressive subclones remain relatively stable in growth pattern. Still, a few of them manage to survive in proximity to the BMSC where there is a higher concentration of the IL-6 and IGF-1 cytokines, while the more aggressive subclones occupy the outer region of the tumor. The different steps of tumor development and intraclonal competition are shown in Figure 5. As the tumor progresses, the concentrations of IL-6 and IGF-1 become stable due to their consumption by the growing number of myeloma cells. As a result, the aggressive subclones expand to the detriment of the less aggressive ones. By the end of the simulation, we see that the tumor size is stable but the competition between clones is still in progress (Figure 5c vs. Figure 5d). The emergence of more aggressive clones not only reduces the populations of the less aggressive ones but also increases the total number of malignant cells. These clones survive in areas farther from the BMSC, making the tumor expand. We show the total number of malignant cells over time in Figure 4b. The proportions of each clone in the total population show that in the end of the simulation, the clone c 1 only represents ≈5% of the total population. The subclone c 2 is predominant, with ≈60% of cells (Figure 6). Overall, the global population of malignant cells increases with the emergence of more resistant subclones. The emergence of the subclone c 2 leads to an increase in the number of malignant cells because the cells belonging to this subclone can survive far from the BMSCs. The subclone c 3 emerges few hours after c 2 but barely manage to survive because of limited resources. It gives rise to the subclone c 4 , which is as aggressive as the subclone c 2 due to the additional IRF4 mutation. As the tumor progresses, these two subclones c 2 and c 4 become predominant because they are better adapted to survive in sites with limited resources. By the end of the simulation, the number of malignant cells oscillates around a stable value as well as the total concentration of cytokines in the domain (Figure 4). We show the percent of the malignant cell population occupied by each clone over time in Figure 6.

Discussion

Bone marrow stromal cells play an important role in the pathogenesis of MM. After extravasation into the bone marrow, MGUS and myeloma cells migrate and home to the areas surrounding BMSCs. This homing process is mediated by the chemokine SDF-1. It promotes the chemotaxis and homing of myeloma cells upon the interaction with its receptor CXCR4 [START_REF] Hideshima | Understanding multiple myeloma pathogenesis in the bone marrow to identify new therapeutic targets[END_REF]. However, it is crucial for myeloma cells to be sufficiently close to BMSCs to be attracted by the secreted SDF-1. Otherwise, they will die if they are left without enough resources to survive and divide. The progression of MGUS to MM with t(11;14) translocation is marked by the emergence of new aggressive subclones which is crucial for the expansion of the tumor because it allows its adaptation to limited resources. As the tumor progresses, the aggressive subclones become predominant because lower concentrations of IL-6 and IGF-1 allow them to survive.

To study the development of MM tumors and its intraclonal heterogeneity, we developed a specific hybrid discrete-continuous model. The model was able to reproduce the experimental results presented in [START_REF] Walker | Intraclonal heterogeneity and distinct molecular mechanisms characterize the development of t(4;14) and t(11;14) myeloma[END_REF]. We have used it to investigate the central role of the BMSCs in myeloma cell homing and the progression of MGUS to MM. BMSCs participate in the homing of myeloma cells and provide them with the necessary cytokines for their survival and proliferation. Numerical simulation results suggest that the initial distance between the BMSC and infiltrating myeloma cells is of paramount importance for the survival of the latter. After their homing, new, more aggressive myeloma subclones emerge due to the acquisition of RAS mutations. Other gene mutations such as the one acquired by the IRF4 gene further increase the aggressiveness of some of these subclones. They compete with each other for the available cytokines, resulting in the predominance of the more fit subclones. To better quantify the heterogeneity of clones during MM progression, we show the kernel density plots at different stages of MGUS to MM progression in Figure 7. The x-axis shows the aggressiveness phenotype function scaled from 0 to 1. These results show the predominating subclones at the different stages of tumor progression. Our findings suggest that the total number of malignant cells oscillates around a stable value after a few weeks of MM development in agreement with the in vivo experiments conducted in [START_REF] Rozemuller | A bioluminescence imaging based in vivo model for preclinical testing of novel cellular immunotherapy strategies to improve the graft-versus-myeloma effect[END_REF]. However, we speculate that MM tumors can further expand due to other mechanisms such as stimulation of IL-6 and IGF-1 production by BMSC, the migration to other BMSCs [START_REF] Manier | Bone marrow microenvironment in multiple myeloma progression[END_REF], or the emegence of more resistant clones due to other mutations. The multi-scale model follows a systems biology approach [START_REF] Kitano | Systems Biology: A brief overview[END_REF] by integrating different interfering biological processes in one model. To understand complex phenomena such as MGUS progression to MM, it is important to properly use available data in describing events at the single cell level where processes such as gene expression and mutations take place as well as the larger tissue level where each cell interacts with its environment. Systems biology approaches focus on the whole system rather than the sum of its components. Similarly, this study is more focused on the behavior of the global model of MM homing and tumor growth than the various individual processes which regulate it.

Although the model presented here was able to reproduce the experiments describing MM intra-clonal heterogeneity, it has several limitations. Firstly, the intracellular regulation network was limited to two pathways while in reality MM is more biochemically complex [START_REF] Morgan | The genetic architecture of multiple myeloma[END_REF]. Furthermore, the number of considered mutations is also reduced compared to those found in MM cases. Interactions of the myeloma cells and BMSCs with other components of the bone marrow such as the osteoblasts and osteoclasts were not included in the present model. Other mechanisms affecting myeloma cell proliferation were introduced implicitly as a random perturbation of the cell cycle in the myeloma cells. The simplification of the myeloma cell transduction pathways and mutations included was due to limitations of computational power and the difficulty of studying complex models. We have limited the model to two pathways of intracellular regulation because we wanted to study the effect of cytokines IL-6 and IGF-1 on the fate of the myeloma cells and how that fate is affected by RAS mutations and the IRF4 mutation that is associated with mutant RAS genes. Also, we assumed that IL-6 and IGF-1 play the same role in the activation of the ERK/RAS and Akt/FKHR pathways. This hypothesis was made because it is difficult to distinguish the interfering actions of the different cytokines in the cell culture experiments. Finally, the study was restricted to a site of the bone marrow containing a single BMSC. Other configurations with more BMSCs and other marrow cell types can be considered as well. In future works, we will introduce the action of treatment and study the role of the intraclonal heterogeneity in MM drug resistance.

We split the first equation of the problem (P 1 ) into two sub-steps as follows: .

     S n+1
Therefore, we can write the first equation of the system (A2) in the form: a i,j S n+1/2 i-1,j + b i,j S n+1/2 i,j + c i,j S n+1/2 i+1,j = f i,j , For each fixed j, j = 1, 2, ..., N y -1, we solve numerically: with the boundary conditions S n i=0 = 0, S n i=Nx = 0. We solve the Equation (A3) using Thomas algorithm. For that, we write the left boundary condition S i=0 = 0 as follows:

0 = L 1/2 S 1 + K 1/2 ,
where L 1/2 = 0 and K 1/2 = 0. Then, from the Equation (A3) for i = 1: . We continue for i = 2, 3, ..., N x -1:

S i = L i+1/2 S i+1 + K i+1/2 , ( A4
)
where

L i+1/2 = -c i b i +a i L i-1/2 , K i+1/2 = f i +a i K i-1/2 b i +a i L i-1/2
. We first obtain the coefficients L i+1/2 , K i+1/2 using the Equation (A3). Next, we find the solution S n+1/2 for the sub-step n + 1/2 by backward sweep using the Equation (A4). We apply the same procedure on the second equation of the system (A2) to compute the next step solution S n .

  clone or the t(11;14)/K-RAS mutant clone.

Figure 1 .

 1 Figure 1. (a)The intracellular regulation of myeloma cells as described in the model. Bone marrow stromal cells (BMSCs) secrete the cytokines insulin-like growth factor 1 (IGF-1), interleukin-6 (IL-6), and chemokine stromal cell-derived factor 1 (SDF-1) which are necessary to the survival, homing and proliferation of myeloma cells. Via their respective receptors, IGF-1 and IL-6 activate the RAS/ERK pathway which promotes the cell proliferation. They inhibit apoptosis through the phosphatidylinositol-3 kinase/protein kinase B/Forkhead in rabdomyosarcoma (Akt/FKHR) pathway. The cell migrates and homes to BMSCs through the SDF-1/CXCR4 axis. The IRF4 mutation, which has been associated with concomitant RAS mutations, promotes survival and proliferation. BMSCs, which are much larger cells than the myeloma cells, are shown in reduced in size in this figure as well as the receptors of both IGF-1 and IL-6; (b) The parallel evolution pattern of multiple myeloma clones resulting in intra-clonal heterogeneity. More aggressive clones result from a more aggressive N-RAS mutation in clone 2 or the acquisition of IRF4 mutation in addition to the less aggressive K-RAS in clone 4. Each clone is shown by its corresponding color in the model.

Figure 2 .

 2 Figure 2. (a) The activation rate of RAS protein as a function of the genotype function z; (b) The ERK threshold for division as a function of the genotype function z, it decreases due to the IRF4 mutation found in the clone c 4 .

Figure 3 .

 3 Figure 3. The sequential steps of myeloma cells homing to a BMSC. Myeloma cells are represented by the cells with the smaller radii and the BMSC is the large cell in the middle. Each clone of multiple myeloma (MM) cells is denoted by a specific color. The red color gradient represents the summed concentrations of the cytokines SDF-1, IL-6, and IGF-1 shown with white (0) to red (1). (a) After their infiltration, the myeloma cells move towards the BMSC; (b) The surviving cells surround the BMSC; (c) The myeloma cells divide and form the tumor niche; (d) The tumor expands and more aggressive subclones start emerging.

Figure 4 .

 4 Figure 4. (a) The total concentration of the IL-6 and IGF-1 cytokines over time in log scale; (b) The total number of malignant cells over time.

Figure 5 .

 5 Figure 5. Snapshots of a simulation showing the competition between clones. (a) The tumor consists mainly of the clone c 1 (shown in yellow) with few cells in the intermediate state (in purple) between clones and the emergence of the clone c 2 (in cyan) in the sides. The size of the tumor remains limited because clone c 1 cells need relatively high concentration of I to survive; (b) Compared to the clone c 1 , the clone c 2 cells expand and survive in areas with lower concentration of cytokines; (c) The clone c 2 cells surround the tumor and crowd out the cells of the clone c 1 leading to the reduction of their population. The clones c 3 (in blue) and c 4 (in magneta); (d) The subclone c 4 is as aggressive as the subclone c 2 due to the additional IRF4 mutation and it manages to coexist with it in the remote areas with fewer cytokines.

Figure 6 .

 6 Figure 6. The proportion of each clone in the total population over time.

Figure 7 .

 7 Figure 7. Gaussian kernel density plots indicating the frequency of cells acquiring each specific mutation at the different stages of the simuation. (a) The initial population composed of cells belonging to the subclone c 1 ; (b) c 2 is the most predominant subclone after 16 hours of the beginning of the simulation; (c) The K-RAS subclones c 3 and c 4 emerge and start expanding; (d) The clones c 2 and c 4 represent the majority of the population because they are equally aggressive.

a i S n+1/ 2 i- 1 + b i S n+1/ 2 i+ c i S n+1/ 2 i+1=

 2122 f i , ∀i = 1, 2, ..., N x -1 (A3)

a 1 S n+1/ 2 0+ b 1 S n+1/2 1 + c 1 S n+1/2 2 = f 1 1 = L 3 / 2 S 2 = -c 1 b 1 , K 3 / 2 = a 1 S 1 ,0 -f 1 -b 1

 12111211322132111 1,j + K 3/2 ,where we denote L 3/

  We solve the first equation for each fixed j to obtain S n+1/2 . Next, we solve the second to obtain S n . Let us consider the first equation:

	/2 δt/2 -S n i,j i,j -S n i,j S n+1 i+1/2,j δt/2	= D = D S n+1/2 i-1,j -2S n+1/2 i,j h 2 S n+1 i-1,j -2S n/2 i,j +S n+1 +S n+1/2 i+1,j i+1,j h 2 + D + D S n+1/2 S n i,j-1 -2S n i,j +S n i,j+1 h 2 i,j-1 -2S n+1/2 i,j i,j+1 h 2 +S n+1/2 + W -σS n+1/2 i,j + W -σS n+1 i,j . ,	(A2)
	S n+1/2 i,j δt/2 -S n i,j	= D	S n+1/2 i-1,j -2S n+1/2 i,j h 2	+ S n+1/2 i+1,j	+ D	S n i,j-1 -2S n i,j + S n i,j+1 h 2	+ W -σS n+1/2 i,j	.
	Rearranging the terms we obtain:
		D h 2	S n+1/2 i-1,j + -	2D h 2 -	1 δt/2	-σ	S n+1/2 i,j	+	D h 2	S n+1/2 i+1,j =
		a i,j						b i,j	c i,j
				-	S n i,j δt/2	-D	S n i,j-1 -2S n i,j + S n i,j+1 h 2	-W
									f i,j

Computation 2017,[START_REF] Mcelwain | Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth[END_REF][START_REF] Shirinifard | 3D Multi-Cell Simulation of Tumor Growth and Angiogenesis[END_REF]

Acknowledgments:

The last author was partially supported by the Russian Science Foundation (Grant no. 15-11-00029).

Author Contributions: All authors of the paper have contributed to the study design. A.B., M.J.K and V.V. conceived and developed the hybrid model. A.B., F.-E.B. and R.A. performed the numerical simulations. A.B., F.-E.B., R.A., M.J.K and V.V. wrote the paper and analyzed the results. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

Appendix A.1. Equations for Cell Motion and Intracellular Regulation

We solve the following problem which describes the motion of each cell:

First, we write the late equation as a system of displacement x i and velocity v i :

We first determine the velocity using the first equation of the system, then we compute the displacement of the cell using the second equation. The forces term were introduced implicitly at each timestep. The equations for intracellular regulation are solved using the Euler explicit scheme.

Appendix A.2. Reaction-Diffusion Equations for Extracellular Cytokines Concentration

We consider a rectangular two-dimensional domain Ω and we denote its boundaries by Γ. We use the alternating direction implicit method to solve the following problem:

where D is the diffusion coefficient, W and σ are the production and degradation rates respectively, and φ(x, y) represents the initial condition condition for I. We consider the two-dimensional grid (x i , y j , t n ) = (ih, jh, nδt), where h and δt are the space and time steps respectively and we discretize it: i = 1, 2, ..., N x and j = 1, 2, ..., N y . We first rewrite the problem (P) in the following form:

Appendix B. Parameter Values

Because of the simplifications made to the regulation network of myeloma cells, the parameters were fitted to reproduce the experimental results. The average cell cycle of the cancerous cells was considered to be stochastic. Both the extracellular and intracellular proteins concentrations are considered to be nondimensional.