A two-phase model of solid tumour growth

Many solid tumour growth models are formulated as systems of parabolic and/or hyperbolic equations. Here an alternative, two-phase theory is developed to describe solid tumour growth. Versions of earlier models are recovered when suitable limits of the new model are taken. We contend that the multiphase approach represents a more general, and natural, modelling framework for studying solid tumour growth than existing theories.     

A mathematical analysis of a model for capillary network formation in the absence of endothelial cell proliferation

An analysis of a parabolic partial differential equation modelling capillary network formation is presented. The model includes terms representing cell random motility, chemotaxis, and haptotaxis due to the presence of chemical stimuli: tumour angiogenic factors and fibronectin. The analysis provides an underlying insight into mechanisms of cell migration which are crucial for tumour angiogenesis. Specific 1 and 2D examples are discussed in detail.     

Stochastic modelling and optimization for environmental management

In modelling and managing complex environmental systems, inherent uncertainties of all relevant natural processes are to be taken into consideration. In the present paper diverse stochastic modelling and optimization approaches for handling such problems (primarily in the field of water quality analysis and control) are highlighted, drawing on the findings of case studies and real-world applications.     

Biologically guided intensity modulated radiation therapy planning optimization with fraction-size dose constraints

Although intensity modulated radiation therapy plans are optimized as a single overall treatment plan, they are delivered over 30–50 treatment sessions (fractions) and both cumulative and per-fraction dose constraints apply. Recent advances in imaging technology provide more insight on tumour biology that has been traditionally disregarded in planning. The current practice of delivering physical dose distributions across the tumour may potentially be improved by dose distributions guided by the biological responses of the tumour points. The biological optimization models developed and tested in this paper generate treatment plans reacting to the tumour biology prior to the treatment as well as the changing tumour biology throughout the treatment while satisfying both cumulative and fraction-size dose limits. Complete computational testing of the proposed methods would require an array of clinical data sets with tumour biology information. Finding no open source ones in the literature, the authors sought proof of concept by testing on a simulated head-and-neck case adapted from a more standard one in the CERR library by blending it with available tumour biology data from a published study. The results show computed biologically optimized plans improve on tumour control obtained by traditional plans ignoring biology, and that such improvements persist under likely uncertainty in sensitivity values. Furthermore, adaptive plans using biological information improve on non-adaptive methods.     

Chartwell Books: A Case Study in Mathematical Modelling

A case is described that has been used successfully many times as a vehicle for teaching mathematical modelling. The case has been closely derived from a consultancy assignment with Chartwell Books. The place of mathematical modelling within decision-support systems is examined, and this is followed by an outline of one such mathematical modelling methodology. The problem situation facing the managing director of Chartwell Books concerning the modelling of labour and material costs is described. A detailed application of the mathematical modelling methodology is worked through, finishing with a discussion of the value of the case for student teaching and of the problems that students have encountered when tackling the Chartwell case. Possible extensions to the basic methodology, such as validation and the human-computer interface, are also briefly considered.     

Mathematical models of tumour angiogenesis

We first study a parabolic–ODE system modelling tumour growth proposed by Othmer and Stevens [Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (4) (1997) 1044–1081]. According to Levine and Sleeman [A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (3) (1997) 683–730], we reduced it to a hyperbolic equation and showed the existence of collapse in [A. Kubo, T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential Integral Equations 17 (2004) 721–736]. We also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously. Next we apply the above result to another model proposed by Anderson and Chaplain arising from tumour angiogenesis and show the existence of collapse. Further we investigate a contact point between these two models and a common property to them.     

Increasing engineering students' awareness to environment through innovative teaching of mathematical modelling

This article presents the results of two studies on using aninnovative pedagogical strategy in teaching mathematical modellingand applications to engineering students. Both studies are dealingwith introducing non-traditional contexts for engineering studentsin teaching/learning of mathematical modelling and applications:environment and ecology. The aims of using these contexts were:to introduce students to some of the techniques, methodologiesand principles of mathematical modelling for ecological andenvironmental systems; to involve the students in solving real-lifeproblems adjusted to their region emphasizing the aspects ofboth survival (short term) and sustainability (long term); toencourage students to pay attention to environmental issues.On one hand, the contexts are not directly related to engineering.On the other hand, the chances are that many graduates of engineeringwill deal with mathematical modelling of environmental systemsin one way or another in their future work because nearly everyengineering activity has an impact on the environment. The firststudy is a parallel study conducted in New Zealand and Germanysimultaneously with first-year students studying engineeringmathematics. The second study is a case study of the experimentalcourse Mathematical Modelling of Survival and Sustainabilitytaught to a mixture of year 2–5 engineering students inGermany by a visiting lecturer from New Zealand. The modelsused with the students from both studies had several specialfeatures. Analysis of students’ responses to questionnaires,their comments and attitudes towards the innovative approachin teaching are presented in the article.     

Model development in discrete-event simulation and system dynamics: An empirical study of expert modellers

An empirical study comparing the model development process followed by experts in discrete-event simulation (DES) and system dynamics (SD) modelling is undertaken. verbal protocol analysis (VPA) is used to study the modelling process followed by ten expert modellers (5 SD and 5 DES). Participants are asked to build simulation models based on a case study and to think aloud while modelling. The generated verbal protocols are divided into seven modelling topics: problem structuring, conceptual modelling, data inputs, model coding, verification & validation, results & experimentation and implementation and then analyzed. Our results suggest that all modellers switch between modelling topics, however DES modellers follow a more linear progression. DES modellers focus significantly more on model coding and verification & validation, whereas SD modellers on conceptual modelling. Observations are made revealing some interesting differences in the way the two groups of modellers tackle the case. This paper contributes towards the comparison of DES and SD.     

Maturity,distance and density (MD<Superscript>2</Superscript>) metrics for optimizing trust prediction for business intelligence

The modelling and management of trust between interacting parties are crucial parts of the overall business intelligence strategy for any organization. Predicting trust values is a key element of modelling and managing trust. It is of critical importance when the interaction is to be conducted at a future point in time. In the existing body of work, there are a few approaches for predicting trust. However, none of these approaches proposes a framework or methodology by which the predicted trust value can be considered in light of its accuracy or confidence level. This is a key element in order to ensure optimized trust prediction. In this paper, we propose a methodology to address this critical issue. The methodology comprises a suite of metrics—maturity, distance and density (MD²) which are capable of capturing various aspects of the confidence level in the predicted trust value. The proposed methodology is exemplified with the help of case studies.     

具有年龄结构和约束的群落系统的最优收获

研究一类有年龄结构和相互作用的两种群构成的群落系统的最优收获问题,要求控制过程结束时的种群状态充分接近预先指定的年龄分布.证明了最优控制的存在性,运用Dubovitskii-Milyutin理论导出了最优性条件.这种处理方法为研究连续年龄分布下种群收获问题提供了统一框架.     

A World in Which Nothing Ever Happens Twice

The paper discusses the difficulties caused by the non-repeatable nature of all operational research studies. The author regrets the anecdotal nature of much of our research and makes a plea for more documentation of case studies rather than word of mouth folklore. Simulation is shown to be a powerful tool for experimental replication and is especially valuable in that it offers a better approach than mathematical modelling. In a final section, doubt is cast on the usefulness of much of the teaching of O.R.     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Introductory business OR cases: successful use of cases in introductory undergraduate business college operational research courses

A case methodology for developing the modelling skills of students in introductory operational research courses taught in undergraduate business programmes is presented. Motivated by a desire to develop such skills in students when the primary objective of such courses was to expose these students to various solution techniques, the approach takes on greater relevance now that many instructors of such courses have shifted their focus to the development of modelling skills. Example case scenarios that have been used by the author in the classroom are presented, and a comparison of the results achieved by the author after implementation of the case methodology is made with those achieved by the author prior to implementation.     

The novel fractional discrete multivariate grey system model and its applications

Fractional order accumulation is a novel and popular tool which is efficient to improve accuracy of the grey models. However, most existing grey models with fractional order accumulation are all developed on the conventional methodology of grey models, which may be inaccurate in the applications. In this paper an existing fractional multivariate grey model with convolution integral is proved to be a biased model, and then a novel fractional discrete multivariate grey model based on discrete modelling technique is proposed, which is proved to be an unbiased model with mathematical analysis and stochastic testing. An algorithm based on the Grey Wolf Optimizer is introduced to optimize the fractional order of the proposed model. Four real world case studies with updated data sets are executed to assess the effectiveness of the proposed model in comparison with nine existing multivariate grey models. The results show that the Grey Wolf Optimizer-based algorithm is very efficient to optimize the fractional order of the proposed model, and the proposed model outperforms other nine models in the all the real world case studies.     

On the usage of qualitative reasoning as an approach towards enterprise modelling

This paper analyzes the computational requirements of enterprise modelling, presents a reasoning framework, and reports about a prototype implementation. Enterprise modelling studies complex systems which are normally only partially known, and focuses on studying the qualitative properties of the underlying system, and thus differs from traditional quantitative approaches. Various modelling approaches from computer science and management science are discussed with respect to their suitability for enterprise modelling. Qualitative reasoning is identified as a promising basis to develop a computational framework for enterprise modelling. A rules-constraints-method of reasoning with imprecise knowledge, which is essentially an interval-based version of difference equations, is proposed as a powerful and versatile tool for enterprise modelling. As an illustrative example, a model of a business organization, derived from the concept of a financial statement, is developed and simulated.     

Modelling LGD for unsecured personal loans: decision tree approach

The New Basel Accord, which was implemented in 2007, has made a significant difference to the use of modelling within financial organisations. In particular it has highlighted the importance of Loss Given Default (LGD) modelling. We propose a decision tree approach to modelling LGD for unsecured consumer loans where the uncertainty in some of the nodes is modelled using a mixture model, where the parameters are obtained using regression. A case study based on default data from the in-house collections department of a UK financial organisation is used to show how such regression can be undertaken.     

Radiofrequency ablation planning: An application of semi-infinite modelling techniques

In radiofrequency (RF) ablation a needle-shaped probe is inserted into the patient’s body in order to heat and subsequently destroy the malignant tissue around the needle tip. The determination of the optimal probe position poses an intricate problem, as it requires the modelling of the tumour destruction depending on the attained temperature as well as the incorporation of constraints that prohibit puncturing bones or other risk structures.In this work we present a new optimization procedure that reflects both aspects adequately. We assess tumour destruction by solving the underlying system of partial differential equations using a finite element method. Next we show how the probe’s position and orientation can be optimized by gradient-based methods. Ensuring that risk structures are not harmed by the probe is easily modelled using semi-infinite constraints in the optimization problem.Techniques to reduce the semi-infinite problem to a standard nonlinear constrained optimization problem are introduced and demonstrated as a proof-of-concept on real clinical data. The results indicate the high potential of this combination of PDE-based simulation and numerical optimization for RF ablation planning.