Dumitru Trucu

Mathematical modelling of cancer invasion : implications of cell adhesion variability for tumour infiltrative growth patterns

Cancer invasion, recognised as one of the hallmarks of cancer, is a complex, multiscale phenomenon involving many inter-related genetic, biochemical, cellular and tissue processes at different spatial and temporal scales. Central to invasion is the ability of cancer cells to alter and degrade an extracellular matrix. Combined with abnormal excessive proliferation and migration which is enabled and enhanced by altered cell-cell and cell-matrix adhesion, the cancerous mass can invade the neighbouring tissue. Along with tumour-induced angiogenesis, invasion is a key component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we explore the spatio-temporal dynamics of a model of cancer invasion, where cell-cell and cell-matrix adhesion is accounted for through non-local interaction terms in a system of partial integro-differential equations. The change of adhesion properties during cancer growth and development is investigated here through time-dependent adhesion characteristics within the cell population as well as those between the cells and the components of the extracellular matrix. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as tumour infiltrative growth patterns (INF).

Inverse time-dependent perfusion coefficient identification

The identification of the time-dependent perfusion coefficient in the transient bio-heat conduction equation is investigated. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution may be a heat flux, interior temperature or mass measurement which is taken permanently along the time interval of interest. A numerical approach based on the boundary element and mollification methods is developed. Numerical results are presented and discussed.

A Multiscale Moving Boundary Model Arising in Cancer Invasion

Cancer invasion of tissue is a key aspect of the growth and spread of cancer and is crucial in the process of metastatic spread, i.e., the growth of secondary cancers. Invasion consists in cancer cells secreting various matrix degrading enzymes (MDEs) which destroy the surrounding tissue or extracellular matrix (ECM). Through a combination of proliferation and migration, the cancer cells then actively spread locally into the surrounding tissue. Thus processes occurring at the level of individual cells eventually give rise to processes occurring at the tissue level. In this paper we introduce a new type of multiscale model describing the process of cancer invasion of tissue. Our multiscale model is a two-scale model which focuses on the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix, and on the microscale dynamics of the MDEs, produced at the level of the individual cancer cells. These microscale dynamics take place at the interface of the cancer cells...

Reconstruction of the Space- and Time-Dependent Blood Perfusion Coefficient in Bio-Heat Transfer

The identification of the space- and time-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. While boundary and initial conditions are prescribed, additional temperature measurements are considered inside the solution domain. The problem is approached both from a global and a local perspective. In the global approach a Crank–Nicolson-type scheme is combined with the Tikhonov regularization method. In the local approach, we compute both the time first-order and space second-order derivatives by means of first kind integral equations. A comparison between the numerical results obtained using the two methods shows that the local approach is more accurate and stable than the global one.

Strategies of eradicating glioma cells: a multi-scale mathematical model with MiR-451-AMPK-mTOR control.

The cellular dispersion and therapeutic control of glioblastoma, the most aggressive type of primary brain cancer, depends critically on the migration patterns after surgery and intracellular responses of the individual cancer cells in response to external biochemical and biomechanical cues in the microenvironment. Recent studies have shown that a particular microRNA, miR-451, regulates downstream molecules including AMPK and mTOR to determine the balance between rapid proliferation and invasion in response to metabolic stress in the harsh tumor microenvironment. Surgical removal of main tumor is inevitably followed by recurrence of the tumor due to inaccessibility of dispersed tumor cells in normal brain tissue. In order to address this multi-scale nature of glioblastoma proliferation and invasion and its response to conventional treatment, we propose a hybrid model of glioblastoma that analyses spatio-temporal dynamics at the cellular level, linking individual tumor cells with the macroscopic behaviour of cell organization and the microenvironment, and with the intracellular dynamics of miR-451-AMPK-mTOR signaling within a tumour cell. The model identifies a key mechanism underlying the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells in response to fluctuating glucose levels in the presence of blood vessels (BVs). The model predicts that cell migration, therefore efficacy of the treatment, not only depends on oxygen and glucose availability but also on the relative balance between random motility and strength of chemoattractants. Effective control of growing cells near BV sites in addition to relocalization of invisible migratory cells back to the resection site was suggested as a way of eradicating these migratory cells.

An inverse coefficient identification problem for the bio-heat equation

In this article, we investigate both analytical and numerical techniques for the identification of the constant perfusion coefficient in the transient bio-heat conduction equation. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution may be a heat flux, an interior temperature or an average temperature measurement at a single instant. Numerical results obtained using the boundary element method combined with an ordinary non-linear minimization are presented and discussed.

A Multiscale Mathematical Model of Tumour Invasive Growth

Known as one of the hallmarks of cancer (Hanahan and Weinberg in Cell 100:57–70, 2000) cancer cell invasion of human body tissue is a complicated spatio-temporal multiscale process which enables a localised solid tumour to transform into a systemic, metastatic and fatal disease. This process explores and takes advantage of the reciprocal relation that solid tumours establish with the extracellular matrix (ECM) components and other multiple distinct cell types from the surrounding microenvironment. Through the secretion of various proteolytic enzymes such as matrix metalloproteinases or the urokinase plasminogen activator (uPA), the cancer cell population alters the configuration of the surrounding ECM composition and overcomes the physical barriers to ultimately achieve local cancer spread into the surrounding tissue. The active interplay between the tissue-scale tumour dynamics and the molecular mechanics of the involved proteolytic enzymes at the cell scale underlines the biologically multiscale character of invasion and raises the challenge of modelling this process with an appropriate multiscale approach. In this paper, we present a new two-scale moving boundary model of cancer invasion that explores the tissue-scale tumour dynamics in conjunction with the molecular dynamics of the urokinase plasminogen activation system. Building on the multiscale moving boundary method proposed in Trucu et al. (Multiscale Model Simul 11(1):309–335, 2013), the modelling that we propose here allows us to study the changes in tissue-scale tumour morphology caused by the cell-scale uPA microdynamics occurring along the invasive edge of the tumour. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as the tumour infiltrative growth patterns discussed in Ito et al. (J Gastroenterol 47:1279–1289, 2012).

Three-scale convergence for processes in heterogeneous media

In this article, we propose a new notion of multiscale convergence, called ‘three-scale’, which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. This generalizes and extends the notion of two-scale convergence, a well-established concept that is now commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences in L 2(Ω) are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterization of three-scale convergent sequences and is then continued with the introduction of the notion of ‘strong three-scale convergence’ whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.

Inverse space-dependent perfusion coefficient identification

The identification of the space-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution is a boundary temperature measurement. A numerical approach based on a Crank-Nicolson finite-difference scheme combined with Tikhonovs regularization methods is developed. Numerical results are presented and discussed.

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with three examples arising from cancer invasion.