Mathematical modelling and simulation can potentially provide insight into the underlying causes of tumour invasion and metastasis, help understand clinical observations, and be of use in designing targeted experiments and assessing treatment strategies. The ultimate goal is to develop individualized therapeutic protocols based on patient-specific tumour characteristics.

Many research groups are expending significant effort to address this target, modelling cancer as a complex system and simulating tumour progression at multiple time and spatial scales. In a newly published paper in the journal Nonlinearity, John Lowengrub from the University of California, Irvine, presents recent results and reviews progress in the modelling of cancer (Nonlinearity 23 R1).

"Models that employ a biologically-founded, multiscale approach could help to quantitatively link critical cellular-scale effects with clinically observed tumour growth and invasion," explained Lowengrub. "By using patient tumour-specific data, this modelling may provide a predictive tool to characterize the complex in vivo tumour physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis for individual patients."

Modelling frameworks
Mathematical models of cancer growth tend to fall into two categories: discrete cell-based models and continuum models. In discrete modelling, individual cells are tracked and updated according to a specific set of biophysical rules. While this method can address detailed biological processes, it is computationally demanding for larger-scale systems. For example, a full simulation of a small 1 mm3 tumour, which may contain several hundred thousand cells, is not currently feasible.

The review paper focuses mostly on the continuum approach, which provides a good alternative for larger-scale systems. Continuum models treat tumours as a collection of tissue, describing densities or volume fractions of cells and other elements. Lowengrub and co-authors - from the University of Texas Health Science Center at Houston and the University of Tennessee Knoxville - begin by examining tumour growth in homogeneous tissue, using a basic model that represents the tumour as a sphere-like structure without direct access to the vasculature.

The authors then evaluate tumour growth in complex, heterogeneous tissue, and move on to examine the transition to the vascularized growth phase through tumour-induced angiogenesis. They also discuss multiphase modelling, in which a tumour is described as comprising at least one solid phase (such as cells or extra-cellular matrix) and one liquid phase (such as interstitial fluid).

The paper also evaluates the relevance of theoretical modelling to patients suffering from brain and breast cancers. One recent study, for example, compared results from a multiphase model of glioblastoma multiforme with clinical data. The model correctly predicted tumour morphology as seen in vivo: regions of viable cells, necrosis in inner tumour areas and a tortuous neovasculature. Excellent agreement was found when comparing the tumour "virtual histopathology" to clinical histopathology.

Lowengrub points out that such encouraging results support the idea that sophisticated multiphase tumour simulators, capable of simulating vascularized tumour growth in three spatial dimensions, and calibrated by in vitro and in vivo data, have the potential to predict cancer behaviour in patients.

Future approach
While continuum modelling is appropriate at the tissue scale where gross tumour behaviour can be quantified, it cannot model individual cells and discrete events. Hybrid-modelling frameworks offer a promising alternative. Here, the tumour tissue is described using both discrete (cell-scale) and continuum (tumour-scale) elements. With the potential to combine the best features of both approaches, such models may provide more realistic coupling of biophysical processes across a wide range of length and time scales.

Lowengrub concludes that while mathematical models and simulation results still lag behind experimentation in studying cancer progression in vivo, there have been significant advances made in modelling the progression of this complex disease. Mathematical modelling, for example, has revealed that parameters controlling the tumour shape may also control its ability to invade, suggesting that tumour morphology could serve as a predictor of invasiveness and treatment prognosis.

Looking ahead, Lowengrub identifies three main challenges in taking this work forward. One is to correctly model the multiscalar biology of cancer for specific locations in the body. Another is the correct calibration of the model parameters so that they have values that are biologically and clinically relevant, and eventually are patient-specific.

"The third challenge is to continue the development of numerical techniques that can enable fast and accurate solutions to the model, so as to yield simulations that are sufficiently detailed and that can faithfully represent clinically-observed tumour evolution," Lowengrub told medicalphysicsweb.

• For the full story, see: Nonlinear modelling of cancer: Bridging the gap between cells and tumours J S Lowengrub et al Nonlinearity 23 R1.