Modelling the Growth and Treatment of Tumours

Inclusion of Imaging Data towards Personalized Predictions

  • Fig. 1: Comparison between standard (top) and hyper-fractioned therapy for the case of an early stage acute responding tumours, e.g. head and neck. For convenience, only one quarter of the tumour is shown in 6mm x 6mm boxes. Colours and transparency: red – necrotic; green – quiescent; blue – proliferating cells.Fig. 1: Comparison between standard (top) and hyper-fractioned therapy for the case of an early stage acute responding tumours, e.g. head and neck. For convenience, only one quarter of the tumour is shown in 6mm x 6mm boxes. Colours and transparency: red – necrotic; green – quiescent; blue – proliferating cells.
  • Fig. 1: Comparison between standard (top) and hyper-fractioned therapy for the case of an early stage acute responding tumours, e.g. head and neck. For convenience, only one quarter of the tumour is shown in 6mm x 6mm boxes. Colours and transparency: red – necrotic; green – quiescent; blue – proliferating cells.
  • Fig. 2: Comparison of the simulated tumour geometry without (left) and with radiotherapy treatment (1.8Gy, middle). The differences of the number of cells (right): the closer the lattice cells are to white the more similar both pictures are. For convenience, only one quarter of the tumour is shown.

Cancer is a leading cause of morbidity and mortality worldwide, with approximately 14 million new cases and 8.2 million cancer related deaths in 2012 [1]. Moreover, the global cancer burden is expected to exceed 20 million new cancer cases by 2025. Understanding the spatial and temporal behaviour of cancer is a crucial precondition to achieve a successful treatment. Because no two cancer cases are the same, every patient should receive a treatment plan designed specifically for their case, in order to improve the patient's survival chances. 

One way of optimizing cancer therapy is given by mathematical cancer models, which have emerged as a powerful tool to explore the interplay of the different biological changes happening before, during and after treatment. To make sense of cancer, these models include information such as physical forces, equations to describe how tumours grow and spread and mathematical approaches to study how cancer cells interact at the microscopic scale. Inclusion of medical information from each patient gained through clinical images (e.g. Magnetic Resonance Images (MRI), Positron Emission Tomography (PET)), blood samples and biopsies, allows for patient specific initialisation, calibration and validation of cancer models.
Modelling Tumour Growth
Modelling tumour development and treatment is an ongoing area of research. The term in-silico oncology has been coined in order to highlight the computational tools used to solve model equations by simulations (readers are referred to [2]). For long time, mathematics of tumour models were divided, following the spatial and temporal scales, into two classes: continuous and discrete. Nowadays, hybrid multi-scale approaches, potentially including cell-cycle dynamics, have emerged as promising tools [3]. An example of the latter is given in [4], where a tumour model able to employ cell-line-specific biological parameters and information derived from pre-therapy imaging data was developed to investigate avascular tumour which consists of motile cells that cannot only proliferate but also migrate. If the number of tumour cells present within a geometric unit exceeds the maximum allowed number, cells are able to move into a nearby vacant unit, i.e.

a unit with less than the maximum allowed number of cells, ultimately resulting in tumour expansion. 

Moreover, avascular tumours are known to be finite in size and to develop a necrotic core which is surrounded by a rim of hypoxic cells, which in turn are surrounded by a rim of proliferating cells. The multiscale cell-cycle model presented in [4] has shown to be able to mirror this macroscopic behaviour as a cumulative result of changes happening in each tumour cell. Because the state of each tumour cell describing its spatial position in the tumour, its internal biological state (including its position in the cell cycle and interaction with the local biological environment) is saved for every time step, it is possible to explore how cells in different parts of the tumour would react under various environmental conditions, e.g. oxygen level and radiation therapy. 
Modelling the Tumour Response to Therapy - The Radiotherapy Case
Given that in a clinical scenario one is interested in understanding how a specific patient will respond to a given radiotherapy treatment, it is important to explore how a specific tumour would react to a chosen therapy plan. To this end, the tumour's response to different fractioning radiation schedules including single doses, standard fractioned (one session radiation every day over six weeks) and hyper-fractioned (two smaller doses a day over six weeks) radiation were studied. For instance, for the case of acute responding tumours, e.g. head-and-neck tumour, the hyper-fractioned approach outperformed the standard fractioned one as depicted in Fig. 1.
The most prominent model to estimate the survival probability (fraction of tumour cells that survives treatment to initial tumour cell number) is the linear-quadratic (LQ) model. This model is empirical, and although derived from clinical data, does not adequately fit all cell survival experiments. For instance, for late responding tumours, the LQ model overestimates the magnitude of cell kill, or, equivalently, underestimates the surviving fraction. This means that by using the LQ model to design the therapy schedule the needed doses to achieve the wanted result is being underestimated, thus potentially leading to treatment failure. 
Using the above mentioned multi-scale tumour model the behaviour of the LQ model at high doses under normoxic and hypoxic conditions has been investigated. For the case of late responding cancer types (e.g. prostate cancer) which have high repair capacity, the LQ model fails to account for recovery of the cells. For instance, using the LQ model, less than 0.09% of the cells were estimated to survive single doses of 14.2 Grey. However, when additionally accounting for cell-cycle regulations, almost 2% of cells would have survived the same doses. These results demonstrate that taking the recovery of the cells into account when planning a treatment schedule can avoid an underestimation of the survival fraction, and consequently, prevent further cancer development [5]. 
Applying the Model to the Clinical Case - Glioma
Although hybrid models serve as powerful tools to better understand interactions happening at various scales, their ultimate goal is to find ways to guide treatment decisions in a clinical environment. To this end, clinical images of the tumour at different time points and acquired by various imaging modalities can be used to calibrate model parameters, such as cell proliferation (PET), number of cells (MRI) and invasion of cells into the tumour's surroundings (MRI). Model predictions at a third time point can be calculated and compared to corresponding images, thus allowing for the model to be evaluated and validated.
The overwhelming majority of tumour growth models have been devoted to tumours in the brain such as gliomas. Those continuous models describe tumour growth at the macroscopic level by means of cell density. A discrete model which includes cell-cycle regulation can shed light on the state of the local tumour cells whilst mimicking the macroscopic growth of the early stage glioma. To explore this possibility, a multiscale stochastic model was applied to longitudinal MRI time series of two clinical low-grade glioma cases [6]. Patient specific values for model parameters were extracted from MRI images at two consecutive time points. Those values were used to run numerical simulations leading to predictions of the spatio-temporal tumour evolution at a third point in time, which were compared to actual patients MRI acquired at the same time point. 
Results showed good agreement in predicting the structure and the development of the tumour over time. Moreover, because one of the patients had received three sessions of radiotherapy before the acquisition of the third MRI scan, the tumour's response to radiotherapy was also included. Simulations were also able to spot differences in the tumour cell number three days after the therapy which would not have been visible in images. The difference in cell number (2,021,801 less cells), represented by a smaller simulated radius (Δr=2.54mm), can be seen in Fig. 2.
Challenges, Outlook and the Vision of Personalised Oncology
One of the biggest challenges in modelling tumour growth and response to treatment is the availability of comprehensive patient specific data. On the other hand, the more data is integrated the more data has to be processed for modelling, which ultimately results in a higher computational cost. Hence, in the future, efforts should be directed towards the development of a multi-scale model of tumour growth built a priori from data derived from various imaging modalities. To this end, those modalities need to offer anatomical, metabolic, and functional information at a high spatial resolution level. Here, treating cells with similar functional characteristics as groups instead of tracking each tumour cell could allow for the model to remain computationally tractable, whilst enabling the inclusion of different biological activities happening at the microscopic level.
Nowadays, patients are used to hearing predictions based on the odds of treatment working for their tumour on a population average basis. Once clinically validated, models of tumour growth and treatment could help change this scenario and allow for each patient to hear a prediction based on their own personal data.
Thaís Roque1 and Waldemar Zylka2
1 Institute of Biomedical Engineering, University of Oxford, England, formerly with Westphalian University 
2 Westphalian University, Campus Gelsenkirchen, Germany
[2] Araujo, RP. and McElwain, DLS.: Bull Math Biol 66, 1039–1091 (2004)
[4] Rudigkeit, N. et al.: Biomed Eng 56, Sup. 1 (2011)
[5] Roque, T. et al.: Biomed Eng 57 Issue SI-1 (2012)
[6] Roque, T. et al.: Biomed Eng 58 Sup. 1 (2013)
Prof. Dr. Waldemar Zylka
Westphalian University
Campus Gelsenkirchen
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