Yao-Li Chuang

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Three-dimensional multispecies nonlinear tumor growth—II: Tumor invasion and angiogenesis

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.

Using mathematical models to understand the time dependence of the growth of ductal carcinoma in situ.

Abstract #1165 Background: Models of cancer growth have been developed that predict tumor size and growth dynamics for invasive tumors. However, it has been difficult to model ductal carcinoma in situ (DCIS) because of the constraints introduced by its containment within the duct system.
 Materials and Methods: We have developed a spherical model of growth of solid type DCIS using chemical engineering models of reaction and diffusion in porous media to represent the spread of DCIS in the duct systems. The model predicts tumor diameter based on four input parameters: the ratio of the apoptosis rate to the proliferation rate (A), the diffusion penetration length for nutrient to sustain the tumor growth (L), the volume fraction that tumor cells occupied within the involved breast tissue (V), and the time taken for a cell to complete mitosis(T). We have estimated L, V, and T from the literature, and then back-calcuated A for a range of diameters. We have used these four parameters as inputs and studied the time dependence of the evolution of DCIS.
 Results: We have found that the range of the values of A that we determined are within an adeqaute physiological range based on rates of proliferation and apoptosis taken from the literature. Using the model, the time to reach at least 95% of the maximum size ranges from less than 30 days for DCIS measuring 0.5 cm to almost 80 days for DCIS measuring 6 cm in diameter.
 Discussion: There has been little understanding of how long it takes for DCIS to grow, and whether it reaches a steady state size. Our simulations show that DCIS can grow to sizes as large as 6 cm in less than 3 months if it has the correct properties, including a high proliferation rate relative to the apoptosis rate and appropriate access to nutrients. This finding may help to explain why many cases of DCIS are not diagnosed before they progress to invasive carcinoma. Citation Information: Cancer Res 2009;69(2 Suppl):Abstract nr 1165.