An Adaptive Multigrid Algorithm for Simulating Solid Tumor Growth Using Mixture Models

In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in (Wise et al., J. Theor. Biol. 253 (2008)) and used to study the development of glioma in (Frieboes et al., NeuroImage 37 (2007) and tumor invasion in (Bearer et al., Cancer Research, 69 (2009)) and (Frieboes et al., J. Theor. Biol. 264 (2010)). The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell-species coupled with reaction-diffusion equations for the substrate components. Numerical solution of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.

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.

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.

Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method

This is the first paper in a two-part series in which we develop, analyze, and simulate a diffuse interface continuum model of multispecies tumor growth and tumor-induced angiogenesis in two and three dimensions. Three-dimensional simulations of nonlinear tumor growth and neovascularization using this diffuse interface model were recently presented in Frieboes et al. [2007. Computer simulation of glioma growth and morphology. NeuroImage S59-S70], but that paper did not describe the details of the model or the numerical algorithm. This is done here. In this diffuse interface approach, sharp interfaces are replaced by narrow transition layers that arise due to differential adhesive forces among the cell species. Accordingly, a continuum model of adhesion is introduced. The model is thermodynamically consistent, is related to recently developed mixture models, and thus is capable of providing a detailed description of tumor progression. The model is well-posed and consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell species coupled with reaction-diffusion equations for the substrate components. We demonstrate analytically and numerically that when the diffuse interface thickness tends to zero, the system reduces to a classical sharp interface model. Using a new fully adaptive and nonlinear multigrid/finite difference method, the system is simulated efficiently. In this first paper, we present simulations of unstable avascular tumor growth in two and three dimensions and demonstrate that our techniques now make large-scale three-dimensional simulations of tumors with complex morphologies computationally feasible. In part II of this study, we will investigate multispecies tumor invasion, tumor-induced angiogenesis, and focus on the morphological instabilities that may underlie invasive phenotypes.

Prototyping virtual cancer therapist (VCT): a software engineering approach

This paper proposes the virtual cancer therapist (VCT), a scalable and robust cancer expert system that makes cancer diagnosis, recommends therapy plans, and simulates therapy plans in silico. This system consists of an evaluation core that makes prognosis and chemotherapy simulations, a biomedical database that supports therapy planning, and an optimizer module that makes cancer diagnosis and produces queries for the optimal therapy plans. With the support of its patient record database and simulation core, VCT can also be used to establish an in silico drug discovery standard that dramatically reduces the drug discovery timeline and cost. The prototype of VCT presented in this paper has not only demonstrated the capability of VCT but also identified problems that need to be addressed in the next cycle of development.

Predictive computer simulations of tumor drug response demonstrate that 3-D hypoxic gradients significantly increase drug resistance

2071

Background: We created a three-dimensional physiologically based computer (in-silico) model of cancer based on a description of biological events at the cellular scale with input variables determined from patient specific information, such as in-vitro drug response experiments and in-vivo tumor imaging, with the long term goal of individualized treatment selection. The central hypothesis is that such a model that incorporates basic tumor growth kinetics information is capable of representing and predicting tumor response to chemotherapy. Methods: We measured in-vitro tumor growth and drug response for Doxorubicin sensitive and resistant MCF-7 breast cancer cells through trypan blue exclusion counts, tridiated thymidine incorporation, and the XTT assay. We used these results of parameter-based statistics to define input variables to our in-silico model of cancer, and ran computer simulations to measure the drug response predicted by the model. Results: The computer model could accurately predict the in-vitro response of drug sensitive and resistant MCF-7 breast cancer cells. The model also predicted that gradients of oxygen and nutrient in a tumor microenvironment, whether naturally occurring or induced by treatment, and which in previous work we found could increase the invasive capability of tumor cells and destabilize tumor morphology, could also contribute to acquired drug resistance by increasing the population of quiescent cells. Conclusions: We demonstrated that a rigorously, experimentally calibrated computer model of cancer is accurately predictive of in-vitro tumor response to chemotherapeutic drugs, and established that this model offers a means to quantitatively study tumor drug response. We did this through a grounds-up physical representation of tumor biology, not by fitting to experimental data. This validation begins the path to computational modeling and more efficient prediction of in-vivo tumor response to chemotherapy.

No significant financial relationships to disclose.

Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method

We present a multi-scale computer simulator of cancer progression at the tumoral level, from avascular stage growth, through the transition from avascular to vascular growth (neo-vascularization), and into the later stages of growth and invasion of normal tissue. We use continuum scale reaction-diffusion equations for the growth component of the model, and a combined continuum-discrete model for the angiogenesis component. We use the level set method for describing complex topological changes observed during growth such as tumor splitting and reconnection, and capture of healthy tissue inside the tumor. We use an adaptive, unstructured finite element mesh that allows for finely resolving important regions of the computational domain such as the necrotic rim, the tumor interface and around the capillary sprouts. We present full nonlinear, two-dimensional simulations, showing the potential of our virtual cancer simulator. We use microphysical parameters characterizing malignant glioma cells, obtained from recent in vitro experiments from our lab and from clinical data, and provide insight into the mechanisms leading to infiltration of the brain by the cancer cells. The results indicate that diffusional instability of tumor mass growth and the complex interplay with the developing neo-vasculature may be powerful mechanisms for tissue invasion.

Two-Dimensional Chemotherapy Simulations Demonstrate Fundamental Transport and Tumor Response Limitations Involving Nanoparticles

Zheng et al. (2004) developed a multiscale, two-dimensional tumor simulator with the capability of showing tumoral lesion progression through the stages of diffusion-limited dormancy, neo-vascularization (angiogenesis) and consequent rapid growth and tissue invasion. In this paper we extend their simulator to describe delivery of chemotherapeutic drugs to a highly perfused tumoral lesion and the tumor cells' response to the therapy. We perform 2-D simulations based on a self-consistent parameter estimation that demonstrate fundamental convective and diffusive transport limitations in delivering anticancer drug into tumors, whether this delivery is via free drug administration (e.g., intravenous drip), or via 100 nm nanoparticles injected into the bloodstream, extravasating and releasing the drug that then diffuses into the tumoral tissue, or via smaller 1-10 nm nanoparticles that are capable of diffusing directly and targeting the individual tumor cell. Even in a best-case scenario involving: constant ("smart") drug release from the nanoparticles; a homogenous tumor of one cell type, which is drug-sensitive and does not develop resistance; targeted nanoparticle delivery, with resulting low host tissue toxicity; and for model parameters calibrated to ensure sufficient drug or nanoparticle blood concentration to rapidly kill all cells in vitro ; our analysis shows that fundamental transport limitations are severe and that drug levels inside the tumor are far less than in vitro , leaving large parts of the tumor with inadequate drug concentration. A comparison of cell death rates predicted by our simulations reveals that the in vivo rate of tumor shrinkage is several orders of magnitude less than in vitro for equal chemotherapeutic carrier concentrations in the blood serum and in vitro, and after some shrinkage the tumor may achieve a new mass equilibrium far above detectable levels. We also demonstrate that adjuvant anti-angiogenic therapy "normalizing" the vasculature may ameliorate transport limitations, although leading to unwanted tumor fragmentation. Finally, our results suggest that small nanoparticles equipped with active transport mechanisms (e.g., chemotaxis) would overcome the predicted limitations and result in improved tumor response.

Near-Contact Motion of Surfactant-Covered Spherical Drops: Ionic Surfactant

A lubrication analysis is presented for near-contact axisymmetric motion of spherical drops covered with an insoluble nondiffusing surfactant. The surfactant equation of state is arbitrary; detailed results are presented for ionic surfactants. The qualitative behavior of the system is determined by the dimensionless force parameter &Fcirc;, the external force normalized by the maximum resistance force generated by Marangoni stresses. For &Fcirc; > 1 drops coalesce on a time scale commensurate with the coalescence time tau0 for drops with clean interfaces. For &Fcirc; < 1, the system evolves on the time scale tau0 until Marangoni stresses approximately balance the external force; thereafter a slow evolution occurs on the Stokes time scale. In the long-time regime a self-similar surfactant concentration profile is attained that scales with the extent of the near-contact region. The gap width decreases exponentially with time but slower than for rigid particles because of surfactant backflow. For &Fcirc; < 1, drop coalescence does not occur without van der Waals attraction. Quantitative results depend only moderately on the surfactant equation of state. Copyright 1999 Academic Press.

[Is mammography useful in the detection of breast cancer in women 35 years of age or younger?]

Breast cancer in women 35 years old or younger is unusual. It accounts for 1-3.6% of all breast cancers but is the leading cause of cancer mortality in women 15-35 years old. The diagnostic delay, with T2 or more advanced cancer at clinical presentation, is due to the patient's age and the opinion of low mammographic reliability for cancer diagnosis in this age group. To assess the usefulness of mammography in breast cancer patients aged 35 years or younger, we reviewed the clinical, mammographic and histologic data of 65 cancers collected in 7 breast diagnosis and counseling centers in Lombardy. Fifty-three patients (81.5%) were referred for a palpable breast mass, which was a T2 or more advanced cancer in 23 cases. Mammography showed malignant patterns (spiculated opacities, clusters of microcalcifications, casting, branching and ductal type calcifications) in 31 patients (47.7%). Mammography was not definitive but correctly suggested further examinations in 30 women and it had only 4 false negatives. Ultrasonography performed in 43 patients was negative in 3 (7%), pathologic and pathognomonic for cancer in 27 (62.8%) and pathologic but not indicative of malignancy in 13 (20.2%). The cytologic or histologic diagnosis of breast cancer was made under US guidance in 24 cases. In women aged 35 years or younger mammography was effective in identifying breast cancers; US and fine-needle aspiration biopsy (FNAB) complete mammography. We believe that mammography can be a valuable screening tool in young women at high risk for breast cancer because of family history.