Full-scale, three-dimensional simulation of early-stage tumor growth: The onset of malignancy

Highlights

• A 3D model to predict vascular tumor growth at the onset of malignancy is proposed.

• We propose a suitable computational method based on isogeometric analysis.

• We present numerical simulations that aim at reproducing in vitro and in vivo setups.

• We use the geometry of a colon histopathological image in macroscopic simulations.

Abstract

Malignant tumors have the ability to trigger the growth of new vasculature towards them through a complex process called tumor angiogenesis. These new blood vessels provide cancerous cells with sufficient nutrients for growth and a means to escape the primary tumor and invade other tissues. This paper proposes a three-dimensional model that aims at predicting how the tumor and its associated vasculature grow during the onset of malignancy. The cellular growth equations of our model are derived using the phase-field method. We use standard reaction–diffusion equations to model the transport of chemicals. The model resolves the tumor-induced vasculature explicitly, considering capillaries with a diameter of up to ∼25  $\mu m$. We propose a suitable computational method based on isogeometric analysis. We present several numerical examples to test the model, including a macroscopic simulation (∼1 cm³) on a geometry taken from a histopathological image of a human colon. The proposed model naturally predicts the angiogenic switch. Our computations show realistic patterns of tumor and vascular growth.

Introduction

Healthy cells produce signals to control their division process. Cancer cells, however, reproduce without the normal restraints. Abnormal cells that proliferate out of control may give rise to a tumor  [1], [2]. In most cases, tumors grow rapidly consuming the nutrient and oxygen delivered by pre-existing blood vessels. However, when the cancerous mass reaches approximately 2–3  ${\mathrm{mm}}^3$   [3], the absence of blood vessels inside the tumor arrests the growth of the lesion. Simplifying significantly the actual biological process, we could say that those cells located at the outermost rim may still obtain nutrients and oxygen from the circulatory system through diffusion mechanisms and continue their proliferation. On the contrary, cells in deep layers of the tumor do not have access to any source of nutrients and die from starvation forming a necrotic core. There is a transition layer between the proliferating rim and the necrotic core, called hypoxic region. The level of oxygen in such region is low due to an imbalance between the oxygen consumption and supply  [4], [5]. Under hypoxic conditions, cancer cells enter a temporary, dormant state. Growth at the proliferating rim and death at the necrotic core balances the global growth of the tumor. This situation may persist for years without presenting a threat to the life of the individual.

Hypoxic tumor cells may be able to promote the growth of new capillaries towards them through a process known as tumor-induced angiogenesis. These cells have gained the ability to release a series of chemical substances to the extracellular matrix, generally termed tumor angiogenic factors or TAFs. Some of the most potent TAFs include vascular endothelial growth factor (VEGF), basic fibroblast growth factor (bFGF), integrins, placenta growth factor (PLGF), or thrombospondin-1 (TSP-1) [6], [7], [8], [9]. TAFs diffuse from the tumor until they bind to the membranes of the cells that line the blood vessels, that is, endothelial cells. At this point tumor-induced angiogenesis starts and may be briefly described as follows (for a detailed explanation the reader is referred to [10], [11], [12]). Initially, TAFs alter the quiescent phenotype of endothelial cells either to a migratory or to a proliferative phenotype  [13], [14]. The former, the tip endothelial cells or TECs, will lead the capillary growth, while the latter, the stalk endothelial cells or SECs, will elongate the sprout through continuous cell division. The selection between TECs or SECs is based on the lateral inhibition mechanism [15]. The first cells reached by TAFs become TECs and subsequently express a protein called delta-like ligand 4 or Dll4. This protein binds to the nearby endothelial cell Notch receptors. Notch-activated endothelial cells become stalk cells instead of tip cells when TAFs reach them. As a result, an incipient sprout starts to develop from the parent vessels. Once liberated from the vascular membrane that envelops the parent vessel, TECs spearhead the migration following gradients of TAF. To enhance the survey of their microenvironment they extend forward membrane protrusions called filopodia  [16]. Behind, the capillary grows as SECs multiply rapidly. The process continues until the leading cell fuses with another vessel (anastomosis) or until the driving stimuli end. The outcome of tumor-induced angiogenesis is a new, although pathologically defective, vascular network that partially satisfies the metabolic demands of the highly-proliferative cancer cells. As a consequence, the previously small, growth-restrained tumor can now grow unbounded becoming malignant. Tumor angiogenesis plays a key role in the onset of malignancy in several solid cancers. In particular, it is critical in the growth of colorectal cancer, in which we focus in this work.

Colorectal cancer is the third most common type of cancer globally and the third leading cause of death in United States [17], [18]. Usually, this disease develops slowly, often showing no symptoms, which makes its detection difficult. Colorectal cancer incidence and death rates have decreased due to the increase of screening frequency through colonoscopy and improvement of treatments. Due to the crucial role of angiogenesis, a number treatments that target the growth of capillaries have been developed, such as bevacizumab, regorafenib, or aflibercept  [19]. Moreover, colorectal cancer develops through a series of differentiated stages, as shown next, being angiogenesis an important predictor for early-stage colorectal cancer  [20].

The colon and the rectum form part of the lower digestive system. Their main functions are to absorb water and nutrients from digested food, to form and store waste and to move waste out of the body. As shown in Fig. 1, the colon is made up of distinct tissue layers, which from innermost to outermost are: mucosa, submucosa, muscularis propria and serosa. The mucosa layer lines the lumen of the colon and is composed by a thin layer of epithelial cells, a layer of connective tissue and a thin layer of muscle. The submucosa is a connective tissue that envelops the mucosa in which blood and lymph vessels, nerves and mucous glands are embedded. The submucosa is in turn surrounded by a thick layer of muscles, the muscularis propria, that powers the movement of the stool through the colon. Finally, the serosa layer enwraps the colon. The main blood and lymph vessels that supply and drain wastes from the colon are located outside these layers in the mesentery. The rectum also presents concentric layered tissues analogous to those of the colon. Due to the ordered colon and rectum layered microstructure, colorectal cancer staging is commonly defined using the TNM staging system  [21], where T stands for the size of the primary tumor, N for its spread to regional lymph nodes and M for the presence of distant metastases. In this system, each letter is followed by a number, such that the higher the number, the more advanced the cancer stage. Thus, in colorectal cancer, at the earliest stage of cancer, that is Tis or carcinoma in situ, the tumor is at the beginning of its development, appears as a budding shape and is confined into the mucosa layer. In the next stage, that is T1, the tumor invades the submucosa layer. At this point, the onset of malignancy, angiogenesis plays a pivotal role as it allows the tumor to spread through subsequent layers. In the next tumor stages, namely T2, T3 and T4, the tumor grows through the muscle layer, the serosa layer and into other organs, respectively. The TNM system also classifies the stages of cancer as a function of regional lymph node invasion, such that N0 indicates no invasion, N1 three or less regional lymph nodes invaded and N2 four or more invaded lymph nodes. Finally, in this system, M0 designates a non-metastasized tumor, while M1 marks the presence of distant metastases. In a more general sense, these classifications are grouped in broader stages, numbered from 0 to IV. In the insets of Fig. 1 we depict several of these stages, highlighting those related to this work. Thus, Fig. 1(A) shows a carcinoma in situ which is still within the mucosa layer. It is only when the tumor triggers angiogenesis, gets vascularized and invades the submucosa that cancer advances to a malignant stage, as shown in Fig. 1(B). Thereafter, the new supply of nutrients and oxygen facilitates the invasion of outer layers (Fig. 1(B)–(E)) and the spread to other organs (Fig. 1(F)).

Arguably, the avascular to vascular tumor switch through angiogenesis is one of the most important events after a tumor has formed. Thus, being able to fully understand and control angiogenesis at the early-stages of cancer development may lead to effective cancer treatments. To this end, the computational mechanics community has developed in the last decades a number of mathematical models of tumor-induced angiogenesis that try to unveil key mechanisms and predict its behavior in silico (for a comprehensive review on this topic see  [22], [23], [24]). A variety of approaches have been developed in two-dimensions, such as the works in  [25] or  [26], in which using the same discretized model for angiogenesis developed in  [27] coupled with tumor growth they study drug delivery or glioma vascularization and invasion, respectively. Another example of a two-dimensional study is  [28], where the authors study the avascular to vascular tumor transition and the formation of a necrotic core in a fully continuous model. More recently, several authors have focused in vascular tumor growth in three dimensions; see for example [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. One critical difficulty of coupling tumor growth with angiogenesis in realistic problems is the scale barrier: while angiogenesis occurs almost at cellular scale, tumor dynamics takes place at the macroscale  [40]. Discrete tumor models are commonly used for cell-scale systems because cellular behavior and cell–cell interactions can be incorporated into the model easily. However, their computational cost becomes prohibitive for tumors that are detectable on usual imaging modalities. Continuum models have the capability of describing tumor dynamics with much less computational effort at macroscale [30], [31], [33], [38], [39]. The scale gap between angiogenesis and tumor growth is one of the reasons why only very few models can predict the switch from avascular to vascular growth in one theory, while keeping a continuous description of capillaries. Among the numerous models for tumor angiogenesis, the work developed by Travasso et al. in  [41] uses a hybrid approach that has shown promising predictive capabilities. The model features agent-based tip endothelial cell chemotactic migration coupled with continuous descriptions of tumor angiogenic factors and stalk and quiescent endothelial cells. The migration of TECs was augmented in  [42] to include haptotaxis. More recently, the model was coupled with a tumor growth model to simulate the angiogenesis switch in  [39], [43].

In this paper, we extend to three dimensions the model that we recently put forth in  [43]. We present a suitable computational method based on isogeometric analysis that permits a simple discretization of the model equations on complex geometries. The aim of this paper is to apply the model to early-stage colorectal cancer to study the role of angiogenesis and the avascular/vascular malignant transition within the mucosa and submucosa layers. To accomplish this, we perform a three-dimensional computation on the geometry of a macroscopic piece of colon tissue taken from a histopathological image.

This paper is organized as follows: Section  2 describes the model for vascular tumor growth including the phase-field models for cellular growth and reaction–diffusion equations for soluble substances. Section  3 presents the numerical method. Four numerical simulations are presented in Section  4. Finally, we draw conclusions in Section  5.