The Tumor Growth Process

The complexity of the tumor growth process involves many different phenomena. There are two main phases in this process; avascular and vascular. Tumor cells need oxygen and nutrients for proliferation, in the first phase, these supplements are provided by diffusion through the surrounding tissue. This is sufficient till the tumor size reaches to 2-3 millimeters and cells will need more supply. In this stage, the cells in the center of the tumor are exposed to the lack of oxygen and nutrients and form a necrotic core6. In order to survive and continue the proliferation, tumor cells need their own blood supply, therefor, they start to secrete some growth factors; The receptors on the nearest blood vessel respond to these factors and new micro
vessels are formed and reach the tumor, now cells are supplied to continue the proliferation. This phenomenon is called angiogenesis which its importance is addressed in some of the literature7–12. Tumor growth occurs at different spatial-temporal scales including processes at the atomic, molecular, microscopic and macroscopic scales respectively13.(fig.1) The most common modeling method used at the atomic scale is molecular dynamics (MD) simulation, in which atoms and molecules interact for a period of time3. Molecular scale represents an average of the properties of proteins' population. Cell signaling mechanisms and the natural regulators of biological systems are usually investigated at this scale14. The next scale is the microscopic scale which is also referred to as the tissue or multi cellular scale, and in some definition, it also includes the cellular scale. Models at this scale describe the transformation of normal cells, considering cell-cell and cell-matrix interactions and tumor environment heterogeneity13. The larger scale is the macroscopic scale that considers tumor as an organ and focuses on properties such as morphology, shape, invasion, etc.15. Mathematical models are used for these scales separately or combined to capture different related phenomena in different scales14,16. Figure 1. Different scales in tumor growth modeling

The goal of mathematical modeling and simulation is to offer a better understanding of tumor growth which will ultimately improve the therapeutic outcome.
When one knows about the outcome of different therapies for a specific patient, they can choose the best plan so that the patients expose to fewer side effects and experience a life with higher quality. Since the information available from image observations are mostly at macroscopic scale, we only review models in macroscopic scale which can mainly be divided into two groups: with and without considering mass effect. Then we will focus on the studies with image-based source of
information.

2.1 Diffusion-reaction models

Most of the models presented for tumor growth in macroscopic scale without considering mass effect 17–21, have used Reaction-Diffusion (proliferation–invasion) formulation as follow19:

∂c/∂t=∇.(D∇c)+R(c,t) (1)
η.(D∇c)=0 (2)

Where c is the tumor cell density and D is diffusion coefficient and R is the reaction (proliferation) term. This model describes the density of tumor cells and reflects the net rates of invasion and proliferation for the whole population of tumor cells22. It is known to be highly non-linear providing a unique profile of invasion at the leading edge of the imageable tumor23. The model considers the expansion of tumor detectable edge like a traveling wave that approaches a constant velocity v=2√Dρ 24.
Diffusion takes place due to tumor growth factor concentration gradient and the reaction term describes cell proliferation. Different models are proposed in the literature that mostly differ by the construction of the D tensor and the form of the proliferation term. There are different expressions for R in literature25–28 as exponential29, Logistic30 or Gompertz 31 (table.1).

Table 1. Different expressions of reaction term
Exponential Gompertz Logistic ρc ρc ln⁡(1⁄c) ρc(1-c)

Equation 2 shows no flux boundary condition, according to which tumor cells do not diffuse toward brain boundaries and ventricles with normal directions η.

2.2 diffusion-reaction-advection models

Two classes of cell motion models are broadly defined: Phenomenological and Mechanical models 32. Phenomenological models assume cells are moving through a combination of chemotaxis, haptotaxis, etc. ignoring mechanical effects32. In contrast, mechanical models focus on the influence of mechanical properties of the tumor and surrounding tissue on tumor growth 33. Bio-mechanical models, on the other hand, take into account the mass-effect of the tumors by modeling the deformation of the surrounding tissues based on their mechanical properties34. Cells live in an extracellular matrix(ECM) that provides structural and biochemical support to the cells35, cell adhesion molecules bind cells surfaces to ECM36, so the pressure produced by tumor growth will drive the ECM to deform, which in turn drifts cells. Deformation of the neighboring tissue induced by tumor growth is commonly referred to as mass-effect37. The importance of its modeling and simulation has been addressed in different publications34,38,39 for the purpose of registration and treatment planning. Most of the models simulate proliferation based on a reaction-diffusion equation and couple this with a bio-mechanical model for the mass-effect40–43.
In order to take to account this important parameter, Hogea et al.42 added mass effect to reaction-diffusion model and modeled glioma via a nonlinear reaction-advection-diffusion equation, with a two-way coupling with tissue elasticity equation. Their represented model is as follow:

∂c/∂t=∇.(D∇c)-∇.(cv)+R(c) (9)

Where the second term is the advection term used to take in to account the mass effect, v is cell drift velocity and R is the reaction term. They have considered the isotropic diffusion in white and grey matter, with diffusion coefficients D_w and D_g, respectively. They regarded the brain as a deformable solid in a restricted region and used the following general set of equations to describe its motion:

■(■(ρv ̇=∇.τ+b momentum@τ=λ∇.u+μ(∇u+〖∇u〗^T) constitutive)@■( v=u ̇ kinematics@m ̇=0 material properties)) (10)

Where v is the velocity field, u is the displacement field, τ is the Cauchy stress tensor and λ and μ are spatially varying Lame’s coefficients; b represents distributed forces and m denotes material properties including diffusion coefficient D and Lame’s coefficients in linear elasticity. Body force b is assumed to be proportional to cell density gradient:

b=-f∇c (11)

In this equation, f is a positive function that is related to tissue deformation. Developing this model they were successful to increase the accuracy of tumor growth prediction.

Role of image modalities in tumor growth modeling In recent years noninvasive imaging techniques have dramatically increased; several imaging techniques are now available to evaluate the tumor status quantitatively.
New functional imaging techniques integrate morphological, pathological and physiological alterations and are used as early predictors of the therapeutic response. They allow earlier assessment of therapy response by observing alterations in perfusion, oxygenation, and metabolism. The imageable information of the tumor depends on the tumor cell distribution and the imaging modality 44,45. In this section a brief description of usual modalities which are used in diagnosis, response assessment and follow-ups in oncology, is
presented.