Time Delays in Solid Avascular Tumour Growth

In the paper the model of early stage of tumour growth is described. Two main cellular processes are considered — proliferation and apoptosis. We focused on the effect of time delays in both processes. Mathematical analysis and computer simulations are presented. INTRODUCTION Many models which study different stages and effects of tumour growth were proposed and studied within last years, e.g. [1] – [7]. The model we study is based on the idea of avascular multicellular spheroids (MCS) modelling, see [3] – [7]. In this paper we focus on the case of uniformly proliferating tumour, i.e., MCS without a hypoxic region and necrotic core inside. We consider the diffusion of nutrient and two basic processes. One of them is a cell proliferation and second one is underlying apoptosis. The aim of this paper is to introduce time delays into both processes. For the case with equal delays some analysis was done in [7]. We consider the more general case, i.e., with two different delays, which is more interesting but also more difficult from the analytical point of view. 2 At the beginning, we formulate the basic model without delays. We assume that the growth of MCS is symmetric and the space co-ordinate is the radius r. We study the changes of two variables • σ(r, t) — the diffusiable chemical (a vital nutrient) concentration at radius r and time t, • R(t) — the outer MCS (tumour) radius at time t. The changes of nutrient (e.g. oxygen or/and glucose) are described by reaction diffusion equation. It is assumed that the nutrient is simply consumed by tumour cells with the consumption rate s. Because the tumour doubling time scale (weeks) is much longer than the nutrient diffusion time scale (minutes or hours) we make the quasi steady approximation in the nutrient equation. Therefore, we assume that the derivative of σ with respect to time is equal to 0 and obtain the following equation 1 r2 ∂ ∂r (r ∂σ ∂r ) = a, (1) where the left hand side of Eq. (1) represents Laplasian in spherical co-ordinates. The changes of MCS volume are governed by the principle of mass balance, i.e., 1 4π d dt ( 4 3 πR(t)) = S(t) −Q(t), (2) where S(t) = ∫ R(t) 0 sσ(r, t)rdr, Q(t) = ∫ R(t) 0 scrdr (3) are the total rate of cell proliferation and the total rate of cell death, respectively. In Eq. (3) s and sc are positive constants and denote the rates of cell proliferation and cell death within the tumour. For simplicity, we assume that s = 1 (if not we can re-scale the coefficients σe, a and c). We close the model be prescribing the following boundary and initial conditions σ(R(t), t) = σe, ∂σ ∂r (0, t) = 0, R(0) = R0, (4) where σe is the constant nutrient concentration external to the tumour. It is reasonable to assume that σe > c. Calculating σ from Eq. (1) under 3 the conditions defined in Eqs. (4) we obtain σ(r, t) = σe − a 6 (R(t) − r). (5) STATEMENT OF THE MODEL WITH DELAYS In this section we study the model with delays in proliferation and underlying apoptosis. Both of these processes incorporate time delays. In the first case, the delay represents the time taken for the cells to undergo mitosis. In the second one, the delay represents the time taken for the cells to modify the rate of cell loss due to apoptosis. We assume that these delays are constant (τ1, τ2 > 0). Hence, instead of Eq. (3) we consider