Modelling Acidosis and the Cell Cycle in Multicellular Tumour Spheroids

Abstract

A partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells within a multicellular tumour spheroid. The rates of cell quiescence and necrosis are assumed to depend upon the local nutrient and acid concentrations. Quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells and a description of anaerobic metabolism by the cells is included. Parameterised with data from the literature, our model predicts that the tumour size is reduced in the presence of acid. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Instead it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis.

1. Introduction

Multicellular tumour spheroids (MCTS) are three dimensional cellular aggregates which mimic many of the characteristics of in vivo tumours [13]. They have been the focus of research for experimentalists and applied mathematicians over the past thirty years [2, 18]. Initial mathematical modelling work in the area focused on simple models describing MCTS growth in the context of nutrient delivery to the tumour [5, 10]. With time and further experimental understanding a number of continuum mathematical models have been developed which have focused on the effect certain biological mechanisms (biochemical and biomechanical) have on MCTS development, [4, 14, 25] are but a few examples. The work presented here involves mathematical modelling of two different aspects of tumour growth; the cell cycle and acidosis and the effects both of these have on MCTS growth.

The cell cycle is a series of tightly regulated biochemical events which control the growth and development of a cell. To summarise; newly divided cells grow during the Gap-1 phase G₁, before entering a period during which their DNA is synthesised (S-phase). Gap-2 (G₂), a short period following S-phase, allows the cell time to prepare for cell division, involving splitting of the DNA spindle and physical division of the cell in two (M-phase). The newly generated cells may continue around this proliferation cycle or may enter a period of extended time in G₁. Such cells are defined to have entered the quiescent G₀ phase. Entering such a phase is usually driven by factors external to the cell, for instance a decrease in growth factors or nutrient deprivation. Cells may undergo two basic forms of cell death; apoptosis or necrosis. Apoptosis is a decision by a proliferating cell to commit cell ‘suicide’. In doing so the cell fragments into smaller pieces which may then passively move throughout the tumour. Physiological events over time, such as decreased nutrient concentration within the tumour or high acidity, can have harmful effects on quiescent cells and may eventually lead to necrosis; the breaking down of the cellular wall and release of cell contents into the extracellular environment.

Recent experimental and mathematical modelling work has elucidated the importance of pH levels on tumour morphology. Noninvasive magnetic resonance (MR) techniques have been developed to measure both intracellular pH (pH_I) and extracellular pH (pH_X) of human and animal tissues [8, 9]. Virtually all tumour pH data to date show an acidic pH_X and alkaline pH_I relative to normal tissue. Moreover, it is found that the pH_X becomes more acidic as the tumour grows, consistent with reduced perfusion [9]. Clinical specimens have shown that these changes have a molecular basis in upregulation of the glucose transporter 1 and the Na⁺/H⁺ exchanger [7]; the distribution of upregulated cells is in excellent agreement with predictions of mathematical models [20].

In this paper we consider a mathematical model of a MCTS which includes a simple model of the cell cycle, where cells are considered to exist in either a proliferating or quiescent state. The model includes a description of the nutrient and acid concentration within the MCTS. In what follows, the effect they have on the distribution of proliferating and quiescent cells and overall tumour size is investigated.

2. Model Formulation

Let P(x, t) and Q(x, t) represent the density of proliferating and quiescent cells per unit volume, respectively, whose mass conservation is described by

$$P_t+\nabla ·(\mathbf{u}P)=(K_B(C)-K_Q(C,H)-K_A)P+K_P(C)Q,$$ (1)

$$Q_t+\nabla ·(\mathbf{u}Q)=K_Q(C,H)P-(K_D(C,H)+K_P(C))Q,$$ (2)

where u(x, t) represents the local velocity of the cells and K_I (I = B, P, Q, D, A) are cell cycle transition rates which we assume are dependent upon the local diffusible nutrient C(x, t) and extracellular hydrogen ion (acid) concentration H(x, t). Spheroid growth is dependent upon oxygen and glucose levels throughout the tumour, but for simplicity in what follows we consider a generic nutrient. Quiescent cells are assumed to be essentially metabolically inactive, and hence produce less hydrogen and use less oxygen than proliferating cells.

Very little quantitative data is available on the cell cycle reaction rates and therefore we will take simple expressions which capture the qualitative behaviour,

$$K_B(C)=k_{B}C,$$ (3)

$$K_P(C)=k_{P}C,$$ (4)

$$K_Q(C,H)=k_Q\times (C_{\infty }-C)+k_Q^{\prime }\times (H-H_{\infty }),$$ (5)

$$K_D(C,H)=k_D\times (C_{\infty }-C)+k_D^{\prime }\times (H-H_{\infty }),$$ (6)

$$K_A=k_A{C}_{\infty }.$$ (7)

Here K_B(C) represents the rate of cell birth, K_P(C) is the rate of cell transfer from the quiescent to proliferating compartments, K_Q(C, H) is the rate at which cells move from the proliferating to quiescent compartment (quiescence), K_D(C, H) is the rate of cell death from the quiescent cell compartment (necrosis), K_A(C) is cell death from the proliferating cell compartment (apoptosis) and C_∞ and H_∞ denote the concentration of nutrient and acid at the tumour boundary, respectively, which are assumed to be constant. We note that C < C_∞.

This acidification leads to death of normal cells due to activation of p53-dependent apoptosis pathways, as well as loss of function of critical pH-sensitive genes. Tumour cells, however, are relatively resistant to acidic pH_X; due to part to mutant p53 genes. Whilst normal cells die in environments with a persistent pH below about 7, tumour cells continue to proliferate in a relatively acidic medium (pH 6.8) [6]. Beyond this point quiescence and eventually necrosis occur [16]. This biological knowledge is reflected in the independence of apoptosis K_A on H, and the monotonic increase of quiescence K_Q and necrosis K_D with H.

Given that the rate of diffusion of nutrient throughout the spheroid is rapid compared to the time scale of growth, we adopt the standard quasi steady-state assumption [25]

$$D_C{\nabla }^{2}C={\sigma }_C(P+{\epsilon }_{C}Q)C.$$ (8)

This equation has two nutrient consumption terms, one relating to proliferating cells (σ_C) and the other to quiescent cells (σ_Cε_C). Here D_C is the nutrient diffusion coefficient.

In the case of acid diffusion throughout the spheroid, we also make a quasi steady-state assumption

$$D_H{\nabla }^{2}H=-(P+{\epsilon }_{H}Q)({\sigma }_H+{\sigma }_H^{\prime }(C_{\infty }-C)),$$ (9)

where the acid diffuses at a rate D_H and σ_H and σ_Hε_H represent the production of acid by proliferating and quiescent cells respectively.

In respect of equations (8) and (9) ε_C and ε_H represent the fact that quiescent tissue is essentially metabolically inactive, consuming significantly less oxygen than its proliferating counterpart and producing significantly fewer hydrogen ions. Tumours rely on anaerobic metabolism and hence produce acid at a rate σ_H under normoxic conditions (the Warburg effect [24]); nonetheless, as oxygen levels decrease, acid production increases linearly at rate ${\sigma }_H^{\prime }$ (the Pasteur effect [17]).

To determine the motion of the cells we assume there is no additional space in the tumour (incompressibility) so that

$$N=P+Q,$$ (10)

where N is a constant. This also assumes that cells have similar size. A mass balance equation can then be found by adding equations (1) and (2) and then using (10) to obtain

$$\nabla ·\mathbf{u}=\frac{1}{N}((K_B(C)-K_A)P-K_D(C,H)Q).$$ (11)

We impose the following boundary and initial conditions. For boundary conditions on C and H we let C(x, t) = C_∞ and H(x, t) = H_∞. At the same point we impose a zero flux condition on the cells, namely ∇ · (uP) = 0. For the initial conditions we assume all cells are initially proliferating, i.e. P = P₀, Q = 0 and the nutrient and acid concentration throughout the MCTS is constant, C(x, t) = C_∞ and H(x, t) = H_∞. For simplicity we assume no cell shedding.

2.1. Non-dimensionalisation

Equations (1), (2), (8), (9) and (11) are non-dimensionalised according to

$$\begin{array}{l}\xi =r/L,\tau =t/T,p=P/N,q=Q/N,\\ h=(H-H_{\infty })/H_0\text{and}\upsilon =T\mathbf{u}/L,\end{array}$$ (12)

where $L=\sqrt{D_C/(N{\sigma }_C)}$ and T = 1/(k_BC_∞) represent length and timescales, respectively and the acid concentration is scaled with respect to H₀ = (σ_HD_C)/(σ_CD_H). In one-dimensional spherical polar co-ordinates, radial symmetry assumed, the model becomes

$$p_{\tau }+\nabla ·(\upsilon p)=({\kappa }_b(c)-{\kappa }_q(c,h)-{\kappa }_a)p+{\kappa }_p(c)q,$$ (13)

$$q_{\tau }+\nabla ·(\upsilon q)={\kappa }_q(c,h)p-({\kappa }_d(c,h)+{\kappa }_p(c))q,$$ (14)

$${\nabla }^{2}c=(p+{\epsilon }_{C}q)c,$$ (15)

$${\nabla }^{2}h=-(p+{\epsilon }_{H}q)(1+{\sigma }_h(1-c)),$$ (16)

$$\nabla ·\upsilon =({\kappa }_b(c)-{\kappa }_a)p-{\kappa }_d(c,h)q$$ (17)

where the non-dimensional rates are given by

$$\begin{array}{c}{\kappa }_b(c)=c,{\kappa }_p(c)=k_{p}c,{\kappa }_q(c,h)=k_q(1-c)+k_q^{\prime }h,\\ {\kappa }_d(c,h)={\kappa }_d(1-c)+{\kappa }_d^{\prime }h\text{and}{\kappa }_a=k_a,\end{array}$$ (18)

and the non-dimensional parameters are

$$k_i=\frac{k_I}{k_B},{\sigma }_h=\frac{{\sigma }_H^{\prime }{C}_{\infty }}{{\sigma }_H}\text{and}{k}_i^{\prime }=\frac{k_I^{\prime }{H}_0}{k_B{C}_{\infty }}$$ (19)

for i = p, q, d, a and I = P, Q, D, A.

The initial and boundary conditions become

$$\begin{array}{c}p(\mathrm{\Gamma }(\tau ),0)=1,q(\mathrm{\Gamma }(\tau ),0)=0,c(\mathrm{\Gamma }(\tau ),0)=1,\\ h(\mathrm{\Gamma }(\tau ),0)=0,\upsilon (0,\tau )=0,p_{\xi }(0,\tau )=q_{\xi }(0,\tau )=0,\\ c_{\xi }(0,\tau )=h_{\xi }(0,\tau )=0,c(\mathrm{\Gamma }(\tau ),\tau )=1,h(\mathrm{\Gamma }(\tau ),\tau )=0,\end{array}$$ (20)

where $\mathrm{\Gamma }(\tau )=\frac{R(t/T)}{L}$ is the non-dimensional spheroid radius with

$$\upsilon (\mathrm{\Gamma },\tau )=\frac{d\mathrm{\Gamma }}{d\tau }\text{and}\mathrm{\Gamma }(0)={\mathrm{\Gamma }}_0,$$ (21)

and ∇ · (υp) = 0 on the boundary.

2.2. Parameter estimation

The model is dependent on ten non-dimensional parameters as set out in Table 2. These may be estimated from parameters available in the literature as shown in Table 1.

Table 2.

Non-dimensional model parameters.

Parameter	Value	Parameter	Value
Γ0	3	kq	0.9
εC	10−2	$k_q^{\prime }$	4 × 10−2
εH	10−2	kd	5 × 10−2
σh	0.6	$k_d^{\prime }$	8 × 10−4
kp	5 × 10−2	ka	7 × 10−2

Table 1.

Dimensional parameter values.

Parameter	Value	Description	Reference
DC	1.5 × 10−5 cm2 s−1	Nutrient (oxygen) diffusion coefficient.	[15]
σC × N × C∞	2.2 × 10−1 mM s−1	Tissue nutrient (oxygen) consumption rate.	[3]
εH	10−2	Quiescent: proliferating metabolic ratio.	[16]
C∞	5 × 10−2	mM Normal nutrient (oxygen) concentration.	[3]
DH	1.1 × 10−5 cm2 s−1	Hydrogen ion diffusion coefficient.	[16]
N × σH	5 × 10−5 mM s−1	Hydrogen ion production rate.	[16]
H∞	5.6 × 10−5 mM	Normal hydrogen ion concentration.	[16]
R0	5–50 μm	Tumour cell radius.	[1]
cQ	10−1	Non-dimensional nutrient-induced quiescence threshold.	[11]
cD	5 × 10−2	Non-dimensional nutrient-induced necrosis threshold.	[1]
HQ	4 × 10−4 mM	Acid-induced quiescence threshold.	[16]
HD	10−3 mM	Acid-induced necrosis threshold.	[16]
σ0:1	1.6	Anoxic: normoxic acid production ratio.	[19]

Values for the cell cycle rates specified in the model are not readily available and as such we have determined these as follows. By definition, each time unit τ corresponds to one full cell cycle period, which we take to be 12 hours. From this and knowing C_∞ we can utilise T to determine k_B = 4.6 × 10⁻⁴ mM⁻¹ s⁻¹.

Previous authors have taken k_p = 0.05 and k_q = 0.9 [21], which assumes a high rate of quiescence and few cells returning from the quiescent to proliferating compartment. Using this value of k_q, and assuming that κ_q(c_Q, 0) = κ_q(1, h_Q) allows us to calculate $k_q^{\prime }=4\times {10}^{-2}$. This assumes that the rate of quiescence at the quiescent threshold is the same, whether due to nutrient deprivation or acid concentration. In order to calculate the non-dimensional parameters h_Q and h_D, we have used equation (19) to obtain H₀ = 1.5 × 10⁻⁵ mM, and then using equation (12) h_Q = 20 and h_D = 60 follow.

To calculate k_d and $k_d^{\prime }$, we use the necrosis thresholds c_D and h_D (see Table 1), and assume that these correspond to the points at which more cells die than resume proliferation, i.e. κ_d = κ_p. This leads to k_d = 5 × 10⁻² and $k_d^{\prime }=8\times {10}^{-4}$. Investigations have shown that widespread apoptosis occurs in cells fixed mid cycle after around one week, or 14 time units [12]. As such, we approximate k_a = 1/14 = 7 × 10⁻².

In order to calculate a value for σ_h we take σ_h + 1 = σ_0:1 as the anoxic to normoxic tumour acid production ratio. Utilising the value of σ_0:1 in Table 1 yields σ_h = 0.6.

3. Model simulations and results

Numerical solutions to the system of equations (13)–(17), with the respective boundary and initial conditions, were found using the weighted average flux (WAF) method for hyperbolic equations [23] and Successive Over Relaxation (SOR) method. The parameters used are those shown in Table 2. Model solutions are plotted at t =1, 4, 10, 20, 30 and 40 days, respectively. Although we solve the dimensionless system (13)–(17), for illustrative purposes our results are presented in dimensional form.

Typical model results are shown in Figure 1((a) – (d)). It can be seen that there is a higher concentration of nutrient (Figure 1(b)), and a lower concentration of acid (Figure 1(d)) around the edge of the spheroid, leading to a focusing of proliferating cells at the boundary of the spheroid (Figure 1(a)). The spatial velocity profile of the cells is shown in Figure 1(c). The results here are similar to those found elsewhere, for instance in [25].

Figure 1.

(a)–(d) numerical simulations typical of the model, (e)–(f) demonstrate the effect of acid on the overall size of the tumour. Parameter values used were as detailed in Table 2.

A comparison of the model with and without acid present is shown in Figure 1((e) and (f)). The proliferating and quiescent cell distributions along with that of their respective velocities, were qualitatively equivalent to Figures 1(a)–(d). However, including acid in the model leads to a smaller spheroid at steady-state and slightly slower rate of growth as demonstrated in Figures 1(e) and (f), respectively. This result is not surprising, in our model decreased nutrient and increased acid concentrations lead to higher quiescence and necrosis, and hence a smaller tumour. Indeed, in a study of a variety of MCTS, final cell counts fell with increasing background acidity or decreasing nutrient (oxygen) levels [7].

3.1. Nutrient consumption and acid production

In the work which follows we wish to understand the effect that different rates of nutrient consumption and acid production by proliferating and quiescent cells, respectively have on the overall growth and development of the tumour.

Figure 2 is the result of varying the value of ε_C, the difference in the rate of nutrient consumption by proliferating and quiescent cells. Figures 2(a) and 2(b) compare the total radius and the radial velocity for four values of ε_C when hydrogen ions are present with ε_H = 0.01, and when no hydrogen ions are present. Increasing the value of ε_C results in a slower rate of growth and a smaller steady-state radius; an increased rate of oxygen consumption by quiescent cells means the nutrient concentration falls more rapidly and more cells become quiescent. Thus there are fewer proliferating cells available to increase the tumour size.

Figure 2.

(a) and (b) comparing the effect of different rates of nutrient consumption by quiescent versus proliferating cells on the overall size of the MCTS. (c)–(f) Results for different values of ε_C, the nutrient consumption ratio of proliferating to quiescent cells. In both cases all other parameters were held constant as defined in Table 2.

However, increasing ε_C leads to a lower concentration of acid throughout the tumour as shown in Figure 2(d), a rather counterintuitive result. We expect that decreasing the nutrient concentration would lead to a higher acid concentration via anaerobic metabolism (the term σ_h(1 − c) in equation (16)). Our results demonstrate, however, that a decrease in nutrient concentration leads to a higher rate of quiescence and therefore fewer proliferating cells. Since the quiescent cells produce less acid than proliferating cells via ε_H, this then leads to a decrease in the acid concentration. This result is demonstrated by setting ε_H = 0; as ε_C increases the acid concentration decreases (results not shown).

Our finding here shows the importance that the cell cycle state of the cells can have on the development of the tumour. By anaerobic metabolism alone, we would expect an increase in the acid concentration throughout the tumour. Including a description of proliferating and quiescent cells and the difference in acid production between them can counteract this effect. When the overall number of proliferating cells is greater than that of quiescent cells we expect a higher acid concentration as dictated by ∊_H ≪ 1. When, however, when the number of proliferating cells is less than the number of quiescent cells, less acid is produced and although the nutrient concentration is low, the effect of anaerobic metabolism on the acid concentration is secondary to that of the cell cycle state. We note that varying σ_h alone had only a marginal affect on the acid concentration, i.e. 0 ≤ σ_h ≤ 1 only resulted in a decrease in the spheroid radius of R ~ 0.025.

3.1.1. Nutrient and acid effect on quiescence and necrosis

The rates of cell quiescence and necrosis are dependent upon both the concentration of nutrient and acid. In this section we wish to quantify and compare the effect that nutrient and acid concentration have on the transition to these cell states.

In order to compare cell necrosis via a reduction in the nutrient or an increase in acid concentration, we define

$$f_d=\frac{{\int }_{\mathrm{\Omega }}{k}_d(1-c)-k_d^{\prime }hd\mathrm{\Omega }}{{\int }_{\mathrm{\Omega }}{k}_d(1-c)+k_d^{\prime }hd\mathrm{\Omega }},$$ (22)

where Ω is the volume of the tumour at steady-state. When f_d = 1 necrosis is dominated by a decrease in nutrient concentration, whilst when f_d = −1 acidosis has a greater effect than nutrient deprivation. We varied k_d and $k_d^{\prime }$ according to 0.01 ≤ k_d ≤ 0.1 and ${10}^{-4}\le k_d^{\prime }\le {10}^{-3}$, respectively. Although $k_d^{\prime }$ is an order of magnitude less than k_d the results of Figure 3(a) show that necrosis is affected more by acidosis than by a decrease in the nutrient concentration.

Figure 3.

The effect of nutrient and acid concentration on cell necrosis (a) and quiescence (b). Here f_d and f_q are as defined by equations (22) and (23), respectively.

Cellular quiesence via either a decrease in nutrient or increase in acid concentration, can be compared in a similar way such that

$$f_q=\frac{{\int }_{\mathrm{\Omega }}{k}_q(1-c)-k_q^{\prime }hd\mathrm{\Omega }}{{\int }_{\mathrm{\Omega }}{k}_q(1-c)+k_q^{\prime }hd\mathrm{\Omega }}.$$ (23)

Here we have varied k_q and $k_q^{\prime }$ according to 0.1 ≤ k_q ≤ 1 and $0.01\le k_q^{\prime }\le 0.1$, respectively and as shown in Figure 3(b) cell quiescence is most sensitive to a change in acid concentration.

4. Summary and Conclusions

A mathematical model describing the growth of a MCTS and the effect that nutrient and acid concentrations have on the distribution of proliferating and quiescent cells throughout the tumour has been formulated and solved.

We have found that the distribution of proliferating and quiescent cells can have important consequences on the amount of acid produced in the tumour. Decreased nutrient levels do not necessarily lead to excessive acid concentration via anaerobic metabolism. Instead, because quiescent cells produce less acid than proliferating cells, when the nutrient concentration is low, there are more quiescent and fewer proliferating cells and hence the increased effect of anaerobic metabolism is negligible. Our result here is dependent upon the rates of anaerobic metabolism and cells moving from the proliferating to quiescent cell compartments, which have been obtained from experimental data in the literature.

Analysis of the model has also quantified the difference in cell quiescence and death due to either the local nutrient or acid concentration. In the case of necrosis we varied the rates of necrosis due to nutrient deprivation and acidosis, respectively, over one order of magnitude. By then comparing the overall number of dead cells produced via each mechanism we were able to determine that acidosis affects necrosis more than a reduction in the nutrient concentration. A similar result was found in the case of quiescence, i.e. cells are more likely to quiesce as a result of acidosis than a reduction in the nutrient concentration (see Figure 3(b)).

Our results have shown that whilst nutrient and acid concentration are independently important in affecting tumour growth, in understanding the ‘bigger picture’ one needs to account for the effect that the cell cycle state of the cells within the tumour has on their concentration. Effectively, the cell cycle state of the cells and respective nutrient and acid concentrations are thus interdependent systems. This result could be taken into account in respect of therapeutic strategies; it is important to quantify the ratio of proliferating and quiescent cells within the tumour in order to understand the effect of decreased nutrient concentrations. A reduction in nutrient concentration may not simply lead to an increase in the acid concentration within the tumour.

In the work presented here we have not considered the effect of acidosis on the rate of apoptosis nor have we accounted for the difference in intracellular and extracellular acid and their subsequent effects. Furthermore a full account of dead cell matter accummulation, via both apoptosis and necrosis, as included in [22], should be considered in future work.