Abstract
The processes underpinning solid tumour growth involve the interactions between various healthy and tumour tissue components and the vasculature, and can be affected in different ways by cancer treatment. In particular, the growthlimiting mechanisms at play may influence tumour responses to treatment. In this paper, we propose a simple ordinary differential equation model of solid tumour growth to investigate how tumourspecific mechanisms of growth arrest may affect tumour response to different combination cancer therapies. We consider the interactions of tumour cells with the physical space in which they proliferate and a nutrient supplied by the tumour vasculature, with the aim of representing two distinct growth arrest mechanisms. More specifically, we wish to consider growth arrest due to (1) nutrient deficiency, which corresponds to balancing cell proliferation and death rates, and (2) competition for space, which corresponds to cessation of proliferation without cell death. We perform numerical simulations of the model and a steadystate analysis to determine the possible tumour growth scenarios described by the model. We find that there are three distinct growth regimes: the nutrient and spatially limited regimes and a bistable regime, in which both growth arrest mechanisms are simultaneously active. Thus, the proposed model has the features required to investigate and distinguish tumour responses to different cancer treatments.
1 Introduction
Solid tumour growth is a complex process involving interactions between multiple different cell types, e.g. healthy, tumour and immune cells, as well as other tissue components such as the extracellular matrix and the vasculature. Each of these, and their interactions, can be affected in different ways by anticancer therapies, which has prompted extensive experimental and theoretical research on tumour growth dynamics. In particular, many mathematical models, of varying complexity, have been proposed to describe tumour growth. These models typically include a single, or a grouping of, growthlimiting process(es) which can significantly influence treatment response. In this paper, we propose an ordinary differential equation (ODE) model of tumour growth that distinguishes between two alternative mechanisms for growth arrest of a tumour population. In this way, our simple model lends itself to future studies of the effectiveness of different combination cancer therapies.
Tumour growth dynamics. Tumour growth is typically separated into two stages: the avascular and the vascular stages (Chaplain 1996). During the avascular phase, all tumour cells initially proliferate by consuming nutrients that diffuse from blood vessels in neighbouring healthy tissue, resulting in an initial exponential growth of the tumour. As the tumourās size increases and cell numbers grow, the outermost cells begin to form a proliferative rim while those in the central regions gradually become quiescent (i.e. alive, but nonproliferative) as their access to crucial nutrients is cut off. This results in linear growth of the tumour. Quiescent cells eventually die due to prolonged nutrient deprivation, forming a necrotic core within the tumour that decays at a certain rate. Thus, an avascular tumour gradually approaches a diffusionlimited, equilibrium size as the rate of growth due to cell proliferation in the welloxygenated outer rim balances with the rate of cell degradation in the nutrientstarved necrotic region. In order to progress, the tumour must develop its own vasculature via angiogenesis (Chaplain 1996), which triggers the growth of new blood vessels towards the tumour. Once vascularised, a tumour can access the nutrients necessary for further, rapid growth and discard metabolic waste. Blood vessels additionally provide a means for tumour cells to travel to other tissues, where they may establish secondary tumours or metastases.
Mathematical models of tumour growth and their growthlimiting mechanisms. Both stages of tumour growth have been modelled, either independently or in combination, using a range of approaches. This includes phenomenological ODE models as well as more detailed, spatially resolved models such as partial differential equation (PDE) models, discrete, individualbased models (IBMs) and hybrid models combining differential equations and IBMs. For detailed reviews of spatially averaged continuum models, see Murphy etĀ al. (2016), and, of spatially resolved continuum, discrete and hybrid models of tumour growth, see Araujo and McElwain (2004), Byrne (2012), Roose etĀ al. (2007), Cristini and Lowengrub (2010), Martins etĀ al. (2007), and Deisboeck etĀ al. (2011), respectively.
A common characteristic of most of these models is their focus on describing either a single growthlimiting process or a group of growthlimiting processes, whose effects cannot be distinguished. For example, some phenomenological ODE models (e.g. Gompertz, logistic, etc. (Murphy etĀ al. 2016)) predict the growth of an avascular tumour to a constant (in time) limiting size, called the carrying capacity. The latter can be defined in different ways: the available free space for the tumour to occupy (Liu etĀ al. 2021), the volume that can be reached given nutrient availability (Zahid etĀ al. 2021), among others (Milzman etĀ al. 2021). Regardless of the precise definition of this carrying capacity, a single mechanism for growth arrest of the tumour is explicitly considered: a tumour reaches steady state as the proportion of proliferating cells converges to zero (with no explicit cell death). This also holds for ODE models that incorporate a timedependent carrying capacity, e.g. the model of vascular tumour growth developed by Hahnfeldt etĀ al. (1999), which assumes a timedependent carrying capacity to account for dynamic changes in the vascular support available to the tumour.
In terms of spatially resolved models, the seminal model developed by Greenspan (1972) describes diffusionlimited (avascular) tumour growth, where the tumour reaches an equilibrium size as the death of cells in nutrientpoor regions balances the birth of cells in nutrientrich regions. This same single growth arrest mechanism is also described by the multiphase model of avascular tumour growth developed by Lewin etĀ al. (2020). Further, the model of vascular tumour growth developed by Panovska etĀ al. (2007), which comprises a coupled system of nonlinear PDEs, also depicts a tumour that reaches an equilibrium size as cell death and cell birth balance one another. However, they consider mechanical constraints to tumour growth (which limit cell proliferation) in addition to those imposed by nutrient availability (which lead to cell death), i.e. they incorporate several growthlimiting processes, but their effects are combined from a mathematical perspective. In contrast to the aforementioned spatially resolved, continuum models, Enderling etĀ al. (2006) extend the model of tumour invasion developed by Anderson (2005) to represent a tumour which attains an equilibrium size as the remaining available space for tumour cells decreases to zero. In particular, they predict growth arrest due to a cessation of proliferation, with no cell death. As a final example, Drasdo and HĆ¶hme (2005) developed a discrete model of in vitro tumour growth which describes the growth of monolayers and multicellular spheroids. The key growthlimiting factor for monolayers is contact inhibition, whereas it is nutrient deficiency for spheroids. Despite the differences between these two growthlimiting processes, they both can lead to a tumour reaching an equilibrium size due to the balance of cell proliferation and cell death.
The above brief and nonexhaustive summary illustrates that a significant amount of research has focused on developing models of solid tumour growth of increasing biological complexity. Yet, they all typically share the common feature of describing a single mechanism by which a tumour population may reach a steady state, i.e. via cessation of cell proliferation without any cell death or via the balancing of cell proliferation and cell death. The question of how to strike the balance between the biological detail of a mathematical model, which constrains it, and its suitability for making clinically relevant predictions (e.g. through data fitting and parameter estimation) therefore poses itself.
Structure of the paper. In the present paper, we aim to derive a model of vascular tumour growth that retains the simplicity of phenomenological models, while providing additional mechanistic insight and capturing the key behaviour of more complex models. We focus on vascular tumours as, in practice, tumours are likely to have reached this stage by the time they are detected. Further, the effect of various treatments on the vasculature can be significant, especially when considering combination treatments. A main feature we seek for our model is that it can distinguish between two alternative growthlimiting mechanisms:

1.
growth arrest in response to nutrient deficiency, which translates into the balancing of cell proliferation and death,

2.
growth arrest in response to competition for space, which translates into a cessation of all proliferation, with no cell death.
As a result, the model provides a means to understand how mechanisms of growth limitation may affect tumour responses to treatment, particularly when both mechanisms are simultaneously active.
This paper is structured as follows. In Sect.Ā 2, we derive a new ODE model of vascular tumour growth. We investigate the model behaviour in Sect.Ā 3 by performing numerical simulations and a steadystate analysis. The paper concludes in Sect.Ā 4, where we discuss our findings and avenues for future work.
2 The Mathematical Model
In this section, we present a new model of vascular tumour growth, which considers the interactions of a tumour with the physical space in which it is growing and a nutrient supplied by the tumour vasculature. Our first key simplifying assumption is that the vascular volume remains constant during tumour growth. In particular, we neglect angiogenesis and vascular remodelling and view the vascular volume, V, as a model parameter (rather than a dynamic variable) which influences the availability of nutrient and space and, thus, the tumourās carrying capacity. We make this simplifying assumption in order to limit the complexity of the ODE model we are proposing, which enables us to focus on our aim of describing two different growthlimiting mechanisms using a model that exhibits rich solution structure despite its simplicity. We will come back to this point in the Discussion.
Now, denoting the tumour cell volume by T and the nutrient concentration by c, we propose the following system of timedependent ODEs:
where
Our model is based on the following assumptions:

There is a fixed amount of physical space, \(S_{\max }\), which can be occupied by T and V; the available free space is \(S = S_{\max }  (T+V)\).

Tumour cells, T, need space and nutrient in order to proliferate. We assume that they proliferate at a rate proportional to the nutrient concentration, c, and the amount of free space available, S, with constant of proportionality \(q^*_2 > 0\).

Below a fixed nutrient concentration, \(c^*_{\min }\), satisfying \(0< c^*_{\min } < c^*_{\max }\), tumour cells die of nutrient starvation at a rate that is a monotonically decreasing function of c for \(0 \le c < c^*_{\min }\) such that (i) no tumour cells die at \(c=c^*_{\min }\) and (ii) when \(c=0\) the death rate of tumour cells attains its maximal value of \( \delta ^*_1 c^*_{\min }\), where \(\delta ^*_1 > 0\) is a constant of proportionality. We refer to \(c^*_{\min }\) as the threshold for severe hypoxia, which is defined as a state of oxygen deficiency.

A nutrient, c, chosen to be oxygen, is supplied by the tumour vasculature, V, at a rate that is a monotonically decreasing function of c for \(0 \le c \le c^*_{\max }\) such that (i) no oxygen is supplied once the oxygen concentration in the tumour attains the value \(c^*_{\max } > 0\) and (ii) the maximal supply rate of oxygen is attained when \(c=0\) and is equal to \(g^* c^*_{\max }\), with \(g^* > 0\). Here, \(c_{\max }^*\) represents the oxygen concentration in the vasculature (note that, from (2), \({c}\le {c}^{*}_{\text {max}}\)) and \(g^*\) represents the rate of oxygen exchange per unit volume area of blood vessels.

Tumour cells consume c for maintenance at a rate proportional to c, with rate constant \(q^*_1 > 0\). They also consume c for proliferation at a rate proportional to the proliferation rate, with a conversion factor \(k^* > 0\) defined such that \(q^*_3 = \frac{q^*_2}{k^*} >0\).
In order to reduce the number of parameters in our model, we nondimensionalise the system (1)ā(3) by introducing the following scalings:
Recalling that \(V \equiv V^*_0\) and dropping hats for notational convenience, we obtain the following system:
where
Given that we defined \(k^* >0\) such that \(q^*_3 = \frac{q^*_2}{k^*}\), we can define \(k > 0\) such that \(q_3 = \frac{q_2}{k}\), where
3 Investigating Model Behaviour
3.1 Numerical Simulations: Tumour Growth Curves
In this section, we present numerical simulations of the tumour growth model (4)ā(5).
Numerical setup. We solve the ODE system (4)ā(5) for \(t \in (0, T]\), where \(T > 0\), using ODE45, a singlestep MATLAB builtin solver for nonstiff ODEs that is based on an explicit RungeāKutta (4,5) formula, the DormandāPrince pair. We impose the initial conditions \((T(0),c(0))= (0.05,1)\). We choose \(T(0)=0.05\) so that the tumour is initially much smaller than the available free space (i.e. \(0 < T \ll 1\)), but also large enough to be vascular with \(V_0 \in (0,0.005]\). The choice of \(c(0)=1\) is so that we consider tumours which are, initially, well oxygenated.
The focus of this work is on characterising the qualitative behaviour of the model and assessing its suitability for benchmarking combination cancer treatments. As such, accurately estimating parameter values by fitting the model to experimental data is beyond the scope of this work. We instead combine parameter values from different sources in the literature and preliminary numerical simulations to fix the values of \(c_{\min }\), g and k. Further, we consider a biologically realistic range of possible values for the remaining parameters. Our dimensionless parameter choices, listed in Table 1, are motivated by the arguments included in āAppendix A1ā.
Remark 1
A key simplifying assumption we make in defining the model parameters is that the dimensionless rate of cell death due to nutrient starvation, \(\delta _1\), is equal to the dimensionless rate of cell proliferation, \(q_2\). This enables us to reduce the number of system parameters, making it more tractable. We motivate this assumption with experimental evidence which suggests that tumour cell proliferation and death rates can be highly correlated for some tumours (Leoncini etĀ al. 1993; Liu etĀ al. 2001; Vaquero etĀ al. 2004). This would suggest that these parameters are proportional to each other. Our numerical and analytical studies reveal that the key results in this paper are not affected by the value of this constant of proportionality (results not shown) and we, therefore, set it equal to 1.
Simulating tumour growth. For fixed values of \(q_1\) and \(q_3\), we observe how tumour growth dynamics change as we vary the initial vascular volume, \(V_0\). The results presented in Fig.Ā 1 show that the model exhibits initial exponential or linear growth followed by a growth slow down as the tumour reaches a limiting size. These dynamics are characteristic of both experimental data and existing phenomenological models (Drasdo and HĆ¶hme 2005; Koziol etĀ al. 2020; Murphy etĀ al. 2016).
We note also that the tumourspecific parameters \(q_1\), \(q_3\) and \(V_0\) appear to determine the tumourās limiting size, which can vary quite significantly from one parameter set to another (Figs.Ā 1c, e). In particular, studying Fig.Ā 1d, f, which represent the evolution of the logarithm of the oxygen concentration in the tumour, c, alongside Fig.Ā 1c, e, suggests that the large differences between tumour limiting sizes arise as a result of whether c is above or below the oxygen threshold for severe hypoxia, \(c_{\min }\). This hints at the existence of different growth regimes, which could be characterised by different growthlimiting mechanisms. We investigate this further by performing a steadystate analysis, which is presented in the following section.
3.2 SteadyState Analysis
We now perform a steadystate analysis for the system (4)ā(5) in order to understand how varying \(q_1, q_3\) and \(V_0\), the oxygen consumption rates for proliferation and maintenance and the initial vascular volume, respectively, impacts the equilibrium tumour volume attained in the long term. The model dynamics are different depending on whether we are in a nutrientrich regime, i.e. \(c_{\min } \le c \le 1\), or in a nutrientpoor regime, i.e. \(0 \le c < c_{\min }\). We therefore investigate the existence and stability of steadystate solutions in each of these regimes separately, referring to steady states satisfying \(c_{\min } \le c \le 1\) as spatially limited (SL), while steady states satisfying \(0 \le c < c_{\min }\) are called nutrientlimited (NL).
3.2.1 SteadyState Solutions: Existence and Multiplicity
Spatially Limited Steady States. In this case, the nutrient is plentiful (i.e. \(c_{\min } \le c \le 1)\) and we find the steadystate solutions by solving:
Since \(0< V_0 < 1\), it is straightforward to show that there are two steady states:
Clearly, \(\mathrm {SS}_1\) is an admissible steady state, and the condition on \(V_0\) implies that \(T_2^*=1V_0 > 0\) for \(\mathrm {SS}_2\). For \(c_2^*\) to lie in the appropriate nutrient regime, we require \(c_2^* \ge c_{\min }\) (as \(c_2^* < 1\) follows trivially). This condition is satisfied when the following relationship holds:
Now, steady state \(\mathrm {SS}_1\) represents a tumourfree state, which exists for all combinations of parameters. In contrast, steady state \(\mathrm {SS}_2\) represents tumours which occupy all of the available free space; their growth is limited by space availability, and at steady state, there is neither cell proliferation nor death. For g and \(c_{\min }\) fixed, the inequality (11) implies that \(\mathrm {SS}_2\) exists if \(V_0\) is bounded below by an increasing function of \(q_1\). In other words, if the tumour vasculature can support the tumourās maintenance needs at equilibrium, then \(\mathrm {SS}_2\) exists.
NutrientLimited Steady States. In this case, there is a nutrient shortage (i.e. \(0 \le c < c_{\min }\)) and we determine the steadystate values by solving:
The trivial steady state for T implies that, as \(V_0>0\), the steady state for c must be 1. However, this is not a valid steady state in the NL regime, since \({ 1 = c > c_{\min }}\). Therefore, we suppose \(T>0\) and find solutions of (12a)ā(12b) by determining the intersection points of the T and cnullclines. Using (12a), we find that the nonzero Tnullcline is given by:
which, given that we assume \(\delta _1 = q_2\), reduces to
To determine the cnullcline, we solve the quadratic equation (12b) for T and find it has two branches:
We can now determine the steadystate values for c by equating (14) and each branch in (15) and solving for \(c_\pm \) corresponding to \(T_\pm \). We find:
where
We note here that we obtain a unique steadystate value for c (i.e. (16b)) when the curve T(c) defined by (14) intersects only one of the branches \(T_\pm (c)\) defined by (15). In particular, T(c) intersects \(T_(c)\).
Finally, this implies that there are up to two NL steady states:
where T(c) is defined by (14) and \(c_\pm \) are defined by (16a)ā(16b).
Similarly to the SL steady states, in order to exist and be physically realistic, \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\) must satisfy \(0 \le T \le 1V_0\) and \(0\le c < c_{\min }\). Given the expression for T(c) provided in (14) and assuming that \(0 \le c < c_{\min }\), we have \(T(c)< 1V_0\) for any parameter combination. Due to the complexity of the expressions for the steady states \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\), extensive algebraic manipulation is required to determine the regions in parameter space in which the remaining conditions hold. We therefore fix \(g= 5\) and \(c_{\min }=0.01\) and use Mathematica to determine the regions in \((V_0,q_3)\)space for different fixed values of \(q_1\) in which \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\) exist as admissible steadystate solutions. We illustrate these regions in Fig.Ā 2 and we also include the regions in which \(\mathrm {SS}_1\) and \(\mathrm {SS}_2\) are admissible, as defined previously.
Remark 2
Figure 2 corresponds to specific values of g and \(c_{\min }\). We can, however, obtain qualitatively similar results for different values of these parameters. In particular, for \(q_1\) fixed, increasing g shifts the regions of existence of each steady state to the left, which decreases the size of the regions in which both \(\mathrm {SS}_1\) and \(\mathrm {SS}_3\) and all steady states exist and increases the size of the region in which both \(\mathrm {SS}_1\) and \(\mathrm {SS}_2\) exist. In contrast, for \(q_1\) fixed, increasing \(c_{\min }\) shifts the regions of existence of each steady state to the right, which increases the size of the regions in which both \(\mathrm {SS}_1\) and \(\mathrm {SS}_3\) and all steady states exist and decreases the size of the region in which both \(\mathrm {SS}_1\) and \(\mathrm {SS}_2\) exist.
Steady states \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\), which always satisfy \(\mathrm {SS}_3 < \mathrm {SS}_4\), represent tumours that are not SL (i.e. \(0< T(c) < 1  V_0\)); their growth is limited by nutrient availability, and at steady state, the rates of cell proliferation and death are nonzero and balance. From Fig.Ā 2, we see that, for fixed g and \(c_{\min }\), \(\mathrm {SS}_3\) exists when (i) there is no nontrivial SL steady state, i.e. \(V_0\) cannot sustain a SL tumourās oxygen requirements for maintenance or (ii) there is a nontrivial SL steady state, i.e. \(V_0\) can sustain a SL tumourās oxygen requirements for maintenance, but the oxygen requirements for proliferation (\(q_3\)) are high. In contrast, for fixed g and \(c_{\min }\), \(\mathrm {SS}_4\) only exists in case (ii).
3.2.2 Stability Analysis
In the previous section, we defined the modelās steadystate values and their regions of existence in \((V_0,q_1,q_3)\)space for fixed g and \(c_{\min }\). We now investigate their stability.
Spatially Limited Steady States. We first perform a linear stability analysis for steady states \(\mathrm {SS}_1\) and \(\mathrm {SS}_2\); we compute the Jacobian of the system (4)ā(5) when \(c_{\min } \le c \le 1\) at each steady state and find the corresponding eigenvalues. The Jacobians of the system evaluated at \(\mathrm {SS}_1\) and \(\mathrm {SS}_2\) are, respectively:
It is straightforward to obtain the eigenvalues \((\lambda _1,\lambda _2)\) and \((\mu _1,\mu _2)\) for the Jacobians evaluated at \(\mathrm {SS}_1\) and \(\mathrm {SS}_2\), respectively:
Since \(q_1, q_2>0\) and \(0<V_0<1\), it is clear that steady state \(\mathrm {SS}_1\) is unstable and steady state \(\mathrm {SS}_2\) is stable. Thus, in the SL regime, tumour elimination is not possible and the tumour persists.
NutrientLimited Steady States. We can perform a similar stability analysis for \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\). The Jacobian of the system (4)ā(5) when \(0 \le c < c_{\min }\) is:
Using the fact that Eqs.Ā (12a)ā(12b) hold at both steady states \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\), it is straightforward to show that \(J_{11} < 0\), \(J_{22} < 0\) and \(J_{12} > 0\) for \(J _{\mathrm {SS}_3}\) and \(J _{\mathrm {SS}_4}\) (for all possible parameter sets). However, determining the sign of \(J_{21}\) for either \(\mathrm {SS}_3\) or \(\mathrm {SS}_4\) is complicated given the algebraic complexity of the expressions of T and c for these steady states. Therefore, we establish the sign of \(J_{21}\) in specific cases by performing phase plane analyses. FigureĀ 3 contains two phase portraits for sets of parameters that, respectively, correspond to a region of parameter space where the only NL steady state is \(\mathrm {SS}_3\) and where the two NL steady states exist (see Fig.Ā 2). We can deduce from these phase portraits that, for these parameter sets, \(J_{21}<0\) for \(\mathrm {SS}_3\), whereas \(J_{21}>0\) for \(\mathrm {SS}_4\). Phase portraits for other parameter sets (not shown) suggest that the sign of \(J_{21}\) remains the same for both steady states, irrespective of the values of the parameters.
Given the signs of each component of the Jacobian evaluated, respectively, at \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\), we now discuss the stability of these two steady states. First of all, since \(J_{11} < 0\) and \(J_{22} < 0\) for \(\mathrm {SS}_3\) and \(\mathrm {SS}_4\), the trace of the Jacobian is negative for both steady states. For \(\mathrm {SS}_3\), the determinant of the Jacobian is positive, which implies that \(\mathrm {SS}_3\) is stable. In contrast, for \(\mathrm {SS}_4\), the determinant could be positive or negative depending on the parameter values (since \(J_{11} J_{22} > 0\) and \(J_{12} J_{21} > 0\)). We therefore cannot confirm whether \(\mathrm {SS}_4\) is stable or not, but the phase portrait in Fig.Ā 3b shows that \(\mathrm {SS}_4\) is unstable for that specific parameter set. Constructing phase portraits for other parameter sets (not shown) suggests that \(\mathrm {SS}_4\) is unstable, irrespective of parameter values. We therefore conclude that \(\mathrm {SS}_3\) is stable, and make the conjecture that \(\mathrm {SS}_4\) is unstable.
3.3 The Existence of Three Tumour Growth Regimes
In the two previous sections, we defined the steadystate solutions for Eqs.Ā (4)ā(5), determining where they exist in parameter space, and we also found their respective stabilities. These results enable us to understand how the choice of parameters influences the equilibrium tumour size(s) that can be attained in the long term. To illustrate this, we constructed bifurcation diagrams which show how the stable and unstable steadystate solutions for T change as \(V_0\) varies, for fixed values of \(q_1\) and \(q_3\). These are presented in Fig.Ā 4.
We see that, if \(q_3\) is smaller than \(q_1=0.1\), then the system evolves to a SL or NL steady state and there is no bistability. In contrast, if \(q_3\) is sufficiently large relative to \(q_1=0.1\), then a bistable region exists in addition to those monostable regions in which T automatically attains either a SL or a NL steady state. As \(q_3\) increases, the region of bistability increases in size. This is consistent with Fig.Ā 2, which presents the regions of existence of each steady state and also shows that bistability only occurs if \(q_1\) is sufficiently small relative to \(q_3\).
Combining all of the preceding results, we conclude that, depending on \((V_0,q_1,q_3)\), there are three possible scenarios for tumour growth, which we illustrate in Fig.Ā 5:

1.
Spatially limited growth: the tumour has access to sufficient nutrient to keep growing until it runs out of physical space to occupy; growth stops due to densitydependent inhibition which stops cell proliferation without any cell death.

2.
Nutrientlimited growth: the tumour grows until the birth of new cells balances the death of cells due to nutrient starvation; growth stops due to nutrientdependent inhibition.

3.
Bistability: the system is bistable and the tumour can grow to a NL or SL steady state, depending on the initial conditions. By considering trajectories in the phase plane, it is possible to show that, given initial conditions that satisfy \(0 < T(0) \ll 1\), the tumour will grow to a NL steady state (for example, see Fig.Ā 5b).
4 Discussion
Understanding tumour growth dynamics is essential for investigating tumour response to treatment. This is why developing tumour growth models and extending them to study treatment response has been at the forefront of mathematical oncology research for many years. As increasing numbers of models are proposed, a key question is how to balance the biological detail of a model with its analytical tractability and suitability for data fitting and parameter estimation. We have attempted to strike that balance by proposing an ODE model of vascular tumour growth which retains the simplicity of phenomenological models while incorporating some mechanistic details. In addition, a key goal was to develop a model that incorporates two alternative mechanisms for tumour growth arrest.
By studying the behaviour of the model (4)ā(5) numerically, we found that the desired qualitative growth trends are preserved: we observed exponential and/or linear growth, followed by convergence to a parameterdependent limiting size. Our steadystate analysis further revealed that, depending on the oxygen consumption rates and the initial vascular volume, the model exhibits three distinct tumour growth regimes. These regimes are characterised by two different growthlimiting processes: growth arrest due to cell proliferation balancing cell death due to nutrient starvation vs. growth arrest without cell death where cells stop proliferating due to space constraints.
Despite the simplicity of the model, we expect that it can provide additional insight compared to other models of tumour growth, which typically only represent a single growthlimiting mechanism (Drasdo and HĆ¶hme 2005; Greenspan 1972; Hahnfeldt etĀ al. 1999; Lewin etĀ al. 2020; Murphy etĀ al. 2016; Panovska etĀ al. 2007). This is because the mechanisms responsible for growth arrest may influence how a tumour responds to a particular treatment. We illustrate this below with two simple examples that motivate how our model can be used to study tumour response to various treatments. We first consider a treatment, here called treatment 1, that causes oxygenindependent vascular damage and damageinduced angiogenesis (e.g. high hyperthermia). In particular, this treatment can significantly alter the vascular volume, V, which was held constant in the tumour growth model presented in this paper. Since the tumour steady state depends on the oxygen consumption rates, \(q_3\) and \( q_1\), and the vascular volume, \(V \equiv V_0\), changes to the vascular volume will affect the steadystate tumour volume. This may, in turn, also change the tumourās growth regime (i.e. nutrientlimited (NL), bistable or spatially limited (SL)) and the mechanism driving growth arrest.
To illustrate this, let us assume that V is a monotonically decreasing function of the dose D of treatment 1. Then, as D increases, the value of V in the model Eqs.Ā (4)ā(5) changes so that the tumour traverses regions of parameter space in ways that depend on the initial vascular volume, \(V_0\), as well as other model parameters (see Fig.Ā 6a). This can result in the system moving through different sequences of steadystate behaviour. We demonstrate this using the bifurcation diagrams in Fig.Ā 6, which show how the tumour steadystate volume depends on the vascular volume, for fixed \((q_3,q_1)\). On the one hand, tumours initially in NL or bistable regimes exhibit an immediate and gradual decrease in tumour steadystate volume as V decreases (Fig.Ā 6b, \(A_0\) to \(A_1\), and Fig.Ā 6c, \(B_0\) to \(B_1\)). In these cases, the steadystate solutions remain on the NL steadystate branch of the bifurcation diagrams in Fig.Ā 6b, c as V decreases. On the other hand, the steadystate size of a tumour initially in a SL regime increases marginally as V decreases towards a threshold value (Fig.Ā 6d, \(C_0\) to \(C_1\)). Once this value is reached, the tumour undergoes a rapid and large reduction in steadystate volume as it switches to the NL regime (Fig.Ā 6d, \(C_1\) to \(C_2\)). As V continues to decrease, the tumour undergoes a more gradual, sustained decrease in steadystate volume, remaining in the NL regime (Fig.Ā 6d, \(C_2\) to \(C_3\)) and behaving similarly to tumours that are initially in bistable or NL regimes. In this case, the steadystate solution can jump from the SL steadystate branch to the NL steadystate branch of the bifurcation diagram in Fig.Ā 6d, provided that treatment 1 elicits a sufficiently large decrease in V.
These possible variations of tumour steadystate values and growth regimes in response to treatment 1 highlight how such a treatment can be effective at reducing longterm tumour burden, especially in the case of SL tumours. In particular, the preceding analysis emphasises that treatment of SL tumours should be designed to drive the system into the NL regime, where the larger spatially limited tumour steady state does not exist and the tumour is guaranteed to evolve to a reduced volume. It would therefore be interesting to consider the effect of combining treatment 1 with another treatment such as chemotherapy that, say, causes tumour cell damage. However, elucidating tumour response to such a combined treatment would require us to consider the dynamics of the tumour cells and vasculature and their interactions. In particular, we would need to extend our model to include a further ODE to account for the dynamics of V: this is beyond the scope of the present work. Even so, we can illustrate the potential of our current model for studying combination treatments using the following example.
Suppose that we combine a treatment, here called treatment 2, which causes tumour cell damage in an oxygendependent manner (e.g. radiotherapy) with another treatment, here called treatment 3, which can reoxygenate the tumour (e.g. mild hyperthermia). Treatment 2 alone causes a decrease in tumour volume, whose magnitude depends on how cell death due to the treatment and nutrient deficiency compares to cell proliferation. By increasing the oxygen concentration in the tumour, treatment 3 has three key effects: (i) increasing the cell kill caused by treatment 2, (ii) increasing the proliferation rate of tumour cells and (iii) decreasing the rate of cell death due to nutrient deficiency (if the death rate is nonzero). Thus, applying treatment 3 before treatment 2, we predict a synergistic benefit of combining the treatments if the magnitude of the decrease in tumour volume is larger than that achieved by treatment 2 alone. We illustrate this scenario in Fig.Ā 7.
Given the effects of reoxygenation previously described, we claim that NL tumours are more likely to respond poorly to this combination treatment than SL tumours. Suppose that we have two tumours at steady state, one NL and one SL. Recall that, at equilibrium, the rates of cell proliferation and cell death balance for NL tumours and nutrient availability is growth rate limiting, whereas for SL tumours equilibrium is achieved when mechanical constraints halt cell proliferation throughout the entire tumour mass, i.e. nutrient is not limiting and there is no cell death. Reoxygenation (treatment 3) perturbs the NL steady state, but not the SL steady state. More specifically, reoxygenation allows cell proliferation to outweigh cell death due to nutrient deficiency, and therefore, the NL tumour can grow past its pretreatment volume before the celldamaging treatment (treatment 2) is applied. As a result of this additional tumour growth due to treatment 3, the increase in cell kill by treatment 2 due to reoxygenation may not be sufficient to enhance the tumourās response compared to treatment 2 alone. Thus, the combination treatment may fail. In contrast, reoxygenating the SL tumour increases the average oxygen concentration in the tumour without affecting the rates of cell proliferation and cell death as they both remain zero. By doing so, treatment 3 only increases cell kill by treatment 2, which increases the efficacy of treatment 2 and ensures that there is a benefit of combining the treatments. We illustrate the expected responses of steadystate NL and SL tumours to treatment 2 alone and to a combination of treatment 3 followed by treatment 2 in Fig.Ā 8.
Treating tumours that are not at steady state, we also expect to observe a percentage increase in tumour growth due to reoxygenation that is larger in NL than SL tumours: while the growth rates of NL and SL tumours both increase due to enhanced proliferation, that of NL tumours is further increased by a decrease in cell loss. As previously explained for NL tumours at steady state, a sufficiently large increase in tumour growth following treatment 3 can negatively impact the success of the combined treatment. All of the preceding considerations imply that, by using the tumour growth model presented in this paper to study tumour response to this combination treatment, we can expect varying treatment outcomes depending on tumour type and growthlimiting mechanism.
As mentioned in the Introduction, our simple model exhibits complex behaviour that lends itself to investigating and distinguishing between the response of different tumours to a range of combination cancer therapies. While we have illustrated this briefly with the two preceding hypothetical scenarios, a detailed investigation into how this model can be exploited to address specific combination treatments will be presented in a future paper. More specifically, we aim to extend the tumour growth model to incorporate tumour response to different treatments and then use the resulting dynamic model to conduct an indepth study of treatment outcome. In particular, we seek to explore whether the response to treatment is sensitive to the mechanisms underpinning tumour growth and to the form of tumour growth assumed. If so, we may also be able to discern the underlying mechanisms from observed responses to treatment. Moreover, our simplifying assumption that vascularisation is constant allowed us to analyse the system in depth and find behaviours that depend critically on the level of vascularisation. Hence, future work will incorporate more realistic vascular dynamics. The extensions of the study presented here will help to identify optimal, patientspecific treatment combinations and to increase our understanding of high variability in treatment response between patients.
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Acknowledgements
We would like to thank Prof. Gail ter Haar and Dr. Sarah BrĆ¼ningk for helpful discussions about the biological realism of our modelling approach.
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C.C. is supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
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Appendix A: Defining the Parameter Regime of Interest
Appendix A: Defining the Parameter Regime of Interest
Here we justify our dimensionless parameter choices, which are summarised in Table 1, by referring to dimensional parameter values.
We first define a range of values for the vascular volume \(V_0\). Assuming that the average diameter of a cell is \(20 \,{\upmu }\hbox {m}\), we estimate that the maximum number of cells that can occupy \(1 \, \hbox { mm}^{3}\) of space is \(1.25 \times 10^5\) (Ardaseva 2020). Now, the Krogh cylinder model shows that oxygen can diffuse up to \(100200 \, {\upmu }\hbox {m}\). Using the average cell size and cylindrical geometry, we estimate that a section of blood vessel of length \(20 \,{\upmu }\hbox {m}\) can supply oxygen to a number of cells in the range [100,Ā 300]. In particular, given that the average diameter of a capillary is \(8 \, {\upmu }\hbox {m}\) (MĆ¼ller etĀ al. 2008) and using cylindrical geometry once again, we further estimate that \(1 \times 10^{6} \, \hbox { mm}^{3}\) of blood vessel can provide oxygen for a volume of cells in the range \([8 \times 10^{4}, 2.4 \times 10^{3}] \, \hbox { mm}^{3}\). Therefore, a biologically realistic range for \(V_0\) such that the tumour may be sufficiently oxygenated is \([10^{4},5 \times 10^{3}]\). Since tumour vasculature is usually less effective than healthy vasculature (Carmeliet and Jain 2000), we consider \(V_0 \in (0, 5 \times 10^{3}]\).
We now determine \(c_{\min }\), k and g. To do so, we first estimate \(S_{\max } = 10^{6} \, \hbox { m}^{3}\) in accordance with the average size of vascular tumours in mice (FaustinoRocha etĀ al. 2013; Wu etĀ al. 2013). Moreover, the average oxygen partial pressure in peripheral normal tissues is approximately \(p_{\max } = 38 \,\hbox { mmHg}\) (OrtizPrado etĀ al. 2019), and therefore, using Henryās Law to convert \(p_{\max }\) into a concentration, it is straightforward to calculate \(c_{\max }=2.1 \times 10^{3} \, \hbox { kg} \hbox { m}^{3}\). Now, hypoxia is attained when oxygen levels fall below \(8\, \hbox { mmHg}\) (McKeown 2014). Unlike normal tissue cells, tumour cells can survive in such hypoxic conditions and only start to die in severely hypoxic conditions, which range from \(0.75\, \hbox { mmHg}\) to \(0.075\, \hbox { mmHg}\) depending on the tumour (McKeown 2014). Hence, we take the average of these two values, and using Henryās Law again, we have \(c^*_{\min } = 2.26 \times 10^{5} \, \hbox { kg} \hbox { m}^{3}.\) This implies that
and we therefore set \(c_{\min } = 10^{2}\).
In addition, we expect that the rate of oxygen exchange per unit volume area of blood vessels, \(g^*\) (\({\hbox { m}^{3} {\min }^{1}}\)), depends on a variety of factors, including the quality of the vascular network, which can lead to highly heterogeneous tumour perfusion rates (Gillies etĀ al. 1999), and surface areas of oxygen exchange. Since our model does not account for these details, we fix \(g^*\) so that, for average values of the oxygen consumption rates, we have SL tumours for \(V_0 \ge 5 \times 10^{3}\) and NL or bistable tumours for \(V_0 \le 1 \times 10^{4}\) (recall that, for \(V_0 \in [10^{4}, 5 \times 10^{3}]\), the vasculature should be sufficient to supply oxygen to the whole tumour). Based on preliminary numerical simulations (not shown), we set \(g^* = 5 \times 10^{6} \, {\hbox { m}^{3} {\min }^{1}}\). Setting the timescale to be \(\tau = 1 \, \hbox { min}^{1}\), we thus have
Finally, \(k^*\) (\(\mathrm {m}^6 \) \(\hbox { kg}^{1}\)) is the parameter that relates the constant for oxygen consumption for proliferation, \(q^*_3\), and the constant for proliferation, \(q^*_2\). Since cells consume oxygen faster than they proliferate, we expect \(q^*_2 < q^*_3\). As the value of \(k^*\) is not readily available in the literature, we performed preliminary numerical simulations (not shown) to find a value of \(k^*\) that gives realistic growth dynamics. As a result, we fix \(k^* = \frac{10^{5}}{2.1} \) \(\mathrm {m}^6 {\hbox { kg}^{1}}\) and this implies that
We now estimate the oxygen consumption rates, the proliferation rate and the death rate. Biological observations suggest that the death and proliferation rates of tumour cells are highly correlated in some tumours: the faster the growth, the larger the death rate (Leoncini etĀ al. 1993; Liu etĀ al. 2001; Vaquero etĀ al. 2004). This suggests that these rates are proportional to each other and we make the simplifying assumption that the dimensionless parameters \(\delta _1\) and \(q_2\) satisfy \(\delta _1 = q_2\) in order to reduce the number of parameters in the model. We note here that this assumption does not affect the qualitative behaviour of the model summarised in the paper, and in particular, the three tumour growth regimes are preserved.
Now, we estimate \(\delta ^*_1\) (\({ \hbox { m}^{3} \hbox { kg}^{1}\hbox { min}^{1}}\)) as
where the numerators of both fractions correspond to death rates found in the literature (Lewin etĀ al. 2018; Schaller and MeyerHermann 2006) and the denominator corresponds to the average oxygen concentration below \(c_{\min }\) (this scaling allows us to obtain the correct units for \(\delta _1^*\)). We then have:
which we nondimensionalise according to (6) to obtain:
Now, since \(q_2=\delta _1\) by assumption, we have
and this gives us the following dimensionless ranges for \(q_3 = q_2/k\):
We further justify the range of values determined for \(q_3\) by considering experimentally determined values for the rate of oxygen consumption by cells. Wagner etĀ al. (2011) conduct their own experiments as well as review results from other authors and they report oxygen consumption rates in the range \([6 \times 10^{5}, 6 \times 10^{3}] \, {\hbox {kg}\hbox { m}^{3} {\min }^{1}}\). These constant consumption rates correspond to the oxygen uptake of cells in a welloxygenated medium, i.e. where the rate of oxygen consumption is at its maximum. Given the volume of the tumours in their experiments and the maximal oxygen concentration they set, we can find the following estimate for the range of \(q_1^*\):
Given (6) and \(S_{\max } = 10^{6}\), this implies that
Here, we determine a range of values for \(q_1\) using consumption rates that correspond to overall oxygen consumption, i.e. they do not distinguish between consumption for proliferation and for maintenance. In practice, \(q_1\) would therefore be smaller than the values in (A10). In view of this, we recover the dimensionless parameter ranges in Table 1 by rounding down to the closest order of magnitude the upper and lower bounds of the parameter ranges we found. We note once more that, since this paper focuses on demonstrating the qualitative behaviour of the model (4)ā(5), defining plausible ranges for parameter values is sufficient.
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Colson, C., Byrne, H.M. & Maini, P.K. Combining Mechanisms of Growth Arrest in Solid Tumours: A Mathematical Investigation. Bull Math Biol 84, 80 (2022). https://doi.org/10.1007/s11538022010342
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DOI: https://doi.org/10.1007/s11538022010342
Keywords
 Tumour growth
 Growthlimiting mechanisms
 Ordinary differential equation model
 Bistability