2.1 Mathematical formulation

The mathematical framework for polymersome uptake in a well-mixed culture model presented here closely follows that of our previously published model. Here we extend the model [15] to include time-varying cell concentration, which we previously assumed to be constant. We use a system of n+ 5 ordinary differential equations to describe the uptake of polymersomes (see Fig 1), where n is the maximum number of bonds that can form between the polymersome and receptors on the cell surface. We use the same assumptions in our model set up as we did previously. We assume that the system is well-mixed, i.e cells and particles are evenly distributed in space which is representative of in vitro conditions, and so ignore any spatial effects. Furthermore we assume that there are a fixed number of available ligands on the polymersome surface that can bind to free cell surface receptors.

Step 1 of RME is the binding of receptors to ligands (see Fig 1a), and therefore the number of free ligands on a polymersome changes over time due to binding with cell surface receptors. We describe the change in the number of free ligands over time, t, by (1) where L is the moles of free vesicle ligands per cm³, k_a is the rate of receptor-ligand binding per minute and k_d is the dissociation rate per minute, F is the moles of cell receptors per cm³ and B₁ is the moles of receptor-ligand complexes bound with one bond per cm³. Here the first term on the RHS of Eq (1) accounts for binding of a free polymersome to a single cell receptor and the second term accounts for dissociation of that bond such that particles are no longer bound to the cell.

From Eq (1) we can calculate the number of free polymersomes per cm³, V, by (2) where l is the fixed moles of free ligands per polymersome. We use the law of mass action to describe the binding kinetics so receptor-ligand complexes with one bond change over time through, (3) where N_A is Avogadro’s constant, B₂ is the moles of receptor-ligand complexes bound with two bonds per cm³, k_in is the polymersome internalisation rate per minute and ρ_l and ρ_f are the fraction of ligands and receptors respectively that are available for 2-D binding, i.e only ligands and receptors within a certain range can form bonds. The term 2-D binding is used to describe the subsequent binding after one initial receptor-ligand bond had formed.

Eq (3) involves terms that are functions of ρ_l and ρ_f. In the third term on the RHS of Eq (3), (ρ_l lN_A − 1) represents the ligands available on the polymersome for subsequent binding after the initial binding event. The term describes that there is one less ligand available for subsequent binding, in which the polymersome would become bound to the cell with two bonds (for example see Fig 1B). The fourth term on the RHS represents dissociation of a bond, meaning that a new receptor and ligand are now free binding at a rate 2k_d (where the factor 2 represents that any one of the two bonds can dissociate). Internalisation of polymersomes occurs at a constant rate k_in; in this well-mixed model we assume that the binding rate and internalisation rate are independent of the number of bonds that have already formed following our previously published model [15]. Bond dependent internalisation is explored later in this paper.

Generalising, the number of receptor-ligand complexes bound by i bonds changes over time by (4) where 2 ≤ i ≤ n − 1 and n is the maximum number of ligand-receptor bonds that can form.

The number of receptor-ligand complexes bound by n bonds change over time by, (5) When a maximal number of bonds, n, is reached, polymersomes are internalised into endosomes (see Fig 1D.ii). The acidic environment of the endosome causes polymersomes to break down and release the encapsulated drug. The rate of change of internalised polymersomes over time is then given by, (6) Internalised polymersomes are lost due to polymersome rupture at an assumed constant rate, the rupture rate d_b. When polymersomes rupture within the cell they release the encapsulated chemotherapy drug (see Fig 1E), giving the rate of change of intracellular drug as, (7) where P is the concentration of intracellular drug per cm³. The release of the drug is proportional to the rupture of the polymersome, ν, and we also include a decay in drug activity which we assume occurs at constant rate, d_p, which corresponds to uptake or removal of the chemotherapy drug.

During the process of binding there is a change in the number of free receptors, F, on the cell surface. This is because once a receptor binds with a ligand it is no longer free to bind to other ligands, and becomes internalised in the process of endocytosis. Once inside the cell, the receptors are recycled back to the cell surface [20] (see Fig 1E). We describe this rate of change of free receptors by, (8) where we have assumed that receptors have a limited life on the cell surface, with the constant linear decay rate d_f. We assume that receptor production occurs on the cell surface and that receptors are recruited to the cell surface at a rate proportional to the number of bound receptors. We base these assumptions on the similar model of Ghaghada et al [21] who investigate receptor recycling and nanoparticle-cell bonds through receptor-mediated targeting for liposomes. We describe receptor recycling using a function of the total number of bound receptors, R(b_tot), where b_tot is the total number of bound complexes per cell. We assume R(b_tot) is constant if there are no bound receptors (b_{tot = 0}), otherwise we assume an increasing saturating function dependent on the total number of bound complexes per cell. R(b_tot) is given by (9) m is the number of tumour cells per cm³, α is the Hill exponent, c₁ is the rate of receptor production and c₂ and c₃ characterise the hill equation. Here c₂ is the maximum recycling rate and c₃ is the value at which the half maximum of the function R(b_tot) is reached.

In an extension to our previous model we describe how the tumour cell density changes over time due to the balance of proliferation and death through (10) where m is the number of cells per cm³, the carrying capacity of the system is K and we assume that the cells undergo logistic proliferation at constant rate, r. We include cells in our model since the number of cells directly correlates to the uptake amount of nanoparticles and therefore the number of remaining free polymersomes. Cell death is described by the function g(ϕ), which we assume is constant when there is no intracellular drug present due to natural cell death, otherwise it is an increasing saturating function of the intracellular drug concentration [22]. Cell death is then described through (11) where describes the mole of drug per cell, P is the moles of drug per cm³, P₀ is the drug concentration that produces a 50% maximal response, d_m is the natural cell death, β is the Hill exponenent and μ is the cell death rate due to drug delivery. The functions of cell death, g(ϕ), and receptor recycling, R(b_tot) are generalised hill equations which are commonly used to model cell kinetics and nutrient consumption, for example see references [22–24].

We further impose the following initial conditions, where i = 1, …, n, M₀ is the initial cell concentration, V₀ is the initial polymersome concentration, F₀ is the initial cell surface receptor concentration, B_i(0) is the concentration of polymersomes bound with i complexes, B_in(0) is the internalised polymersome concentration and P(0) is the released drug concentration, all at time t = 0. These conditions state that initially there are no bound or internalised polymersomes present when the cells and particles are mixed together and therefore there is no released drug.

2.2 Parameterisation

The parameterisation of the system proved difficult since the majority of the parameters for specific polymersome binding are unknown due to a lack of relevant experimental data. Hence we look to a similar system of liposomal targeting which has similar dynamics to those of polymersome targeting and is more well understood experimentally. Specifically we look to use parameters from a liposomal modelling study by Ghaghada et al. [21], in which targeted liposomes bind to folate receptors of C6 glioma cells. The parameter values used in the subsequent analysis are shown in Table 1. Further information on how the parameters were chosen is provided in the S1 File.

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Table 1. Model parameter values.

Where no data were available in the literature we estimated parameters values to match experimental observations. See S1 File.

https://doi.org/10.1371/journal.pone.0254208.t001

2.3 Nondimensionalisation

We non-dimensionalise the model in order to understand the dominant terms in the system and to enable prioritisation of the parameters needed for future experiments. We rescale the system as follows: where i = 1, …, n and the non-dimensional variables are denoted by hats. Removing the hats for notational convenience the rescaled system then reads (12) (13) (14) (15) (16) (17) (18) (19) with (20) where The rescaled parameters, denoted by tildes, are defined as The rescaled system is then subject to the following initial conditions, (21) for i = 1, …, n. The values of the rescaled parameters are given in Table 1. We see that in the rescaled system the larger parameters (on the order of or close to 10³) are the internalisation rate, k_in, the polymersome rupture rate, d_b, the drug decay rate, d_p, the natural cell death rate, d_m, and cell death due to drug, μ, so we expect that these will have large influences on the system. On the other hand the initial binding rate, k_a, and the dissociation rate, k_d, are small (less than one) so we expect these to have a negligible impact on the system.

We solve the system numerically using the ODE solver ode45 in Matlab, which is based on the explicit Runge-Kutta 45 method. Fig 3 shows representative results for a typical set of parameter values, given in Table 1. Initially there is a quick decrease in free receptors due to the fast initial binding with ligands on the polymersome surfaces, then a more gradual decline is observed as the number of tumour cells, and hence, cell surface receptors decrease. With the maximum number of complexes being set to twenty, the majority of polymersomes only create a small number of bonds with the cell surface before being internalised, with the maximum number of internalised polymersomes occurring after around one hour. The intracellular drug concentration exhibits a similar behaviour as the internalised polymersomes. This is to be expected as the drug release is assumed to be proportional to the rate of internalisation of polymersomes. The simulations show that, for this specific set of parameter values, the tumour cell population decreases to zero over time after polymersomes are introduced. The decrease is rapid over approximately the first 10 hours and then slows as the population approaches zero.

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Fig 3. Predictions of the model (Eqs 12–19) for the parameter values given in Table 1.

The nondimensional system and parameters were used for the simulations and the results were converted to dimensional values. The maximum number on bons, n = 20.

https://doi.org/10.1371/journal.pone.0254208.g003

2.4 Singular perturbation analysis

The results of the numerical system show rapid changes in the system at early time points. This is to be expected based on the large coefficients we find in our nondimensionalisation. To get an analytical handle on this we carry out a singular perturbation analysis of early time points in the system. We use the non-dimensional system given in the previous section 2.3 with the hats and tildes dropped for convenience.

For simplicity, we first consider a single binding interaction, where only one complex between a ligand and receptor has to form for the drug-loaded polymersome to be internalised (n = 1). We justify this by observing that the binding rate is much smaller than the internalisation rate so that a polymersome will likely be internalised after one bond has formed and before other bonds form. We do this by setting ρ_l = l⁻¹ and ρ_f = 0 so that B_i(t) = 0 for 2 ≤ i ≤ n.

For further simplification of the model we look at the parameters involved in the kinetics of receptors and ligands. Note that, with the parameter values described in the previous section, we have the dimensionless values, k_d ∼ O(10⁻²), and ηk_d ∼ O(10⁻⁴) and so we set the dissociation rate k_d = 0. Hence we can write , and where ε ≪ 1 is a scaling parameter. We also look to simplify the saturating functions for receptor recycling, R(b_tot), and cell death, g(ϕ), given by (20). The total cell death rate, g(ϕ), can be linearised since is neglible thus reduces to . To simplify , we assume that c₃ is small compared to , so that R(b) ∼ c₁ + c₂ ≔ R.

Before performing the analysis we make one further assumption in order to simplify the system, assuming t = τε where τ represents short time behaviour. Using these assumptions the system then becomes (22) (23) (24) (25) (26) (27) whilst initial conditions remain the same as those given in Eq (21).

Using this reduced system we look to find the inner solutions of the system. We propose asymptotic solutions of the form P(τ; ε) = ∑_{n = 0} εⁿ Pⁿ, m(τ; ε) = ∑_{n = 0} εⁿ mⁿ, V(τ; ε) = ∑_{n = 0} εⁿ Vⁿ and F(τ; ε) = ∑_{n = 0} εⁿ Fⁿ and equate coefficients of powers of epsilon to look for the inner solutions for free polymersomes and free receptors at early time points. For coefficients of ε⁰, the solutions for m and P are m = 1 and P = 0, as given by the initial conditions. These solutions allow us to solve the leading order solution for free receptors, F, and free polymersomes, V, with the equations for bound polymersomes (B) and internalised polymersomes (B_in) decoupling from the rest of the system. For coefficients of order zero the equation for free polymersomes, (23), yields (28) Solving this with the initial condition V(0) = V₀, the leading order solution is then given simply by (29) Using this solution, we now look for the leading order solution for free receptors. Eq (24) becomes (30) which is easily solved using the integration factor method with initial condition F⁰(0) = F₀ to give (31) where (32) and C₁ is a constant of integration, (33)

During this time period free polymersomes become bound to cells (see Fig 3). Thus the number of free polymersomes V₀(t) should not be equal to a constant. Equating coefficients of powers of ε for the free polymersome equation gives (34) which can be easily solved to give (35) where C₂ is a constant of integration, given by (36)

Next we find solutions for bound and internalised polymersomes. Recalling the solutions for free polymersomes and receptors, (29) and (31), and the equation for bound polymersomes (25) we have (37) Again we can solve for B(t) using the integration factor method to give, (38) where C₃ is a constant of integration, (39) We now use the solution for bound polymersomes, (38), to solve for internalised polymersomes.

By combining the equation for B_in(τ) (26) and by using (31), leads to (40) where the following parameter groupings have been used, (41) (42) (43) Again this can be solved using the integration factor method, which yields (44) where C₄ is a factor of integration, (45)

The numerical and analytical results for bound and internalised nanoparticles are shown in Fig 4(A) and 4(C) along with the relative error between the solutions (B,D). We find a good match between the dimensionless numerical and analytical solutions at early time points which is demonstrated by the relative error between the solutions. The numerical and analytical solutions are shown for up to seven minutes in order to capture the key behaviour at early time points, that is that the maximum number of bound polymersomes occurs at a turning point, but the relative error is shown for a longer time period to demonstrate the validity of the results.

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Fig 4. Numerical and analytical solutions for (A) bound and (C) internalised polymersomes for up to seven minutes and (B,D) the relative error in the numerical and analytical solutions for bound and internalised polymersomes respectively for up to one hour.

The numerical solution is the original model but with some simplifying assumptions, given by Eqs 25 and 26, and the analytical solutions are derived from these via a perturbation analysis. The analytical solutions are given by Eqs 38 and 44. The nondimensional system and parameters were used for the simulations and the results were converted to dimensional values. The parameters used are given in Table 1.

https://doi.org/10.1371/journal.pone.0254208.g004

The relative error for bound and internalised polymersomes is shown in Fig 4 for the time period 0 ≤ t ≤ 60 minutes. For the bound polymersomes we see that for the first twenty minutes of the simulation the relative error is a few percent, after which the error grows with time. For the internalised polymersomes the relative error remains below 0.5%. Therefore, we take the analytical solutions to be acceptable for early time periods, less than ten minutes, in the model.

2.5 Maximising polymersome uptake

In Fig 4 we observe that the maximum number of bound polymersomes occurs at a turning point so now we look for an approximated analytical value of this maximum.

Differentiating (38) and solving at zero, the turning point occurs at t = t*, where (46) therefore the maximum of number bound polymersomes, B^max, is then given by (47)

Similarly, we can find the maximum of internalised polymersomes. We see from Fig 4 that at early time points the concentration of internalised polymersomes increases with time. Therefore we find the solution for internalised polymersomes as τ → ∞ (48)

We use this result to investigate the importance of different parameters on polymersome internalisation which subsequently affects the delivered chemotherapy drug concentration.

2.6 Parameter dependency of maximum bound and internalised polymersomes

Fig 3 reveals rapid system dynamics at early time points. To understand this behaviour in more detail we performed a singular perturbation analysis, the predictions for which are shown in Fig 4. From the analytical results we derived expressions for the maximum bound and internalised polymersomes. Fig 5 shows the behaviour of these quantities when we vary key parameters. In Fig 5, we observed that B^max is an increasing function of l, the fixed number of ligands per polymersome. The internalised polymersomes initially increases as l increases to reach a peak at approximately 500 ligands per polymersome before decreasing. As l continues to increase the internalisation rate by tumour cells slows, indicating that there is an optimal number of ligands to maximise polymersome internalisation. This is in agreement with the work of Ghaghada et al. [21] who also found that a specific number of ligands maximises polymersome uptake. B^max and are both increasing saturating functions of k_a, meaning with increasing k_a more polymersomes become internalised. On the other hand B^max is a decreasing saturating function of the scaling factor, η. The maximum number of internalised polymersomes increases rapidly with η before decreasing, indicating an optimal value.

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Fig 5. Sensitivity of the analytical solution of the maximum bound polymersomes and maximum internalised polymersomes, given by B^max (first column) and (second column), to variations in ligand number, l (first row), binding rate, k_a, (second row) and the dimensionless scaling parameter, η, (third row).

The nondimensional system and parameters were used for the simulations and the results were converted to dimensional values. The parameters used are given in Table 1.

https://doi.org/10.1371/journal.pone.0254208.g005

Next we apply our model to a spheroid system in order to incorporate spatial effects of polymersome delivery and to explore the impact of various internalisation factors on tumour growth, as well as tissue properties such as permeability.

3.1 Mathematical formulation

We develop the model to describe radially symmetric avascular tumour growth with infiltrating polymersomes. The space is multiphase, accounting for cells and the surrounding material such as water and cell debris, which is reflective of experimental conditions. The model incorporates the binding kinetics and release of polymersome carried chemotherapy drug and so we can predict its associated impact on tumour growth.

In this model we do not account for certain physiochemical aspects of the nanoparticles, such as the variability in size, and the electric or magnetic properties. Sorrell et al [15] accounted for the variable sizes of the polymersomes, which occurs as a result of the production process, through the use of a stochastic model. The coating of polymersomes is chosen specifically for biological environments such that they do not react with the environment in any adverse way and so investigating these properties is beyond the scope of this work.

To setup the model we use dimensional parameters. The spheroid is a spherically symmetric sphere of radius, R. We assume the spheroid is comprised of two continuous phases, tumour cells and cellular material, as is common in the literature [17, 22, 28]. These phases are described by the volume fraction per unit control volume of tumour cells, m(r, t) and cellular material, w(r, t), which are functions of the distance from the spheroid centre, r, and time, t. We also include transport of oxygen, c(r, t), via diffusion from the culture media surrounding the spheroid, as well as distribution and binding of polymersomes. In order to simplify the model we neglect the chemotherapy drug components and instead calculate cell death as a function of polymersome concentration.

We apply a no-void condition within the spheroid, representing conservation of mass, hence m + w = N_o, where N_o is a constant, given that m and w are normalised to a control volume of tumour cells.

The system is augmented via transport equations for the distribution of polymersomes, giving (49) (50) (51) (52) (53) (54) (55) where J_m, J_w, J_c and J_v are the fluxes of tumour cells, cellular material, oxygen and free polymersomes, respectively. The fluxes are given by (56) where v is the common advection velocity (the velocity of all constituents in the model due to bulk motion) and D_m, D_w, D_c, D_v are the diffusivities of the tumour cells, cellular material, oxygen and free polymersomes respectively. We assume that once the polymersomes have bound to the cell then they move with the cell. Using the no-void condition and by summing Eqs (49) and (50) we can deduce the following advection velocity, (57)

Next, we discuss the functional forms of the terms in the model. The framework we are using here has been adapted from similar work which models the use of therapeutic macrophages by Webb et al [17]. The function p_m(w, c) denotes tumour proliferation, which we take to be an increasing saturating function of both the cellular material and oxygen concentration, (58) and the death rate due to apoptosis, d_m(c), is a decreasing function of the oxygen concentration, (59) We capture oxygen metabolism through the function d_c(m, w, c) which increases with tumour cell density, (60) where (61) c_p is the oxygen threshold for proliferation, c_c is the oxygen concentration for half maximal cell death, is the maximum uptake rate of oxygen by cells, is the maximum uptake rate of oxygen due to proliferation and is the maximum cell growth rate under nutrient rich conditions. S(c, c_p) is a scaled Hill function with maximal value S = 1 when c = c_∞ (which we assume is the maximum value of oxygen at the tumour boundary), with the steepness of the curve dependent on the value of the exponent α > 0.

For simplicity the death of cells is a calculated as function of bound particles, reducing the number of calculations needed in the model. We assume that the death rate of tumour cells, g(B), is proportional to the total number of bound polymersomes, (62) where γ is the potency of the polymersomes. To further simplify the framework, we assume a fixed number of receptors per cell, f₀. So that, at any time, the free receptors per cell is given by this fixed number minus those involved in polymersome binding, namely (63)

Using this model set up, we can solve Eqs (53)–(55) independently to give numerical equations for polymersomes with different bond numbers, i.e for each B_i, (where i = 1, …, n), in terms of the remaining model variables. It is useful to solve for each B_i because this allows us to perform investigations into how the number of bonds between polymersomes and cells effects possible outcomes of spheroid growth. The values of the parameters that are introduced in this section are given in Table 2, the rest of the parameters which were also used in the previous section remain the same and are given in Table 1.

3.2 Initial and boundary conditions

Next we set the initial and boundary conditions for the model. Initially, the tumour is allowed to grow without the presence of therapeutic polymersomes, as the spheroid would in experiments, therefore at t = 0 (64) (65) where V_∞ is the polymersome surface concentration, M₀ is the tumour cell density, c₀ is the oxygen concentration, B(r, 0) is the bound polymersome concentration and V(r, 0) is the polymersome concentration at distance r from the spheroid centre and R₀ is the initial radius of the spheroid. At some time, t*, we introduce the polymersomes on the surface of the tumour spheroid by setting the surface concentration of polymersomes, to some non-zero value (previously V_∞ = 0 when 0 < t < t*).

Assuming the tumour is symmetric about the origin, we have (66) On the tumour boundary r = R(t), we fix the nutrient concentration, c = c_∞. Both cellular material and polymersomes can move across the tumour boundary with their flux across the boundary proportional to (w_∞ − w(R, t)) and (V_∞ − V(R, t)), respectively, where w_∞ is the external concentration of cellular material. The tumour boundary moves with the velocity v_m and cellular material moves across the boundary at velocity v_w, hence the flux boundary conditions for cellular material and unbound polymersomes on r = R(t) are given by, (67) (68) where h_w and h_v are the positive permeabilities of the cellular material and free polymersomes, respectively, across the tumour boundary. The velocities of tumour cells, v_m, cellular material, v_c, and free polymersomes, v_v, are given by

The boundary conditions for bound polymersomes, B_i’s, on r = 0 and r = R(t) can be found assuming the bound polymersomes are in quasi steady state (discussed in section 3.4).

The spheroid boundary moves with the tumour velocity v_m, (69) Using (57) we can re-write (69) as (70) Similarly to the well-mixed system we nondimensionalise due to the large variation in the magnitude of parameter values and to allow us to identify dominant terms in the model.

3.3 Nondimensionlisation

In this section, the system of Eqs (49)–(55) is nondimensionalised alongside the counterpart initial and boundary conditions to evaluate the dominant balance of different mechanisms. We use the following re-scalings to nondimensionalise the system, where the hats denote the nondimensional variables, where v_m is the volume of a tumour cell, R_m is the radius of an individual tumour cell and c_∞ is the room temperature atmospheric oxygen pressure. On dropping the hats we obtain (71) (72) (73) (74) (75) (76) where and

The non-dimensional advection velocity then is as follows (77) The parameter re-scalings read as, For convenience in the remaining analysis we drop the tildes.

3.4 Numerical solutions

Due to the fast timescale of oxygen diffusion (D_w = 1 × 10⁻⁶cm²s⁻¹) compared to the timescale of tumour growth (0.05cm/day) [19] we make the assumption that the oxygen concentration is at quasi -steady state, as is common in the literature [19, 30]. The binding kinetics of the receptors and ligands are fast compared to that of tumour cell growth, the binding rate, k_a, is on the order of 10⁸mol⁻¹min⁻¹cm³ whereas cell division is on the order of 10⁻⁴min⁻¹. This allows us to make the assumption that bound polymersomes are in quasi steady state compared to cell growth. We also note that the flux of cells is much lower than that of free nanoparticles and oxygen, so comparatively the cells are stationary on the tissue scale of cell movement, growth and decay. Thus the equations describing ligand-receptor complexes, Eqs (74)–(76), become (78) (79) (80) and the equation for oxygen, Eq 72, becomes (81)

We can now solve analytically for each B_i. First, we take n = 2, where the maximum number of bonds per polymersome is 2. Eqs (78)–(80) then give (82) (83) We can express the solution for B₁ and B₂ analytically in terms of m and V, namely, (84) (85) where (86) Note that the algebraic system (78)–(80) with n = 3 is analytically intractable so we need to make an additional simplifying assumption by introducing an additional approximation to F. To do this we assume that there are many more free receptors than bound, so that , where i = 1, …, n, and we can make the approximation (87) which allows us to calculate B_i for n > 2. We solve the system for up to n = 5 (equations not shown here for brevity).

With the solutions for B_i(r, t) we solve the nondimensionalised system (Eqs 71–73) with the boundary conditions given by (88) On the spheroid boundary, r = R(t), the oxygen concentration is c = 1 and we have (89) (90) The spheroid boundary (r = R(t)) now moves with the tumour velocity so that (91) The system is solved numerically using NAG routine D03PHF which uses a finite difference approach to integrate over one spatial variable and the method of lines to reduce the PDEs to a system of ODEs. The resulting system is solved using a backward differentiation formula method.

3.5 Spheroid growth before, during and after polymersome application

With the system solved numerically we can investigate impact of applying polymersomes to tumour spheroid growth. Fig 7 shows the model predictions in space and time for tumour cell density, oxygen concentration and free and total bound polymersomes for a constant internalisation function and a maximum number of bonds, n = 5. The spheroid grows initially in the absence of oxidative stress. During this period we observe linear growth of the spheroid radius, with a high concentration of proliferating cells at the surface which rapidly decays to no proliferating cells towards the spheroid centre at r = 0, indicating a necrotic core due to lack of oxygen. At time t = 40 days polymersomes are applied to the surface of the spheroid, which are then removed at t = 50 days. During the period of polymersome application, we observe a noticeable decrease in spheroid radius. The polymersomes are confined to the boundary of the spheroid, where the chemotherapy drug is unloaded into proliferating cells, causing the spheroid to shrink. As would be expected, the total bound polymersomes follows a similar trend to free polymersomes. Once the polymersomes are removed the cells at the edge of the spheroid become re-oxygenated due to the smaller radius allowing for effective diffusion. This then provides a good environment for growth, hence the spheroid re-grows linearly for t > 50 days.

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Fig 7. Evolution of a tumour spheroid and associated variables in time, including the tumour cell, oxygen, free and bound polymersome concentrations for n = 5.

The cellular concentration is given as the volume fraction of live tumour cells. The rest of the concentrations are dimensionless. Initially growth occurs with no free polymersomes on the tumour surface up to a radius of 175μm, then we introduce polymersomes by setting V_∞ = 10 when 40 ≤ t ≤ 50 days. The tumour growth rate decreases and the spheroid shrinks for the period of time when treatment is administered, but returns to linear growth (indicating travelling wave solutions) afterwards. The simulations were conducted using the dimensionless framework and the time and distance converted to dimensional form afterwards. The parameter values used are give in Tables 1 and 2.

https://doi.org/10.1371/journal.pone.0254208.g007

Next we investigate the model predictions using two different internalisation functions.

3.6 Effects of constant and bond dependent internalisation functions

It has been assumed so far in this paper, and in our previous work, that internalisation is constant. However, it is possible that internalisation is bond dependent so we also define a bond dependent internalisation function, in addition to the constant function. The bond dependent function is a simple step-wise Hill function (92) where ε₁ is a small number. In this scenario polymersomes are not internalised until they reach the bond number specified by the activator, j*.

In Fig 8A we show the model predictions for tumour radius using the constant internalisation function for n = 1 ⋯ 5. As before the tumour is allowed to grow until t = 40 days at which point polymersomes are applied until t = 50 days. We see very little variation in the predictions which implies internalisation happens quickly before multiple bonds can form. This is in agreement with our predictions in the well-mixed system (see Fig 3). In Fig 8B and 8C, we show the predictions for the case of the bond dependent function. Fig 8 indicates that when the activator, j* ≤ 4, there is a decrease in spheroid radius during the period of treatment comparable to that of constant internalisation, but if j* > 4 there is no reduction in spheroid growth during the treatment time. Therefore, with this set of parameters j* = 5 appears to be a threshold. That is, by forcing the polymersome to bind to at least 5 complexes before internalisation we slow down the internalisation of the polymersome sufficiently so that treatment is then not effective within this time frame. Fig 8 also shows longer term solutions, for 0 ≤ t ≤ 2500 days. We find that over long time periods the model predictions tend to the same solution.

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Fig 8. The impact of polymersomes on the spheroid dynamics using the numerical solutions for bonds.

(A) Tumour radius dependency on the maximum number of bonds, n, for the constant internalisation function. The maximum bond number is varied from n = 1 ⋯ 5. (B) (C) The tumour radius with time for a bond dependent internalisation function for various values of the activator, j*. Polymersomes are applied to the tumour surface at 40 ≤ t ≤ 50 days with the parameters V_∞ = 10, the rest of the parameter values used are give in Tables 1 and 2. (B) Polymersome application and growth over 80 days (C) Polymersome application and growth over a much longer time period of 2500 days.

https://doi.org/10.1371/journal.pone.0254208.g008

3.7 Large time behaviour

In the absence of polymersomes the tumour spheroid appears to grow linearly (see Fig 7). In order to find exact solutions for this behaviour we look to the asymptotic solutions of the system as t → ∞ and hence we carry out a travelling wave analysis by assuming that the tumour grows with constant speed, u > 0.

The rate of change of the spheroid radius will then be (93) so that R ∼ ut as t → ∞. We redefine our system with the travelling wave coordinate, z = r − ut, and rewrite the Eqs (49), (51) and (52) in terms of z: (94) (95) (96) where the prime denotes the derivative with respect to z. We use the quasi-steady state expressions for bound polymersomes as described in Eqs 84 and 85. The advection velocity can be written as (97) The system is represented by the following first order ODEs, (98) (99) (100) (101) (102) (103) for i = 1, …, n. Note that we have neglected the terms containing r⁻¹ since they are when R → ∞.

We solve this system of equations numerically using AUTO (a bifurcation and continuation software) [31]. To facilitate the solver we truncate the semi-finite domain to where R > 0 and taken to be sufficiently large for AUTO to solve the boundary value problem.

We rescale z (104) so that .

The wavespeed u can be written using 91 (105)

The system is subject to the following boundary conditions on the truncated domain : at (106) at , (107) (108) which follow from (88)–(90). The oxygen concentration is fixed at c = 1 when (chosen arbitrarily). We impose an additional boundary condition which fixes the wave speed u at , given by u = h_w(w_∞ + m − 1). This extra boundary condition allows us to use the bifurcation and continuation AUTO software [31] to solve (98)–(103) to calculate the wave speed, u.

3.8 Travelling wave solutions of spheroid growth

Fig 9 shows the wavespeed, u, against potency, γ, and external polymersome concentration, V_∞. We can see that by increasing the potency of the polymersomes, γ, the wavespeed decreases to zero (A), indicating a bifurcation from travelling waves (linear growth) to steady-state where tumour growth is confined. As V_∞ increases we observe similar dynamics, i.e. a greater concentration of polymersomes results in a more effective treatment (B).

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Fig 9. Travelling wave and steady state solutions for the tumour spheroid model.

(A) The travelling wave velocity, u, with varying potentcy, γ, and fixed polymersome concentration at the spheroid boundary, V_∞ = 2, and (B) travelling wave velocity with varying V_∞ and fixed γ = 1 × 10⁻⁴. (Middle and bottom) Travelling wave: steady state bifurcation curves. (Middle row) Bifurcation curves for γ with (C) various external polymersome concentrations, V_∞ and fixed external nutrient concentration w_∞ = 1 ⋯ 5, and (D) with various cellular material concentrations, w_∞, and a constant polymersome concentration, V_∞ = 5. In (E) we see the bifurcation in the polymersome diffusivity and spheroid permeability parameter space. The rest of the parameter values used are give in Tables 1 and 2.

https://doi.org/10.1371/journal.pone.0254208.g009

In Fig 9, we also show the travelling wave:steady-state bifurcation in (V_∞,γ) parameter space, with a varying external cellular material concentration, w_∞ (C). We can see that, with increasing concentration of applied polymersomes, a lower polymersome potency is required for steady state solutions. We would expect that by increasing w_∞ the tumour growth would increase resulting in a larger parameter region for travelling waves. We have found that this is true but only to a certain threshold value of w_∞, (D).

In Fig 9(E) we explore the travelling wave:steady-state bifurcation in relation to the diffusion of the polymersomes and the permeability across the spheroid boundary. We notice that for a fixed value of h_v (for example, if h_v = 2.2) the solutions can be either travelling waves for very small or very large values of D_v, but steady state solutions occur for intermediate values of D_v.

Finally, we examine the relationship between γ, V_∞ and w_∞ further by fixing V_∞ and following the travelling wave, steady state bifurcation in (w_∞,γ) parameter space. In Fig 10, we examine the behaviour of the tumour cells and the internal velocity field within the tumour spheroid for these parameter values. Typically the advection velocity within the spheroid is negative, which means that material is advected into the centre of the spheroid. However, when increasing w_∞ over a certain threshold, we see that the advection velocity becomes positive near the spheroid boundary, this change in the direction of the velocity results in the polymersomes being kept at the tumour boundary where the tumour density is highest. As a consequence more viable cells are being targeted and subsequently a lower polymersome potency (γ) is required to give the same reduction in tumour size. We also show how the free polymersome and oxygen density vary over the spheroid radius for various value of w_∞.

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Fig 10.

In (A, B) and (C) are the solution profiles for tumour cells, free polymersomes and oxygen, respectively, at varying values of external nutrient concentration, w_∞ = 0.2 (blue),0.4,0.6,0.8 (purple). (D) Advection velocity with w_∞ = 0.2, 0.4, 0.6, 0.8. We see a positive velocity at the tumour boundary with larger values of w_∞. The rest of the parameter values used are give in Tables 1 and 2.

https://doi.org/10.1371/journal.pone.0254208.g010