miR-451-AMPK-mTOR dynamics

We refer to the interactions represented by edges in Fig. 3B as the core miR-451-AMPK-mTOR control system. See Figure S1 in Supplementary File for the proposed role of miR-451 in the regulation of LKB1/AMPK-mTOR signaling in response to high and low glucose levels. By convention, the kinetic interpretation of arrows and hammerheads in the network represents induction (arrow) and inhibition (hammerhead). miR-451 level and activity of its target complex (CAB39/LKB1/AMPK), and mTOR levels are represented by ‘M’, ‘A’, and ‘R’, respectively. By simplifying the network as shown in Fig. 3A, we obtain the following dimensionless version of the model based on Fig. 3B (1) (2) (3) where G is the signaling pathways from glucose to miR-451, S₁, S₂ are the signalling pathways to AMPK complex and mTOR, respectively, λ₁, λ₃, λ₅ are the autocatalytic enhancement parameters for miR-451, AMPK complex, mTOR, respectively and λ₂, λ₄, λ₆ are the Hill-type inhibition saturation parameters from the counter part of miR-451, AMPK complex, mTOR, respectively, α is the inhibition strength of miR-451 by the AMPK complex, β is the inhibition strength of the AMPK complex by miR-451, and finally γ is the inhibition strength of the mTOR by the AMPK complex. The parameter ϵ₁ is given by the ratio of the degradation rates of M and A, respectively. Similarly, the parameter ϵ₂ is given by μ₁/μ₃, the ratio of the degradation rates of M and R, respectively.

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Figure 3. A simplified model of the network shown in Fig. 1.

(A) Conceptual model of regulation of miR451, AMPK complex, and mTOR in GBM cell migration and proliferation [6]. (B) Cartoon model (extended from [41]): miR-451 level and activity of its target complex (CAB39/LKB1/AMPK), and mTOR levels were represented by ‘M’, ‘A’, and ‘R’, respectively.

https://doi.org/10.1371/journal.pone.0114370.g003

The parameters for the equations (1)–(3) are given in Table 1 and are referred to as essential control parameters. See Supplementary materials for the derivation of the model based on a dimensional cartoon model in Figure S2 in Supplementary File.

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Table 1. Parameters used in the intracellular miR-451-AMPK-mTOR system.

https://doi.org/10.1371/journal.pone.0114370.t001

Mechanical effects on tumor growth: The cell-based component

Cell-mechanics is believed to play a large role in tumor growth and invasion. Mechanical stresses and the signaling pathways influence growth in a phenomenologically-specified manner in our model.

Active force and equations of motion.

The active force is the force that a cell generates via complex biochemical and bio-mechanical processes in order to move. These processes include generation of traction forces on the adhesion sites [19], tension, and activation of the acto-myosin machinery [24], in particular myosin II for glioma cells infiltrating brain tissue [25]. In the original lattice-free cell-based models [19, 26], the traction force exerted by a cell in the interior of a cell aggregate is exerted on one of its neighboring cells. For a cell in contact with a flat 2-dimensional substrate, the ‘pulling’ process on the substrate gives rise to the component of the force in the plane of the substrate. If a cell is not in contact with the substrate, the entire force is obtained by pulling on the neighboring cell whose center is closest to the line of the desired motion. (See Dallon and Othmer [19] for more detail.) In this work, we do not consider the detailed acto-myosin machinery nor collective cell migration which are key players in a certain stage of many cancers [24]. Here, we have used a simplified description, in which we assume that the traction force for active cell migration is generated for only the cells (i) that are in contact with the substrate (i.e., without physical constraints) and (ii) that receive the correct migratory signal from the miR-451-AMPK-mTOR control system. Therefore, the migratory cells do not depend on neighboring cells for active migration but they can transmit the traction force directly to the substrate in brain tissue without pulling its neighboring cells. The model can be extended to incorporate traction forces generated by all cells, but we consider only migratory individual cells due to their rather dispersal migration patterns in glioma invasion [27, 28].

The traction force $T_i^a$ for migratory cell i is given by (9) where d_r is a unit vector of the moving direction from random motion, G, C are the concentrations of glucose and a chemoattractant, respectively (described in Section above), ψ₁, ψ₂, ψ₃ are scaling factors of weight distribution favoring random motion, glucose and other chemoattractants, respectively (ψ₁, ψ₂, ψ₃ ∊ [0, 1]; ψ₁+ψ₂+ψ₃ = 1), A is the level of AMPK complex at the ith cell site. We allow a small randomness in the magnitude of traction force. Then, the indicator function ϕ(A) is given by (10) where F₀ is the basal magnitude of the traction force ($0\le \left|T_i^a\right|\le F_0$) and ϕ_r is a random number (ϕ_r ∊ [0.8, 1.2]). Therefore, the traction force is completely turned off for cells in the proliferative phase (M > th_A, A < th_A, R > th_R) or cells under physical constraints. If there are other cells 90° in any direction (T_{i, g}) of motion of cell i, the cell i is defined to be under physical constraints. For a more precise algorithm, let j, the cell index for the cell whose center is in the closest direction to the direction of T_{i, g}, be given by $j=\{k:ma{x}_{k\in {\mathcal{N}}_i^a}\frac{x_k-x_i}{|x_k-x_i|}\cdot {\widehat{T}}_i\}$ where x_n is the location of the center of the n-th cell, ${\mathcal{N}}_i^a=\{k:0<|x_k-x_i|\le d_n\}$, the neighboring cells of the i-th cell, and ${\widehat{T}}_i=\frac{T_{i,g}}{\left|T_{i,g}\right|}$. The cell i is defined to be under physical constraints if j > 0, i.e., when another cell is blocking the way in the migration direction ${\widehat{T}}_i$. For example, when a cell is completely surrounded by neighboring cells, such as a cell within the growing tumor core, the traction force is turned off. From the formulation in the equation (9), it would be turned off when gradients of glucose (∇G) and chemoattractants (∇C) are zero in the absence of random motility (ψ₁ = 0), i.e., no traction force is generated in the absence of chemotactic signals.

In view of these assumptions above, the force balance on migratory cells involves the reaction $T_i^{a,*}=-T_i^a$ to the traction force $T_i^a$, adhesion forces between cells (A_{i, j}), the drag due to the fluid acting on the glioma cell, internal forces (R_{j, i}), and the passive reactive force due to cell-substrate ($R_{0,i}^{*}$) and cell-cell deformation ($R_{j,i}^{*}$). In summary, by using equations (4)–(10) and neglecting acceleration due to slow movement of cells, Newton’s law for the ith cell reduces to (11) where v_i is the velocity of cell i, ${\mathcal{N}}_i$ is the neighborhood of cell i, μ_cell (resp., μ_s, μ_f) is the degree of adhesiveness between the cells (resp., between the substrate and the cells, and the fluid viscosity), r_ib = u_b+b₀, and A_ij = A_ij(t), A_if = A_if(t), A_is are the areas of contact regions between cell i and cell j, cell i and the interstitial fluid or matrix, and cell i and the substrate at time t respectively, A = A(t) is the total area of an undeformed cell. The solution of this equation (11) provides a trajectory of cell i. For more details see Dallon and Othmer [19]. Parameters in the cell-based component are listed in Table 2.

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Table 2. Parameters for the cell-based component of the model.

TW = this work. *dimensionless value.

https://doi.org/10.1371/journal.pone.0114370.t002

Dynamics of biochemical players

The macroscopic dynamics of concentrations of relevant biochemical players (oxygen, glucose, chemoattractants, ECM, MMPs, and chemotherapeutic) is modelled using a suitable partial differential equation incorporating vessels as sources within the simulation domain. Let K(x, t), G(x, t), C(x, t), ρ(x, t), and P(x, t) denote the concentration of oxygen, glucose, chemoattractants, ECM, and MMPs at position x at time t, respectively. Their rate of change can be expressed as (12) (13) (14) (15) (16) where D_K(x), D_G(x), D_C(x), D_P(x) are the diffusion coefficients of oxygen, glucose, chemoattractant, and MMPs, respectively, μ_C, μ_P are the natural decay rates of chemoattractant and MMPs, respectively. The third terms in the equations (12)–(13) are a function describing the consumption of oxygen and glucose by tumour cells at rates $l_c^K,l_c^G$, respectively. The fourth terms in the equations (12)–(13) represent the consumption of oxygen and glucose by other cells in the brain tissue at rates μ_K, μ_G. Here, I_CT(⋅) is an indicator function on the tumor sites C_T (including both proliferative and migratory cells) (17) Similarly, I_B(⋅) is an indicator function on the set B that represents the blood vessel: (18) Thus the terms r_K I_B(⋅) and r_G I_B(⋅) in the equations (12)–(13) describe the supply of oxygen and glucose via blood vessel at rates r_K and r_G, respectively. We assume that the oxygen and glucose are supplied through the pre-existing blood vessels, and then diffuse throughout the tissue feeding the cells within its diffusion limit. As the tumour cells grow, some of the cells within the tumour mass will be starved of oxygen and glucose which in turn affect their intracellular processes, enabling them to adapt to the changing microenvironment. These changes due to spatial and temporal heterogeneity of oxygen and glucose levels is incorporated into the model using the intracellular dynamics described in the previous section. The second term in the equation (14) represents injection of chemoattractants at a rate $l_{in}^C$ on a set Ω_R near the resection cavity after conventional surgery. These injections are administered over the time intervals $[{\tau }_j^C,{\tau }_j^C+{\tau }_d^C]$ ($j=1,\dots ,N_C;{\tau }_1^C<{\tau }_2^C<\dots <{\tau }_{N_C}^C$) where ${\tau }_d^C$ is an equal duration of injection and N_C is the total number of injections. Due to its injection location, this injection in addition to the diffusion effect and natural decay would create a gradient of chemoattractants, which would serve as a localization of migratory glioma cells back to the area near the periphery of the resection cavity for second surgery or other anti-tumor treatments. Even though the ECM provides a structural support for the guidance of cell migration, high density of ECM also blocks cell migration and a migratory cell need to produce proteinases such as MMPs in order to infiltrate the brain tissue. The first and second terms in the equation (15) represent the degradation of the ECM by MMPs secreted by invasive tumor cells at a rate l₁ and release/reconstruction of the ECM at a rate l₂, respectively. MMPs are secreted by invasive tumor cells at a rate l₃ in the equation (16) when cell motility is necessary, therefore, its concentration is localized at the location (I_Cm) of glioma cells in the migratory phase (M < th_M, A > th_A, R < th_R). Here, I_Cm(⋅) is an indicator function on the set C_m that represents invasive cell sites: (19) where ${\mathbb{T}}_m$ is the migratory phase, which will be defined precisely later. (see equations (20)–(21) in Results section.) We assume no flux (Neumann) boundary conditions for the system (12)–(16). Parameters in the cell-based component are listed in Table 3.

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Table 3. Parameters that are used in the reaction-diffusion equations.

https://doi.org/10.1371/journal.pone.0114370.t003

Dynamics of the core control (miR-451-AMPK-mTOR) system

We recall that low levels of miR-451 (up-regulated AMPK complex and down-regulated mTOR) induce reduced cell proliferation and increased cell motility while over expression of miR-451 (down-regulation of AMPK complex and up-regulation of mTOR) leads to elevated cell proliferation and reduced migration in experiments [6, 18]. In order to take into account the effect of glucose conditions in our model on phenotypic changes (proliferative and migratory cells), we first test how the glucose level (G) affects the levels of key players (M, A, R) in our core control system.

When the core miR-451-AMPK-mTOR system (1)–(3) is in equilibrium, we can solve miR-451 levels (steady state (S.S) value M^s) as a function of the extracellular glucose level (G). In a similar fashion, we can also obtain the bifurcation curve of steady state values of AMPK activity (A^s), and mTOR levels (R^s) w.r.t. external glucose signal (G). Fig. 4A shows the graphs M = M(G) (blue), A = A(G) (red), R = R(G) (green) as a S-shaped curve (hysteresis) with reversed direction of the A-curve. While the upper and lower branches of those curves are stable, the middle branch is unstable. Under glucose withdrawal conditions, the system (1)–(3) travels along the lower branch (M low, A high, R low) of the miR-451 curve and the cells are in the migratory phase. The cell continues to migrate as G is increased until it reaches the right knee point of the bifurcation curve (∼0.6). Around this point, both miR-451 and mTOR levels jump to the upper branch, with elevated levels of miR-451 and mTOR and down-regulated AMPK, and the cells are put in the proliferative phase (migration switch is turned off). Theofore, the effect of glucose is history dependent. The size of the bi-stability window ($W_b=[b_m^w,b_M^w]$) depends on other parameters and may even disappear under the perturbation of some parameters. By taking the thresholds, th_M (= 2.0) of miR-451 levels, th_A (= 2.0) of AMPK complex, and th_R (= 3.0) of mTOR, we shall define the migratory region ${\mathbb{T}}_m$ (red box in Fig. 4B) by (20) and the proliferative region ${\mathbb{T}}_p$ (blue boxes in Fig. 4) by (21)

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Figure 4. Bifurcation curve of the miR-451-AMPK-mTOR module.

(A) High and low glucose levels (G) provide an on-off switch of miR-451 over-expression and determine the dichotomous behavior: cell proliferation or migration [41]. Y-axis = steady state (SS) of miR-451, AMPK, and mTOR. $W_b=[b_m^w,b_M^w]=a$ window of bi-stabilty. (B) Characterization of proliferation and migration of glioma cells in miR-451/mTOR-AMPK domain. The proliferative region (blue box) is defined as the region where the miR-451 level is above a threshold, th_M (M > th_M), the AMPK level is below a threshold, th_A (A > th_A), and the mTOR level is above a threshold, th_R (R > th_R), while the levels of miR-451, AMPK, and mTOR in the migratory region (red box) satisfies M < th_M, A > th_A, R < th_R. We set th_M = 2.0, th_A = 2.0, th_R = 3.0. (C, D) Dynamics of the core control system in response to low (G = 0.1 in (C)) and intermediate (G = 0.45 in (D)) glucose levels. When G = 0.1, there is only one stable steady state (red dot, (M^s, R^s, A^s) = (0.3, 1.4, 4.0)) in the migratory region while, for G = 0.45, there are three steady states: two stable steady states (red dot ((M^s, R^s, A^s) = (1.8, 1.7, 2.7)) in the migratory region and blue dot ((M^s, R^s, A^s) = (3.0, 4.1, 0.6)) in proliferative regions) and one unstable steady state (black circle (((M^s, R^s, A^s) = (2.1, 3.3, 0.9)) in the middle). Red and blue curves indicate the trajectories of solutions for two pairs of very close initial conditions: (M, A, R)(0) = (2.1, 4.0, 5.0) and (2.2, 4.0, 5.0); (M, A, R)(0) = (2.1, 0.1, 0.1), and (2.2, 0.1, 0.1).

https://doi.org/10.1371/journal.pone.0114370.g004

For an illustration of the dynamics, Figs. 4C, 4D show the dynamics of the core control system in response to low (G = 0.1; Fig. 4C) and intermediate (G = 0.45; Fig. 4D) glucose levels. For the low glucose level (G = 0.1; Fig. 4C), there is only one stable steady state (red dot; (M^s, R^s, A^s) = (0.3, 1.4, 4.0)) in the migratory region ${\mathbb{T}}_m$ where miR-451 (M) and mTOR (R) expressions are low but AMPK activity (A) is high. On the other hand, for high glucose levels (G > 0.6), there is only one stable steady state in a region ${\mathbb{T}}_p$ where miR-451 (M) and mTOR (R) expressions are high but AMPK activity (A) is low (data not shown; cf. Fig. 4A). For an intermediate glucose level in the bi-stability window G = 0.45 ∊ W_b in Fig. 4D), there are three steady states: one stable S.S. (red dot; (M^s, R^s, A^s) = (1.8, 1.7, 2.7)) in the high AMPK activity region ${\mathbb{T}}_m$, another stable S.S (blue dot; (M^s, R^s, A^s) = (3.0, 4.1, 0.6)) in the high miR-451 activity region ${\mathbb{T}}_p$), and one unstable steady state in the middle (black circle; (M^s, R^s, A^s) = (2.1, 3.3, 0.9)). In this case, starting from two very close initial conditions (either (M, A, R)(0) = (2.1, 4.0, 5.0), (2.2, 4.0, 5.0) or (M, A, R)(0) = (2.1, 0.1, 0.1), (2.2, 0.1, 0.1)), the dynamics lead to either ${\mathbb{T}}_p$ or ${\mathbb{T}}_m$ phases. This dichotomous behaviors of the miR-451 (M), AMPK complex (A), and mTOR (R) modules in our mathematical model in response to high, intermediate, and low glucose levels are in good agreement with biological observations in [6, 18] (and other references therein), which leads us to the characterization of up- and down-regulation of these molecules relative to the threshold values (th_M, th_A, th_R) for cell proliferation and migration in our modeling framework.

For simplicity, we assume that miR-451-AMPK-mTOR is the only signaling network that determines the cell proliferation and set G₀ = 0 for the rest of sections. In the next section, we incorporate the core control system into the full hybrid model and analyze the full dynamics of glioma cell behavior in the presence of BVs.

Dynamics of the hybrid model

Figs. 5A–5D show tumor growth and associated invasion patterns in the computational domain Ω = [0, 1]² at t = 0, 80, 160, 240 h after surgical resection of the main tumor mass at the center of the domain at t = 0 h. Invisible invasive cells (red circles) respond to the gradient of glucose (∇G) and randomly move with a probability of ψ₁ = 0.6. Most of cells settle in one of BV sites (for instance, see black arrow in Fig. 5D) while some cells are still in the migratory phase (black arrowhead in Fig. 5D). See Figure S5 in Supplementary File for the corresponding spatial profiles of concentrations of diffusive molecules. Figs. 5E–5H show spatial profiles of those cells near one of the BVs in a smaller frame ([0.57, 0.63] × [0.39, 0.45]; blue solid box in Figs. 5A–5D) at the same time frame. Once settled in the BV (marked in green @) region, these cells respond to high glucose levels from the BV and become proliferative cells (blue circles). Fig. 5K shows time courses of concentrations of miR-451 (blue solid line), AMPK complex (red dotted line), mTOR (black square), and glucose (green circle) for a growing cell near the BV (cellid = 6; black arrowhead in Figs. 5E–5H). Initially, low miR-451 level is increased and maintained above the threshold value (th_M = 2.0) in response to fluctuating but high glucose concentrations at the cell site as the cell is getting close to the BV site. This leads to down-regulation of AMPK complex (A < th_A = 2.0) and up-regulation of mTOR (R > th_R = 2.0; th_R was marked in dotted line). Moreover, a highly adaptive response of AMPK complex and mTOR can be observed in response to fluctuating high miR-451 levels. This over-expression of mTOR leads to a regular cell cycle and proliferation. Our simulation results are in good agreement with up-regulation of miR-451 expression among aggregates of tumor cells near BVs in biological experiments [6, 18].

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Figure 5. Dynamics of cell migration and proliferation in the hybrid model.

Some diffusely infiltrating glioma cells in the ${\mathbb{T}}_m$-phase are attracted to the glucose-rich BV sites and switch to the ${\mathbb{T}}_p$-phase for fast growth in the favorable microenvironment while others are still moving around in the heterogeneous environment. (A-D) Tumor growth/invasion patterns in the computational domain Ω = [0, 1]² at t = 0, 80, 160, 240 h after surgery. Arrowhead = migratory cell, arrow = growing cells near a BV. @ = BV. (E-H) Profiles of tumor cells in a small region [0.57, 0.63] × [0.39, 0.45] (blue solid box in (A-D)) of the domain. Arrowhead = 6th cell, arrow = 94th cell. @ = BV. (I, J) Migration routes of two cells (cellid = 6 in (I), 94 in (J)). Red arrowhead = initial position, Black arrow = final position at t = 240 h. (K, L) Concentrations of miR-451 (blue solid line), AMPK complex (red dotted line), mTOR (black circle), and glucose level at the cell site of two cells considered in (I, J). Threshold value of mTOR (Th_R = 2.0) was represented in black dots. Legends in (L) is same as one in (K).

https://doi.org/10.1371/journal.pone.0114370.g005

At the later time (t > 175h), the glucose level, and consequently miR-451 is decreased. There are two possible reasons for this: (i) the cell is mechanically pushed away from the BV site (source of glucose) through mechanical interactions with other cells in the local neighborhood as the number of growing tumor cells is increased; (ii) as the tumor cells grow, the consumption of glucose by tumor cells is increased. This imbalance between supply and consumption of glucose as the tumor grows near the BV and translocation of the cell from mechanical interactions with other cells as it grows cause fluctuating glucose levels at the cell site overall. Random motility of the cell (ψ₁ = 0.6) before it enters the BV region also contributes to the irregular glucose levels. See Fig. 5I for trajectory of the cell which was initially positioned away from the BV (red arrowhead) and passively positioned near the outer boundary of the growing tumor mass near the BV at the final time (t = 240 h; black arrow). When free space is still available (Fig. 5F), motile cells and growing cells are still moving around the BV. However, as the growing tumor cell density is increased (Fig. 5F→Fig. 5G→Fig. 5H), active migration of cells inside the growing tumor mass is completely turned off due to physical constraints. Growth of these cells surrounded by neighboring cells in the core is also partially suppressed due to growth function (cf Figure S3 in Supplementary Information) [20–23]. On the other hand, invasive cells which were initially positioned far away from BVs, did not settle down at one of those BVs during the simulation. (For instance, a cell (cellid = 94) marked with black arrow in Fig. 5H.) Fig. 5J shows the relatively long route of the traveling glioma cell (cellid = 94) from initial position (red arrowhead) to (black arrow). Despite the random motion occurring most of time, the cell migrate toward the same BV site as in Figs. 5E–5H in a consistent fashion. Fig. 5L shows concentrations of variables in the core miR-451-AMPK-mTOR system. As the cell approaches the BV, the glucose level at the cell site is slowly increased and the biochemical signal in migratory phase (M < th_M, A > th_A, R < th_R) switches to the signal in proliferative phase (M > th_M, A < th_A, R > th_R) around t = 110 h. However, this cell is still in the migratory mode at the final time (t = 240 h) since it did not reach the BV site yet. Therefore, a biomechanical switch for cell fate, i.e., proliferation and migration, may play a significant role in developing an anti-cancer drug or determining tumor-healthy tissue margins for any followup surgical resection of the recurrent tumors.

In the next section, we explore the glioma growth and invasion patterns for various random cell motility parameters (ψ₁) and different BV densities, and investigate the effect of conventional chemotherapy drugs on killing migratory cells under these conditions.

Glioma cell invasion and efficacy of chemotherapeutic after surgery

Figs. 6A–6C show spatial profiles of migratory cells (red dots) and growing cells near BVs (blue dots) at t = 164 h for various BV density when the random motility parameter was set ψ₁ = 0.2. Here the number of BV sites was increased (from N_b = 31 in (A), to N_b = 47 in (B), and to N_b = 88 in (C)) with relatively uniform densities in the computational domain. Figs. 6G–6H show time courses of populations of growing cells near BVs (in (G)) and invasive cells away from BVs (in (H)) for these three cases (N_b = 31 (red solid line), 47 (blue square), 88 (black circle)). As the BV density is increased, population of growing cells near BVs is increased and the number of invasive cells away from BV sites is decreased since the chance of being attracted to BVs is increased. At the final time (t = 164 h), there are no invasive cells left for these three cases (red squares in Fig. 6I) but the number of cells at BVs is increased as N_b is increased. When the random motility parameter (ψ₁) is set to be an intermediate value (ψ₁ = 0.6), the overall dynamics of changes in populations of migratory cells and proliferative cells near BVs is similar to the case in ψ₁ = 0.2. See Figs. 6J–6K. However, the chance of settling in BVs is decreased due to the increased random motility (from ψ₁ = 0.2 to ψ₁ = 0.6). For example, a migratory cell (black arrow in Fig. 6D) did not settle in one of BVs when ψ₁ = 0.6 and blood density is relatively low (N_b = 31). However, these migratory cells eventually settle in one of BVs when blood density is high (N_b = 88 in Fig. 6F).

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Figure 6. Effect of random motility and blood density on regulation of migration patterns.

The tumor population is increased regardless of the relative ratio of the random motility (ψ₁) as the number of blood vessels (N_b) is increased because more invasive cells settle down near BVs, leading to faster tumor growth. On the other hand, fewer infiltrative cells switch to the growth phase for the relatively low BV density in the case of greater cell motility, reducing overall tumor growth but increasing further infiltration potential. (A-C) Spatial patterns of migratory cells (red dots) and growing cells near BVs (blue dots) at t = 164 h for a low (ψ₁ = 0.2) random motility in the presence of various densities of blood vessels (N_b = 31, 47, 88) in the brain tissue after surgery. @ = BV sites. (D-F) Same as (A-C) but for an intermediate level of random motility (ψ₁ = 0.6). (G-I) Time courses of populations of growing cells at BVs (G) and invasive cells distant from BVs (H) for three cases in (A-C) with the low (ψ₁ = 0.2) random motility: (N_b = 31 (red solid line), 47 (blue square), 88 (black circle)). Populations of proliferative (blue empty circles) and invasive (red filled squares) tumor cells at final time t = 164 h for various N_b (N_b = 31, 47, 88) are shown in (I). (J-L) Same as (G-I) but for an intermediate level of random motility (ψ₁ = 0.6). The legend is same as one in (G-I).

https://doi.org/10.1371/journal.pone.0114370.g006

In Fig. 7 we illustrate the effect of chemotherapy on killing cancerous cells for all cases in Fig. 6. A S-phase targeting chemo-drug was given intravenously at t = 150 h. Figs. 7A–7C and 7D–7F show spatial profiles of cancerous cells in the case of different BV densities at t = 164 h for low (ψ₁ = 0.2) and intermediate (ψ₁ = 0.6) random motilities, respectively. Figs. 7G–7I show time courses of populations of growing tumor cells near BVs (Fig. 7G), invasive tumor cells away from BVs (Fig. 7H), and apoptotic tumor cells responding to the chemo-drug (Fig. 7I). Figs. 7J–7L show the time courses of populations when the random motility is strengthened (ψ₁ = 0.6). Despite different numbers of tumor cells and subgroups near BVs, significant number of cancer cells near BVs for all six cases are killed already for all six cases due to relatively easy access of the drugs near BVs (green dots in Figs. 7A–7F; Figs. 7G, 7J). However, the killing rates of invasive cells (red dots Figs. 7A–7F) away from BV depend on the random motility (ψ₁) and BV densities. When the random motility is low (ψ₁ = 0.2), all migratory cells become growing cells at the BV sites and therefore respond to chemotherapeutic regardless of BV densities (Figs. 7A–7C). On the other hand, when the random motility is increased (ψ₁ = 0.6) and BV density is low (N_b = 31), some of the migratory cells do not settle in a BV site and are not killed due to low accessibilities of chemotherapeutic far from BV sites (red dots in Fig. 7D; red solid curve in Fig. 7K). As the BV density is slightly increased (N_b = 47), the number of drug-free invasive cells is decreased but these survived cells are sources of potential recurrence even after chemotherapy, decreasing overall longterm survival rate of the glioma patients. However, the BV density is further increased (N_b = 88), all tumor cells respond to the chemo-drug and are killed since all migratory cells are localized near BVs. These results imply that depending on the changes in microenvironment near the origin of the main glioma mass, different appropriate strategies of chemotherapy should be applied to glioma patients. The blood brain barrier (BBB), a separation of circulating blood from the brain extracellular fluid (BECF) in the central nervous system (CNS), also plays a significant role in regulation of diffusion of the chemo-drugs at BV sites, i.e., transport of the drugs to brain tissue. Therefore, the BBB in addition to spatial distribution of BVs near the tumor site would affect the efficacy of chemotherapeutic after surgical resection of the main glioma core.

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Figure 7. Effect of chemo-drug on killing migratory cells.

Chemotherapy is highly effective for growing cells near BVs but the treatment efficacy is low for highly infiltrative glioma cells (high ψ₁) due to low permeability at BVs (BBB). (A-C) Spatial patterns of glioma cells at t = 164 h for a low (ψ₁ = 0.2) random motility in the presence of various densities of blood vessels (N_b = 31, 47, 88) after intravenous injection of S-phase targeting-Chemo-drugs at t = 150h. Red = migratory cells, blue = growing cells near BV sites, green = apoptotic cells due to the chemo-drug. @ = BV sites. (D-F) Same as (A-C) but for an intermediate level of random motility (ψ₁ = 0.6). (G-I) Time courses of populations of growing cells at BVs (G), invasive cells away from BVs (H), and killed tumor cells near BV sites (I) for three cases in (A-C) with the low (ψ₁ = 0.2) random motility: (N_b = 31 (red solid line), 47 (blue square), 88 (black circle)). (J-L) Same as (G-I) but for an intermediate level of random motility (ψ₁ = 0.6). The legends in (H-L) are same as one in (G).

https://doi.org/10.1371/journal.pone.0114370.g007

Tumor recurrence after conventional surgery is a major obstacle in treating glioblastoma. In the next section, we propose various localization strategies as a way of eradicating the invisible migratory cancer cells in combination with chemotherapy after surgery.

Predictions of the model for a possible therapeutic approach

In this section, we developed therapeutic strategies for eradicating invisible migratory glioma cells in the brain tissue after conventional surgery where only visible parts of the tumor mass are removed. Here we assume that invasive cells in the surrounding tissue can sense and respond to the chemoattractant gradient (∇C) as well as glucose gradients created from BV sites.

We first test the efficacy of the localization strategy without chemotherapy on killing invasive glioma cells. Figs. 8A–8D show spatial profiles of the invasive tumor cells at t = 0, 100, 200, 240 h, respectively. After first surgery (region inside black solid circle in Fig. 8A) at t = 0, a chemoattractant was injected at the center (0.5, 0.5) of the resected area. See Fig. 9B for spatial profiles of chemoattractant at the corresponding time frames t = 0, 100, 200, 240 h. Even though migratory cells (red circles) do the random walk (ψ₁ = 0.2), they begin to migrate back to the tissue near the resection site. On the other hand, high glucose levels trigger the intracellular switch from the migratory phase to the proliferative mode for invasive cells near the BV site (Fig. 9A). BV sites were marked in @. Figs. 8E–8H show close-up profiles of movement of glioma cells in smaller subframes ([0.05, 0.2] × [0.55, 0.65] in (E-F); [0.14, 0.16] × [0.58, 0.61] in (G-H)) in Figs. 8A–8D. while most of other invasive cells (red circles) are migrating toward (black arrows) the resection site due to injection of a chemoattractant, a glioma cell (cellid = 33; black arrowheads) settles in a BV site on the routes of the migratory path (Fig. 8F). The cell near a BV (green @) then settles in a comfortable environment with enough glucose and begins to grow again. Figs. 8G–8H show growth of cancerous cells near the BV in a zoom-up subframe (blue box in Fig. 8F). Blue cells in Figs. 8G–8H show progenies of the first immigrant cell to arrive at the BV (gray cell; black arrowheads).

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Figure 8. A proposed localization strategy of eradicating invisible migratory cells after first surgery.

Some infiltrative glioma cells move back to the periphery of the resection site for the second surgery in response to the injected chemoattractant at the resection site after conventional surgery. However, some growing cells near BVs remain near BVs and do not respond to the chemotactic forces, reducing the efficacy of the localization strategy. (A-D) Spatial profile of cancerous cells at t = 0 (A), 100 (B), 200 (C), 240 (D) h after surgical resection at t = 0h followed by an injection of a chemoattractant at the resection site. The visibly detectable tumor core was surgically removed at t = 0 h (black solid circle in (A)). The localized invasive (red circles) glioma cells form a mass of tumor in (D) for a possible secondary surgery after 10 days, leading to eradication of invisible migratory cells. However, some of the cells (blue circles) attracted to BVs can still grow. Domain size = [0, 1]². (E-H) Spatial profiles of a subset of migratory cells. Red circles = migratory cells, blue circles = growing cells, @ = BV. (I-J) A time course of miR-451-AMPK-mTOR (I) and cell volume in a growing glioma cell (arrowheads in (E-H); cellid = 33) near a BV in response to high glucose from the BV.

https://doi.org/10.1371/journal.pone.0114370.g008

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Figure 9. Dynamics of diffusible molecules.

(A-D) Spatial profiles of glucose (G in (A)), chemoattractant (C in (B)), ECM (ρ in (C)), and MMP (P in (D)) at t = 0, 100, 200, 240 h that corresponds to Figs. 8A–8D. Glucose fluxes are provided at BV sites. The chemoattractant was injected at the center of the domain (0.5, 0.5). Domain size = [0, 1]². Glucose is continuously supplied through BVs while chemoattractants diffuse through the domain after strategic injections at the center to attract infiltrative glioma cells. ECM is degraded and undergoes constant remodeling via MMPs secreted by migratory tumor cells toward the center during the localization process.

https://doi.org/10.1371/journal.pone.0114370.g009

Figs. 8I–8J show time courses of the core control module (miR-451, AMPK, and mTOR) and cell volume at a cell site (cell id = 33; arrowheads in Figs. 8E–8H) in response to fluctuating glucose levels due to its proximity to BV on the way back to the resection site. At the beginning of the simulation (t = 0h), when a glioma cell is randomly searching for a better environment, both miR-451 and mTOR levels are low and the AMPK level is high due to the relatively low glucose level in the environment. As the cell approaches the BV site, the high glucose level (green line with black marker) induces over-expression of miR-451 levels (blue solid line) and decrease in AMPK activities (red dotted line) leading to up-regulation of mTOR levels (black square). The threshold value of mTOR (th_R = 2.0) was set in a black dotted line. The glucose level is increased during its migratory phase toward the BV site and maintains a relatively plateau state after stalling the migration and initial cell divisions but slightly decreased after t = 200 h due to translocation from the mechanical interactions with the neighboring daughter cells and increased consumption of glucose by the growing mass of these tumor cells. This switch to the growth phase leads to downstream signals to generate the regular cell cycle for this growing cell (Fig. 8J). As the number of daughter cells increases the original cell (black arrowhead) is surrounded by other cells and the duration of the cell cycle is slightly increased due to partial growth arrest from mechanical constraints after t > 200 h. Over-expression of miR-451 and proliferation of glioma cells near BVs were observed [6]. While infiltrative tumor cells after surgery may not be detected visually, a localized mass of invasive tumor cells may be detected by conventional screening tools such as MRI and could be surgically removed or eliminated by radiotherapy. This would increase the probability of eliminating invasive cells. However, localization of these cells to the resection site may not be enough and growing cells near BV sites may have to be removed as well in this case. Fig. 9 shows spatial profiles of glucose (G in Fig. 9A), chemoattractant (C in Fig. 9B), ECM (ρ in Fig. 9C), and MMP (P in Fig. 9D) at t = 0, 100, 200, 240 h that corresponds to Figs. 8A–8D. Glucose fluxes are provided at BV sites. The chemoattractant was injected at the center (0.5, 0.5) of the domain [0, 1]².

Random motility of glioma cells affect treatment strategies since unpredictabilities of migration direction might affect localization. Figs. 10A–10D shows different patterns of glioma cell localization with various random motility of glioma cell migration (ψ₁ = 0.1 (Fig. 10A), 0.3 (Fig. 10B), 0.5 (Fig. 10C), 1.0 (Fig. 10D). As the motility parameter (ψ₁) is increased, the efficacy of localization of invasive cells back to the resection site is decreased (Fig. 10E) but the population of growing cells near BV sites also is decreased. Figs. 10G–I show trajectories of an invasive cell (cell id = 1) with a starting point (0.32, 0.89) when ψ₁ = 0.1 (10G), 0.8 (10H), 1.0 (10I).

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Figure 10. Effect of random motion on the efficacy of localization strategies.

(A-D) Patterns of cell migration under the localization strategy with various random motility rates: ψ₁ = 0.1 (A), 0.3 (B), 0.5 (C), 1.0 (D). (E) Population of localized invasive cells at day 10 within a narrow strip (Ω_p) on the periphery of the resected area (Ω_s) from the first surgery in the whole domain (Ω). Ω_p = {P = (x, y) : dist(P, Ω_s) < w^th} where w^th = 0.2, 0.7, 1.2 mm. (F) Population of proliferative cells at BV sites after chemoattractant injection as in (E) at final time (t = 10 days). (G-I) Trajectories of migratory cells in the brain tissue when ψ₁ = 0.1 (G), 0.8 (H), 1.0 (I). Arrowhead = initial position, arrow = end position. As random motility ψ₁ is increased, the relative chemotaxis strength is decreased and infiltrative glioma cells do not respond to the chemoattractant signals, reducing the treatment efficacy of localization strategies.

https://doi.org/10.1371/journal.pone.0114370.g010

To test the validity of the model, we measure cell speeds and patterns of migration direction changes for various random cell motility parameters (ψ₁) in order to compare them with experimental data. Fig. 11A shows time courses of the cell speed and direction change (degree) of a migratory cell responding to the chemotactic signal near the resection site (cellid = 1; Figs. 10G–I) with various random motility levels: ϕ₁ = 0.1 (red solid line), ϕ₁ = 0.5 (blue square) and ψ₁ = 1.0 (black circle). When the random motility is small (ψ₁ = 0.1), the cell speed fluctuates between values in the range of (18–36) μm/h until the cell speed begins to drop down to zero around∼212 h due to cell aggregation near the resection site. For the intermediate case (ψ₁ = 0.5), the cell speed still fluctuates in the lower range (16–32 μm/h) but the cell did not arrive at the resection site and still maintains the high speed at the final time (t = 10 days). When the random motility ratio is 1, the cell does not respond to the chemoattractant signal at all and persists the random walk motion with the relatively lower cell speed (8–14 μm/h). Fig. 11B shows direction changes (degree) of the migratory cells with various random motilities (ψ₁ = 0.1, 0.5, 1.0) in Fig. 11A. Direction changes become more frequent as the random motility parameter (ϕ₁) is increased. Figs. 11E–11F show cell speeds and direction changes in degrees of the same cell (cellid = 1) over a time interval [140, 160] (in hours) for the base random motility in the model (ψ₁ = 0.2). The cell speed fluctuates between low value (18 μm/h) and high value (31 μm/h) and has the low value when the direction change (degrees) is high (black arrows in Figs. 11E–11F). These fluctuating speeds of migratory glioma cells and patterns of migration direction changes are consistent with experimental data in a study by Farin et al. [27] where cell speeds and trajectories of glioma cells were measured after injection of the eGFP and DsRed-2 labeled cells into neonatal rat forebrains. In this study the highest cell speed was 100 μm/h and the average speed of all cells was measured to be 24.7±0.8 μm/h. Figs. 11C–11D show distributions of the average speed of all cells (Fig. 11C) and migratory cells (Fig. 11D) for small (ψ₁ = 0.2; blue) and large (ψ₁ = 1.0; red) random motility. High localization of very low speeds in Fig. 11C accounts for proliferative cells near BVs, the progenies of small number of invasive cells that settled in BV sites. The low cell speed is because of the movement due to growth, not cell movement. While large random motility induces relatively high speeds (18–23 μm/h), small random motility leads to relatively lower cell speeds. A wide range of cell speeds of glioma cells in various biochemical and biomechanical conditions have been reported: 39–45 μm/h (2D barrier-free culture condition) and 15–20 μm/h (3D culture in the absence/presence of EGF-stimulation) [29], 15–25 μm/h (with/without α-actinin isoforms) [30], 5–80 μm/h (neonatal rat forebrain) [27], 15–48 μm/h (in collagen I matrix) [31]. Therefore, the cell speeds in our model simulations are in good agreement with experimental data.

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Figure 11. Effect of the random motion on cell speeds and directions.

(A, B) Cell speed and moving direction of a migratory cell (cell id = 1; Figs. 10G–10I) for small (ψ₁ = 0.1; red solid line), intermediate (ψ₁ = 0.5; blue square) and large (ψ₁ = 1.0; black circle) random motility. (C, D) Distributions of average speeds of all cells (C) and invasive cells (D) for small (ψ₁ = 0.2; blue) and large (ψ₁ = 1.0; red) random motility. (E, F) Cell speed and direction changes of the cell (cell id = 1) over time interval [140, 160] in hours for the base random motility (ψ₁ = 0.2). The black arrows indicate the location and corresponding times at which the cell changed direction. Fluctuating speeds within a biologically observed range [27, 29–31] and saltatory patterns in the migration direction are consistent with experimental data in a study by Farin et al. [27].

https://doi.org/10.1371/journal.pone.0114370.g011

Efficacy of the localization method would also heavily depend on practical injection strategies of chemoattractants and therefore we next investigated the optimization of injection strategies. Figs. 12A–12D show localization patterns of cancer cells 10 days after injections of chemoattractants with the various injection rates ($l_{in}^C=5.67\text{e}2$ (A), 2.67e2 (B), 5.67e1 (C), 9.67 (D); N_C = 1). A chemoattractant was injected at four sites ((0.58, 0.5), (0.5, 0.58), (0.42, 0.5), (0.5, 0.42)) as point sources on the periphery of the resection site after conventional surgery. We notice that when large amount of a chemoattractant was introduced (as in Figs. 12A–B, 13A), dispersed invasive cells in the brain tissue bio-mechanically react to the injection of chemoattractants and migrate back to the injection sites on the periphery of the resection site at the center of the domain. However, invasive cells do not respond to the intervention in the case of low injection rates ($l_{in}^C=9.67$) as in Fig. 12D. Fig. 12E shows the population of localized invasive cells at day 10 within a narrow strip (Ω_p) on the periphery of the resected area (Ω_s) from the first surgery in the whole domain (Ω). Here, Ω_p = {P = (x, y) : dist(P, Ω_s) < w^th} where w^th = 0.2, 0.7, 1.2 mm. As the injection rate of chemoattractants ($l_{in}^C$) is increased, the number of localized cells is increased. For instance, most invasive tumor cells except cells trapped in BV sites migrated back to the resection site in response to the injection at a high rate ($l_{in}^C=567$; Fig. 12A). None of the invasive cells respond to the weak chemotactic signal in the case of the low injection rate ($l_{in}^C=567$; Fig. 12D). Figs. 12G–12I show population dynamics of localized cells (12G), invasive cells outside the target (12H), proliferative cells in BV sites (12I) at t = 0, 120, 240 h. We notice that an increase in $l_{in}^C$ leads to an increase in localized cells near the target area (resection site for the second surgery) and a decrease in the number of invasive cells in the brain tissue (thus potentially aggressive). However, the localization strategy via the chemoattractant injection does not prevent aggressive proliferation of glioma cells at several BV sites in the neighborhood. Therefore, an increase in injection rates of chemoattractants near the resection site leads to an increase in the number of aggressively growing cells at BV sites as well as shown in Fig. 12F. This indicates that a simple injection of chemoattractants near the resection site may not be sufficient and further additional strategies is necessary in order to prevent this unwanted rapid growth of tumor cells near BV sites. Fig. 13A shows a spatial profile of the chemoattractant when injected at a high rate ($l_{in}^C=567$). Fig. 13B shows concentrations of the chemoattractant in the radial direction for four injection rates ($l_{in}^C=567$, 267, 56.7, 9.67) corresponding to the cases in Figs. 12A–D.

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Figure 12. Localization efficacy in response to various chemoattractant injection rates.

(A-D) Localization patterns of cancer cells at the final time point (t = 10 days) in response to injections of chemoattractants on the periphery of the resection site with the various injection rates ($l_{in}^C=5.67\text{e}2$ (A), 2.67e2 (B), 5.67e1 (C), 9.67 (D)). N_C = 1. (E) Population of localized invasive cells at day 10 within a narrow strip (Ω_p) on the periphery of the resected area (Ω_s) from the first surgery in the whole domain (Ω). Ω_p = {P = (x, y) : dist(P, Ω_s) < w^th} where w^th = 0.2, 0.7, 1.2 mm. (F) Population of proliferative cells at BV sites after chemoattractant injection as in (E) at final time (t = 10 days). Parameters: ψ₁ = 0.2, ψ₂ = 0.0, ψ₃ = 1.0. (G-I) Populations of localized cells (G), invasive cells outside the target (H), proliferative cells in BV sites (I) at t = 0, 120, 240 h. While the localization efficacy is increased in part as chemoattractant injection rate ($l_{in}^C$) is increased, there are still growing cells near BVs without additional treatment.

https://doi.org/10.1371/journal.pone.0114370.g012

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Figure 13. Spatial profiles of the chemoattractants near the surgical site.

(A) Spatial profile of the chemoattractant at a high injection rate ($l_{in}^C=567$) in Fig. 12A. (B) Concentrations of the chemoattractant in the radial direction for four different injection rates ($l_{in}^C=567$, 267, 56.7, 9.67) in Figs. 12A–D. Relatively strong chemoattractant injections are necessary in order to create a sufficient gradient of chemoattractants to attract infiltrative glioma cells.

https://doi.org/10.1371/journal.pone.0114370.g013

Now, we test our new hypothesis: intravenous administration of chemo-drugs in addition to localization of invasive cells would lead to better clinical outcomes in eradicating glioma cells. For this purpose, we introduce a chemo-drug that targets S-G2-M-phase in the cell cycle and diffuses in the brain tissue with sources at BVs after intravenous injection of the drug. Therefore, we introduce the following additional governing equation for the chemo-drug (22) where D_D is the diffusion coefficient of the drug, $l_b^D$ is the intravenous infusion rate of the chemo-drug, and μ_D is the natural decay rate of the drug. In Fig. 14 we develop strategies to improve anticancer efficacy by intravenous infusion of a chemo-drug that targets S–G₂–M-phase at t = 150 h. Figs. 14D, 14E show the population of survived and dead tumor cells, respectively, for various $l_b^D$ ($l_b^D=0.127$, 22.7, 127). With the low infusion rate ($l_b^D=0.127$; black empty circle), the efficacy is low and surviving cells near BVs will generate potential danger to the patients with regrowth or invade brain tissue as the re-formed tumor grows. For example, none of tumor cells near one of the BVs (marked in @) are responding to the chemo-drug at t = 165 h for the case of $l_b^D=0.127$ (Fig. 14A) while most of cells were dead with high infusion rate ($l_b^D=127$; gray cells in Fig. 14C; see red curve in Fig. 14E for dead cell population). As the drug infusion rate is increased, the anti-cancer efficacy is increased (Fig. 14F). Therefore, these results illustrate that the eradication strategy of invisible migratory cells by localization to the resection bed is not good enough and a combined strategy of intravenous infusion of chemo-drug for growing tumor cells near BVs and localization for migratory cells should be used in order to eradicate all the tumor cells.

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Figure 14. Effect of S-phase-targeting chemo-drugs on eradicating cancer cells.

(A-C) Tumor migration-proliferation-apoptosis patterns at a BV site (@) at time t = 165 h in response to various intravenous infusion of a S-phase-targeting chemo-drug at t = 150 h in addition to localization of invasive tumor cells. While the intravenous infusion of the drug with a low infusion rate ($l_b^D=0.127$ in (A)) does not kill the cancer cells, larger amounts of drugs remove glioma cells in the S-phase. Blue ellipse = proliferative cells, gray ellipse = apoptotic cells. Frame size = [0.488, 0.512] × [0.89, 0.91] ⊂ [0, 1]². (D, E) Time courses of populations of surviving actively growing cells (in (D)) and dead cells (in (E)) in response to three different dose rates ($l_b^D=0.127$ (dotted black circle), 22.7 (solid blue square), 127 (solid red line)). The legend in (E) is the same as one in (D). (F) Efficacy of killing cancerous cells by the S-phase-targeting drug: populations of proliferative (blue solid circle) and dead cells at the BV sites with various drug infusion rates ($l_b^D=1.27e-01(1)$, 5.27e-01(2), 1.27e+00(3), 5.27e+00(4), 1.27e+01(5), 2.27e+01(6), 3.27e+01(7), 4.27e+01(8), 5.27e+01(9), 6.27e+01(10), 7.27e+01(11), 1.27e+02(12), 5.27e+02(13), 1.27e+03(14), 3.27e+03(15)). As the drug infusion rate ($l_b^D$) is increased, the drug effectively kills the actively proliferative tumor cells at the BV sites. Parameters: ψ₁ = 0.2, ψ₂ = 0.0, ψ₃ = 1.0. Intravenous infusion of a strong S-phase-targeting chemo-drug in addition to localization of cells may effectively kill growing cells near BVs, leading to complete eradication of tumor cells: infiltrative individual cells in the brain tissue and proliferative cells in a growing mass near BVs.

https://doi.org/10.1371/journal.pone.0114370.g014

Many biochemical factors can induce the stiffening of tissue; these include fibrosis and brain tumor growth [32], inhomogeneous endothelial transformation [33], and reorganization of ECM fibrils by differentiation and highly contractile stromal cells [34]. Hetergeneity in the tumor microenvironment leads to drug resistance [35], and reduced efficacy of tumor-targeting agents [36]. Therefore, it is important to observe the effect of microenvironmental factors on the localization strategies for patient-specific treatment options. In Fig. 15 we investigate the role of microenvironment in regulating the localization of the migratory glioma cells in response to chemoattractants injected near the resection bed at the center of the domain. In order to test the effect of tissue stiffness on the efficacy of the localization strategy, we prescribe lower diffusion coefficients (D_L) on the left half domain (Ω_L = [0, 0.5] × [0, 1]) while keeping the normal (control) diffusion coefficients (D_R = D) in the right half domain (Ω_L = [0.5, 1] × [0, 1]): D_L = 0.025 * D (Fig. 15A), 0.075 * D (Fig. 15B), 0.1 * D (Fig. 15C), D (control; Fig. 15D). As D_L is decreased, some of migratory cells (black arrow in Figs. 15A, 15B) do not respond to the chemoattractant signals in the left half domain due to weak transport of the chemoattractant by diffusion (the lower panel in Figs. 15A, 15B). Figs. 15E, 15F show the population of invasive cells within a narrow strip in the whole domain (Ω) and subdomains (Ω_L, Ω_R), respectively, at day 10 for various diffusion coefficients in the left domain (D_L = 0.01 * D, 0.025 * D, 0.05 * D, 0.075 * D, 0.1 * D, D). Here the narrow strip (Ω_p) is located on the periphery of the resected area (Ω_s) from the first surgery in the whole domain (Ω) at day 10: Ω_p = {P = (x, y) : dist(P, Ω_s) < w^th} where w^th = 0.2, 0.7, 1.2 mm. As D_L is decreased, the population of localized tumor cells is significantly decreased in the left half domain (solid curves in Fig. 15F) while there is no significant difference in the right half domain (dotted curves in Fig. 15F), thus there is a significant decrease in the whole plane overall (Fig. 15E). This illustrates that the efficacy of the localization strategy also depends on the microenvironmental factor and the clinical outcomes will depend on tissue composition and location within the brain.

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Figure 15. Role of the tumor cell microenvironment in efficacy of the treatment.

(Top; A-D) Profiles of cancer cells at day 10 in response to injected chemoattractants on the periphery of the resection site in different tissue environment. (Bottom; A-D) Profiles of chemoattractants that correspond to each case in top panels. Diffusion coefficients of all variables were reduced on the left half side of the domain (Ω_L = [0, 0.5] × [0, 1]) while they were fixed on the right half domain (Ω_L = [0.5, 1] × [0, 1]) relative to control (D) in whole domain (in (D); Ω = [0, 1]²): D_L = 0.025D in (A), D_L = 0.075D in (B), D_L = 0.1D in (C), D_L = D in (D). As the diffusion coefficient D_L on the left half plane is decreased (from (D) to (A)), i.e., brain tissue is getting tougher, the efficacy of bringing those invasive cancerous cells back to the resection site is decreased. (E) Population of localized invasive cells within a narrow strip (Ω_p) on the periphery of the resected area (Ω_s) from the first surgery in the whole domain (Ω) at day 10. Ω_p = {P = (x, y) : dist(P, Ω_s) < w^th} where w^th = 0.2, 0.7, 1.2 mm. (F) Population of localized invasive cells within Ω_p in the left (Ω_L) and right (Ω_R) half domain at day 10. Parameters: ψ₁ = 0.2, ψ₂ = 0.0, ψ₃ = 1.0. Efficacy of localization strategies depends on microenvironmental factors such as stiffness of tissue and brain tissue composition/geometry even with chemo-drugs since effective transport of key molecules (chemoattractants and chemo-drugs) by the diffusion process depends on these active microenvironmental players, requiring a careful priori assessment of patient-specific data.

https://doi.org/10.1371/journal.pone.0114370.g015