Cell behaviors

Before a description of each theoretical evolution, we first enumerate the several stochasticized behaviors taken into account in our model.

• Since we aim to describe the competition between the tumor nodule and the CTL, we consider a longitudinal section of the tumor nodule evolving on a compact two-dimensional domain described as a grid Ω, in parallel with a population of motile CTL. The dimension of the considered grid Ω is constant. At the beginning of the evolution, Ω is much larger than the tumor. This assumption can become false along the temporal evolution when the population of CTL fails to eradicate the tumor, leading to uncontrolled tumour nodule growth.
• Each CTL moves according to a symmetric random walk on Ω and is reflected when it eventually reaches the boundaries of the grid. This is an important assumption for our stochastic model. We believe that this assumption is legitimate since it describes the fact that even if some CTL can exit Ω in a real biological framework, other CTL can enter, keeping the size of the CTL population constant. This last assumption could be amended by considering a CTL population whose size can be modified over time, through an unbalance between CTL entering and exiting Ω, however, we have not pushed further our investigations in this direction.
  Hence, this shortcut roughly means that we will consider a population of CTL of constant size. When a CTL encounters a tumor cell, it automatically stops its walk and starts the killing process of the tumor cell, which is also stochastic according to our model. We propose two displacement models in what follows: the first one is a pure symmetric random walk already described in [13]; the second is a self-interacting re-inforced random walk, whereby CTL are attracted towards scout siblings that have encountered the tumor nodule.
• All along the computational time, a CTL becomes “exhausted” within the tumor environment when it reaches a maximal number of 5 killing rounds. Hence, an exhausted CTL can kill no more tumor cells. This value is based on our expertimental observations under in vitro conditions in which human CTL kill ∼ 5 target cells over prolonged time. Exhausted CTL also evolve in Ω according to a random walk, bouncing back both when encountering the boundaries of the grid and when contacting the tumor mass. CTL exhaustion is irreversible since CTL are confined within the tumor microenvironment and therefore they cannot reacquire an activation status. This parameter has been inserted in the model in order to mimic the in vivo situation in which tumor infiltrating CTL exhibit an exhausted phenotype and are functionally impaired [5].
• The tumor nodule is composed of 2 different areas: the proliferative shell composed of actively dividing cells and the central core composed of non-proliferative cells (See Fig. 1A). This concentric (but not necessarily exactly spherical structure) mimics the structure of tumor clusters as defined by confocal laser scanning microscopy (See Fig. 1B). Without CTL intervention, the proliferative shell will stochastically grow, with a dynamic described by a Galton-Watson branching process (each cell of the proliferative shell is divided in 2 cells after a stochastic geometrical time, see details below).
• In the model we implement the possibility that, along each division, tumor cells could acquire mutations that might render them progressively more resistant to CTL attack (as part of an immunoediting process) or even invisible to CTL (mimicking MHC Class I loss described in tumor cells during immunoediting [6]). Hence, for the acquisition of resistance, daughter tumor cells can become stochastically and progressively more resistant than their mother cells.

Discretization of the time and of Ω

CTL and tumor cells are spatially parameterized by their coordinates (x, y) on a discrete grid (See Fig. 1A), where the step size movement Δ of each CTL is this value corresponds to the mean diameter of a melanoma cell as measured by confocal laser scanning microscopy. The evolution is discretized according to a time step δ_t, so that the time sequence (t_n)_{n ≥ 0} is equally spaced: δ_t = t_n − t_n−1. The time step δ_t is fixed to 1.44 minutes (see material and methods) according to experimental measurements of the mean CTL velocity (the mean recorded velocity is about 8.66μm/min). Such settings make it possible to handle CTL movements on the grid (discretized on the basis of the size of a tumor cell) with a velocity that reproduces experimental measurements.

Movement of the CTL population

The number of CTL (N_CTL) is fixed for each numerical simulation, assuming that cells entering in the field or generated by cell division replace CTL leaving the field or dying. For CTL displacements, positions are parameterized by their planar coordinates at time $t_n:z_n^i:=(x_n^i,y_n^i)$, for i = {1, …, N_CTL} and these coordinates still belong to the regularly spaced grid. Without any interaction with the tumor nodule or another CTL, each CTL evolves according to a symmetric random walk moving from one location to another to its 1-neighbors (see Fig. 1C). Equation (1) describes the displacement probability of each CTL i in the grid in the four directions of the space: (1)

We also propose a second displacement model based on a self-interacting re-inforced random walk.

Biased displacement of CTL through chemotactism

In the attempt to optimize the number of CTL collisions with tumor cells, we introduced a bias for CTL movement. The updated model postulates that when a CTL collides with the tumor nodule, it releases chemo-attractants that guide the trajectories of other CTL towards the nodule (see Fig. 1D for a graphical illustration). We thus consider a self-interacting particle system, inspired from population dynamics already described in other biological fields (e.g. ant colonies [15]). The principle can be described in this way: when one (or several) CTL hits the boundary of the nodule, other CTL may be attracted to the position of this scout CTL if they are located close enough (the distance between scout cells and attracted ones should be less than a parameter D). In these conditions, the model postulates that a given CTL i (whose position $z_n^i$ does not belong to the frontier of the nodule at time t_n) is attracted towards the barycenter of the scout CTL $B_n^i$ (see Fig. 1D). If ${\theta }_n^i$ denotes the planar angle between the two vectors $\overrightarrow{B_n^i{z}_n^i}$ and $\overrightarrow{Ox}$ (as shown in Fig. 1D), the attraction towards the barycenter is provided by ${\nu }_n^i$ that belongs to [0, ν], which quantifies the attraction strength. This parameter ${\nu }_n^i$ depends on the distance between $B_n^i$ and $z_n^i$, and on the number of scout CTL, denoted $K_n^i$. The model postulates a fixed maximal scout CTL number K (corresponding to 10 CTL) and a maximal attraction strength ν. To define the impact that the attraction of CTL towards the tumor nodule might have on tumor eradication, we let vary the parameter ν in the different numerical simulations. Finally, we use the following formula which stands that the maximal attraction number is reached when $K_n^i=K=10$ CTL and the attracted CTL $z_n^i$ is very close to the barycenter $B_n^i$.

Thus, the position of $z_{n+1}^i$ (CTL moves from $z_n^i$ to its 1-neighbors) follows the mixture law $(1-{\nu }_n^i)\times$ symmetric random walk $+{\nu }_n^i\times$ drifted attraction towards $B_n^i$. Precisely, if ϵ(a) denotes the algebraic sign of any real number a, we define and as well as

CTL killing

A single CTL detects the nodule through a contact process: when one CTL hits the frontier of the nodule, the CTL stops moving instantaneously and starts the killing of the tumor cells in its immediate $\sqrt{2}-$neighborhood (Fig. 1A). We assume that before exhaustion, a CTL kills a tumor cell with a constant rate μ, which is transposed in our model by a continuous exponential distribution for killing time. According to our time discretization, such a probabilistic distribution needs to be slightly modified. The distribution of the stochastic killing time is thus converted to a stochasticly geometrical law 𝓖(μ*) which is the equivalent distribution for discrete time to an exponential law in a continuous setting.

The mean measured time in minutes for detection of apoptosis induction in melanoma cells in vitro [14] is 1/μ = 26.4min. To obtain the inverse of the mean time required for killing one target cell according to the discretized time used in simulations, it is necessary to compute μ* = μ ⋅ δ_t (see material and methods). Hence, the probability that a CTL i kills a tumor cell at time t_n is After killing a first tumor cell, the CTL restarts a symmetric random walk according to the rules defined in (1). A CTL becomes “exhausted” when it reaches a maximal number of killing rounds fixed to 5.

Tumor nodule growth

In the proliferating shell, tumor cells are divided according to a stochastic time that follows a geometric distribution. The average measured time of duplication of melanoma cells in vitro is 16.66 hours (1000min, as we experimentally measured). This is used to fix the tumor division rate parameter via λ* = λ × δ_t, where 1/λ = 1000 (see material and methods). The model assumes that the probability for a tumor cell i to divide at a time t_n is

The tumor nodule growth is also described by the parameter E (E = thickness of the proliferating shell) (see material and methods). It means that per unit time, the dividing rate of tumor cells in the proliferative shell (corresponding to the cells for which the distance from the nodule frontier is lower than E) is λ*. Each division yields the addition of a new tumor cell. Two cases are possible:

1. Some free location is available in the immediate neighborhood ($\sqrt{2}$ -neighborhood) of the mother cell and the new cell chooses its place uniformly among all free neighborhood;
2. The new cell chooses an occupied location in its $\sqrt{2}$-neighborhood and pushes another cell located in the proliferating shell towards the exterior of the nodule. Recursively, if the pushed cell encounters another tumor cell, it pushes this cell towards the exterior of the nodule and so on. Equivalently, when a cell in the interior of the proliferative part divides, it adds a new cell on the closest point of the frontier of the nodule.
   Both cases of such dynamics are illustrated in Fig. 1A. After each division event, the proliferative shell is instantaneously updated.

Immunoediting

As pointed above, tumor cells could acquire mutations and then become resistant to CTL lysis. We consider that after each duplication of a tumor cell, a daughter cell could become progressively more resistant with a rate p_res. In this case a CTL will kill daughter cells within a time longer than that required to kill mother cells. The model postulates several levels of resistance R ∈ {1, 2, 3, …}. A tumor cell with R = 1 is a tumor cell which did not increase its resistance upon division. A tumor cell with R_d = R_m + 1 is a tumor cell which increases its resistance upon one division (R_d is the resistance of the daughter cell, R_m is the resistance of the mother cell). As a result, each tumor cell acquiring resistance upon division requires on average twice as long as its mother cell to be killed, yet it can still be killed. In the model, a tumor cell becomes invisible to CTL with the rate p_inv. We assume that the probability for a tumor cell to become invisible (p_inv) is equal to 0 if the tumor cell has not acquired resistance (R < 3). This probability is about $\frac{1}{12}$ if R = 3 and increases with the resistance of the mother cell up to $\frac{1}{4}$. See Supporting Information S1 Table for the values of p_res and p_inv.

Stochastic dynamical agent based model or differential equations?

Computational works on biological dynamical systems consider either infinitesimal agent based model described by a stochastic particles system or partial differential equations (PDE) that describe the mean chronological evolution of a macroscopic number of interest (density of CTL, size of the tumor nodule, …). Indeed, these two alternative ways of describing the biological system are generally equivalent. On the one side it is easier to amend the dynamical model with particles systems because of its flexibility, on the other side, partial differential equations are easier to handle from a mathematical point of view.

Agent based models are slightly more natural than differential approaches since they mimic more closely the nature of the real phenomenon. The proportion density of the CTL has a questionable existence since CTL are positioned into the systems and evolve stochasticly.

In this work, we aim to describe a possible interacting effect in the population of CTL due to chemotactism for a specific subpopulation of CTL. This phenomenon can be handled with stochastic particles without any difficulty, by the addition of a drift to the symmetric random walk described by (1). This leads to a re-inforced random walk whose simulation is simple and easy to reproduce or modify. By no means this chemotactism can be described so easily with PDE. Recent advances in this direction relies on Keller-Segel models (see for instance [16]) but these models are far from being trivially solved by PDE. Especially, these models need to be enhanced to implement a chemotactism towards a specific zone of the population of CTL. We are currently working on an improvement of a Keller-Segel type description of our setting with the view to obtain simpler equations.

Experimental parameters measurement

The mean velocity of CTL displacement (8.66 μm/min) was measured by time-lapse microscopy using a confocal laser-scanning microscope (either a Zeiss LSM-510 or a Zeiss LSM-710 microscope, Zeiss Germany). Measurements were performed on non-stimulated CTL that were free to move on poly-D-lysine coated LabTeck-chambers.

The mean time required for killing a melanoma cell (26 minutes) was estimated by measuring, in a significant number of CTL/target cell conjugates, the time occurring between the initial CTL/target cell contact and the beginning of blebbing in target cells, as reported in I. Caramhalo et al. by time-lapse confocal microscopy [14]. Experimental measurements of the mean time required for melanoma cell killing by CTL were employed to estimate μ.

Melanoma cell mean diameter (12.5 μm) was estimated on paraformaldehyde fixed melanoma cell spheroids using a confocal microscope.

The time required for melanoma cell division was measured in standard melanoma cell 2-D cultures by counting the cell number at fixed time intervals over a total period of three days.

Melanoma cell spheroids were generated using the ‘hanging-drop’ method. Briefly 1000 cells/25μl were seeded in the wells of a Terasaki culture plate. The plate was then inverted to form suspended droplets. The measurement of the diameter of the spheroids was performed using a Zeiss LSM-710 microscope using a 20x objective. The mean diameter of a spheroid was estimated by calculating the mean value of 4 diameters measured using the LSM software (Zeiss). To visualize the alive/dead fraction of cells in the spheroid at day 6 of culture the spheroids were incubated for 2 hours with 5nM Ethidium-1-HomoDimer (Eth-1-HD, red) and 1 μM calcein (green) at 37^o C. Staining was visualized using a Zeiss LSM-710 using and 20x ojective.

For the measurement of CTL/tumor cell cluster formation, HLA-A2⁺ melanoma cells (D10) were loaded for 2 hours at 37^o C, either with NLVPMVATV or VLAELVKQI (two different epitopes of the human cytomegalovirus protein pp65), washed and conjugated either with the CTL clone VLA-E2 or with the CTL clone NLV-2 as indicated in S4 Fig. legend. After 30 minutes co-culture the same number of CMFDA-loaded VLA-E2 CTL (loaded with CMFDA-green) were added. After additional 48 hours cells were analyzed using a confocal laser scanning-microscope (Leica SP8). z-stacks of confocal images were acquired and analyzed using the Image J software.

CTL were purified from peripheral blood of healthy donors and were kept in culture by periodic re-stimulation with irradiated peripheral blood mononuclear cells. Peripheral blood mononuclear cells (PBMC) were isolated from buffy coats of healthy donors obtained through the Établissement Français du Sang (EFS Midi-Pyrénées, Purpan University Hospital, Toulouse, France). Blood samples were collected and processed following standard ethical procedures (Helsinki protocol), after obtaining written informed consent from each donor and approval for this study by the local ethical committee (Comité de Protection des Personnes Sud-Ouest et Outremer II).

Time and Scale calibration: grid definition

In our grid model the distance between two neighbor cells is fixed according to the mean melanoma cell size, which is denoted Δ. The value Δ = 12.5μm was measured by confocal microscopy. The time-step along our iterations is strongly related to the velocity of CTL since it is the time necessary for one CTL to move from its position to one of its 1-neighborhood. In order to obtain an immediate velocity of each CTL that corresponds to the one observed on real data, we fix the time-step δ_t to be the mean time necessary for one CTL to cover a distance that corresponds to Δ. Experimental measurements of the displacements of CTL establish a mean velocity of v = 8.66 μm/min. We then choose δ_t = 1.44 min, which means that there is about 1 minute and 30 seconds between two consecutive iterations of the random walks.

Estimation of μ: the inverse of the time required for killing one target cell

We compute an estimation of μ in an independent context of our stochastic dynamical model. Experimental measurement of the time required for melanoma cell killing by CTL were employed to compute μ estimation [13]. For a number N of observations, we observe several lysis duration ${(t_{lysis}^i)}_{i\in \{1,\dots ,N\}}$, and standard estimation on exponential law yields an estimator of the rate ${\widehat{\mu }}_N^{-1}$, which is the inverse of the mean time of lysis. To obtain the rate according to our time step δ_t, we then compute ${\widehat{\mu }}_N^{*}$ as

Estimation of λ: rate of division

The estimation of the division rate λ of tumor cells is slightly more complex than the estimation of μ owing to the nature of available biological observations. Measurements were performed in culture conditions in which tumor cells grew in 2-D without forming cellular spheroids. In these conditions, the replication of the tumor cells evolves without any geometric constraints. The number of cells was kept fixed at time 0 (denoted n₀) and counted at time T so that we obtained this final number n_T. Value of tumor cell growth obtained in N individual culture plates corresponds to the observation of $(n_T^1,\dots ,n_T^N)$.

The model can be seen as a Pure Birth Process (Yule-Furry process) from a probabilistic point of view: when one tumor cell is replicated, the population increases of one individual cell. If we denote X_t the number of cells at time t, the probability that {X_t = n} given by P_n(t) = P(X_t = n) satisfies Since the probability that one birth occurs during the interval [t, t + h], up to the condition {X_t = n}, is nλh + o(h). Then (1 − nλh) + o(h) is the probability that no division appears during [t, t + h] if {X_t = n}. Then (P_n(t))_{t ≥ 0,n ∈ ℕ} satisfies the differential system (2) Since the starting number of cells is n₀, we have P_n0(0) = 1, and for all n ≠ n₀, P_n(0) = 0. We remind the notation of the binomial coefficients $\left(\begin{array}{c}n\\ k\end{array}\right)$, whose valued are defined as According to this definition, One may check that the unique solution of (2) is given by (3) Let us briefly recall the proof of (3), which is obtained by induction. In this view, we denote (H_n0+k) the induction assumption for the integer n₀ + k and remark that an equivalent formulation of (3) is (4) Let us remark that (2) implies that P_n0 satisfies $P_{n_0}^{\prime }(t)=-n_0\lambda P_{n_0}(t)$, whose solution is This proves that (H_n0) is true. Now, assume that (H_n0+k−1) holds and we want to establish (H_n0+k). Remark that (2) leads to From (H_n0+k−1), we deduce that this last equation is equivalent to As a consequence, we get Since the population is initialized with n₀ cells, P_n0+k(0) = 0 and we deduce that (H_n0+k) holds (see Equation (4)).

Hence, X_t follows a negative binomial distribution 𝓝𝓑(n₀, e^−λt) initialised with n₀ cells and a succeed parameter e^−λt.

We aim to estimate λ and by denoting p = e^−λT, we can use the Maximum Likelihood Estimation (M.L.E.) to compute ${\widehat{p}}_N$, and then ${\widehat{\lambda }}_N$. Using our observations $(n_T^1,\dots n_T^N)$, the log-likelihood is given by We use now (3) to obtain We can optimize the last expression with respect to p to obtain that Since we have set p = e^−λT, we deduce that ${\widehat{\lambda }}_N$ is given by ${\widehat{\lambda }}_N=T^{-1}\log (1+\sum _{i=1}^N\frac{n_T^i}{N{n}_0}).$ As in the estimation of μ, to obtain the rate according to time step, we then compute ${\widehat{\lambda }}_N^{*}={\widehat{\lambda }}_N\times \delta t.$

Estimation of E: thickness of the proliferative part of the nodule

The size of the proliferative part is an important feature for the growth of the tumor nodule. Hence, E should be carefully estimated: we assume in the sequel that E is constant over the time evolution of the nodule. In order to fix a plausible value of E, the global diameter of the nodule was observed over the time. We denote (d_t)_{0 ≤ t ≤ T} such a diameter from the initial time t = 0 to the ending one t = T (see S3 Fig.) and design a mathematical study that describes the effect of E on the average evolution of the diameter of the nodule. The main difficulty to estimate E relies on the nature of our observations, that are dependent on the temporal evolution of the diameter of the nodule.

For the sake of simplicity, we assume here a circular structure of the tumor with an inner necrotic part, whose radius is ${\tilde{r}}_t$, and an outer proliferative part between ${\tilde{r}}_t$ and $r_t={\tilde{r}}_t+E$ (see S2 Fig.). The observed diameter is then d_t = 2r_t. From the tumor growth model, the division of a tumoral cell yields an addition of a new cell in the proliferative part if the mother cell does not belong to the frontier with the necrotic part of the nodule (whose radius is ${\tilde{r}}_t$). If Γ_t is the average number of tumoral cells in the proliferative part, we then have (5) where p_Δ,E is the proportion of mother cell in the proliferative part that does not encounter the necrotic one. This probability can be approximated as soon as $\max (\Delta ,E)<<{\tilde{r}}_t$:

• The area of the cells that belong to the frontier of the necrotic part is given by
• The area of the proliferative part is given by

As a consequence, If we denote $\tilde{\lambda }=\lambda (1-\frac{\Delta }{E})$, the solution of Equation (5) is (6) We can use the observed sequence (d_t)_{0 ≤ t ≤ T} of the diameter of the nodule to obtain a second equation on Γ(t) since Γ(t) is related to d_t by the following formula: (7) (8) Using this last equation at time t = 0 and t = T, and using (7) in (6), E should satisfy: (9) It is then possible to numerically solve this equation, but as we can only consider entire parts of cell sizes, we choose $E^{*}=\Delta \lceil \frac{E}{\Delta }\rceil$, where E* is the size of the proliferative part in our simulations. We obtained with our data E* = 2Δ. As a goodness of fit testing for our value E*, S4 Fig. compares the experimental observed diameter with the evolution of the theoretical diameter (d_t)_{0 ≤ t ≤ T} obtained with (7) (for E* = Δ and E* = 2Δ: $d_t=\frac{{\Delta }^2\Gamma (t)}{4E}+E,$ for all 0 ≤ t ≤ T).

Early productive CTL/tumor collisions determines CTL success in tumor eradication

Having established the basic parameters of the model, we employed numerical simulations to compute the probability of success of CTL in eradicating the tumor nodule without chemotactism, i.e. meaning that CTL motility is driven by a pure symmetric random walk. In a first approach, we investigated the number of collisions between CTL and tumor cells that were required to grant tumor eradication. For this analysis we considered a set of numerical simulations performed by varying the number of CTL ranging from 600 to 1100. This CTL range was chosen on the basis of initial observations indicating that 600 CTL exhibit a low chance of tumor eradication while 1100 exhibit a high chance of tumor eradication, within the defined size of the grid. Numerical simulations showed that, for CTL success, it was important that a minimum number of CTL/tumor cell productive collisions would occur during the early time points of CTL/tumor cell dynamic confrontation. More precisely, in our mathematical simulation lasting 72 hours, we computed N_E, which was the number of collisions during the first 5 hours (defined as early collisions): Simulations showed that N_E was significantly larger in the case of CTL success than in the case of CTL incapacity to eradicate the tumor (Fig. 2A). Fig. 2A also shows that

• the average value of N_E corresponding to CTL inefficacy was 𝔼[N_E] ≃ 112 collisions/5 hours
• the average value of N_E corresponding to tumor eradication was 𝔼[N_E] ≃ 137 collisions/5 hours.

Taken together the above results indicate that in a cancer immunoediting scenario, the larger the number of early collisions N_E between CTL and target cells, the higher the probability of tumor eradication.

Early productive collisions depend on CTL population size

In order to define the parameters that might ensure a sufficient number of early collisions leading to tumor eradication, we initially varied the size of the CTL population. CTL follow a pure symmetric random walk for their displacement. The number of CTL used in the simulations should not be considered as absolute, but it should be related to the dimension of the grid. We varied the CTL number for two reasons. First, because it has been previously described that the number of available killer cells is a crucial parameter in defining the success of a killer cell population over tumors [17]. Second, because it has been experimentally demonstrated that the capacity to eliminate tumor target cells increases with the size of the CTL population [14]. In line with these reported data, our numerical simulations showed that the probability of success in tumor nodule eradication was improved with the increase of CTL number while the mean time required for nodule eradication decreased (Fig. 2 B–E and S1 and S2 Movies). This probability of success is denoted below P_S and corresponds to the empirical probability of success of the CTL population over 50 Monte-Carlo runs. We observe that

• P_S ≥ 0.95 as soon as the CTL number is higher than 900 (for the estimated value of mean killing time $\frac{1}{\mu }$),
• P_S ≤ 0.15 when the CTL number is lower than 600 (for the estimated value of mean killing time $\frac{1}{\mu }$).

An alternative way to graphically represent the impact of CTL number on the probability of success in tumor nodule eradication is shown in Fig. 3 in which three simulations performed with high (1200, A), intermediate (400, B) and low (200, C) CTL number are presented. The plots represent the number of cells in the tumor nodule, the number of killed tumor cells, the number of exhausted CTL and the number of invisible tumor cells over time. They show that, while at high and low CTL numbers the balance between CTL success and tumor resistance is rapidly in favor of CTL or of tumor respectively (A and C), at intermediate cell numbers a more complex behavior is observed (B). A first phase in which CTL manage to reduce tumor size is followed by a period of equilibrium in which the size of tumor nodule remains stable. These phases are followed by a phase in which the number of tumor cells starts to grow in parallel with the stochastic generation of tumor cells not visible by CTL. Interestingly, these three phases in tumor/CTL confrontation observed at intermediate CTL number are reminiscent of the three phases described in cancer immunoediting: elimination, equilibrium and escape [6].

To investigate if the increase in the number of killer cells have an impact on the number of early CTL/target cell productive collisions, we measured the distribution of N_E (the number of collisions during the first 5 hours) under different conditions. This analysis showed that when the number of CTL was increased from 400 to 1600 a sharp increase of the productive early collisions was observed (S1 A Fig., compare black and cyan histograms, mean values of 𝔼(N_E) = 85 collisions/5 hours, black versus 𝔼(N_E) = 186 collisions/5 hours, cyan).

Taken together these results indicate that the CTL population size strongly affects CTL/tumor nodule early collisions and, in turn, tumor eradication.

Early productive collisions do not depend on the time required for killing

We next investigated the impact that CTL time of killing (defined as the time required by a single CTL to annihilate a target cell) might have on tumor eradication (without chemotactism). To this end, numerical simulations in which the time required to kill target cells was reduced to less than 10% of the experimentally measured time were performed [14]. Interestingly, a sharp reduction of the time required for killing of individual tumor cells (down to 3 minutes from the initial 26 minutes) did not significantly affect neither the efficacy of tumor nodule eradication by CTL nor the mean time required for nodule eradication (Fig. 2 D–E). Conversely, when the time of killing was increased 4 times (up to about 2 hours) a clear effect on the efficacy of CTL-mediated cytotoxicity was observed. Indeed, the probability of success went down to P_S ≤ 0.25 (for 1200 CTL), as compared to P_S = 1 measured for a killing time of 26 minutes.

We next investigated whether early CTL/tumor nodule productive collisions would depend on the time required for killing. As shown in S1 B Fig., when the time required to kill a target cell was reduced of 30% or of 90% we observed a similar moderate increase in the number of initial productive collisions (N_E), indicating that it is not possible to surpass an upper limit of productive collision number by modulating the time of killing. The mean collision value was 𝔼(N_E) = 130 (instead of 85) when the time of killing was reduced of 30%. A similar mean collision value of about 130 was observed when the time of killing was reduced of 90%.

Taken together, the above results support the finding that the number of productive contacts between CTL and tumor cells is an important parameter influencing the success of a CTL population. They show that, while an augmentation of the time required for killing of individual target cells strongly affects CTL efficacy, the sharp reduction of the time required for target cell killing does not significantly affect the probability of CTL success in tumor eradication.

CTL chemotactism towards scout cells having detected the tumor strongly augments early productive collisions

In the above-described numerical simulations the model postulated a scenario in which CTL are not preferentially directed towards the growing tumor and follow as symmetric random walk. In these conditions, only CTL stochasticly colliding with a cognate tumor cell exhibit cytotoxicity, thus only a low percentage of CTL participate to the killing. As a consequence, the CTL population, in its whole, mostly ignores the growing tumor. We now consider the situation of a self-interacting population of random walks influenced by chemotactism, as described in the model.

Numerical simulations based on the above-described biased movement show that the probability of success in tumor nodule eradication increases with the increase of CTL attraction towards the scout cells. As shown in Fig. 4 and S3 Movie, with a relatively low number of CTL, the increase of the attraction of CTL towards scouts having detected the tumor nodule, strongly augments the probability of tumor nodule eradication. For instance for 400 CTL with no attraction there was no chance of success (see Fig. 4A), while a relatively moderate attraction of ν = 0.1 guarantees almost 100% probability of success. Lower attraction strengths (ranging between 0.01 to 0.075) still yielded significant impact on the probability of tumor eradication. Fig. 4A also shows that attraction reduces the range between the minimal number of CTL required for 100% eradication and success (exhibiting probability of tumor eradication equal to 1, denoted N_S) and the maximal number of CTL failing in tumor eradication (exhibiting probability of tumor eradication equal to 0, denoted N_F). For example, when κ = 5, we can show that:

• when ν = 0 (no attraction through chemotactism), the maximal failing number N_F ≃ 600 and the minimal success number N_S ≃ 1200, leading to N_S − N_F ≃ 600,
• when ν = 0.1, N_F ≃ 200 and N_S ≃ 350, leading to N_S − N_F ≃ 150.

These results indicate that the attraction “sharpens” the phase of transition in the number of CTL that are determinant for tumor eradication in the sense that the difference between N_S and N_F becomes smaller when the attraction strength ν is increased. Finally, the stronger the attraction ν, the earlier is the phase transition of the probability of CTL success.

• when ν = 0, N_S ≃ 1200,
• when ν = 0.1, N_S ≃ 350,
• when ν = 0.3, N_S ≃ 180.

The mean time required for nodule eradication (denoted T_S) decreased accordingly (Fig. 4B). When we have a number of 400 CTL, we observe that:

• when ν = 0, the averaged time for tumor eradication is infinite (no tumor eradication),
• when ν = 0.1, 𝔼(T_S) ≃ 21 hours,
• when ν = 0.3, 𝔼(T_S) ≃ 12 hours.

To verify whether the increase of CTL attraction would result in an augmentation of productive early collisions, we measured the distribution of the number of collisions during the first 5 hours in conditions in which a moderate attraction was applied. As shown in S1 C Fig. the distribution of early collisions for a given number of CTL (400 CTL) was sharply increased in the presence of attraction (cyan histogram, 𝔼(N_E) = 188 collisions/5 hours) when compared to the distribution of the same number of CTL with no attraction (black histogram, 𝔼(N_E) = 85 collisions/5 hours). Taken together, the above results show that in a cancer immunoediting scenario the rapid recruitment of killer cells helps to reach a sufficient number of early productive collisions leading to tumor eradication.