Detection and characterization of chemotaxis without cell tracking

Abstract

Swarming has been observed in various biological systems from collective animal movements to immune cells. In the cellular context, swarming is driven by the secretion of chemotactic factors. Despite the critical role of chemotactic swarming, few methods to robustly identify and quantify this phenomenon exist. Here, we present a novel method for the analysis of time series of positional data generated from realizations of agent-based processes. We convert the positional data for each individual time point to a function measuring agent aggregation around a given area of interest, hence generating a functional time series. The functional time series, and a more easily visualized swarming metric of agent aggregation derived from these functions, provide useful information regarding the evolution of the underlying process over time. We extend our method to build upon the modelling of collective motility using drift–diffusion partial differential equations (PDEs). Using a functional linear model, we are able to use the functional time series to estimate the drift and diffusivity terms associated with the underlying PDE. By producing an accurate estimate for the drift coefficient, we can infer the strength and range of attraction or repulsion exerted on agents, as in chemotaxis. Our approach relies solely on using agent positional data. The spatial distribution of diffusing chemokines is not required, nor do individual agents need to be tracked over time. We demonstrate our approach using random walk simulations of chemotaxis and experiments investigating cytotoxic T cells interacting with tumouroids.

1. Introduction

Chemotaxis refers to directed cell movement driven by individual cellular responses to a biochemical gradient. Chemotaxis is a critical driver of an array of physiological processes; it plays a crucial role in embryonic development, directing migratory behaviour of cells in growing tissues [1], formation of new capillaries to sites of ischaemia via angiogenesis [2], and orchestrates cell dynamics in wound healing [3,4]. In addition, the self-amplifying clustering of immune cells driven by chemotaxis, which we term chemotactic swarming, has been observed during responses to injury where neutrophils swarm to sites of inflammation and recently in cytotoxic T cells interacting with tumour cells [5–7]. These concepts are emerging as highly relevant in the engineering field of swarm robotics, where the spatial coordination of a collective of autonomous vehicles is concerned.

Developments in live-cell imaging techniques increasingly allow for precise localization of individual cells over time [8–11]. As such, a significant amount of imaging data capturing cell motility is becoming available. Probing these data to detect the presence of chemotactic swarming presents a unique challenge, due to the fact that visualizing chemoattractant gradients in conjunction with cell movements is very difficult, particularly without disturbing the underlying dynamics. While individual cell tracking can be used to analyse chemotaxis, tracking individual cells requires a high sampling rate (with associated photo-toxicity), a smaller field of view than would otherwise be available, a low cell density so that consecutive positions of cells are non-ambiguous, and imaging capacity that is capable of imaging in three dimensions if the cells are not confined to a two-dimensional (2D) substrate [10,12–14]. New statistical approaches need to be developed to detect and characterize chemotactic swarming using cell position data alone.

In this paper, we set out a method for detecting and characterizing the chemotaxis of agents towards a known area of interest, e.g. a tumour. We first present a general framework for the analysis of aggregation of agents for any type of agent-based processes, independent of the underlying rules governing individual agent movements. To do this, we adapt existing spatial statistical approaches to analyse the positional data generated by an agent-based process. Our methodology involves adapting Ripley’s K function from spatial statistics [15–18] to analyse the distribution of agents around the given area of interest. We refer to these adapted functions as focal K functions. Similarly to the analysis of static spatial data, these focal K functions can be used to analyse the spatial aggregation of agents around an area of interest. We then use these functions to produce a scalar metric of agent aggregation, the time series of which provides an easily visualized measure of agent aggregation over time.

We next provide a statistical method for the analysis of the drift and diffusivity governing agent movement under the assumption that the evolution of the agent population is well approximated by the Fokker–Planck equation. The Fokker–Planck equation is an advection–diffusion type partial differential equation (PDE) that allows for spatial and temporal dependence with respect to drift and diffusivity terms [19–21]. Our statistical approach uses a novel combination of spatial statistics and functional data analysis. The empirical estimates for focal K functions generated from an agent-based process form a functional time series. By modelling the evolution of focal K function using functional linear models we are able to estimate the drift and diffusivity of the approximating Fokker–Planck equation. We introduce a moving window approach to produce time-varying estimates for drift and diffusivity. Our approach is in contrast to other techniques for estimating PDE parameters [22–34]. An advantage of our approach is that it deals specifically with positional data from agent-based processes, and via the use of a simple functional linear model our approach is easily implemented and interpretable, allowing for fast and accurate estimation of drift and diffusivity parameters that may vary with respect to space and time.

The Fokker–Planck equation is commonly used to model collective cell movement and chemotaxis. Keller–Segel models of chemotaxis, popular and well studied continuum models for chemotaxis, use coupled Fokker–Planck type equations for cell position and chemokine concentration [21,35–37]. Developing techniques to fit Keller–Segel models to experimental data is an active area of research [26]. Fokker–Planck type equations have been shown to provide close approximations to the evolution of cell density among agent-based models of cell movement [38–47].

We develop our methodology for motile agents in two dimensions, but we expect that our methodology can be easily extended to one-dimensional and three-dimensional (3D) processes. Our formulation of this approach for 2D analysis is motivated by T cell–tumour experiments detailed in a later section.

We provide an open source implementation of our tool in R [48], along with code for producing and analysing the presented simulations and experiments, available at https://github.com/JackHywood/Chemotacticswarming.

2. Agent-based processes with clustering

We consider systems of agents (e.g. cells, animals, robots) that change their position over time and analyse their arrangements within some observation region, denoted by R. With a particular focus on chemotactic swarming, we are interested in establishing whether agents are attracted or repulsed from some central region of interest, B, which lies in R. For any given time t, the realization of the dynamic agent-based process results in a spatial point pattern within R. Let X(t) be the spatial coordinate data for agents at a given time t, such that X(t) = {x₁, x₂, …, x_n(t)}, with x_j being the spatial coordinates of the jth agent, and n(t) being the total number of agents observed within the observation region R at time t. Likewise, let the time points at which observations of an agent-based process occur be given as T = {t₁, t₂, …, t_N}, with N giving the total number of time points observed.

For simplicity, we here consider the case where the number of agents within R does not change with time, i.e. n(t) = n is constant. In particular, agents cannot cross the boundary of R. We consider cases in which n(t) varies with time in later sections.

We continue by considering the following idealized spatial geometry for such agent-based processes. For simplicity, we continue by working in 2D with the assumption that R is circular with radius r_R, and also that B is a circle with radius r_B, with B occurring at the centre of R. We also assume that the distribution of agents in R is isotropic, i.e. that agents are distributed uniformly with respect to direction from the centre of R. We note that this framework allows for the case in which B is a point, i.e. setting r_B = 0.

Using polar coordinates and assuming distribution of agents within R is isotropic, we define the density of agents as

$$\lambda (r,t)=\underset{|\mathrm{d}s|\to 0}{lim}\frac{\mathbb{E}[Y(\mathrm{d}s,t)]}{|\mathrm{d}s|},0\le r\le r_R$$ 2.1

with ds defining a small annulus with inner radius r from the centre of R, |ds| giving the volume of ds, Y(ds, t) giving the number of agents within ds at time t, and $\mathbb{E}[\cdot ]$ denoting expectation with respect to the distribution of a given process evolving stochastically from a given set of initial conditions.

Using a common continuum model for cell movement [20,21,26,35–37], we assume that the evolution of λ(r, t) is governed by the Fokker–Planck equation:

$$\frac{\mathrm{\partial }\lambda }{\mathrm{\partial }t}=-\mathrm{\nabla }\cdot (\mathbf{f}\lambda )+\mathrm{\nabla }\cdot (\mathbf{D}\mathrm{\nabla }\lambda ).$$ 2.2

It is well established that under certain conditions the evolution of agent density for discrete agent-based models can be well approximated by PDEs [20,43,46,49].

Working in polar coordinates, and using the assumption of isotropy, we have

$$\frac{\mathrm{\partial }\lambda }{\mathrm{\partial }t}=\frac{1}{r}\frac{\mathrm{\partial }}{\mathrm{\partial }r}(r(-f(r,t)\lambda (r,t)+D(r,t){\lambda }^{\prime }(r,t))),$$ 2.3

where f(r, t) is the drift term, D(r, t) the diffusivity term, and λ′(r, t) = ∂λ/∂r.

For a given time t, if the movement of agents positioned at a distance r from the origin is unbiased then f(r, t) = 0. If there is bias towards the centre of R then f(r, t) < 0. If there is bias away from the centre of R then f(r, t) > 0. The diffusivity term D gives a measure of the component of movement that can be modelled as simple diffusion.

We note that agent-based processes may evolve such that agent behaviour within B is of no interest, i.e. we are only interested in the behaviour of agents outside of B. For example, an agent-based process might evolve such that agents that reach or enter B stop moving and as such their behaviour and distribution within B are of no interest. In such circumstances, the range of r values analysed can be truncated accordingly, for example, r ∈ [r_B, r_R], for the procedures demonstrated in the following sections.

2.1. Focal K functions for agent-based processes

K functions are a well-known summary spatial statistic used for detecting and characterizing the deviation of spatial point patterns from complete spatial randomness [15,17,18]. A crucial fact that we make use of is that certain spatial K functions can be both easily estimated based on sequential observations of agent positions alone, and then related to the function f defined in equation (2.3) allowing one to estimate f. We here adapt K functions for the analysis of agent-based processes, introducing what we refer to as focal K functions.

We define the overall density of agents within R as

$${\lambda }_R=\frac{n}{|R|},$$ 2.4

with n giving the number of agents within R, and |R| giving the volume of R.

We define focal K functions as follows:

$$\begin{array}{c}K(r,\hspace{0.17em}t)=\frac{1}{{\lambda }_R}\mathbb{E}\hspace{0.17em}[\mathrm{Number}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\mathrm{agents}\hspace{0.17em}\mathrm{within}\hspace{0.17em}r\hspace{0.17em}\mathrm{of}\hspace{0.17em}\mathrm{the}\hspace{0.17em}\mathrm{origin}\hspace{0.17em}\mathrm{at}\hspace{0.17em}\mathrm{time}\hspace{0.17em}t]\\ =\frac{2\pi }{{\lambda }_R}\underset{0}{\overset{r}{\int }}s\lambda (s,t)\mathrm{d}s,\hspace{0.17em}0\le r\le r_R.\end{array}$$ 2.5

For each time point t_i we can use the associated spatial point pattern X(t_i) to produce an empirical $\hat{K}(r,t_i)$ function as an estimate of K(r, t_i). We assume observation times are regularly spaced, so that for a positive constant τ, t_i+1 − t_i = τ for all i ≥ 1, and hence we express $\hat{K}(r,t_i)$ as ${\hat{K}}_i(r)$. We define ${\hat{K}}_i(r)$ as

$${\hat{K}}_i(r)=\frac{1}{{\lambda }_R}\sum _{k=1}^n\mathbb{1}(\delta (k)\le r),0\le r\le r_R,$$ 2.6

where δ(k) is the distance from the origin for the kth agent in R, and $\mathbb{1}(\cdot )$ is the indicator function. Since our analysis of agents is limited to the behaviour of agents within R we do not require an edge correction term in equation (2.6) as is typical when producing K function estimates in spatial statistics, though we expect that one could easily be introduced if required (see [18]).

2.2. Focal L functions and the swarming metric

Analogous to spatial statistics, we can transform focal K functions to produce focal L functions that are linear with respect to r and visually easier to interpret, especially for smaller r values. Such focal L functions can then be used to produce what we refer to as a swarming metric, giving a single value that measures the level of agent aggregation.

As noted previously, we may only be interested in the distribution of agents outside of B, i.e. their distribution within B is not of interest. In this circumstance, we can compare a given distribution of agents against a uniform distribution of agents within the annulus R − B, with the agent density within B equal to 0. As such, the agent density associated with this distribution is

$$\lambda (r,t)=\{\begin{array}{cc}0,& 0\le r<r_B\\ \frac{{\lambda }_R{r}_R^2}{r_R^2-r_B^2},& r_B\le r<r_R.\end{array}$$

It then follows from equation (2.5) that in this case the focal K function takes the form

$$K(r,t)=\{\begin{array}{cc}0,& 0\le r<r_B\\ \frac{\pi r_R^2(r^2-r_B^2)}{r_R^2-r_B^2},& r_B\le r<r_R.\end{array}$$ 2.7

Given equation (2.7), we adjust the usual definition of the L function found in spatial statistics accordingly, defining focal L functions as

$$\begin{array}{c}L(r,t)=\sqrt{\frac{((r_R^2-r_B^2)/r_R^2)K(r,t)+\pi r_B^2}{\pi }}-r_B,\hfill \\ r_B\le r\le r_R.\hfill \end{array}$$ 2.8

As such, for a uniform distribution of agents within the given annulus, with the associated focal K function in equation (2.7), we have L(r, t) = r − r_B for r ∈ [r_B, r_R]. Equation (2.8) reverts to the normal L function of spatial statistics by setting r_B = 0.

Focal L functions can be used to further simplify the measurement of agent aggregation by producing a simple scalar metric of aggregation as detailed below. By producing a time series of this metric, we are able to demonstrate how the distribution of agents changes with time in a simple and easily visualizable manner. We define this swarming metric as M(t), and, in order to give a formal definition, we proceed by defining a perfectly aggregated distribution of agents and a perfectly dispersed distribution of agents.

Under this formulation, a given number of agents is perfectly aggregated within R if they are all within B. The focal L function associated with this distribution of agents is

$$L(r)=r_R-r_B,r_B\le r\le r_R.$$ 2.9

The perfectly dispersed configuration of agents will consist of agents being localized on the outer edge of R. The associated focal L function for this distribution of agents is

$$L(r)=\{\begin{array}{cc}0,& r_B\le r<r_R\\ r_R-r_B,& r=r_R.\end{array}$$

The swarming metric M(t) can then be produced for a given time point t by transforming the associated focal L(r, t) function as follows:

$$M(t)=\frac{2}{{(r_R-r_B)}^2}{\int }_{r_B}^{r_R}(L(s,t)-(s-r_B))\mathrm{d}s.$$ 2.10

As such, M(t) has a range of [ − 1, 1], with −1 being equivalent to agents being perfectly dispersed, and 1 being equivalent to agents being perfectly aggregated. Representative configurations of these are shown in figure 1. The empirical metric ${\hat{M}}_i$ is produced using ${\hat{L}}_i(r)$ in equation (2.10).

Figure 1.

Representative diagrams indicating agent arrangements and associated swarming metric, M, values. (a) M = −1, (b) M = 0, (c) M = 1.

As a scalar summary, M(t) does not contain the amount of information relating to agent aggregation as the functions K(r, t) or L(r, t). However, we suggest that a scalar time series of ${\hat{M}}_i$ values may be easier to interpret than the functional time series ${\hat{K}}_i(r)$ or ${\hat{L}}_i(r)$. Furthermore, we suggest that the time series of ${\hat{M}}_i$ may be most useful in comparing realizations of different experimental conditions in which one expects a divergence between M(t) for each condition.

2.3. The relationship between f, D and focal K functions

The functions f and D can be related to the focal K functions as follows. From equation (2.5), we have

$$\frac{\mathrm{\partial }K}{\mathrm{\partial }t}=\frac{2\pi r}{{\lambda }_R}(-f(r,t)\lambda (r,t)+D(r,t){\lambda }^{\prime }(r,t)).$$ 2.11

We introduce the function γ(r, t) = −(λ_R/2πr)∂K/∂t, such that

$$\gamma (r,t)=f(r,t)\lambda (r,t)-D(r,t){\lambda }^{\prime }(r,t).$$ 2.12

Equation (2.12) shows that, under the above assumptions, there is a simple linear relationship between γ(r, t) and the functions λ(r, t) and λ′(r, t). Given estimates ${\hat{\gamma }}_i(r)$, ${\hat{\lambda }}_i(r)$ and ${\hat{\lambda }}_i^{\prime }(r)$ at each time point t_i, one can employ functional linear regression to produce estimates for f and D. This can be achieved using B-spline (or other) smoothing of empirical focal K functions as outlined in electronic supplementary material, 1. The major functions used to produce smoothed empirical focal K functions and to perform functional linear regression are found in the fda R package [50].

Below we apply two methods, again detailed in electronic supplementary material, 1, to estimate f and D: one in which it is assumed both are time-invariant, so that f(r, t) = f(r) and D(r, t) = D(r), and one in which these functions are allowed to be time-varying. Time-invariant estimates are denoted $\hat{f}(r)$ and $\hat{D}(r)$, whereas the time-varying estimates produced at each discrete viewing time are denoted $\hat{f}(r,t_i)={\hat{f}}_i(r)$ and $\hat{D}(r,t_i)={\hat{D}}_i(r).$

It may be known a priori whether f and D are time-invariant or time-varying due to experimental conditions, e.g. due to the induction of a constant chemokine gradient. Also, if K_i(r) strongly suggests that f(r, t) = 0, and D is assumed to be time-invariant, time-invariant estimates may be suitable. In general however, we suggest that most processes will likely demonstrate a degree of time dependence for f and D.

We suggest initially analysing data using time-varying f and D, and considering whether the functional estimates exhibit any obvious trends. If no obvious trends are present, it may be appropriate to proceed to using a time-invariant estimate for f and D. We suggest that even for cases that demonstrate trends in the time-varying estimates, the time-invariant estimates $\hat{f}(r)$ and $\hat{D}(r)$ may provide a useful measure of overall drift and diffusivity.

3. Simulations of simple two-dimensional agent-based models of chemotaxis

We here apply the above approaches to simple agent-based models (ABMs) of chemotaxis. The models are based on other off-lattice ABMs of cell movement [46,51].

Each ABM consists of 1000 individual agents moving within an enclosed circular region, R, with radius r_R = 100, with a circular central region B, with radius r_B = 20, located at the centre of R.

Simulations commence with all agents randomly distributed within R under a uniform distribution. The models are discrete-time processes with time steps of constant duration, τ. We set τ = 1 for all simulations. Each simulation runs for 200 time steps.

Each ABM evolves via a similar underlying process. A random sequential update method is used to perform simulations [43,52]. That is, at each time step, 1000 sequential random selections of the 1000 agents are made, with selected agents performing a movement as described below. As such, an agent may move more than once, or not at all, for a given time step. If an agent is selected to move, they move in a direction, θ, drawn from a probability density function that is dependent on an idealized chemokine gradient. Once a direction is selected the agent moves a constant distance, Δ, in that direction. We use a reflecting boundary at the edge of R, such that if an agent attempts to move out of R it is reflected back into R. These processes are not exclusion processes and agents can occupy the same position in space. We set Δ = 1 for all simulations.

Zero chemotaxis. For the ABM of zero chemotaxis, agents are not biased to move in any particular direction; θ is drawn from a uniform distribution over [0, 2π].

Chemotaxis. To model chemotaxis, we use a similar concept as in [51,53,54]. An idealized attractive chemokine gradient is established radiating from the origin. Agents are more likely to move up the gradient with the likelihood determined by the steepness of the local gradient at the position of the agent.

We take the idealized chemokine concentration, v(r, t), to be the product

$$v(r,t)=V(r)T(t),$$ 3.1

with V(r) giving the distribution of chemokine with respect to r, and T(t) giving a time dependent function that allows the chemokine concentration to change with time.

The steepness of v(r, t) determines how biased agent movements are. Thus we use the absolute value $|\frac{\mathrm{\partial }v}{\mathrm{\partial }r}|$ to parametrize the distribution for θ.

For each agent, the angle of movement is drawn from a von Mises distribution [20,55]. Specifically, for an agent at position (r, α) in polar coordinates, at time t, θ(r, α, t) is drawn using the probability density function

$$p(\theta ;r,\alpha ,t)=\frac{\exp (|\mathrm{\partial }v/\mathrm{\partial }r|\cos (\theta -(\pi +\alpha )))}{2\pi I_0(|\mathrm{\partial }v/\mathrm{\partial }r|)},$$ 3.2

with I₀( · ) giving the modified Bessel function of the first kind of order 0. As such, the expectation of agent movements is α + π, i.e. towards the origin, with the variability around this value determined by the steepness of the chemokine gradient, |∂v/∂r|.

For our simulations, we set V(r) to be a Gaussian function:

$$V(r)=\beta \hspace{0.17em}{\mathrm{e}}^{(-r^2)/2{\sigma }^2},$$ 3.3

with the parameter β > 0 increasing the steepness of the gradient for larger values and σ increasing the ‘diffusivity’ of the chemokine, providing a measure of how far the chemokine signal is effective. For all chemotaxis simulations, we set β = 40 and σ = 30.

3.1. Simulation results

For each case, we visualize the empirical functional time series produced using rainbow plots [56,57]. Rainbow plots are a useful approach in visualizing functional time-series data; the order in which functions occur in time is indicated by colour with early functions red, followed by orange, yellow, green, blue, indigo, with the latest functions violet.

For each simulation, we then produce estimates $\hat{f}$ and $\hat{D}$ using functional linear modelling as described in electronic supplementary material, 1, using the fda R package [50]. For each simulation, we produce the smoothed empirical focal K functions using 40 splines and a smoothing parameter of 1 for time-invariant simulations and 10⁻² for the time-varying simulation. We use order 6 splines, since this will ensure that the second derivatives of the smoothed focal K functions, which are used to produce the functional covariate ${\hat{\lambda }}^{\prime }(r)$, will be smooth [58]. The number of splines used and the smoothing parameter employed for the estimated f and D must also be selected, and we use 20 splines and smoothing parameters of 10 for both f and D for all simulations.

In addition, we note that we find estimates for f and D can sometimes be inaccurate around r = 0 and r = r_R, due the the inherent instability of higher order derivatives of splines at boundaries [58]. We find that estimating f and D over a truncated region, [r₁, r₂], with r₁ > 0 and r₂ < r_R, but still close to these values, can produce better estimates. For all simulations, we use r₁ = 5, and r₂ = 95.

For a discussion regarding implementation and robustness of estimates for f and D with respect to smoothness of the focal K functions, temporal resolution, agent density and number of time steps, see electronic supplementary material, 2. We find that estimates are robust to even small agent densities and limited numbers of time steps. As demonstrated in electronic supplementary material, 2, we note that there is a dependence of the estimates for D on the smoothness of the focal K functions, and suggest that using relatively smooth focal K functions that still capture the variability of the underlying step functions.

We are able to compare $\hat{f}$ and $\hat{D}$ against approximations of the true drift and diffusivity functions associated with each ABM calculated using the first and second moment of agent movements [20,43,49,55,59]. See electronic supplementary material, 3, for details.

3.1.1. Zero chemotaxis

For the case of zero chemotaxis, we run three replicates with different initial starting agent positions. For simplicity, we present ${\hat{K}}_i(r)$, ${\hat{L}}_i(r)$, ${\hat{M}}_i$ and ${\hat{\gamma }}_i(r)$ for only the first replicate; see figure 2. The estimates $\hat{f}(r)$ and $\hat{D}(r)$ obtained using functional linear modelling for each of the three simulations are plotted together on the relevant panels to allow for comparison. We see that for the given simulation the ${\hat{K}}_i(r)$ and ${\hat{L}}_i(r)$ functional time series do not demonstrate an obvious trend. This is reflected in the stable trajectory of the swarming metric ${\hat{M}}_i$. We note that ${\hat{M}}_i>0$ as agents are initially distributed uniformly throughout R, with some agents placed within B, and ${\hat{M}}_i$ compares the distribution of agents to the uniform distribution of agents within the region R − B, i.e. with 0 agents within B. The estimates $\hat{f}(r)$ vary around 0, strongly indicating that no bias of agent movement towards B is detected for this model. The estimates $\hat{D}(r)$ perform well in estimating the true D(r) function. The estimates $\hat{f}(r)$ and $\hat{D}(r)$ are similar between the three simulations.

Figure 2.

Zero chemotaxis: (a) ${\hat{K}}_i(r)$; (b) ${\hat{L}}_i(r)$; (c) ${\hat{M}}_i$; (d) ${\hat{\gamma }}_i(r)$; (e) $\hat{f}(r)$; and (f) $\hat{D}(r)$. The estimates $\hat{f}(r)$ and $\hat{D}(r)$ for the first replicate are given by solid black lines, with the dotted black lines giving the 95% confidence intervals for these estimates, the red solid lines giving the true f(r) and D(r) functions, and the blue lines giving 0. Estimates for $\hat{f}(r)$ and $\hat{D}(r)$ associated with the two additional replicates using different initial positions are added to panels (e) and (f) as solid grey and dashed grey lines.

3.1.2. Time-invariant chemotaxis

For the model of time-invariant chemotaxis, we set T(t) = 1, such that attraction exerted on agents is time-invariant. We run three replicates with different initial starting agent positions. Similarly to above we present ${\hat{K}}_i(r)$, ${\hat{L}}_i(r)$, ${\hat{M}}_i$ and ${\hat{\gamma }}_i(r)$ for only the first replicate; see figure 3. We can see that the ${\hat{K}}_i(r)$ and ${\hat{L}}_i(r)$ functions associated with the first experiment increase over time for all r values. This is reflected in the positive trend in the swarming metric ${\hat{M}}_i$. The estimates $\hat{f}(r)$ for each simulation perform very well in estimating the true f(r) function. The estimates $\hat{D}(r)$ likewise perform well in estimating the true D(r) function.

Figure 3.

Time-invariant chemotaxis: (a) ${\hat{K}}_i(r)$; (b) ${\hat{L}}_i(r)$; (c) ${\hat{M}}_i$; (d) ${\hat{\gamma }}_i(r)$; (e) $\hat{f}(r)$; and (f) $\hat{D}(r)$. The estimates $\hat{f}(r)$ and $\hat{D}(r)$ for the first replicate are given by solid black lines, with the dotted black lines giving the 95% confidence intervals for these estimates, the red solid lines giving the true f(r) and D(r) functions, and the blue lines giving 0. Estimates for $\hat{f}(r)$ and $\hat{D}(r)$ associated with the two additional replicates using different initial positions are added to panels (e) and (f) as solid grey and dashed grey lines.

3.1.3. Linearly increasing chemotaxis

To model chemotaxis with linearly increasing strength, we set

$$T(t)=1+0.02t.$$ 3.4

We perform and analyse only one simulation for this model; see figure 4. The functional time series ${\hat{f}}_i(r)$ for i = 15, …, 186 is produced using a moving window-type estimator, as discussed in electronic supplementary material, 1, using a window width of 29 time points. We observe that the functional time series performs well in estimating the functional time series of the true f(r, t_i) functions for i = 15, …, 186, with strength of attraction increasing with time for this simulation. We note similar results, with slightly reduced accuracy, are obtained for the estimates ${\hat{D}}_i(r)$.

Figure 4.

Linearly increasing chemotaxis: (a) ${\hat{L}}_i(r)$; (b) ${\hat{M}}_i$; (c) ${\hat{f}}_i(r)$ for i = 15, …, 186; (d) true f(r, t_i) for i = 15, …, 186; (e) ${\hat{D}}_i(r)$ for i = 15, …, 186; (f) true D(r, t_i) for i = 15, …, 186.

4. T cell motility experiments

Cytotoxic T cells (CTLs) are specialized immune cells that seek out and destroy cancer cells, and they are the principal mediators of adoptive cell transfer immunotherapies [60]. These burgeoning therapies have proven revolutionary in the treatment of blood-borne cancers, yet have thus far not shown similar success when applied to solid tumours, which leukocytes infiltrate poorly [61,62]. Therefore, understanding the spatial movements of T cell populations with respect to tumour masses is key to the development of effective immunotherapies targeting solid malignancies.

Recently, CTLs engaging tumour targets have been demonstrated to engage in chemokine signalling, attracting distant CTLs [7]. Here, we analysed ex vivo experiments of CTLs engaging tumour cells, presented in [7] (see fig. 1 in [7]). The experimental details, along with movies of the experiments, and additional related results, can be found in [7].

In brief, the experiments occur within a single well of a 96-well optical plate, and consists of a central dense collection of tumour cells and extra-cellular matrix (referred to as tumouroid) surrounded by fluorescent effector CD8+ T cells embedded in a 3D collagen matrix. We consider two different experimental conditions. One we refer to as the pre-embedded-cognate experiment, in which the tumouroid consists of cognate tumour cells and pre-embedded, non-fluorescent, CTLs, i.e. the tumouroid already contains CTLs at the start of the experiment. The cognate tumour cells in this tumouroid present the activating antigen on their surface, thus leading to T cell arrest, activation and tumour cell clearance. The other experimental condition we refer to as the non-cognate experiment, in which the tumouroid contains non-cognate tumour cells only. Non-cognate tumour cells do not activate CTLs, which hence do not arrest on contact and thus continue their search.

We hypothesized that in the pre-embedded-cognate experiment the peripheral CTLs surrounding the tumouroid would be attracted to the tumouroid via chemotaxis. Specifically, we expected that CTLs initially pre-embedded within the tumouroid, and peripheral CTLs that made contact with the cognate cells making up the tumouroid during the experiment, would both become activated and secrete a chemokine to recruit more distant CTLs. In comparison, we hypothesized that in the non-cognate experiment CTLs would not become activated upon contact with the non-cognate cells making up the tumouroid, and hence there would be no detection of chemotaxis of CTLs towards the tumouroid.

Images were taken of the entire closed well every 5–6 min over approximately 16 h and the positional data for the fluorescing CTLs, i.e. those initially dispersed around the tumouroid, were recorded during the experiment. We note that in contrast the non-fluorescent CTLs pre-embedded in the tumouroid were not imaged. Owing to the resolution required and the comparatively large area being imaged (greater than 6 mm in diameter), imaging was constrained to a 2D slice through the x–y plane of the entire well. Images of the first and last time point from both experiments are presented in figure 5.

Figure 5.

Images of non-cognate and pre-embedded-cognate experiments: (a) image from first time point of non-cognate experiment; (b) image from final time point of non-cognate experiment; (c) image from first time point of pre-embedded-cognate experiment; (d) image from final time point of pre-embedded-cognate experiment. CTLs are blue, with tumouroids red. Scale bars: 500 μm. Time stamps indicate time of image acquisition. Images are from experiments reported in [7].

For each experiment, we need to define the regions R and B. In 2D, the tumoroids are approximately circular. We choose to set r_B equal to the average of longest and shortest distances from the centre of geometric mass to the edge of the tumouroid. Owing to the technicality of sample preparation, the tumouroid cannot be precisely positioned such that its centre of mass is at the exact centre of the well. We set r_R to the shortest distance from the tumouroid centre to the well edge. The radius of the base of the well is approximately 3150 μm. For the non-cognate experiment, we set r_B = 1076.75 μm and r_R = 2855 μm. For the pre-embedded-cognate experiment, we set r_B = 1099 and r_R = 2985.

While the imaged system is closed to the entry and egress of cells in the horizontal plane due to the walls of the well, it is open with respect to cell movements in the vertical axis due to the 3D nature of the collagen matrix. We expected that the net effect of this vertical movement would be negligible. We noted that for both experiments, the trend in cell number is relatively stable for the first 10 h of the experiments. Since we expect any evidence for the existence of chemotaxis to be evident during this period, we continued by analysing only the first 10 h of each experiment. In addition, a relatively sharp decrease in cell number was noted to occur over the first hour of the pre-embedded-cognate experiment. Since we expected evidence of chemotaxis to still be evident after the first hour of the experiment we continued by removing the first hour of the pre-embedded-cognate experiment from our analysis. See figure 6 for the associated cell numbers and ${\hat{K}}_i(r)$ functions for the remaining time points.

Figure 6.

Cell number dynamics: (a) total number of cells imaged within R for each time step analysed for the non-cognate experiment; (b) associated ${\hat{K}}_i(r)$ functions for the non-cognate experiment; (c) total number of cells imaged within R for each time step analysed for the pre-embedded-cognate experiment; (d) associated ${\hat{K}}_i(r)$ functions for the pre-embedded-cognate experiment.

The approach presented in previous sections requires that the number of agents observed does not change with time. In electronic supplementary material, 4, we consider the case where agent numbers can change with time as a result of an agent birth–death process (in addition, we also consider the unrelated case where R has an open boundary). We demonstrate that if it can be assumed that the birth–death rate is spatially invariant and linear with respect to λ(r, t), then it has no impact on the interpretation of the change in K and L functions, the swarming metric M, or the approach to estimating f and D. For the purposes of this paper, we continue with this assumption for the analysis of the T cell motility experiments.

4.1. T cell experiment results

For both experiments, we set the number of splines to 200 and a smoothing parameter of 10⁸. We use 20 splines for estimating both f and D, with smoothing parameters 10⁻¹. We estimate f and D over [20, r_R − 20].

Relevant plots for the non-cognate experiment are displayed in figure 7. The functional time series for ${\hat{L}}_i(r)$ has no apparent trend and appears nearly linear, suggesting that agents are approximately uniformly distributed outside the tumouroid. The associated time series for the swarming metric ${\hat{M}}_i$ varies around 0 for the duration of the experiment. Since ${\hat{L}}_i(r)$ and ${\hat{M}}_i$ do not demonstrate any trend, we continue by producing time-invariant functional estimates for f and D. We note that the estimate $\hat{f}(r)$ is approximately 0 for all r values. This result was consistent with our expectation that CTLs would not be attracted towards the tumouroid for this experiment.

Figure 7.

Non-cognate experiment: (a) rainbow plot of ${\hat{L}}_i(r)$; (b) ${\hat{M}}_i$; (c) time-invariant $\hat{f}(r)$ with 95% confidence intervals (dotted black lines); (d) time-invariant $\hat{D}(r)$ with 95% confidence intervals (dotted black lines).

Relevant plots for the pre-embedded-cognate experiment are displayed in figure 8. The functional time series for ${\hat{L}}_i(r)$ demonstrates a positive trend, with ${\hat{L}}_i(r)$ appearing to increase for all r values over time. The associated time series for the swarming metric ${\hat{M}}_i$ demonstrates an associated increase. Since these time series demonstrate obvious trends we first estimate time-varying f and D. The associated estimates ${\hat{f}}_i(r)$ and ${\hat{D}}_i(r)$ using a window width of 19 appear relatively stable over time. As such, we proceed to produce time-invariant estimates $\hat{f}(r)$ and $\hat{D}(r)$. The time-invariant $\hat{f}(r)$ is negative for all r values, and approaches 0 as r increases towards r_R. This result is consistent with our expectation that CTLs would be attracted to the tumouroid via chemotaxis for this experiment. We note that $\hat{f}(r)$ is strikingly similar in form to that in our simple ABM of chemotaxis, suggesting that the chemokine concentration in the experiment may be well approximated by a Gaussian distribution and that CTLs respond to the steepness of the chemokine gradient. The estimate for $\hat{D}(r)$ is similar to that for the non-cognate experiment, though we note that it is decreased for r values between r_B and approximately 1500.

Figure 8.

Pre-embedded-cognate experiment: (a) rainbow plot of ${\hat{L}}_i(r)$; (b) ${\hat{M}}_i$; (c) rainbow plot of ${\hat{f}}_i(r)$; (d) rainbow plot of ${\hat{D}}_i(r)$; (e) time-invariant $\hat{f}(r)$ with 95% confidence intervals (dotted black lines); (f) time-invariant $\hat{D}(r)$ with 95% confidence intervals (dotted black lines).

5. Discussion

In this paper, we have presented a novel approach for detecting and characterizing directional bias in agent movements solely from positional data for agent-based processes. Our approach does not require the tracking of agents over time, meaning that our approach can be used when temporal tracking of individual agents is not available. For instance, in comparison to collecting positional data alone, individual cell tracking typically requires several undesirable features, including a high sampling rate with associated phototoxicity, a small field of view, and a relatively low cell density [10,12–14].

Techniques from spatial statistics, such as the pair correlation function, are increasingly being applied to agent-based processes [54,63–73]. Our approach employs a novel combination of both spatial statistics and functional data analysis. There is great scope to combine spatial statistical methods and functional data analysis to quantify the dynamics in agent-based processes. In previous work, we combined such approaches in the analysis of spatially homogeneous agent-based processes [54]. As advances in microscopy, and also GPS tracking, yield detailed observations of dynamic and complex processes such as immune cell responses, animal behaviour, or search and rescue robot swarms, more advanced statistical metrics are increasingly needed.

The approach presented here introduces focal K functions that provide a measure of agent aggregation around an area of interest for a given time point. We use these functions to define a related scalar, M, which we refer to as the swarming metric. The functional and scalar time series associated with the focal K functions and the swarming metric, respectively, provide useful information regarding the evolution of agent-based processes with respect to agent aggregation. These methods are general, and do not rely upon any assumptions regarding the underlying agent behaviour. We envisage that the swarming metric in particular may provide a useful means of comparing different experimental conditions of agent-based processes.

If the evolution of population density with time is well approximated by the Fokker–Planck equation, an advection–diffusion PDE, then these focal K functions can be analysed using a functional linear model to estimate the associated drift and diffusivity terms. Importantly, this allows for any bias in agent movements to be detected and characterized. Our approach allows for the analysis of agent-based processes with both time-invariant as well as time-varying attraction or repulsion. In addition, there are no requirements regarding whether the system is or is not in a steady state.

We demonstrate our approach using simulations, analysing a set of simple 2D random walk agent-based models. We find that our approach performs well for these sets of simulations. Moreover, we find that estimates appear to be robust with respect to the density of agents and the number of observations. We find that estimates for diffusivity are dependent on the degree of smoothing used in smoothing the focal K functions, and that this dependence is accentuated by reduced temporal resolution. While we suggest that using a relatively large level of smoothing can ensure accurate estimates for diffusivity, determining an explicit approach for selecting the most appropriate degree of smoothing represents a useful extension.

We have here only analysed simulations of agents undergoing a random walk process without persistence or exclusion. Both these properties may be relevant in considering specific agent-based processes, such as bacterial chemotaxis, or cellular processes with high agent density. Recent results have demonstrated that drift–diffusion PDEs can be produced to approximate the evolution of agent-based processes exhibiting bias and persistence in the setting of modelling bacterial chemotaxis [74]. In addition, drift-diffusion type PDEs can be produced to approximate the evolution of a lattice-based agent-based process that includes bias, persistence, and exclusion, with the drift and diffusion terms being dependent on agent density [72]. Determining whether our approach can be employed to analyse similar off-lattice processes represents a useful extension of this work.

We apply our approach to the analysis of ex vivo experiments of T cells interacting with a central tumouroid consisting of tumour cells. These represent a portion of novel experimental results used to demonstrate that CTLs swarm to tumours [7]. Consistent with these findings, our analysis of these experiments identifies an attractive signal that biases cell movements towards the tumouroid. As outlined in [7], the attraction exhibited by T cells represents homotypic signalling; that is, the activated CTLs in contact with cognate tumour cells produce the chemoattractant that induces chemotaxis in other CTLs. As such, this process represents what we refer to as agent-driven swarming, such that the swarming behaviour exhibited is driven, at least in part, by the presence of other agents at the area of interest. This differentiates agent-driven swarming from other types of processes in which the attractive signal is derived from sources other than the agents. It can be of considerable interest to researchers whether observed swarming is agent–driven or otherwise. If it is known a priori that the agents within a given process represent the only possible source of attractant, or this can be proved experimentally, then detecting that agent movements are biased towards a given area is enough to determine agent-driven swarming. However, it may represent a useful extension of this work to produce a statistical technique for detecting agent-driven swarming in circumstances in which this is not known and cannot be demonstrated experimentally.

Data accessibility

All relevant code and data are publicly available at https://github.com/JackHywood/Chemotacticswarming.

Authors' contributions

J.H. contributed to development of statistical methods, analysis and writing of manuscript. G.R. and M.R. contributed to developing statistical methods. M.B. and S.P. conceived of and performed experiments.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by Australian Research Council Discovery Project grant DP180102458 to M.R. and M.B. M.R. acknowledges support from the University of Sydney Centre for Advanced Food Enginomics. M.B. is additionally supported by funding via EMBL Australia.