In this paper we demonstrate an application of the unscented Kalman filter in the context of cell movement, using a model defined in terms of stochastic differential equations (SDEs)

Giurghita, Diana

Husmeier, Dirk

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Statistical Modelling of Cell Movement

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Statistical modelling of cell movementDiana Giurghita1, Dirk Husmeier11 University of Glasgow, United KingdomE-mail for correspondence: d.giurghita.1@research.gla.ac.ukAbstract: In this paper we demonstrate an application of the unscented Kalmanfilter in the context of cell movement, using a model defined in terms of stochasticdifferential equations (SDEs).Keywords: Chemotaxis, Cancer, Stochastic Differential Equations, UnscentedKalman Filter.1 IntroductionMany important biological processes, such as wound healing, tissue devel-opment and cancer cell invasion, are based on the collective movement ofcells. One of the main mechanisms for directed cell movement is chemo-taxis, where cells follow chemical gradients (chemoattractants) present intheir environment. These gradients might arise from the presence of a localsource of chemoattractant or due to local depletion of the chemical in theenvironment (Tweedy et al. 2016). An example of the former scenario isthe migration of breast tumour cells that respond to the epidermal growthfactor released by macrophages. In an attempt to acquire a deeper under-standing of the mechanisms behind cell movement many population basedmodels have been formulated using partial differential equations, with veryfew of them attempting to fit these models to actual data.In this paper, we propose a model that describes the movement of anyindividual cell being driven by an external resource gradient using SDEs ofthe form:dXt = σdBXt , dYt =αβ exp[−β(Yt − γt)]{1 + exp[−β(Yt − γt)]}2 dt+ σdBYt (1)Equations (1) describe the evolution in time of the x and y coordinates ofa cell in 2D space. σdBXt and σdBXt are Brownian motion terms which inThis paper was published as a part of the proceedings of the 32nd Interna-tional Workshop on Statistical Modelling (IWSM), Johann Bernoulli Institute,Rijksuniversiteit Groningen, Netherlands, 3–7 July 2017. The copyright remainswith the author(s). Permission to reproduce or extract any parts of this abstractshould be requested from the author(s).2 Statistical modelling of cell movementthis model represent the intrinsic randomness in a cell’s movement. Thecoordinate in the y direction has a drift term that is described by threeparameters: α - the amplitude of the resource gradient, β - the steepness ofthe gradient and γ which indicates how fast the gradient changes over time.The strength of the random component in the cell movement equations isindicated by the diffusion coefficient σ.FIGURE 1. Three frames from the video recording Dictyostelium cells movement.In this paper, we present our approach to fitting this model to cell move-ment data using a non-linear Bayesian filter. We provide some insight intothe particularities of this model using simulated data and we discuss theresults of this analysis from a real data set, describing the movement ofDictyostelium cells (see Figure 1).2 MethodsInference in non-linear dynamical systems poses numerous challenges dueto the stochastic nature of the data, intractable likelihoods and uniden-tifiable parameters. Recent developments have tackled this problem usinglikelihood-free methods (sequential Monte Carlo ABC) or computationalmethods (particle Markov Chain Monte Carlo) (Golightly & Wilkinson,2011), however these can become too computationally expensive as thenumber of time points or parameters increases. The unscented Kalman fil-ter (UKF) is an online Bayesian filtering method that can easily be scaledup to higher dimensions (Julier & Uhlmann, 1997). Intuitively, the UKFstarts from the initial distribution of the state vector, drawn from a mul-tivariate normal distribution, which is then iterated through a predictionand updating step for each measurement available using the transition andobservation models.We introduce the UKF by referring to a general state-space model:xt = f(xt−1, t), yt = g(xt,νt) (2)where xt represents the vector of the hidden states, yt are the measure-ments, t is the process noise at time t, νt is the observation noise at timet and the functions f and g represent the transition and, respectively, theGiurghita and Husmeier 3observation models. The model parameters θ can be included as dynamicalvariables in the hidden states vector xt, which means they will be estimatedat every time point along with the observed system states (Sitz et al., 2002).The advantage of the method comes from the fact that the probability dis-tribution of the predictor step: p(xt|y1:t−1) and the probability distributionof the updating step p(xt|yt,y1:t−1) can be obtained in closed form usingproperties of the Gaussian distribution (Julier & Uhlmann, 1997):p(xt|y1:t−1) ≈ N (xt|µ¯t, Σ¯t) (3)p(xt|yt,y1:t−1) ≈ N (xt|µt,Σt) (4)where µ¯t and Σ¯t are the prediction mean and covariance at time t andµt and Σt are the update mean and covariance at time t (see Julier &Uhlmann, 1997 for full derivations). Therefore, the algorithm essentiallyupdates the mean and covariance of the Gaussian distribution of the statevector at each iteration. The approximation of the Gaussian distribution ismade using the unscented transform, which consists of a set of determinis-tically chosen sigma-points that are passed through the non-linear functionand weighted to obtain the mean and covariance of the Gaussian. The un-scented transformation is used twice for each iteration of the algorithm: inthe prediction and respectively in the update step.3 Simulation resultsWe apply the Euler-Maruyama discretisation to bring the system in Equa-tion (1) into the standard state-space model described in Section 2:Xt = Xt−1 + σ∆BXt (5)Yt = Yt−1 +αβ exp[−β(Yt−1 − γt)]{1 + exp[−β(Yt−1 − γt)]}2 dt+ σ∆BYt (6)Here ∆BXt and ∆BYt are just sums of random normal increments betweentime t−1 and t. Equations (5) and (6) thus define the transition function ffrom Equation 2. In this scenario, we assume the process is observed witha small amount of Gaussian noise νt ∼ N (0, 0.12), so g from Equation 2 isjust the identity function.We then fit the UKF to a synthetic data set using the following parameters:α = 5, β = 2, γ = 0.5, σ = 0.1. The results summarised in Figure 2 indicategood agreement between the estimated UKF path and the true cell path.The UKF also provides good estimates for the parameters: βˆ = 1.88, γˆ =0.51, σˆ = 0.04 with relatively small standard errors: 0.17, 0.83, 0.06, exceptfor αˆ where the estimates indicate a more substantial deviation from thetrue parameter (bias: 0.44 and standard error is 0.35).4 Statistical modelling of cell movement0 2 4 6 8 10Time-0.200.2Xx-coordinate0 2 4 6 8 10Time0246810Yy-coordinate0 2 4 6 8 10468α0 2 4 6 8 10135β0 2 4 6 8 10024γTrue valueUKF estimate+/- 1 std dev0 2 4 6 8 100123σ3 4 5 6 7 8α-2000-1900-1800-1700-1600-1500-loglikelihoodProfile Likelihood-loglikmin(-loglik)α=5FIGURE 2. Simulation results: UKF tracking of cell coordinates: x and y, andparameters: α, β, γ, σ for time interval [0, 10]. Parameter estimates include ±1standard error bounds. On the right hand side, negative log profile likelihoodplot for α, obtained by fixing the other three parameters at their true values.A potential source of bias as the one observed in Figure 2 can be investi-gated by looking at the likelihood i.e.: marginal likelihood with respect tothe hidden states. In order to do that, we first derive the probability of theobserved system at time t conditional on the state of the system at timet− 1 by integrating out the latent variable xt:p(yt|yt−1) =∫p(yt|xt)p(xt|yt−1)dxt (7)=∫N (yt|xt,Rt)N (xt|µ¯t, Σ¯t)dxt (8)Using the Gaussian convolution integral results (Bishop, 2006) we simplify(8) to p(yt|yt−1) = N (yt|µ¯t, Σ¯t + Rt), where µ¯t and Σ¯t are the predictedmean and covariance at time t, and Rt is the measurement noise covariancematrix at time t. The log likelihood is then:L = log∏tp(yt|yt−1) =∑tlogN (yt|µ¯t, Σ¯t + Rt) (9)∝∑t{log det(2piΣt) + 0.5(yt − µ¯t)TΣ−1t (yt − µ¯t)}, (10)Giurghita and Husmeier 5Where Σ¯t + Rt = Σt. We evaluate the marginal log likelihood in (10) byconsidering a grid of values for each parameter in the model and fittingthe UKF with each parameter combination. The results summarising theprofile likelihood for the α parameter in Figure 2 can be used to calculatethe Cramer-Rao lower bound, which provides an indication of the intrinsicuncertainty specific to the problem. In this case, the minimum standarddeviation attainable by an estimator of α is 0.14. Considering the standarderror obtained from the UKF estimation for α is 0.35, this then indicatesthat the estimated value of the parameter is reasonably close to the truevalue.300 350 400 450 500 550 600x-coordinate0200400y-coordinateUKF estimate of two cell pathsCell 1 pathUKF estimateCell 2 path0 50 100 150 200450525600x-coordinate0 50 100 150 2000300600y-coordinateCell 10 50 100 150 200Time320360400x-coordinateCell 20 50 100 150 200Time050100150y-coordinateFIGURE 3. UKF tracking of two cells with different movement patterns.4 Real data applicationDictyostelium cells are widely used in experiments as proxies for under-standing the mechanisms of human disease because of their similarities toimportant human cells (leukocytes and cancer cells) in terms of biologyand response to chemotaxis (Tweedy et al., 2016). The data consists of twocell paths corresponding to Dictyostelium cells locations extracted from atime series of high resolution microscopy images (see Figure 1). We em-phasise that the main interest for biological applications is the inference of6 Statistical modelling of cell movementthe parameters. However, since the true parameters for the real data areunknown, we use the tracking of the cell trajectories as a proxy for assess-ing the accuracy of inference (see Figure 3). As can be seen from the leftpanel of Figure 3, we have picked two cells with very different behaviour(one dominated by drift, the other dominated by diffusion). In both cases,the path reconstructed with the UKF is very accurate.5 Conclusions and future workIn this paper, we demonstrate the application of the UKF, a Bayesian fil-tering technique that adequately trades off accuracy versus computationalefficiency, to a real-world problem potentially relevant to cancer research:the movement of Dictyostelium cells, which has not been tackled at individ-ual cell level before. Our results indicate that the UKF can be successfullyused for parameter inference and tracking cells displaying various move-ment patterns. Future work will extend this work by applying the UKFto a population of cells. Additionally, we plan to fit models describing al-ternative movement mechanisms, such as the self-induced gradient modeldescribed by Tweedy et al. (2016) and employ model selection criteria tochoose the best model.Acknowledgments: The research described in this article is part of theresearch programme of SoftMech, the Centre for multiscale soft tissue me-chanics with application to heart & cancer, funded by the Engineering andPhysical Sciences Research Council (EPSRC) of the UK, grant referencenumber EP/N014642/1.ReferencesBishop, C. (2006) Pattern Recognition and Machine Learning. SpringerGolightly, A. & Wilkinson, D. J. (2011) Bayesian parameter inference forstochastic biochemical network models using particle Markov chainMonte Carlo. Interface focus, 1(6), 807 – 820Julier, S. J. & Uhlmann, J. K. (1997) A New Extension of the Kalman Fil-ter to Nonlinear Systems. AeroSense’97, 182 – 193Sitz, A., Schwarz, U., Kurths, J. & Voss, H. U. (2002) Estimation of pa-rameters and unobserved components for nonlinear systems fromnoisy time series Phys. Rev. E, 66(1), 016 – 210Tweedy L., Knecht D.A., Mackay G.M. & Insall R.H. (2016) Self-GeneratedChemoattractant Gradients: Attractant Depletion Extends the Rangeand Robustness of Chemotaxis. PLoS Biol, 14(3)

Diana Giurghita

Dirk Husmeier

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Statistica Neerlandica

Statistical modelling of cell movement

Collective cell movement affects vital biological processes in the human organism such as wound healing, immune response, and cancer metastasis. A better understanding of the mechanisms driving cell movement is then essential for the advancement of medical treatments. In this paper, we demonstrate how the unscented Kalman filter, a technique used extensively in engineering in the context of filtering, can be applied to estimate random or directed cell movement. Our proposed model, formulated using stochastic differential equations, is fitted on data describing the movement of Dictyostelium cells, an amoeba routinely used as a proxy for eukaryotic cell movement

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Statistical Modelling of Cell Movement

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