Carlo Bianca

Julien Riposo

Mimic Therapeutic Actions Against Keloid by Thermostatted Kinetic Theory Methods

Keywords: 02, 30, Jr Partial differential equations -02, 50, Cw Kinetic theory -02, 60, Nm Integral and integrodifferential equations

Mimic therapeutic actions against keloid by thermostatted kinetic theory methods

Carlo Bianca, Julien

Introduction

Wound healing is a dynamic process consisting of four continuous, overlapping, and precisely programmed phases: Hemostasis, the inflammation phase, the proliferation phase, and the remodelling phase, see the review paper [1]. The events of each phase must happen in a precise and regulated manner. Interruptions, aberrancies, or prolongation in the process can lead to delayed wound healing or a non-healing chronic wound, such as keloid. Keloid is a hyperproliferative response of connective tissue in response to skin trauma. The causes which trigger this phenomenon are poorly understood and currently no successful treatment have been developed [2]. In particular the defective control mechanisms in keloid result in the excessive cell proliferation and extracellular matrix synthesis: keloid-derived fibroblasts have a greater proliferative capacity than normal derived fibroblasts [3]. Even if fibroblasts have a major role, other cells like keratinocytes and melanocytes can be involved [2], and the causes can be also the presence of a virus [4] and a generic susceptibility [5]. The possible therapy for keloid strictly depends on the location, size, depth of the lesion and the age of the patient. Therapeutic treatment includes occlusive dressings, compression therapy, intralesional corticosteroid injections, cryosurgery, excision, radiation therapy, laser therapy, interferon therapy, 5-fluorouracil, doxorubicin, bleomycin, verapamil, retinoic acid. Other promising therapies include antiangiogenic factors, including vascular endothelial growth factor inhibitors, phototherapy, tumor necrosis factor (TNF)-alpha inhibitors, and recombinant human interleukin, which are directed at decreasing collagen synthesis.

Recently the defective induction of stress-induced premature senescence during wound healing has been proposed as a possible mechanism of keloid formation [6]. Specifically keloid fibroblasts undergo senescence at a rate lower than that of normal scar, thus depositing collagen and other extracellular matrix proteins beyond that expected in normal wound healing. The proposed mechanism could lead to new treatment possibilities for keloid, e.g. a therapy that induce senescence could be used to prevent the formation of keloid, and consecutively enable the formation of a normal scar.

From the modeling point of view, the fibro-proliferative disease has been widely studied at macroscopic level, but insight at cellular and sub-cellular level scale is still lacking, see the review paper [7] and references cited therein. Therefore, there is a remarkable need of developing the necessary modeling approach that can handle the full range of time and length scales required Send offprint requests to:

a Present address: bianca@lptmc.jussieu.fr, riposo@lptmc.jussieu.fr to model complex biological systems, with emphasis on spatially distributed systems, and apply this framework to fibrosis disease such as keloid.

The keloid formation has been recently modeled in [8] and further analyzed in [9]. The model, which refers to the keloid formation triggered by a virus and its possible malignant effects under the immune system surveillance, is based on the kinetic theory for active particles [START_REF] Bianca | Towards a Mathematical Theory of Multiscale Complex Biological Systems[END_REF]. At the basis of the methods there is the decomposition of the whole system into different functional subsystems composed by particles, called active particles, able to express a function or strategy. Therefore the microscopic state of the active particles includes, in addition to the classical geometrical and mechanical variables, also the activity variable. The description of each functional subsystem is obtained by the definition of a distribution function over the microscopic state of interacting particles. The evolution of the system is determined by microscopic nonlinear interactions, which take into account not only mechanical rules but also modifications of the biological state. It is worth stressing that the kinetic theory for active particles has been also employed for the modeling of other biological systems, see, among others, papers [11][12][13][14].

This paper is concerned with a further refinement of the mathematical model proposed in [8]. Specifically in order to take into account possible therapeutical actions, the model [8] is generalized by introducing different external forces, which mimic a therapy at the macroscopic scale. The introduction of an external force field moves the system out of equilibrium and in order to have a steady state a Gaussian thermostat is coupled to the external force field. The Gaussian thermostat is thus a damping term adjusted to control the energy into the system, the interested reader in a more deeper understanding of the thermostat and its applications in molecular dynamics simulations is referred to the book [START_REF] Evans | Statistical Mechanics of Nonequilibrium Fluids[END_REF] and to the recent review paper [START_REF] Jepps | Deterministic thermostats[END_REF]. The resulting framework is thus a thermostatted kinetic model where the excess of energy introduced by the external force field is removed by the thermostat. This new framework acts as a general paradigm for the derivation of a specific model for keloid. In particular the thermostatted kinetic model proposed in the present paper is the object of a computational analysis to test a recent hypothesis of therapeutic action against keloid formation and the possible development of a cancer. The model depicts the competition of the immune system cells with a virus, the mutated fibroblast cells, and the cancer cells by means of the density function and the distribution function of cells. Three different external forces, mimic therapeutic actions, are taken into account: a vaccine for the virus, the PUVA therapy for the keloid and a vaccine for the cancer cells. The results are in agreement with the evidence that the sole action of the immune system is not sufficient to obtain a total depletion of keloid, thus the introduction of a therapy is an essential step.

The contents of the present paper are outlined as follows: After this introduction, Section 2 is concerned with the presentation of the thermostatted kinetic theory for active particles and specifically with the derivation of a mathematical structure that acts as a general paradigm for the derivation of specific models. The framework includes an external force field coupled with a Gaussian thermostat that allows the control of the global activation energy. Section 3, after a short phenomenological analysis with the aim of understanding who are the main actors that are responsible for the keloid formation, is devoted to the derivation of our model for keloid formation and specifically to the definition of the functional subsystems, the interactions among the cells and the introduction of the external forces. Section 4 deals with a computational analysis on a test case obtained by setting a value to the different parameters of the model. The computational analysis consists of a sensitivity analysis on the magnitude of the external forces when the rate of heterogeneity of keloid cells is let vary. Finally Section 5 concludes the paper by suggesting possible further refinements and developments of the model, to be regarded as future research directions.

The thermostatted kinetic framework

This section is concerned with the underlying thermostatted kinetic framework that acts as a general paradigm for the derivation of specific models. Specifically the framework aims at modeling a complex biological system composed of a large number of cells (active particles) that interact in nonlinear manner. A constant force field F is assumed to extent an action on the cells.

The microscopic state of a cell is the variable u ∈ D u ⊂ R, which means that the cell at time t ∈ [0, ∞) is able to express a strategy modeled by the variable u (activity variable). Cells expressing the same function are grouped into a subsystem, called functional subsystem. The evolution of each functional subsystem is depicted by the distribution function f i (t, u) : [0, ∞) × D u → R + , for i ∈ {1, 2, . . ., n}, and such that, for any fixed time t, the quantity f i (t, u)du represents the density of cells in the elementary volume du centered at u. In general, the macroscopic variables are defined as moments weighted of the distribution function f i . Specifically the pth-order moment of f i reads:

E p [ f i ](t) = D u u p f i (t, u) du, (2.1)
where, in particular, the density (mass), the activation-momentum (linear-momentum), and the activation energy (kinetic energy) of the functional subsystem f i is obtained for p = 0, p = 1 and p = 2, respectively. Let f = f(t, u) = ( f 1 (t, u), . . . , f n (t, u)) be the vector whose components are the distribution functions of the functional subsystems and

f (t, u) = n ∑ i=1 f i (t, u).
(2.

2)

The evolution equation for the ith functional subsystem is obtained by equating the time derivative of f i to the balance of the inlet and outlet flows in the elementary volume du. Accordingly we have:

∂ t f i (t, u) + ∂ u F i 1 -u D u u f (t, u) du f i (t, u) = J i [f](t, u), (2.3)
where F i denotes the external force acting on the cells of the ith functional subsystem, J i [f](t, u) denotes the following operator that models the gain-loss of cells due to transitions in the activity variable:

J i [f](t, u) = n ∑ j=1 D u ×D u η i j A i j (u * , u * , u) f i (t, u * ) f j (t, u * ) du * du * -f i (t, u) n ∑ j=1 D u η i j f j (t, u * ) du * . (2.4)
where

• η i j models the probability that a cell of the ith functional subsystem, with activity u * , interacts instantaneously with a cell of the jth functional subsystem, with activity u * ;

• A i j = A i j (u * , u * , u) : D u × D u × D u → R +
is the density function modeling the probability that cells of the ith functional subsystem, with activity u * , interacting with cells of the jth functional subsystem, with activity u * , reach the activity u. In particular A i j (u * , u * , u) satisfies the following identity:

D u A i j (u * , u * , u) du = 1, ∀ u * , u * ∈ D u .
In what follows, the transport term in (2.3) that models the introduction of the external force coupled to the Gaussian thermostat [17,18], will be denoted as

T F i [f](t, u) := ∂ u F i 1 -u D u u f (t, u) du f i (t, u) . (2.5)
In particular (2.5) is a damping operator adjusted to control the global activation energy:

E 2 [f](t) = n ∑ i=1 D u u 2 f i (t, u) du = D u u 2 f (t, u) du.
Remark 21 If the external force depends on the activity variable, namely F i = F i (u), then the thermostatted term (2.5) reads:

T F i [f](t, u) = ∂ u F i (u) -u D u u F i (u) f (t, u) du f i (t, u) .
(2.6)

The framework (2.3) can be further generalized by introducing the role of nonconservative processes. Specifically, interactions among the cells may generate proliferation/destruction of other cells (birth-death process). This type of interaction is modeled by the following operator:

N i [f](t, u) = f i (t, u) n ∑ j=1 D u η i j µ i j f j (t, u * ) du * , (2.7)
where µ i j denotes the net proliferative/destructive rate.

Cell mutations can occur because of DNA corruptions, thus generating a new functional subsystem. Accordingly, mutations are modeled by the following operator:

M i [f](t, u) = n ∑ h=1 n ∑ k=1 D u ×D u η hk ϕ i hk f h (t, u * ) f k (t, u * ) du * du * . (2.8)
where ϕ i hk denotes the net mutative rate into the ith functional subsystem, due to interactions that occur with rate η hk between the cells, with activity u * , of the hth functional subsystem and the cells, with activity u * , of the kth functional subsystem. Bearing all above in mind, the thermostatted kinetic framework with proliferative/destructive and mutative interactions reads:

∂ t f i (t, u) + T F i [f](t, u) = J i [f](t, u) + N i [f](t, u) + M i [f](t, u).
(2.9)

It is worth stressing that the parameters of the model can be function of the activity, namely η i j (u * , u * ), µ i j (u, u * ) and ϕ i hk (u * , u * , u). However in order to simplify our model these functions will be assumed as constants.

From the mathematical point of view, the Cauchy problem related to the general framework (2.9) has been analyzed and the existence and uniqueness of the solution has been proved, including the existence of stationary solutions [19]. Therefore we are allowed to develop the appropriate computational methods to obtain simulations of a specific model.

A thermostatted kinetic model for the treatment of keloid

This section is concerned with the derivation of a specific thermostatted kinetic model for the treatment of keloid by means of a therapy. According to the general framework (2.9), the first step is the definition of the functional subsystems. Following [8], we assume that keloid formation involves the following five interacting functional subsystems (see Table 1):

1. Fibroblast cells (Fc): The activity variable represents the proliferation ability and the distribution function is denoted by f 1 (t, u); 2. Virus (V): The activity variable is a magnitude of their aggressiveness related to their proliferation ability. The distribution function is denoted by f 2 (t, u); 3. Keloid-fibroblast cells (KFc): The activity variable represents the proliferation ability. The keloid-fibroblast cell is a fibroblast cell that, because of a mutation, has acquired a significant advantage with respect to the proliferation rate. The distribution function is denoted by f 3 (t, u); 4. Cancer cells (Cc): The activity variable is a magnitude of their progression ability and the distribution function is denoted by f 4 (t, u); 5. Immune system cells (ISc): The activity variable u is a magnitude of the activation and thus of the response to foreign agents. The distribution function is denoted by

f 5 (t, u).
In what follows the cells of the functional subsystems 2, 3, and 4 will be called non-self cells.

Bearing formula (2.1) in mind, the local density and the local activation of the ith functional subsystem, for i ∈ {1, 2, 3, 4, 5}, read respectively:

E 0 [ f i ](t) = ∞ 0 f i (t, u) du, E 1 [ f i ](t) = ∞ 0 u f i (t, u) du . (3.10)
The term A i j defined in (2.4) is assumed to be defined by a delta Dirac function (deterministic output d i j (u * , u * ) of a pair interaction) depending on the microscopic state of the interacting pairs. Specifically, let α be a positive parameter which models the heterogeneity rate of the KFc, and ε < 1 a scale parameter that is introduced to evaluate the difference among the heterogeneity rates of the V and the Cc with respect to the KFc, then A i j reads:

A i j (u * , u * , u) = δ(u -d i j (u * , u * )), (3.11)
where

d i j (u * , u * ) =      u * + εα if j = 1 and i = 2, u * + α if j = 2 and i = 3, u * + ε 2 α if j = 2 and i = 4, u *
otherwise.

(3.12)

FUNCTIONAL SUBSYSTEMS ACTIVITY DISTRIBUTION FUNCTION Fibroblasts cells (Fc) Proliferation f 1 (t, u) Virus (V) Aggressiveness f 2 (t, u) Keloid-fibroblasts cells (KFc) Proliferation f 3 (t, u) Cancer cells (Cc) Progression f 4 (t, u) Immune system cells (ISc) Activation f 5 (t, u)
Table 1. The Functional subsystems, the activity variable, and the distribution functions.

The assumption (3.12) means that the functional subsystems V, KFc, and Cc are the only subsystems subject to transitions into the activity variable, namely

J 1 [f](t, u) = J 5 [f](t, u) = 0. The terms J 2 [f](t, u), J 3 [f](t,
u), and J 4 [f](t, u) are derived under the assumption that the non-self cells have a tendency to increase their microscopic state. Bearing all above in mind we have

J i [f](t, u) =                  E 0 [ f 1 ](t) [ f 2 (t, u -ε α) -f 2 (t, u)] if i = 2, E 0 [ f 2 ](t) [ f 3 (t, u -α) -f 3 (t, u)] if i = 3, E 0 [ f 2 ](t)[ f 4 (t, u -ε 2 α) -f 4 (t, u)] if i = 4, 0 otherwise. (3.13)
Let β be the proliferation rate of the KFc, and β i the proliferation rate of the ISc. The modeling of proliferative events is based on the following assumption for the proliferation rate of cells:

µ + i j (u, u * ) =                  ε 2 β if j = 1 and i = 1, εβ if j ∈ {1, 5} and i = 2, β if j ∈ {1, 2} and i = 3, εβ if j = 2 and i = 4, β i if j ∈ {2, 4} and i = 5, ε 2 β i if j = 3 and i = 5, 0 otherwise. (3.14)
According to (3.14), the V proliferates because of encounters with the Fc and the ISc; the KFc proliferate because of encounters with the Fc and the V; the Cc proliferate because of encounters with the V; the ISc proliferate because of encounters with V, KFc, and Cc. It is worth stressing that according to (3.14), the proliferation rate of the KFc is greater than the proliferation rates of Fc, V, and Cc; the proliferation rate of the ISc, when encounter the V and the Cc, is greater than the proliferation rate of the ISc when encounter the KFc. The proliferation term thus writes:

P i [f](t, u) =                          ε 2 β f 1 (t, u) E 0 [ f 1 ](t) if i = 1, εβ f 2 (t, u) [E 0 [ f 1 ](t) + E 0 [ f 5 ](t)] if i = 2, β f 3 (t, u) [E 0 [ f 1 ](t) + E 0 [ f 2 ](t)] if i = 3, εβ f 4 (t, u) E 0 [ f 2 ](t) if i = 4, β i f 5 (t, u) [E 0 [ f 2 ](t) + ε 2 E 0 [ f 3 ](t) + E 0 [ f 4 ](t)] if i = 5.
(3.15)

Let δ i be the inhibition (destruction) rate of the ISc because of encounter with the V and the Cc, and δ the destruction rate of the V and the Cc because of the action of the ISc. In order to model the interactions that have as result the destruction of cells, we define the following destruction rate:

µ - i j (u, u * ) =              -εδ if j = 2 and i = 1, -δ if j = 5 and i ∈ {2, 4}, -ε 2 δ if j ∈ {2, 5} and i = 3, -ε 2 δ i u * if j = 3 and i = 5, -δ i u * if j ∈ {2, 4} and i = 5, 0 otherwise. (3.16)
According to (3.16), the Fc are destroyed because of interactions with the V; the V and the Cc are destroyed because of interactions with the ISc; the KFc are destroyed because of interactions with the V and the ISc; the ISc are destroyed because of interactions with V, KFc, and Cc; the ISc are able to eliminate the V and the Cc more efficiently than the KFc; the latter are destroyed by the V less efficiently than the Fc.

The assumption (3.16) takes into account that the non-self cells with a high level of activity have the ability to inhibit or destroy the ISc (immune suppression or immune-subversion). In particular we assume that the Cc and the V have a more efficiently action of inhibition on the ISc with respect to the KFc (inhibited immune cells do not play a relevant role in the competition and may be equivalently assumed as eliminated). Accordingly the destructive terms read:

D i [f](t, u) =                          -εδ f 1 (t, u) E 0 [ f 2 ](t) if i = 1, -δ f 2 (t, u) E 0 [ f 5 ](t) if i = 2, -ε 2 δ f 3 (t, u) [E 0 [ f 2 ](t) + E 0 [ f 5 ](t)] if i = 3, -δ f 4 (t, u) E 0 [ f 5 ](t) if i = 4, -δ i f 5 (t, u) [E 1 [ f 2 ](t) + ε 2 E 1 [ f 3 ](t) + E 1 [ f 4 ](t)] if i = 5.
(3.17)

Bearing all above in mind, we have

µ i j = µ + i j + µ - i j , N i [f] = P i [f] + D i [f].
Let γ be the mutation rate of a fibroblast cell, which measures the possibility that a Fc undergoes a mutation and becomes a keloid-fibroblast cell, and λ the mutation rate that measures the possibility that a keloid-fibroblast cell undergoes a mutation and becomes a cancer cell. The genetic mutations are modeled by defining the following mutative rate: Accordingly to (3.18), we assume that the Fc may mutate into KFc because of encounters with the Fc; the Fc may mutate into KFc because of encounter with the V; the KFc may mutate into Cc because of encounters with the V. In particular, we have assumed that is more likely that the Fc become KFc when they encounter the V. It is worth noting that (3.18) implies that the microscopic state of the cells does not change during the mutation. The mutation term thus writes:

ϕ i hk (u * , u * , u) =      εγ δ(u -u * ) if h = 1, k = 1, and i = 3, γ δ(u -u * ) if h = 1, k = 2,
M i [f](t, u) =          γ f 1 (t, u) [ε E 0 [ f 1 ](t) + E 0 [ f 2 ](t)] if i = 3, λ f 3 (t, u) E 0 [ f 2 ](t) if i = 4, 0 otherwise. (3.19)
Finally we assume that a constant external force field F i , for i = {2, 3, 4}, acts on the ith functional subsystem and mimics a specific therapy.

The thermostatted kinetic model, which consists in a system of evolution equations for each distribution function f i , for i ∈ {1, 2, 3, 4, 5}, thus reads:

                                                                                                 ∂ t f 1 = ε ε β ∞ 0 f 1 (t, u) du -δ ∞ 0 f 2 (t, u) du f 1 (t, u) , ∂ t f 2 + ∂ u F 2 1 -u D u u f (t, u) du f i (t, u) = εβ ∞ 0 [ f 1 (t, u) + f 5 (t, u)] du -δ ∞ 0 f 5 (t, u) du f 2 (t, u) -f 2 (t, u) ∞ 0 f 1 (t, u) du + f 2 (t, u -ε α) ∞ 0 f 1 (t, u) du , ∂ t f 3 + ∂ u F 3 1 -u D u u f (t, u) du f i (t, u) = β ∞ 0 f 1 (t, u) du -(1 -β + ε 2 δ) ∞ 0 f 2 (t, u) du f 3 (t, u) -ε 2 δ f 3 (t, u) ∞ 0 f 5 (t, u) du + f 3 (t, u -α) ∞ 0 f 2 (t, u) du +γ ε ∞ 0 f 1 (t, u) du + ∞ 0 f 2 (t, u) du f 1 (t, u) , ∂ t f 4 + ∂ u F 4 1 -u D u u f (t, u) du f i (t, u) = (εβ -1) ∞ 0 f 2 (t, u) du -δ ∞ 0 f 5 (t, u) du f 4 (t, u) +λ f 3 (t, u) ∞ 0 f 2 (t, u) du + f 4 (t, u -ε 2 α) ∞ 0 f 2 (t, u) du , ∂ t f 5 = β i ∞ 0 [ f 2 (t, u) + f 4 (t, u)] du + ε 2 ∞ 0 f 3 (t, u) du f 5 (t, u) -δ i ∞ 0 u [ f 2 (t, u) + f 4 (t, u)] du + ε 2 ∞ 0 u f 3 (t, u) du f 5 (t, u). (3.20)
The model (3.20) is characterized by 11 parameters which are positive constants (eventually equal to zero), small with respect to unity and having the following biological meaning:

• α is the heterogeneity rate of the KFc;

• β is the proliferation rate of the KFc; • β i is the proliferation rate of the ISc;

• δ is the destruction rate of the V and the Cc by the ISc; • δ i is the destruction rate of the ISc by the V and the Cc;

• γ is the mutation rate of the Fc into KFc; • λ is mutation rate of the KFc into Cc; • ε is the scale factor.

• F i , for i = {2, 3, 4}, is the external force that acts on the ith functional subsystem and mimics a therapy.

It is worth pointing out that the α-parameter refers to transition into the activity variable, the β-parameters refer to proliferative events, the-δ parameters refer to destructive interactions, the parameters γ and λ refer to mutations, and ε is a scale parameter, see Table 2. The all parameters have to be tuned by suitable experiments.

Computational analysis: Mimic therapeutic actions

This section is concerned with the computational analysis of the model (3.20) and specifically with the evolution of the different functional subsystems when an external action acts on the system as a therapeutic action. The main aim is to simulate the prompt response against the formation and evolution of keloid and the possible onset of cancer. The computational analysis is thus addressed to analyze the effects of three different therapeutic actions: An action which mimics a vaccine against the virus, an action which mimics a vaccine against cancer cells, and an action which can also mimic surgery on keloid. It is important to note that, according to our model, the onset of cancer cells is a consequence of mutations in the keloid cells because of the virus, the latter in part also responsible for mutations in the fibroblasts cells which generate the keloid (remember that according to our assumptions, the genetic susceptibility is also responsible for keloid formation). Therefore the main role of the external actions

Interactions Fc V KFc Cc ISc Fc Proliferative ε 2 β Destructive -εδ Mutative εγ γ V Conservative εα Proliferative εβ εβ Destructive -δ KFc Conservative α Proliferative β β Destructive -ε 2 δ -ε 2 δ Mutative λ Cc Conservative ε 2 α Proliferative εβ Destructive -δ ISc Proliferative β i ε 2 β i β i Destructive δ i -ε 2 δ i δ i
Table 2. The functional subsystems, the interaction terms, and the related parameters of the model (3.20).

should be to act on the functional subsystems of virus and keloid cells. However in order to have a more global view of what the model developed in the present paper is able to reproduce, we will consider also an external action on the cancer cells.

It is worth stressing that our model is an exploratory model. Thus, at this stage, we are only interested in the emerging phenomena that the model is able to reproduce and not in the tuning with an in-vivo or in-vitro experiment or empirical data, which can be considered as subject of further investigations. Accordingly, the computational analysis focuses on the model (3.20) when only one external action is applied. The dynamics of the model when the three actions can act at the same moment will be straightforward. However to think that the three different actions may be applied at the same time can not be suitable then we believe that our model will fit well if a combination of the three different actions will be performed at different times. Specifically the first step is to act against the virus (F 2 = 0, F 3 = F 4 = 0); when the virus is killed, the reached nonequilibrium stationary state will be the initial state for the model (3.20) with the action of a therapy for keloid (F 3 = 0, F 4 = 0); when keloid cells are eliminated, the new reached nonequilibrium stationary state will be the initial state for the model (3.20) with the therapeutic action against the cancer cells (F 4 = 0). In particular the intermediate step can be performed thanks to the introduction of the thermostat that allows the existence of a nonequilibrium stationary state.

The computational analysis will be addressed by fixing nine of the eleven parameters and performing a sensitivity analysis on the parameter α and on the therapeutical action F i , for i ∈ {2, 3, 4}. In the analysis we will show the evolution of the density, the behavior of the distribution function and the activation energy. The computational scheme is that of the well-known generalized collocation method where the variable u is discretized into a suitable set of collocation points. The integral terms are approximated by means of algebraic weighted sums in the nodal points of the discretization. The particularization of the evolution equations in each node and the enforcing of the initial conditions transform the model that is a system of integro-differential equations into a systems of ordinary differential equations, describing the evolution of the values of the distribution functions in the node of the collocation, see Section 2 of paper [9] for all the details. The choice of the distribution functions at time t = 0 is based on the assumption that, before the formation of keloid and the related onset of cancer, the virus infects the Fc. Our analysis starts when the number of Fc in the wound is equal to the number of V and a number of immune system cells have reached the wound. Accordingly we assume nonzero initial conditions only for the functional subsystem of Fc, V, and ISc.

The test case is based on the following choice of the parameters: γ = 0.4 (the rate of mutation of a fibroblast cell into a keloid-fibroblast cell is not negligible), δ = 0.3 (the ability of the immune system cells to inhibit the non-self cells is quite low), δ i = 0.5 (the non-self cells have an intermediate ability to inhibit the response of the immune system), β = 0.4 (the non-self cells have an intermediate ability to proliferate), β i = 0.35 (the rate of proliferation of the ISc is quite low), λ = 0.5 (the rate of mutation of a keloid-fibroblast cell into a cancer cell is not negligible) and ε = 0.5.

It is worth stressing that the computational analysis performed in the present paper does not cover the whole variety of conceivable dynamics but represents a useful test case.

Simulating the effects of a vaccine against the virus

This subsection deals with the computational analysis for the model (3.20) when a constant external force, mimic a vaccine against the virus, is introduced. Following the interaction rules proposed into the model, the onset of keloid is a consequence of the virus, then it is expected that if the vaccine is able to reduce the action of the virus then keloid formation and cancer can be prevented. Accordingly we have F 2 = 0, F 3 = F 4 = 0 and we let the magnitude of the parameter α vary from low to higher values, namely from low to higher heterogeneity. The results of the computational analysis are summarized as follows.

Analysis for low values of α. The effect of the external action is evident by comparing the dynamics depicted by the model (3.20) when F 2 = 0 with the dynamics when F 2 = 0.

The dynamics of the model (3.20) for F 2 = F 3 = F 4 = 0 has been widely debated in [8], therefore the new computational analysis will be focused on the case F 2 = 0 and F 3 = F 4 = 0, and more precisely F 2 ∈ {0, 0.0005, 0.01}. Looking at the left panel of Figure 1, for α = 0.2 (low rate of heterogeneity) the heterogeneity of the virus is bounded, then the main effect of the external force (vaccine for the virus) is to decrease the maximum of the density of V when the magnitude of the vaccine increase. Therefore, thanks also to the action of the immune system, we can see a faster depletion of the V and consequently of the KFc and the Cc. However, as the right panel of Figure 1 shows, the keloid cells start to increase again; this is a consequence of the genetic susceptibility that thus requires the definition of a therapeutical action for keloid. The global energy of the system even in the presence of the external action is preserved thanks to the thermostat. Analysis for intermediate values of α. Setting α = 0.5, the heterogeneity of the non-self cells becomes less negligible with respect the previous case. Therefore the role of the external force is now fundamental for avoiding keloid formation and onset of cancer. The computational analysis shows that high values of the vaccine are required in order to reduce the number density of keloid cells with high values of activity. Consequently the number density of the Cc is reduced. The vaccine for the virus has an important role in the proliferation of the immune system cells. Indeed as Figure 2 shows, in absence of a vaccine for the V the immune system is inhibited; the vaccine helps the immune system cells to proliferate again. Analysis for high values of α. The heterogeneity of the non-self cells in now very high thus keloid formation and cancer need to be inspected. As Figure 3 shows the vaccine is able to control the evolution of the virus and the depletion of the virus with high levels of aggressiveness depends on the magnitude of the vaccine. However high values of the vaccine are necessary to eliminate keloid and cancer, see Figure 4, then the definition of a therapeutic action against keloid formation and cancer is now fundamental.

Simulating the effects of a therapy against the keloid

This subsection deals with the computational analysis for the model (3.20) when a constant external force, mimic a therapeutical action against the keloid, is considered. Following [6], a therapy that induces senescence could be used to prevent the formation of a normal scar. Specifically the treatment proposed in [6] is based on photodynamic and PUVA therapy, which are capable to induce cell senescence. The therapy, based on a combination of a photosensitizer and a light source, produces oxidative stress and thus produce higher quantity of senescent cells with the minor apoptotic and necrotic effect. Bearing all above in mind, we assume that F 3 = 0, F 2 = F 4 = 0 and again we let the magnitude of the parameter α vary from lower to higher values. Specifically for low values of α, the therapy and the immune system are capable to perform a prompt action, see the left panel of Figure 5 where the density of the KFc is depicted for different values of the external force. Moreover, as the right panel of Figure 5 shows, the global activation energy is controlled by thermostated. The effects of the therapy are visualized in Figure 6, where the distribution function of the KFc is depicted in absence of therapy and in the presence of the therapy. As Figure 6 shows, the therapy allows the depletion of cells at different stages of mutation and a bounded magnitude of therapy is required in order to have the total depletion of keloid. Moreover, differently from the case of a vaccine for the virus, in this case the increase of new keloid cells, after the elimination of keloid, is not observed. For intermediate values of α, the general effect of the therapy is that of acting homogeneously on the KFc and the Cc and thus to obtain a distribution more uniform with respect to the degree of mutation. In particular the Fc are able to proliferate again. For high values of α, the rate of mutation allows the onset of keloid cells with high proliferation rate and of cancer cells with high values of progression. In particular even if the therapy allows the immune system to have a prompt response (see Figure 7) against the cancer cells (and virus), the immune system cells are not able to prevent tumor formation at tissue scale. Therefore a high magnitude of the therapy is required. This is the case where surgery is the only therapy that can work.

Simulating the effects of a vaccine against the cancer

This subsection deals with the computational analysis for the model (3.20) when a constant external force, mimic a therapeutical action against the cancer, acts on the system. In the case of cancer, the vaccine strictly depends on the type of tumor developed. Therefore the main aim of this subsection is to show that our model is able to consider the introduction of a vaccine and to simulate the evolution of the cancer. In particular we will focus on the case of high values of α.

According to our model, the development of cancer is the final result of the keloid formation, then the vaccine against the cancer has a fundamental action only on the Cc and the ISc. As Figure 8 shows, the vaccine is able to inhibit the formation of a tumor at the macroscopic scale. Moreover the immune system is able to proliferate again, see the right panel of Figure 9. Finally, as the right panel of Figure 9 shows, the action of the therapy allows to maintain bounded the global activation energy of system (thanks to the introduction of the thermostat).

Conclusions and research perspectives

The present paper has been devoted to test the capability of a new thermostatted kinetic framework for the active particles to model a therapy against keloid formation and the possible development of cancer. By employing different therapies on the different functional subsystems that compose the system, we have shown that some therapies proposed in the pertinent literature can be well described by our model. The analysis performed in this paper has been of computational kind and has been focused on the reaching of the emerging phenomena that are typical of keloid formation. The role of the immune system has been taken into account as a whole system without specifying who are the cells involved in the competition; this assumption is based on the fact that its action on keloid is limited by the genetic susceptibility that does not allow to the immune system to recognize these cells as foreign cells and thus acting for a possible total depletion.

The model developed and analyzed in the present paper is based on the introduction of an external force field, which mimics a therapy, at the macroscopic scale but in the absence of external interactions at the microscopic scale. The therapy acts directly on the number of foreign cells without performing an interaction at the microscopic level (in the activity variable). The modeling of external therapy at the microscopic (cellular) scale can be performed by representing the external therapy as a functional subsystem that have the ability to modify the state u of the cell by a particular action related to the variable ω ∈ D u . Assuming that the ith inner functional subsystem interacts with the rth external agent, for r ∈ {1, 2, . . . , m}, and denoting by g ir = g ir (t, ω) : [0, ∞) × D u → R + the related distribution function (known function of its arguments), the microscopic external actions are modeled by the following operator: where g i = (g i1 , . . . , g ir ) and

η e ir is the inner-outer encounter rate between the rth external agent, with state ω * , and the cell of the ith population, with state u * .

-B ir (u * , ω * , u) is the inner-outer transition probability density which describes the probability density that a cell of the ith population, with state u * , falls into the state u after an interaction with the rth external agent whose state is ω * .

The density B ir satisfies, for all r ∈ {1, 2, . . . , m} and i ∈ {1, 2, . . . , n}, the following condition: Bearing all above in mind, the thermostatted kinetic framework for open systems with proliferative/destructive and mutative interactions now reads: ∂ t f i (t, u) + T F i [f](t, u) = J i [f](t, u) + N i [f](t, u) + M i [f](t, u) + Q i [f, g i ](t, u). It is worth stressing that usually the interaction domain of the cell with state u * is not the whole domain D u but a subset Ω u * ⊆ D u , which contains the cells with activity u * ∈ Ω u * that are able to interact with the particles with activity u * . This is a phenomenon that is typical in tumor dynamics and specifically when the immune system is not capable to interact with all the tumor cells (tumor escape), see the review paper [20]. Specifically the immune response fails to completely eliminate the tumor, and the interact process results in the selection of tumor cell variants that are able to resist, avoid, or suppress the antitumor immune response, leading to the escape phase. Accordingly, a positive function ω(u * , u * ) can be introduced to weight the interactions among the cells; this function is assumed normalized with respect to integration over u * and it has a compact support in the domain of influence Ω u * ⊆ D u of the interactions. Moreover: (5.24)

The development of a model which includes a therapy against the keloid and that takes into account the above raised issues is object of future research directions. Moreover the possibility to perform an asymptotic analysis that allows the derivation of the dynamics at the tissue scale is a further research perspective. The asymptotic analysis can be performed by employing the methods developed in papers [21,22]. The final goal is the identification of the parameters with the aim to compare the emerging phenomena with the experimental data.

  and i = 3, λ δ(uu * ) if h = 3, k = 2, and i = 4

Fig. 1 .

 1 Fig. 1. The time evolution of the density of V (left panel) and of the density of KFc (right panel) for α = 0.2 and F 2 = 0 (solid line), F 2 = 0.0005 (dashed line), F 2 = 0.01 (dot line).

Fig. 2 .

 2 Fig. 2. The time evolution of the density of ISc for α = 0.5 and F 2 = 0 (solid line), F 2 = 0.0005 (dashed line).

Fig. 3 .

 3 Fig. 3. Distribution function of V for α = 0.8 and for F 2 = 0 (left panel) and for F 2 = 0.0005 (right panel).

Fig. 4 .

 4 Fig. 4. Distribution function of KFc for α = 0.8 and for F 2 = 0 (left panel) and for F 2 = 0.0005 (right panel).

Fig. 5 .

 5 Fig. 5. The time evolution of the density of KFc (left panel) and of the global activation energy of the system (right panel) for α = 0.2 and F 2 = 0 (solid line), F 2 = 0.0005 (dashed line), F 2 = 0.01 (dot line).

Fig. 6 .

 6 Fig. 6. Distribution function of the KFc for α = 0.2 and for F 2 = 0 (left panel) and for F 2 = 0.01 (right panel).

Fig. 7 .

 7 Fig. 7. The time evolution of the density of ISc for α = 0.8 and F 2 = 0 (solid line) and for F 2 = 0.001 (dashed line).

  Q i [f, g i ](t, u) = m ∑ r=1 D u ×D u η e ir B ir (u * , ω * , u) f i (t, u * ) g ir (t, ω * ) du * dω *f i (t, u) m ∑ r=1 D u η e ir g ir (t, ω * ) dω * ,(5.21)

B

  ir (u * , ω * , u) du = 1, ∀ u * , ω * ∈ D u .(5.22)

Fig. 8 .

 8 Fig. 8. Distribution function of the Cc for α = 0.8 and for F 2 = 0 (left panel) and for F 2 = 0.0005 (right panel).

Fig. 9 .

 9 Fig. 9. The time evolution of the density of ISc (left panel) and of the global activation energy (right panel) for α = 0.8 and F 2 = 0 (solid line), F 2 = 0.0005 (dashed line).

  * , u * ) du * = Ω u * ω(u * , u * ) du * = 1.


Shashwat Mishra
email: shashwat.mishra@imag.fr

Vincent Leroy
email: vincent.leroy@imag.fr

Sihem Amer-Yahia
email: sihem.amer-yahia@imag.fr

Discovering Characterizing Regions For Consumer Products

Consumer behaviour holds special importance in the retail industry. Consumer location impacts consumer behaviour by dictating purchase trends. This paper investigates the problem of examining product sales across a chain of stores to extract the geographic regions that characterize a product. Characterizing region for a product is a coherent geographic region where the consumers actively consume the said product. We introduce DICE, a diffusion-based technique to uncover all such regions for a given product, when they exist. In contrast to current state of the art, DICE involves minimal usage of parameters and shows remarkable tolerance to noise. We present experiments conducted on real datasets from a general commercial supermarket in France. Empirical evaluation and user-studies establish that the presented method significantly outperforms its natural baseline and previous state of the art approaches.

I. INTRODUCTION

Customer receipt analysis is a common task in the retail industry and has proven to be very effective for recommending products, implementing loyalty programmes for customer rewards and discounts, and optimizing stocks for retailers. Uncovering areas that mark active consumption for consumer products is key to understanding how to provision stores in different locations. In this paper, we adopt a product-centric analysis across multiple stores that contrasts sales of a given product in different stores to identify regional trends. In order to do that, we introduce the notion of characterizing region for a product, and address the challenge of discovering them. To the best of our knowledge, the presented body of work is the first to address such a problem.

Understanding consumer behaviour is of vital importance for marketers and consumer-goods companies. Much work has thus been devoted to analyze retail data from consumer-centric and store-centric viewpoints. While consumer-centric analyses are used to drive product discounts and recommendations, store-centric analyses focus on shelf space management and relative product placement in the store to increase revenue.

In this paper, we complement these approaches by introducing a product-centric analysis that combines in-store sales with geographic proximity to identify regional trends for a product. Such analysis is critical to product marketing. Popular shopping trends do indeed affect a customer's decision of which product to buy. In addition, geographic proximity plays an important role in a customer's decision to shop at one place or another. While some customers choose a store solely based on geographic proximity, others will travel the extra mile to find some products. The ability to combine trends with geo proximity can be very useful for cross-store management and provisioning.

Consider Figure 1a which shows the heatmap computed using the sales of a certain product 1 in various stores across France. At each store, the total units sold for the product is normalized by the total units sold for all products. The hot zones for the product can be immediately identified as the southern and western parts of France. These regions, where the consumers actively consume the product, characterize the said product. Our aim is to identify and isolate all such regions that exist for a product. Such characterizing regions are vital sources of information. Besides being a holistic cross-store view of a product's sales, they offer valuable information to the retailers and manufacturers. One application with great potential is the decision to expand or not a product's availability in some regions. Most importantly, a key application that is hardly enabled by other analyses, is crossstore advertising: a store may advertise that some products could be found in neighboring stores in order to balance its overall supply/demand chain. Characterizing regions could also be used by government instances that go beyond a single food supply chain to study the consumption of some products or product types and correlate that information with health indicators for different regions.

A key challenge in the identification of characterizing regions is to account for both, in-store sales of a product and geo proximity of stores. The basic tasks involved in uncovering characterizing regions are spotting areas deemed as hot zones, and identifying boundaries for the characterizing regions within the hot zones. While it is straightforward to spot such areas for the product in Figure 1a, obtaining them via manual inspection may not always be trivial. Even if one could spot the hot zones, realizing a boundary could be difficult as there may not be a clear contour demarcating sharp fall in product sales. Automated techniques that can examine large volume of products to uncover quality characterizing regions, when they exist, are thus highly lucrative.

A natural way to address the problem is to return the regions constituted by the k stores that exhibit highest (normalized) sales for the product. Figure 1b shows the result obtained using such (TopK) scheme. While the resulting regions are consistent with the heatmap (Figure 1a) they offer a fragmented view of the product's regional trends. Product sales across Figure 1a shows the heatmap for a product where darker color implies higher sales. Figures 1b and1c show the characterizing regions extracted using the natural baseline (TopK) and our method (DICE). The proposed method returns smoother regions while preserving "holes".

stores show high variance (noise) leading to several stray maximas or outliers i.e. stores that lie far from hot zones but have sales higher than some stores in the hot zone.

TopK erroneously selects such outliers over stores within the characterizing regions. Thus, it lacks the necessary smoothing required to obtain coherent regions.

Uncovering colloquial regions corresponding to entities from geo-tagged data has recently drawn attention of the research community [START_REF] Rattenbury | Towards automatic extraction of event and place semantics from flickr tags[END_REF] [2] [3] [START_REF] Intagorn | Learning boundaries of vague places from noisy annotations[END_REF]. The current state of the art is established in [START_REF] Thomee | Uncovering locally characterizing regions within geotagged data[END_REF] where the authors study tagged photographs to extract regions for geo-entities. The key idea is to represent the tag's frequency distribution across the map as an image and identify contours that demarcate sharp changes in the frequency. These contours are further subjected to a series of complex image processing operations to obtain the characterizing region. Region discovery for retail products is not amenable to such a solution as it assumes the existence of a sharp boundary (in terms of sales) between the characterizing region and its surroundings. As we see later in Section IV, this assumption seldom holds true for retail products. This analysis suggests that both, the natural baseline and the current state of the art for related problems, are insufficient for uncovering characterizing regions for retail products. There is a clear need for a solution that is robust to the noise in the data and effectuates the necessary smoothing required to obtain coherent regions.

To address the insufficiencies of existing methods, we propose a new approach, coined Diffusion based Iterative Characterizing region Exploration (DICE), that combines store sales and geographic proximity to find a set of coherent regions for a given product. Intuitively, the proposed technique starts with a seed set of stores that exhibit high sales for the product and iterates over neighboring stores to find the largest contiguous regions covering all high-sales stores or stores that are contiguous to high density ones. DICE models the diffusion of product endorsements to neighboring stores that delivers enhanced smoothing. As demonstrated in Figures 1b and1c, the obtained results are significantly better than the natural baseline. In Section IV, we present our experiments that conclusively establish the significant gains achieved over the natural baseline and the current state of the art.

To summarize our findings, the performance studies con-ducted on real datasets from retail domain suggest that a large number of consumer products have associated characterizing regions. Further, the proposed technique offers high quality results when compared to the natural baseline and the previous state of the art. We validate our results via both, empirical comparisons that report > 25% gain over previous methods and user studies that establish that majority of the users prefer DICE over competing methods and show wide agreement (> 70%) with characterizing regions returned by DICE. This paper makes the following contributions:

1) We define the problem of finding characterizing regions of a product and argue for its significance. 2) We design and implement a novel diffusion-based algorithm that, given a product and its sales in different stores, finds characterizing regions of that product. 3) We run performance experiments that include empirical evaluation as well as a user study with novice and expert users that validate our findings on real datasets from a large French supermarket. We compare ourselves to the natural baseline (TopK) and the previous state of the art (SSR). Our experiments establish that DICE significantly outperforms both.

Our paper is organized as follows. In Section II, we formalize our data model and state the problem of identifying the set of characterizing regions of a product. Section III describes the design process for DICE. Section IV is dedicated to our experiments and findings. The related work is summarized in Section V. We conclude in Section VI.

II. DATA MODEL AND PROBLEM

A. Model

We have a set of products, P = {p 1 , p 2 ...}, which represent distinct items available in a typical departmental store. Examples of products are tuna sauce, red wine, detergent, and cat food. For confidentiality purposes, we do not provide specific product names and brands in this paper.

We also have a set of stores, S = {s 1 , s 2 ...}, where each store s has a pair of geo-coordinates (s.lat, s.lon) using which we define the distance between two stores s i and s j as below g(s i , s j ) = (s i .lat -s j .lat) 2 + (s i .lon -s j .lon) 2 (1)

We define a bijective mapping from the set of stores to a set of cells. The cell for each store is simply a polygon enclosing that store, with the property that the closest store from any point within the cell is the enclosed store. Such a set of cells is easily obtained by computing the Voronoi map over store locations. The resulting Voronoi polygons form the cells. We show an example in Figure 2.

We say that two stores are connected if corresponding Voronoi polygons are adjacent. For a given set of stores, the region constituted by those stores may range from being completely fragmented (when no two stores are connected) to completely unified (when all stores are contiguous). We call the latter a coherent region.

A transaction is a purchase of some (non-zero) units of certain products at a store. We assume the existence of a set of transactions, T = {t 1 , t 2 ...}, such that each transaction t ∈ T is a 2-tuple indicating the store at which the transaction occurred and the set of products purchased in the transaction. For example, the transaction t = s k , {p l , p m } indicates that products p l and p m were purchased at store s k .

Using T s to denote the set of transactions occurring at store s, the total units of product p sold at s is computed as:

sales(p, s) = t∈Ts f (p, t) (2)
Here, f is a function that gives the units of p sold in transaction t. We assume f (p, t) = 0 if p was not sold in transaction t.

We can now define the density of a product p at a store s, d p (s), as follows:

d p (s) = sales(p, s) p ∈P sales(p , s) (3)
We compute the density of a product p at all the stores in S. This is referred to as the density distribution, D p , of the product.

D p = {d p (s 1 ), d p (s 2 )...d p (s |S| )} (4)

B. Problem

Our aim is to identify, given a product p, a set of stores, S p ⊆ S, such that the stores in S p have high product density and result in coherent regions. Note that a product may have more than one coherent region, as shown in Figure 1c. We refer to each such coherent region as a characterizing region and S p as the set of stores that constitute the characterizing regions.

A product may not have any meaningful characterizing region. For a product p, the set of its characterizing regions, S p is well defined if the product exhibits localized overconsumption. If the consumption is relatively uniform across all stores, or the stores with high density are scattered, the product may not be coupled to a geographic region and S p is thus undefined. That is, for example, the case for generic products that are consumed uniformly across all stores, such as chewing gum, toilet paper etc.

The technique we describe in this paper assumes the existence of characterizing regions for the product. Subsequently, we show how our solution offers out-of-the-box support for segregation of products that do have characterizing regions from those that do not, in Section IV.

III. METHODOLOGY

Our objective is to analyze the density distribution in order to identify characterizing regions for the product. We focus on geographically coherent regions, defined in Section II as a large connected component in the associated Voronoi map. Intuitively, a coherent region should initially contain several neighboring stores exhibiting a high density for the product, as well as other stores with relatively high densities. Conversely, isolated high-density stores should not cause the emergence of a region. Thus, our aim is to design a smoothing process that can expand dense regions. The main challenge in this context is to balance the smoothing, that allows DICE to include some regions, while retaining characteristics of the original data distribution, as some "holes" within a region represent valuable marketing information.

As stated previously, the natural approach of selecting top k stores works well for products with sharply defined characterizing regions but it is not a viable solution for majority of products. Further, it enforces that all products have the same number of stores (k) in the final result, which may not reflect true behaviour.

We observe that the popularity of a product at a store is an endorsement of the product by that store. This endorsement may affect the product's popularity in neighboring stores as stores may influence other stores. Modeling this diffusion gives us a more refined method to achieve the desired smoothing.

Diffusion over graphs has been well studied in different version of the influence maximization problem first introduced by Kempe et al. [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. The problem involves a scenario where a user initially activates certain nodes in a social graph. The activated nodes may then influence neighboring nodes and activate them. The process stops when no nodes can be activated. The objective is, given k, to find the set of k initial nodes to activate to cause maximum spread. To solve the problem, one has to assume an Influence Propagation (IP) model. The IP model specifies how the influence spreads from an active node to an inactive node. Relevant literature offers a choice between two popular IP models, Independent Cascade (IC) and Linear Threshold (LT), which characterize different types of social interactions [START_REF] Chen | Scalable influence maximization in social networks under the linear threshold model[END_REF] [START_REF] Zhou | UBLF: an upper bound based approach to discover influential nodes in social networks[END_REF]. While the IC model assumes independent interaction for each pair of nodes, the LT model uses threshold-based behaviour.

[8] [9] [10] [11]
To design DICE, we select the linear threshold model of influence propagation as it allows us to incorporate influence from multiple influencing agents simultaneously and does not require to assume independence between stores. We now give an overview of our algorithm and discuss its two core processes.

Overview of the algorithm: For a given product p, DICE starts with its (normalized) density distribution D p . It selects an initial seed-set of stores (Figure 3b). These stores are marked as active and the remaining stores as inactive. The activated stores cast an influence on inactive ones which in turn may become active. The algorithm iterates and stops when no inactive stores can be activated (Figure 3c). DICE then extracts coherent regions as significant connected components covering the activated stores (Figure 3d). We now go over the salient features of IP models and discuss them in the context of our problem in the following subsections.

A. Seeds selection

In influence propagation, the seeds represent the initial active entities, i.e. the entities that initiate the spread of a trend. Traditionally, influence propagation models are used to compute, given a social network, an optimal seed-set to maximize overall influence spread [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF]. DICE however aims at analyzing past sales records. Hence, the seed-set of a product p is selected from the input data using D p . Seed selection plays a crucial role in influence propagation. The primary goal of seeds is to bootstrap the process of diffusion. DICE selects the seeds as the stores that, given D p , appear to be the most characterizing. Seeds(p, m) is defined as the set of m stores having the highest density for product p.

Seeds(p, m) ⊆ S, |Seeds(p, m)| = m d p (s) ≥ d p (s ) ∀ s ∈ Seeds(p, m), s ∈ S \ Seeds(p, m) (5)
Our aim is to design an algorithm that uses as few parameters as possible. The formalization of seed-set selection introduces m, number of desired seeds, as the only parameter admitted by DICE. In practice, the method is robust and only needs to select a sufficient amount of seeds to ensure that each characterizing region of p contains a few of the selected seeds. As long as this condition is met, influence spreads in those regions resulting in their discovery by the method. Minor increase in number of seeds will not have any negative impact. However, selecting too many seeds tends to add noise to DICE's results as the resulting large number of outliers may unify into erroneous regions. Our experiments in Section IV reveal that stable results are returned by DICE for large range of values for m.

B. Influence spread

Influence propagation recursively marks entities as active as they get influenced by a trend. Active entities spread influence to other entities, which in turn may become active. Propagation ends when no new entity gets activated. In our model, given a product p, the set of active stores S p is initiated to Seeds(p, m). A non-seed store s becomes active and is added to S p if the influence it receives exceeds its threshold θ p s . We define the value of this threshold as the difference between the product density at s and the lowest density of a seed.

θ p s = min x∈Seeds(p,m) d p (x) -d p (s) (6)
DICE assigns a weight factor w s to each store s. Since larger stores attract more customers, and therefore constitute a stronger source of evidence for finding characterizing regions, w s scales with the total number of sales at s, with a logarithmic damping factor.

w s = log 1 +
(e -1) p∈P sales(p, s)

max x∈S p∈P sales(p, x ) (7)
We ran experiments to evaluate different choices for weighting the stores (including w s = 1). Logarithmic formulation, as in Eq. 7, delivered best results. Intuitively, Eq. 7 assigns higher weights to stores with greater sales thus allowing them to spread more influence. The store with greatest number of sales is assigned a weight of 1.

Here, e is the base of the natural logarithm. Upon its activation, a store s i influences another store s j by b sj ,si , which contributes to the possibility of recursively adding more stores to S p , and is defined as follows:

b sj ,si = w si .d p (s i ) g(s i , s j ) 2 x∈S, x =si 1 g(si,x) 2 (8)
At any point of the recursive influence spread, the total influence received by a store s is evaluated as x∈Sp b s,x , and is compared against the threshold θ p s . DICE focuses on regional trends, so we postulate that a store with high density (for a product p) influences other stores and attempts to induce the popularity of the product in its neighboring stores. We set the influence of a store on another to decay with the squared distance between these stores. As Figure 2a shows, stores are far from being evenly geographically distributed: populated regions, around major 3b). The activated stores influence other stores potentially activating them (Figure 3c). The region constituted by the set of activated stores is reported as the characterizing region (Figure 3d).

cities, have a lot of stores, while rural areas are much sparser.

The distance between neighboring stores varies significantly, which is amplified by the squared factor in the decay model. This requires DICE to be able to take into account the surroundings of a store s to parameterize b * ,s , such that isolated stores may still spread significant amount of influence, while limiting it on populated regions to avoid snowball effects. The chosen formulation (Eq. 8) essentially amounts to selecting the total influence that a store can spread (w si .d p (s i )), and distributing it among stores according to their distance. This ensures that the maximum influence cast by any store s i over any store s j is bounded to w si .d p (s i ).

We remark that DICE does not completely eradicate the fragmentation issue incurred by the natural baseline (TopK). S p may still contain stores that are clear outliers, i.e. lie far from hot zones. However, the fraction of such stores is very small as compared to the stores that form connected components. As we see in Section IV, DICE significantly mitigates the fragmentation incurred by the natural baseline (TopK).

C. Detailed algorithm

The details of DICE are presented in Algorithm 1. S p is first initialized to be the set of seeds following the approach described in Section III-A, line 1. I is the set of remaining stores that can potentially become active through propagation (line 2). The accumulated influence for each store in I is initialized to 0 (line 4). DICE then loops until I converges by monitoring A prev , the set of stores activated in the last iteration (line 7). DICE updates the influence accumulated by inactive stores of I with the influence they receive from stores activated in last iteration A prev (line 11) following the update rules defined in Section III-B. If the influence received by an inactive store exceeds its threshold (line 13) it becomes active and is added to A cur , the set of stores activated in this current iteration. I, A prev and S p are then updated at the end of each influence propagation iteration. After convergence, DICE returns the set of active stores S p (line 21).

IV. RESULTS

In this section we report on experiments conducted to study the behaviour of DICE and evaluate its performance in comparison to its competitors. All experiments are conducted on a real dataset that consists of retail transaction logs. Our s.inf← 0 5: end for 6: A prev ← S p 7: while A prev = ∅ do 8:

A cur ← ∅ 9:
for all s ∈ I do A prev ← A cur 19:

S p ← S p ∪ A prev 20: end while 21: return S p performance experiments conclusively establish that DICE significantly outperforms both TopK and SSR. According to our user studies, the percentage of users that prefer DICE, TopK and SSR are 47%, 21% and 32% respectively. The percentage gains achieved by DICE over TopK and SSR, using empirical measure, are 25% and 99% respectively.

In the following headings, we first describe the dataset, (Section IV-A) and the test-bench (Section IV-B). This is followed by a succinct description of the competing methods, natural baseline (TopK) and the previous state of the art (SSR) in Section IV-C. Subsequently we discuss the various experiments in Section IV-D.

A. Data

We conduct our experiments over logs generated at a popular general commercial supermarket that owns over 1800 outlets in mainland France. The logs were gathered over a period of 27 months from May 1 st , 2012 to July 31 st , 2014. The data consists of a large number of csv files in different domains such as stores, transactions, products, customers. Each line in stores corresponds to a store and consists of a unique store id and associated meta data (number of employees, geocoordinates, opening date etc). In similar fashion, each line in transactions and products contains an entry for unit transaction and unit product respectively. The data model described in Section II fits perfectly to the dataset.

We subjected the raw data to some cleaning operations which we describe next.

1) Stores:

Stores with fewer transactions may result in superfluous densities. We filter the set of stores to retain those that have a sufficient number of transactions (|T s | ≥ 20k). This results in more than 1400 stores. In Figure 2, we show the location of obtained stores on the map of France and the associated Voronoi map.

2) Products: Availability of certain products was limited to a select few stores. Our focus in this study is on products that are widely available yet over-consumed in certain locales. We filter the original set of products to retain only those that are available in at least η fraction of the stores (coverage). In Figure 4, we show the number of products for different values of η. We fix η to 0.9 obtaining roughly 10k products. We remark at this point that not all products are expected to have characterizing regions.

Table I reports the final cardinalities of the sets. We examine over 1 billion transactions across more than 1400 stores for more than 10k products.

B. Platform

All experiments were run on a Intel E5-2650L machine with 32 GB RAM, running CentOS 6.5. Each algorithm was implemented in Java and executed on JRE 1.6.0 31.

C. Competitors

In the following, We introduce (in succinct detail) the natural baseline (TopK) and the current state of the art approach (SSR) to uncover characterizing regions for products.

1) TopK: TopK is the trivial algorithm for discovering characterizing regions given product's density distribution. It returns the stores with the k highest densities as the result S p . The choice of k is non-trivial. By design, the baseline lays no emphasis on obtaining stores that result in coherent regions. If the characterizing regions are sharply defined i.e. the density of stores that comprise the regions is greater than all stores outside the regions, TopK delivers considerably good results given suitable k. However, for most products the assumption does not hold good and resulting characterizing regions are fragmented and erroneous.

Note that TopK shares certain aspects with DICE as the latter uses the former for seed-selection which bootstraps the diffusion process.

2) Scale Space Representation (SSR): SSR [START_REF] Thomee | Uncovering locally characterizing regions within geotagged data[END_REF] establishes the current state of the art for characterizing region discovery. The authors attempt to uncover colloquial region boundaries for location related Flickr tags (such as "germany", "poland") from a large volume of tagged photographs.

The method starts by gridding the map and computing frequency of the given tag in each cell of the grid. This gives a representation of the tag's frequency-distribution over the map in the form of an image. In the next step, the method identifies all contours in the image across which the frequency changes abruptly. This is done by subtracting a high blurred version of the image from a low blurred version of the image. This step results in a binary map that highlights the aforementioned contours.

In the final step, the binary map is subjected to contour filling and morphological closing to arrive at closed connected blobs. Of potentially several resultant blobs, the authors retain the largest. The grid cells that constitute this blob are returned as the characterizing region. The approach thus returns a single characterizing region per product which may not be true for all products.

Note that the data model above closely resembles the one described in Section II. A photograph corresponds to a transaction and the tags within the photograph correspond to products. The two models differ slightly in that photographs can manifest at any arbitrary point on the map, our transactions are constrained to occur at fixed locations (stores).

Note further that the instantiation of smoothing and imageprocessing operations in SSR requires specification of multiple parameters. The approach is thus, highly parametric (requiring specification of up to 4 parameters). The quality of obtained regions is sensitive to the choice of parameters, thus it is required that the system be trained on some supplied ground truth to obtain optimal parameter combination.

Experiments

We conduct three different types of experiments. The first experiment studies the impact of parameter m, the seedsize for DICE. The remaining two experiments compare the performances of DICE, TopK, and SSR using an empirical approach and a user-study respectively.

1) Impact of seed-size parameter: DICE admits a single parameter, the initial seed-set size, m. |S p | is guaranteed to increase with increase in m because a higher value of m implies a bigger seed-set (which by design is part of the result). Furthermore, a bigger seed-set implies more influence on other stores. We study the impact of seed-set size on the characterizing regions for different products.

We observe that for the same choice of m, certain products diffuse more than others. We illustrate this with the help of an example. Figure 5 shows the characterizing regions, obtained via DICE, for two different products for two different values of m. While the first product is a peculiar meat, the other is a common chewing gum. We mark the stores that form the initial seed-set in red. Note that the stores that comprise the seed-set are more localized for peculiar meat than for chewing gum. In Table II we show the size of the final characterizing regions which establishes that the former product grows significantly more than the latter. This behaviour allows us to segregate products that have meaningful characterizing regions from those that do not by using a suitable threshold. For the following discourse, where we evaluate the quality of extracted regions, we restrict ourselves to products that resulted in meaningful characterizing regions.

2) Empirical Comparison: Evaluation of the quality of returned characterizing regions is difficult in the absence of ground truth. Note that relevant previous studies [START_REF] Intagorn | Learning boundaries of vague places from noisy annotations[END_REF] [5] have exclusively dealt with extraction of colloquial boundaries for geographic concepts (such as "germany"). It can be easily assumed that the region extracted for tag "germany" overlap with the international boundaries of the eponymous country. However, for consumer products, it is difficult to affirm with certainty that the over-consumption of a certain product will be restricted to certain locales.

Ground Truth: We attempt to fill the void created by the absence of ground-truth with the aid of domain experts. We construct a sequence of 2-tuples consisting of products and French administrative regions i.e. every tuple is of the form p, r such that p ∈ P, r is an administrative region of France, and p is expected to be characterizing in r. Experts were instructed to report only such pairs which could be argued for with high confidence. On account of absence of data, the administrative region of Corsica was omitted. For each of the remaining 21 administrative regions of France, we obtained 2 products that were expected to be characterizing in the said region. This resulted in 42 ground truth pairs of the form p, r . Evaluation Scheme: Each of DICE, TopK, and SSR is evaluated against each pair p, r of ground truth by measuring the Jaccard overlap (in terms of area) between the characterizing regions for p (as returned by the method) and the French administrative region r. Subsequently, the Jaccard overlap is averaged over all the pairs. This measure, Average Jaccard Overlap, serves as the primary empirical measure for comparing the three methods where higher values are desirable. We also compute and report the maximum and minimum over all ground truth pairs.

Parameterization: A key benefit of DICE is its dependence on a single parameter , the initial seed-set size m. We observed that for most products, there exists a range of m for which the resulting regions are nearly the same. This is expected because if the supplied value of m is less than, but close to, the optimal value, the iterative diffusion-based design of DICE would ensure the discovery of all relevant stores Thus, we keep the seed-set size fixed at 100 for all subsequent experiments as it returned quality results for a wide choice of products.

TopK suffers from a serious limitation when compared to SSR and DICE. The characterizing regions for all products contain the same number of stores, k, as fixing k fixes the number of stores in the returned regions. Single value of k may not be suitable for several products simultaneously. Thus, choosing k becomes a non-trivial task. DICE does not suffer from this limitation because though the seed-set size m is same for all products, the number of stores added during diffusion phase are different and product specific.

We give TopK the flexibility to return characterizing regions of different sizes in the following manner. For each product, TopK uses the number of stores in the characterizing regions returned by DICE as the value of k i.e. k = |S DICE p |. Thus, TopK is executed with different values of k for different products. One could argue that this gives undue advantage to TopK by giving it a calculated guess for k. As we show later, despite the added advantage, TopK is outperformed by DICE.

In contrast, SSR requires proper specification of up to four parameters. Of the four parameters listed in Table III, two, Gaussian kernel size and , are fixed to values specified in [START_REF] Thomee | Uncovering locally characterizing regions within geotagged data[END_REF]. Optimal values for the remaining two parameters are obtained using Recursive Random Search [START_REF] Ye | A recursive random search algorithm for large-scale network parameter configuration[END_REF]. We explore the range [0, 1000] for σ and [0, 200] for ρ. In Figure 6, we show the variation in Jaccard overlap achieved by SSR (averaged over all ground-truth pairs) w.r.t its parameters, in vicinity of the optimal values. The optimal aggregate Jaccard overlap achieved by SSR is 0.18. We list the complete set of parameters for all three algorithms with their default values in Table III. We abstain from listing the default value for TopK since instantiating DICE instantiates TopK. Result: Finally, we present the results of the empirical comparison in Figure 7. DICE achieves significantly higher values for Average Jaccard overlap (Average) than TopK and SSR. The relative gains are 19.6% and 99.02%. DICE also achieves the highest value for maximum Jaccard overlap (Max) and minimum Jaccard overlap (Min) against a ground truth pair.

For deeper examination, we explicitly present and discuss the characterizing regions returned for two anonymized products, detergent and canned tomatoes, in Figures 8 and9.

In Figure 8, results from TopK, while consistent with the corresponding heatmap, Figure 8d, are expectedly more fragmented. Whereas SSR (Figure 8c) picks up high frequency variation within the characterizing region and results in a characterizing region that is too small. As compared to the product in Figure 8, the product in Figure 9 has less sharply defined characterizing region (i.e. the product density in the characterizing region is not significantly higher than elsewhere). This can be ascertained from their respective heatmaps (Figure 8d and Figure 9d). This increased noise has adverse effects on both TopK and SSR. For TopK, Figure 9b, the severity of incurred fragmentation increases substantially. SSR, Figure 9c, results in too large a region, covering almost all of France. This occurs as the high frequency variations give rise to several contours (demarcating sharp fall in product density) in the binary map of SSR which unify into a single connected component post morphological closing.

Note that DICE is also adversely affected by increased noise as established by Figures 9a and8a. However, the mitigating effect of diffusion based design counters the introduced noise enabling DICE to deliver best results for both products. The Jaccard overlap as achieved by DICE, TopK, and SSR are 0.41, 0.31, 0.02 for detergent and 0.36, 0.17, 0.04 for canned tomatoes.

We also remark that SSR was observed to consistently result in regions that are either too small or too large. Figures 8c and9c present the two different extremes achieved by SSR.

To substantiate our argument about fragmentation, we quantified the incurred fragmentation (as the average pairwise distance between stores in S p ) and measured it for different choices of seed-set size (m). We show the obtained result in Figure 10. TopK incurs more fragmentation than DICE. In either case, the fragmentation increases with an increase in seed-set size, since larger seed-sets result in larger characterizing regions that are spread over a large area. At higher values of m, both methods return similar regions and thus incur the same amount of fragmentation. 3) User Study: We conduct two different types of user studies. The first study (comparative evaluation) is done to perform a human assessment of the quality of the results obtained by the three methods. The second study (independent evaluation) captures the utility of results returned by DICE.

Each study was independently conducted on two different groups of respondents, Users and Experts. The group Users consisted of 30 novice users from France, while the group Experts consisted of 8 experts from the retail domain who had in-depth knowledge of products and their consumption trends. We now present the two user studies.

Comparative Evaluation: The first study provides a comparison of the goodness of the three methods as perceived by average users and marketing experts. We generated, randomly, a list of 20 products and the corresponding characterizing regions returned by all three methods. For each product, the respondents were tasked with indicating which of the three methods best reported the characterizing regions for the product. The maps were anonymized and randomly permuted to eliminate any bias. In Table IV we present the votes received by each method averaged over all user and products. DICE establishes itself as a clear favorite among both Users and Experts. While SSR outperforms TopK for Users, the opposite holds for Experts. We attribute this observation to the fact that Users exhibit an aesthetic bias that influences them to vote for the method that returns aesthetically pleasant characterizing regions. SSR, due to its region selection step, returns a single contiguous region for each product which is appealing to Users. Experts, being more informed about the products' consumption trends, do no suffer from this bias.

Independent Evaluation: The aim of the second user study is to perform an independent evaluation of DICE. For each product in a randomly generated list of 10, users were asked to express their level of agreement that the shown map (obtained via DICE) correctly highlights all parts of France expected to be characterizing for the product. Agreement levels were expressed on a scale of 1 to 5. Results were averaged over all users and all products.

We present the final figures in Table V. More than 70% of Users and 80% of Experts show some level of agreement with the results. Experts show greater agreement than Users which may again be attributed to the aesthetics bias as Experts are more likely to show agreement with non-trivial shaped regions.

V. RELATED WORK

Uncovering characterizing regions for geotagged data is a relatively new problem. Most works have dealt with examining different types of user generated content, such as photographs, blogs, tweets etc. to unearth colloquial boundaries [START_REF] Rattenbury | Towards automatic extraction of event and place semantics from flickr tags[END_REF] [2] [3] [4] [START_REF] Thomee | Uncovering locally characterizing regions within geotagged data[END_REF].

The influence maximization problem [START_REF] Kempe | Maximizing the spread of influence through a social network[END_REF] [7] [8] [14], due to its generic formulation, has been widely studied under various use-cases. Consequently, the IC and LT influence propagation models have been successfully applied to address viral marketing [START_REF] Aslay | Viral marketing meets social advertising: Ad allocation with minimum regret[END_REF] [10], recommendation systems [START_REF] Ye | Exploring social influence for recommendation: a generative model approach[END_REF], identification of influential peers in social graph [START_REF] Zhou | UBLF: an upper bound based approach to discover influential nodes in social networks[END_REF] [15] etc. However, no previous work has investigated the applicability of such propagation models towards discovering prominent regions for retail products.

To the best of our knowledge, the presented body of work is the first to address the problem of uncovering characterizing regions for consumer products. The application of influence propagation models to uncover such regions has not been investigated in past works.

VI. CONCLUSION AND FUTURE WORK

In this paper, we present DICE, an algorithm that extracts characterizing regions for a given consumer product using retail transaction logs. We investigate the application of decay-based, weighted, influence propagation models to achieve enhanced smoothing. Intuitively, DICE starts from a seed set of stores with high sales for a product and iterates over neighboring stores to find the largest contiguous regions covering all high-sales stores or stores that are contiguous to high density ones. As a consequence of this study we have developed a tool that can be used to extract and analyze colloquial characterizing regions for consumer products.

Our empirical evaluation and user studies on customer receipts of a large food supply chain in France show that DICE discovers meaningful, coherent, product-centric characterizing regions and outperforms its competitors. This opens several avenues for future work. In particular, we seek to explore the following two directions. We intend to combine pattern mining techniques with characterizing regions to extend our focus from single product to set of products. We also seek to correlate found regions with health indicators for different regions to gather information on customers' wellbeing. Both directions open challenging computational questions that necessitate the development of scalable algorithms to enable the production of such analytics efficiently.

Fig. 1 .

 1 Fig.1. Inadequacy of TopK for uncovering characterizing regions. Figure1ashows the heatmap for a product where darker color implies higher sales. Figures1b and 1cshow the characterizing regions extracted using the natural baseline (TopK) and our method (DICE). The proposed method returns smoother regions while preserving "holes".

2 .

 2 Distribution of stores across France. Fig 2a shows the exact location of all stores while Figure 2b shows the computed Voronoi cells. Each cell corresponds to a single store. Color of each cell is governed by the total number of transactions that occur at the store. Darker color implies more transactions.

Fig. 3 .

 3 Fig. 3. Illustration of different stages in DICE. Normalized density distribution (represented as heatmap in Figure 3a) forms the starting point. Selection of the desired size for the seed-set (m) results in selection of top-m stores as seeds (Figure3b). The activated stores influence other stores potentially activating them (Figure3c). The region constituted by the set of activated stores is reported as the characterizing region (Figure3d).

Algorithm 1

 1 Diffusion based Iterative Characterizing region Exploration (DICE) Input: p: product Input: m: seed-set parameter Output: S p : Set of stores that form the characterizing regions 1: S p ← Seeds(p, m) 2: I ← S \ S p 3: for all s ∈ I do 4:

Fig. 4 .

 4 Fig. 4. Distribution of products w.r.t. fraction of stores covered (η). The X-axis shows the different coverage values. The Y-axis shows the number of products with coverage ≥ specified value.

Fig. 5 .

 5 Fig. 5. Impact of seed-size (m) on two different products, peculiar meat and chewing gum. Initial seed-set stores are indicated in red while those added during diffusion in grey. The seed-set stores are more localized for product peculiar meat than for chewing gum. Consequently the former encounters more growth (larger number of grey stores) at both values of m during the diffusion step.

Fig. 6 .

 6 Fig.6. Parameter space exploration for SSR[START_REF] Thomee | Uncovering locally characterizing regions within geotagged data[END_REF]. Optimal Average Jaccard Overlap is obtained at Kernel Width=74 and Morphological Radius=16.

Fig. 7 .

 7 Fig. 7. Performance evaluation of TopK, SSR and DICE against the 42 ground truth pairs. Seed-set size for DICE (m) was fixed at 100. For TopK, k = |S DICE p

Fig. 8 .

 8 Fig.8. Illustration of characterizing regions obtained for the ground-truth pair (detergent, Languedoc-Roussillon). While TopK (Figure8b) significantly overlaps with Languedoc-Roussillon, it includes several distant stores. SSR (Figure8c) picks up the sharp density variation within the characterizing region and results in a region that is too small. DICE delivers the best result (Figure8a). Jaccard overlap with the boundaries of Languedoc-Roussillon region of France are 0.41 (DICE), 0.31 (TopK) and 0.02 (SSR).

Fig. 9 .

 9 Fig.9. Illustration of characterizing regions obtained for the ground truth pair (canned tomatoes, Picardie). TopK (Figure9b) results in high fragmentation and SSR (Figure9c) results in a very large region. Jaccard overlap achieved by the 3 approaches with the French administrative region of Picardie are 0.36 (DICE), 0.17 (TopK) and 0.04 (SSR).

Fig. 10 .

 10 Fig. 10. Mitigation of fragmentation by DICE. Y-axis shows the average pairwise distance between stores in Sp. X-axis shows the seed-size parameter (m) for DICE. S T opK p

TABLE I .

 I STATISTICS OF RETAIL DATASET.

		Cardinality
	Number of products (|P|)	10,650
	Number of stores (|S|)	1,426
	Number of transactions (|T |)	1,108,748,098

TABLE II .

 II IMPACT OF SEED-SIZE ON SIZE OF CHARACTERIZING REGIONS (|Sp|).

	Product	|Sp| (m = 50)	|Sp| (m = 100)
	peculiar meat	71	175
	chewing gum	56	121

TABLE III .

 III DEFAULT PARAMETERS FOR EXPERIMENTS.

	Algorithm	Parameter	Value
	DICE	Seed-set size (m)	100
	SSR	Kernel width (σ)	74
	SSR	Morphological radius (ρ)	16
	SSR	Gaussian kernel size	45
	SSR		5 × 10 -4

TABLE IV .

 IV USER-STUDY: COMPARATIVE EVALUATIONNot agree Weak agree Moderate agree Agree Strong agree

				DICE		SSR
			Users	46.55	21.13	32.32
			Experts	47.46	34.19	18.35
	TABLE V.	USER-STUDY: INDEPENDENT EVALUATION.
	Users	24.05	23.53		28.47	18.33	5.62
	Experts	10.24	34.58		27.62	19.52	8.04

The product in example is a type of canned tuna sauce that is hidden for confidentiality purposes.