domain: math sdv nlin

text:
Raluca Eftimie
email: r.a.eftimie@dundee.ac.uk

Charlotte Barelle

Mathematical investigation of innate immune responses to lung cancer: the role of macrophages with mixed phenotypes

Keywords: Mathematical modelling, Non-small cell lung cancer, M1 and M2 macrophages, Macrophages with mixed phenotypes, 2020 MSC: 92C50

Macrophages' role in the evolution of solid tumours is a well accepted fact, with the M1-like macrophages having an anti-tumour role and the M2-like macrophages having a pro-tumour role. Despite the fact that some clinical studies on lung tumours have emphasised also the presence of macrophages with mixed M1 and M2 phenotypes in addition to macrophages with distinct phenotypes, the majority of studies still use the distinct M1-M2 classification to predict the evolution of tumours and patient survival. In this theoretical study we use a mathematical modelling and computational approach to investigate the role of macrophages with mixed phenotype on growth/control/elimination of lung tumours. We show that tumour control in the presence of M2→M1 re-polarising treatments is mainly the result of macrophages with mixed phenotypes (due to the assumption of short half-life of M1-like macrophages). We also show that the half-life of various macrophage phenotypes (distinct M1 or mixed M1/M2 phenotypes) impacts the outcome of various therapeutic strategies targeting tumour-associated macrophages. All these results suggest the need for a better experimental understanding of the kinetics of macrophages inside solid tumours.

Introduction

The non-small cell lung cancer is the most common type of lung cancers, and the leading cause of cancer-related deaths in the world [1,2]. Due to the absence of clinical symptoms, most cancers are diagnosed only when they reach an advanced stage and when treatments are not effective anymore. For a long time it was thought that NSCLC is non-immunogenic, but over the last decade a number of studies have shown that this lack of immunity is the result of immuneevasive mechanisms employed by the tumour cells [1]. A recent study [3] on the immune cell composition of human NSCLC has shown that T cells represent the most abundant immune cell population (≈ 47%), followed by B cells (≈ 16%), macrophages (≈ 4.7%) and NK cells (≈ 4.5%). However, in [START_REF] Kargl | Neutrophils dominate the immune cell composition in non-small cell lung cancer[END_REF] the authors found that neutrophils are the most prevalent immune cells in NSCLC, while in [START_REF] Dai | The number and microlocalisation of tumour-associated immune cells are associated with patient's survival time in non-small cell lung cancer[END_REF] the authors identified more macrophages than T cells inside tumour islets and stroma. A very recent study [START_REF] Tamminga | Immune microenvironment composition in non-small cell lung cancer and its association with survival[END_REF] that analysed publicly available raw microarray expression data on immune composition of NSCLC concluded that the majority of the immune infiltrates inside these tumours is represented by the macrophages, followed by T cells and B cells. Moreover, in regard to the infiltrating macrophages, the authors in [START_REF] Tamminga | Immune microenvironment composition in non-small cell lung cancer and its association with survival[END_REF] concluded that the majority of these cells have a M2-like phenotype (i.e., alternatively activated, anti-inflammatory and pro-tumour cells [START_REF] Mantovani | Macrophage polarization: tumor-associated macrophages as a paradigm for polarized m2 mononuclear phagocytes[END_REF]), with the next abundant cells having M1-like phenotypes (i.e., classically activated, pro-inflammatory and anti-tumour cells [START_REF] Mantovani | Macrophage polarization: tumor-associated macrophages as a paradigm for polarized m2 mononuclear phagocytes[END_REF]) or M0 (non-activated) phenotypes. It is possible that some of these contradictory results might be the result of the type of cell markers used to classify the cells, including the different types of macrophages, as discussed below in more detail.

Because many clinical studies [START_REF] Ma | The m1 form of tumorassociated macrophages in non-small cell lung cancer is positively associated with survival time[END_REF][START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF][START_REF] Cao | M2 macrophage infiltration into tumor islets leads to poor prognosis in non-small-cell lung cancer[END_REF][START_REF] Jackute | Distribution of m1 and m2 macrophages in tumor islets and stroma in relation to prognosis of non-small cell lung cancer[END_REF][START_REF] Rakaee | Prognostic value of macrophage phenotypes in resectable non-small cell lung cancer assessed by multiplex immunohistochemistry[END_REF] have focused on the prognostic value of the numbers/percentages of various macrophage sub-populations that infiltrate various tumour areas (islets, stroma) in NSCLC patients, in this study we focus on the role of macrophages on the growth, control, and elimination of lung tumours. Again, we need to emphasise that all these clinical studies on NSCLC show different results, some being reproduced and summarised in Figure 1. One reason for these differences could be the various markers used to classify the various cell types; e.g., CD68/iNOS [START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF][START_REF] Jackute | Distribution of m1 and m2 macrophages in tumor islets and stroma in relation to prognosis of non-small cell lung cancer[END_REF] and CD68/HLA-DR [START_REF] Ma | The m1 form of tumorassociated macrophages in non-small cell lung cancer is positively associated with survival time[END_REF][START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF] for M1 cells; CD68/CD163 [START_REF] Ma | The m1 form of tumorassociated macrophages in non-small cell lung cancer is positively associated with survival time[END_REF][START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF][START_REF] Jackute | Distribution of m1 and m2 macrophages in tumor islets and stroma in relation to prognosis of non-small cell lung cancer[END_REF], CD68/CD204 [START_REF] Rakaee | Prognostic value of macrophage phenotypes in resectable non-small cell lung cancer assessed by multiplex immunohistochemistry[END_REF], CD68/CD206 [START_REF] Xu | The M2 macrophage marker CD206: a novel prognostic indicator for acute myeloid leukemia[END_REF] for M2 cells). In addition, a number of studies that profiled human NSCLC showed that a small percentage of macrophages (i.e., between 0-11%) have markers characterising M2 cells as well as markers characterising M1 cells [START_REF] Ma | The m1 form of tumorassociated macrophages in non-small cell lung cancer is positively associated with survival time[END_REF][START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF][START_REF] Rakaee | Prognostic value of macrophage phenotypes in resectable non-small cell lung cancer assessed by multiplex immunohistochemistry[END_REF]; see also Figure 1(a),(b). However, in a very recent study [START_REF] Singhal | Human tumor-associated monocytes/macrophages and their regulation of T cell responses in early-stage lung cancer[END_REF] on early-stage lung cancer the authors showed experimentally that tumourassociated macrophages from NSCLC expressed both M1 and M2 markers (e.g. HLA-DR, CD206, CD163), sometimes at levels higher than the in vitro differentiated M1 and M2 macrophages. They also showed that approximately 40% of CD14 + cells identified inside the NSCLC tissue had high levels of both HLA-DR and CD163 markers, and more than 50% had high levels of both HLA-DR and CD206 markers. (Note that CD14 + is a human protein produced mostly by macrophages.) Returning to Figure 1, we also note that since some of the markers can be expressed by both M1 and M2 cells, the percentages shown in some of the sub-panels, e.g., sub-panel (b), do not add up to 100%. We can conclude from here that there are still many contradictory results about the proportion of macrophages with mixed phenotypes inside lung tumours (probably due to Figure 1: Data on macrophage percentages in tumour islets (isl.) and stroma (str.) as we approximated it from the following clinical studies on NSCLC: (a) the study in [START_REF] Ma | The m1 form of tumorassociated macrophages in non-small cell lung cancer is positively associated with survival time[END_REF]. Here we show data for M1 (blue) and M2 (green) percentages in long-term patients survival -to compare it with the study in [START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF]. Percentages were calculated using the data from Table 2 in [START_REF] Ma | The m1 form of tumorassociated macrophages in non-small cell lung cancer is positively associated with survival time[END_REF], by taking the ratio of M1 (or M2) cell numbers in either islets or stroma to the total number of M1 (or M2) cells inside both islets and stroma. The authors also mentioned the existence of a small percentage of cells with mixed M1/M2 phenotype (between 1.2%-8.1% with a median of 3.1%; see the cyan-coloured bar). (b) the study in [START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF]. Here we show data for long-term patients survival. Percentages were taken directly from Table 2 in [START_REF] Ohri | Macrophages within nsclc tumour islets are predominantly of a cytotoxic m1 phenotype associated with extended survival[END_REF]. The authors also mentioned the existence of a small percentage of cells with mixed M1/M2 phenotype (between 2.5%-10.2% with a median of 6.45%; see the cyan-coloured bar). (c) the study in [START_REF] Cao | M2 macrophage infiltration into tumor islets leads to poor prognosis in non-small-cell lung cancer[END_REF], where there was no mention on whether the data was for long-term or short-term patients. Percentages were taken directly from Table 2 in [START_REF] Cao | M2 macrophage infiltration into tumor islets leads to poor prognosis in non-small-cell lung cancer[END_REF]; (d) the study in [START_REF] Jackute | Distribution of m1 and m2 macrophages in tumor islets and stroma in relation to prognosis of non-small cell lung cancer[END_REF], where again there was no mention on whether the data was for long-term or short-term patients. We calculated the percentages using the data from Table 3 in [START_REF] Jackute | Distribution of m1 and m2 macrophages in tumor islets and stroma in relation to prognosis of non-small cell lung cancer[END_REF], by taking the ratio of the total number of M1 (or M2) cells (from patients with stage IIA-IIB cancers) in either islets or stroma over the total number of M1 (or M2) cells inside both islets and stroma. Similar results were obtained for macrophage data from patients with stage IA-IB or IIIA-IIIB cancers.

the fact that the classification of macrophages subsets is still in its infancy [START_REF] Quatromoni | Tumour-associated macrophages: function, phenotype and link to prognosis in human lung cancer[END_REF] and still poses many challenges [START_REF] Jayasingam | Evaluating the polarisation of tumour-associated macrophages into M1 and M2[END_REF]).

To address various questions related to tumour-macrophage interactions (as well as questions about the effect of different immuno-and chemo-therapies on tumour-macrophage interactions), the last few decades have seen the development of a large variety of mathematical models [START_REF] Almuallem | Oncolytic viral therapies and the delicate balance between virus-macrophage-tumour interactions: A mathematical approach[END_REF][START_REF] Breems | The re-polarisation of m2 and m1 macrophages and its role on cancer outcomes[END_REF][START_REF] Eftimie | Modelling and investigation of the CD4 + T cellsmacrophages paradox in melanoma immunotherapies[END_REF][START_REF] Eftimie | Investigation into the role of macrophages heterogeneity on solid tumour aggregations[END_REF][START_REF] Li | Computational modeling of the crosstalk between macrophage polarization and tumour cell plasticity in the tumour microenvironment[END_REF][START_REF] Leonard | Macrophage polarization contributes to the antitumoral efficacy of mesoporous nanovectors loaded with albumin-bound paclitaxel[END_REF][START_REF] Leonard | Nonlinear response to cancer nanotherapy due to macrophage interactions revealed by mathematical modeling and evaluated in a murine model via CRISPR-modulated macrophage polarisation[END_REF][START_REF] Louzoun | A mathematical model for pancreatic cancer growth and treatments[END_REF].

We note that all these models focus on the two extreme types of macrophages, M1-like and M2-like cells, and ignore the possible role of macrophages with mixed phenotypes.

In this paper we use a mathematical modelling and computational approach to shed some light on the following questions:

• What biological mechanisms influence the level of macrophages with mixed M1/M2 phenotypes?

• What is the importance of macrophages with mixed M1/M2 phenotypes on tumour growth/control/elimination?

To this end, we derive a new mathematical model (an extension of a simple model introduced in [START_REF] Eftimie | Investigation into the role of macrophages heterogeneity on solid tumour aggregations[END_REF]) that considers the interactions between a homogeneous NSCLC population and three macrophage sub-populations: the M1 cells, the M2 cells, and cells with mixed M1/M2 phenotypes. Despite the human macrophage data shown in Figure 1, there are very little other human studies that would allow us to obtain a better understanding of the mechanisms involved in the macrophage-tumour interactions in human lungs. For this reason, in this study we focus on murine data to parametrise this new mathematical model for tumour-macrophage interactions. Numerical simulations are performed to understand the role of macrophage polarisation rates (M1→M2) and re-polarisation rates (M2→M1) on the reduction/growth in tumour size, as well as the interactions between these rates and cell kinetics on tumour evolution.

The paper is structured as follows: in Section 2 we introduce the new mathematical model and parametrise it with murine data; in Section 3 we discuss the dynamics of this new model as we vary the initial conditions and different model parameters. Due to the uncertainty in the parameter values, we also perform local and global sensitivity analysis to shed some light on the important parameters. We conclude in Section 4 with a summary and discussion of the results.

Model Description

The following mathematical model is used to describe and investigate the interactions between tumour cells and macrophages. This model focuses on the temporal evolution of: tumour cells (u T ), macrophages with a M1-like phenotype (u M 1 ), macrophages with a M2-like phenotype (u M 2 ), and macrophages with a mixed M1-M2 phenotype (u M 12 ); see also Figure 2.

du T dt =p t u T 1 - u T K t (1 + r m2 u M 2 + r m12 u M 12 ) -d t1 u T u M 1 -d t2 u T u M 12 , (1a)
du M 1 dt =p m1 u M 1 1 - u M 1 + u M 12 + u M 2 K m -d m1 u M 1 -α m1 u M 1 u T u T + K * t + α m21 u M 12 , (1b)
du M 12 dt =p m12 u M 12 1 - u M 1 + u M 12 + u M 2 K m -d m12 u M 12 + α m1 u M 1 u T u T + K * t -α m12 u M 12 u T u T + K * t -α m21 u M 12 + α m2 u M 2 , ( 1c
)
du M 2 dt =p m2 u M 2 1 - u M 1 + u M 12 + u M 2 K m -d m2 u M 2 + α m12 u M 12 u T u T + K * t -α m2 u M 2 . ( 1d
)
The above equations incorporate the following biological assumptions:

• Equation (1a) describes the dynamics of tumour cells, which are assumed to grow logistically at a rate p t up to a carrying capacity K T . This logistic growth models the phenomenological observation that tumour growth slows down as size increases, due to lack of nutrients [START_REF] Benzekry | Classical mathematical models for description and prediction of experimental tuour growth[END_REF]. The macrophages with a M2-like phenotype (u M 2 and u M 12 ) contribute to the proliferation of tumour cells [START_REF] Chen | Macrophages reprogrammed by lung cancer microparticles promote tumour development via release of IL-1β[END_REF][START_REF] Mantovani | Macrophage polarization: tumor-associated macrophages as a paradigm for polarized m2 mononuclear phagocytes[END_REF][START_REF] Jayasingam | Evaluating the polarisation of tumour-associated macrophages into M1 and M2[END_REF]], which we model using coefficients r m2 and r m12 . The macrophages with a M1-like phenotype reduce tumour growth [START_REF] Mantovani | Macrophage polarization: tumor-associated macrophages as a paradigm for polarized m2 mononuclear phagocytes[END_REF] at rates d t1 (the u M 1 cells) and d t2 (the u M 12 cells).

• Equation (1b) describes the dynamics of macrophages with a dominant M1 phenotype (u M 1 ). Since tissue-resident macrophages are maintained via self-proliferation with minimal monocyte input [START_REF] Patel | The fate and lifespan of human monocyte subsets in steady state and systemic inflammation[END_REF][START_REF] Soucie | Lineage-specific enhancers activate self-renewal genes in macrophages and embryonic stem cells[END_REF][START_REF] Poh | Targeting macrophages in cancer: from bench to bedside[END_REF], and since in [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF] the authors showed that macrophages have an exponential growth followed by a stationary phase, here we consider a logistic growth at a rate p m1 , up to a carrying capacity K m . These cells die naturally [START_REF] Italiani | From monocytes to M1/M2 macrophages: phenotypical vs. functional differentiation[END_REF] at a rate d m1 . In the presence of tumour cells, the M1 macrophages can polarise (at a rate α m1 ) towards a mixed M1/M2 phenotype. External factors (e.g., anti-tumour treatments [START_REF] Heusinkveld | Identification and manipulation of tumour associated macrophages in human cancers[END_REF]) can re-polarise (at a rate α m21 ) the macrophages with a mixed M1/M2 phenotype towards an anti-tumour M1-dominant phenotype.

• Equation (1c) describes the dynamics of macrophages with mixed M1/M2 phenotype (u M 12 ). As before, since tissue-resident macrophages are selfproliferating [START_REF] Patel | The fate and lifespan of human monocyte subsets in steady state and systemic inflammation[END_REF][START_REF] Soucie | Lineage-specific enhancers activate self-renewal genes in macrophages and embryonic stem cells[END_REF] and their proliferation has an exponential phase followed by a stationary phase [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF], here we assume a logistic growth at a rate p m12 , up to a carrying capacity K m . Moreover, these cells die naturally at a rate d m12 . Finally, we assume that these cells with mixed M1/M2 phenotypes can polarise (at a rate α m12 ) in the presence of tumour cells towards a more distinct M2 phenotype, or can re-polarise (at a rate α m21 ) following external treatment [START_REF] Heusinkveld | Identification and manipulation of tumour associated macrophages in human cancers[END_REF] towards an M1 phenotype. In the presence of tumour cells, M1 macrophages can polarise (at a rate α m1 ) towards a mixed M1/M2 phenotype. Anti-tumour treatments [START_REF] Heusinkveld | Identification and manipulation of tumour associated macrophages in human cancers[END_REF] can re-polarise (at a rate α m2 ) the M2 macrophages towards a mixed M1/M2 phenotype.

• Equation (1d) describes the dynamics of macrophages with a dominant M2 phenotype (u M 2 ). Again, we assume that this population grows logistically at a rate p m2 , up to a carrying capacity K m . These cells die naturally at a rate d m2 [START_REF] Italiani | From monocytes to M1/M2 macrophages: phenotypical vs. functional differentiation[END_REF]. Anti-tumour treatments [START_REF] Heusinkveld | Identification and manipulation of tumour associated macrophages in human cancers[END_REF] can re-polarise the M2 macrophages (at a rate α m2 ) towards a mixed M1/M2 phenotype.

In the presence of tumour cells, the macrophages with a mixed M1/M2 phenotype can polarise (at a rate α m12 ) towards M2 cells.

Parameter estimation

Even if our biological questions were triggered by human NSCLC data (see Figure 1), to identify some of the model parameters we focus on murine experiments for which there is more data (compared to humans). Below we summarise the data we used to estimate different model parameters.

• The study in [START_REF] Sato | Ex vivo model of non-small cell lung cancer using mouse lung epithelial cells[END_REF] investigated the growth of tumours resembling nonsmall cell lung cancer (NSCLC) in mice lungs. In Figure 3 we reproduce the murine tumour growth data from [START_REF] Sato | Ex vivo model of non-small cell lung cancer using mouse lung epithelial cells[END_REF], together with the solution of a logistic equation for tumour growth. The best fit of the numerical solution to the data (obtained using the classical least square method) was obtained for p t = 0.23 and K t = 1400. This proliferation rate is consistent with the doubling time (i.e., 2.97 days) of NSCLC cells inoculated into nude mice, as calculated experimentally in [START_REF] Yamaji | Correlation between interleukin 6 production and tumour proliferation in non-small cell lung cancer[END_REF] .

(a) (b)

Figure 3: (a) Reproduction of tumour growth data from [START_REF] Sato | Ex vivo model of non-small cell lung cancer using mouse lung epithelial cells[END_REF] (black circles) together with the solution of a logistic growth equation for tumour growth (red curve), as given by eq. (1a) in the absence of any macrophages. Here Kt = 1400 and pt = 0.23, and the initial condition for the numerical simulation of the tumour logistic growth is u T (0) = 4mm 3 . (b) Reproduction of macrophage growth data from [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF] (open circles and black squares) -where data was transformed from number of cells to potential volume occupied by these cells (see discussion below) -together with the solution of a logistic equation for macrophage growth (red curves), as given by the sums of of eqns. (1b)+(1c)+(1d) when p m1 = p m12 = p m2 := pm and d m1 = d m12 = d m2 = 0. The continuous curve was obtained for pm = 0.88 and Km = 6.72, while the dotted curve was obtained for pm = 0.483 and Km = 6.72. The initial condition for the numerical simulation of macrophage logistic equation is u M (0) = 0.006mm 3 (where u M describes the total macrophage size).

• In [3] the authors have calculated that macrophages represent ≈ 4.8% of the total immune infiltrates into human NSCLC. For our murine model, we assume that immune cell infiltrates represent up to 10% of tumour mass, and the macrophages represent 4.8% of these immune infiltrates.

(Note that in [START_REF] Cao | M2 macrophage infiltration into tumor islets leads to poor prognosis in non-small-cell lung cancer[END_REF] it was estimated that macrophages represent ≈ 15.84% of all cells inside tumour tissue, and thus our assumption is not completely unrealistic.) Thus, for a maximum tumour volume of K t = 1400mm 3 we obtain a maximum macrophage volume of K m = 6.72mm 3 .

• In [START_REF] Haley | Comparative morphology and morphometry of alveolar macrophages from size mammalian species[END_REF] the authors calculated the diameter of an alveolar macrophage at ≈ 19µm. In [START_REF] Monte | Does the cell number 10 9 still really fit one gram of tumour tissue?[END_REF] the author suggested that a volume of 1000mm 3 can contain up to 9.39 × 10 7 cells of diameter 22µm, or up to 2.44 × 10 8 cells of diameter 16µm. In this study, we assume that a volume of 1000mm 3 can contain ≈ 10 8 macrophages. Thus a carrying capacity K m = 6.72mm 3 can contain ≈ 6.72 × 10 5 macrophages. These numbers are consistent with the experimental study in [START_REF] Redente | Tumour signalling to the bone marrow changes the phenotype of monocytes and pulmonary macrophages during urethane-induced primary lung tumorigenesis in a/j mice[END_REF], where the authors showed that the number of macrophages from control mice ranged from 8 × 10 4 cells/mouse to 2.4 × 10 5 cells/mice.

In [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF], the authors measured macrophage growth, and calculated a proliferation rate between 0.487/day and 0.88/day (in different mice). In Figure 3 we approximated the two macrophage data sets from [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF] (Fig.

14.20.1 in [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF], which shows cell numbers), where we transformed cell numbers into cell volumes (using the assumptions and calculations above).

Since in [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF] the authors showed that macrophages grow logistically, we fitted a logistic growth curve with rate p m and carrying capacity K m = 6.72mm 3 . The continuous red curve in Figure 3 shows macrophage growth for p m = 0.88, while the dashed red curve shows macrophage growth for p m = 0.487. Throughout this study we will consider an average macrophage proliferation rate of p m1 = p m12 = p m2 = 0.7.

• The experimental study in [START_REF] Patel | The fate and lifespan of human monocyte subsets in steady state and systemic inflammation[END_REF], which focused on the adoptive transfer of human mononuclear phagocytes into mice, showed that classical (M1) macrophages circulate for a mean of 1.01 days, whereas intermediate (M12) and nonclassical (M2) macrophages have longer mean lifespans of 4.30 and 7.41 days, respectively. In [START_REF] Italiani | From monocytes to M1/M2 macrophages: phenotypical vs. functional differentiation[END_REF] the authors stated that the murine M1-like macrophages (involved in phagocytosis) have a half-life between 18-20hrs, while the murine M2-like macrophages (involved in tissue repair) have a half-life between 5-7 days. Therefore, in this study we assume that:

the M1-like macrophages have a half-life of ≈ 0.8 days (i.e., ≈ 19. All other parameter values that appear in model (1) are unknown. In the following, we discuss some of the assumptions we consider when choosing the ranges over which we vary these parameters:

• We assume that the presence of 1% of max tumour is enough to trigger a u M 1 → u M 12 polarisation, and further a u M 12 → u M 2 polarisation. Thus, we consider K * t = 1%K t .

• We assume that macrophages with mixed M1/M2 phenotypes exhibit half the anti-tumour effect of the M1 macrophages. Thus, we consider d t1 ∈ (10 -5 , 10 -1 ) (with a baseline value of d t1 = 0.01) and d t2 ∈ (5 × 10 -6 , 5 × 10 -2 ) (with a baseline value of d t2 = 0.005).

• We assume that macrophages with mixed M1/M2 phenotype have half the pro-tumour effect of M2 macrophages. Thus, we consider r m2 ∈ (10 -2 -10 0 ) (with a baseline value r m2 = 0.1) and r m12 ∈ (5 × 10 -3 , 0.5) (with a baseline value r m12 = 0.05).

• In humanised murine experiments [START_REF] Patel | The fate and lifespan of human monocyte subsets in steady state and systemic inflammation[END_REF] it has been found that that only a small proportion of M1 macrophages will re-polarise to an intermediate M12 phenotype, but most M12 macrophages will re-polarise to an M2 phenotype during their lifespan. Throughout this study we assume α m1 < α m12 . For the re-polarisation rates we assume α m2 = α m21 . Thus, we consider α m1 ∈ (10 -4 , 10 -1 )/day (with a baseline α m1 = 10 -3 /day), α m12 ∈ (10 -3 , 10 -1 )/day (with a baseline of α m12 = 10 -2 /day), and α m2 = α m21 ∈ (10 -3 , 10 0 )/day (with baselines of 0.0, since we assume that initially there is no treatment to force a macrophage re-polarisation towards an M1-like phenotype).

All these estimated parameter values and ranges which appear in model ( 1) are summarised in Table 2.1. For simplicity, and to avoid numerical problems caused by a stiff system, we rescale the tumour cells by K t and the macrophages by K m . This leads to unitary carrying capacities (K t = 1, K m = 1), and to a rescaling of the following five parameters:

d t1 = d t1 × K m , d t2 = d t2 × K m , r m2 = r m2 × K m , r m12 = r m12 × K m , K * t = K * t K t . (2)
These rescaled parameter values are listed in the fourth column in Table 2.1 -where we removed the bars for simplicity. These values are used for the numerical simulations performed throughout this study.

Numerical results

The initial conditions for the numerical simulations of system (1) were obtained from the rescaled initial conditions in Figure 3:

u T (0) = 4 K t = 4 1400 = 0.002857, u M 1 (0) = 0.006 K m = 0.006 6.72 = 0.000893, u M 12 (0) = 0.0, u M 2 (0) = 0.0. (3)
Thus, we assume that when the tumour is introduced into the system, it elicits a pro-inflammatory immune response characterised only by the presence of a non-zero u M 1 population. The numerical solution is propagated in time using a classical fourth order Runge-Kutta method. 2.1. We see that initially (t < 10 days) the immune response is dominated by macrophages with a M1 phenotype. Then a transient increase in the cells with a mixed M1/M2 phenotype (for 10 < t < 30) is associated with a reduction in tumour size. Tumour relapse is associated with an increase in the cells with a M2 phenotype (for t > 40). We emphasise here that this baseline case assumes α m2 = α m21 = 0 (see Table 2.1), i.e., no external immune treatment to induce a re-polarisation of M2-like macrophages towards an M1-like phenotype [START_REF] Heusinkveld | Identification and manipulation of tumour associated macrophages in human cancers[END_REF].

Baseline system dynamics

In Figure 4(b) we show the effect of externally-inducing a macrophage repolarisation on model (1); i.e., α m2 = α m21 = 0.01. We observe a slight reduction in tumour size in the long term (i.e. t > 80 days), which is associated with

Table 1: Summary of the parameter values used in this study. The 2 nd column shows the dimensional parameters, the 3 rd column shows their units, and the 4 th column shows in bold the rescaled parameter values; see eq. (2) (If there is no difference between the values in the 2 nd and 4 rd column, it means that the parameter was not rescaled). For most of the parameter values we show a whole range, with the value inside the parentheses being the baseline value used for the simulations. The time unit is "day", cells are described by cell volume "vol".

Param. Original values

Original Units

Rescaled values

Description (original values) 10 -4 -10 -1 (0.01)

p t 0.
1 day 10 -4 -10 -1 (0.01)
Polarisation rate of macrophages with a mixed M1/M2 phenotype towards a M2-dominant phenotype α m2 0-10 0 (0.0) 1 day 0 -10 0 (0.0) Re-polarisation rate of M2 cells towards a mixed M1/M2-phenotype α m21 0-10 0 (0.0) 1 day 0 -10 0 (0.0) Re-polarisation rate of macrophages with a mixed M1/M2 phenotype towards a M1-dominant phenotype a higher percentage of macrophages with mixed M1/M2 phenotype (and a very small but non-zero population of M1-like macrophages). The effect of further increasing α m2 , α m21 will be discussed below, in Figure 7(d).

u u u u M12 % u % u % u % u M1 u T u M12 u M2 (a) M1 M2 T M1 M12 M2 % u % u M1 M12 M2 u (ii) Cell volumes (i) (b) (i)
Cell volumes (ii)

Sensitivity analysis

Since many parameter values were estimated within certain ranges, in the following we evaluate the robustness of model (1), by investigating the sensitivity of tumour size to small perturbations in some model parameters and in initial conditions. We start in Section 3.2.1 with a local sensitivity analysis to investigate how small perturbations in the initial conditions and macrophage polarisation/re-polarisation rates impact tumour dynamics. However, since we do not know if there is interaction between these polarisation/re-polarisation rates and all other estimated parameters that appear in the model, in Section 3.2.2, we perform a global sensitivity analysis.

Local sensitivity analysis

In the following we focus on tumour size at t = 40 (while tumour is decreasing following an increase in macrophages with mixed phenotypes; see Figure 4) and at t = 60 (while tumour relapses and is close to its carrying capacity; see Figure 4). The change in tumour size is calculated using the discrete formula for the derivative of the output (tumour size) with respect to the input (e.g., initial condition u j (0), where j ∈ {T, M 1, M 12, M 2}) [START_REF] Zi | Sensitivity analysis approaches applied to system biology models[END_REF]:

L S = u new T (40) -u baseline T (40) |u new j (0) -u baseline j (0)| . ( 4
)
Remark 1. We decided to focus on time t = 60, when the relapsing tumour would approach the carrying capacity (for the baseline parameter values) since we would like to understand the mechanisms that could impact the late-stage tumours, as many lung tumours are discovered when they have already reached a late stage. Moreover, for this local sensitivity analysis we decided to ignore the earlier times (e.g., t < 30) since the immune response was too weak at these times, and changes in immune-related parameters had almost no impact on tumour size. We chose to focus on t = 40 because at this time the level of immune response (and in particular the macrophages with mixed phenotype) is high enough to have a significant impact on tumour.

Remark 2. In eq. ( 4) we do not normalise the sensitivity index [START_REF] Kirch | The effect of model rescaling and normalization on sensitivity analyis on an example of a MAPK pathway model[END_REF]. This is mainly because when we investigate local sensitivity to the initial conditions, there are two zero baseline initial conditions for u M 2 and u M 12 and therefore we cannot divide by these values for normalisation (i.e., the ratio |u new j (0)u baseline j (0)|/u baseline j (0), j ∈ {M 12, M 2}, does not make sense). A similar problem is encountered when we perform the sensitivity to α m2 and α m21 (which have zero baseline values; see Table 2.1). For this reason, in this study we decided to work with equation ( 4) and not with a normalised version of this equation.

Tumour sensitivity to initial conditions. In Figure 5 we show the magnitude of changes (i.e., L S given by eq. ( 4)) in tumour size on days (a) t = 40 and (b) t = 60, as we increase/decrease the initial conditions for the tumour and M1 macrophage populations by a factor of 10 2 from their baseline values. The M12 and M2 initial macrophage levels are increased from 0 to 10 -3 . We see in Figure 5 that the initial conditions for all macrophage sub-populations have a significant impact on the early stages of tumour growth (i.e., t < 50), and only the mixed phenotype macrophages seem to play an important role also in the later stages (t > 50). Moreover, increasing the initial tumour size by a factor of 10 2 does not seem to have an impact on either early or later tumour stages, which suggests that the baseline initial tumour size is already large enough.

Only a decrease in the initial tumour size has an impact on tumour levels on both t = 40 and t = 60.

Tumour sensitivity to polarisation/re-polarisation rates. In Figure 6 (panel (b)), as we vary separately each of the rates α m1 , α m12 , α m21 , α m2 . The rates α m1 , α m12 are increased/decreased by 90% from their baseline levels, while rates α m2 , α m21 are increased from their zero baseline levels to 0.01. Panels (c),(d),(e) show the time evolution of tumour (t ∈ [0, 100]) as we vary these rates polarisation/re-polarisation rates. As expected, when we decrease α m12

or we increase α m2 we see a decrease in tumour size. However, unexpectedly, a decrease in α m1 (which should reduce the M1→M2 polarisation), seems to cause a significant increase in the tumour population.

In Figure 7 we show the effect of varying at the same time and by the same amount the polarisation rates α m1 and α m12 (where we assume that α m1 = α m12 =: α 1 ), and the re-polarisation rates α m2 and α m21 (where we assume that α m2 = α m21 =: α 2 ). Panels (a) and (b) show the magnitude of the changes in tumour size on days t = 40 and t = 60, while panels (c) and (d) show the timevariations in u T (t) as we vary simultaneously α m1 = α m12 and α m2 = α m21 , respectively.

• Figure 7(c): when we decrease α m1 = α m12 = α 1 to α 1 = 0.00001 we first observe a tumour increase (for t ∈ (30, 50)) followed by a large decrease (for t ∈ (50, 80)). The initial tumour increase is unexpected, since a reduction in α 1 should keep the macrophages in a M1-like phenotype.

• Figure 7(d): when we increase α m2 = α m21 = α 2 , we observe that the tumour can be reduced and eventually eliminated for very large repolarisation rates (e.g., α 2 = 0.2). However, in the short time (t ∈ (30, 50))

the tumour grows to large sizes; it can even reach its carrying capacity before being killed by the M1-like macrophages. Therefore, these results suggests that there might be a range for the re-polarisation rates that would lead to optimal treatment. (For murine experiments mice are killed for humane reasons when the tumours grow too large, and thus a potential tumour decay for t > 80 would not be observed).

To further investigate the unexpected effect of α m1 on tumour growth, in Fig- Only in this particular case, a decrease in α m1 leads to lower tumour sizes -although the tumour will eventually relapse and grow to its carrying capacity. This temporary decrease in tumour size seems to be the result of higher %u M 12 (see Figure 8(c)(ii)-(ii')).

Since macrophages death rates seem to play an important role in macrophage re-polarising treatments, in Figure 9 we explore the region in the (d m12 , d m1 )

-90% of baseline parameter +90% of baseline parameter increased from 0.0 to 0.01 4)) on days t = 40 and t = 60, as we vary at the same time the two polarisation rates α m1 = α m12 =: α 1 by ±90% (from a common baseline value of α 1 = 0.0001) and α m2 = α m21 =: α 2 (from their baseline value 0.0 to 0.01). Since the increase in α 2 has a very little effect on tumour size on day t = 60, to improve the visualisation we multiplied the value L S from eq. ( 4 space for which a decrease in α m1 (from 10 3 to 10 4 ) leads to a decrease or an increase in tumour size. The thick red curve in the top-left panel corresponds to a stationary tumour (i.e. the dot and dash-dot curves overlap, and the tumour is neither increasing nor decreasing). For three cases corresponding to three corners of this main panel, we show both the time-evolution of the tumour, and the percentage of macrophages corresponding to some of these tumours. We observe that tumour reduction/control for d m12 = 0.14, d m1 = 0.45 is associated with a large percentage of macrophages with mixed M1/M2 phenotypes. In contrast, tumour reduction/control for d m12 = 0.35, d m1 = 0.45 is associated with a very low percentage of macrophages with mixed M1/M2 phenotypes. Hence, the death rates of macrophages with M1-like phenotype or mixed M1/M2-like phenotype seem to play an important role on tumour growth/control/decay, and on the percentage of mixed macrophages during tumour growth/control/decay. We will discuss these results in more detail in Section 4.

Global sensitivity analysis

To determine the impact that possible interactions between multiple uncertain parameters have on overall tumour dynamics, next we perform a global sensitivity and uncertainty analysis using the classical LHS/PRCC (Latin Hypercube Sampling/Partial Rank Correlation Coefficient) approach [START_REF] Blower | Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example[END_REF][START_REF] Marino | A methodology for performing global uncertainty and sensitivity analysis in systems biology[END_REF].

In Figure 10 show the PRCC values. For the PRCC analysis, we note that the parameters with large PRCC absolute values are the most important. In particular, values closer to ±1 indicate parameters that influence strongly the outcome variable, while the sign indicates the qualitative relation between input parameters and output variables (with " + " sign indicating that the parameter is directly proportional to the outcome, and the "-" sign indicating that the parameter is inversely proportional to the outcome). In Figure 10 we see that:

• The parameters with the largest impact on u T are α m21 , α m2 , d t1 , p m2 and d m12 . Interestingly, these parameters correspond to the four tumourassociated macrophage targeted therapeutic strategies discussed in [START_REF] Joseph | Tumour-associated macrophage polarisation and re-education with immunotherapy[END_REF]:

antibody-mediated elimination of tumour cells (d t1 ), blockage of monocyte recruitment to tumours (p m2 ), re-polarisation to an M1-like phenotype (α m21 , α m2 ), and suppression of macrophage survival (d m12 ).

• The parameters with the largest impact on u M 1 are α m2 , d m12 , and p m2 .

• The parameter with the largest impact on u M 12 is α m21 , followed by α m2 , d m12 and p m2 .

• The parameter with the largest impact on u M 2 are α m2 , followed by p m2 .

Among the polarisation rates, the largest impact is provided by α m12 .

The above results identified the proliferation/recruitment rate of M2 macrophages (p m2 ) as a parameter important not only for tumour dynamics, but also for the dynamics of all macrophage phenotypes, including u M 1 and u M 12 . Therefore, reducing p m2 will impact tumour growth both directly (through the direct pro-tumour effect of M2 macrophages [START_REF] Mantovani | Tumourassociated macrophages as treatment targets in oncology[END_REF][START_REF] Chen | Macrophages reprogrammed by lung cancer microparticles promote tumour development via release of IL-1β[END_REF]) and indirectly (through the M1 macrophages and macrophages with mixed M1/M2 phenotypes).

Discussion

In this study we developed and investigated numerically a new mathematical model for the temporal dynamics between non-small cell lung cancer and macrophages in the lung, with the ultimate goal of shedding some light on the importance of macrophages with mixed phenotypes.

After showing the baseline dynamics of this new model (Figure 4), we started performing a local sensitivity analysis, to gain some understanding of the effects of changes in the initial conditions (Figure 5), as well as changes in those parameters describing macrophage polarisation/re-polarisation rates (Figures 6789).

In regard to the initial conditions, our local sensitivity analysis showed that the most important role is played by u M 12 (0). For the baseline parameter values investigated in this study (see Table 2.1) we observed that a decrease in the polarisation rate α m12 (i.e., u M 1 → u M 12 polarisation) led to larger tumours, even when we increased the initial level of M1 macrophages (u M 1 (0)) or the M1 macrophages phagocytosis rate (d t1 ). We discovered that this was the result of short half-life of M1 cells (i.e., large d m1 ; see Figures 89). The results presented in this study depend heavily on the parameters used for the simulations. Some of these parameter values were obtained from in-vitro and ex-vivo experiments [START_REF] Chitu | Measurement of macrophage growth and differentiation[END_REF][START_REF] Sato | Ex vivo model of non-small cell lung cancer using mouse lung epithelial cells[END_REF], and therefore they could be different from the in-vivo murine parameters and even more from the in-vivo human parameters.

Unfortunately, we do not have in-vivo data to parametrise these mathematical models, and our best approach was a sensitivity and uncertainty analysis to understand the extent of variations in model outcomes. Global sensitivity analysis (Figure 10) revealed the parameters with the largest impact for tumour dynamics (α m21 , α m2 , d t1 , p m2 and d m12 ), and interestingly these parameters were also the parameters involved in four of the macrophage-targeted treatment approaches for cancer as identified in [START_REF] Joseph | Tumour-associated macrophage polarisation and re-education with immunotherapy[END_REF]: re-polarisation of macrophages towards an M1-like phenotype (α m21 , α m2 ), suppression of tumour-associated macrophages survival (d m12 ), blockade of macrophage recruitment (p m2 ), and antibody-mediated elimination of tumour cells by macrophages (d t1 ).

Overall, the results of this study emphasise the need for a better experimental understanding of the kinetics (doubling time, half lives) of macrophages with different phenotypes that can be found inside solid tumours (especially the macrophages with mixed phenotypes). Most of the experimental studies in the literature focus on the kinetics of T cells [START_REF] Boer | Quantifying T lymphocyte turnover[END_REF][START_REF] Macallan | Current estimates of T cell kinetics in humans[END_REF], but given the importance of tumour-associated macrophages on tumour evolution, more experimental studies are necessary to better understand the macrophage kinetics.

Unfortunately, the lack of robust macrophage markers can lead to inaccurate macrophage counts [START_REF] Jayasingam | Evaluating the polarisation of tumour-associated macrophages into M1 and M2[END_REF], which further impacts our hope of reliable data on macrophage kinetics. Until more data will become available, we have to continue using modelling and computational approaches to propose hypotheses regarding the macrophage dynamics and their interactions with various components of the tumour microenvironment. , 0, 0), which exists only if p m1 > d m1 .

• Tumour-Free M2-cells Free (TFM2F) state: (u * T , u Stability of steady states. The linear stability of the above steady states is controlled by the eigenvalues of the Jacobian matrix associated with the system

(1):

J(u

Figure 2 :

 2 Figure 2: Caricature description of the cell-cell interactions depicted by model (1).

  2hr), corresponding to a death rate d m1 = 0.87/day; the mixed M1/M2 macrophages have a half-life of ≈ 3 days, corresponding to a death rate d m12 = 0.23/day; the M2-like macrophages have a half-life of ≈ 5.14 days, corresponding to a death rate d m2 = 0.09/day.

Figure 4 (

 4 Figure 4(a) shows the dynamics of model (1) for the baseline parameter values given in Table2.1. We see that initially (t < 10 days) the immune

Figure 4 :

 4 Figure 4: (a) Dynamics of model (1), for the baseline parameter values listed in Table 2.1, when α m2 = α m21 = 0. (b) Dynamics of model (1), when we assume α m2 = α m21 = 0.01 > 0. Sub-panels (i) show the time-evolution of the tumour cells and macrophages; Sub-panels (ii) show the time-evolution of the percentage of macrophage composition.

Figure 5 :

 5 Figure 5: Changes in tumour size (see L S in eq. (4)) on days (a) t = 40 and (b) t = 60, as we vary the initial conditions (ICs) for each of the four variables from their baseline values (in red colour) to the lower values (cyan colour) and the upper values (black colour) of the indicated IC ranges. Since the baseline values for u M 12 (0) and u M 2 (0) were zero, we only increased these values to 10 -3 .

Figure 6 :

 6 Figure 6: Changes in tumour size (see Ls in eq. (4)) on days (a) t = 40 and (b) t = 60, as we vary separately α m1 , α m12 by ±90% from their baseline values (black and cyan colours), while α m21 and α m2 are increased separately from 0.0 to 0.01 (magenta colour). In (c),(d) we show the time-evolution of the tumour as vary separately by ±90% (c) α m1 and (d) α m12 . In (e) we show the time-evolution of the tumour as we vary separately α m21 and α m2 : each rate is increased from 0.0 (baseline) to 0.01. The dotted vertical lines indicate the times t = 40 and t = 60.

ure 8

 8 we show the tumour cell volume as we vary: (a) d t1 (to verify whether tumour increase for low α m1 is the result of low M1 phagocytosis); (b) u M 1 (0) (to verify whether tumour increase for low α m1 is the result of low initial M1 population); (c) d m1 (to verify whether tumour increase is the result of a continuously low M1 population due to high death rate). It is clear that while an increase in d t1 and u M 1 (0) alone, or a decrease in d m1 alone, leads to a temporary reduction in tumour size, when we combine them also with a decrease in α m1 the tumour grows back. A more significant tumour reduction is observed for very low d m1 values (i.e., d m1 ≤ 0.5).

Figure 7 :

 7 Figure 7: Panels (a),(b) show changes in tumour size (i.e., L S value given by eq. (4)) on days t = 40 and t = 60, as we vary at the same time the two polarisation rates α m1 = α m12 =: α 1 by ±90% (from a common baseline value of α 1 = 0.0001) and α m2 = α m21 =: α 2 (from their baseline value 0.0 to 0.01). Since the increase in α 2 has a very little effect on tumour size on day t = 60, to improve the visualisation we multiplied the value L S from eq. (4) by a factor of 10. Panel (c) shows the time-evolution of u T when we assume α M 2 = α M 21 = 0.0 and we vary α 1 := α M 1 = α M 12 . Panel (d) shows the time-evolution of u T when we assume α M 1 = α M 12 = 0.0001 and we vary α 2 := α M 2 = α M 21 . The dotted vertical lines indicate the times t = 40 and t = 60.

  Figure 7: Panels (a),(b) show changes in tumour size (i.e., L S value given by eq. (4)) on days t = 40 and t = 60, as we vary at the same time the two polarisation rates α m1 = α m12 =: α 1 by ±90% (from a common baseline value of α 1 = 0.0001) and α m2 = α m21 =: α 2 (from their baseline value 0.0 to 0.01). Since the increase in α 2 has a very little effect on tumour size on day t = 60, to improve the visualisation we multiplied the value L S from eq. (4) by a factor of 10. Panel (c) shows the time-evolution of u T when we assume α M 2 = α M 21 = 0.0 and we vary α 1 := α M 1 = α M 12 . Panel (d) shows the time-evolution of u T when we assume α M 1 = α M 12 = 0.0001 and we vary α 2 := α M 2 = α M 21 . The dotted vertical lines indicate the times t = 40 and t = 60.

Figure 8 :

 8 Figure 8: Time-evolution of tumour cell volume as we vary different parameters related to M1 cells. The "baseline" case (continuous curve) corresponds to the parameter values listed in Table2.1, where α m1 = 10 -3 . (a) Vary d t1 alone (dotted curve) or in combination with α m1 (dash-dot curve). To see some difference in tumour dynamics, we had to increase d t1 by a factor of 30: from d t1 = 1.344 to d t1 = 201.6 (b) Vary the initial condition u M 1 (0) alone (dotted curve) or in combination with α m1 (dash-dot curve). Here we increased the initial condition by a factor of 100: from u M 1 (0) = 0.000892 to u M 1 (0) = 0.0892. (c) Vary d m1 alone (dotted curve) or in combination with α m1 (dash-dot curve). Here we show the effects of medium vs. low values of d m1 : (i) d m1 = 0.6, (ii) d m1 = 0.5. For this panel (c) we also show the percentage of various types of macrophages: (i') for the case d m1 = 0.6, (ii') for the case d m1 = 0.5.

Figure 9 :

 9 Figure9: Bifurcation diagram in the (d m12 , d m1 ) parameter space showing the regions where a decrease in α m1 from 10 -3 (dot curves) to 10 -4 (dash-dot curves) leads to a decrease in tumour size (white region) or an increase in tumour size (cyan region). To make it clear what we mean by stationary tumour (i.e. the dot and dash-dot curves overlap), we also show the time-evolution of the tumour and macrophage percentage at three corner points in this parameter space: (i) d m12 = 0.14, d m1 = 0.45; (ii) d m12 = 0.37, d m1 = 0.45; and (iii) d m12 = 0.37, d m1 = 0.67. To clarify that the delay in tumour relapse observed for case (ii) is the result of reduced tumour levels, in this sub-figure we also show a zoom-in of tumour growth curves for u T < 0.05.

Returning to Figure 1 Figure 10 :

 110 Figure 10: Global sensitivity and uncertainty analysis for variables (a) u T , (b) u M 1 , (c) u M 12 , (d) u M 2 as we vary 14 parameters within the ranges specified in Table2.1. Sub-panels (i) show the mean+standard deviation, together with max/min values of these variables as we vary t. Sub-panels (ii) show the PRCC values corresponding to each of the parameters varied in Table2.1.

  r m2 u * M 2 + r m12 u * M 12 ) -d t2 u * M 12 .(A.3)• Tumour-Present Immune-Present (TPIP) steady state:(u * T , u * M 1 , u * M 12 , u * M 2 ).Proposition 2. For the case (ii) when a macrophage re-polarisation treatment is considered (i.e., α m2 , α m21 > 0), model (1) can exhibit the following tumourfree and tumour-present steady states:• Tumour-Free Immune-Free (TFIF) state:(u * T , u * M 1 , u * M 12 , u * M 2 ) = (0, 0, 0, 0),which exists for all parameter values;• Tumour-Free M1-cells Present (TFM1P) state: (u * T , u * M 1 , u * M 12 , u * M 2

α

  m2 (-d m1 p m2 + p m1 (α m2 + d m2 ))(α m2 + d m2 -p m2 )K m ((-d m1 -α m21 )α m2 -d m1 (α m21 + d m12 ))p 2 m2 -p m1 p m12 (α m2 + d m2 ) 2 + (α + m2p m1 + (α m21 + d m12 )p m1 + d m1 p m12 )(α m2 + d m2 )p m2 and u * M 2 = [p m1 (α m2 + d m2 ) -d m1 p m2 ] (α m2 + d m2 -p m2 ) • ((-α m21 -d m12 )p m2 + p m12 (α m2 + d m2 ))K m p m2 (((-d m1 -α m21 )α m2 -d m1 (α m21 + d m12 ))p 2 m2 -p m1 p m12 (α m2 + d m2 ) 2 ) + p m2 (((α m2 + α m21 + d m12 )p m1 + d m1 p m12 )(α m2 + d m2 )p m2 ) . • Tumour-Present Immune-Free (TPIF) state: (u * T , u * M 1 , u * M 12 , u * M 2 ) = (K t , 0, 0, 0),which exits for all parameter values;• Tumour-Present M1, M12 and M2 Immune response Present (TPIP) state:(u * T , u * M 1 , u * M 12 , u * M 2 ) = (u * T , u * M 1 , u * M 12 , u * M 2 ).Remark 3. Both cases investigated above, namely (i) no re-polarising treatment (α m2 = α m21 = 0) and (ii) re-polarising treatment (α m2 , α m21 > 0) had a tumour-only steady state (TPIF) and coexistence steady states (TPIP). However, case (i) can exhibit two other steady states characterised by the presence of tumour cells and the absence of M1-like cells (u * M 1 = 0): the TPM2P and TPM12M2P states. This suggests that in the absence of any external treatment to re-polarise the macrophages towards an M1-like phenotype, the tumours are always formed of M2-like macrophages or macrophages with a mixed M1/M2 phenotype.

  [START_REF] Leonard | Nonlinear response to cancer nanotherapy due to macrophage interactions revealed by mathematical modeling and evaluated in a murine model via CRISPR-modulated macrophage polarisation[END_REF]

			1 day	0.23		Proliferation rate of tumour cells
	K t	1400	vol	1		Tumour carrying capacity
	K * t	14	vol	0.01		Tumour level that triggers M1→M2
						macrophage polarisation
	r m2	10 -2 -10 0	1 vol	6.72 × 10 -2 -	Contribution of M2 macrophages to
		(0.1)		6.72 (0.672)	the proliferation of tumour cells
	r m12	0.005 -0.5	1 vol	0.0336	-	Contribution of macrophages with
		(0.05)		3.36 (0.336)	mixed M1/M2 phenotype to the
						proliferation of tumour cells
	d t1	10 -2 -10 0	1 day•vol	6.72 × 10 -2 -	Tumour killing rate by M1
		(0.2)		6.72 (1.344)	macrophages
	d t2	5 × 10 -3 -	1 day•vol	3.36 × 10 -2 -	Tumour killing rate by macrophages
		0.5 (0.1)		3.36 (0.672)	with mixed M1/M2 phenotype
	p m1	0.487 -0.88	1 day	0.487 -0.88	Proliferation rate of M1 cells
		(0.7)		(0.7)
	p m12	0.487 -0.88	1 day	0.487 -0.88	Proliferation rate of macrophages
		(0.7)		(0.7)		with mixed M1/M2 phenotype
	p m2	0.487 -0.88	1 day	0.487 -0.88	Proliferation rate of M2 cells
		(0.7)		(0.7)
	K m	6.72	vol	1		Macrophages carrying capacity
	d m1	0.83-0.924	1 day	0.83 -0.924	Natural death rate of M1 cells
		(0.87)		(0.87)
	d m12	0.14-0.83	1 day	0.14 -0.83		Natural death rate of cells with
		(0.23)		(0.23)		mixed M1/M2 phenotype
	d m2	0.09-0.14	1 day	0.09 -0.14		Natural death rate of M2 cells
		(0.1)		(0.1)
	α m1	10 -5 -10 -2	1 day	10 -5 -10 -2	Polarisation rate of M1 cells towards
		(0.001)		(0.001)		a mixed M1/M2-phenotype
	α m12

  α m21 K m (α m21 + d m12 -p m12 ) (p m1 -p m12 )α m21 -d m1 p m12 + d m12 p m1 , (d m12 + α m21 -p m12 )(-d m1 p m12 + p m1 (d m12 + α m21 )) p m12 (p m12 (d m1 + α m21 ) -p m1 (d m12 + α m21 )).α m2 α m21 K m p m2 (d m2 + α m2 -p m2 ) ((-d m1 -α m21 )α m2 -d m1 (α m21 + d m12 ))p 2 m2 -p m1 p m12 (d m2 + α m2 ) 2 + (d m2 + α m2 )(p m1 α m2 + (α m21 + d m12 )p m1 + d m1 p m12 )p m2

	* M 1 , u * M 12 , u * M 2 ) = (0, u * M 1 , u * M 12 , 0),
	with
	u * M 1 =
	u * M 12 K m • Tumour-Free, Immune response Present (TFIP) state: (u * = T , u * M 1 , u * M 12 , u * M 2 ) = (0, u * M 1 , u * M 12 , u * M 2 ), with
	u * M 1 =

  11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44

		 a
	* T , u * M 1 , u * M 12 , u * M 2 ) =	  

Appendix A. Spatially-homogeneous steady states

We have seen in Figures 7(d) and 8(c),(d) that changes in some parameters can lead to lower tumour sizes in the long term (for α m21 , α m2 > 0), in contrast to the case α. To understand better this long-term dynamics of model (1), in the following we summarise the steady states exhibited by this model by focusing on two case: (i) the baseline case characterised by α m2 = α m21 = 0, and (ii) the treatment case characterised by α m2 , α m21 > 0.

• The TFIF state is always unstable.

• The TFM2P state is always unstable when it exists (i.e., for p m2 > d m2 ).

• The TFM12P state is stable provided that dm12 pm12 < min{ dm2 pm2 , dm1 pm1 }, and

• The TFM1P state is stable provided that dm1 pm1 < min{ dm2 pm2 , dm12 pm12 } and

• The TPIF state is stable provided that p m2 < d m2 and p m1 < d m1 and

• The TPM2P state is stable provided that dm2 pm2 < min{ dm1 pm1 , dm12 pm12 }.

The stability of TPM12M2P and TPIP states (for the case

is more difficult to investigate for general parameters. Also the stability of the steady states corresponding to the case α m2 , α m21 > 0 is difficult to investigate for the general parameters (the exceptions being the TFIF and TFM1P states, whose stability is given by the same conditions as in Proposition 3). For these reasons, the following Remark summarises the stability of all steady states for the baseline parameter values in Table 2.1.

Remark 4. For the baseline parameter values given in Table 2.1 (with α m2 = α m21 = 0), the steady states TFM1P, TPM12M2P and TPIP do not exist.

The stability of the existent steady states is as follows: the TFIF state is unstable (saddle), the TFM2P state is unstable (saddle), the TFM12P state is unstable (saddle), the TPIF state is unstable (saddle), and the TPM2P state is stable (node). These results explain the dynamics observed in Figure 4, where the solutions approach the only stable steady state: the TPM2P state.

For the case α m2 = α m21 = 0.1 (and the rest of parameters as in Table 2.1), the steady state TFM1P does not exist.

The stability of the existent steady states is as follows: the TFIF state is unstable (saddle), the TFM2F state is unstable (saddle), the TFIP state is unstable (saddle), the TPIF state is unstable (saddle), and TPIP is stable (node). This explains the dynamics observed in Figure 7(d), where the solutions approach the only stable steady state: the TPIP state.

[1] D. Carbone, D. Gandara, S. Antonia, C. Zielinski, L. Paz-Ares, Non-small cell lung cancer: role of the immune system and potential for immunotherapy, J. Thorac. Oncol. 10 (7) (2015) 974-984.

[2] C. Gridelli, A. Rossi, D. Carbone, J. Guarize, N. Karachaliou, T. Mok, F. Petrella, L. Spaggiari, R. Rosell, Non-small-cell lung cancer, Nature

Reviews 1 (2015) 15009.

[3] B. Stankovic, H. Bjøhovde, R. Skarshaug, H. Aamodt, A. Frafjord, E. Müller, C. Hammarström, K. Beraki, E. Baekkevold, P. Woldbaek,


Kia Dashtipour
email: k.dashtipour@napier.ac.uk

Mandar Gogate
email: m.gogate@napier.ac.uk

Alexander Gelbukh
email: gelbukh@gelbukh.com

Amir Hussain
email: a.hussain@napier.ac.uk

Adopting Transition Point Technique for Persian Sentiment Analysis

Keywords: Sentiment Analysis, Persian, Machine Learning

Sentiment analysis is used to analyses people's opinions, views and emotions towards different entities such as products, organizations, companies and events. People's opinions are important for most others during their decision-making process. For example, if someone wants to buy a product, they might want to know more about that product and the experiences of others with that product. Sentiment analysis is able to classify the reviews based on their polarity; even if reviews are expressed in a sentence or document, sentiment analysis is used to classify it into positive, negative or neutral reviews. In this paper, we proposed a framework using TF-IDF and transition point to detect polarity in Persian movie reviews. The proposed approach has been evaluated using different classifiers such as SVM, Naive Bayes, MLP and CNN. The experimental results show the transition point is more effective in comparison with traditional feature such as TF-IDF.

I. INTRODUCTION

With the emergence of Internet, people is capable to share their comments and opinion using social network such as Twitter, Facebook, etc. Internet allow people to debate and share their opinions on different topics [START_REF] Gogate | Cochleanet: A robust language-independent audio-visual model for real-time speech enhancement[END_REF]- [START_REF] Ahmed | Offline arabic handwriting recognition using deep machine learning: A review of recent advances[END_REF]. There has been interest from companies and organisations in classifying opinion to obtain useful information about their product and services. In other words, it identifies whether a sentence is positive or negative [START_REF] Dashtipour | Multilingual sentiment analysis: state of the art and independent comparison of techniques[END_REF]- [START_REF] Adeel | Lip-reading driven deep learning approach for speech enhancement[END_REF]. The sentiment analysis is useful for companies to automatically determines the polarity of the sentence. Sentiment analysis can identify the overall polarity of thousands of comments on short time of period. The evaluation of comments allows companies to improve their product and services. Thus, government used sentiment analysis to understand public's opinion about services which government provides for people . Sentiment analysis has been classified into predicted class such as subjective or objective, predict polarity which can be positive, negative or neutral or level of classification such as phrase, sentence or document and different approach can be applied such as supervised or unsupervised [START_REF] Jiang | Robust visual saliency optimization based on bidirectional markov chains[END_REF]- [START_REF] Cambria | Senticnet 6: Ensemble application of symbolic and subsymbolic ai for sentiment analysis[END_REF].

Sentiment analysis consist of two different approaches, the first approach is based on machine learning classifiers. It is used to trained classifier based on the predefined classes, this method is called supervised classification. The second method is based on the words along with their polarities (+1 to -1), the model calculates the polarity of the sentence from polarity of the word and assign final polarity into sentence, this is called unsupervised method (Joshi et al. 2017). The supervised classification has received high performance; however, it has some shortcomings. It required to build a trained dataset and labelled documents by human expert. The process is very time consuming and it is difficult to train sarcasm and idioms. Moreover, the model can be built for multi-domain purposed, in this case, the performance of the approach is become lower because trained data is not applied for different domain [?], [START_REF] Dashtipour | Exploiting deep learning for persian sentiment analysis[END_REF], [START_REF] Ahmed | Deep neural networkbased contextual recognition of arabic handwritten scripts[END_REF], [START_REF] Dashtipour | A comparative study of persian sentiment analysis based on different feature combinations[END_REF].

The lexicon based approach can be used to identify the polarity of the sentence, the lexicon based can be built manually which requires lots of effort or automatically which is fast and less human effort required [START_REF] Dashtipour | An ensemble based classification approach for persian sentiment analysis[END_REF]- [START_REF] Dashtipour | Public perception towards fifth generation of cellular networks (5g) on social media[END_REF]. The main challenges of sentiment analysis research are most of current approaches is used for English language. However, with growth of the Internet, people share their opinions and comments in different languages. Therefore, sentiment analysis is valuable for other languages such as Persian.

It is a difficult task for SA to understand the text structure for Persian language. For example, the following sentence, ("I do not like the movie direction, but I really like the acting"), contains positive and negative sentiment about the movie. Most of the current approaches are not able to detect the overall polarity of the sentence, the traditional system is unable to understand the different sentiment expressed in the sentence. Additionally, the presence of negation can flip the polarity from positive into negative. In addition, The online users can discuss various topics. For example, if they are talking about one product [START_REF] Dashtipour | Adaptation of sentiment analysis techniques to persian language[END_REF], [START_REF] Ozturk | A novel deep learning driven, low-cost mobility prediction approach for 5g cellular networks: The case of the control/data separation architecture (cdsa)[END_REF]. Alternatively, they can discuss about other product. Additionally, they can discuss about different features of the product.

, ("The charger of the Apple mobile is very good, but I do not like the charger for Apple laptop"). The sentence aspect is charger and the sentiment of the sentence for mobile is positive however, the second part of the sentence has a negative sentiment about the laptop [START_REF] Alqarafi | Toward's arabic multi-modal sentiment analysis[END_REF]- [START_REF] Dashtipour | Persent 2.0: Persian sentiment lexicon enriched with domain-specific words[END_REF].

There are lots of meaningful information is behind the text. For example, , "It was not a bad movie, but if they are using different cast it could be a better movie". The natural language processing is used to understand the text. The use of keywords, punctuation and frequency of words are very useful to understand the text. However, the increase of web content in different languages make the English algorithm inefficient. In the order to overcome these challenges in this thesis we focus on Persian SA. Most of the previous studies have only focused on English SA and therefore most of the tools and resources are available in English. Hence the tools and research, in the most of the other languages such as Persian, are comparatively less developed [START_REF] Dashtipour | A novel context-aware multimodal framework for persian sentiment analysis[END_REF]- [START_REF] Churcher | An experimental analysis of attack classification using machine learning in iot networks[END_REF]. Persian SA has the following main challenges: Lack of tools and resources: To the best of our knowledge, there is no valuable tools nor any online lexicon available for Persian language.

Utilizing many informal words: There are lots of informal words available in Persian language. Specifically, when online users share their opinion and comments online, they are using more of these informal words and phrases.

Lack of comprehensive approaches:

There is no valuable comprehensive approach available in Persian language. Most of the current approaches utilize the available approaches of English and translate the Persian dataset into English. Therefore, In this paper, the supervised machine learning approach used to identify the sentiment classification of Persian dataset and TF-IDF has been used with n-gram features, this information is more effective than frequency evaluated by our experiment results and transition point is used on n-gram features. The SVM, Naïve Bayes, MLP and CNN classifiers used to evaluate the performance of the proposed approach. This paper is organised as follows: Section II is related work of English and Persian sentiment analysis approaches, Section III is present methodology of supervised approach for Persian movie reviews, Section IV provides results and discussion and Section V is conclusion of the paper

II. RELATED WORK

With the growth of Internet, there are various types of online reviews are available such as product and movie reviews, these reviews are very helpful for companies. Sentiment analysis aim to classify and detect vital information from these reviews. In this section, the current Persian and English sentiment analysis has been summarised.

A. English

Agarwal et al. [START_REF] Agarwal | Conceptlevel sentiment analysis with dependency-based semantic parsing: a novel approach[END_REF] proposed concept parser based on the dependency relationship of words in the sentence, after filtering the words Mrmr feature selection has been used to train feature for machine learning classifiers, the proposed model used to classify the document into positive and negative. The movie reviews have been used to evaluate the performance of the approach, the overall accuracy is 88.9%.

Da Rocha et al. [START_REF] Da Rocha | Polarity classification on web-based reviews using support vector machine[END_REF] proposed a hybrid approach to identify the sentiment of the sentence, it consists of two phase feature selection method, the main step is pre-processing, then use of feature selection method, in order to evaluate the performance of the approach the movie review dataset has been used and SVM classifier has been trained. The overall performance of the approach is 81%.

Dashmukh et al. [START_REF] Deshmukh | Entropy based classifier for crossdomain opinion mining[END_REF] proposed a multidomain lexicon to predict the polarity of the sentence, the proposed algorithm used entropy with modified quantity instead of traditional entropy algorithm, the dataset contains different domains of product reviews, there are multiple experiments have been carried out to analyse the proposed approach, the overall accuracy of the proposed approach is 81.66% Sentiment analysis is used to classify document and sentence.

Hung et al. [START_REF] Hung | A performance comparison of feature extraction methods for sentiment analysis[END_REF] proposed an approach to classify the document based on the polarity, the document has been transferred into set of features such as unigram, bigram, trigram and part-of-speech tag and SentiWordNet has been used to assign polarity to features, the result of experiment display the bigram and trigram is performed better in comparison with other features.

Poria et al. [START_REF] Poria | Deep convolutional neural network textual features and multiple kernel learning for utterance-level multimodal sentiment analysis[END_REF] proposed approach to extract features from short texts, based on the inner layer of convolutional neural network, the extracted features of multimodal sentiment analysis have been used and then the feature vectors of text, audio and video has been combined, the proposed approach improved the performance by 14% and the overall accuracy of the approach is 88.60%.

B. Persian

Saraee and Bagheri et al. [START_REF] Saraee | Feature selection methods in persian sentiment analysis[END_REF] proposed a model sentiment classification of reviews documents in Persian, the model is based on pre-processing and feature selection and Naïve Bayes has been used to evaluate the performance of the approach, the mobile reviews have been used and overall performance of the approach is 81.02%.

Ebrahimi et al. [START_REF] Rashed | A supervised method for constructing sentiment lexicon in persian language[END_REF] proposed an approach to detect polarity in Persian sentences,the linguistic features have been extracted and mutual information has been used to identify the polarity of the sentence using SentiStrength of words. To evaluate the performance of the approach online reviews has been used, the result of SentiStrength has been compared with SentiWordNet and the overall accuracy is 80%.

Bagheri et al. [START_REF] Bagheri | Persian sentiment analyzer: a framework based on a novel feature selection method[END_REF] proposed sentiment classification approach for document level in Persian language, there are various feature types such as document frequency, term frequency has been used, and Naïve Bayes has been trained, the mobile reviews have been used to evaluate the performance of the approach, the overall accuracy of the approach is 0.907%.

III. METHODOLOGY

Pre-processing: The pre-processing step consists of cleaning text from unnecessary data, for example )It is really goodddd movie), the normalisation of data used to clean the data, removing stop-words is useful to improve the performance of the dataset, for example some words can be removed because these words do not contain any polarity such as , ) ,From, to, etc.).

N-grams:

The n-gram is sequence of n items in the text. The n-gram size one is called unigram, size two is bigram and size three is trigram. For example, "I went to school". The unigram is "I", "went", "to", "school", the bigram is "I went", "went to", "to school" and trigram is "I went to", "went to school".

TF-IDF: TF-IDF is stand for term frequency-inverse document frequency and it is statistical measure to calculate the importance of words in document, the importance of the words will be calculating with number of times which words appear in the document [START_REF] Chen | Turning from tf-idf to tf-igm for term weighting in text classification[END_REF].

The TF-IDF is consist of two parts, first part calculates the term frequency which is number of word appear in document divided by total number of words in the document and second term is inverse document frequency calculate the logarithm of number document divided by number of document which the term appears in them.

F requency = N umberof document totalnumberof term (1)
IDF: Inverse document frequency is measure the importance of the words in the whole documents, the following equation is used to weight the term.

F requency = N umberof document numberof documenttermappear (2)
Transition Point: The transition point is a frequency values to split the words in document into two different sets low and high. The transition point used to identify the medium frequency which is closely related to the conceptual content of the document.

IV. RESULTS AND DISCUSSIONS

Persian movie reviews Dataset: In order to evaluate the performance of the proposed framework the Persian movie reviews dataset has been used, the movie review contains 1000 positive and 1000 negative. The movie review has been collected from www.caffecinema.com and www.cinematicket.org, the movie reviews consist of movie reviews from 2014 to 2016.

In order to evaluate the performance of the proposed approach, the dataset is pre-processed, after normalisation and removing stop-words. The features such as unigram, bigram and trigram is extracted, then frequency, tf-idf and transition point used to train the SVM, Naïve Bayes, MLP and CNN. The Table 1 shows the result on TF-IDF and frequency (Freq) using different features such as unigram, bigram and trigram. The 10-fold cross validation used to train SVM, Naïve Bayes, MLP and CNN. The result shows the TF-IDF received better results in comparison with frequency.

In order to evaluate the performance of classification techniques, the following four evaluation metrics are used: precision, recall, f-measure and accuracy.

P recision = T P T P + F P

(3)

Recall = T P T P + F N (4)
F measure = 2 * P recision * Recall P recision + Recall (5)

Accuracy = T P + T N T P + T N + F P + F N (6)
The table V, VI, VII and VIII show the result for IDF transition point using n-gram features. There are SVM, Naïve Bayes, MLP and CNN classifiers is trained. The trigram features received better performance and unigram and bigram

V. CONCLUSION

In this paper, a supervised machine learning approach has been proposed for Persian using unigram feature with TF-IDF and frequency to identify the overall polarity of the Persian movie reviews. The experimental result show the TF-IDF is performed more effectively in comparison with frequency. The IDF transition point outperformed TF-IDF and frequency. In our future work, we will developed an approach to identify the overall polarity of multilingual sentences in Arabic, Persian and English.

TABLE I COMPARISON

 I TF-IDF AND FREQUENCY NAÏVE BAYES

	Features Precision	Recall Fscore Accuracy	Features Precision Recall Fscore Accuracy
	Unigram Freq	0.65	0.66	0.66	66.21	Unigram Freq Bigram	0.79	0.78	0.79	79.3
	Bigram Freq	0.66	0.65	0.66	66.94	Bigram Freq Trigram	0.80	0.80	0.80	81.96
	Trigram Freq	0.67	0.67	0.67	67.85	Unigram Freq Trigram	0.82	0.82	0.82	82.59
	Unigram TF-IDF	0.69	0.69	0.69	69.62	Unigram TF-IDF Bigram	0.83	0.83	0.83	83.95
	Bigram TF-IDF	0.70	0.71	0.72	71.39	Unigram TF-IDF Trigram	0.84	0.83	0.84	84.81
	Trigram TF-IDF	0.75	0.75	0.75	75.64	Bigram TF-IDF Trigram	0.85	0.84	0.85	85.91
					TABLE II
				COMPARISON TF-IDF AND FREQUENCY SVM
	Features Precision	Recall Accuracy	Features Precision Recall Fscore Accuracy
	Unigram Freq	0.69	0.68	0.69	69.8	Unigram Freq Bigram	0.81	0.80	0.81	81.46
	Bigram Freq	0.71	0.70	0.71	71.09	Bigram Freq Trigram	0.82	0.81	0.82	82.94
	Trigram Freq	072	0.71	0.72	72.52	Unigram Freq Trigram	0.83	0.82	0.83	83.63
	Unigram TF-IDF	0.72	0.72	0.72	72.51	Unigram TF-IDF Bigram	0.84	0.83	0.84	84.59
	Bigram TF-IDF	0.75	0.74	0.75	75.94	Unigram TF-IDF Trigram	0.86	0.85	0.86	86.93
	Trigram TF-IDF	0.78	0.77	0.78	78.67	Bigram TF-IDF Trigram	0.89	0.88	0.89	89.76
					TABLE III
				COMPARISON TF-IDF AND FREQUENCY MLP
	Features Precision	Recall Fscore Accuracy	Features Precision Recall Fscore Accuracy
	Unigram Freq	0.75	0.75	0.75	75.69	Unigram Freq Bigram	0.84	0.83	0.84	84.69
	Bigram Freq	0.77	0.76	0.77	77.92	Bigram Freq Trigram	0.85	0.86	0.85	85.67
	Trigram Freq	0.77	0.78	0.79	77.5	Unigram Freq Trigram	0.86	0.85	0.86	86.43
	Unigram TF-IDF	0.79	0.79	079	79.53	Unigram TF-IDF Bigram	0.88	0.84	0.86	86.59
	Bigram TF-IDF	0.80	0.79	0.80	79.82	Unigram TF-IDF Trigram	0.89	0.89	0.89	89.61
	Trigram TF-IDF	0.81	0.81	0.81	81.79	Bigram TF-IDF Trigram	0.91	0.91	0.91	91.84

Funding: Amir Hussain was supported by the UK Engineering and Physical Sciences Research Council (EPSRC)-Grants Ref. EP/M026981/1, EP/T021063/1, EP/T024917/1.


Kia Dashtipour
email: k.dashtipour@napier.ac.uk

Mandar Gogate
email: m.gogate@napier.ac.uk

Alexander Gelbukh
email: gelbukh@gelbukh.com

Amir Hussain
email: a.hussain@napier.ac.uk

Persian Sentence-level Sentiment Polarity Classification

Keywords: Sentiment Analysis, Persian, Machine Learning

Assigning positive and negative polarity into Persian sentences is difficult task, there are different approaches has been proposed in various languages such as English. However, there is not any approach available to identify the final polarity of the Persian sentences. In this paper, the novel approach has been proposed to detect polarity for Persian sentences using PerSent lexicon (Persian lexicon). For this, we have proposed two different algorithms to detect polarity in the sentence and finally SVM, MLP and Naïve Bayes classifier has been used to evaluate the performance of the proposed method. The SVM received better results in comparison with Naïve Bayes and MLP.

I. INTRODUCTION

Analysing of web content is becoming very important due to augmented communication via Internet, and the number of online opinions has been increased. Therefore, the sentiment analysis has been introduced to classify the data based on their polarity. The sentiment analysis is automated application that can be used in different field to structure the data [START_REF] Dashtipour | Persian named entity recognition[END_REF]- [START_REF] Shiva | Complex-valued computational model of hippocampal ca3 recurrent collaterals[END_REF]. Analyse of sentiment has been spread into different fields such as marketing, sport, movies, news etc. There are lots of opinions available online about different product and service, and sentiment analysis can help users to make better decision by recommending pros and cons of a product. For example, if someone decide to buy a laptop, they can read these online opinions and decide which laptop is better [START_REF] Gogate | Cochleanet: A robust language-independent audio-visual model for real-time speech enhancement[END_REF]- [START_REF] Nisar | Cognitively inspired feature extraction and speech recognition for automated hearing loss testing[END_REF]. It is noted that 90% of users' reads online reviews and opinions before they make buying decision [START_REF] Dashtipour | Exploiting deep learning for persian sentiment analysis[END_REF]- [START_REF] Dashtipour | A novel context-aware multimodal framework for persian sentiment analysis[END_REF]. The recent studies focused on issues in different types of opinions to determine the knowledge gaps and discover the future challenges [START_REF] Dashtipour | A hybrid persian sentiment analysis framework: Integrating dependency grammar based rules and deep neural networks[END_REF]- [START_REF] Dashtipour | Multilingual sentiment analysis: state of the art and independent comparison of techniques[END_REF].

There is huge amount of data available online in different languages and in unstructured form that need to be converted into useful form. To overcome this challenge most of approaches has been proposed in English language and other languages e.g. Persian. Where Persian language is less developed to overcome this challenge, therefore, this paper focused on identifying the polarity for Persian sentences [START_REF] Cambria | Senticnet 6: Ensemble application of symbolic and subsymbolic ai for sentiment analysis[END_REF]- [START_REF] Liaqat | Non invasive skin hydration level detection using machine learning[END_REF]. In this paper, the novel framework has been proposed to detect Persian sentences polarity. For these two algorithms has been proposed to detect polarity in the Persian sentences. The first algorithm detects part-of-speech (POS) tag in Persian sentences, it detects tag such as adjective, adverb, noun and verb in the sentence, the second algorithm is used to detect negation in the sentence, PerSent lexicon has been used to assign polarity to the detected POS features. Section II provides the related work, section III is provides the methodology, section IV is experiment and result and finally section V concludes the paper.

II. RELATED WORK

In this section, the related work for sentiment analysis will be discussed, there are various approaches to detect polarity in the sentence.

• Document level: In the document level, the task is to understand the overall polarity of the whole document. For example, the document contains various positive and negative comments about different product, the final polarity of the document will be identified. • Sentence level: The task is determining whether the expressed sentence is positive, negative or neutral [START_REF] Dashtipour | An ensemble based classification approach for persian sentiment analysis[END_REF], [START_REF] Alqarafi | Toward's arabic multi-modal sentiment analysis[END_REF]. • Aspect level: The task in aspect level is to identify the different aspect of the sentence, for example, ("I like the restaurant service but I did not like the food"), the tone of sentence is negative but some aspect is not negative, the aspect level extract these features from sentence, the "restaurant service" and "food" will be extracted from sentence.

Ikeda et al, [START_REF] Ikeda | Learning to shift the polarity of words for sentiment classification[END_REF] proposed an approach for detect polarity in English for movie reviews. First the data is collected, after pre-processing, there are different features such as unigram, bigram and trigram are selected. The proposed approach achieved accuracy of 76%.

Choi and Cardie etl at, [START_REF] Choi | Adapting a polarity lexicon using integer linear programming for domain-specific sentiment classification[END_REF] proposed an approach to identify the polarity of the sentence using novel technique which is used lexicon to find the characteristic of data directly. This method is used to find the relation and characteristic between words and sentences. The proposed system received better performance to detect negative sentences in compare to positive sentence. Alamoudi et al, [START_REF] Alamoudi | Sentiment classification and aspect-based sentiment analysis on yelp reviews using deep learning and word embeddings[END_REF] proposed a novel model based on the rating-based feature and evaluate features using English document sentiment analysis. There are n-gram features has been extracted to evaluate the performance of the system. The overall performance of the system was 93.24%.

Basiri et al. [START_REF] Basiri | Sentence-level sentiment analysis in persian[END_REF] addressed the problem of lack of resources in Persian by developing a lexicon and an automatically labelled sentence-level corpus. A Naive Bayes classifier was trained on the collected corpus to determine the polarity in short sentences. However, the proposed approach is lack of comparison results with the state-of-the-art approaches. In addition, Ebrahimi et al. [START_REF] Rashed | A supervised method for constructing sentiment lexicon in persian language[END_REF] proposed a method to detect polarity in Persian online reviews based on adjectives extracted from the sentence and translated SentiWordNet lexicon. However, the proposed method is limited to adjectives and it does not exploit nouns, adverbs and verbs that provide extra information on the underlying sentiment. Moreover, Razavi et al. [START_REF] Razavi | Word embedding-based approach to aspect detection for aspect-based summarization of persian customer reviews[END_REF] detect polarity based on extracted nouns and adjectives and a Persian lexicon. However, the proposed approach do not exploit the word order and hierarchical semantic dependency.

Recently, Zhao et al. [START_REF] Zhao | Weakly-supervised deep embedding for product review sentiment analysis[END_REF] proposed a model to detect sentiment in product reviews using deep learning classifiers. The words are first converted into an embedding vector representation and then a deep learning classifier is used to classify sentiments. The experimental results showed the effectiveness of their proposed CNN (87%) as compared to LSTM (82%). Sohangir et al. [START_REF] Sohangir | Big data: Deep learning for financial sentiment analysis[END_REF] proposed a framework to detect sentiment in the financial market. The StockTwits dataset was used, and feature selection method such as chi-square was used to select features. Finally CNN classifiers were used to evaluate the performance of the approach. The proposed framework obtained an accuracy of 70.88%. Wagh et al. [START_REF] Wagh | A twitter sentiment analysis using nltk and machine learning techniques[END_REF] proposed an approach to identify the sentiment in tweets and twitter API utilized to collect the tweets. Different classifiers such as Naive Bayes and Logistic Regression were used to evaluate the performance of the approach. Experimental results showed that logistic regression achieved better accuracy as compared to Naive Bayes. Pitsilis et al. [START_REF] Pitsilis | Detecting offensive language in tweets using deep learning[END_REF], proposed a framework to discern hateful content in social media using LSTM classifier. The authors collected data from Twitter and fed extracted features into an LSTM classifier. The results demonstrated the effectiveness of LSTM as compared to SVM.

The table I summarise some of the existing approaches of sentiment classification polarity for various languages.

III. METHODOLOGY

In this section, we will discuss about proposed framework to detect polarity in Persian language sentences. The figure below display the proposed framework to identify the polarity of Persian language sentences.

The algorithm shows in Fig 2 has been proposed to identify the final polarity of Persian sentence. The JHAZM part-ofspeech tag of tag has been used to identify the tag for Persian sentence.

The algorithm shows in Fig 3 has been proposed to identify the negation in the Persian sentence, if word is appeared in the sentence with positive polarity, change the overall polarity into negative and if this word is appearing with negative polarity change polarity into positive.

Pre-processing: The pre-processing step is consisting of tokenisation, normalisation and stop words removal. The tokenisation is processing of breaking texts into single tokens, normalisation is processing of removing noised in the text. For example, (The movie was greattttt) will be changed into (The movie is great) and stop word is removing unnecessary words such as (from, to etc.). PerSent lexicon: In order to identify the polarity of Persian sentences, the PerSent lexicon will be used, the lexicon contains 1500 Persian words along with their part-of-speech tag and their polarity [START_REF] Dashtipour | Persent 2.0: Persian sentiment lexicon enriched with domain-specific words[END_REF]- [START_REF] Hussien | Comparison of sentiment analysis approaches using modern arabic and sudanese dialect[END_REF].

Persian polarity: In order to calculate the overall polarity of the sentence, the noun, adjective, adverb and verb are extracted from the sentence. The tag such as determiners are discarded. The following equation has been used:

Sp = N p + V p + AV p + ADJp n (1)
Sp denote the sentence polarity, N-p denote the noun polarity, Vp denote verb polarity, AV-P is adverb polarity and ADJ-P denotes adjective polarity. For example, (The movie was great), NOUN, ADJECTIVE, -VERB. The adjective and verb is extracted and the PerSent lexicon will used to identify the polarity of the sentence, the formula number one has been used to calculate the overall polarity of the sentence. Once the polarity of the sentence has been identified the following rules has been used to identify the whether sentence is positive, negative or neutral.

If sentence polarity is greater than 0 then the polarity of the sentence is positive, if it is less than 0 is negative and equal to 0 is neutral.

IV. RESULTS AND DISCUSSIONS

Dataset: The Persian movie review dataset has been used to evaluate the performance of the proposed approach, the table below display the result of proposed approach. The JHAZM is Python tools for Persian NLTK contain part-of-speech tag, dependency parser, normalisation and stemming. The movie reviews contain 1000 positive and 1000 negative sentences. The movie has been collected from www.caffecinema.com and The adjective is performed better in comparison with other part-of-speech tag because the adjective is identified the polarity of the sentence. The weakness of proposed approach for Persian sentence polarity detection is not able to handle sarcasm in the sentence, it is required to develop system to detect sarcasm in the sentence, the further study is required to detect ironic and sarcasm in the sentence to improve the performance of the approach. The Table 4 display the example of Persian sentence polarity detection approach, the input sentences, English translation and final polarity of the sentence.

V. CONCLUSION

In this paper, a novel approach has been proposed to identify the polarity of the Persian sentences, the approach has been proposed to extract POS tag of the Persian sentence, then PerSent lexicon (Persian lexicon) has been used to assign polarity to extracted features and SVM, MLP and Naïve Bayes classifiers has been used to evaluate the performance of the approach. In the future work, we will develop an approach of multilingual polarity detection for Persian, Arabic and English.

Funding: Amir Hussain was supported by the UK Engineering and Physical Sciences Research Council (EPSRC)-Grants Ref. EP/M026981/1, EP/T021063/1, EP/T024917/1.

Fig. 1 .

 1 Fig. 1. Proposed Framework for Persian Sentiment Analysis