A phenomenological model for cell and nucleus deformation during cancer metastasis
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Abstract
Cell migration plays an essential role in cancer metastasis. In cancer invasion through confined spaces, cells must undergo extensive deformation, which is a capability related to their metastatic potentials. Here, we simulate the deformation of the cell and nucleus during invasion through a dense, physiological microenvironment by developing a phenomenological computational model. In our work, cells are attracted by a generic emitting source (e.g., a chemokine or stiffness signal), which is treated by using Green’s Fundamental solutions. We use an IMEX integration method where the linear parts and the nonlinear parts are treated by using an Euler backward scheme and an Euler forward method, respectively. We develop the numerical model for an obstacleinduced deformation in 2D or/and 3D. Considering the uncertainty in cell mobility, stochastic processes are incorporated and uncertainties in the input variables are evaluated using Monte Carlo simulations. This quantitative study aims at estimating the likelihood for invasion and the length of the time interval in which the cell invades the tissue through an obstacle. Subsequently, the twodimensional cell deformation model is applied to simplified cancer metastasis processes to serve as a model for in vivo or in vitro biomedical experiments.
Keywords
Cell deformation Nucleus deformation Monte Carlo simulations Cancer metastasis Cellbased model1 Introduction
Cell locomotion is closely involved in various physiological and pathological processes. For example, migration of leukocytes is important for the inflammatory response and movement of fibroblasts and also vascular endothelial cells are essential for wound healing (Lauffenburger and Horwitz 1996). On the contrary, cell migration can play a detrimental role during cancer metastasis, where the dissemination of cancer cells initializes the invasionmetastasis cascade as introduced by Chambers et al. (2002b), Fidler (2003), Lambert et al. (2017).
The diversity of cancers exceeds 200 distinct disease entities, which have differences in the normal cells of origin and similarities in subsequent cancer metastasis. Compared to primary tumors, metastatic cancers cause the overwhelming majority of cancerassociated deaths as high as 90% (Lambert et al. 2017; Seyfried and Huysentruyt 2013; Gupta and Massagué 2006). During the metastatic spreading of tumors, cancer cells can undergo transitions between two forms of movement, which are the amoeboid mode and the mesenchymal mode to optimize their invasiveness (Paňková et al. 2010; Sahai and Marshall 2003). Moreover, Pinner and Sahai (2008) observe that cancer cells are able to move quickly (up to 15 \(\upmu \mathrm{m/min}\)) like some leukocytes and rapidly change their shapes and directions of migration in an amoeboid manner with intravital confocal microscopy technology. Amoeboid movement could happen in the absence of matrix protease (Wolf et al. 2003; Wyckoff et al. 2006) where cancer cells alternatively generate large contractile force pushing fibers of matrix away and squeeze between small paths. However, if the contractile force is insufficient to deform the stiff extracellular matrix (ECM), the matrix metalloproteases (MMP’s) will be secreted by cancer cells to degrade the ECM and thereby invade further (Kalebic et al. 1983; Wolf et al. 2013). In summary, cancer cells frequently chemically and/or mechanically ‘dig’ their ways through ECM in order to reach the distinct parts of the body.
When a single cancer cell is metastasizing through a narrow cavity, it must deform its morphology by extending its membrane into an elongated protrusion; this is often driven by external signals such as chemotaxis, durotaxis or tensotaxis. Large cell deformations will also induce changes in the nucleus morphology. Extensive deformation of the nucleus can induce damage, and reduce the nuclear envelope integrity, see for instance the work by Denais et al. (2016). However, the cancer cell is also capable of repairing its ruptured nuclear envelope and damaged DNA after the penetration. Then, the cell may be able to further promote cancer development. Thus, as noted by Denais et al. (2016), the stage of nuclear envelop rupture could represent a particularly fragile point, thereby providing an opportunity to develop new antimetastatic cancer drugs to inhibit DNA repair and increase cell death. Cell deformation during cancer metastasis has been difficult to study in detail both in vivo and vitro, and further understanding of cell deformation mechanisms is crucially important. In cases where the pore sizes are much smaller than the size of the nucleus, the nucleus mostly arrests and fails to penetrate the pore due to a defective nuclear deformability. On the contrary, with pore diameters above a threshold, e.g., 7 \(\upmu \mathrm{m}\) in the work (Wolf et al. 2013), MMPindependent migration in dense ECM relies on the hourglassshaped deformation of the nucleus. Hence, in our current work, we develop a mathematical model to investigate the correlation between the deformation of a cell and of its nucleus, and show the dynamic changes in cell mechanostructure that occur during the invasion process.
Mathematical modeling has been shown to be an important tool to quantify the relations in many biomedical processes such as wound healing, cell migration and tumor progression in various scales. Cell deformation and migration models exist in the colony scale, e.g., in the works by Rey and GarciaAznar (2013), Byrne and Drasdo (2009), and Vermolen and Gefen (2012), where the cell geometry is fixed to be circular or spherical, respectively, in two and threedimensional simulations. On a smaller scale, one looks at the deformation of individual cells, and to this extent, cellular automata models have been developed and combined with finiteelement strategies by Borau et al. (2014) and Oers et al. (2014). Other cell deformation models are based on phasefield models, like in the work by Marth and Voigt (2014), or on viscoelasticity with moving meshes as in (Madzvamuse and George 2013). A phenomenological approach to cell migration and deformation is proposed in Vermolen and Gefen (2013) and Vermolen et al. (2014), wherein the latter work cell migration and deformation have been modeled in relation with the immune response system where white blood cells migrate out of the venules and transmigrate through the venule walls to chase and engulf pathogens. Moreover, Odenthal et al. (2013) introduce a deformable cell model to describe the mechanical communication among the interacting cells and between the cell and its environment. Another deformable model regarding the interactions with emphasis on the relationship between varying matrix geometries and adhesion, contractility as well as cell velocity can be found in (Tozluoğlu et al. 2013). In terms of the nucleus deformable models, MoussaviBaygi et al. (2011) establish a coarsegrained model of the nuclear pore complex to simulate the nucleocytoplasmic transport. As the increasing attention in the cell mechanics, agentbased models are booming, see (van Liedekerke et al. 2015), where three types agentbased models are described.
Cao et al. (2016) develop a chemomechanical model to investigate the impacts of transmigration through confined interstitial spaces on the geometrical and mechanical features of the cell nuclei. In their model, the shape alterations of the cell and nucleus during the transendothelial migration driven by actomyosin contraction force can perturb genomic organization, which in turn affects the behavior of the cells. More nuclear profiles regarding chromatin deformations and nuclear envelope deformations during transmigration are further investigated. This mechanical model successfully predicts the morphological evolution when one cell transmigrate an endothelial gap (Cao et al. 2016). In comparison, our model extends the process and behavior of cell transmigration driven by a chemical/stiffness signal during cancer metastasis, whereas the most inner cellular mechanical properties are neglected for sake of simplicity.
None of the aforementioned studies, however, have taken into account the Monte Carlo uncertainty quantification in the cell deformation modeling. Our work aims at modeling the interaction between cell deformation (due to migration) and the deformation of the nucleus as well as quantitative analysis of unknown parameters by Monte Carlo simulations. We quantify the correlation between nuclear deformation relaxation and the cell’s ability to penetration through narrow passages, which is important in the context of metastatic invasion. Section 2 describes the mathematical model in terms of the equations, subsequently, the numerical method is presented in Sect. 3, which is followed by the description of the results in Sect. 4. Finally, conclusions are drawn in Sect. 5.
2 The mathematical model
This section introduces the model in terms of the mathematical relations. We start with the deformation of cell and its nucleus in two dimensions and extend the formalism to three spatial dimensions subsequently. Moreover, the model is applied to simplified physiological transmigration of cancer cells and six parameters are studied by Monte Carlo simulations.
2.1 The model in two dimensions
Comparison of CPU time and the cell penetration time \(\tau \)
N  10  30  50  100 

CPU time (s)  2.43  5.07  7.81  14.85 
\(\tau \) (h)  0.3771  0.3735  0.3812  0.3906 
2.2 Extension to three spatial dimensions
2.3 The application to cancer metastasis
3 The numerical method
3.1 Time integration
3.2 Cell shape
3.3 The Monte Carlo simulations
In our model, most experimental data is difficult or even impossible to collect, therefore, we refer to other literature data or estimate the input data and thereby evaluating the quantification of the propagation of uncertainty in the variables is very important. To investigate the output influence and correlation among variables, Monte Carlo simulations are carried out based on the model of cancer metastasis. There, a cell transmigrates through a narrow rough tubular path to get from one part of the surrounding tissue to another part. Passage through the tube requires deformation of the cells’ cytoplasm and nucleus and affects the corresponding penetration time \(\tau \) which is quantified under different conditions.
3.4 Error analysis
Parameter values
Constant  Notation  Value  Unit  Source 

Radius of a circular cell in 2D  R  12.5  \(\upmu \mathrm{m}\)  (Champion and Mitragotri 2006) 
Radius of a spherical cell in 3D  R  10  \(\upmu \mathrm{m}\)  (Champion and Mitragotri 2006) 
Radius of a circular nucleus in 2D  \(R^{ n}\)  5  \(\upmu \mathrm{m}\)  (Friedl et al. 2011) 
Radius of a spherical nucleus in 3D  \(R^{ n}\)  8  \(\upmu \mathrm{m}\)  (Friedl et al. 2011) 
Cell deformation relaxation  \(\alpha \)  250  \(\mathrm{h}^{1}\)  estimated 
Nucleus deformation relaxation  \(\alpha ^n\)  2500  \(\mathrm{h}^{1}\)  estimated 
Diffusivity of the chemokine  D  3600  \(\upmu \mathrm{m}^{2}/\mathrm{h}\)  (Jayaraman et al. 2001) 
Mobility of points on cell membrane  \(\beta \)  60  \(\mathrm{h}^{1}\)  (Vermolen and Gefen 2012) 
Secretion rate of the chemokine  \(\gamma _\mathrm{s}\)  1.2 \(\times 10^6\)  \(\mathrm{mol/h}\, \upmu \mathrm{m}^3\)  (Savinell et al. 1989) 
Time step in 2D  \(\Delta \)t  0.0001  h  (Pinner and Sahai 2008) 
Time step in 3D  \(\Delta \)t  0.01  h  – 
Number of nodes on a 2D cell  N  30  –  – 
Number of circles on a 3D cell  \(N_\mathrm{c}\)  30  –  – 
4 The numerical simulations
First, we describe the simulations in which one cell migrates toward the gradient of an increasing stimulus along obstacles in 2D and 3D. Subsequently, this deformation model of cell and its nucleus is applied to a simplified cancer metastasis phenomenon. Furthermore, six parameters are studied and analyzed by Monte Carlo simulations.
4.1 Parameter values
Most often the experimental parameter values are not available to us, therefore estimating input values based on experimental literature is essential. For example, we use 10 \(\upmu \mathrm{m}\) in 2D and 16 \(\upmu \mathrm{m}\) in 3D for diameters of the nucleus referring to the work by Friedl et al. (2011), where the diameter of the nucleus varies from 10–20 \(\upmu \mathrm{m}\) in 2D and 5–15 \(\upmu \mathrm{m}\) in 3D. Analogously, other default input values are listed in Table 2, as well as the sources from the literature whenever possible.
4.2 Cell migration along a rigid object in 2D and 3D
4.2.1 One cell migrating along a rigid object in 2D
The cell moves according to the gradient of chemokine. Snapshots at different stages of the migration are shown in Fig. 3, where the red, green and gray objects visualize the cell, nucleus and a rigid obstacle, respectively. Furthermore, the signal source location is represented by an asterisk. To pass a stiff barrier or overcome an obstacle, the migrating cell has to reshape and adapt the mechanostructure of the cytoplasm and the membrane. That is done via exerting contractile forces or withstanding the stresses from neighbor cells, which are mediated by the cell cytoskeleton (Brunner et al. 2006). According to the experimental observation of Brunner et al. (2006), one migrating cell could push a small obstacle upward by exerting forces and crawl underneath this obstacle. Given a larger obstacle in our simulation, the cell and nucleus are more likely to crawl along the rigid boundary by morphological adjustments to different extents. Ultimately, the cell and nucleus are able to return to their initial shapes due to cell polarity once the source is no longer active.
4.2.2 One cell moving along a rigid object in 3D
In general, dimensionality does not affect the expected numerical result in this case. Furthermore, the computational time of a 2D model is much shorter as a result of the need for fewer gridpoints on the boundaries of the cell and nucleus, and thereby we use 2D model for further application and analysis in this work.
4.3 Application to cancer metastasis in 2D
There are preexisting openings (pores, fiberlike or channellike tracks) in ECM that enable cancer cells to migrate with an independence of MMP’s (Paul et al. 2017). In this section, we apply the model to the transmigration of cancer cells through pores and channels to migrate from one part to another part of the tissue without degrading ECM.
4.3.1 Simulation on penetration of a cell through a cavity
4.3.2 Simulation on penetration of a cell through a tube channel
Mechanical boundaries could regulate some biomedical processes and Mak et al. (2013) demonstrate that if the confined dimensional modulation of a microfluidic device has a mechanical barrier smaller than the cell nucleus, then metastatic breast adenocarcinoma cells likely deform in elongated morphological states and invade distinct sites. Here, taking mechanical boundaries into account, we use the trigonometric function (from equation (15)) to simulate the different roughnesses through changing the value of parameter \(\epsilon \) and \(\omega \). A highly rough boundary of the channel is defined if the perturbation [see Eq. (15)] has a high frequency or/and a big amplitude, which is determined by the surface of the endothelial cells. Whereas a lower frequency (also a lower amplitude) as we depict in Fig. 6 could show where each cell is located. The discrepancy between the endothelial cellular surfaces and the channel through which the cancer (or immune) cell migrates, could be a consequence of the extracellular matrix around the cells. Then the boundary of the channel can have various roughnesses, which combined with other parameters, are analyzed by using Monte Carlo simulations based on this model. Moreover, this model is incorporated with Poisseuille flow to simulate the micro blood flow referring our work (Chen et al. 2018). To investigate how the cell speed changes in the current scenario, the speed evolution with the respect of time is plotted in Fig. 8a without the perturbation of vector Wiener process. As we expected, the cell speed slows down when it starts to squeeze the opening and subsequently accelerates to move toward the emitting source. When the \(\tau \) equals approximate 0.37 hour, the instantaneous speed reaches a peak and drops to zero after the engulfment of the source and cell shape recovery. During the transmigration in the tube, the cell migrates with a speed vibrating up and down at 200 \(\upmu \mathrm{m/h}\), which is in the range 1–5 \(\upmu \mathrm{m/min}\) for the typical speed of amoeboid movement observed in vivo in the work (Pinner and Sahai 2008). Moreover, the cell speed can be controlled under various conditions, like the number of emitting sources, the diffusion coefficient, cell mobility.
4.3.3 Simulation on cancer metastasis
4.4 Parameter study with Monte Carlo simulations
If certain input values contain uncertainties, Monte Carlo simulations could be a way to evaluate the impacts of output. This method enables us to estimate of the impact from variables ranging from various statistical distributions like Pareto, uniform, normal, lognormal, Chisquare, exponential (Mooney 1997). Furthermore, Monte Carlo simulations have been used over a spectrum of systems, which is typically concluded in following four steps, (1) generate the input random values based on their probability distribution functions; (2) calculate samples; (3) repeat the abovementioned steps with a number of trials \(N_\mathrm{s}\); (4) calculate the mean and construct a relative frequency distribution of the simulated results (Mooney 1997; Mahadevan 1997). Furthermore, one can estimate the correlation between the various input and output parameters.
The model introduced in Sect. 4.3.2 is used in Monte Carlo simulations, with the channel boundary of 60 \(\upmu \mathrm{m}\) in length and approximately 10 \(\upmu \mathrm{m}\) in width. The transit time interval that starts once one of the cell’s boundary points enters the channel and lasts until the last point exits the channel is defined as the penetration time \(\tau \). In this section, the influences of several parameters on the penetration time \(\tau \) are investigated.
As we discussed in Sect. 3.4, the accuracy of the simulation result depends on the number of samples. To achieve an accurate approximation, the number of samples is tested that is shown in Fig. 9.
Note that the axes represent the logarithm of sample count and the mean of transit time, respectively. If the sample count in the Monte Carlo simulations is too small, then the average penetration time has not yet converged (see Fig. 9 for \(N_\mathrm{s} < 200\)). We observe that using 10,000 samples only gives very small fluctuations of the average penetration time (see Fig. 9). The result has converged sufficiently to approximate 0.356 h. However, to evaluate the uncertainty of input data quantitatively, 10,000 samples are chosen in our simulation which give acceptable computation times in the order of hour. Using equation (31), the Monte Carlo error is estimated by \(\Vert E_{\mathrm{mc}}\Vert = \Vert \hat{\tau }^{\Delta t}\hat{\tau }_{N_\mathrm{s}}^{\Delta t}\Vert \simeq \frac{S_n}{\sqrt{N_\mathrm{s}}}\).
4.4.1 Monte Carlo simulations on parameters D, \(\beta \), \(\alpha \), \(\alpha ^n\)
We start with the Monte Carlo simulations on four input parameters which are the diffusion coefficient of the chemokine D, cell point mobility \(\beta \), cell deformation relaxation \(\alpha \) and the nucleus deformation relaxation \(\alpha ^n\). We sample them from the normal distribution, then they can be generated by Eq. (28) with the default values in Table 3.
Parameter values
D  \(\beta \)  \(\alpha \)  \(\alpha ^n\)  

Value  \(N \sim (3600, 30^2)\)  \(N \sim (60, 3^2)\)  \(N \sim (250, 40^2)\)  \(N \sim (2500, 125^2)\) 
4.4.2 Monte Carlo simulations on parameters \(\epsilon \) and \(\omega \)
Analogously, scatter diagrams about \(\epsilon \) and \(\omega \) with penetration time \(\tau \) indicating their correlations are shown in Fig. 14. With the increase in roughness, one cell travels a longer time to penetrate the channel in most cases. Furthermore, the increment of \(\omega \) makes a contribution to the total travel time of one cell. This is also reflected by the correlations of \(r = 0.4310\) and \(r = 0.7100\) between the penetration time and \(\epsilon \) and \(\omega \), respectively.
4.4.3 Monte Carlo simulations on parameters D, \(\beta \), \(\alpha \), \(\alpha ^n\), \(\epsilon \) and \(\omega \)
To investigate the impacts of variables on output results and analyze the correlations of each variable with penetration time \(\tau \), a couple of scatter plots are shown in Fig. 16, respectively. Adding some control variables that are statistically distributed yields more uncertainty to the system. The increase in uncertainty generally decreases the correlation. Therefore, in current simulation of six parameters, the correlation of parameters D, \(\beta \), \(\alpha \), \(\epsilon \) and \(\omega \) with time \(\tau \) decrease slightly compared with the simulations with the variation of four parameters. The correlation between \(\tau \) and \(\alpha ^n\) is still negligible. Further, Fig. 16 shows that the roughness (\(\epsilon \) and \(\omega \)) dominantly influences the cell travel time.
5 Discussion and conclusions
In this work, we develop a cellbased model to describe the morphological evolution of the cell and nucleus in a phenomenological way. The cell cytoskeleton spanning between the nucleus and the cell membrane is simulated by 30 springs. As we expected, an immune cell or a single cancer cell can deform according to the specific obstacles or paths when it encounters a stiff obstacle in a 2D or 3D environment. Compared with some existing models, e.g., a model investigating the role of nucleus deformation in the cell deformation under different geometrical and fluid flow conditions (SerranoAlcalde et al. 2017) and a threedimensional model describing nucleus mechanics during cell migration and deformation (Giverso et al. 2018), one of the major advantages of our modeling is its efficiency regarding CPU time, which enables to carry out Monte Carlo simulations for evaluation of parameter sensitivity. A further merit of the current model is its simplicity. If one is able measure the velocity of points on the surface of the cell under the influence of (the gradient of) a generic (being a concentration or a stiffness for instance) signal, then the \(\beta \)parameter can be determined. If one further is able to measure the retraction speed on the boundaries of the cell and the nucleus once the signal has disappeared, then it can fit the \(\alpha \) parameters.
The uncertainties in the input values necessitate us to study the impact of uncertainty by carrying out Monte Carlo simulations. With 10,000 samples, the correlations of each variable D, \(\beta \), \(\alpha \), \(\alpha ^n\), \(\epsilon \) and \(\omega \) with cell penetration time \(\tau \) are analyzed. The results show that \(\alpha ^n\) has no significant correlation with the penetration time in current situation, where the reason probably is the low range of parametric values in our simulations. A larger range, with variations over a lognormal distribution could give a higher correlation. The use of very high values of \(\alpha ^n\) in the model when the cell is penetrating through an aperture needs more investigation. Moreover, SerranoAlcalde et al. (2017) state that a small cell nucleus does not play a crucial role in cell deformabilitybased experiments under a fluid flow. Therefore, the deformability of the nucleus could be impacted by the size of the nucleus, and thereby influence the penetration time. Whereas, other variables influence the cell penetration time \(\tau \) to varying degrees, where the correlation of roughness is the most significant.

Compared to a 2D model, a 3D model is more physiological, however, there is no significant qualitative difference in terms of expected numerical results. Moreover, taking the Monte Carlo simulations into account, the CPU time for simulating the 2D model is much more reasonable. However, a 3D model will still be an interesting research direction in the future.

Amoeboid and mesenchymal movement, as the two basic forms of cell locomotion, mutually transform and participate in the process of cell migration. The former is also called pseudopodia movement including lamellipodia and filopodia, which normally takes place close to the cell front as a result of cell polarization (Lauffenburger and Horwitz 1996; Lämmermann and Sixt 2009; Paul et al. 2017). Since the interconversion between the amoeboid model and the mesenchymal model due to the cytoskeleton rearrangement happens during cancer cell migration (Zhao et al. 2011), the filopodia that is an extension of active membrane of cell front and rear might be considered in future work.
 In the current work, we define constant values for the cell deformation relaxation \(\alpha \) and cell mobility \(\beta \) everywhere, while they in general depend on chemokines. Therefore, to introduce surfaceresident chemical species, some surface partial differential equations can be incorporated such that it describes the evolution of the chemical signals over the membrane surface. This amounts to solvingThis is an interesting and relevant research direction, which will be taken into consideration in future work.$$\begin{aligned} \begin{aligned}&{\underline{a}}_t + \nabla _{\Gamma } \cdot (\mathbf{v} {\underline{a}})  D_a \Delta _{\Gamma } {\underline{a}} = {\underline{f}} ({\underline{a}}), \\&\mathbf{v} = \frac{\mathrm{d}}{\mathrm{d}t} {\underline{x}}(t), \qquad (t,\mathbf{x}(t)) \in \mathbb {R}^+ \times \Gamma (t). \end{aligned} \end{aligned}$$(36)
 A tumor is typically surrounded by a dense network of collagen fibers, which are normally utilized by motile cancer cells to guide their paths (Sahai 2007). Furthermore, mutated cancer cells are capable of remodeling the normal ECM around them, abnormal ECM or the density of fibers preferably reshapes aligned direction in a parallel arrangement, which forms an anisotropic medium and thereby has a significant impact on cell migration. If we formalize this directional dependence through the socalled orientation tensor \(\Psi \). Then we get the following revision on the response to the external signal of the migration equations:where \(\beta _0\) and \(\beta _1\) are two constants and \(\Psi \) can be obtained by$$\begin{aligned} \begin{aligned} \mathrm{d}{} \mathbf{x}_i(t) =\,&(\beta _0 \mathbf{I} + \beta _1 \Psi ) \nabla c(t,\mathbf{x_i}(t)) \mathrm{d}t + \alpha ( \mathbf{x}_i^n(t)\\&+ \hat{\mathbf{x}}_i  \mathbf{x}_i(t)) \mathrm{d}t +\eta \mathrm{d}{} \mathbf{W}(t), \\&i \in \{1,\ldots ,N\}, \end{aligned} \end{aligned}$$(37)For the formalism, one can refer to the work by Cumming et al. (2010) and a further application in the work (Chen et al. 2018).$$\begin{aligned} \Psi (t,\mathbf{x}) = \begin{pmatrix} \Psi _{xx} &{} \Psi _{xy} \\ \Psi _{xy} &{} \Psi _{yy} \end{pmatrix}. \end{aligned}$$(38)

We note that the relaxation parameter of the nucleus has little correlation with the transmigration time. This finding seems counterintuitive. According to the studies of SerranoAlcalde et al. (2017), the stiffness of the nucleus hardly plays a role in cell deformability experiments if the nucleus is relatively small. However for larger sizes, this deformability of the nucleus may become more important.
Notes
Compliance with ethical standards
Funding
This study is funded by China Scholarship Council.
Conflict of interest
The authors declare that they have no conflict of interest.
Supplementary material
References
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