Systematic simulation of tumor cell invasion and migration in response to time-varying rotating magnetic �eld

Cancer invasion and migration play a pivotal role in tumor malignancy, which is a major cause of most cancer deaths. Rotating magnetic field (RMF), one of typical dynamic magnetic field, can exert substantial mechanical influence on cells. However, studying the effects of RMF on cell is challenging due to its complex parameters, such as variation of magnetic field intensity and direction. Here, we developed a systematic simulation method to explore the influence of RMF on tumor invasion and migration, including a finite element method (FEM) model and a cell-based hybrid numerical model. Coupling with the data of magnetic field from FEM, the cell-based hybrid numerical model was established to simulate the tumor cell invasion and migration. This model employed partial differential equations (PDEs) and finite difference method to depict cellular activities and solve these equations in a discrete system. PDEs were used to depict cell activities, and finite difference method was used to solve the equations in discrete system. As a result, this study provides valuable insights into the potential applications of RMF in tumor treatment, and a series of in vitro experiments were performed to verify the simulation results, demonstrating the model's reliability and its capacity to predict experimental outcomes and identify pertinent factors. Furthermore, these findings shed new light on the mechanical and chemical interplay between cells and the ECM, offering new insights and providing a novel foundation for both experimental and theoretical advancements in tumor treatment by using RMF.


Introduction
Invasion and migration are essential issues in tumor malignancy, particularly for breast cancer.In general, tumor cells migrate from primary tumor site to surrounding tumor stroma, ultimately transforming into circulating tumor cells (CTCs), leading to the occurrence of malignant tumor, which is a major cause of tumor deaths (Woodhouse et al. 1997).Triple-negative breast cancer (TNBC), the most malignant subtype of breast cancer, is characterized by high metastasis and recurrence (Sun et al. 2012).The five-year disease-free survival rate of TNBC patients following surgery is notably below 30%, significantly lower than other subtypes (Kim et al. 2018).Therefore, it is imperative to delve into the intricate mechanisms underlying cancer invasion and migration, and it is important for the development of new therapies or targeted drugs for TNBC treatment.
Extracellular matrix (ECM) is a complex mechanically-sensitive macromolecular network composed of collagen, fibenectin, laminin, proteoglycan and integrin, which is critical important for the cancer invasion and migration (Zhang et al. 2017).At the initial stage of tumor invasion, tumor cell or fibroblast secrete proteolytic enzymes to degrade ECM, such as matrix metalloproteinases (MMPs) and urokinase-type plasminogen activator (uPA), thus tumot cells enable to proliferate and migrate through the tumor stroma and spread through the blood or lymphatic vessels, which leads to the invasionmetastasis cascade, resulting in the metastasis of cancer to distant locations in the body from its primary site (Chaffer and Weinberg 2011).Unravelling the interaction between tumor cells and ECM is critical important for better understanding the mechanism of cancer invasion and migration.
Studies have shown that magnetic field, as a non-interventional physical technology, can effectively impede tumor growth.Magnetic fields can be classified into two main categories: steady-state magnetic field and dynamic magnetic field.Dynamic magnetic field are characterized by variations in both intensity and direction over time, with the rotating magnetic field (RMF) serving as a notable example.Compared to static magnetic field, RMF is considered safer and capable of providing periodic mechanical stimulation, resulting in more substantial mechanical effects on cells, which has attracted more attention in tumor therapy in recent years (Ren et al. 2017;Sun et al. 2012;Zha et al. 2018).However, RMF is characterized by complex parameters, multi-dimension and multi-effects, which make it difficult to measure its magnetic strength and force.Furthermore, there is limited research exploring the impact of RMF on tumor cell invasion and migration from the perspective of ECM-cell interactions.Therefore, the development of novel methods to investigate the properties of RMF is imperative for understanding its effects on tumor cells, and bring new prospect of tumor therapy.
Recently, there has been a growing interest in employing numerical models to simulate biological processes, including tumor growth and invasion.Numerical simulation models offer a valuable approach for investigating the impact of these physical  and Lolas 2005;Marchant et al. 2006;Perumpanani et al. 1996;Swanson et al. 2000;2003).Continuum models are valuable for simulating macroscopic scale, such as overall tumor growth and metastasis.However, they often overlook microscopic factors, like cell heterogeneity and intercellular interactions, as they tend to regard cell invasion model to simple, heat-like diffusion processes.To couple the microscopic scale of cancer invasion, discrete models were developed.Following the discrete approach, it becomes straightforward to achieve quantification and visualization of cell growth, deformation, movement, and interactions during cancer invasion (Chen et al. 2020;Smeets et al. 2016;Valentim et al. 2023).Nevertheless, solving the discrete model, particularly with largescale populations, demands substantial computing resources and time.In pursuit of enhancing the efficiency of numerical models, hybrid discrete-continuum models for tumor migration and invasion have been introduced, which combine the characteristics of both continuum model and discrete model, and demonstrate their utility in simulating cancer invasion across multiple scales (Anderson et al. 2000;Liu et al. 2022;Pahle 2009).
In the realm of cancer invasion research, hybrid modeling has elevated our comprehension of intricate details underlying the migration and invasion processes, since the level of abstraction in the discrete models is often lower (Colombi et al. 2015;Mohammadi and Dehghan 2020;Scianna and Preziosi 2012).Hybrid modeling typically entails the formulation of rules and parameters based on experimental observations or measurements, facilitating both qualitative and quantitative comparisons between experimental and simulated outcomes (Franssen et al. 2019).
In this study, we initially employed finite element method (FEM) analysis to simulate the magnetic field generated by the RMF device.Subsequently, we developed a cell-based hybrid numerical model to comprehensively simulate the biological processes and mechanical interactions during cancer invasion and migration under RMF.A series of PDEs were used to describe the proliferation, migration and MMP secretion of tumor cells.Furthermore, the simulation results were validated by a series of experiments under RMF to verify the model.It is helpful to clarify the suppressing effect of RMF on the occurrence and progression of tumor, and also provide important theoretical data for application of RMF on tumor treatment.

RMF device
The RMF device for cell experiments was built by Haina Machinery Co., Ltd, (Shanxi, China).The magnetic field was generated by four cylindrical NdFeB permanent magnets (diameter 35 mm, height 30 mm), and the relative face of the upper and lower magnets was N pole to N pole.The sample was placed in the middle of the permanent magnets, and the experimental group and the control group were separated by an iron plate, which was used as the magnet vane.

Physical simulation of RMF device
Finite element analysis (FEA) of RMF device was performed using AC/DC Module of COMSOL Multiphysics 5.6 (COMSOL AB, Stockholm, Sweden ) .A parametric sweep calculation was utilized to simulate the quasistatic state of the RMF device during rotation.The computations were conducted on a PC: Intel(R) Core™ i7-4790K CPU @4.00GHz; NVIDIA GeForce GTX 2060 GPU; 32.0 GB RAM.

Framework of the model
The entire model framework consisted of four primary modules, as illustrated in Figure 1: (1) cell module, (2) magnetic field module, (3) MMP module, (4) ECM module.Two primary types of interactions between elements were considered: mechanical interaction and chemical interaction.The mechanical interaction was characterized as linear elastic contact which occurred between cell and cell.The chemical interaction, which was modeled by a set of PDEs to simulate the ECM degradation and haptotaxis between cells.The cells in the model were simplified as circular elements with mass but without thickness.The basic attributes of cells were defined by radius and positions (xcoordinate and y-coordinate).For each discrete cell, and its activities were described as follows: Proliferation: cell growth, and resulting in the generation of a new cell with the same radius as the original one, at a random angle around it.This process is described as follows: where n is tumor cell density, D denotes the cell diffusion coefficient;  represents cell proliferation rate; x signifies the one-dimensional coordinate.
Migration: Cells underwent migration within the tissue.The direction and distance of migration was determined by random motion and ECM concentration difference.
Since the ECM can act as an obstacle to cell movement, it is assumed that cells tend to migrate towards areas with lower ECM concentrations which has been extensively studied (Price and Thompson 2002).The assumption of migration was described by a reaction-diffusion equation: where n represents tumor cell density, f denotes ECM density,  > 0 is the (constant) haptotactic coefficient, (, ) can be a constant or a function of either MMPs or ECM concentration, which indicate the chemokinetic response.Here (, ) was set as a constant D n referred to as the cell migration coefficient.The prior and latter term of equation represents random motility and haptotaxis, respectively.
Cell-cell interactions: Throughout the migration and proliferation processes, cell may contact with each other to induce solid extrusion, where relative movement between the two cells will happen.In this model, the overlap between cells was used to simplify the deformation, and the overlap would generate force on the cells.For each cell, the resultant force was determined by vector summation of all the forces acting upon it, and then induced an acceleration, propelling the cell along the same direction until the forces reached equilibrium.To iteratively compute all the elastic contacts of cell in the population and bring the whole system into mechanical equilibrium (where the resultant forces of all cells are zero), simulated annealing (SA) algorithm was employed.Further details regarding the SA algorithm was described in the Supplement S1 (Steinbrunn et al. 1997).
MMP secretion: matrix metalloproteinase (MMP) is a kind of enzymes secreted by tumor cell to degrade ECM, facilitating the migration of tumor cells.The Formula 5 is used to describe this process: where m denotes MMP concentration,   > 0 is the (constant) MMP diffusion coefficient,  represents the MMP production coefficient,  is the MMP decay coefficient.
ECM-cell interaction: the interaction between cells and ECM is established by the haptotaxis and MMP degradation.The MMP degradation is described as follows: where f represents ECM density, m denotes MMP concentration,  > 0 (constant) is the degradation coefficient.The initial condition of ECM density is defined according to the experiment to be simulated.

Coupling the effect of magnetic field with tumor cell
The gradient magnetic field has a significant impact on both of cell and medium, and the force acting on the cells within the medium is calculated as follows: where  and   is the magnetic susceptibility of cell and medium,  0 is magnetic permittivity, V represents the volume of cell (we used area S to replace V in 2-D model), B stands for magnetic flux density.It is assumed that the force exerted on cells by magnetic field is directly proportional to the square of the gradient of magnetic flux density, which will influence cell migration direction and cell activities (Kauffmann et al. 2011).The direction and intensity of magnetic field was assessed by using COMSOL.
Thus, the magnetic field density matrix was constructed and integrated into the numerical model.By using the Formula 7, the effect of magnetic field on tumor cells can be simulated.

Numerical simulation
To apply the continuum system to the cell-based hybrid numerical model, finite difference method (FDM) was used to solve the density of tumor cells, ECM and MMP within the discrete system, which is described as: The parameters used in the model and SA algorithm was listed in Supplement S2.
The model analyzed the dynamic evolution of tumor growth morphology and mechanical condition from a randomly distributed tumor cell population or a specific distributed of tumor cell population through global iterative calculation.The invasion behavior of tumor cells and mechanical environment of tumor can be simulated and anal.The iterative algorithm of the model is described as follows (Supplement S1: Fig. S4): 1. Initialization of tumor growth area: a blank tissue (a 1000 × 1000 square) was established and the density of ECM was set; 2. A certain number of tumor cells were seeded on the tissue; 3. Tumor cells initiated to migrate, proliferate and secrete of MMP to facilitate invasion.Whenever cells changed their positions, the collisions between cells were checked and SA algorithm was used to equilibrate the mechanical condition of whole system; 4. Iteratively execute step 3 for a predefined simulation duration or until a specified termination criterion is met.

Scanning electron microscopy of cell morphology
The frequency and intensity used here is 5 Hz and 0.4 T according to our previous results.110 5 MDA-MB-231 cells were seeded in 35 mm dishes to expose RMF for 48 h, and then cells were fixed with 4% paraformaldehyde for 15 min and rinsed with PBS.
Cells were then dehydrated using a series of graded alcohol solution (30%, 50%, 70%, 80%, 90%, 95% and 100%) for 15 min in each gradient concentration, and air-dried at least 2 h.The desiccated samples were scanned using a scanning electron microscope (SEM) (VEGA 3 SBH, Tescan, Czech Republic) with 10.0kV HV.To quantify the deformation of cell morphology, both of long axis and short axis of each cell were recorded.Image J software 2.1.0(National Institutes of Health, Maryland, USA) was used to analyze.Polarization coefficient = long axis / short axis.

Cell viability assay
0.5×10 4 MDA-MB-231 cells/well were seeded in 96 well cell plate.After cell adhesion, the cells after exposure were cultured under 5 Hz, 0.4 T RMF, and the cell viability was detected by CCK8 kit (Biosharp, Anhui, China) at day 1, 2 and 3, respectively.Gen5 Microplate Reader (BioTek Instruments, Winooski, United States) was used to determine absorbency at 450 nm, and a reference wavelength was with 630 nm.
2.9 Wound-healing assay 1×10 6 MDA-MB-231 cells were seeded on 35 mm dish.A wound was scratched by using sterile 20 μL pipette tip. 2 mL serum-free medium was gently added to each well, and cells were cultured under RMF at 5 Hz, 0.4 T. Photos were taken after exposuring for 12 h and 24 h, respectively, and the distances between scratch were recorded.Image J software 2.1.0(National Institutes of Health, Maryland, USA) was used to analyze.Cell migration area = scratch area (0 h) -area between scratches (12 h/24 h).Scratch healing rate = cell migration area / scratch area (0 h).

Statistical analyses
All the statistical analyses were performed by GraphPad Prism 9 (GraphPad Software Inc., San Diego CA, USA).The sample number for each group was ≥3, and numerical data were presented as mean ± standard deviation (SD).p values were considered statistically significant at *p < 0.05, **p < 0.01, and ***p < 0.001.

RMF device generates complex magnetic and mechanical environment on tumor cell
We conducted a simulation of the magnetic field distribution within the RMF apparatus under an N-N configuration, as shown in Figure 2A and 2B.The density of the magnetic flux was presented in Figure 2A, where the maximum magnetic flux density reached 3.13 Tesla (T) in the vertical direction and 0.17 T in the horizontal direction.The arrows in the Figure 2B indicate the direction of the magnetic flux, with varying shades representing the different densities of the magnetic flux.These results were obtained using COMSOL Multiphysics 5.6 (COMSOL AB, Stockholm, Sweden), reflecting the complexity and gradient variations of the magnetic field between two permanents magnets.The dynamic variation of magnetic scalar potential in the RMF device is illustrated in Figure 2E.The magnetic field distribution between two permanent magnets resembles an hourglass shape, with a range extending up to 10 4 amperes.in which the range can reach 10 4 A. The force acting on the cells was internally computed as an integral of the surface stress tensor across all sphere boundaries.The formula for the stress tensor is as follows: In this equation,  1 represents the normal vector pointing outward from the sphere's boundary, T2 corresponds to the stress tensor of the surrounding air.As depicted in Figure 2F, the spheres experienced a continually varying force along the magnetic flux, resulting in a substantial stress range on the cell membrane, leading to deformation and reconfiguration of the cell membrane and cytoskeleton.with rotation degrees (degree=15, 25, 35, 40, 45, 50, 55, 65 and 75, respectively).These panels highlight the alternations in magnetic properties and stress distribution under varying rotational influences.

Simulation of RMF on tumor cell and its growth after exposure to RMF
The effects of RMF on growth of tumor cell were simulated over time is showed in Figure 3.In Figure 3A, the bright red discs represented active cells, whereas the dark red discs are inactive cells.Initially, five tumor cells were seeded on the tissue at 0-time step.The model performed the iterative calculation for 100-time steps and the tumor cells proliferated with a preset probability every 10-time steps.Notably, the tumor's morphology did not exhibit the expected circular growth pattern.Instead, it exhibited "raised" or "sunken" structures on its periphery, as observed in Figure 3B, and MMP gathered at specific positions within the tumor, as depicted in Figure 3C.
Compared with the control group, the exposed group exhibited a noteworthy reduction in the total cell count (p < 0.0001), signifying the inhibitory impact of RMF on tumor cell proliferation (Figure 3D). Figure 3E showed that the proportion of inactive cells in exposed group was larger than that of control group (p = 0.0101).
Figure 3F showed the number of total cells and inactive cells were changed over time during the tumor growth.Inactive tumor cells tent to migrate to locations with minimal mechanical stress, where they reactivated and continued to proliferate.To further analyze the mechanical effect of RMF on tumor growth, the scatter plot was used to illustrate the mechanical condition at 100-time step.
The results are depicted in Figure 3G, where the colors in the main plot correspond to the magnitude of the force acting on the tumor cells (with deeper red indicating stronger forces), while the upper and right bar plots display the cell distribution along the axes.Compared with the control group, the cells in exposed group with subjecting to high force (red blocks) were more concentrated, and the overall cell distribution was more clustered.This suggests that the magnetic forces applied to the cells resulted in increased internal stress within the tumor, ultimately inhibiting its growth.

Simulation of tumor cell invasion and migration after exposure to RMF
To assess the impact of RMF on the migratory capacity of tumor cells, a simulated wound-healing experiment was conducted.In this experiment, cells were initially seeded on both the left and right sides of the tissue, creating a wound with a length of 160 μm.The proliferation coefficient for this experiment was set to 0.1 (in contrast to the normal condition of 0.7) to mimic a serum-free medium.indicating the inhibiting effect of RMF on the migratory capacity of tumor cell.
To elucidate the intricate processes that occur during wound healing, the migration of individual cells was analyzed using a cell-based hybrid model.Since the cells are distinct, the focus can be directed towards the analysis of the movement of a single cell.
Here, two methods of calculation were defined (Figure 4C): (1) total individual cell migration distance: summing up all the distances that an individual cell has migrated; (2) average individual cell migration distance: this is computed by dividing the total individual cell migration distance by the total migration time (s).Thus, the individual cellular migration can be analyzed.Figure 4 D and F show that, in control group, the total migration distance of a single cell during 200 TS was about 458.17 m, with an average step-wise migration of approximately 15.47 m.In exposed group, the total migration distance of a single cell during 200 TS was about 232.24 m, with an average step-wise migration of about 7.16 m.The total and average individual cell migration distance were summarized in the box plot (Figure 4F and 4G).Within each group, the distribution of migration distances exhibited similarity, and notably, the distribution of migration distance in control groups was larger than the exposed group, indicating the applicability and stability of this model.

Full factor correlation analysis in simulative migration
The full factor correlation analysis (FFCA) was performed to further investigate the key factors and their interrelationships during cancer invasion in this model.Ten factors were taken into account in FFCA and the results were shown in Figure .5 (the full names of all ten factors were depicted in Supplement S5).The distribution of all factors over time was clustered and depicted in Figure 5A.The values of all factors were normalized using the formula (  − ̅ )/ (where ̅ and  is the mean and standard derivation of the factor, respectively), where green denoted high value and blue denoted low value.
Factors were clustered into three cluster, which indicated the similarity of each factor.
Three clusters were identified among factors (row), and two clusters were found in time (column), in which the same cluster primarily representing similar characteristics.
However, the average migration distance of the exposed group significantly deviated from the force factors, suggesting a weak relationship between migration distance and the force factors in the exposed group.Principal Component Analysis (PCA) was then employed to reduce the dimensionality of all factors and identify the key factor, as illustrated in Figure 5B According to the results, Dim-1 (with cell number as the main contributor) explained 50.3% variance, while Dim-2 (with force as the main contributor) explained 19.5% variance.The correlation matrices of exposed and control group were shown in

The simulation results were verified by in vitro experiments
The in vitro experiments were designed to verify the appropriateness of cell-based hybrid model.Motion trajectory of magnetic nanoparticles under 1.5 Hz and 0.4 T RMF was depicted in Figure 6A.The snapshots revealed that magnetic nanoparticles formed several rod-like structures with varying lengths, and the direction of these magnetic nanoparticles was indicted by red bars.After exposed to RMF, these magnetic rods were aligned, swam in tandem with the rotation of magnetic field.The schematic diagram of the motions was generated by stacking all 6 snapshots, and shapes of magnetic rods was displayed in one single plot.The blue bars represented the magnetic rods, and gradient colors were used to denote different time interval of snapshots.The red dotted lines are the evolving paths of the magnetic rods over time.It confirmed that RMF generated a complex magnetic and mechanical environment.
Considering the potential mechanical effects of RMF on cell membrane, we conducted cell morphology detection using SEM.As shown in Figure 6B, the cells exposed to RMF were stretched along the axis compared with that in control group, which was consisted with the simulation results as shown in Figure 2F.Polarization coefficient was used to quantify the deformation of cell morphology, and it was significant increased after exposure (p = 0.025 < 0.05).As subjected to RMF, the magnetic force caused the cell membrane to deform along the direction of magnetic field, resulting in excessive stress on the cell membrane, which might potentially damage cell structure.In Figure 6C, it is clear that cell viability was significantly decreased after 48 h and 72 h RMF exposure, aligning with the simulation results in Figure 3F.Cell migration ability was always assessed by wound-healing assay.Figure 6D demonstrated that cell migration in the exposed group decreased to 32.46% (p = 0.0009, < 0.001) after 12 h treatment compared to the control group, further declining to 43.89% (p = 0.002, < 0.01) in the RMF group after 24 h.This result suggested that RMF could significantly impede the wound healing capability of MDA-MB-231 cells, corroborating the simulation results in Figure 4A.
Cellular invasion is a multiple pathological process, where cells interact with ECM.
It's assumed in the model that cells could interact with ECM by secreting MMP to promote cell invasion and migration by facilitate the chemokinetic and haptotactic response, thus the invadopodia-mediated matrix degradation assay was performed.In Figure 6E, the degraded sites of gelatin were darker than the surrounding.F-actin and cortactin which serve as marker proteins localized to invadopodia, was found at the sites where gelatin degradation occurred, confirming that the formation of invadopodia was associated with matrix degradation.We quantified formation of cell invadopodia, and the results indicated a significant reduction in the colocalization of F-actin with cortactin in the RMF group compared to the control group (Figure 6F) (p < 0.0001).
This finding suggests that RMF has the potential to diminish the invasive and matrix degradation capabilities of MDA-MB-231 cells by inhibiting the formation of invadopodia, thus suppressing tumor invasion and migration.
and physiological factors on tumor growth, yielding instructive and predictive insights for subsequent experiments and treatment strategies [9].Compared with in vitro experiment, the repeatability and operability of numerical model presents irreplaceable advantages, simplifying the comprehension of the mechanisms underlying tumor occurrence and progression.Generally, these numerical models can be divided into continuum model, discrete model and hybrid model based on different modeling methodologies.Continuum models often employ ordinary differential equations (ODEs) or partial differential equations (PDEs) to depict the progression of tumorigenesis and screen the time-varying factors, for instance, the dynamic change of cell population, ECM density and chemical concentration during tumor growth and invasion (Carrasco-Mantis et al. 2023; Chaplain

Fig. 1
Fig. 1 The schematic diagram of the model.

Fig. 2
Fig. 2 Simulated results of RMF device with N-N situation.(a) The density of the magnetic flux, depicting the variation in intensity across the field.(b) The direction of

Fig. 3
Fig. 3 Simulated results of tumor growth and morphology.(a) Snapshots of tumor growth at 50 TS and 100 TS, (b) the 'raised' and 'sunken' structure formed on the tumor Fig. 4A presents snapshots of the experiment at 1-, 70-, and 140-time steps.The quantified results of the relative wound healing area are presented in Figure 4B.They reveal a notable reduction in the wound healing area in the exposed group (70 TS: p = 0.0037; 140 T: p < 0.0001),

Fig. 4
Fig. 4 Simulation results of wound-healing experiment.(a) simulation effect of RMF on wound healing at 1-, 70-and 140-time steps, (b) relatively area of wound healing

Figure
Figure 5C and 5D.Force factors exhibited a negative correlation with cell number

Fig. 5
Fig.5 Correlation analysis of variables in simulative migration.(a) Heatmap displaying the distribution of all factors over time.(b) PCA plot of all factors;

Fig. 6
Fig. 6 The effect of RMF on MDA-MB-231 cell morphology and viability,