A variational model for finger-driven cell diffusion in the extracellular matrix

Cells spreading within tissues, such as cancer metastases or fibroblasts, engage in interactions with both the cellular matrix, and other cells. In recent times, considerable attention has been focused on elucidating the mechanics of cell movement resulting from their interactions with the extracellular matrix (ECM). In [10‐12, 33] an in-depth description of the physiological mechanisms underlying these motions is provided.

The mobility of cells is influenced by a multifaceted interplay of factors, encompassing not only biochemical signaling within the cell and subsequent cytoskeletal rearrangements, but also the biophysical attributes of the surrounding environment. Central to these environmental factors is the ECM, made of intricate network of insoluble structural fibrous proteins like collagen type I, glycoproteins, and soluble glycosaminoglycans. Collectively, these components serve as microstructural guidance cues and biochemical stimuli, influencing the movement of individual entities.

Cell migration occurs through two main mechanisms. The first mode is characterized as mesenchymal migration, where cells penetrate the matrix by generating a pathway around them, relying on proteolysis and actin polymerization. The second mode involves amoeboid migration, where cells move through any available pathway by squeezing through, utilizing membrane protrusions known as blebs. We mainly focus on the former mechanism, although in a very simplified framework. In this kind of migration, cells degrade the surrounding matrix by breaking it down after creating protrusions, subsequently remodeling it along their trajectory; in particular cells exhibit the ability to alter the orientation of collagen fibers, aligning them in the direction of their motion, through a mechanism called proteolysis [18].

Significantly, the ECM undergoes considerable remodeling, profoundly impacting cell spreading, motility, and differentiation. Fiber reorientation within the ECM plays a pivotal role in guiding the migration of both normal and cancer cells. Understanding the mechanical behavior of the ECM is paramount for deciphering the intricate interplay between fibrous materials and cellular responses.

The mechanical remodeling of the ECM is marked by noteworthy spatial patterns of densification and fiber alignment, localized within finger-like bands connecting distant cell clusters. Remarkably, cancer cells display a preference for invading regions of the ECM with higher density and greater fiber alignment, with specific ECM density and alignment patterns identified as biomarkers for breast cancer [6].

Cell-induced forces bring about significant changes in ECM structure. Along the axis of each tether, the ECM undergoes substantial straining, with deformations reaching up to \(40\%\) of its original length [14]. Conversely, in the transverse direction, the ECM undergoes compression, resulting in a reduction of thickness to as little as half or even a quarter of its initial value. The unexpected localization of densification within the tethers highlights the intricate interplay between cellular forces and ECM mechanics [14].

Fig. 1 visually represents these processes, displaying a cartoon of a cell cluster surrounded by ECM, illustrating the contraction and deformation of the ECM, summarizing the experimental findings from [14]. In this work, it is posited that tether formation is induced by the buckling instability of network fibers under cell-induced compression.

The association with instability, as discussed in [8, 14], was initially established through experiments and modeling on fiber networks, open-cell foams, and fibrin [19, 20]. Individual fibers offer support against tensile forces and may become stiffer as tension increases [17, 32]. However, when subjected to compression, these fibers buckle, leading to a loss of stiffness. These microscopic buckling instabilities give rise to the formation of macroscopic bands characterized by intense compressive deformation and high density: within these bands, the fibers are predominantly buckled and compacted. These dense regions alternate with areas of standard density and minimal compressive strain, where the fibers are generally straight and loosely organized [14].

The observed mechanical behavior of the ECM during cellular remodeling underscores the dynamic nature of fibrous materials and their pivotal role in cellular processes. Understanding the mechanisms underpinning these intricate deformation patterns and their implications for cell behavior holds significant promise across various fields, including tissue engineering, regenerative medicine, and cancer research.

In this work, we present a simple variational formulation of a chemo-mechanical model within a phase-field framework, elucidating fundamental aspects of the phenomena mentioned previously. This includes (i) the emergence of anisotropy through a macroscopic measure of fiber stiffness in the direction of transverse anisotropy, (ii) the manifestation of micro-buckling leading to material softening, (iii) the emergence of localized field patterns reminiscent of experimental findings on finger-like structures formation, and (iv) a diffusion of a phase representing cells that spread around the tissue. To maintain clarity in fundamental concepts without introducing unnecessary complexities, we deliberately refrain from incorporating large deformations. Upon reaching a critical strain, an abrupt jump in the response results in the emergence of a sudden anisotropy. Physically, we interpret this as microscopic-scale buckling. In terms of modeling, this categorizes our model alongside materials with a domain of reversibility, resembling those with an elastic range.

In our prior work [8], we addressed a comparable issue, emphasizing solely mechanical aspects while overlooking cell diffusion and its interplay with the matrix. Specifically, we examined the manifestation of localized anisotropy in processes involving loading and unloading, and introduced a memory effect to depict cyclic behavior. In this study, we solely focus on the loading phase, yet integrate chemical diffusion into the analysis. We finally notice that, despite out previous model, which was rate-independent, in the present contribution rate-independent processes are coupled to the rate-dependent one of cell diffusion; this requires a different framework, although governing equations of the dissipative behavior are also proposed in a variational form, using an energetic formulation of rate-dependent materials [23‐26].

The paper is organized as follows. In Sec. 2 we describe the elastic energy and the material parameters of a transversely isotropic material, with the aim to elucidate the transition from an initially isotropic material to one with transverse isotropy. In Sec. 3 we introduce the relevant fields entering in our formulation, lay down the general balance equations and the thermodynamic restrictions, propose a variational formulation of the evolution equations; therefore, we suggest specific constitutive choices and derive the governing equations. In Sec. 4 some numerical results are shown and discussed.

As detailed in the introduction, due to micro-buckling, the ECM suddenly undergoes a transition from isotropy to anisotropy. In this section, we delve into understanding how the stiffness of a plane transversely isotropic material depends on its four key material constants: E (Young modulus), \(\nu \) (Poisson coefficient), \(c_1\), and \(c_{2}\) (transversely isotropic constants), as discussed in [7, 30]. Our focus is on highlighting a material that not only exhibits a softer response in a specific direction but also maintains consistent stiffness in the orthogonal direction. The primary objective is to elucidate the transformation from an initially isotropic material to one with transverse isotropy, offering insights into admissible anisotropic material parameters within the space ensuring the positivity of the elastic energy. In this realm, we opt for a trajectory that highlights a decrease in stiffness aligned with transverse isotropy. For more in-depth information, readers are referred to [7].

The elastic energy density is assumed to bewhere \({\textbf {E}}({\textbf {u}})=\text {sym\,}\nabla {{\textbf {u}}}\) is the linearized strain measure expressed in terms of the displacement field \({\textbf {u}}\) and \(\mathbb {C}\big ({\textbf {n}}(\vartheta )\big )\) is the elasticity tensor of a linearly elastic transversely isotropic material, with respect to the unit direction \({\textbf {n}}(\vartheta )\), parametrized by the angle \(\vartheta \). This elasticity tensor, in a two-dimensional setting, can be represented throughwhere the first term on the right-hand-side represents the elastic energy of an isotropic material, while the elastic moduli \(c_1\) and \(c_2\) modulate the anisotropic response. We define a softening material when the ratio between the Young modulus tested along the parallel and orthogonal directions with respect to the fiber \({\textbf {n}}(\vartheta )\) is less than 1:

To match the goal of this section, we can consider a material with \(c_1=0\). Indeed, the mechanical effect of the constant \(c_2\) is visualized in polar plots of the Young modulus (see Fig. 2) as a function of the angle between the testing and fiber directions. To ensure the positiveness of the energy density (1) the constants \(c_1, c_2\) must belong to the yellow region represented in Fig. 2 whereOne can nondimensionalize the constant \(c_2\) by defining \(\alpha :=-c_2/c_{2m}\) with \(0\le \alpha \le 1\), consequently the elastic energy density takes the form:It is worth noticing, with the aid of Fig. 2, that while \(\alpha \) tends to 1 the stiffness in the fiber direction \({\textbf {n}}(\vartheta )\) tends to 0.

Looking at the bigger picture, the decrease in stiffness along the direction of buckled fibers reveals micro-buckling at the macroscopic level. Consequently, following our previous contribution [8], we suggest interpreting \(\alpha \) in (3) not as a constant material parameter but as a field undergoing evolution.

We suppose that cells are able to diffuse within the matrix, and denote by c the cell density field, so that \( \int _\Pi c \) represents the net mass of the diffusing cells in an arbitrary part \(\Pi \subset \Omega \) (at time t). We further introduce the vectorial species flux \({\textbf {j}}\). The cell mass balance is the requirement thatfor every region \(\Pi \subset \Omega \). The first term in the right-hand side of (4) gives the rate at which the cells are transported to \(\Pi \) by diffusion across \(\partial \Pi \), being \(\varvec{\nu }\) the outward unit normal; the minus sign renders this term non-negative when the flux \({\textbf {j}}\) points into \(\Pi \). Eq. (4) yields the local cells mass-balanceHere and henceforth a superimposed dot denotes a time derivative.

Drawing on [5, 13, 16], our proposed model relies on internal state variables that characterize the evolution of the system, which depends on two fundamental energy aspects: an internal energy density and a dissipation potential. Let \(\mathbb {S}=\{{\textbf {E}}({\textbf {u}}), \vartheta , \alpha , c, \nabla {\alpha }\}\) be the list of the state variables:We consider the second law of thermodynamics as an a priori constraint, expressed for isothermal processes through the Clausius-Duhem inequality:Here \(\psi \) is the Helmholtz free energy density, \({\textbf {S}}\) is the Cauchy stress tensor and \(\mu \) represents the chemical potential, allowing cells to migrate from the boundary into the domain. In this context, we assume chemical equilibrium between cells and the substrate.

Exploiting the dissipation inequality within the framework of materials with internal variables needs some caution, because they are a priori constrained to obey a constitutive evolution law, as detailed in the seminal work by Coleman and Gurtin [5]. More specifically, the material, at each point, is characterized by the response functionsSince \(\dot{\psi }=\partial _\mathbb {S}\widehat{\psi }\cdot \dot{\mathbb {S}}\), on requiring that (6) be satisfied whatever the local continuation of any conceivable process, one easily finds the following standard constitutive relationsfor the stress and the chemical potential. The presence of \(\nabla \alpha \) in the list of state variables produces some difficulties, as detailed in [22]. Indeed, the presence of a non-local term would likely affect the entropy production, and then the dissipation; more specifically, according to [22], an extra entropy flux is in order, which would result in the absorption of some terms in the reduced dissipation inequality. In particular, the term \(\partial _{\nabla \alpha }\widehat{\psi }(\mathbb {S})\cdot \nabla \dot{\alpha }\) appearing when evaluating \(\dot{\psi }\) in (6) kind of evaporate according to this reasoning (see [22] for more details). Now it remains to discuss the residual dissipation related to the variation of \(\vartheta \), \(\alpha \), and due to \({\textbf {j}}\). As to \(\vartheta \) we choose not to associate a dissipation. As to the latter, it is customarily accepted that the dissipation rate \(\delta \) reveals different entropic effects having distinct origins, so that the positiveness of the dissipation rate is assumed separately for the local and diffusive contributions (see also [22]). For this reason, we assume that the following relations hold true, which are stronger than (6) and not its direct consequence:or ratherwhere the flux \({\textbf {j}}\) is in need of further constitutive prescriptions.

In order to ensure the inequality (10)\(_2\), we introduce a dissipation potential \(\varphi _{\textrm{diff}}\), a 2-homogeneous convex function of the flux \({\textbf {j}}\), such thatFor the following purposes, we also introduce the dual diffusive dissipation potential, a 2-convex homogeneous function of \(\nabla \mu \), as a suitable Legendre transform, such thatWith this, we establish the connection between the cell flux and the gradient of the chemical potential, expressed as:where the positive semidefinite second order \({\textbf {M}}(\mathbb {S})\) represents the mobility tensor that, for the time being, we admit being dependent on all the state list \(\mathbb {S}\). Eq. (12) is a generalized version of the classical Fick’s law. A specific dependence of the mobility tensor on the state variable list \(\mathbb {S}\) will be presented in the following.

This section delves into the examination of the scenario outlined in Fig. 1: a hollow domain experiences a progressively increasing internal negative pressure p at given and constant chemical potential \(\mu _0\). The hole stands for a cell cluster, whose effects on the ECM are represented by its capability to contract the tissue and to maintain chemical equilibrium. Our aim is to showcase that our simple model captures, in qualitative terms,⁵ the localization phenomenon described in the introduction and the subsequent finger-driven diffusion of the cell phase into the ECM.

In Fig. 3a, due to the symmetry of the problem, is reported the effective domain of the analysis and its discretization.

The energetic properties and the fulfillment of overall conditions within the framework of incremental energy minimization strongly suggest the appropriateness of the alternative minimization strategy as the preferred solution algorithm [27]. In this specific case, it involves four sequential steps repeated over each time increment: The exponential choice of the time evolution is dictated by the typical exponential solution of the diffusion-like equation as (37). The pseudocode presented in Algorithm 1 offers more details in the process of minimization described just above.

The code, [29], implemented in python as frontend of FEniCSx, a renowned open-source computing platform for solving partial differential equations [1, 2, 21], utilizes an unstructured mesh to discretize the domain, refined in the vicinity of the hole, as shown in 3a. The displacement field \({\textbf {u}}\), along with the anisotropic gauge fields and chemical potential \(\alpha \) and \(\mu \), are considered in piecewise affine scalar finite element spaces over the domain (using FEniCSx P1 elements). Discontinuous Galerkin approximations suffice for the field \(\vartheta \) (DG elements), due to the absence in derivatives of the orientation field.

We present the outcomes of a numerical experiment involving a progressively increasing loading process in pressure p with a constant chemical potential \(\mu \) applied at the boundary \(\partial _p\Omega \), as depicted in Fig. 3b. The experimental setup is illustrated in Fig. 3a and is tailored to explore the mechanical and diffusive characteristics of the material.

In the initial loading phase, the pressure undergoes a linear increase, inducing compressive hoop strain \(\varepsilon \) across the material, with maximum compression occurring at the hole’s boundary. Until the activation criterion (40)\(_1\) is satisfied, the material maintains isotropy (i.e., \(\alpha =0\) throughout) as well the diffusion process. Upon further pressure increments, the criterion (40)\(_1\) is met simultaneously along the entire internal boundary. At this juncture, a thin layer with \(\alpha >0\) undergoes a sudden bifurcation through a snap-back instability, transforming into a series of radial tethers where \(\alpha \rightarrow 1\). Subsequently, cells diffuse in preferred directions dictated by the anisotropy gauge field. It must be noticed that the initiation points of nucleation are determined by the mesh, whereas in the actual problem, they are influenced by small imperfections, which are inherent to such problems. Similar phenomenon is observed in the fracturing process of ceramic slabs subjected to thermal shocks [4] and in periodic array of crack in thermal shock problem [31].

Fig. 4 tracks the evolution of \(\alpha \) and \(\mu \) at point P\(_1\) of the domain. Additionally, snapshots at three specific time points during the simulation for \(\alpha \) and \(\mu \) are depicted in Fig. 5 and Fig. 6, illustrating the preferential diffusion along the formed fingers. In Fig. 7 we present a magnified view of the dashed area in Fig. 6, illustrating the directions of the flux \({\textbf {j}}\) with white arrows.