## Introduction

Genomic instability is the primary driver of tumor initiation, however disease progression is increasingly recognized to be promoted by pathologic interactions between a tumor and its microenvironment, including exchange of mechanical forces with a remodeled extracellular matrix (ECM)1,2. Biomechanical alterations observed in tumors include accumulation of solid stresses, ECM stiffening, and interstitial fluid pressurization3. In particular, build-up of solid stresses within the primary tumor is caused by its growth within the host tissue4, but also by activation of stromal cells (e.g., cancer-associated fibroblasts) and ECM remodeling3,5. Remodeling, in turn, is directly related to ECM stiffening6,7, while fluid pressurization is due to a combination of leaky blood vessels, dysfunctional lymphatics, and a dense interstitial ECM8. Altered ECM structure and mechanics due to remodeling represents a distinguishing feature of desmoplastic cancers, that is cancers that contain high levels of ECM constituents, particularly hyaluronan and collagen type I. The ECM is actively remodeled by tumor and stromal cells by means of biochemical (deposition/removal) and biomechanical (tension/compression) interactions9. As a result, ECM remodeling involves alterations not only in content, but also spatial organization, of matrix components. In particular, accumulation of collagen in the interstitial space between cancer cells is mainly due to increased deposition10,11,12, whereas alignment of collagen occurs primarily at the tumor-stromal interface and is due to mechanical interactions with the surrounding matrix13. In breast cancer, collagen accumulation and alignment are powerful predictors of disease progression and survival14,15, and such fibrotic remodeling has been shown to induce a malignant phenotype and play a causative role in invasive behaviors6. Indirect evidence of mechanical engagement of collagen by primary tumors is provided by the identification of tumor-associated collagen signatures (TACS): collagen fibers have been found to be aligned parallel (TACS-2) or perpendicular (TACS-3) with respect to the tumor boundary16. While the presence of TACS-3 has been associated with the presence of tensional forces generated by contractile metastatic cells13, it can be reasoned that the presence of TACS-2 is instead associated with an earlier stage of disease in which compressive forces are applied to the ECM by uncontrolled proliferation, although little quantitative evidence has been provided so far. Physically, compression of collagen can lead to densification and stiffening without the need for de novo production. Biologically, reaction forces generated by the compressed ECM on the growing tumor can increase the metastatic potential of cancer cells17,18. Furthermore, compression of the ECM can affect its fluid transport properties, thus leading to fluid pressurization in the peri-tumoral stroma. While tensile ECM remodeling has received considerable attention, ECM compression needs improved biomechanical characterization and definition of how it affects tumor progression in the early stages of avascular growth. Albeit computational models have established a link between tumor growth, mechanical compression of collagen, and increased peri-tumoral interstitial fluid pressure via a reduced hydraulic permeability of the matrix19,20, the mechanisms underlying the structural and functional consequences of such compressive remodeling in collagen networks remain unknown. To fill this gap, here we used spheroids of epithelial breast cancer cells embedded in fibrous collagen to provide quantitative evidence of mechanical compression upon spheroid growth, and investigated the mechanical basis of compressive remodeling of collagen by combining biomechanical experiments with biophysical modeling of collagen network mechanics.

Multicellular tumor spheroids embedded in collagen represent a physiologically relevant 3D model that recapitulates key features of the tumor microenvironment, including biomechanical interactions within a 3D fibrous matrix, and biochemical gradients of nutrients and oxygen21. Spheroid growth has been linked with the generation of compressive stresses in agarose gels22, and has been implicated with the presence of TACS-2 in collagen gels23. However, while agarose is a cellularly inert material endowed with a linearly elastic behavior over a wide range of tensile and compressive strains, collagen I is a pathophysiologically relevant material characterized by complex mechanical properties. Typical features of collagen network mechanics include: material nonlinearity over finite strains, anisotropy due to fiber alignment along the direction of principal strain, time-dependent behavior due to inherent viscoelasticity or interstitial fluid flow, compressibility, and plasticity24. Collagen I monomers self-assemble in vitro at neutral pH and generate fibrils which further associate into networks of fibril bundles25, which here we will refer to as fiber networks. These networks are weakly cross-linked, thus presenting poor structural stability and mechanical function with respect to excised connective tissues. The mechanical properties of collagen gels can be significantly enhanced via cross-linking26, and they are commonly tested via shear rheometry27 and tensile testing28, while compressive properties are still poorly understood29,30. Hence, we employed multiphoton microscopy and confined compression testing – a well-established technique in the field of cartilage mechanics – to characterize the structural, mechanical, and fluid transport properties of collagen networks at various concentrations, before and after chemical cross-linking. A comparison between control and cross-linked collagen networks allowed us to separate poroelastic effects (due to interstitial fluid movements) from viscoplastic effects (due to fiber network dynamics) by modeling the experimentally observed behaviors via continuum biphasic and discrete network models. Overall, our results suggest that tumor growth induces compressive remodeling, a localized densification of collagen caused by fiber bending and cross-link rupture, which in turn reduces the hydraulic permeability of the peri-tumoral matrix.

## Results

### Compression of collagen by breast cancer spheroids

Breast cancer spheroids were generated from MCF-10A human mammary epithelial cells by seeding ~103 cells in low attachment wells in the presence of a small volume fraction of Matrigel (Methods). In this way, we created compacted spheroids rather than growth-arrested acinar structures31, which were embedded in collagen I hydrogels. MCF-10A cells possess many features of a normal mammary epithelium despite presenting genetic and epigenetic abnormalities32, and were therefore chosen to model an early tumor. Spheroid growth in collagen was monitored via time-lapse DIC imaging over the course of the successive 48 hours (Fig. 1a, Supplementary Video 1). Despite differences in ligand density, doubling the collagen concentration from 2 to 4 mg/mL did not significantly impact the time course of radial expansion (Fig. 1b), thus suggesting that MCF-10A spheroid growth is primarily driven by proliferation. The steady increase in spheroid radius led to visible deformations in the surrounding collagen, which estimated by modeling analytically the spheroid as an expanding spherical inclusion (Supplementary Methods). Matrix deformations, measured as stretch ratios in the principal directions $$({\lambda }_{r},{\lambda }_{\theta },{\lambda }_{\varphi })$$ evaluated at the spheroid-collagen boundary, were found to be radially compressive and tangentially tensile (Fig. 1c,d). More importantly, multiphoton imaging showed that such deformations do not propagate smoothly into the collagen matrix but seem to localize around the spheroid edge (Fig. 1e,f). In fact, a dense circumferential layer of collagen – which was not present immediately after embedding – is clearly evident in both collagen concentrations examined herein and reveals itself as a prominent peak in the radial profiles of SHG signal intensity (Fig. 1g), thus indicating that compressive forces localize around the spheroid-collagen boundary. Although active collagen deposition by cells may play a role in the observed densification, we hypothesized that this phenotype is caused by the mechanical forces generated by expanding tumor spheroids.

### Local densification and plasticity are caused by cross-link rupture

The lower hydraulic permeabilities quantified in control gels can be explained by examining the morphology of collagen hydrogels after removal of compressive loads. SHG images acquired using a long working distance objective captured the entire gel thickness and revealed the presence of a dense layer of collagen in correspondence of the gel surface that underwent compression (Fig. 6a). Collagen densification was associated with remnant plastic deformations in control gels of all concentrations, as shown by the significantly lower thicknesses after compression (Fig. 6b and Supplementary Table 3). Plasticity was also observed in low density cross-linked gels but it was less marked and not associated with localized densification of collagen. Mean SHG intensity decays with increasing depth of penetration are due to scattering and absorption, however control gels displayed a sharp peak in SHG intensity in correspondence of the dense collagen layer with subsequent decay to intensity values comparable to cross-linked gels (Fig. 6c). The densification of collagen on the surface of the gel after compression and the sharp peak in SHG axial intensity profiles were reminiscent of the densification observed around the spheroid boundary after 48 hours of proliferation (Fig. 1e–g). The presence of localized plastic deformations, along with the multiple relaxation mechanisms identified by the relaxation time spectrum analysis (Fig. 3d), clearly indicated that fluid flow is not the only mechanism underlying time-dependent responses of collagen networks during compression. Viscoplastic effects due to interactions between fibers likely underlie the localized densification of collagen upon compression identified by our experiments. The majority of continuum models used to model collagen network mechanics are purely elastic37 or model viscoplasticity via phenomenological relations38 which do not account for the physical mechanisms underlying the observed behaviors. In order to address this gap, we developed a discrete network model to gain further insights into the mechanisms of compressive matrix remodeling. A cubic 3D network of collagen fibers was generated to reproduce a 4 mg/mL gel and compressed at one of the boundaries while maintaining all the others fixed, in order to simulate the kinematics of confined compression. Individual fibers were modeled as elastic rods capable of storing elastic energy in both stretching and bending, based on their Young’s modulus E. Forces generated upon deformation within each fiber are transmitted through the network via cross-liking: a fiber pair exchanges stretching and bending forces through the cross-linked node until the total force acting on the cross-linked node exceeds a critical force fbreak, in correspondence of which the cross-link breaks and the two fibers move independently. Discrete fiber network simulations allowed us to reproduce qualitatively experimentally observed behaviors and gain insights into the underlying mechanisms (Fig. 7, Supplementary Video 2). In fact, rate-dependent behaviors were observed by compressing the network boundary at different speeds (Fig. 7a) and the stress decay after a step of compression was reminiscent of our experimental results (Fig. 2d). By simulating the fiber network response in absence of interstitial fluid flow, the model shows that stress relaxation in the network is due to a combination of fiber stretching and bending. The former primarily generates peak forces during compression while the latter dominates the force decay at constant deformation (Fig. 7b,c). Such decay minimizes the overall bending energy, which is one order of magnitude higher with respect to the stretching energy and represents the primary source of elastic energy storage in the network upon compression. Network architectures visualized before and after compression (Fig. 7d,e) show that fibers densify on the compressed boundary in a fashion that is remarkably similar to what we observed in control gels after compression (Fig. 6a). In fact, the sharp peak in node density after compression is similar to the peaks in SHG signals observed experimentally (Fig. 7f). The localized densification is associated with reorientation of fibers in the direction perpendicular to the applied deformation (Fig. 7g), likely caused by fiber buckling under compression.

The role played by cross-linking in the mechanics of fiber networks was explored by varying E and fbreak, the two key parameters that control the mechanical behavior of the network. Interestingly, the cross-link density has little impact on the simulated stress responses (Supplementary Fig. 7). While Young’s moduli of individual collagen fibrils can be measured experimentally39, little is known about the cross-link strength. To address this gap, we performed a parameter sweep in which E was varied between 10 MPa and 250 MPa while fbreak was varied between 1 pN (10−12 N) and 1 µN (10−6 N) to cover a broad and realistic range of values. The results of such parametric study (Supplementary Fig. 8) allowed us to gain insights into the mechanical changes occurring in a fiber network after treatment with GA. Figure 8 shows simulation results for $$E=50\,MPa,\,{f}_{break}={10}^{-9}\,N$$ displaying negligible equilibrium stresses which are reminiscent of the responses observed in control gels. More importantly, such responses are associated with modest bending energies and rupture of cross-links at each compression step (Fig. 8). The time course of the stress was moderately affected by a 5-fold increase in the fiber Young’s modulus alone ($$E=250\,MPa,\,{f}_{break}={10}^{-9}\,N$$) despite the overall bending energy increased considerably. A higher fiber stiffness caused an even higher rupture of cross-links which explains the lack of stiffening at the network level. Higher peak and equilibrium stresses, similar to the responses observed in GA cross-linked gels, were observed only by increasing both E and $${f}_{break}$$ ($$E=250\,MPa,\,{f}_{break}={10}^{-8}\,N$$). The stiffer responses are associated with lower stretching and higher bending energies and, more importantly, with virtually no cross-link rupture (Fig. 8c). Overall, our model shows that increasing fiber stiffness alone is not sufficient to achieve stiffer network properties, but it requires increased cross-linking strength. We conclude that both factors are affected by collagen cross-linking with GA, with stronger intrafibrillar bonds leading to higher fiber stiffness and creation of interfibrillar bonds leading to cross-linking strengthening. Conversely, our modeling results suggest that cross-link rupture and low bending energies – due to fiber buckling under compression – underlie the matrix remodeling observed in collagen hydrogels after compression and, likely, at the boundary of proliferating tumor spheroids. Such compressive remodeling alters the hydraulic permeability properties of the collagen matrix, therefore suggesting impaired convective fluid transport in the tumor microenvironment.

## Discussion

Remodeling of tissues occurs by means of biochemical activity and biomechanical forces generated by resident cells interacting with the structural and mechanical properties of the surrounding ECM. In desmoplastic cancers, ECM components can accumulate either within the interstitial space via increased production, or alternatively align along the tumor-stromal interface via expansion of the tumor mass. The existence of these distinct modalities of remodeling is confirmed by stress alleviation therapy3 outcomes: anti-fibrotic drugs, such as the angiotensin receptor blocker Losartan, significantly lower interstitial accumulation of ECM while leaving nearly intact the peri-tumoral fibrous capsule10. Such fibrous capsule, mostly made of collagen, is likely initiated in the stage of avascular growth and accumulates at the periphery as the tumor increases in size, thus generating a dense, tangentially aligned layer of collagen fibers (TACS-2). The growth-induced compressive forces responsible for peri-tumoral remodeling are also present in the interior of the tumor, where they combine with interstitial remodeling to cause interstitial fluid pressurization and intra-tumoral vessel collapse40. The role of interstitial ECM remodeling in the tumor interior has been documented by several studies examining the effect of ECM depletion on the physical properties of desmoplastic tumors10,11,12,40. Compared to interstitial remodeling, little work has been done to estimate peri-tumoral compressive forces in realistic environments and to characterize structural and functional changes associated with compression of the tumor microenvironment. Computational studies have shown that compression causes densification and tangential alignment of peri-tumoral collagen, while providing correlations between tumor stress, matrix permeability, and fluid velocity19,20. Based on the observation of similar TACS-2 phenotypes in vivo16 and in vitro23, we hypothesized that structural and functional effects of tumor-driven collagen remodeling can be isolated by employing realistic in vitro models, such as the multicellular spheroid embedded in a 3D collagen gel. Therefore, we created spheroids from MCF-10A cells, an epithelial cell line derived from a human mammary tumor expressing abnormal protein expression profiles32. We embedded MCF-10A spheroids into collagen I gels to investigate the mechanical remodeling of fibrous collagen, without confounding contributions from hyaluronan or other ECM components. Contrarily to the findings documented by Helmlinger et al.22 in agarose, varying collagen concentration does not impact significantly the time course of spheroid radial expansion (Fig. 1b), which suggests that collagen networks cannot generate enough solid stress to inhibit spheroid growth. Instead, they undergo radially compressive and tangentially tensile deformations (Fig. 1c,d) consistent with the stress distributions observed in tumors5. Since tumor growth induces radial compression of collagen, we refer to TACS-2 as a compressive remodeling phenotype.

We conclude that not only overall content, but also spatial organization of collagen represents a key determinant of transport properties in fibrous networks. Compressive remodeling – that is compression-induced localized densification of collagen – is caused by fiber buckling and cross-link rupture and is associated with lower hydraulic permeability and higher interstitial fluid pressure. Our findings suggest that both solid mechanics and fluid transport properties of the tumor microenvironment can be altered by a growing tumor compressing the ECM into a fibrous capsule. In breast tumors, presence of a fibrous capsule is a common indicator of a benign lesion, despite the fact that in 10–20% of cases such well-defined tumors are carcinomas46. Moving forward, combined experimental and theoretical models of ECM remodeling will play a key role in providing a better understanding of how microenvironmental solid and fluid mechanics impact tumor progression.

## Methods

### Collagen and spheroid preparation

Collagen solutions were prepared by mixing high concentration, acid-solubilized rat tail collagen I (Corning Life Sciences, Bedford, MA) with equal volumes of a neutralizing buffer, consisting of a 100 mM HEPES solution in 2x PBS adjusted to pH 7.3 via addition of NaOH47,48. The desired collagen concentrations (1–4 mg/mL) were reached by adding adequate volumes of neutralized collagen to cold 1x PBS. Collagen solutions were prepared on ice and allowed to self-assemble into networks inside an incubator (37 °C) for 1 hour. Acellular gels for biomechanical testing were formed by aliquoting 200 µL of liquid collagen inside cylindrical PDMS molds with a diameter of 9 mm (Supplementary Methods). Cross-linking was achieved by adding glutaraldehyde (GA) diluted in deionized water (dH2O) to acellular gels. For each collagen concentration, gels were randomly divided into two groups after polymerization: one was incubated with 4 mL of dH2O (control) while the other was exposed to the same volume of 0.2% v/v GA (Sigma-Aldrich, St. Loius, MO) at room temperature for 12 hours (cross-linked). Control and cross-linked gels were then washed-out twice using 2 mL of 1x PBS. Multicellular spheroids were cultured and embedded in collagen as follows. MCF-10A cells (ATCC, Manassas, VA) were cultured in DMEM/F-12 supplemented with 5% horse serum, 20 ng/mL EGF, 0.5 mg/mL hydrocortisone, 100 ng/mL cholera toxin, 10 μg/mL insulin, and 1% penicillin/streptomycin. Cells were maintained at 37 °C and 5% CO2 and, before reaching confluence, were trypsinized, counted, and resuspended at a concentration of 106 cells/mL. Spheroids were generated by seeding ~103 cells per well in a 96-well ultra-low attachment plate and allowed to form for 48 hours in presence of 2.5% v/v Matrigel49. Once formed, spheroids were transferred individually into glass bottom 6-well plates (MatTek, Ashland, MA) and embedded within 150 μL of liquid collagen at concentrations of 2 and 4 mg/mL. Collagen self-assembled within an incubator for 1 hour and the cell culture plates were carefully rotated every minute for the first 10 minutes in order to guarantee that the spheroids were fully embedded within the collagen matrix and were not in touch with the glass coverslip.

### Time-lapse imaging and analysis

Spheroid growth in collagen was imaged using a customized spinning disk confocal setup consisting of a Leica DMI 6000B microscope (Leica, Wetzlar, Germany) equipped with an ImagEM CCD camera (Hamamatsu Photonics, Hamamatsu, Japan), a FW-1000 high speed filter wheel (Applied Scientific Instrumentation, Eugene, OR), and a LiveCell environmental chamber (Pathology Devices, San Diego, CA). 2 mL of culture media were added to each well and the plate was maintained at 37 °C, 5% CO2, and 80% humidity for the entire duration of the imaging study. For each spheroid within a 6-well plate, differential interference contrast (DIC) images were collected every 10 minutes for 48 hours as 3 × 3 tiled, 200 μm z-stacks. A 10x objective was used to image an area of 1450.3 × 1450.3 μm2 with a resolution of 1.126 μm/pixel. Image acquisition was controlled using the MicroManager 1.4 software (https://micro-manager.org). Minimum intensity projections were used to visualize the 3D data sets as 2D movies and to ensure that the spheroid equator was captured in subsequent analyses. The spheroid radius at each time frame was measured using a modified version of an established algorithm50. Briefly, the center of the spheroid was found by minimization of the Mumford-Shah functional while determination of the spheroid radius was carried out by calculating the spherically symmetric image “graininess” – a first-order approximation to the square of the image gradient50 – and by determining the furthest point where the radial graininess drops below 75% of its maximum. The distance between such point and the center represents the spheroid radius, which approximates well the spheroid size under the assumption of spherical symmetry.

### Multiphoton microscopy and analysis

After 48 hours of DIC imaging, spheroids embedded in collagen were fixed overnight using cold 4% PFA and stained using 2 μM DAPI. Tumor spheroids and fibrous collagen were imaged using a Bruker Ultima Investigator multiphoton microscope (MPM). A laser beam (Insight DeepSee+, Spectra Physics, Santa Clara, CA) with an excitation wavelength of 880 nm was focused onto the samples through either a 16x water objective (Nikon, 0.8 N.A., WD = 3 mm) or a 60x oil objective (Olympus, 1.42 N.A., WD = 0.15 mm). Two-photon fluorescence (TPF) from DAPI stained nuclei and second harmonic generation (SHG) signal from the collagen matrix were collected, respectively, using 550/50 nm and 440/40 nm bandpass filters. Spheroids were imaged using a 16x objective (1x optical zoom) and the following settings: 1024 × 1024 pixels at a resolution of 0.805 µm/pixel, a stack size of 600 µm with 5 µm steps. Matrix densification upon spheroid growth was visualized via radial profiles of SHG intensity. Similarly, plastic remodeling of acellular gels after biomechanical testing was visualized via axial profiles of SHG intensity. The overall gel thickness was measured as the distance between the gel surface and the glass coverslip, and for each sample we report the mean thickness from three measurements at separate x-y locations. Microstructural features of acellular collagen networks were imaged using a 60x objective (2x optical zoom) and the following settings: 1024 × 1024 pixels at a resolution of 0.076 µm/pixel, a stack size of 10 µm with 1 µm steps. High-resolution imaging of microstructural details was consistently carried out 5 µm away from the coverslip to avoid artifacts. Image stacks were thresholded and the mask from each SHG image was applied to the corresponding TPF image, which varies in intensity as a function of collagen cross-linking51. Following Raub et al.51, the threshold was defined as the mean plus twice the standard deviation of the maximum SHG intensity measured in a small area containing no discernible collagen fibers. The largest threshold among all experimental groups was used as the global SHG intensity threshold and used to generate a mask for each image. These SHG masks were then used to segment collagen fibers and quantify mean TPF intensity of collagen as well as mean fiber area. Due to the random orientation of collagen fibers, the volumetric porosity equals the areal porosity52 which was quantified directly from the masks. Finally, mean fiber structure was extracted using CT-FIRE, a validated algorithm for segmentation of microscopy images and extraction of fiber geometry and alignment35. Analysis of images with CT-FIRE led to quantification of fiber orientation (deg), diameter (nm), length (µm), and density (µm−3).

### Biomechanical testing

Custom components (Supplementary Fig. 1) were developed to perform confined compression tests on acellular collagen gels using a commercial DHR-2 rheometer (TA Instruments, New Castle, DE). A precompression of ~144 µm was applied to cylindrical gels to bring them to a thickness H = 3 mm, which was thereupon regarded as the unloaded thickness. Contact between the indenter and the hydrogel surface was confirmed by a spike in the axial force (f) registered by the rheometer. The baseline force (foff) was used to offset the axial force experimentally measured (fexp) during compression. Unless otherwise stated, compression steps of 3% were applied at a constant rate of 1%/s and followed by hold periods of 180 s, during which the deformation remained constant while the force decreased to reach a plateau. Experimental data on gap height and axial force were sampled at a frequency of 100 Hz. The deformed gel thickness h was obtained subtracting the glass coverslip thickness to the value of gap height controlled by the rheometer. The axial stretch ratio was thus calculated as $$\lambda =h/H$$. The axial stress was instead calculated as

$$-{\sigma }^{\exp }=\frac{{f}^{\exp }-{f}_{off}}{\pi {D}^{2}/4},$$
(1)

where D = 8 mm represents the diameter of the porous indenter. Herein we follow the convention that the Cauchy stress (σ) is positive under tension and negative under compression. Equilibrium values of stretch and stress were calculated based on the average of the last 60 seconds in each hold phase. Data analysis and numerical modeling were carried out using Matlab R2018a (Mathworks, Natick, MA), unless otherwise specified. Stress relaxation data were analyzed using a generalized Maxwell model, in which the time evolution of the Cauchy stress is described as

$$\sigma (t)={\sigma }_{e}+{\int }_{-\infty }^{+\infty }H(\tau ){e}^{-t/\tau }d\,\mathrm{ln}\,\tau ,$$
(2)

where H(τ) is the continuous relaxation time distribution function and τ represents the relaxation time. A solution for H(τ) was found by solving a minimization problem (Supplementary Methods). The relaxation time was varied from 10−2 to 104 seconds, for both control and cross-linked gels. For each sample, the area under the entire spectrum at each compression step was calculated using the Matlab function trapz().

### Continuum biphasic modeling

Collagen hydrogels represent mixtures consisting of a solid phase (collagen fibers) and a fluid phase (interstitial liquid). Based on the observed expulsion of interstitial fluid during compression, we implemented a biphasic model of collagen hydrogel mechanics – rooted in the theory for biphasic mixtures developed in the field of cartilage mechanics53,54 – to fit experimental data from confined compression. The observed bulk mechanical behavior under compression was separated into a deformation-dependent stress generated by the solid collagenous matrix and a transient pressurization generated by the fluid filling the interstitial space (Supplementary Methods). Briefly, the material behavior of the solid constituent was described using the Yeoh strain energy function36

$$W={c}_{1}({I}_{C}-3)+{c}_{2}{({I}_{C}-3)}^{2}+{c}_{3}{({I}_{C}-3)}^{3},$$
(3)

where $${I}_{C}=tr\,{\bf{C}}$$ represents the first invariant of the Cauchy-Green tensor, while c1, c2, and c3 are material parameters subjected to the constraints $${c}_{1},{c}_{3} > 0$$ and $${c}_{2} < 0$$. The fluid transport properties of collagen hydrogels were instead described by a hydraulic permeability tensor k, which – according to Darcy’s law – regulates hydraulic flow in porous media in response to pressure gradients52. Here, we assumed an isotropic and strain-independent hydraulic permeability, that is

$${\bf{k}}=k{\bf{I}},$$
(4)

this assumption is due to the fact that collagen gels are highly porous materials made of randomly oriented fibers. Based on the assumed constitutive behaviors, the theoretically-calculated axial stress in the mixture was indicated as $${\sigma }_{ZZ}(Z,t)$$ (Supplementary Methods), where Z is the position of a material point along the gel height (with Z = 0 at the interface with the porous indenter and Z = H at the bottom of the confining chamber) while t represents time. The unknown model parameters were determined via nonlinear least squares minimization of the following objective function

$$e=\mathop{\sum }\limits_{i=1}^{N}{[{\sigma }_{ZZ}(0,t)-{\sigma }^{\exp }(t)]}_{i}^{2},$$
(5)

where $${\sigma }^{\exp }(t)$$ represents the experimentally-measured Cauchy stress from Eq. (1), N represents the number of experimental data points, from multiple steps of compression, that were included in the regression. The objective function was minimized using the Matlab function lsqnonlin(), which was used within a two-step fitting procedure55: material parameters from the Yeoh model (c1, c2, c3) were determined by fitting equilibrium data, while the hydraulic permeability (k) was determined by fitting transient responses.

### Discrete network modeling

The mechanisms underlying compressive remodeling of collagen gels after compression were investigated computationally using a discrete network model. The dynamic evolution of a 3D network of collagen fibers was simulated using the framework originally developed by Kim et al.56 to model cytoskeletal networks, with the main difference being that herein we disregarded Brownian forces due to the athermal nature of collagen mechanics57,58. Briefly, collagen fibers with a fixed diameter of 155 nm39,59, a length sampled from an experimentally measured distribution (Supplementary Fig. 4), and random orientations were generated within a 50 × 50 × 50 µm3 cubic volume to achieve a concentration of 4 mg/mL (Supplementary Methods). Each fiber was discretized using small segments of length $${l}_{0}=1\,{\rm{\mu }}{\rm{m}}$$ which, being shorter than the persistence length of collagen $${l}_{p}=5\mbox{--}10\,{\rm{\mu }}{\rm{m}}$$59, could therefore be assumed to behave like linear elastic rods. The mechanical behavior of individual fiber segments was assumed to be governed by stretching and bending potentials which are characterized by the following equations

$$\begin{array}{c}{U}_{s}=\frac{1}{2}{k}_{s}{(l-{l}_{0})}^{2},\\ {U}_{b}=\frac{1}{2}{k}_{b}{(\theta -{\theta }_{0})}^{2},\end{array}$$
(6)

where l and l0 represent the current and equilibrium lengths of an individual fiber segment, while θ and θ0 represent the current and equilibrium angles formed between two adjacent segments. The stretching and bending stiffness constants are, respectively, given by $${k}_{s}=EA/{l}_{0}$$ and $${k}_{b}=EI/{l}_{0}$$, where E is the Young’s modulus of a collagen fiber, $$A=\pi {R}^{2}$$ is the cross-sectional area, and $$I=\pi {R}^{4}/4$$ is the second moment of inertia of a fiber of radius R. Rigid cross-links were formed between adjacent fibers and the cross-link density was varied by changing the maximum distance between fibers allowed to form a cross-link (Supplementary Methods). Using this approach, the forces developed upon deformation in one of the fibers that are part of a rigid cross-link are transferred to the other fiber until the total force at a cross-link exceeds a breaking force $${f}_{break}$$, in correspondence of which the covalent bond ceases to exist and the two fibers are free to move independently. The parameters E and $${f}_{break}$$ characterize the mechanical behavior of the fiber network and were therefore varied parametrically to reproduce qualitatively the various behaviors observed experimentally. The Young’s modulus of collagen fibrils is reported to be on the order of hundreds of MPa39, hence we varied E between 10 MPa and 250 MPa. On the other hand, little is known about the cross-link breaking force which was therefore varied between 1 pN (10−12 N) and 1 µN (10−6 N) to cover a broad range of values. It should be noted that this range was chosen to include the force needed to break intrafibrillar bonds between collagen monomers60. Discrete fiber network simulations were implemented in Matlab and visualized using Visual Molecular Dynamics (VMD, https://www.ks.uiuc.edu/Research/vmd).

### Statistical analysis

Experimental data are presented as mean ± standard error of the mean (SEM), based on fixed or variable sample sizes depending on the type of experiment. Due to the unequal sample sizes, differences between control and cross-linked samples were determined using a Welch’s t-test, which assumes unequal variances. A one-way ANOVA was used to test differences due to collagen concentration, and post-hoc pairwise comparisons were performed using the Bonferroni correction. Differences were considered significant for p < 0.05.