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Annals of Biomedical Engineering ( 2015) DOI: 10.1007/s10439-015-1416-2 A Comparative Study of Collagen Matrix Density Effect on Endothelial Sprout Formation Using Experimental and Computational Approaches AMIR SHAMLOO,1 NEGAR MOHAMMADALIHA,1 SARAH C. HEILSHORN,2 and AMY L. BAUER3 1 Center of Excellence in Energy Conversion (CEEC), School of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran; 2Department of Materials Science & Engineering, Stanford University, Stanford, CA 94305, USA; and 3Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Received 8 March 2015; accepted 4 August 2015) Associate Editor Michael Gower oversaw the review of this article. Abstract—A thorough understanding of determining factors in angiogenesis is a necessary step to control the development of new blood vessels. Extracellular matrix density is known to have a signiﬁcant inﬂuence on cellular behaviors and consequently can regulate vessel formation. The utilization of experimental platforms in combination with numerical models can be a powerful method to explore the mechanisms of new capillary sprout formation. In this study, using an integrative method, the interplay between the matrix density and angiogenesis was investigated. Owing the fact that the extracellular matrix density is a global parameter that can affect other parameters such as pore size, stiffness, cell– matrix adhesion and cross-linking, deeper understanding of the most important biomechanical or biochemical properties of the ECM causing changes in sprout morphogenesis is crucial. Here, we implemented both computational and experimental methods to analyze the mechanisms responsible for the inﬂuence of ECM density on the sprout formation that is difﬁcult to be investigated comprehensively using each of these single methods. For this purpose, we ﬁrst utilized an innovative approach to quantify the correspondence of the simulated collagen ﬁbril density to the collagen density in the experimental part. Comparing the results of the experimental study and computational model led to some considerable achievements. First, we veriﬁed the results of the computational model using the experimental results. Then, we reported parameters such as the ratio of proliferating cells to migrating cells that was difﬁcult to obtain from experimental study. Finally, this integrative system led to gain an understanding of the possible mechanisms responsible for the effect of ECM density on angiogenesis. The results showed that stable and long sprouts were observed at an intermediate collagen matrix density of 1.2 and 1.9 mg/ml due to a balance between the number of migrating and proliferating cells. As a result of weaker connections between the cells and matrix, a lower collagen matrix density (0.7 mg/ml) led to unstable and Address correspondence to Amir Shamloo, Center of Excellence in Energy Conversion (CEEC), School of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran. Electronic mail: [email protected] broken sprouts. However, higher matrix density (2.7 mg/ml) suppressed sprout formation due to the high level of matrix entanglement, which inhibited cell migration. This study also showed that extracellular matrix density can inﬂuence sprout branching. Our experimental results support this ﬁnding. Keywords—Endothelial sprout, Matrix density, Microﬂuidic device, Cellular Potts Model, Multi-scale model. INTRODUCTION The formation of new blood vessels via angiogenesis is a rate-limiting step for normal development and physiology, as well as numerous pathologic processes, including cancer and ophthalmologic disorders such as macular degeneration.20 In cancer, the concept of an ‘‘angiogenic switch’’, whereby neovascularization both precedes and is necessary for tumor progression and metastasis, was proposed by Folkman and Hanahan.27 The development of tumor angiogenesis is mechanistically quite complex, involving vessel growth into an initially avascular tumor mass,21 the early recruitment of vasculature from neighboring tissue,30 and the contribution of circulating endothelial stem cells.3 Given the anti-cancer therapeutic potential of an angiogenesis blockade, signiﬁcant efforts have been devoted toward understanding the molecular mechanisms underlying blood vessel formation.19,29,60,63 Potential therapies for numerous other diseases (e.g., peripheral arterial disease, diabetic wound healing, macular degeneration, stroke, myocardial infarction) include either the prevention or promotion of angiogenesis. Many biochemical factors have been identiﬁed that either deter or encourage blood vessel formation.41 However, very little is known quantitatively about the interrelationship between the cell and the biochemical 2015 Biomedical Engineering Society SHAMLOO et al. and biomechanical environment. Although signiﬁcant strides have been made, a comprehensive and multiscale understanding of the relationships involved in angiogenesis and tumor growth, and thus an effective cure or treatment, is still a work in progress. What is known is that blood vessel formation and cohesive vascular network structures require the coordinated motion of a multitude of cells, favorable environmental factors, and successful signal transduction. These complex vascular networks exhibit dynamic behavior including endothelial cell migration, proliferation, elongation, and branching.14 As endothelial cells migrate into the extracellular matrix (ECM), they undergo cell-type speciﬁc specialization, with tip cells forming ﬁlopodia to sense and migrate toward chemoattractants such as vascular endothelial growth factor (VEGF), while stalk cells proliferate and form vascular tubes.15,23,49 Tip cells express receptors including VEGFR2 and VEGFR3 that are not expressed by stalk cells and that are functionally required for angiogenesis.46,59 These molecular level differences between tip cells and stalk cells result in different migration and proliferation rates.23 While observations at and below the single cell length scale have led to advancements in our understanding of tissue formation, this process cannot be truly described and studied without observing the systems of collective cell phenomena that occur across much larger time and length scales. It is this system of cellular communication and interaction that ultimately results in coordinated cell motion and cohesive tissue formation. Recently, several groups have developed in vitro models of angiogenesis that have produced new insights into the mechanisms of coordinated endothelial cell movement.10,12,24,34,38,40,47,51–53,56,57,61,62 Although various in vitro and in vivo assays have been developed to provide precise insight into the ﬁeld of angiogenesis,31,44 microﬂuidic devices show a considerable potential for angiogenesis studies because they allow precise and quantitative control over the extracellular microenvironment, thereby enabling direct testing of hypotheses that cannot be achieved using in vivo models alone.64 For example, a variety of microfabricated culture devices apply quantiﬁed concentration gradients of soluble factors to two-dimensional (2D) monolayer cultures and/or three-dimensional (3D) sprouting morphogenesis cultures of endothelial cells.10,18,51,52 These quantitative in vitro results were validated through design of an optimized VEGF-delivery system for in vivo murine hind leg ischemia,10 but there remains a need for a high-throughput approach with the ability to test the simultaneous effects of biochemical and biomechanical factors and to explain the mechanisms behind these effects. Furthermore, Mathematical investigations of angiogenesis have employed continuous, discrete, and mechanical models to describe a variety of dynamics believed to inﬂuence angiogenesis.2,9,13,16,17,36,42,50,58 However, despite clear evidence that the ECM is crucial to cellular behavior and vascular patterning, most models of angiogenesis neglect the dynamic interaction between endothelial cells and the ECM. Until recently, no single mathematical model existed that (a) couples multiple time and length scales, (b) generates realistic capillary structures, including branching and anastomoses, without a priori prescribing rules and probabilities to these events, and (c) considers the complex biochemical and mechanical interactions that occur between endothelial cells and the ECM.4,5 Computational techniques can be applied to simulate a wide range of biological phenomena including intracellular processes, formation of multi-cellular structures and distribution of biochemical factors within cellular microenvironments.4–6,26,28,32,33,35,45 These computational methods, when combined with experimental veriﬁcations, have the potential to help explain the mechanisms behind biological processes that cannot be understood from in vivo or in vitro techniques alone.8,48 It was shown that maximal cell migration happens at an intermediate binding level of the cells to the matrix, but more compliant matrices are more favorable for 3D cell movement when keeping the cell–matrix binding level constant.65 Another group applying both experimental and computational techniques to study leukocyte rolling found that in addition to the ligand surface density the convective ﬂux of the bonds and the dissociation rate at the back of a cell’s contact zones determines the rolling velocity of the cells.39 In addition, a recent analysis on neuronal axon path-ﬁnding integrating experiment and simulation has found that different mechanisms may be responsible for neuronal axon path-ﬁnding.43 It was shown that at steeper gradients axon turning is the more possible response of neuronal cells to neurotrophic factors whereas at shallower gradients biased axonal growth rate towards biochemical gradients is a more probable cause for directional path-ﬁnding of neurons.43 In this study, we have developed experimental and computational platforms capable of testing multiple variables and explaining the interplay between biochemical and biomechanical factors in a study of the eﬀects of matrix density on endothelial sprout morphogenesis. Our simulation results were veriﬁed on our experimental platform. Both computational and experimental observations illuminated collagen ﬁbril density as a critical factor in the morphology and characteristics of sprouts within matrices of varying stiﬀness. Our studies elucidate the role of collagen ﬁbril density as a determinant factor aﬀecting the formation A Comparative Study of Collagen Matrix Density Effect of multi-cellular structures from single endothelial cells during angiogenesis. Comparing the results from experimental and computational methods introduced the binding aﬃnity between the cells and the collagen ﬁbers as the main factor aﬀecting endothelial sprout formation within collagen matrices of varying densities. MATERIALS AND METHODS In Vitro Endothelial Sprouting Assays Microﬂuidic devices were fabricated in PDMS (Sylgard 184, Midland, MI) using soft lithography as previously described51,52 (Fig. S1). Transparency masks were generated from CAD ﬁles of ﬂuidic channels using negative photoresist (SU-8). Sharpened needles (20 gauge) were used to punch inlets and out- lets of the cell culture chamber and reagent channels. To form an irreversible seal between the PDMS and the glass substrate, both surfaces were exposed to oxygen plasma treatment for 2–3 min and brought immediately into contact. This microﬂuidic device enables quantiﬁable measurements of endothelial cell chemotaxis in response to known gradients of soluble factors that remain stable from 1 h to weeks (Fig. 1a). The microﬂuidic device also enables real-time imaging of cellular dynamics in both 2D and 3D culture models. A stable concentration gradient across the cell culture chamber was generated as shown in Fig. S2. Rheological tests on the collagen/ﬁbronectin matrix samples were performed using an oscillatory rheometer (Anton-Paar) in parallel-plate geometry. Adult human dermal microvascular endothelial cells (HMVEC, Lonza, Walkersville, MD) were ﬁrst grown as monolayer cultures on the surfaces of collagen- FIGURE 1. The diffusion-based microfluidic gradient generator device (a); Three dimensional monolayer cultures of adult human dermal microvascular endothelial cells (HMVEC) on the surfaces of collagen-coated dextran beads were embedded within collagen I/fibronectin matrices. (b); Collective cell migration and endothelial sprout formation in response to VEGF gradient (c); the equilibrium profile of VEGF concentration within the cell culture chamber; see also Fig. S1 (d); the initial VEGF field in the computation domain (e); the systems integrated mathematical multi-scale model (f). SHAMLOO et al. coated dextran beads (Cytodex, GE, d ~ 170 lm) to form endothelial sprouts (Figs. 1b, 1c). These beads were then suspended within a 3D matrix containing rat tail collagen Type I (BD Biosciences) and ﬁbronectin (ﬁnal concentration of 5 lg/ml, 10% total volumetric mixture) and consequently cultured inside the microﬂuidic device (~20 beads/device) that generated precise and stable VEGF gradients (Fig. 1d). Endothelial Growth Medium-2MV (EGM-2MV) supplemented with various concentrations of VEGF (VEGF-A isoform VEGF (165), R&D Systems, Minneapolis, MN) was continuously supplied to the source reagent channel, while EGM-2MV with no VEGF was supplied to the sink reagent channel. The injection ﬂow rate was 40 nl/min. Upon VEGF stimulation, ECs underwent sprouting morphogenesis, i.e., collective cell movement to migrate away from the bead surface and into the collagen gel matrix as a cohesive, columnar structure (Fig. 1c). More details about the microﬂuidic system are presented in the supplementary section (S1). Collagen matrices of varying densities ranging from compliant gels to stiffer matrices were used to study the effect of matrix density on the sprout morphology, dynamics and branching. To obtain different matrix densities, the collagen concentration was altered while the ﬁnal ﬁbronectin, microcarrier bead, endothelial cell, and media concentrations were kept uniform. The formation of endothelial sprouts was detected and recorded using time-lapse imaging. Characterization of Endothelial Cell Sprouts and Statistical Analysis Individual beads were imaged every 24 h for 3 days using phase contrast and ﬂuorescence microscopy. ImageJ software (NIH freeware) was used to measure the number of cells per sprout, the sprout average thickness, the sprout length and the sprout elongation rate. The average thickness of each sprout was determined by dividing the projected sprout surface area by the length of the sprout centerline. For each condition, between three and six independent experiments were performed, such that the number of beads imaged for each condition totaled 60–120. The one-tailed, nonpaired, Student’s t test was used to determine the statistical signiﬁcance of differences between pairs of conditions. Cell Fluorescent Staining Sprouts within collagen gels were ﬁxed by injecting 4% paraformaldehyde into the source and sink channels and incubating at 4 C for approximately 1 h. The samples were washed at least four times by injecting PBS into the source and sink channels. The cells were blocked with 10% normal goat serum in PBST (0.3% Triton X-100 in PBS solution) for 3 h. The cellular actin cytoskeleton was stained overnight at 4 C using Alexa Fluor 555-conjugated phalloidin (Invitrogen) while the cell nuclei were stained using DAPI. Samples were washed with PBS at least four times and then incubated with PBS at 4 C for 12 h prior to imaging. An inverted ﬂuorescence microscope was used to acquire images. Computational Model of Angiogenesis We also developed and validated a 2D (x, y, t) cellbased, multi-scale model of angiogenesis that integrates tissue, cellular, and molecular scale dynamics to analyze cell–cell and cell–matrix interactions. Using this model, a gradient of VEGF concentration was generated in the simulation domain that is shown in Fig. 1e. This spatio-temporal mathematical model combines a lattice-based cellular Potts model describing individual cellular interactions, a partial differential equation to describe the spatio-temporal dynamics of VEGF, and the Boolean model of intracellular signaling pathways critical to angiogenesis7 (Fig. 1f). A summary of the computational model is presented in the supplementary section (S2). Since the VEGF ﬁeld (V(x, y, t)) exerts an inﬂuence on cell chemotaxis, the discrete and continuous parts of the model feedback on each other at the cell membrane through the chemotaxis term in the total energy equation. Considering the eﬀect of cell phenotype on the chemotaxis potential is one of the notable features of this multi-scale mathematical model. This model distinguishes between the cells that are proliferating and the ones that are migrating. Cells migrate toward increasing concentrations of VEGF from the parent vessel to facilitate sprout extension. The ﬁbrous structure of the ECM is another feature of this model. ECM collagen ﬁbers are randomly distributed at random discrete orientations between 290 and 90. The mesh-like anisotropic structure of the extracellular matrix is modeled by the heterogeneous and random distribution of these ﬁbers. The cell membrane interacts with these ECM ﬁbers through the adhesion term in the total energy equation. The VEGF and cellular distributions and the geometry of the computational model have been initialized to approximate the in vitro microcarrier bead sprouting morphogenesis experiment. Our computational model captures realistic vascular structures including branching as well as contact guidance and inhibition. Computational simulations and experimental observations of soluble factor gradients are in excellent agreement and allow exact speciﬁcation of the A Comparative Study of Collagen Matrix Density Effect soluble factor concentration at every location within the device (comparison of Figs. 1d, 1e). Quantiﬁcation of Collagen Fibril Density Scanning electron microscopy (SEM) was used to determine collagen ﬁbril distribution within collagen matrices of varying densities. To quantify the collagen ﬁbril density, an image analysis algorithm was used to predict the vacant spaces in collagen matrix and the spaces ﬁlled with collagen ﬁbrils using an in-house MATLAB image analysis code. Using this code, we categorized the image pixel counts based on their normalized intensity. By estimating the minimum normalized intensity of the pixels that represent collagen ﬁbers on the SEM images, we determined a criterion showing ﬁbrillar and non-ﬁbrillar spaces in the collagen matrix. We deﬁne a ratio of ﬁbrillar spaces to the total matrix space as the number of pixels having intensity greater than the threshold divided by the total number of pixels. This ratio was used as an input for ﬁbril density, or fractional ﬁlled area, in the simulation studies. More details about this innovative method are presented in the supplementary section (S3). RESULTS Quantiﬁcation of Collagen Fibril Density for Varying Collagen Gel Concentration As previously described, scanning electron microscopy (SEM) was used to determine collagen ﬁbril distribution within collagen matrices of varying densities: 0.7, 1.2, 1.9 and 2.7 mg/ml (Figs. 2a–2d). A model was also made to simulate the distribution of collagen ﬁbers in the in silico study. The architecture of the ECM is anisotropic, with regions of varying densities. A single collagen ﬁbril is ~300 nm long and 1.5 nm wide and is substantially smaller than an EC, which is ~10 lm in diameter.1 Many individual FIGURE 2. Scanning electron microscopy (SEM) shows the distribution of collagen fibers within collagen matrices of varying densities (0.7, 1.2, 1.9 and 2.7 mg/ml) (a–d); The anisotropic structure of ECM of various densities is effectively generated in the mathematical model by randomly distributing 1.1 lm thick bundles of individual collagen fibrils at random orientations (e–h); Histogram of normalized pixel intensity analysis of SEM is presented for the collagen densities of 0.7, 1.2, 1.9 and 2.7 mg/ml (i); Plot (j) shows the relation between the SEM analyzed fractional filled area (i.e., portion of the matrix that is filled with collagen fibrils) and the collagen density used to prepare the matrices. A significant increase in the fractional filled density is observed as a result of an increase in the experimental collagen matrix density. Statistical analysis showed that the fibril densities are significantly different within varying collagen matrices. SHAMLOO et al. FIGURE 3. Experimentally and computationally observed sprouting morphogenesis at varying matrix densities over a time period of 72 h. Within matrices of 0.7 mg/ml density unstable or broken sprouts are observed due to minimal cell–cell interactions. Arrows show minimal cell contact zones. (a). At intermediate collagen densities (1.2 and 1.9 mg/ml) stable sprouts are formed (c and e); however, thicker sprout morphologies formed in 1.9 mg/ml density matrices (e) compared to those formed in 1.2 mg/ml density matrices (c). High matrix density (2.7 mg/ml) inhibited sprout elongation and yielded thicker sprouts (g). Predictions from the respective comparable computational simulations are remarkably close to in vitro observations (b, d, f, and h). The ratio of proliferating to migrating cells of sprouts formed within matrices of varying density (i), *p < 0.05, **p < 0.01. A Comparative Study of Collagen Matrix Density Effect FIGURE 4. Fluorescent inverted (a, b, and c) and confocal (d) images of sprouts formed within matrices of various collagen densities. Red and white colors represent a phalloidin stain of the F-actin cytoskeleton and a DAPI stain of nuclei, respectively. collagen ﬁbrils and other matrix proteins are bound together constituting larger cords or bundles of matrix ﬁbers estimated to be between 100 and 1000 nm thick.22 Our mathematical model accurately captures the heterogeneity and random distribution of a typical ECM ﬁber matrix, such as a Type I collagen matrix. We model this mesh-like anisotropic structure of the ECM by randomly distributing 1.1 lm thick bundles of collagen ﬁbrils at random orientations ranging between 290 to 90 until the desired percentage of total ECM space occupied by collagen ﬁbers is achieved (Figs. 2e–2h). Using an innovative approach, we calculated the ﬁbril density of ECM by means of an image analysis algorithm that is described in supplementary section S3 (Fig. 2i). According to image analysis data which was shown in Fig. 2j, the collagen ﬁbril density is very low in the lowest collagen gel density (0.7 mg/ml yields a ﬁbril density of ~0.13). Analyzing the ﬁbril density at higher collagen gel densities reveals that ﬁbril density increases signiﬁcantly from ~0.13 to ~0.38 when the collagen gel density is increased from 0.7 to 1.2 mg/ml (Fig. 2j). Fibril density increases from ~0.38 to ~0.68 when we increased collagen gel density from 1.2 to 1.9 mg/ml (Fig. 2j). The collagen matrix with 2.7 mg/ ml density yielded a ﬁbril density of ~0.89. These collagen matrix densities approximately correspond to the ﬁbril densities used in the matrices in the computational model. The sprout characteristics from the experimental and simulation platforms were compared and results are discussed below. Comparison of Sprout Morphology and Cell Count in Varying Collagen Densities Using Experimental and Computational Platforms Endothelial cells cultured on surfaces of microcarrier beads within collagen matrices of varying densities demonstrated a distinctive dependency between sprout morphology and matrix density. Tracks of endothelial cells were observed over a 72-h time period within collagen gel matrices of 0.7, 1.2, 1.9 and 2.7 mg/ml density and simulated computationally on matrices with corresponding ﬁbril densities of <0.1, 0.4, 0.7, and >0.9. On the corresponding 0.7 mg/ml collagen matrix density, endothelial cell tracks show minimal cell contact resulting in unstable or broken sprouts (Fig. 3a). This was predicted by the numerical simulations of sprouting angiogenesis on a matrix with a low ﬁbril density (Fig. 3b). In contrast, Fig. 3c shows narrow but stable sprout formation using the experimental platform within a collagen matrix of density of 1.2 mg/ml. The cells forming these sprouts are well connected with each other. Narrow, stable sprout formation was independently predicted using the mathematical model on a matrix with a ﬁbril density of 0.4 (Fig. 3d). Sprouts forming on 1.9 mg/ml density matrices show thicker morphologies in the SHAMLOO et al. FIGURE 5. Comparison of experimental and simulated sprout thickness (a), and sprout length (b), number of endothelial cells per sprout length (c) and number of cells per sprout (d) formed within different collagen densities, *p < 0.05, **p < 0.01. Since broken and unstable sprouts formed within collagen matrix density of 0.7 mg/ml, we were not able to report observed data of sprout thickness for this density. Therefore, only the simulated data is presented for this matrix density (a). experimental platform (Fig. 3e). Greater levels of cell contact for these sprouts make them more capable of forming stable multi-cellular structures. Thicker sprout morphologies on matrices with higher ﬁbril density was also borne out in simulations using ﬁbril density of 0.7 (Fig. 3f). Interestingly, cell migration was inhibited in the stiffest collagen gel matrices (2.7 mg/ml) (Fig. 3g). Suppression of sprout formation was also observed in simulation studies on matrices of associated collagen ﬁbril density (>0.9) (Fig. 3h). The inability of the tip cells to penetrate, elongate, and migrate into matrices of very high density indicates the critical role of biomechanical effectors during new blood vessel formation. As can be seen in Fig. 3i, a balance between proliferation and migration was observed on matrices with 1.2 and 1.9 mg/ml collagen density. A higher ratio of proliferation to migration within collagen density of 1.9 mg/ml led to thicker and more stable sprouts compared to the ones within 1.2 mg/ml collagen density, whereas broken and unstable sprouts formed on 0.7 mg/ml density due to the low ratio of proliferating to migrating cells. Furthermore, no signiﬁcant migration of the cells was observed within matrices of 2.7 mg/ml collagen den- sity, so the ratio of proliferation to migration was not reported in this case. These results suggest that the density of ECM affects sprout morphogenesis by regulating the proliferation and migration of endothelial cells. (As can be seen in Fig. 3i, the ratio of proliferating cells to migrating cells are 0.4 and 0.55 for intermediate collagen densities within which stable sprouts are formed). Sprout morphology was further characterized by calculating the average sprout thickness measured along the length of each sprout formed within the experimental and simulation platforms. The number of cells per sprout and per sprout length was obtained in the experimental platform by staining the cell nuclei and actin cytoskeleton followed by manual counting (Figs. 4a–4d). Figure 5 compares sprout thickness, sprout length, and number of cells per sprout and per sprout length showing quantitative agreement between the experimental and computational results. Quantiﬁcation of observed and simulated results conﬁrms that sprout thickness increases by increasing matrix density (Fig. 5a). It was also observed that the length of sprouts within collagen matrix of 2.7 mg/ml density is signiﬁcantly lower compared to the sprouts within A Comparative Study of Collagen Matrix Density Effect FIGURE 6. Time-lapse imaging of sprout morphology within 1.2 mg/ml (a, b, and c) and 1.9 mg/ml collagen densities (d, e, and f). TABLE 1. Elongation rate of sprouts within various collagen densities. is needed for sprouts to start elongation within stiffer matrices. Elongation rate (lm/h) Collagen density (mg/ml) 24–48 h 48–72 h Tip Cell Filopodia Extension and Sprout Branching 1.2 1.9 8.1 ± 2.4 2.2 ± 0.7 0.5 ± 0.2 4.5 ± 1.5 Figure 7a shows sprout formation on a 1.2 mg/ml collagen matrix. The tip cell of this sprout has long, thin protrusions, called ﬁlopodia. Filopodia have been found to play a critical role in the path ﬁnding and directional navigation of endothelial sprouts.23 Interestingly, sprout branching can be observed frequently at this collagen density (1.2 mg/ml) (Fig. 7b). An increased incidence of branching within a speciﬁc range of ﬁbril densities was predicted by our computational model (ﬁbril density of 0.4) (Fig. 7c). By quantifying the percentage of branches per sprout formed within varying collagen matrix densities, it was observed that branching occurs signiﬁcantly within a speciﬁc range of matrix density (0.7–1.2 mg/ml). The incidence of branching was less likely on other densities considered (Fig. 8). In the present study, our experimental model veriﬁed the hypothesis of our numerical model that a speciﬁc range of matrix density (ﬁbril densities of 0.1 and 0.4) can cause higher incidence of sprout branching compared to the higher matrix densities (Fig. 8). other matrix densities conﬁrming that this matrix density is preventive for sprout elongation (Fig. 5b). Analyzing the experimental and simulation results also showed that the number of cells per sprout and per sprout length increases by increasing collagen matrix density. This is an evidence corroborating higher ratios of cell proliferation to migration within collagen matrices of higher densities (Figs. 5c, 5d). Dynamics of Sprout Elongation Within Matrices of Varying Densities Time lapse imaging of sprouts within varying collagen densities was performed to determine the rate of sprout elongation. Observations obtained from the experimental platform showed that by increasing collagen density, the rate of sprout elongation decreases (Figs. 6a–6f). Sprout lengths were quantiﬁed at three consistent time points for the two intermediate collagen densities. Comparing the sprout morphogenesis at consistent time points across different matrix densities shows that sprouts within 1.2 mg/ml matrix density elongate more quickly compared to the ones within 1.9 mg/ml matrix density over a time period of 48 h (Table 1). However, thicker sprouts form within 1.9 mg/ml matrix densities. As can be seen in Table 1, sprout elongation occurs prominently between 24 and 48 h within 1.2 mg/ml collagen matrices and between 48 and 72 h within 1.9 mg/ml collagen matrices. Table 1 results demonstrate that the elongation rate was higher on lower matrix densities. Additionally, more time DISCUSSION AND CONCLUSIONS The development of complex vascular networks is a critical process during embryonic development, adult tissue remodeling, cancer progression, and for regenerative medicine therapies. The goal of this research was to synergistically and synchronously develop theoretical, computational models and experimental, laboratory models to predict the fundamental biophysics and biochemistry regulating vascular networks. SHAMLOO et al. FIGURE 7. Sprout initiation and branching is shown within 1.2 mg/ml collagen matrix in vitro (a, b).The computational model shows sprout branching within matrix with ~0.4 fibrillar density (c). FIGURE 8. Comparison of experimental percentage of branches per sprout formed within different collagen densities in vitro. Branching is more likely on the range of 0.7–1.2 mg/ ml collagen matrices (related to fibrillar densities of 0.1–0.4) compared to higher gel densities, **p < 0.01. Previous studies have demonstrated the dependency of cell morphology and function to matrix density within various tissues of the body.11,25,37,51,54,55 However, the mechanism responsible for the effect of matrix stiffness on cellular responses has not yet been entirely elucidated. Complex interplay between biochemical and biomechanical factors has made these studies more challenging. Unfortunately, the implementation of techniques that can facilitate the control of key variables are complicated by the difﬁculty of testing various and simultaneous hypotheses. However, experiments and computation can be integrated to achieve precise control of measured variables and to test multiple hypotheses at low cost to potentially explain some of the mechanisms controlling biological phenomena. For example, although previous separate studies had identiﬁed collagen density as playing a role in endothelial sprout formation,5,51 there was still a demand for a joint study of this problem since the experimental work was not able to isolate the primary cause of this dependency and the simulation predictions could not be veriﬁed without related experiments. In the present study the effect of ECM in the computational model was only considered through the key interactions between cell membranes and ECM ﬁbers at random orientations. Good agreement between the results of the computational and experimental model leads us to conclude that the density of ECM is likely to exert an inﬂuence over sprout morphology and stability by regulating the proliferation and migration of endothelial cells through cell–matrix adhesion. It could be stated that the speciﬁc ranges of extracellular matrix density lead to an optimum degree of interactions between cells and extracellular matrix and consequently stable sprouts are formed. Synchronizing the experimental and computational studies, we demonstrated that the ECM density mainly affects sprout morphology by adjusting cell–cell and cell–matrix interactions and a balanced ratio of proliferating cells to migrating cells. This joint computational and experimental study of endothelial sprout formation characterized the matrix density dependence of cell migration and proliferation in endothelial sprout formation and morphology. Various collagen matrix densities were selected to study a relatively wide range of matrix storage moduli. The plateau storage moduli of gels with the selected range of densities (0.7–1.2– 1.9–2.7 mg/ml) that represents the stiﬀness of gels was 30– 80–200 and 700 Pa respectively.51 Quantiﬁcation of scanning electron microscopy (SEM) of collagen ﬁbers revealed a correlation between increasing matrix density and collagen ﬁbril formation and entanglement. The most compliant collagen matrix studied had a very sparse distribution of collagen ﬁbrils with a low degree of entanglement and large mesh sizes. Increasing collagen density increased the entanglement of collagen ﬁbrils considerably, resulting in smaller mesh sizes of the gel. A very high A Comparative Study of Collagen Matrix Density Effect level of entanglement with tiny mesh sizes was observed for collagen matrices with a density of 2.7 mg/ml. Using our platforms, sprout formation of dermal microvascular endothelial cells was studied across a range of collagen concentrations (0.7–2.7 mg/ml) within an equilibrium VEGF concentration gradient. These experiments revealed that endothelial sprouting is stabilized at the intermediate collagen matrix densities of 1.2 and 1.9 mg/ ml, suggesting that close cooperation between biochemical and biomechanical factors is necessary for successful angiogenesis. As suggested by our in silico modeling results, we found that increasing collagen matrix ﬁbril density increased the duration of cell–cell contacts, cell proliferation, and the width of the sprouts. At low matrix densities (0.7 mg/ml), cells migrate as single entities; at intermediate matrix densities (1.2, 1.9 mg/ml), cells undergo sprouting morphogenesis to form long, stable sprouts; while at high matrix densities (2.7 mg/ml), cells cluster and are unable to elongate and penetrate into the ECM to form viable sprouts. It can be stated that within intermediate gel densities, a balance between the rate of proliferation and migration of endothelial cells results in the formation of stable and long sprouts. This balance can determine the stability, dynamics, and morphology of sprout morphogenesis within collagen matrices of varying densities. Within stiffer matrices, higher ratio of proliferating cells to migrating cells was observed to cause the formation of thicker sprouts. As predicted computationally, our experiments also showed that a speciﬁc range of matrix density (0.7 and 1.2 mg/ml) was conductive to endothelial sprout branching. The dynamics of sprout elongation and the number of cells per sprout unit length were also studied. The results of this study can be very helpful for developing new strategies for enhancing or suppressing angiogenesis such as developing tunable biomaterials with speciﬁed biomechanical and biochemical properties. 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