Med Biol Eng Comput DOI 10.1007/s11517-015-1277-8
ORIGINAL ARTICLE
Structural biology response of a collagen hydrogel synthetic extracellular matrix with embedded human fibroblast: computational and experimental analysis Sara Manzano1,2,3 · Raquel Moreno‑Loshuertos1,2 · Manuel Doblaré1,2,3 · Ignacio Ochoa1,2,3 · Mohamed Hamdy Doweidar1,2,3
Received: 3 December 2013 / Accepted: 16 March 2015 © International Federation for Medical and Biological Engineering 2015
Abstract Adherent cells exert contractile forces which play an important role in the spatial organization of the extracellular matrix (ECM). Due to these forces, the substrate experiments a volume reduction leading to a characteristic shape. ECM contraction is a key process in many biological processes such as embryogenesis, morphogenesis and wound healing. However, little is known about the specific parameters that control this process. With this aim, we present a 3D computational model able to predict the contraction process of a hydrogel matrix due to cell–substrate mechanical interaction. It considers cell-generated forces, substrate deformation, ECM density, cellular migration and proliferation. The model also predicts the cellular spatial distribution and concentration needed to reproduce the contraction process and confirms the minimum value of cellular concentration necessary to initiate the process observed experimentally. The obtained continuum formulation has been implemented in a finite element framework. In parallel, in vitro experiments have been performed to
Electronic supplementary material The online version of this article (doi:10.1007/s11517-015-1277-8) contains supplementary material, which is available to authorized users. * Mohamed Hamdy Doweidar [email protected] 1
Group of Structural Mechanics and Materials Modelling (GEMM), Aragón Institute of Engineering Research (13A), University of Zaragoza, Zaragoza, Spain
2
Mechanical Engineering Department, School of Engineering and Architecture (EINA), University of Zaragoza, C/María de Luna s/n, Edificio Betancourt, 50018 Zaragoza, Spain
3
Biomedical Research Networking Center in Bioengineering, Biomaterials and Nanomedicine (CIBER-BBN), Zaragoza, Spain
obtain the main model parameters and to validate it. The results demonstrate that cellular forces, migration and proliferation are acting simultaneously to display the ECM contraction. Keywords Mechano-biology · Synthetic matrix · Hydrogel contraction · FE computational model · Fibroblast · Mechano-sensing · Cell–substrate
1 Introduction This study presents a novel 3D computational model capable of simulating the behavior of hydrogels with embedded adherent cells based on the theory presented by Oster et al. [34, 36]. To our knowledge, this is the first 3D mechano-sensing model that considers migration, proliferation and cytoskeleton exerted forces as an interlinked process, through the mechanical interaction between cells and extracellular matrix (ECM). In particular, this model predicts the cellular distribution and concentration needed to reproduce our own hydrogel deformation observed experimentally together with those reported in the literature [38]. In addition, it is capable of quantifying essential parameters such as minimum number of cells to start the process and provides insight into the influence of hydrogel stiffness and viscosity in the final shape and volume contraction. In the early 1970s, it was discovered that human fibroblasts cultured in or on hydrated collagen gels produce gel contraction [9]. This contraction due to embedded cells has been suggested as a good in vitro model to understand many biological processes such as angiogenesis [31], wound healing [49], pathological fibrosis [45], metastasis of tumors [19], morphogenesis [4] and development of artificial skin [42] in which ECM contraction is observed.
13
Pioneer computational models have been generated based on the role of three phenomena: migration [25, 29, 30], proliferation [8] and actin–myosin traction mechanism [24] that may drive to ECM deformation and formation of new structures. Cell migration, occurs not only during embryological development, but also at maturity in processes such as wound healing (cells migrate into the wound to renew the tissue) or in case of infections (cells of the immune system transmigrate from the vascular system across the vessel wall toward the infected tissues). It is an essential feature in cancer metastasis [50] and in angiogenesis [37]. Importantly, it is well known that cells sense biochemical cues such as gradients of chemotactic agents and/or the physical properties of the ECM to determine their migration behavior [15, 28]. Cell proliferation is another relevant aspect in most biological processes (wound healing, body growth, morphogenesis, tissue regeneration, etc.) [41, 44, 46, 47]. Quintana et al. [38] and Bell et al. [4] described accurately the morphological changes suffered by a hydrogel after 24 days of culture seeded with embryonic fibroblasts. They observed that during the morphogenetic event, the cell density increases not only because of matrix contraction, but also because of cell proliferation. On the other hand, cell cytoskeleton contains a network of fibers that provides support and helps to maintain its shape. Moreover, it is a dynamic system that interacts with other cellular components to generate a traction mechanism [35]. The cytoskeleton is mainly composed of three main elements: microtubules, intermediate filaments and microfilaments or actin filaments. This last component together with certain proteins, such as myosin II, provides molecular basis for contraction process of adherent cells. The generated forces due to this contraction process are transmitted to the ECM through inter-membrane proteins, socalled integrins, whose main function is to anchor the cell to its substrate [2, 5]. Besides to matrix degradation and/or production by cells, the transmitted forces produce physical rearrangement of the hydrogel collagen fibers, which explains the morphological changes that are observed experimentally. Once the deformation is produced, it is believed that cells readapt to their new surrounding environment and generate a new response [13, 26]. Despite the existence of previous computational models aiming to simulate the cell-induced ECM contraction process [8, 11, 24, 25, 29], to date, no reports exist that combine the effect of the three above-mentioned parameters (proliferation, migration and cytoskeleton exerted forces) and its cross-linking in a 3D model. Moreover, many questions regarding this contraction process remain unsolved. What are the main phenomena involved? Do they interact to conform the final
13
Med Biol Eng Comput
ECM deformation? What is the influence of the substrate mechanical properties? To gain insight into these questions, we have generated the present 3D mechano-sensing model capable of predicting cell-based hydrogel deformation and quantifying cellular distribution and concentration at each stage of hydrogel contraction process. Our own observations experimentally validated these predictions, together with those reported in the literature. Studies involving ECM remodeling such as morphogenesis or wound healing mechanisms will benefit from this model for basic research or in silico predictions of the potential effect of phenomena controlling these processes. Also it can be helpful for clinical applications, from the use of the maximal hydrogel deformation as a mechanical marker to indicate the stage of healing in wounds to measure the cellular exerted forces in different morphogenetic pathologies.
2 Materials and methods Understanding biological processes involved in hydrogel deformation as well as measurement of the main parameters is essential for the computational model design. Under these circumstances, we carried out our own experimental approach to qualitatively describing the hydrogel shape and quantitatively determining the input parameters that are needed to the computational model (see Table 1). Moreover, with this test, we measured the diameter of the hydrogel along the process and compared these data with computational results, which enabled us to validate the model. 2.1 Experimental data The experimental test newly designed basically consisted of culturing human fibroblasts within a collagen hydrogel that is freely suspended in phosphate-buffered saline 1×. The mechanical parameters, diameter and morphological changes due to cell contraction forces were monitored in 12 hydrogels (n = 12) during 21 days which are enough to reach its minimum size. The experiment duration was selected according to observation of Quintana et al. [38] and corroborated by our own experiments. After 21 days of cells in culture, the hydrogel exhibits a stable shape with no reduction or expansion. 2.1.1 Hydrogel samples preparation Collagen hydrogels were prepared at a concentration of 3 mg/ml using collagen type I from rat tail (GIBCO) and following the manufacturer’s instructions with small modifications. Briefly, collagen was mixed with fibroblasts
Med Biol Eng Comput Table 1 Computational parameters coincident with the presented experiments of hydrogel contraction process Parameter
Description
Value
Units
Reference
Initial conditions
co
Initial fibroblast concentration in the hydrogel
Hydrogel dimensions
∅
Initial hydrogel diameter Initial hydrogel thickness
cells/mm3 cells/mm3 cells/mm3 mm mm
[38] [38] [38] Measured Measured
Substrate parameters
e ρ
1.0 × 103 2.0 × 103 4.0 × 103 6.0 1.5
Collagen concentration in the ECM Young’s modulus of the collagen hydrogel
0.003 24.5 × 10−6
mg/mm3 MPa
Measured Measured
h
Poisson’s ratio Haptotaxis parameter (migration)
0.2 0.25
cells/cell
Measured [25]
r
Proliferation parameter
0.25
–
[25]
E ν Migration and proliferation parameters
culture medium (see below) in the correct proportion to have the desired collagen concentration. Once mixed, the diluted collagen was neutralized by adding 0.1 N NaOH (0.025 times the collagen volume), placed into a correct mold and incubated at 37 °C in a 5 % CO2 incubator until a firm gel was formed. Dimensions of gels were 9 mm of diameter and 2 mm of thickness (120 µl of final volume). 2.1.2 Human fibroblasts culture Human fibroblasts (IMR90) [33] were cultured in Hepesmodified MEM (SIGMA) supplemented with 10 % (v/v) fetal bovine serum (FBS, SIGMA), 100 µM nonessential amino acids (Lonza), 2 mM l-glutamine (SIGMA), 1 mM sodium pyruvate (Lonza), 100 U/ml penicillin and 0.1 mg/ ml streptomycin (SIGMA). For 3D culture in collagen hydrogels, cells were trypsinized and suspended in culture medium at a final concentration of 4 × 106 cells/ml, whereas collagen was prepared at a concentration of 0.4 % (w/v). Immediately after neutralization, three volumes of collagen were mixed with one volume of cell suspension to obtain a final cell density of 106 cells/ml in 0.3 % collagen and poured inside handmade molds of 9 mm of diameter which were placed in six-well culture plates. This suspension was incubated at 37 °C in a 5 % CO2 incubator until gels were formed. Briefly after gelling, culture medium was poured inside the plate wells, and molds were removed to let gels float in the medium. Cell culture was performed at 37 °C with 5 % CO2, and medium was replaced every 2–3 days.
2.1.4 Mechanical characterization of hydrogel samples To determine the mechanical properties of the hydrogels (Young’s modulus, E and Poisson’s coefficient, ν), oscillatory rheological experiments were performed on a Rheometer System Gemini HR nano (Malvern Instruments, Malvern, UK) using stainless steel cone/plate geometry (4˚ cone angle, 40 mm cone diameter) with the gap set at 150 µm. The torque range of the rheometer was 10 nNm, and the torque resolution was better than 1 nNm. Temperature control was achieved by a Peltier temperature controller, with an accuracy of ±0.1 °C. Dynamic stress sweeps were conducted prior to the frequency sweeps to ensure operation within the linear viscoelastic region. Each sample was equilibrated for 2 min before measurement. The shear modulus (G) was recorded as function of frequency to obtain Young’s modulus (see Eq. 14). Ten measurements were performed immediately after gel formation with embedded cells (time 0) in each of the 12 total hydrogels. Poisson’s coefficient was directly related to hydrogel collagen concentration [40]. 2.1.5 Confocal microscopy The contraction process of the 12 hydrogels was followed using a Nikon eclipse Ti confocal microscope [1]. Digital images were recorded every 1–2 days until 21 days; diameters were measured using the software NIS-Elements D 3.0 from Nikon. Samples were observed at 10× magnification.
2.1.3 Nutrients deprivation experiment
2.2 Computational modeling
To verify the viscoelastoplastic behavior of the hydrogel, cells were killed in a gel of 6 mm diameter and 2 mm thickness. For this purpose, nutrient deprivation was used and the average hydrogel diameter was monitoring in days 11 and 13.
For the proposed model, the hydrogel is considered as a viscoelastoplastic material deformed by cell exerted forces. Cells are described by its population density that is distributed and changes along time. Besides, cell proliferation and migration are considered simultaneously. Following
13
Med Biol Eng Comput
the work of Quintana el al. [38], the computational model considers these three phenomena as the main responsible for the hydrogel contraction. As a result, inside the hydrogel, a heterogeneous distribution of cells concentration is obtained. So, the resulting cellular concentration gradient will translate to an irregular spatial distribution of forces. This process plays a key role in the spatial reorganization of the ECM. 2.2.1 Mechano‑sensing model As in other mechano-biological models [1, 3, 22], two main species have been considered, cells and ECM, which are characterized by cellular concentration (c) and ECM density (ρ), respectively. Note that ECM density is referred to the real density of the collagen matrix before adding cells. A continuum approach has been adopted, so we consider a sufficient number of cells to average their individual effect and only macroscopic results have been monitored. Therefore, the mathematical model has been established by describing the dynamics of those two variables that is represented by the fundamental conservation law for the concentration of each species. 2.2.1.1 Law of conservation of species Q(x, t) Local changes in the concentration of each species (ECM or cell density) can be expressed as [10]:
∂Q + ∇ · J Q = fQ , ∂t
(1)
2.2.1.2 ECM density ρ(x, t) During development, cells generate and degrade collagen ECM [27]. In in vivo experiments as well as in healthy tissue, there is a continuous process of collagen synthesis and collagen degradation, and both are precisely balanced to maintain normal tissue or biomaterial architecture [6, 12, 16]. Thus, in the present model, it is considered that the amount of collagen produced by cells is balanced with those degraded, so the net rate of ECM production is considered null [13], fρ = 0. Thus, local changes in ECM are only influenced by cell deformation of the gel that is by passive convection. Therefore, Jρ can be defined by
∂u , ∂t
(2)
where u denotes the displacement vector of each point of the ECM.
13
Jdiff = −D∇c.
(3)
Here, Jdiff denotes the flux of cells passing through one cm3 of hydrogel and D is the isotropic diffusion coefficient that constitutes a measure of cells motility. Aggregation phenomenon is related to traction forces of the cells, including passive convection and haptotaxis. Aggregation phenomenon is opposite to the previous dispersive cell movement. In this case, there exist two effects that tend to organize the cell population, passive convection (Jconv) and haptotaxis (Jhapt). For passive convection, cells can be passively dragged along by the contraction of the substrate which is deformed by the contractions of distant cells. The expression for convective cell motion is given by
Jconv = c
where JQ denotes the flux (rate of outgoing or incoming flux per unit volume) of each species Q (ECM density (ρ) or cell concentration (c)), fQ is the rate of net production of Q and t is the time.
Jρ = ρ
2.2.1.3 Cellular concentration c(x, t) The cellular flux, Jc (number of outgoing or incoming cells per unit volume), related to cell migration, takes place as a result of two important phenomena: diffusion and aggregation. Diffusion phenomenon is a random, non-directed cell movement, which tends to create a homogeneous spatial distribution of cells, analogous to diffusion of particles in a gas–liquid. So, the same mathematical approach can be used to describe both phenomena. Hence, random motions are conventionally modeled as a diffusion process that can be governed by the law of Fick. It states that this flux is proportional to the gradient of cell concentration (∇c). So,
∂u , ∂t
(4)
where ∂u ∂t is the average cell velocity, that is identified with the local velocity of the matrix in which cells are anchored (no relative movement is allowed since only passive convection is considered in this term). In case of haptotactic phenomenon, motile cells move from less adhesive to more adhesive regions of the substrate [7, 17]. This movement results from the competition between opposite sides of individual motile cells. Each side of the cell has adhesions with the substrate and engages in a tug-of-war with net displacement occurring along the direction of the side with strongest pull and firmest attachments to the substrate. This effect can be modeled by defining the cell flux as, Jhapt = c mean cell velocity . (5) To express the net cell velocity within the substrate adhesiveness gradient, we assume that the mean velocity of migration is proportional to this gradient. That is, Jhapt ∼ c gradient of adhesiveness . (6) In addition, the local density of adhesive areas is proportional to the density of the matrix material [37]. Therefore, we can write the above equation as,
Jhapt = hc∇ρ,
(7)
Med Biol Eng Comput
where the parameter h is the haptotactic coefficient that measures the tendency of cells to migrate following the cell density gradient. So, the total cellular flux can be defined as
Jc = −D∇c + c
∂u + hc∇ρ. ∂t
(8)
For cell proliferation modeling fc, we can express the cell population rate (without migration) as
dN = birth − death. dt
(9)
This simple equation represents a conservation equation of the cell population. The birth and death terms are considered proportional to the total number of cells, N. That is,
dN = (b − d)N = rN, dt
(10)
where b and d are positive constants. Thus, if b > d the population grows, while if b
ORIGINAL ARTICLE
Structural biology response of a collagen hydrogel synthetic extracellular matrix with embedded human fibroblast: computational and experimental analysis Sara Manzano1,2,3 · Raquel Moreno‑Loshuertos1,2 · Manuel Doblaré1,2,3 · Ignacio Ochoa1,2,3 · Mohamed Hamdy Doweidar1,2,3
Received: 3 December 2013 / Accepted: 16 March 2015 © International Federation for Medical and Biological Engineering 2015
Abstract Adherent cells exert contractile forces which play an important role in the spatial organization of the extracellular matrix (ECM). Due to these forces, the substrate experiments a volume reduction leading to a characteristic shape. ECM contraction is a key process in many biological processes such as embryogenesis, morphogenesis and wound healing. However, little is known about the specific parameters that control this process. With this aim, we present a 3D computational model able to predict the contraction process of a hydrogel matrix due to cell–substrate mechanical interaction. It considers cell-generated forces, substrate deformation, ECM density, cellular migration and proliferation. The model also predicts the cellular spatial distribution and concentration needed to reproduce the contraction process and confirms the minimum value of cellular concentration necessary to initiate the process observed experimentally. The obtained continuum formulation has been implemented in a finite element framework. In parallel, in vitro experiments have been performed to
Electronic supplementary material The online version of this article (doi:10.1007/s11517-015-1277-8) contains supplementary material, which is available to authorized users. * Mohamed Hamdy Doweidar [email protected] 1
Group of Structural Mechanics and Materials Modelling (GEMM), Aragón Institute of Engineering Research (13A), University of Zaragoza, Zaragoza, Spain
2
Mechanical Engineering Department, School of Engineering and Architecture (EINA), University of Zaragoza, C/María de Luna s/n, Edificio Betancourt, 50018 Zaragoza, Spain
3
Biomedical Research Networking Center in Bioengineering, Biomaterials and Nanomedicine (CIBER-BBN), Zaragoza, Spain
obtain the main model parameters and to validate it. The results demonstrate that cellular forces, migration and proliferation are acting simultaneously to display the ECM contraction. Keywords Mechano-biology · Synthetic matrix · Hydrogel contraction · FE computational model · Fibroblast · Mechano-sensing · Cell–substrate
1 Introduction This study presents a novel 3D computational model capable of simulating the behavior of hydrogels with embedded adherent cells based on the theory presented by Oster et al. [34, 36]. To our knowledge, this is the first 3D mechano-sensing model that considers migration, proliferation and cytoskeleton exerted forces as an interlinked process, through the mechanical interaction between cells and extracellular matrix (ECM). In particular, this model predicts the cellular distribution and concentration needed to reproduce our own hydrogel deformation observed experimentally together with those reported in the literature [38]. In addition, it is capable of quantifying essential parameters such as minimum number of cells to start the process and provides insight into the influence of hydrogel stiffness and viscosity in the final shape and volume contraction. In the early 1970s, it was discovered that human fibroblasts cultured in or on hydrated collagen gels produce gel contraction [9]. This contraction due to embedded cells has been suggested as a good in vitro model to understand many biological processes such as angiogenesis [31], wound healing [49], pathological fibrosis [45], metastasis of tumors [19], morphogenesis [4] and development of artificial skin [42] in which ECM contraction is observed.
13
Pioneer computational models have been generated based on the role of three phenomena: migration [25, 29, 30], proliferation [8] and actin–myosin traction mechanism [24] that may drive to ECM deformation and formation of new structures. Cell migration, occurs not only during embryological development, but also at maturity in processes such as wound healing (cells migrate into the wound to renew the tissue) or in case of infections (cells of the immune system transmigrate from the vascular system across the vessel wall toward the infected tissues). It is an essential feature in cancer metastasis [50] and in angiogenesis [37]. Importantly, it is well known that cells sense biochemical cues such as gradients of chemotactic agents and/or the physical properties of the ECM to determine their migration behavior [15, 28]. Cell proliferation is another relevant aspect in most biological processes (wound healing, body growth, morphogenesis, tissue regeneration, etc.) [41, 44, 46, 47]. Quintana et al. [38] and Bell et al. [4] described accurately the morphological changes suffered by a hydrogel after 24 days of culture seeded with embryonic fibroblasts. They observed that during the morphogenetic event, the cell density increases not only because of matrix contraction, but also because of cell proliferation. On the other hand, cell cytoskeleton contains a network of fibers that provides support and helps to maintain its shape. Moreover, it is a dynamic system that interacts with other cellular components to generate a traction mechanism [35]. The cytoskeleton is mainly composed of three main elements: microtubules, intermediate filaments and microfilaments or actin filaments. This last component together with certain proteins, such as myosin II, provides molecular basis for contraction process of adherent cells. The generated forces due to this contraction process are transmitted to the ECM through inter-membrane proteins, socalled integrins, whose main function is to anchor the cell to its substrate [2, 5]. Besides to matrix degradation and/or production by cells, the transmitted forces produce physical rearrangement of the hydrogel collagen fibers, which explains the morphological changes that are observed experimentally. Once the deformation is produced, it is believed that cells readapt to their new surrounding environment and generate a new response [13, 26]. Despite the existence of previous computational models aiming to simulate the cell-induced ECM contraction process [8, 11, 24, 25, 29], to date, no reports exist that combine the effect of the three above-mentioned parameters (proliferation, migration and cytoskeleton exerted forces) and its cross-linking in a 3D model. Moreover, many questions regarding this contraction process remain unsolved. What are the main phenomena involved? Do they interact to conform the final
13
Med Biol Eng Comput
ECM deformation? What is the influence of the substrate mechanical properties? To gain insight into these questions, we have generated the present 3D mechano-sensing model capable of predicting cell-based hydrogel deformation and quantifying cellular distribution and concentration at each stage of hydrogel contraction process. Our own observations experimentally validated these predictions, together with those reported in the literature. Studies involving ECM remodeling such as morphogenesis or wound healing mechanisms will benefit from this model for basic research or in silico predictions of the potential effect of phenomena controlling these processes. Also it can be helpful for clinical applications, from the use of the maximal hydrogel deformation as a mechanical marker to indicate the stage of healing in wounds to measure the cellular exerted forces in different morphogenetic pathologies.
2 Materials and methods Understanding biological processes involved in hydrogel deformation as well as measurement of the main parameters is essential for the computational model design. Under these circumstances, we carried out our own experimental approach to qualitatively describing the hydrogel shape and quantitatively determining the input parameters that are needed to the computational model (see Table 1). Moreover, with this test, we measured the diameter of the hydrogel along the process and compared these data with computational results, which enabled us to validate the model. 2.1 Experimental data The experimental test newly designed basically consisted of culturing human fibroblasts within a collagen hydrogel that is freely suspended in phosphate-buffered saline 1×. The mechanical parameters, diameter and morphological changes due to cell contraction forces were monitored in 12 hydrogels (n = 12) during 21 days which are enough to reach its minimum size. The experiment duration was selected according to observation of Quintana et al. [38] and corroborated by our own experiments. After 21 days of cells in culture, the hydrogel exhibits a stable shape with no reduction or expansion. 2.1.1 Hydrogel samples preparation Collagen hydrogels were prepared at a concentration of 3 mg/ml using collagen type I from rat tail (GIBCO) and following the manufacturer’s instructions with small modifications. Briefly, collagen was mixed with fibroblasts
Med Biol Eng Comput Table 1 Computational parameters coincident with the presented experiments of hydrogel contraction process Parameter
Description
Value
Units
Reference
Initial conditions
co
Initial fibroblast concentration in the hydrogel
Hydrogel dimensions
∅
Initial hydrogel diameter Initial hydrogel thickness
cells/mm3 cells/mm3 cells/mm3 mm mm
[38] [38] [38] Measured Measured
Substrate parameters
e ρ
1.0 × 103 2.0 × 103 4.0 × 103 6.0 1.5
Collagen concentration in the ECM Young’s modulus of the collagen hydrogel
0.003 24.5 × 10−6
mg/mm3 MPa
Measured Measured
h
Poisson’s ratio Haptotaxis parameter (migration)
0.2 0.25
cells/cell
Measured [25]
r
Proliferation parameter
0.25
–
[25]
E ν Migration and proliferation parameters
culture medium (see below) in the correct proportion to have the desired collagen concentration. Once mixed, the diluted collagen was neutralized by adding 0.1 N NaOH (0.025 times the collagen volume), placed into a correct mold and incubated at 37 °C in a 5 % CO2 incubator until a firm gel was formed. Dimensions of gels were 9 mm of diameter and 2 mm of thickness (120 µl of final volume). 2.1.2 Human fibroblasts culture Human fibroblasts (IMR90) [33] were cultured in Hepesmodified MEM (SIGMA) supplemented with 10 % (v/v) fetal bovine serum (FBS, SIGMA), 100 µM nonessential amino acids (Lonza), 2 mM l-glutamine (SIGMA), 1 mM sodium pyruvate (Lonza), 100 U/ml penicillin and 0.1 mg/ ml streptomycin (SIGMA). For 3D culture in collagen hydrogels, cells were trypsinized and suspended in culture medium at a final concentration of 4 × 106 cells/ml, whereas collagen was prepared at a concentration of 0.4 % (w/v). Immediately after neutralization, three volumes of collagen were mixed with one volume of cell suspension to obtain a final cell density of 106 cells/ml in 0.3 % collagen and poured inside handmade molds of 9 mm of diameter which were placed in six-well culture plates. This suspension was incubated at 37 °C in a 5 % CO2 incubator until gels were formed. Briefly after gelling, culture medium was poured inside the plate wells, and molds were removed to let gels float in the medium. Cell culture was performed at 37 °C with 5 % CO2, and medium was replaced every 2–3 days.
2.1.4 Mechanical characterization of hydrogel samples To determine the mechanical properties of the hydrogels (Young’s modulus, E and Poisson’s coefficient, ν), oscillatory rheological experiments were performed on a Rheometer System Gemini HR nano (Malvern Instruments, Malvern, UK) using stainless steel cone/plate geometry (4˚ cone angle, 40 mm cone diameter) with the gap set at 150 µm. The torque range of the rheometer was 10 nNm, and the torque resolution was better than 1 nNm. Temperature control was achieved by a Peltier temperature controller, with an accuracy of ±0.1 °C. Dynamic stress sweeps were conducted prior to the frequency sweeps to ensure operation within the linear viscoelastic region. Each sample was equilibrated for 2 min before measurement. The shear modulus (G) was recorded as function of frequency to obtain Young’s modulus (see Eq. 14). Ten measurements were performed immediately after gel formation with embedded cells (time 0) in each of the 12 total hydrogels. Poisson’s coefficient was directly related to hydrogel collagen concentration [40]. 2.1.5 Confocal microscopy The contraction process of the 12 hydrogels was followed using a Nikon eclipse Ti confocal microscope [1]. Digital images were recorded every 1–2 days until 21 days; diameters were measured using the software NIS-Elements D 3.0 from Nikon. Samples were observed at 10× magnification.
2.1.3 Nutrients deprivation experiment
2.2 Computational modeling
To verify the viscoelastoplastic behavior of the hydrogel, cells were killed in a gel of 6 mm diameter and 2 mm thickness. For this purpose, nutrient deprivation was used and the average hydrogel diameter was monitoring in days 11 and 13.
For the proposed model, the hydrogel is considered as a viscoelastoplastic material deformed by cell exerted forces. Cells are described by its population density that is distributed and changes along time. Besides, cell proliferation and migration are considered simultaneously. Following
13
Med Biol Eng Comput
the work of Quintana el al. [38], the computational model considers these three phenomena as the main responsible for the hydrogel contraction. As a result, inside the hydrogel, a heterogeneous distribution of cells concentration is obtained. So, the resulting cellular concentration gradient will translate to an irregular spatial distribution of forces. This process plays a key role in the spatial reorganization of the ECM. 2.2.1 Mechano‑sensing model As in other mechano-biological models [1, 3, 22], two main species have been considered, cells and ECM, which are characterized by cellular concentration (c) and ECM density (ρ), respectively. Note that ECM density is referred to the real density of the collagen matrix before adding cells. A continuum approach has been adopted, so we consider a sufficient number of cells to average their individual effect and only macroscopic results have been monitored. Therefore, the mathematical model has been established by describing the dynamics of those two variables that is represented by the fundamental conservation law for the concentration of each species. 2.2.1.1 Law of conservation of species Q(x, t) Local changes in the concentration of each species (ECM or cell density) can be expressed as [10]:
∂Q + ∇ · J Q = fQ , ∂t
(1)
2.2.1.2 ECM density ρ(x, t) During development, cells generate and degrade collagen ECM [27]. In in vivo experiments as well as in healthy tissue, there is a continuous process of collagen synthesis and collagen degradation, and both are precisely balanced to maintain normal tissue or biomaterial architecture [6, 12, 16]. Thus, in the present model, it is considered that the amount of collagen produced by cells is balanced with those degraded, so the net rate of ECM production is considered null [13], fρ = 0. Thus, local changes in ECM are only influenced by cell deformation of the gel that is by passive convection. Therefore, Jρ can be defined by
∂u , ∂t
(2)
where u denotes the displacement vector of each point of the ECM.
13
Jdiff = −D∇c.
(3)
Here, Jdiff denotes the flux of cells passing through one cm3 of hydrogel and D is the isotropic diffusion coefficient that constitutes a measure of cells motility. Aggregation phenomenon is related to traction forces of the cells, including passive convection and haptotaxis. Aggregation phenomenon is opposite to the previous dispersive cell movement. In this case, there exist two effects that tend to organize the cell population, passive convection (Jconv) and haptotaxis (Jhapt). For passive convection, cells can be passively dragged along by the contraction of the substrate which is deformed by the contractions of distant cells. The expression for convective cell motion is given by
Jconv = c
where JQ denotes the flux (rate of outgoing or incoming flux per unit volume) of each species Q (ECM density (ρ) or cell concentration (c)), fQ is the rate of net production of Q and t is the time.
Jρ = ρ
2.2.1.3 Cellular concentration c(x, t) The cellular flux, Jc (number of outgoing or incoming cells per unit volume), related to cell migration, takes place as a result of two important phenomena: diffusion and aggregation. Diffusion phenomenon is a random, non-directed cell movement, which tends to create a homogeneous spatial distribution of cells, analogous to diffusion of particles in a gas–liquid. So, the same mathematical approach can be used to describe both phenomena. Hence, random motions are conventionally modeled as a diffusion process that can be governed by the law of Fick. It states that this flux is proportional to the gradient of cell concentration (∇c). So,
∂u , ∂t
(4)
where ∂u ∂t is the average cell velocity, that is identified with the local velocity of the matrix in which cells are anchored (no relative movement is allowed since only passive convection is considered in this term). In case of haptotactic phenomenon, motile cells move from less adhesive to more adhesive regions of the substrate [7, 17]. This movement results from the competition between opposite sides of individual motile cells. Each side of the cell has adhesions with the substrate and engages in a tug-of-war with net displacement occurring along the direction of the side with strongest pull and firmest attachments to the substrate. This effect can be modeled by defining the cell flux as, Jhapt = c mean cell velocity . (5) To express the net cell velocity within the substrate adhesiveness gradient, we assume that the mean velocity of migration is proportional to this gradient. That is, Jhapt ∼ c gradient of adhesiveness . (6) In addition, the local density of adhesive areas is proportional to the density of the matrix material [37]. Therefore, we can write the above equation as,
Jhapt = hc∇ρ,
(7)
Med Biol Eng Comput
where the parameter h is the haptotactic coefficient that measures the tendency of cells to migrate following the cell density gradient. So, the total cellular flux can be defined as
Jc = −D∇c + c
∂u + hc∇ρ. ∂t
(8)
For cell proliferation modeling fc, we can express the cell population rate (without migration) as
dN = birth − death. dt
(9)
This simple equation represents a conservation equation of the cell population. The birth and death terms are considered proportional to the total number of cells, N. That is,
dN = (b − d)N = rN, dt
(10)
where b and d are positive constants. Thus, if b > d the population grows, while if b
