The role of cell lysis and matrix deposition in tumor growth modeling
 R Santagiuliana^{1}Email authorView ORCID ID profile,
 C Stigliano^{2},
 P Mascheroni^{1},
 M Ferrari^{3, 4},
 P Decuzzi^{2, 3, 5} and
 B A Schrefler^{1, 3}
https://doi.org/10.1186/s403230150040x
© Santagiuliana et al. 2015
Received: 14 May 2015
Accepted: 22 July 2015
Published: 8 August 2015
Abstract
Background
The multiphase model for tumor growth, proposed by the authors in previous works, is here enhanced. The original model includes a solid phase, the extracellular matrix (ECM) and three fluid phases: living and necrotic tumor cells (TCs), host cells (HCs), and the interstitial fluid (IF).
Methods
We introduce the mathematical model for deposition (remodeling) of ECM during the TCs growth, and lysis. Differently from the previous version of the model we take into account that TCs growing in vitro depose their own ECM not present at the beginning. The lysis retransforms the necrotic cells into IF. The updated mathematical formulation is discretized by means of the finite element method and implemented in a general purpose code.
Results
First we reproduce new experimental data of multicellular tumor spheroid (MTS) growth in vitro. The free boundary conditions used in this simulation together with necrosis and lysis allow following the tumor growth curve up to the final steadystate. The second example, by comparing results of tumor growth in an ECMfree medium and in an ECM remodeling medium highlights how ECM deposition affects tumor growth. In an initially ECMfree medium the tumor is unobstructed and can proliferate more rapidly both without ECM and in case of ECM deposition. The third example shows the effect of lysis: it redirects some tumor cells toward the necrotic core of the MTS and produces outflow of the IF from the tumor mass.
Conclusions
The introduction of lysis and ECM deposition allows capturing different aspects of the avascular tumor growth not yet comprised in the original model: the MTS growth seems to be influenced by ECM deposition and the lysis seems to be a cause of an outflow of the IF from the tumor mass.
Keywords
Background
Much research went in the last two decades into the problem of tumor growth and drug delivery [1]. An extensive literature review [2] has shown that the most recent models for tumor growth are multicomponent models with or without diffuse interfaces among the constituents [3–10]. They all consider a malignant mass (tumor cells: TCs), host cells (HCs) and the interstitial fluid (IF) as homogeneous, viscous fluids and employ reaction–diffusion–advection equations for predicting the distribution and transport of nutrients. In general, these models contain limitations on the evolution of the volume fractions and on the velocities of the different cell populations and the IF is uncoupled from the rest. All models, with the exception of [10], disregard the contribution of the extracellular matrix (ECM). However experiments [11] clearly demonstrate that the tumor ECM is a source of resistance to the delivery of bloodborne drugs to cancer cells, especially for macromolecules and monoparticles [13]. The penetration of macromolecules correlates with ECM elasticity (deformability) and hydraulic conductivity (flow) [11, 13]. Further, TCs tend to deposit new ECM: as the mass of tumor cells increases; the ECM within the malignant tissue undergoes extensive rearrangements with increased deposition of collagen fibers, making the resulting tissue thicker and more difficult to penetrate [12, 13] as compared to host tissue.
We have developed a very general growth model for avascular tumors which is a multiphase flow model in a porous solid (ECM) [14–16]. The system is modeled at macroscopic scale making use of TCAT (thermodynamic constrained averaging theory). This model comprises TCs, which partition into living cells and necrotic cells, healthy cells (HCs), extracellular matrix (ECM) and interstitial fluid (IF). The ECM is a porous solid, while all other phases are treated as fluids. The IF transports chemical species such as nutrients, Tumor Angiogenic Factor (TFA), cytokines, etc. The interaction between the different compartments is fully taken into account through the stress tensor which involves the phase pressures, through the volumetric deformation appearing in the mass balance equations and through several constitutive equations. The flow aspects have been investigated in [17, 18]. The model has been validated with respect to in vitro experiments from literature [19–21] and own experiments [18].
The role of the deformable ECM has been addressed in [22]. The adopted constitutive model is of the Greenelastic and elastoviscoplastic type within a large strain approach. Truesdell objective stress measure is adopted together with the deformation rate tensor. The importance of including a deformable ECM in the model has been highlighted in an example of growth of a melanoma where different stiffness of the ECM leads to a different shape of the tumor. [22, 23].
We extend here the model to allow for observed ECM deposition and ECM remodeling which affect the rate of growth of tumor spheroids. Further we introduce lysis, i.e. the retransformation of necrotic cells into IF.
The main objective is to explain some aspects of MTS growth, such as the possible causes of a more or less strong growth, outflow of the IF from the MTS, the final steady state in the growth curve, by means of the introduction of lysis and ECM deposition in the model. Also the effect of a moving boundary at the border of the MTS is investigated. It will be shown among other aspects that the combination of necrosis and lysis, together with a deformable ECM allows for reproducing outflow of IF from a tumor spheroid, as observed experimentally in [13].
The outline of the paper is the following: the updated mathematical formulation of the model allowing for ECM deposition and lysis are described in the “Methods” Section together with the necessary constitutive equations for fluids and the ECM. Numerical reproduction of experimental results of a multicellular tumor spheroid (MTS) growing in vitro and two examples of MTS growth are presented in Section “Results and discussion”: the first of the two examples refers to the comparison between the growth of a MTS in an ECM deposited by TCs, in an ECM free culture medium and in a remodeling ECM scaffold; the second one shows the influence of lysis on the growth of a MTS. Conclusions and perspectives of the presented multiphase model complete the paper.
Methods
The multiphase tumor growth model
The updates introduced into our tumor growth model [14, 15, 17, 22] are now summarized. It is recalled that tumors are modeled at the macroscopic scale, being the domain of interest too large and the phase distributions too complex for modeling at the microscale. The TCAT approach [24–26] is used to transform known microscale relations to mathematically and physically consistent macroscale relations. These macroscale relations are adequate for describing the tumor behavior while filtering out high frequency spatial variability. The governing equations of the model are closed by introducing constitutive relations into the macroscale equations.
The multiphase system is comprised of the following phases: (1) tumor cells (TCs), which partition into living cells (LTC) and necrotic cells (NTC); (2) healthy cells (HCs); (3) extracellular matrix (ECM); and (4) interstitial fluid (IF); see [15, 17]. The ECM is a porous solid, while all other phases are fluids. The tumor cells may become necrotic upon exposure to low nutrient concentrations or excessive mechanical pressure [27]. The IF, transporting nutrients, is a mixture of water and biomolecules, such as nutrients, oxygen and waste products. In the following mass and momentum balance equations, α denotes a generic phase, t the tumor cells (TCs), h the healthy cells (HCs), s the solid phase (ECM), and l the interstitial fluid (IF).
A short overview of the mathematical model is given next; additional details are available in [22]. We use both vectorial and indicial notation where convenient.
General governing equations
As in the previous work [22], the primary variables of the model are: differential pressures p ^{ hl } and p ^{ th }, IF pressure p ^{ l }, nutrient mass fraction \(\omega^{{\overline{nl} }}\), and displacement u ^{ s } of solid phase (ECM).
The macroscopic mass and momentum balance equations of phases and species have been derived in [15] and their transformation to take into account of the differential pressures as primary variables has been obtained in [22]. The final form of the governing equations shown below are obtained from the general forms (see [22]) by introducing some simplifications and closure relationships (e.g., a Fickian type equation for diffusion of species, a generalized Darcy’s equation for flow of the fluid phases, etc.).
New terms with respect to the previous works are present in the governing equations to introduce the two new aspects of this paper: the ECM deposition by the tumor cells and lysis. For the ECM deposition we take into account that the tumor cells consume more liquid to deposit their own matrix, hence the choice of a mass exchange from the liquid phase directly to the solid phase seems to be the easiest way:\(\mathop {\mathop M\limits_{ECM} }\limits^{l \to s}\). The lysis is the conversion of the necrotic tumor cells in liquid, hence the introduction of a mass exchange from the tumor phase to the liquid phase is required:\(\mathop {\mathop M\limits_{lysis} }\limits^{t \to l}\).
The new mass balance equation of HCs is
From this equation, after the introduction of (3) and taking into account of the differential pressures as primary variables, see [22], we have obtained Eq. (4).
Solid phase behavior
Since an ECM is present in the model its configuration is used as reference configuration whereby the velocities of the fluid are relative to the ECM which is described in a Lagrangian system. This is customary in multiphase flow models within deforming porous media [28].
A large deformation regime is assumed for the ECM (solid phase). As far as the stress and strain measures are concerned, objectivity (invariancy with respect to coordinate transformations, particularly rotations) and workconjugacy (which guarantees energy consistency, i.e. correct expressions for the secondorder energy increment) should be conserved. The satisfaction of the second requirement guarantees that of the first one, but not the opposite. Among the several objective stress rates [29, 30] we take the Truesdell stress rate.
Constitutive relationships
The constitutive relationships for the fluid phases have been extensively dealt with in [17, 18]. The constitutive equations for growth, necrosis, and nutrient consumption are given in [15, 17] and only the new relationships are listed here.
The solid phase behavior is either Greenelastic where the elastic coefficients are derived from the strain energy function or elastoviscoplastic where the stress–strain relation is defined incrementally. The absence of a potential in the second case requires the adoption of objective stress rates mentioned above. Elastoviscoplasticity is appropriate for remodeling of the ECM, see Preziosi et al. [31]. The elastoviscoplastic model has been described in [22] in detail hence only the principal points are summarized here together with the introduction of a new expression of the elastic modulus.
For viscoplastic analysis the constitutive tangent matrix D _{s} should be such that all material symmetries are preserved, in accordance with the associative character of the model. The matrix will generally depend on the current state variables and on the direction of loading.
The elastic range is defined as the set of all possible absolute values of the effective stress that are less than or equal to the frictional constant \(\text{t}_{eff,y}^{\text{s}}\), i.e. the yield limit (this value allows to define the boundary of elastic domain) [22]. Until the absolute value of the effective stress is contained in the elastic range, the rate of change of the viscoplastic strain is zero while beyond this limit it is different from zero.
In this equation \(E_{f}\) is the final value of the Young modulus, \(\varepsilon_{fin}^{s}\) is the final mass fraction of solid and \(\varepsilon_{i}^{s}\) is the mass fraction of solid below which the ECM has negligible mechanical properties. This function has been chosen with reference to the variable Young modulus in [33].
The particular elastoviscoplastic model used to describe the ECM behavior was introduced by Perzyna [34, 35]. Here it is used by introducing the von Mises yield condition as limit yield function, above which the ECM has viscoplastic behavior. The algorithm used to model this constitutive behavior is the “radial return mapping algorithm” by Simo and Hughes [35] and has been implemented in the finite elements code CAST3 M (http://www.cast3m.cea.fr) of the French Atomic Energy Commission together with our model.
An overview of the finite element implementation and solution process is given in the “Appendix”.
Results and discussion
This section shows numerical results of the model used to simulate the multicellular tumor spheroid (MTS) growing in vitro. At the beginning, a comparison with experimental data validates the model and then two cases of MTS growth are analyzed. In these examples the focus is on the ECM deposition and lysis effects.
MTS growth in vitro: comparison with experimental data
Hence the boundary conditions are updated at every step and have fixed values on the moving boundary: zero IF pressure and oxygen concentration equal to 7.0 × 10^{−6} Pa.
Initial conditions
Zone  p ^{ l } [Pa]  S ^{ h } [−]  S ^{ t } [−]  \(\omega^{{\overline{nl} }}\)[−] 

Red zone (up to 280 µm)  0.0  0.0  0.01  7.0 × 10^{−6} 
Parameters depending on cells’ type and IF
Parameter  Symbol  Value  Unit 

Density of the three fluid phases (α = h, t and l)  ρ ^{ α }  1,000  kg/m^{3} 
Dynamic viscosity of IF [22]  μ ^{ l }  1 × 10^{−3}  Pa s 
Dynamic viscosity of TC [22]  \(\mu_{0}^{t}\)  36  Pa s 
Dynamic viscosity of HC [22]  \(\mu_{0}^{h}\)  36  Pa s 
Adhesion of TC [22]  ψ ^{ t }  0  Pa/m 
Adhesion of HC [22]  ψ ^{ h }  0  Pa/m 
\(\omega_{crit}^{{\overline{nl} }}\)  6.0 × 10^{−6}  –  
\(\gamma_{growth}^{t}\)  1.3 × 10^{−2}  kg/(m^{3} s)  
\(\gamma_{necrosis}^{t}\)  1.0 × 10^{−2}  kg/(m^{3} s)  
\(\gamma_{growth}^{{\overline{nl} }}\)  5 × 10^{−4}  kg/(m^{3} s)  
\(\gamma_{0}^{{\overline{nl} }}\)  6 × 10^{−5}  kg/(m^{3} s)  
HCIF interfacial tension [22]  σ _{ hl }  72  mN/m 
TCHC interfacial tension [22]  σ _{ th }  36  mN/m 
TCIF interfacial tension  σ _{ tl }  108  mN/m 
Lysis parameter  λ  0.001  – 
Parameters related to oxygen diffusion
Parameters for ECM taken from [22]
Parameter  Symbol  Value  Unit 

Density of the solid phase  ρ ^{ s }  1 × 10^{3}  kg/m^{3} 
Poisson’s ratio of the ECM  ν  0.4  – 
Young’s modulus of the ECM  Efin  1.6 × 10^{2}  Pa 
Volume fraction of ECM (initial)  \(\varepsilon^{s}\)  0.0  – 
Coefficient a  a  590  Pa 
Intrinsic permeability  k  1.8 × 10^{−15}  m^{2} 
Yield effective stress limit  \(\text{t}_{\text{eff}}^{\text{s}} ,_{y}\)  0.5 × 10^{1}  Pa 
Viscosity  \(\eta\)  5  Pa s 
Hardening modulus  H  1.0 × 10^{2}  Pa 
The simulation captures both the first exponential phase of the growth and the second phase: the plateau where the tumor mass reaches an equilibrium. The equilibrium, in the simulation, is reached thanks to the freeboundary conditions and to the growth limiter due to pressure. Lysis has here only a minor influence, contrarily to the last example. The moving boundary with a Dirichlet condition for the nutrient concentration at the MTS surface allows simulating growth without such a conditions posed at some point in the culture medium that may influence the proliferation and the movement of tumor cells.
A steady radius, as documented in several experiments (see for example [36–38]), characterizes the final phase of spheroid growth. This behavior is described by different models in the literature [39–42]. In [39] Byrne and Preziosi develop a biphasic model of avascular growth based on mixture theory. They report a steady radius for long times which depends on tumor compression and different model parameters. Another example is given by the work of Wise et al. [42], where a porous media model for tumor growth is derived by energetic considerations. Their solution displays the equilibrium radius as a function of cell adhesiveness and other parameters describing the growth of the tumor. In our previous work [22], we validated the model against experimental data describing the first stages of spheroid growth without considering the final phase. In this paper we extend the previous validation and take into account the complete growth pattern.
Comparison between MTS growth in an ECM deposited by TCs, in an ECMfree medium and in an ECM scaffold
In this example we compare the growth of (a) a multicellular tumor spheroid immersed in the interstitial fluid and deposing its own deformable ECM, (b) a multicellular tumor spheroid immersed in the interstitial fluid without ECM and (c) a multicellular tumor spheroid growing in a deformable and decellularized ECM scaffold. Tumor cells can in fact be grown successfully in a decellularized ECM of an organ as shown in Mishra et al. [43]. This allows mimicking the in vivo environment and has been done successfully for an ex vivo 3D lung model where it was possible to grow perfusable lung nodules [43].
The simulations are limited to the avascular stage. In this comparison we show the influence of the ECM presence on the growth.
Boundary conditions are imposed as indicated in Fig. 4. To allow IF flux at the outer boundary the IF pressure is fixed there to zero Pa. Due to the symmetry of the problem there is no flux normal to the radius of the sphere segment. Oxygen is the sole nutrient species and its mass fraction is fixed to \(\omega_{env}^{{^{{\overline{nl} }} }} = 4.2 \times 10^{  6}\) at the outer boundary and throughout the computational domain at initial time. This mass fraction of oxygen corresponds to the average of the dissolved oxygen in the plasma of a healthy individual.
Initial conditions
Zone  p ^{ l } [Pa]  S ^{ h } [−]  S ^{ t } [−]  \(\omega^{{\overline{nl} }}\)[−] 

Red zone (up to 30 µm)  0.0  0.0  0.02  4.2 × 10^{−6} 
Blue zone (up to 1,000 µm)  0.0  0.0  0.0  4.2 × 10^{−6} 
Parameters depending on cells’ type and IF
Parameter  Symbol  Value  Unit 

Density of the three fluid phases (α = h, t and l)  ρ ^{ α }  1,000  kg/m^{3} 
Dynamic viscosity of IF [22]  μ ^{ l }  1 × 10^{−3}  Pa·s 
Dynamic viscosity of TC [22]  \(\mu_{0}^{t}\)  36  Pa·s 
Dynamic viscosity of HC [22]  \(\mu_{0}^{h}\)  36  Pa·s 
Adhesion of TC [22]  ψ ^{ t }  0  Pa/m 
Adhesion of HC [22]  ψ ^{ h }  0  Pa/m 
\(\omega_{crit}^{{\overline{nl} }}\)  3.0 × 10^{−6}  –  
\(\gamma_{growth}^{t}\)  1.8 × 10^{−2}  kg/(m^{3} s)  
\(\gamma_{\text{necrosis}}^{t}\)  9.6 × 10^{−3}  kg/(m^{3} s)  
\(\gamma_{growth}^{nl}\)  2 × 10^{−4}  kg/(m^{3} s)  
\(\gamma_{0}^{{\overline{nl} }}\)  3 × 10^{−4}  kg/(m^{3} s)  
HCIF interfacial tension [22]  σ _{ hl }  72  mN/m 
TCHC interfacial tension [22]  σ _{ th }  36  mN/m 
TCIF interfacial tension  σ _{ tl }  108  mN/m 
Lysis parameter  λ  0.0001  – 
Parameters related to oxygen diffusion
Parameters for ECM taken from [22]
Parameter  Symbol  Value  Unit 

Density of the solid phase  ρ ^{ s }  1 × 10^{3}  kg/m^{3} 
Poisson’s ratio of the ECM  ν  0.4  – 
Young’s modulus of the ECM in case of Fig. 4a  Efin  1 × 10^{2}  Pa 
Young’s modulus of the ECM in case of Fig. 4b  E  1 × 10^{2}  Pa 
Volume fraction of ECM (initial) in case of Fig. 4a  \(\varepsilon^{s}\)  0.0  – 
Volume fraction of ECM (initial) in case of Fig. 4b  \(\varepsilon^{s}\)  0.2  – 
Coefficient a [22]  a  590  Pa 
Intrinsic permeability  k  1.8 × 10^{−15}  m^{2} 
Yield effective stress limit  \(\text{t}_{\text{eff}}^{\text{s}} ,_{y}\)  0.5 × 10^{1}  Pa 
Viscosity  \(\eta\)  5  Pa s 
Hardening modulus  H  1.0 × 10^{2}  Pa 
Recently we have also included the effect of fluid–fluid interfacial tension taking into account explicitly HCIF, σ _{ hl }, TCHC, σ _{ th }, and TCIF interfacial tensions, σ _{ tl } [17]. The values of interfacial tensions in Table 6 have been chosen respecting order of magnitude of experimental measurements [47, 48].
Figure 5c shows the evolution of the oxygen mass fraction in a culture medium with deposited ECM, ECMfree culture medium and remodeling ECM scaffold (solid, dotted and dashed lines respectively). The oxygen decreases from the original mass fraction of 4.2 × 10^{−6} because of its consumption made by living tumor cells. Oxygen is the sole nutrient species considered here. Once the oxygen concentration decreases below the critical value fixed in Table 6 cell necrosis begins. In Fig. 5d we can see a larger necrosis in the MTS growing in deposited ECM and ECMfree culture medium (solid and dotted lines) with respect to the MTS growing in remodeling ECM scaffold. Indeed, TCs grow faster and consume more in the first two cases.
It is recalled that finite displacements and a Lagrangian updated formulation together with the objective Truesdell rate of Cauchy stress are adopted for the simulations, see section “Solid phase behavior”. The Young modulus for the deformable deposited ECM during the MTS growth is described in Eq. (24) with Efin = 8.0 × 10^{1} Pa; E = 8.0 × 10^{1} Pa for the deformable ECM scaffold. Hence when the deposited ECM reaches the volume fraction of \(\varepsilon_{fin}^{s}\) = 0.2 the Young modulus is the same as that of the ECM scaffold with \(\varepsilon_{{}}^{s}\) = 0.2. The constitutive behavior of the deformable ECM is elastic until the yield limit, after that the behavior becomes viscoplastic. The viscoplastic parameters for the Perzyna type model (see [22]) are described in Table 8.
Comparison between MTS growth in a deposited ECM medium with and without lysis
In this section the focus is on the effect of the lysis in the MTS growth in a ECM deposited by TCs. We recall that lysis transforms a part of necrotic tumor cells in the core of MTS into liquid phase increasing the liquid pressure in the center of MTS; remind that with growing tumor density the permeability reduces hindering the outflow through the viable rim, see the constitutive law for permeability in [15]. Since the IF pressure is higher inside the tumor than at its outer boundary, the IF flows towards the boundary while tumor cells move towards the center of the MTS as required by the mass balance. The MTS growth reaches an equilibrium phase in the tumor cell flux between the center and the border of the MTS.
Lysis increases the liquid in the MTS, causing an increase of the IF pressure as shown in Fig. 7b. The red line represents the IF pressure at the last step of the analysis in the case of \(\lambda\) equal to 0.01; the black line represents the IF pressure at last step in the case of \(\lambda\) equal to 0.0001. The increment of IF pressure happens in the necrotic core of MTS, where a part of necrotic cells become liquid and the outflow through the outer rim is reduced because of reduced permeability in the viable rim. The IF pressure passes from negative values of the black curve to positive ones in the red curve in the center of MTS. The IF pressures at the center of MTS, in both black and red curves are higher than at the border of the MTS; hence the IF flows from the center to the border. With the lysis parameter \(\lambda\) = 0.01, this phenomenon is more visible.
In conclusion lysis produces outflow of the IF from the tumor mass. The tumor cells pressure however has an inverse behavior due to lysis: it decreases in the necrotic core as shown in Fig. 7c. Hence it redirects some of the tumor cells towards the interior. Figure 7d shows the volume fraction of living tumor cells at the last calculation step: with \(\lambda\) equal to 0.01 more tumor cells come back to the center of the MTS than with \(\lambda\) equal to 0.0001. The tumor cell flux resulting from proliferation has been conjectured also in [49].
Conclusions
The original model for tumor growth has been enhanced by the introduction of ECM deposition during the TC growth and of lysis, i.e. the retransformation of necrotic cells into IF.
The ECM has been modeled as a porous solid matrix with Greenelastic and elastoviscoplastic material behavior within a large strain approach. Truesdell objective stress measure is adopted together with the deformation rate tensor. An updated Lagrangian formulation has been used for the numerical simulations. A freeboundary simulation of MTS growth with the enhanced mathematical formulation of the model has allowed to reproduce new experimental data, carried out at HMRI. Both the first exponential growth and the plateau when the tumor mass reaches equilibrium have been replicated.
ECM deposition, ECM free growth and ECM remodeling have then been investigated: it appears that the MTS growth seems to be faster in the first two cases. The larger availability of IF to MTS growth and the major growth freedom in a ECMfree medium with respect in a remodeling ECM medium can explain this behavior. The last example shows how lysis affects the results. Lysis transforms a part of the necrotic tumor cells into IF. This increases the liquid in the center of MTS, causing an increase of the IF pressure with respect to the border of the MTS. Hence the IF flows out from the center to the border and produces an outflow of the IF from the tumor mass. The tumor cell pressure instead results lower in the center of MTS because of lysis and this leads some of the tumor cells migrating towards the interior.
The introduction of lysis and ECM deposition allows capturing different aspects of the avascular tumor growth not yet comprised in the original model. We are now extending this more complete multiphase model to include the vascular stage of tumor growth.
Nomenclature
 eqn.:

equation
 eqs.:

equations
 REV:

representative elementary volume
 TCAT:

thermodynamically constrained averaging theory
 a :

coefficient in the pressure–saturations relationship
 \({\mathbf{C}}_{ij}\) :

non linear coefficient of the discretized capacity matrix
 \({\mathbf{d}}^{{\overline{\overline{\alpha }} }}\) :

rate of strain tensor
 \(D_{eff}^{{\overline{il} }}\) :

effective diffusion coefficient for the species i dissolved in the phase l
 \({\mathbf{D}}_{s}\) :

tangent matrix of the solid skeleton
 \({\mathbf{e}}_{{}}^{{\bar{\bar{s}}}}\) :

total strain tensor
 \({\mathbf{e}}_{el}^{{\bar{\bar{s}}}}\) :

elastic strain tensor
 \({\mathbf{e}}_{vp}^{{\bar{\bar{s}}}}\) :

viscoplastic strain tensor
 \({\mathbf{f}}_{v}\) :

discretized source term associated to the primary variable v
 \({\mathbf{K}}_{ij}\) :

non linear coefficient of the discretized conduction matrix
 \(K_{i}\) :

compressibility of the phase i (i = s, t, h and l)
 \({\mathbf{k}}\) :

intrinsic permeability tensor of the ECM
 \(k_{rel}^{\alpha }\) :

relative permeability of the phase α
 \({\mathbf{N}}_{v}\) :

vector of shape functions related to the primary variable v
 \(p^{\alpha }\) :

pressure in the phase α
 \(p^{ij}\) :

pressure difference between fluid phases i and j
 \({\mathbf{R}}^{\alpha }\) :

resistance tensor
 \(S^{\alpha }\) :

saturation degree of the phase α
 \({\mathbf{t}}_{eff}^{{\overline{\overline{s}} }}\) :

effective stress tensor of the solid phase s
 \({\mathbf{t}}_{{}}^{{\overline{\overline{s}} }}\) :

total stress tensor of the solid phase s
 \({\mathbf{t}}_{eff,y}^{{\bar{\bar{s}}}}\) :

yield limit of the solid phase which defines the boundary of elastic domain
 \({\mathbf{u}}^{s}\) :

displacement vector of the solid phase s
 \({\mathbf{v}}^{{\bar{\alpha }}}\) :

velocity vector of the phase α
 \({\mathbf{x}}\) :

solution vector
 \(\bar{\alpha }\) :

Biot’s coefficient
 \(\gamma_{growth}^{t}\) :

growth coefficient
 \(\gamma_{necrosis}^{t}\) :

necrosis coefficient
 \(\gamma_{growth}^{{\overline{nl} }}\) :

nutrient consumption coefficient related to growth
 \(\gamma_{0}^{{\overline{nl} }}\) :

nutrient consumption coefficient not related to growth
 ε :

porosity
 \(\varepsilon^{\alpha }\) :

volume fraction of the phase α
 \(\eta\) :

viscosity parameter of the solid phase
 \(\mu^{\alpha }\) :

dynamic viscosity of the phase α
 \(\rho^{\alpha }\) :

density of the phase α
 \( \sigma_{ij} \) :

interfacial tension between fluid phases i and j
 \( \psi^{\alpha } \) :

adhesion of the phase α
 \( \omega^{{N\overline{t} }} \) :

mass fraction of necrotic cells in the tumor cells phase
 \( \omega^{{\overline{nl} }} \) :

nutrient mass fraction in the interstitial fluid
 \( \omega_{crit}^{{\overline{nl} }} \) :

critical nutrient mass fraction for growth
 \( \omega_{env}^{{\overline{nl} }} \) :

reference nutrient mass fraction in the environment
 \( \mathop M\limits^{\kappa \to \alpha } \) :

interphase mass transfer
 \( \varepsilon^{\alpha } r^{i\alpha } \) :

reaction term i.e. intraphase mass transfer.
Subscripts and superscripts
 crit :

critical value for growth
 n :

nutrient
 h :

host cell phase
 l :

interstitial fluid
 s :

solid
 t :

tumor cell phase
 α :

phase indicator with α = t, h, l, or s
Declarations
Authors’ contributions
BAS contributed the theoretical framework and helped in drafting the manuscript. RS developed the ECM deposition equations and implemented them in the code, performed all simulations and drafted the manuscript. PM developed and implemented in the code the equations for lysis. CS carried out the experimental data at HMRI. PD gave suggestions to clarify the work and MF contributed to the background of this work. All authors read and approved the final manuscript.
Acknowledgements
RS acknowledge the University of Padua for financial support (project n. CPDR121149). MF acknowledges the financial supports from NCI Physical ScienceOncology Centers (NIH U54CA143837), and from The Methodist Hospital Research Institute, including the Ernest Cockrell Jr. Presidential Distinguished Chair. PD acknowledges partial support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/20072013)/ERC Grant agreement No. 616695 and the Houston Methodist Research Institute.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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