Biomech Model Mechanobiol (2017) 16:1141–1157 DOI 10.1007/s10237-017-0876-8
ORIGINAL PAPER
Modeling tumor growth with peridynamics
Emma Lejeune1 · Christian Linder1
Received: 23 November 2016 / Accepted: 11 January 2017 / Published online: 26 January 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract Computational models of tumors have the poten- 1 Introduction tial to connect observations made on the cellular and the tissue scales. With cellular scale models, each cell can be Neoplasms, commonly referred to as tumors, are defined as treated as a discrete entity, while tissue scale models typi- abnormal growth of tissue (Deisboeck et al. 2011). Tumor cally represent tumors as a continuum. Though the discrete growth is problematic because tumors can disrupt organ approach often enables a more mechanistic and biologically function and infiltrate and invade healthy tissue (Gatenby driven description of cellular behavior, it is often compu- and Gawlinski 1996). Tumor growth is also the precursor tationally intractable on the tissue scale. Here, we adapt for metastasis, the spread of cancer from the primary tumor peridynamics, a theoretical and computational approach location to secondary locations, where patient outcomes sig- designed to unify the mechanics of discrete and continuous nificantly worsen (Poste and Fidler 1980). On the cellular media, for the growth of biological materials. The result is scale, tumor growth is driven by mitosis, the process by which a computational model for tumor growth that can represent a single-parent cell divides into two daughter cells (Bel- either individual cells or the tissue as a whole. We take advan- lomo et al. 2008). This differs from other forms of biological tage of the flexibility provided by the peridynamic framework growth, such as the progression of cardiovascular disease, to implement a cell division mechanism, motivated by the fact which are driven by changes in cell size and shape (Kuhl that cell division is the mechanism driving tumor growth. et al. 2006). Despite significant scientific interest and study, This paper provides a general framework for implementing tumor growth is still far from fully understood (Wang et al. a new tumor growth modeling technique. 2015). A better understanding of tumor growth would aid in predictive modeling of tumor progression and contribute to Keywords Peridynamics · Tumor growth · Morphogenesis · more robust clinical decision making, and the development Cell division of new therapies (Clatz et al. 2005). The focus of this paper is on developing a computational model to aid in bridging Mathematics Subject Classification 92C10 · 74L15 the gap between the cellular and tissue scale. In computational modeling of tumor growth, there is an inherent trade-off between continuous and discrete approaches. Continuum modeling is typically used to under- stand tumors on the tissue scale, where tumors are treated as either a volumetrically growing solid (Ambrosi and Mol- lica 2002) or as a constituent in a mixture theory approach B Christian Linder (Byrne 2003; Preziosi and Tosin 2008). Discrete modeling, [email protected] on the other hand, views tumors as a collection of cellular or Emma Lejeune subcellular components (Drasdo et al. 2007; Sandersius and [email protected] Newman 2008). Discrete modeling, which is often favored 1 Department of Civil and Environmental Engineering, by the biophysics community (Norton et al. 2010), allows Stanford University, Stanford, CA 94305, USA a mechanistic description of tumor cell growth and divi- 123 1142 E. Lejeune, C. Linder sion, but is computationally intractable on the tissue scale, which severely limits potential applications in the clinical (a) setting such as modeling the macroscale mechanical interac- tions between tumors and healthy tissue or organs (Frieboes et al. 2007; Lowengrub et al. 2010). To capture the benefits of both discrete and continuum modeling, hybrid modeling approaches have been proposed (Stolarska et al. 2009). For example, one approach treats the active cells on the perime- ter of a tumor as discrete particles and the necrotic cells (b) at the center of the tumor as a continuum captured by a finite element mesh (Kim et al. 2007). In addition, there has been significant effort to formulate continuum mod- els that phenomenologically reflect processes occurring on the cellular scale (Ambrosi et al. 2012; Araujo and McEl- wain 2004). However, current approaches are imperfect. Therefore, developing tumor models that can overcome the Fig. 1 Fundamentally, tumors are composed of biological cells and trade-off between the continuous and discrete approaches tumor growth is driven by cell division. Schematic (a) shows a tumor remains an active area of research (Bellomo et al. 2008; Byrne growing around a blood vessel, and Schematic (b) shows an idealized discrete approximation of the cells in a tumor. In this paper, we present and Drasdo 2008). Here, we present a new approach to mod- a framework for modeling tumor growth that approximates tumors with eling tumor growth. Our aim is to capture the behavior of the idealization shown in (b) tumor cells as mechanistically as possible while creating a framework that can ultimately be implemented on the tissue scale. The starting point for our model is peridynamics. peridynamics. Section 3 discusses the implementation of Peridynamics, a framework that is designed to unify the volumetric biological growth effects using the dual-horizon mechanics of discrete and continuous media, is a technique peridynamic framework. Section 4 introduces cellular scale originally developed for fracture mechanics applications growth mechanisms, specifically a cell division mechanism, (Silling and Lehoucq 2010). Rather than using partial dif- to the peridynamic framework. Section 5 demonstrates our ferential equations to formulate the equations of motion, numerical implementation of peridynamics with volumetric as is done in classical continuum mechanics, peridynam- growth and a cell division mechanism, first by comparing ics uses integral equations which exist on crack surfaces our results to two benchmark problems, second by using our and other singularities (Silling 2000; Silling et al. 2007). framework to examine macroscale isotropic and anisotropic The first paper introducing peridynamics was published growth as an emergent property of cell division behavior, and in 2000 and presents peridynamics as a methodology for third by showing an example of how our framework can be modeling discontinuities and long range forces using a con- used to study tumor morphogenesis. We conclude the paper stitutive relation based on pair-wise interactions between in Sect. 6. particles (Silling 2000). Since then, peridynamic theory has been extended to include more complex constitutive models (Silling et al. 2007; Warren et al. 2009), developed numeri- 2 Background: dual-horizon peridynamics cally (Bobaru and Ha 2011), and applied to model a variety of engineered systems (Kilic and Madenci 2010b). Peridynam- Peridynamics is a formulation that writes the classical bal- ics is typically implemented as a mesh-free method where ance equations as integrals rather than partial differential a block of material is discretized into an array of nodes that equations. In a conventional peridynamic formulation, a each interact with their neighbors (Silling and Askari 2005). given point x interacts with other points within its horizon Here, we will extend peridynamics to capture volumetric Hx where Hx is a line (1D), circle (2D), or sphere (3D), growth and cellular growth mechanisms such as cell division. defined by radius δx, as illustrated in Fig. 2. The difference In our implementation, a node can represent either a single between the conventional formulation of peridynamics and cell or a collection of cells. We use the node as a single-cell dual-horizon peridynamics is that dual-horizon peridynamics interpretation, illustrated in Fig. 1, to guide our formulation allows non-uniformity in horizon size across different points of a cell division mechanism. However, it is important to note (Ren et al. 2016). Dual-horizon peridynamics is equipped to that while that interpretation can be helpful, the framework handle the case where some point x is within the horizon we propose here is not limited to it. of x,butx is not within the horizon of x. Implementing The remainder of the paper is organized as follows. variable horizon size is an important feature for adaptive Section 2 provides a general background on dual-horizon refinement and multi-scale modeling (Bobaru et al. 2009; 123 Modeling tumor growth with peridynamics 1143
(a) (b) (c)
(d) (e)
H H Fig. 2 This figure illustrates the terminology used in dual-horizon peri- x is in the horizon x and dual horizon x of x; Schematic (d)shows Ω Ω H H dynamics. Schematic (a) shows domain 0 transformed to t , points x the force densities that arise when x is in x but not x ; Schematic and x transformed to y and y and bond vector ξ transformed to ξ + η; (e) shows the force densities that arise when x is not in Hx but is in H δ H Schematic (b) defines horizon x as all points within distance of x; x . The corresponding formal definitions are given in Sect. 2.1 Schematic (c) shows the peridynamic force densities f that arise when
Bobaru and Ha 2011). And, variable horizon size is a required define the current configuration. The relative displacement feature for implementing the cellular scale growth mecha- of a bond is defined as η = u − u, described in the dis- η = − nisms presented in Sect. 4 of this paper. crete setting as jk uk uj. Finally, in peridynamics, it is This section will begin by introducing the kinematics and typical to use state notation (Silling et al. 2007). State nota- terminology of dual-horizon peridynamics (which are nearly tion defines a state of order m as a function A· that maps identical to that of conventional peridynamics) and subse- the vector in angle brackets · to a tensor of order m.For quently present the balance laws and the ordinary state-based example, the relative position of bond ξ can be written as constitutive equation. It is worth noting that the notation used yξ= y(x, t) − y(x, t) = ξ + η. In this paper, states are to present the peridynamic equations is not consistent across written with an underline, and angle brackets are used to indi- the literature. We base our notation on the notation for dual- cate the quantity which the state function is acting on. State horizon peridynamics (Ren et al. 2016) and recommended notation is convenient in defining the peridynamic constitu- notation for peridynamic software implementation (Little- tive relations and is amenable to numerical implementation. wood 2015) and use the symbol “∗” in Sect. 3 to indicate the Fundamental to peridynamics is the concept of horizon. terms that we introduce to capture the effects of growth. The bonds between point x and the points inside the horizon of x are assumed to be existent and non-negligible. The hori- 2.1 Kinematics and terminology zon of point x is defined geometrically as all points x such that x ∈ Hx, where Hx is In peridynamic theory, the equation of motion is formulated as an integral of interaction forces between points on the body Hx ={x |||x − x|| <δx} (1) Ω. To formulate the balance of linear momentum, presented in Sect. 2.2, it is first necessary to define parameters. Here, we with δx as the parameter that dictates horizon size. Physi- introduce the terminology required to define the interaction cally, Hx is defined as the domain where any particle will H between material points in the initial configuration Ω0 as x experience force exerted by x. The concept of horizon x and x, illustrated in Fig. 2a. The bond vector between points was introduced for conventional peridynamics, dual-horizon x and x in the initial configuration is defined by the term peridynamics inherits this concept and expands on it (Ren ξ = x − x. In the discrete setting, suitable for numerical et al. 2016). In dual-horizon peridynamics, the dual horizon H implementation, point x is referred to as node j, point x is x is defined as the union of points whose horizons include referred to as node k, and the bond between them is defined x, written as ξ = − as jk xk xj. And, as illustrated in Fig. 2a, the displace- ( , ) ( , ) = + ( , ) H ={ | ∈ H } . ment vector u x t and position vector y x t x u x t x x x x (2) 123 1144 E. Lejeune, C. Linder H For all points x within x , x acts on x. And, unlike the hori- ρ ¨( , ) = (η, ξ) u x t f xx dVx ∈H zon which has a circular (2D) or spherical (3D) shape, the x x dual horizon has an arbitrary shape. If the horizon size δ does − (−η, −ξ) + ( , ) f x x dVx b x t (5) not vary across points, then Hx = H and the dual-horizon x x ∈Hx peridynamics formulation will be identical to that of conven- tional peridynamics. Moving forward, we will define force where ρ is density, u¨ is acceleration, and b is body force (Ren density and the equation of motion in the context of dual- et al. 2016). Internal force is computed as a function of the horizon peridynamics with the knowledge that it reduces to force density vectors defined in Sect. 2.1. Integrating over conventional peridynamics if horizon size is identical for all H the dual horizon x contributes the direct force term acting points. on point x, while integrating over the horizon Hx contributes In dual-horizon peridynamics, the force between points x the reaction force term. In the conventional version of peri- and x is defined in two distinct steps, first using the dual H = H dynamics, where horizon size is uniform, x x and the horizon and then using the horizon. We define force density expression for internal force can be written as a single inte- (η, ξ) vector f xx as the force per volume acting on particle gral. The discrete form of the balance of linear momentum x due to particle x . This implies that x is the location of the is written as force and x is the source of the force, illustrated in Fig. 2c, (−η, −ξ) e. Likewise, f x x is the force density vector acting ρ ¨( , ) = (η , ξ )Δ u xj t f jk jk jk Vk on particle x due to particle x, illustrated in Fig. 2c, d. Each k∈H force density is accompanied by a reaction force density, j − − (−η , −ξ )Δ + ( , ) i.e., f xx is a direct force density acting on x and f xx is a f kj jk jk Vk b xj t (6) ∈H reaction force density acting on x . The direct force density k j at x is computed from points in the dual horizon of x, while the reaction force density at x is computed from points in the where the integral in Eq. (5) is simply replaced by a sum- horizon of x. The net force density acting on point x due to mation. In the numerical implementation of dual-horizon bond x-x is a sum of the direct force density and reaction peridynamics, this equation can be assembled easily by loop- force density written as ing through the horizon of each node and subsequently inferring each node’s dual horizon (Ren et al. 2016). For H − Δ every node k in j, calculate f kj and add the term f kj Vk f (η, ξ) − f (−η, −ξ). (3) xx x x to the horizon summation term of node j and add the term f ΔVj to the dual horizon H summation term of node kj k Similarly, the net force density acting on a point x due to k. With this procedure, the dual horizon is never computed bond x-x is explicitly.
(−η, −ξ) − (η, ξ). 2.3 Constitutive equations f x x f xx (4)
∈/ H The first version of peridynamics, bond-based peridynam- When x is outside the horizon of x (x x), as illus- = ics, treats bonds as independent springs (Silling 2000). In trated in Fig. 2d, direct force f xx 0 and corresponding − = bond-based peridynamics, Poisson’s ratio is limited to 1/3 in reaction force f xx 0. Likewise, in the reverse scenario two dimensions and 1/4 in three dimensions which makes illustrated in Fig. 2e where x ∈/ H f = 0 correspond- x x x it unsuitable for modeling many engineering materials. In ing reaction force − f = 0. By differentiating between x x response to the limitations of bond-based peridynamics, ordi- the contributions from H and H (and subsequently direct x x nary state-based and non-ordinary state-based peridynamics and reaction forces) the antisymmetry of net force density is were introduced (Silling et al. 2007). The qualification “state- preserved even in the case of variable horizon size. based” comes from the fact that the constitutive law utilizes the state notation introduced in Sect. 2.1. In state-based 2.2 Balance of linear momentum peridynamics, bond force is a function of the collective deformation of all bonds that act on the same points as The defining feature of peridynamics is that the classical the bond in question. In state-based peridynamics, Poisson’s balance equations are formulated as integrals rather than par- ratio is not fixed. When modeling soft biological tissue, it tial differential equations (Silling 2000). In the peridynamic is typical to assume incompressibility or near incompress- version of the balance of linear momentum, inertial force, ibility (ν ≈ 0.5); therefore, we do not present the equations body force, and internal force terms at point x and time t are for bond-based peridynamics in this paper. The maximum equated as flexibility in constitutive modeling is introduced with the 123 Modeling tumor growth with peridynamics 1145 non-ordinary state-based peridynamics approach (Warren point x and eξ defined for each bond associated with x,we et al. 2009); however, an adaptation of non-ordinary state- compute the dilation at x, θx as based peridynamics for the cell division aspect of biological growth discussed in Sect. 4 is beyond the scope of this work. n θx = ωξ||ξ|| eξ dVξ (11) In this section, we describe the constitutive equations for mx Hx ordinary state-based peridynamics that will be used for the n n ∈{ , } remainder of the paper. where is the dimension number, 2 3 . This equation In dual-horizon peridynamics, a distinction is made is discretized at node j as between forces that arise at point x from the horizon and θ = n ωξ ||ξ || ξ Δ . forces that arise at point x from the dual horizon. In this sec- j jk jk e jk Vk (12) mj ∈H tion, we define the constitutive relation based on bond x − x k j assuming that x is in the dual horizon of x (x is in the hori- Then, the deviatoric extension state for bond ξ from the per- zon of x). This means that for a given bond x − x we present spective of point x can be computed as the force density corresponding to the action force applied at point x due to x and the subsequent reaction force applied at θ ||ξ|| d x point x. The distinction between which point is x and which e ξ=eξ− . (13) n point is x is arbitrary; therefore, in computing force density at all points x, bond x − x will eventually be treated as bond Using these terms, the scalar force state that defines the linear x − x. The constitutive equations are presented this way elastic ordinary state-based constitutive law for each bond ξ because it is convenient for numerical implementation. from the perspective of point x is written as Fundamentally, peridynamics is a non-local theory mean- κθ ( + )μ ing that non-adjacent points interact. The degree to which = n x ωξ||ξ|| + n n 2 ωξ d ξ t x x e (14) non-local forces come into play is controlled by two parame- mx mx ters, horizon size δ and influence function ωξ. The influence where κ and μ are the Lamé parameters bulk modulus and function can be chosen to weight the effect of certain bonds shear modulus, respectively (Littlewood 2015). Given the more heavily or it can be set to a constant. For example, ωξ scalar force state t, the force density vectors corresponding can be equal to with each bond are computed based on the direction of the ||ξ||2 deformed bond vector as ωξ=exp − or ωξ=1(7) δ2 −η − ξ f (η, ξ) = t x x x x ||η + ξ|| (15) or another user-defined function (Ren et al. 2016; Littlewood 2015). Given a chosen influence function, we compute the which is the action force applied at point x , and influence function weighted volume of the horizon at point H η + ξ x, mx, by integrating over x as − f (η, ξ) = t x x x x ||η + ξ|| (16)
mx = ωξ ξ · ξ dVξ (8) which is the reaction force applied at point x. To compute the H x total force at each node, required for solving Eq. (6), force which is written discretely at node j as density vectors are summed over all bonds in the horizon. = ωξ ξ · ξ Δ 2.4 Bond breaking mj jk jk jk Vk (9) k∈H j Although the focus of this paper is not on fracture, we first describe bond breaking as is typical for solving a peridynamic with ΔV representing the nodal volume associated with the k fracture problem. For modeling bond breaking to simulate node located at node k. In 2D, some constant thickness h is fracture, one approach is to compute the work required to assumed and ΔV = hΔA. In addition, we define extension break a single bond as a function of bond stretch, state eξ as ||ξ + η|| − ||ξ|| s = eξ=||ξ + η|| − ||ξ|| (10) ||ξ|| (17) based on the difference in separation between bonds in the compute the total amount of work required to generate a deformed and reference configurations. Using mx defined at fracture surface, equalize the total amount of work with the 123 1146 E. Lejeune, C. Linder
(a) (b) (c)
Fig. 3 Growth is modeled as an increase in nodal radius which causes shows the new, growth induced, equilibrium position. This process is nodes to shift to a new equilibrium position. Schematic (a)shows driven by step (i) where growth is applied, and step (ii) where the peri- nodes at equilibrium prior to applied growth; Schematic (b)showsthe dynamic framework is used to reach a new equilibrium position non-equilibrium state immediately after applied growth; Schematic (c) strain energy release rate, and solve for critical bond stretch or explicit integration with adaptive dynamic relaxation at s0 (Silling and Lehoucq 2010). The value of s0 will be a every load step (Kilic and Madenci 2010a). In a relatively function of material and discretization defining parameters. straightforward manner, isotropic volumetric growth can be Given some value for critical bond stretch s0, we define captured by including a growth term g in the constitutive γ αΔ history variable where equation analogous to the term T in a one-way coupled thermomechanical formulation (Kilic and Madenci 2010b). 1ifs(t , ξ)
Δ ∗ ||ξ|| as Vx . The parameters , defined for the bond between influence function in the manner defined by Eq. (19), and in node j and node k, are written as Sect. 3.2 we provide an alternative definition for γ that allows for bond breaking and forming. Using these new parameters, 1 ∗ ∗ g( xj + (xk − xj )s ) |xk − xj| ds we redefine horizon weighted volume at a node j as m .The ||ξ || = 1 + 0 j jk 1 | − | updated version of Eq. (9) is written as 0 xk xj ds ||ξ || jk (23) ∗ = ω∗ξ ∗ ||ξ ∗ ||2 Δ ∗ . mj jk jk Vk (29) k∈H for some positionally dependent growth function g(x). When j nodes are interpreted as individual cells, it often makes sense ∗ The expression for elongation state including growth e mod- to think of g as piece-wise constant function where each cell ifies Eq. (10)as is attributed some level of growth. If growth g is homoge- neous, this equation reduces to ∗ξ ∗ =|| − || − ||ξ ∗ || e jk yk yj jk (30) ||ξ ∗ || = ( + )||ξ || . jk 1 g jk (24) ξ + η − where we again replace jk jk with the expression yk y . Putting these expressions together, the new equation for Δ ∗ j The equation for V at node i is defined similarly by inte- dilation θ ∗ at node j, an update of Eq. (12), is written as grating position dependent growth over either the volume or the area associated with node i defined by node radius r .For ∗ n ∗ ∗ ∗ ∗ ∗ ∗ i θ = ω ξ ||ξ || e ξ ΔV . (31) every node, we can define g in spherical coordinates centered j m∗ jk jk jk k j k∈H on the node and compute ΔV ∗ as j
d ∗ π 2π r0 3 The modified deviatoric extension state e , an update of Δ ∗ = 1 (φ) ( + ( ,θ,φ)) θ φ. Vi sin 1 gi r dr d d Eq. (13), is 0 0 3 0 (25) θ ∗||ξ ∗ || ∗ ∗ ∗ ∗ j jk ed ξ =e ξ − . (32) jk jk n In two dimensions, we instead define g in polar coordinates ∗ and compute ΔV as Scalar force state t∗ is then expressed as 2π r 2 ∗ ∗ 1 0 κθ Δ = ( + ( ,θ)) θ. ∗ n j ∗ ∗ ∗ n (n + 2)μ ∗ ∗ ∗ ∗ Vi 1 gi r dr d (26) t = ω ξ ||ξ || + ω ξ ed ξ 0 2 0 kj ∗ jk jk ∗ jk jk mj mj (33) For a constant level of growth gi at each node i, the equation for ΔV ∗ is reduced to replacing Eq. (14). Finally, the force density vectors includ- Δ ∗ = ( + )nΔ . ing growth are computed as Vi 1 gi Vi (27)
∗ −( y − y ) For n = 2, ΔV represents an area multiplied by some f ∗ ( y , y , ξ ∗ ) = t∗ k j kj j k jk kj || − || and constant thickness h = 1. In both Sect. 4 and Sect. 5,we yk yj assume homogeneous growth associated with each node, thus ( y − y ) − f ∗ ( y , y , ξ ∗ ) = t∗ k j kj j k jk kj || − || (34) Eq. (27) is sufficient. yk yj The influence function can also be re-written to account for ω∗ξ ∗ ξ + η growth, , though if the influence function is defined as and inserted into Eq. (21). Again, jk jk is replaced with − a constant, as in (7), it will remain identical. Growth modified the expression yk yj. stretch is computed as 3.2 Peridynamics with bond breaking, forming and || − || − ||ξ ∗ || ∗ yk yj jk re-forming s = jk ||ξ ∗ || (28) jk In the typical formulation of peridynamics, Eq. (18) indicates where a bond with identical levels of growth and deformation that once bonds are broken they will not re-form or “heal”. ∗ will have zero stretch. Here, in addition to introducing ξ ,we This is consistent with our understanding of most materials: ξ + η − replace with the term yk yj because this alternative fracture is irreversible. However, this is not a requirement of notation will be helpful in Sect. 3.2.Thetermγ will enter the the peridynamic constitutive relation. In fact, peridynamics 123 1148 E. Lejeune, C. Linder has been used to model fracture and subsequent healing of nonlinearity, we apply loading (in this case applied growth cortical bone through a modified version of Eq. (18)(Deng g) incrementally as et al. 2008). Biological materials can form and re-form bonds inc 1/2 gt+1 = (1 + gt )(1 + g ) − 1inn = 2 because unlike standard engineering materials, biological a (38) inc 1/3 materials are able to adapt to their surroundings through gt+1 = (1 + gt )(1 + gv ) − 1inn = 3 both passive and active processes (Ambrosi et al. 2012). For example, in many types of tissue adjacent cell membranes are where gt+1 is defined as growth at step t + 1 and com- attached by adhesion links that can either break apart or form puted as a function of growth at step t and some incremental due to external stimuli (Drasdo and Höhme 2005). Therefore, scaling magnitude. Several small increments of growth are in addition to introducing growth into the constitutive equa- applied, and the quasi-static solution is obtained following tion, we present three additional considerations required for each increment. In the original implementations of peri- modeling biological materials and growth with peridynam- dynamics, where a node’s horizon Hx remains the same ics. First, bonds can form and re-form. Second, despite the throughout the simulation (i.e., it does not grow), the list fact that the material model is linear elastic, bond breaking of nodes within the horizon and dual horizon of x are deter- and forming can lead to a nonlinear material response. Third, mined by Eqs. (1, 2) at the start of the simulation. In the case when nodes are changing their connectivity it may be neces- where bond forming is allowed, the horizon must be updated sary to redefine stretch and elongation. at the start of every load increment. Since the horizon of each node changes, it is necessary to re-compute the horizon 3.2.1 Bond forming and re-forming between each step. Furthermore, it is logical to make hori- zon size δ dependent on growth g. We implement the simple As stated previously, Eq. (18) prohibits bond healing and new dependence bond formation. There are many possible different ways to δ∗ = ( + g )δ0 approach formation of new bonds, here we present the simple i 1 i i (39) modification where δ0 is horizon size without any growth. And, rather ( , ξ )< 1ifsjk t jk sinter than determining the horizon based on position in the initial γjk(t, ξ ) = (35) jk 0 otherwise . configuration, the horizon of node i is determined based on position at the end of the previous time step as where a bond automatically exists if two nodes are within H ={ ||| − || <δ∗} i k yk yi i (40) a certain distance defined by interaction stretch sinter.Ina biological context, sinter is some distance below which cells will develop adhesive connections between them. We also where y refers to the calculated position at the end of the last update Eq. (35) to include growth g consistent with Eq. (28) quasi-static step. Alternative relationships may also be appro- as priate; however, nuance in determining horizon size is not necessarily critical because the influence function ω can also ∗ ∗ ∗ ∗ 1ifs (t, ξ ) 2 1 gd = (1 + g ) − 1inn = 2 4 Implementing cellular scale growth mechanisms 2 0 in the peridynamic framework (42) 3 1 gd = (1 + g0) − 1inn = 3. Thus far, we model biological growth by applying load steps 2 and repeatedly solving a quasi-static problem. Based on the Then, given some division axis unit vector, m, which is a implementation of growth and cellular reorganization pre- function of φ in n = 2 and φ and φ in n = 3, a new node sented in Sect. 3, the procedure contains three steps: (i) a b position is assigned to each of the daughter cells. The node determining nodal connections using δ and ω, (ii) applying position is selected such that the nodal areas/volumes do not growth to each node, and (iii) using established peridynamic overlap. If y is the position of the parent cell, and vector m equations to relax to mechanical equilibrium. In this section, 0 defines the angle on which the node divides, then y and y we provide a framework for implementing additional cellular j k are the positions of the daughter cells written as scale mechanisms of growth. Section 4.1 discusses the imple- mentation of a cell division mechanism, and then Sect. 4.2 = + ( + ) contains a general implementation algorithm summarizing yj y0 1 gd r0 m (43) = − ( + ) . the procedures introduced in Sects. 2, 3 and 4.1. yk y0 1 gd r0 m 123 1150 E. Lejeune, C. Linder (a) (b) (c) (d) (e) (f) (g) Fig. 4 Cell division is implemented as a multi-step process where a rium node position (c), non-equilibrium grown position (d), equilibrium parent node splits into two daughter nodes on an axis defined in two grown position (e), non-equilibrium divided position (f), and final equi- dimensions by angle φ. Schematic (a) shows cell division with p = 4 librium position (g) where each parent node has split into two daughter defined by division angle φ; Schematic (b) illustrates parameters di , nodes. Nodes will grow according to a prescribed growth rule until they ri and θi used in Eq. (45); Schematics (c–g) illustrate initial equilib- exceed a threshold size that triggers the division algorithm i = π[( + i ) ]2 During implementation, the node number of y0 is preserved A 2 1 gd r0 in either yj or yk. Immediately after the division step, the π − 2θ (45) − 2[(1 + gi )r ]2(π − sin θ cos θ). cells will not be at mechanical equilibrium. Their position d 0 2π and the position of their neighboring cells will subsequently adjust to attain mechanical equilibrium. i i i Using A , gd is solved for by equating A and area before In the numerical setting, splitting cells over multiple steps 2 2 division procedure A0 = π(1 + g0) r as and allowing mechanical relaxation between each step may 0 yield more stable results. We implement a cell division algo- rithm inspired by the ellipse/barbell shape that cells take on π − θ i = 1 /[ π − (π 2 − θ θ)]− . gd A0 2 2 sin cos 1 as they divide (Gillies and Cabernard 2011). This algorithm, r0 2π illustrated in Fig. 4 and detailed in this section in 2D, starts (46) with overlapping daughter cells with a small separation dis- tance and increases the separation distance until the end of The number of steps p to complete cell division is a matter of the cell division procedure where cells will no longer over- discretion, typical values used in our simulations are p = 2 lap. Given p, the number of steps to complete cell division, to p = 5. Furthermore, in the numerical setting multiple divi- we define separation distance di at step i as sion events occurring in unison may cause stability problems. To remedy this, we introduce an additional random parameter ( + p) that represents probability of division and decrease the step i 2 i 1 gd r0 d = . (44) size when multiple cell division events are about to occur and p staggers the cell division events such that they occur within some interval [gth − τ,gth + τ] rather than at the thresh- During the subsequent mechanical relaxation step, di will old gth exactly. Making this adjustment means only a small ∗ replace ||ξ || when determining the force between parti- fraction of cells divide in unison, and numerical stability is cles mid-division. During the stepped division process, the preserved. Beyond cell division, cellular scale mechanisms particles are overlapped. In the two-dimensional case, an such as cell necrosis and apoptosis may be implemented in a i < equivalent level of growth at each step gd is determined such similar fashion by shrinking (g 0) or removing nodes when that the area of the two daughter cells minus the area of the g falls below some threshold value. In Sect. 4.2, where we overlap remains equal to the area of the parent cell. Angle summarize the overall implementation procedure, we refer i i θ , illustrated in Fig. 4b is defined as θ = arcsin di /2ri , and to these potential algorithms via the general phrase “cellular used to define total area Ai as scale mechanisms”. 123 Modeling tumor growth with peridynamics 1151 Algorithm 1: Biological growth with peridynamics: anisotropy may emerge due to directional cell division. These numerical implementation procedure. concepts are further explored in Sect. 5. Input : Initial node positions and properties, prescribed loading, prescribed cell division parameters Output: Final node positions, final peridynamic forces. for Load step t in prescribed loading do 5 Numerical studies for Node j in node list do increment growth at each node gj according to prescribed Numerical implementation of the peridynamic framework growth rule (ex: eqn. (38)) end for modeling biological growth follows directly from the for Node j in node list do algorithm presented in Sect. 4.2. In this section, we begin if gj > gcrit then by comparing the numerical framework with growth and no begin cell division algorithm: separate parent node division to two benchmark problems with known analyti- into two daughter nodes with φ eqn. (43) end cal solutions. Then, we examine isotropic and anisotropic end growth as emergent properties of the division angle. Finally, while Nodes are not at mechanical equilibrium (i.e., we look at an example problem where growth rate varies ≥≈ acceleration 0) do based on location. In this section, we show results from a for Node j in node list do Compute horizon Hj using eqn. (40) two-dimensional numerical implementation. Table 1 lists all end the parameters required to implement the simulations aside for Node j in node list do ∗ from the prescribed loading conditions which are specific to Compute ω∗ξ using eqn. (37) for every bond with jk each subsection. node k in Hj end for Node j in node list do ∗ 5.1 Benchmark comparison Compute mj using eqn. (29) by looping through the nodes k in Hj end In order to validate our implementation of growth in the peri- for Node j in node list do dynamic framework and ensure that our numerical solution Compute θ ∗ using eqn. (31) by looping through the j scheme is functioning, we chose two simple benchmark prob- nodes k in Hj end lems with an analytical solution for comparison. The first for Node j in node list do problem is quasi-static homogeneous growth of a rectangu- ∗ ∗ − ∗ Compute tkj and subsequently fkj and fkj using eqn. lar block oriented perpendicular to the x-y plane where the H (34) by looping through the nodes k in j displacement along cuts perpendicular to the x and y axis end for Node j in node list do can be computed as Compute acceleration, velocity and displacement using an adaptive dynamic relaxation algorithm = ( + ) = ( + ) , end ux 1 g xuy 1 g y (47) end end Table 1 All of the parameters used to implement the numerical exam- ples are as follows Parameter Value 4.2 Implementation procedure including cellular scale Dimension n 2 mechanisms Initial radius r0 0.5 Initial growth g 0.0 The integrated implementation of the work detailed in 0 δ . Sects. 2–4 is summarized in Algorithm 1. This algorithm Initial horizon 0 1 15 . is the foundation of the numerical implementation used to Breaking stretch sinter 1 15 . generate the results presented in Sect. 5. Overall, the imple- Young’s modulus E 1 0 ν . mentation procedure follows a pattern where the amount of Poisson’s ratio 0 45 ω∗ξ ∗ . growth at each node at each step is prescribed according to Influence function ij 1 0 user-defined rules, cellular scale mechanisms are prescribed Number of division steps p 4 according to user-defined rules, and the quasi-static solu- Division interval τ 0.025 inc . tion to the peridynamic equation of motion is subsequently Incremental area growth ga 0 01 solved to reach mechanical equilibrium. In this implemen- Every node in the domain is assigned these parameters. In addition, at tation, growth on the node-level is always isotropic. Growth the start of every simulation nodes are equally spaced in a hexagonal rate may differ at each node depending on position, and global configuration. If the units are not 123 1152 E. Lejeune, C. Linder (c) (a) (d) (e) ux uy ρ 15 analytical 20 analytical numerical g =0 g =0 80 analytical 10 =012 =012 g . 10 g . =056 =056 5 g . g . 60 (b) 0 0 40 −5 −10 −10 20 −15 −20 −20 0 −10 −5 05x −15 −10 −5 0510y 0.05 0.10 0.15 g2 Fig. 5 For the first benchmark test (a), we compare numerical results position y. For the second benchmark test (b), we compare numerical to the analytical solution for a rectangular (20 × 25 unit length) block, results to the analytical solution for a beam where bending is driven centered at the origin, growing without constraints. Plot (c) shows agree- by differential growth between layers. Plot (e) shows curvature ρ with ment between the analytical and numerical results for x-displacement, respect to lower layer growth g2. Here, the numerical results only deviate where displacement ux is plotted with respect to initial position x;Plot from the analytical solution when the beam is subject to large deforma- (d) shows agreement between the analytical and numerical results for tion y-displacement, where displacement u y is plotted with respect to initial respectively, where ux and u y are displacement, and x and y the initial area, will trigger division along some axis defined are coordinates in the reference coordinates (Madenci and by angle φ. At one extreme, φ may be equal to a constant Oterkus 2014). Excellent agreement between the numeri- value, which would produce highly anisotropic macroscopic cal and analytical solutions is shown in Fig. 5. The second growth, while at the other extreme φ may be uniformly ran- benchmark problem is illustrated in Fig. 5 where change in dom and produce, in an average sense, isotropic growth. curvature is induced by homogeneous quasi-static growth of In order to quantify the influence of the division angle, we the bottom half of a beam. Beam curvature ρ is computed compute an approximate deformation measure F by track- as ing the position of the nodes in our simulation. Because our block of material is growing homogeneously without any 2h ρ = (48) constraint, a single “best-fit” growth induced deformation 3g2 measure Fg can be identified to quantify the results of a sin- gle simulation. As illustrated in Fig. 6a, we define λ as the where h is the beam depth illustrated in Fig. 5b and g 0 2 set of m vectors connecting adjacent nodes at the start of the is the isotropic growth of the bottom half of the beam. simulation and λ as the set of m vectors connecting the iden- This equation was derived for a bimetal thermostat under- tical nodes at the step of the simulation in question. Then, going uniform temperature change where the upper and we construct matrices Λ and Λ as lower layers have different coefficients of thermal expansion 0 (Timoshenko 1925). In our version of the analytical equa- 1 2 3 m 1 2 3 m Λ0 =[λ λ λ ...λ ] Λ =[λ λ λ ...λ ] (49) tion, g2 replaces the term α2ΔT . Curvature with respect 0 0 0 0 to growth is plotted in Fig. 5e where there is agreement g between the numerical and analytical solutions until the and solve for the approximate version of F using the expres- large deformation regime is reached and some deviation is sion observed. g F Λ0 = Λ (50) 5.2 Isotropic and anisotropic growth as emergent properties of division angle solved through the normal equation g = ΛΛT (Λ ΛT )−1 . One major benefit of the framework presented in this paper F 0 0 0 (51) is that the influence of cell division angle can be studied. To do this, we consider a simple problem where homoge- Including a random component in division angle φ means that neous growth is applied to a block of material. When the cell simulation results will be stochastic; therefore, it is necessary division mechanism described in Sect. 4.1 is implemented, to conduct tens to hundreds of simulations to get a meaningful growth at a node exceeding a threshold, defined as double picture of the results. 123 Modeling tumor growth with peridynamics 1153 (b) (c) (a) g φ = (− π , π ) Fig. 6 We compute the growth induced deformation measure F by able, U 2 2 , the average deformation over several simulations tracking the change in position of initial particles and computing the is isotropic. Plot (b) shows a histogram and an approximate probability g g g best-fit value for F using Eq. (51). Schematic (a) shows stretch vectors density function for Fxx and Fyy computed from 500 simulations with at the initial position λ0 defined by relative position of each node, (a-i) 61 hexagonally arranged initial nodes and three division events; Plot g g shows λ when growth has occurred without division, while Schematic (c) compares the average percent difference between Fxx and Fyy for (a-ii)showsλ constructed by tracking the position of only the nodes 500 simulations with a different initial number of nodes, therefore a present at the start of the simulation, the nodes with gray area are different initial number of stretch vectors, and a different number of added when division occurs. The hexagonal initial configuration in (a) division events. As the initial number of nodes increases and/or more g g is convenient because the stretch vectors are all the same length and division events occur, the difference between Fxx and Fyy decreases distributed isotropically. When division angle is a uniform random vari- The first analysis that we conduct is designed to demon- tative plot and the qualitative illustrations of nodal position, strate that uniformly random division, defined as φ = it is clear that β controls the degree of anisotropy. A lower π π g g g U(− , ), produces an isotropic deformation measure F , value of β will correspond to a higher value of Fxx/Fyy. 2 2 g g where we define isotropic deformation as Fxx = Fyy with Overall, Fig. 7 verifies that anisotropic growth is the result g g Fxy = Fyx = 0. The results of this analysis, concisely sum- of division on a fixed angle. marizing 6000 simulations, with 500 simulations with initial node numbers of 7, 19, 37, and 61 and either one, two or 5.3 Cell division as a factor in morphogenesis three division events each, are shown in Fig. 6b, c. Every simulation has a hexagonal initial configuration like the con- As a toy example of how cell division may influence morpho- figurations plotted in Fig. 7. The plot in Fig. 6b is a histogram genesis, we define a growth law as a function of y-coordinate g g of Fxx and Fyy over 500 simulations with 61 initial nodes, with the expression 3 division events, and A = 9A at the end of the simula- ⎧ 0 ⎪ < tions. Although results vary from simulation to simulation, it ⎨0ify yL g g ( ) = F = F f y y/(yU − yL ) if yL ≤ y ≤ yU (52) is clear that on average xx yy. The plot in Fig. 6cshows ⎩⎪ that the average percentage difference between the two terms 1ifyU < y is small and decreases as both the initial node number and the number of division events increase. This indicates that where the increment of growth applied at each node at each for both large populations and populations that divide multi- time step gt is modified as gt := gt f (y). In this example, g ple times the average F converges toward isotropy. Overall, we set the lower bound yL at the base of the simulated block φ = (− π , π ) these results verify that U 2 2 produces isotropic of material and the upper bound yU just√ above the top of growth in an average sense. the simulated block with h/(yU − yL ) = 3/2 where h is Next, we examined the influence of a division angle with initial block height. The differential growth caused by this the form φ = α + U(−β,β) where α is some fixed value, variation in growth rate causes the block to curve, as illus- and φ is a function of α and some random variable scaled by trated in Fig. 8. In the upper row of the figure, the results when g g β.InFig.7, we look at the ratio Fxx/Fyy as radial growth g no cell division is occurring are illustrated. It is interesting progresses for different values of β. In-between cell division to qualitatively compare the results with random division, g g events, Fxx/Fyy is constant. Every time cell division occurs, vertical division, and horizontal division to the case where g g Fxx/Fyy steps to a value influenced by β. From the quanti- no division occurs. Qualitatively, two major differences can 123 1154 E. Lejeune, C. Linder (a) (b) Fig. 7 Our simulations show that anisotropic growth is an emer- results from 10 simulations. The vertical lines highlight each of three gent property of oriented cell division. We define division angle φ division events that occur during the simulation. The simulation results as a function of scaling parameter β and a uniform random variable: (b) illustrate node position between each division event for β = 0, φ = . + (−β,β) β = π β = π β = π 0 0 U . Then, we compare simulation results for different 4 and 2 .When 2 , growth is, in an average sense, β g / g g g values of .Plot(a)showstheratioFxx Fyy with respect to growth g isotropic with Fxx/Fyy = 1 for 10 simulations at each value of β.Thethick lines are the average (a) (b) (c) (d) Fig. 8 The orientation of cell division may also play a part in mor- the lower row (d) φ = 0.0 corresponding to a horizontal orientation. phogenesis. In these simulations, growth g is prescribed as a function Qualitatively, different cell division rules yield different curvature ρ of vertical displacement defined in Eq. (52). In the upper row (a), no and different levels of “fingering” or roughness, illustrated by a dashed cell division occurs; in the second row (b) cell division is random with line, on the upper layer of cells. A quantitative comparison of average φ = (− π , π ) φ = π U 2 2 ; In the third row (c) 2 or vertically oriented; in curvature in the lower layer is made in Fig. 9 be observed across these four cases. First, including the cell et al. 2002; Cristini 2005). Although this simple example division mechanism leads to an uneven upper surface. In par- is not particularly physical, it does lead to further questions ticular, the case where cell division occurs in the vertical because fingering and rough edges are correlated with inva- direction seems to be exhibiting some form of “fingering” sive behavior and nutrient concentration gradients may cause where certain parts of the surface protrude outward (Cristini gradients in growth rate similar to the one tested here. The 123 Modeling tumor growth with peridynamics 1155 dual-horizon peridynamics, we introduce growth into the dual-horizon peridynamics framework in Sect. 3, and we describe a method for implementing cellular scale growth mechanisms such as cell division in Sect. 4. In Sect. 5,we show numerical results that validate our approach, show how isotropic and anisotropic growth can be captured in our framework, and present a simple problem to show that considering cell division can be an important factor in mor- phogenesis. Given these results, it is also interesting to attempt to phe- nomenologically categorize the cell division mechanism. In our initial framework, cell division is triggered by applied growth and occurs in part via cell reorganization. A more complex trigger of cell division, for example, if cell divi- sion was induced by local tension, could be interpreted as Fig. 9 Average curvature of the bottom layer of nodes ρ is used as a a way for cells to reorganize to relieve stress phenomeno- metric to compare multiple simulations. In the left plot, a histogram of logically similar to plasticity. A benefit of the peridynamic μ curvature for 500 different simulations shows that mean curvature ρ framework presented here is that the cell division mechanism is only 0.05% different from curvature computed when there is no divi- sion. By this metric, isotropic division and no division are equivalent. does not required a formal phenomenological classification This plot also shows that when division is always horizontal, ρ is sig- to be implemented. nificantly lower than the no division case and when division is always As stated in Sect. 1, the physical interpretation of the dis- ρ vertical is slightly higher than the no division case cretization can view nodes as either individual cells or as a collection of cells. When nodes are viewed as individual cells, this framework can be used to interpret experiments flexibility provided by the framework in this paper is a good performed on small colonies of cells. When nodes are viewed fit for studying problems where mechanically driven instabil- as simply a means to discretize a block of tissue, this method- ity and nutrient concentration gradient driven instability may ology is suitable for understanding macroscale tumor growth. be coupled. Second, the radius of curvature of the block ρ The main purpose of peridynamics, to unify the mechan- (measured reproducibly using the nodes on the lower layer) ics of continuous and discontinuous media, can be taken appears to be influenced by the cell division mechanism. The advantage of to create multi-scale models with high levels influence of the division mechanism on ρ is presented quan- of discretization at areas of activity and change and low lev- titatively in Fig. 9. els of discretization in necrotic zones where a continuous In Fig. 9, the average radius of curvature ρ of the bot- representation is acceptable. tom layer of nodes in each simulation is used as a metric The work presented here is only the beginning of what to compare multiple simulations. We plot a histogram of ρ can be done with this framework. Moving forward, it will with results from 500 simulations with random division angle be interesting to implement and study different constitutive which shows a relatively large spread. A notable result is that laws, particularly a viscoelastic constitutive law (Linder et al. the mean value of ρ from all the simulations is only 0.05% 2011), the influence of nutrient concentration or cell-signal different from the value of ρ that corresponds to the case driven growth rates, the influence of stress during growth with no cell division, indicating that by this metric the two (Helmlinger et al. 1997), the influence of boundary con- scenarios are, on average, equivalent. We also note that a ditions, additional cellular scale mechanisms, fracture and horizontal division angle will significantly lower ρ, while reattachment of tumor cells, particularly as it may relate to a vertical division angle will slightly raise it. These quanti- cancer metastasis, coupling between cellular scale mecha- tative metrics are in line with the qualitative results, where nisms and geometric instability (Lejeune et al. 2016a, b, c), Fig. 8 clearly shows a relatively large change in curvature and numerical implementation techniques that can take corresponding to the horizontal division angle. advantage of regions of cell necrosis to increase computa- tional efficiency. And, further advances in the theoretical aspects and numerical implementation of peridynamics can 6 Conclusion be translated to improvements in our techniques (Foster et al. 2009). Fundamentally, we believe that this method is a In this paper, we present a methodology for implement- promising way to bridge scales in the study of tumor growth ing biological growth and a cell division mechanism with by allowing a convenient way to realize cellular scale mech- peridynamics. We begin in Sect. 2 with a background on anisms on the tissue scale. 123 1156 E. Lejeune, C. Linder Compliance with ethical standards Gatenby R, Gawlinski E (1996) A reaction–diffusion model of cancer invasion. Cancer Res 56(24):5745–5753 Funding This work was supported by the National Science Foundation Gillies TE, Cabernard C (2011) Cell division orientation in animals. Graduate Research Fellowship under Grant No. DGE-114747. Curr Biol 21(15):R599–R609 Helmlinger G, Netti PA, Lichtenbeld HC, Melder RJ, Jain RK (1997) Conflict of interest The authors declare that they have no conflict of Solid stress inhibits the growth of multicellular tumor spheroids. interest. Nat Biotechnol 15:778–783 Kilic B, Madenci E (2010a) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53(3):194–204 References Kilic B, Madenci E (2010b) Peridynamic theory for thermomechanical analysis. IEEE Trans Adv Packag 33(1):97–105 Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Kim Y, Stolarska MA, Othmer HG (2007) A hybrid model for tumor Int J Eng Sci 40:1297–1316 spheroid growth in vitro I: theoretical development and early Ambrosi D, Preziosi L (2008) Cell adhesion mechanisms and results. Math Models Methods Appl Sci 17:1773–1798 stress relaxation in the mechanics of tumours. Biomech Model Kuhl E, Maas R, Himpel G, Menzel A (2006) Computational modeling Mechanobiol 8(5):397–413 of arterial wall growth. Biomech Model Mechanobiol 6(5):321– Ambrosi D, Ateshian G, Arruda E, Cowin S, Dumais J, Goriely A, 331 Holzapfel G, Humphrey J, Kemkemer R, Kuhl E et al (2011) Per- Lejeune E, Javili A, Linder C (2016a) An algorithmic approach to multi- spectives on biological growth and remodeling. J Mech Phys Solids layer wrinkling. Extreme Mech Lett 7:10–17 59(4):863–883 Lejeune E, Javili A, Linder C (2016b) Understanding geometric Ambrosi D, Preziosi L, Vitale G (2012) The interplay between stress instabilities in thin films via a multi-layer model. Soft Matter and growth in solid tumors. Mech Res Commun 42:87–91 12:806–816 Araujo RP, McElwain DLS (2004) A linear-elastic model of anisotropic Lejeune E, Javili A, Weickenmeier JE, Kuhl LC (2016c) Tri-layer tumour growth. Eur J Appl Math 15(3):365–384 wrinkling as a mechanism for anchoring center initiation in the Bellomo N, Li NK, Maini PK (2008) On the foundations of cancer developing cerebellum. Soft Matter 12:5613–5620 modelling: selected topics, speculations, and perspectives. Math Linder C, Tkachuk M, Miehe C (2011) A micromechanically motivated Models Methods Appl Sci 18(4):593–646 diffusion-based transient network model and its incorporation into Bobaru F, Ha YD (2011) Adaptive refinement and multiscale modeling finite rubber viscoelasticity. J Mech Phys Solids 59(10):2134–2156 in 2D peridynamics. J Multiscale Comput Eng 9(6):635–659 Littlewood D (2015) Roadmap for peridynamic software implementa- Bobaru F, Yang M, Alves LF, Silling SA, Askari E, Xu J (2009) Conver- tion. SAND Report, Aandia National Laboratories, Albuquerque, gence, adaptive refinement, and scaling in 1D peridynamics. Int J NM and Livermore, CA Numer Meth Eng 77(6):852–877 Lowengrub JS, Frieboes HB, Jin F, Chuang YL, Li X, Macklin P, Wise Byrne H (2003) Modelling solid tumour growth using the theory of SM, Cristini V (2010) Nonlinear modelling of cancer: bridging the mixtures. Math Med Biol 20(4):341–366 gap between cells and tumours. Nonlinearity 23:R1–R91 Byrne H, Drasdo D (2008) Individual-based and continuum models of Madenci E, Oterkus E (2014) Peridynamic theory and its applications. growing cell populations: a comparison. J Math Biol 58(4–5):657– Springer, NewYork 687 Mitchell JA (2011) A nonlocal, ordinary, state-based plasticity model Clatz O, Sermesant M, Bondiau PY, Delingette H, Warfield SK, for peridynamics. SAND report 3166 Malandain G, Ayache N (2005) Realistic simulation of the Norton KA, Wininger M, Bhanot G, Ganesan S, Barnard N, Shinbrot T 3-D growth of brain tumors in MR images coupling diffu- (2010) A 2d mechanistic model of breast ductal carcinoma in situ sion with biomechanical deformation. IEEE Trans Med Imaging (DCIS) morphology and progression. J Theor Biol 263(4):393– 24(10):1334–1346 406 Cristini V (2005) Morphologic instability and cancer invasion. Clin Poste G, Fidler I (1980) The pathogenesis of cancer metastasis. Nature Cancer Res 11(19):6772–6779 283(5743):139–146 Cristini V, Lowengrub J, Nie Q (2002) Nonlinear simulation of tumor Preziosi L, Tosin A (2008) Multiphase modelling of tumour growth and growth. J Math Biol 46:191–224 extracellular matrix interaction: mathematical tools and applica- Deisboeck TS, Wang Z, Macklin P, Cristini V (2011) Multiscale cancer tions. J Math Biol 58(4–5):625–656 modeling. Annu Rev Biomed Eng 13:127–155 Ren H, Zhuang X, Cai Y,Rabczuk T (2016) Dual-horizon peridynamics. Deng Q, Chen Y, Lee J (2008) An investigation of the microscopic Int J Numer Meth Eng 108(12):1451–1476 mechanism of fracture and healing processes in cortical bone. Int Sandersius SA, Newman TJ (2008) Modeling cell rheology with the J Damage Mech 18(5):491–502 subcellular element model. Phys Biol 5(1):015002 Drasdo D, Höhme S (2005) A single-cell-based model of tumor growth Silling S, Askari E (2005) A meshfree method based on the peridynamic in vitro: monolayers and spheroids. Phys Biol 2(3):133–147 model of solid mechanics. Comput Struct 83(17–18):1526–1535 Drasdo D, Höhme S, Block M (2007) On the role of physics in the Silling SA (2000) Reformulation of elasticity theory for discontinuities growth and pattern formation of multi-cellular systems: What can and long-range forces. J Mech Phys Solids 48:175–209 we learn from individual-cell based models? J Stat Phys 128(1– Silling SA, Lehoucq RB (2010) Peridynamic theory of solid mechanics. 2):287–345 Adv Appl Mech 44:73–168 Foster JT, Silling SA, Chen WW (2009) Viscoplasticity using peridy- Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic namics. Int J Numer Methods Eng 81:1242–1258 states and constitutive modeling. J Elast 88(2):151–184 Frieboes HB, Lowengrub JS, Wise S, Zheng X, Macklin P, Bearer EL, Stolarska MA, Kim Y, Othmer HG (2009) Multi-scale models of cell Cristini V (2007) Computer simulation of glioma growth and mor- and tissue dynamics. Philos Trans R Soc A Math Phys Eng Sci phology. Neuroimage 37:S59–S70 367(1902):3525–3553 123 Modeling tumor growth with peridynamics 1157 Timoshenko S (1925) Analysis of bi-metal thermostats. JOSA Warren TL, Silling SA, Askari A, Weckner O, Epton MA, Xu J (2009) 11(3):233–255 A non-ordinary state-based peridynamic method to model solid Wang Z, Butner J, Kerketta R, Cristini V,Deisboeck TS (2015) Simulat- material deformation and fracture. Int J Solids Struct 46(5):1186– ing cancer growth with multiscale agent-based modeling. Semin 1195 Cancer Biol 30:70–78 123