IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 1

Multi-level computational modeling andquantitative analysis of bone remodeling

Nicola Paoletti, Pietro Lio, Emanuela Merelli and Marco Viceconti

Abstract —Our work focuses on bone remodeling with a multiscale breadth that ranges from modeling intracellular andintercellular RANK/RANKL signaling to tissue dynamics, by developing a multi-level modeling framework. Several importantfindings provide clear evidences of the multiscale properties of bone formation and of the links between RANK/RANKL and bonedensity in healthy and disease conditions. Recent studies indicate that the circulating levels of OPG and RANKL are inverselyrelated to bone turnover and bone mineral density and contribute to the development of osteoporosis in post-menopausal women,and thalassemic patients. We make use of a spatial process algebra, the Shape Calculus, to control stochastic cell agents thatare continuously remodeling the bone. We found that our description is effective for such a multiscale, multi-level process andthat RANKL signaling small dynamic concentration defects are greatly amplified by the continuous alternation of absorption andformation resulting in large structural bone defects. This work contributes to the computational modeling of complex systems witha multi-level approach connecting formal languages and agent-based simulation tools.

Index Terms —Osteoporosis, multi-level, Shape Calculus, bone remodeling, multiscale, RANK/RANKL, agent-based simulation

✦

1 INTRODUCTION

O STEOPOROSIS is a skeletal disease characterizedby low bone density and structural fragility,

which consequently leads to frequent micro-damagesand spontaneous fractures. This disease affects pri-marily middle-aged women and elderly people andat present its social and economic impact is dra-matically increasing, so much that the World HealthOrganization considers it to be the second-leadinghealth-care problem. Osteoporosis and many otherbone pathologies are attributable to disorders in themechanism of bone remodeling (BR), the process bywhich aged bone is continuously renewed in a ba-lanced alternation of bone resorption and formation.Bone remodeling plays an essential role in repairingmicro-damages, in maintaining mineral homeostasisand in the structural adaptation of bone in responseto mechanical stress. In healthy conditions the balancebetween the resorption phase, performed by cellscalled osteoclasts, and the formation phase, performedby osteoblasts, ensures the “good quality” of the bone.On the contrary, several diseases can be ascribedto the imbalance between resorption and formation:osteoporosis is an example of negative remodelingthat is, when the resorption process prevails on the

• N. Paoletti and E. Merelli are with the School of Science and Technol-ogy, University of Camerino, Italy.Email: name.surname@unicam.it

• P. Lio is with the Computer Laboratory, University of Cambridge,United Kingdom.Email: pl219@cam.ac.uk

• M. Viceconti is with the Department of Mechanical Engineering,University of Sheffield, United Kingdom.Email: m.viceconti@sheffield.ac.uk

formation one.An important factor affecting bone metabolism

is RANK/RANKL/OPG signaling, depicted in Fig. 1.RANK is a protein expressed by osteoclasts; RANKis a receptor for RANKL, a protein produced by os-teoblasts. RANK/RANKL signaling triggers osteoclastdifferentiation, proliferation and activation, thus itprominently affects the resorption phase during boneremodeling. Osteoprotegerin (OPG) is a decoy recep-tor for RANKL. It is expressed by mature osteoblastsand it binds with RANKL, thus inhibiting the produc-tion of osteoclasts. Several important findings provideclear evidence of the multiscale properties of boneformation and of the links between RANK/RANKLand bone density in healthy and disease conditions.However, there is a lack of knowledge of the ge-netic and environmental factors responsible for ageand gender specific differences in bone fragility andfracture rates. Recent studies indicate that the cir-culating levels of RANKL are inversely related tobone turnover and Bone Mineral Density (BMD) andcontribute to the development of osteoporosis in post-menopausal women [1], and of thalassemia-inducedosteoporosis [2].This paper focuses on modeling the multiscale

dynamics of bone remodeling, showing that smallchanges in RANKL concentration at molecular levelcan lead to significant disruptions at tissue level, andto important pathologies like osteoporosis. In parti-cular we are interested in assessing the emergenceof osteoporosis in older patients, that are typicallycharacterized by a reduced cellular activity, in termsof lower bone formation and resorption rates; lowergrowth rates; and higher death rates. Two differentclasses of patients are compared:

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 2

Fig. 1. RANK/RANKL/OPG pathway in bone remod-eling involving osteoclasts (Oc), osteoblasts (Ob), andtheir precursors (POc and POb, resp.). RANK/RANKLsignaling triggers osteoclasts’ differentiation and prolif-eration. RANKL/OPG inhibits osteoclasts’ recruitmentand induces osteoblasts’ maturation. The processspans different scales: molecular, cellular and tissue.

• healthy patients, with regular RANKL levels andcellular activity, and

• osteoporotic patients, with an overproduction ofRANKL and a reduced cellular activity.

From a methodological point of view, we define amulti-level modeling framework (depicted in Fig. 2)combining a high-level process-algebraic specifica-tion, and a low-level stochastic agent-based simula-tion [3], [4]. The specification language is the ShapeCalculus, a spatial process algebra for describing bio-logical systems [5], [6] and more generally, suitable forsystems characterized by a strong interplay betweenphysical and computational features, also known ascyberphysical systems. We show how the Shape Calcu-lus allows us to express in a uniform and straightfor-ward way the multiscale nature of bone remodelinginvolving the molecular, the cellular and the tissuelevel. While we aim to keep the specification languageas generic as possible, the agent-based model is re-fined with additional features such as stochasticity, en-riching agents’ actions with stochastic rates modelingthe propensity of the action; and perception, that is thecapability to communicate at distance and sense theneighborhood. The agent-based level provides an ex-ecutable implementation of the process-algebraic one,and it can be formally derived from the specificationby applying a set of translation functions. This levelenables the analysis of qualitative and quantitativeresults from the simulation of the model such astabular data, plots, and 3D views of the agents andtheir environment.Our work gives new biological insights into the

-Agent based systemSimulation and analysis of execution traces

Shape CalculusFormal specification

SOc[B ]Oc SOb[B ]Ob

SOy[B ]Oy

Translation

Fig. 2. Multilevel modeling framework. The ShapeCalculus algebraic specification is translated into anexecutable agent-based model. In the upper box, bonecells are drawn as Shape Calculus processes. Thelower box shows a snapshot of the simulated system.

bone remodeling process, regarded as a multiscalecomplex biological system [7], and in particular intothe link between imbalances in the molecular con-centration of RANKL and the occurrence of defectivebone pathologies. We show that the combination ofhigher RANKL levels with a lower cellular activityleads to negative remodeling phenomena and afterseveral remodeling cycles, to severe bone loss, micro-structural degradation and osteoporosis. In addition,the proposed computational framework couples a for-mal specification level with an agent-based simulationlevel. The former provides a better expressivenessof the BR process, from the intercellular signalingto tissue dynamics. The latter provides a simulationenvironment for Shape Calculus specifications anda diagnostic tool for comparing different classes ofpatients, and for analyzing the bone mass and struc-ture; the individual evolution of bone cells; and theconcentration levels of RANKL. We believe that theproposed framework could help in understanding themultiscale dynamics of bone remodeling and possiblycontribute to the application of predictive compu-tational models as diagnostic tools in personalizedmedicine and in everyday clinical practice.

The paper is organized as follows. Section 2 intro-duces the process of bone remodeling and how it isrelated to bone pathologies. In Section 3 we presentthe high-level specification of bone remodeling inthe language of the Shape Calculus. In Section 4we describe the stochastic agent-based simulator. InSection 5, we analyze and validate the results ofthe simulation, comparing healthy and pathologicalscenarios. Conclusions are given in Section 6.

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 3

2 DYNAMICS OF BONE REMODELING

Bone remodeling is a process which iterates through-out life and it consists of two main phases: boneresorption and bone formation. Bone homeostasis isone of the most important requirements in the BR sys-tem, since pathologies typically arise when resorptionand formation are not well-balanced: osteoporosis isan example of negative remodeling where resorptionprevails on formation. In this situation even smallnegative changes in bone density are greatly amplifiedafter several remodeling iterations. The BR systemconsists of multiple dynamic and interacting com-ponents, the bone cells. Osteoclasts (“the diggers”)and osteoblasts (“the fillers”), form together the so-called Basic Multi-cellular Units (BMUs). Another typeof cells, the osteocytes, plays a relevant role in theremodeling process. They are located in the bonematrix and are interconnected through the canalicularnetwork. Osteocytes control the BMU activity, servingas mechanosensors: load-induced signals from thetissue level are transmitted at cellular level to activatebone remodeling. This process is generally referredas mechanotransduction. Therefore osteocytes’ signalingand RANK/RANKL/OPG signaling introduced inSection 1 represent the main communication protocolsbetween the components of the BR system.

There are two main types of bone: compact (orcortical) tissue that forms the outer shell of bones,consisting of a very hard mass of bony tissue arrangedin concentric layers (Haversian systems); trabecular(also known as cancellous or “spongy”) tissue thatis located beneath the compact bone and consistsof a meshwork of bony bars (trabeculae) with manyinterconnecting spaces containing bone marrow. Thekey events occurring during a regular BR cycle are:

• Origination.After a micro-crack, or as a responseto mechanical stress, the osteocytes in the bonematrix start producing biochemical signals to-wards the surface cells around the bone, termedlining cells. The lining cells pull away from thebone matrix, forming a canopy which mergeswith the blood vessels.

• Osteoclast recruitment. Stromal cells divide anddifferentiate into osteoblasts precursors. Pre-osteoblasts start to express RANKL, attractingpre-osteoclasts, which have RANK receptors ontheir surfaces. RANK/RANKL signaling triggerspre-osteoclasts’ proliferation and differentiation.

• Resorption. The pre-osteoclasts enlarge and fuseinto mature osteoclasts. In cortical BMUs, os-teoclasts excavate cylindrical tunnels in the pre-dominant loading direction of the bone, whilein trabecular bone they act at the bone surface,digging a trench rather than a tunnel. After theresorption process has terminated, osteoclasts un-dergo apoptosis.

• Osteoblast recruitment. Pre-osteoblasts mature

into osteoblasts and start producing OPG. OPGinhibits RANKL expression, and consequentlyprotects bone from excessive resorption since itavoids other osteoclasts from being recruited byRANK/RANKL binding.

• Formation. Osteoblasts fill the cavity by secretinglayers of osteoid. Once the complete mineral-ization of the renewed tissue is reached, someosteoblasts can go into apoptosis, while other canturn into lining cells or remain trapped in thebone matrix and become osteocytes.

• Resting. The damage has been repaired and theinitial situation is re-established.

The RANK/RANKL/OPG pathway (Fig. 1) promi-nently affects the outcome of the BR process. In-deed, RANK/RANKL signaling induces osteoclasts’differentiation and proliferation, so that higher con-centrations of RANKL lead to a higher productionof osteoclasts and in turn to a higher resorptionactivity. In principle an overproduction of RANKL,which is indicative of an inflammation or of hormonalimbalance, does not directly give rise to osteoporosis:in young patients an unexpected high resorption ac-tivity can be well-balanced by a proper number ofosteoblasts. However, in older people cellular activ-ity is typically less effective in terms of resorptionand formation rates. When combining high RANKLvalues with aging factors, a higher resorption activitypromoted by RANKL cannot be balanced by the avail-able osteoblasts. This configuration would give riseto negative remodeling phenomena and after severalremodeling cycles during which the bone mass keepsconstantly decreasing, to defective bone diseases likeosteoporosis.

2.1 Related work

In the last twenty years, a variety of mathematicalmodels has been proposed in order to better under-stand the dynamics of bone remodeling (reviewedin [8], [9], [10]). Early models were focused on theorgan level, and they typically describe bone as acontinuum material only characterized by its den-sity. Bone micro-structure is not taken into account,thus ignoring structural adaptation mechanisms aswell as complex cellular-level dynamics. Therefore incontinuum-based organ-level models the differencesbetween cortical and trabecular bone were just in theirapparent density. In order to address the lack of micro-structural information, later models started looking atthe biomechanical properties of the bone tissue thatis, how mechanical loading affects the tissue structureand consequently its function. Models describing themechanical response and adaptation of bone typicallyrely on Finite Element Methods (FEM) [11], a techniqueborrowed from the engineering community. In thiscase, bone adaptation is often modeled as a structuraloptimization problem: a FEM analysis computes the

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 4

mechanical stresses at each location of the bone ma-trix; then, structural changes are driven in order tooptimize the homogeneous distribution of mechanicalloading, until a remodeling equilibrium is reached.Another class of models describe the single-cellularand multi-cellular level and the interactions occurringamong the different types of bone cells involved inthe BR process. Most of these models analyze thecontinuous variations in the number of bone cells, andthe bone density is usually calculated as a function ofthe number of osteoblasts and osteoclasts and theirformation and resorption rates. These models canalso incorporate intracellular signaling pathways andmechanosensing mechanisms, that is the process bywhich mechanical stimuli are transduced into cellularsignals.Despite most of the current efforts in the modeling

of the BR process rely on continuous mathematics (i.e.differential equations models), in this work we exploitcomputational models, regarding the bone remodel-ing system as a reactive concurrent system [12]. Bonecells are modeled as processes that continuously reactwith each other by emitting and sensing molecularsignals (RANK/RANKL/OPG signaling), and concur-rently operate and access to shared resources (i.e. thebone).Computational bone remodeling is a research field

of increasing interest and there are several recentworks worth mentioning. In [13] the mechanism ofbone remodeling inspires a structural optimizationalgorithm which combines a FEM analysis with Cel-lular Automata; [14] defines a Petri Net model for BR;in [15] a particular class of membrane systems areused to describe the BMU dynamics; finally in [16]probabilistic verification techniques are employed inorder to assess bone pathologies over a stochasticmodel of bone remodeling.In the next section we will discuss the importance

of capturing spatial and physical features of biologicalsystems, and we will provide a computational modelfor bone remodeling, described in the process alge-braic language of the Shape Calculus.

3 BONE REMODELING IN THE SHAPECALCULUS

In this section we firstly introduce the Shape Calcu-lus [5], [6] a spatial process calculus for modelingbiological and cyberphysical systems, in which theclassical notion of process is enriched with featureslike mass, position, velocity, shape, collision, binding,and splitting. Secondly we give a high-level formaldescription of the spatial and multiscale dynamics ofbone remodeling, by employing the Shape Calculusas a specification language.Current research trends in cellular biology are giv-

ing a crucial role to spatial organization and loca-tion, in other words to the spatial cell biology. Indeed

spatial organization and location prominently affectthe behavior, the development and the evolution ofbiological entities: they occupy a certain space; theymove and interact with their spatial neighbors; theyexpress signals which propagate within a particulardistance; and their internal spatial organization andtheir shape regulates their functionalities. The dynam-ics of bone remodeling exhibits spatial and geometri-cal properties. For instance, the amount of resorbedbone depends on the surface extension of osteoclasts,which have ruffled borders functional to resorption;osteocytes sense micro-fractures due to the apoptosisof the cells close to it; and abnormalities in bone sizeand shape may underlie pathological conditions.Many computational formalisms equipped with

spatial capabilities have been proposed so far (see [17]for an exhaustive review). In the field of processalgebras, compartmentalization is a common way torepresent process localization [18], [19], [20]. Recentworks address the problem of expressing also morecomplex spatial features [21], [22], [23], by consider-ing physical information, position and movement indiscrete or continuous coordinate systems, geometryand geometrical transformations. The Shape Calculusfeatures a rich set of spatial and physical primitivesand being particularly suitable for describing cellularmechanisms as well as molecular signaling, it couldhelp in giving a deeper understanding of the spatialaspects involved in the process of bone remodeling.

3.1 Overview of the calculus

The building blocks of the Shape Calculus are the so-called 3D processes which move, collide and interactin the 3D space. A 3D process is characterized by aninternal behavior B specified in a timed variant of CCS(Calculus of Communicating Systems) [24]; and by a 3Dshape S, that defines the geometry and the physicalproperties of the process. S[B] denotes a 3D processwith behavior B and shape S. As in [3], [4], the syntaxof the calculus, given in Table 3.1, includes additionalterms for expressing repeating behaviors (iteration),removal of a 3D shape (thanatos), and time-boundedbehaviors (duration).The set S of 3D shapes is composed of basic shapes

σ, defined as a tuple σ = 〈V,m,p,v〉 (geometry,mass, position, velocity); and by compound shapesof the form S1〈X〉S2, resulting from the binding oftwo 3D shapes S1 and S2 on a common surface X .Shapes move according to their velocities that aredetermined by a general motion law - for instanceas in a force field or Brownian motion - and by thecollisions occurring among shapes.The internal behavior of a 3D process supports bind

actions, with which processes can bind together form-ing a compound new process; and split actions forbreaking bonds. The binding between 3D processesis modeled in a similar way to CCS communication

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 5

S ::= 3D shape (S-term)

σ = 〈V,m,p,v〉 Basic shape

S〈X〉S Composed shape

B ::= Behavior (B-term)

nil Null behavior

〈α,X〉.B Bind

ω(α,X).B Weak split

ρ(L).B Strong split

ǫ(t).B Delay

B +B Choice

(B)k.B Iteration

Θ.B Thanatos

δ(t,B).B Duration

K Process name

P ::= 3D process (3DP-term)

S[B] S ∈ S, B ∈ B

P 〈a,X〉P Composed 3D process

N ::= 3D network (N-term)

nil Empty network

P P ∈ 3DP

N‖N Parallel 3D processes

TABLE 1Shape Calculus syntax

on complementary channels; in the Shape Calculus,channels act as binding sites and are of the form〈α,X〉, where α is a channel name, and X is theactive surface. The set of channels is denoted with C.Binding between two processes occurs if they exposecompatible channels. Two channels c1 = 〈α,X〉, c2 =〈β, Y 〉 ∈ C are said compatible (written c1 ∼ c2), ifthey have complementary channel names: α = β; andthey share a common surface of contact: X ∩ Y 6= ∅.In addition, the Shape Calculus supports two basicactions for splitting an established bond: the weak-split ω(α,X) (not-urgent, can be postponed) and thestrong-split ρ(L), L ⊆ C (urgent, must be performed assoon as it is enabled). The sets of weak-split actionsand strong-split actions are denoted with ω(C) ={ω(α,X)|〈α,X〉 ∈ C}, and ρ(C) = {ρ(α,X)|〈α,X〉 ∈C}, respectively.The term (B)k is a syntactical construct for express-

ing behaviors repeating for a finite (Bn, n ∈ N) orinfinite (B∞) number of iterations. The behavioralterm Θ, or Thanatos, removes a 3D shape from thenetwork. Even when the behavior B of a 3D processS[B] reaches the termination state nil, the associatedshape S would keep existing, moving and (elastically)colliding. By performing a Θ before the behavior

termination (Bdef= B1.Θ.B2), we ensure that the asso-

ciated 3D shape will not affect the other processes thatare concurrently executing. The term δ(t, B) calledbehavior duration allows us to express time-bounded

behaviors, for instance an osteoblast that continuouslymineralizes bone until its life time has passed. It hasthe following meaning: while t > 0, δ(t, B) ≡ B; oncet = 0, δ(t, B) ≡ nil.The set 3DP of 3D processes is composed of basic

processes of the form S[B], which are characterizedby a behavior B ∈ B encapsulated in a 3D shapeS ∈ S; and by composed processes P1〈a,X〉P2, whereP1, P2 ∈ 3DP, a is the name of the two compatiblechannels, and X is the common surface of contact.Finally, we define a network of 3D processes (3D

network, for short) as the parallel composition of zeroor more 3D processes. Given a finite set of indexesI = {1, . . . , n}, we can alternatively write (‖Pi)i∈I or(‖Pi)

ni=1, with Pi ∈ 3DP, to denote the network that

consists of all Pi with i ∈ I : P1‖ . . . ‖Pn.In detail, the semantics of the calculus is given in

Appendix A. The language presented in this workincludes some new features with respect to the orig-inal formulation of the Shape Calculus. In this waywe substantially improve the expressiveness of thecalculus, with terms of practical usefulness (Iteration)and terms driven by biological evidence (Thanatosand Duration). We refer the interested reader alsoto papers [5], [6] for a broader view of the originalformulation of the calculus.

3.1.1 A key applicationIn this part we give an idea of the semantics ofthe Shape Calculus through the example in Fig. 3:the OPG inhibition of RANKL, also discussed in theprevious sections.Consider two 3D processes: a pre-osteoblasts

Pbdef= SOb[BPb], and a mature osteoblast Ob

def=

SOb[BOPG], where BPbdef= 〈rankl,X1〉.ρ(rankl,X1).BOPG;

BOPGdef= 〈rankl,X2〉.ρ(rankl,X2).BOb; and BOb is the

behavior of an active osteoblast which formsnew bone. Pb expresses RANKL by exposingthe output channel 〈rankl,X1〉; Ob expresses thedecoy receptor OPG, here implemented as theinput channel 〈rankl,X2〉. We assume that theexposed surfaces X1 and X2 correspond to the wholesurface of SOb. If X1 and X2 come into contact, i.e.X1 ∩ X2 6= ∅, the channels 〈rankl,X1〉 and 〈rankl,X2〉will become compatible. Consequently, the twoprocesses bind together mimicking the RANKL/OPGbinding, and the resulting compound process isSOb[ρ(rankl,X1).BOPG]〈rankl,Y〉SOb[ρ(rankl,X2).BOb],where Y = X1 ∩ X2. At this point, both processescan perform a split on the established bond,

written Pbρ(rankl,X1)−−−−−−→ and Ob

ρ(rankl,X2)−−−−−−→. Whenthe split is fired, the resulting 3D network willbe SOb[BOPG] ‖ SOb[BOb]. In a few words theabove described model of RANKL/OPG bindingimplements a reaction from a pair (pre-osteoblast,mature osteoblast) to a pair (mature osteoblast, activeosteoblast).

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 6

<rankl,X1>

�(rankl,X2)

SOb

SOb

SOb

SOb SOb

BOb<rankl,X2>

SOb

<rankl,X2>

<rankl,Y>

SOb[〈rankl,X1〉.ρ(rankl,X1).BOPG] ‖ SOb[〈rankl,X2〉.ρ(rankl,X2).BOb]↓

SOb[ρ(rankl,X1).BOPG]〈rankl,Y〉SOb[ρ(rankl,X2).BOb]↓

SOb[BOPG] ‖ SOb[BOb]

Fig. 3. RANKL/OPG signaling in the Shape Calculus.The pre-osteoblast expresses RANKL (〈rankl,X1〉) andthe mature osteoblast expresses OPG (〈rankl,X2〉). Bybinding and splitting on 〈rankl, ·〉, the mature osteoblastinhibits the pre-osteoclast which subsequently startsproducing OPG as well. The example models a re-action (pre-osteoblast, mature osteoblast) → (matureosteoblast, active osteoblast).

3.2 Formal specification of bone remodeling

In Table 2, the high-level Shape Calculus specifi-cation of bone remodeling is presented. Employingthe Shape Calculus as a specification language has atwofold benefit:

• Multiscale expressiveness. The algebraic approachis able to span different scales in a compositionalway: a BMU is defined in terms of a networkof bone cells, and the bone tissue in terms of anetwork of BMUs.

• Biological meaningfulness. The specification followsthe paradigm of Processes as cells and Channelsas Molecular Signals. As a matter of fact, cellscommunicate with each other via direct contact(or over short distances). Similarly in the ShapeCalculus, 3D processes communicate by bindingon a common surface of contact. FurthermoreRANK, RANKL and OPG are surface-boundproteins and consequently can be faithfully de-scribed by Shape Calculus channels of the form〈a,X〉, where X is a portion of the surface of theassociated process (i.e. a cell).

In this context, the model does not define particularbinding surfaces for the channels exposed by the 3Dprocesses, but we assume that their whole surface isavailable for binding for two main reasons. Firstly,there is no clear experimental evidence of the posi-tioning of the RANK and RANKL (implemented asShape Calculus channels) molecules at the cell surface.Secondly, the definition of a precise geometry formolecular channels would have restricted the bindingcapabilities of the bone cells, without any biologicalmotivation. Let B(S) be the boundaries of a shape S;for the sake of succinctness, in our specification wewill denote the channel 〈α,B(S)〉 with 〈α,X〉.In addition there are two particular actions not

encoded as behavioral terms, resorb and form. They

are timed action responsible for decreasing/increasingthe density of the part of bone which the osteo-clast/osteoblast is attached to, according to deter-mined resorption/formation rates. These two termshave been left unimplemented at this level in order toavoid unnecessary and inaccurate modeling artifacts.Indeed it would be quite inopportune to express thebone mass as a 3D process, and moreover the activi-ties of resorption and formation cannot be properlydescribed by a binding operation. The quantitativeparameters of the model are summarized in Table 3.

4 STOCHASTIC AGENT-BASED SIMULATOR

Traditionally, Systems Biology has been described byusing continuous deterministic mathematical modelslike ODEs and PDEs. A growing amount of experi-mental results is nowadays showing that biochemicalkinetics at the single-cell level are intrinsically stochas-tic, suggesting that stochastic models are more effec-tive in capturing the multiple sources of heterogeneityneeded for modeling a biological dynamical systemin a realistic way. In particular biological systemshave also developed strategies for both exploiting andsuppressing biological noise and heterogeneity.Similarly the process of bone remodeling exhibits

stochastic features. The statistical fluctuations inRANKL concentrations in the blood micro-circulationproduce changes in the chemotaxis, i.e. the processby which cells move toward attractant molecules,of osteoclasts and osteoblasts (for example, the celldifferentiation, number and arrival time). Other im-portant sources of variability are related to the timeand space organization of the BMU; the availabilityof molecules required for building the bone; the regu-latory feedback control and the mechanical boundaryconditions [25]. In this part we address the need forsuch quantitative information in order to enrich thequalitative description of bone remodeling providedwith the Shape Calculus in Section 3. So far fewefforts have been made in characterizing the stochasticdynamics of bone remodeling, and we have recentlyinvestigated this research direction in [16], wherea stochastic model for cellular bone remodeling ispresented and probabilistic verification techniques areemployed to assess particular bone pathologies.In this section we describe the implementation level

of our modeling framework (as introduced in Sec-tion 1), that is an agent-based simulator for Shape Cal-culus specifications featuring stochastic actions andperception. In particular we have developed a libraryto support the definition of agents as 3D processes,providing a rigorous translation of Shape Calculusspecifications into agent code (see Appendix B forthe details on the translation function). Parameters forthe Shape Calculus model and for the simulator arelisted in Table 3. Table 4 shows an example wherethe algebraic specification of an osteocyte is translatedinto agent code.

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 7

Tissue, BMU

T issuedef= (‖ABMUi)ai=1

‖ (‖QBMUj)qj=1

BMUadef= (‖Oyi)

nOy

i=1‖ (‖Ocj)

nOcj=1

‖ (‖Obk)nObk=1

BMUqdef= (‖Oyi)

nOy

i=1

Bone tissue is structured in a active BMUs (BMUa) partic-ipating in the remodeling process, and in q quiescent BMUs(BMUq). Each active BMU is in turn composed by nOy osteo-cytes, nOc osteoclasts and nOb osteoblasts. An inactive BMU ismodeled as a network of only osteocytes.

Osteocyte

Oydef= SOy[(〈can,X〉+ 〈can,X〉)kOy + 〈oySig,X〉.Θ]

An osteocyte can bind with other kOy osteocytes through thechannel 〈can, ·〉 and form the network of canaliculi. Alterna-tively, if they are not near enough to communicate and bind witheach other, e.g. osteocytes near the bone surface, they expose thechannel 〈oySig, ·〉 modeling the osteocytes’ signaling that willactivate the resorption phase and the remodeling process. Afterhaving performed a bind on 〈oySig, ·〉, the osteocyte dies sinceit has been consumed by the attached osteoclast.

Osteoclast

Ocdef= SOc[BOc]

BOcdef= δ(tOc ,BAOc).Θ.〈deathF,X〉kOc

BAOcdef= (〈oySig,X〉+ 〈rankl,X〉.ρ(rankl,X)).resorb

During its lifetime tOc, an osteoclast behaves as an activeosteoclast (BAOc): it continuously absorbs bone, induced byosteocytes’ signaling (channel 〈oySig, ·〉) or by RANKL (channel〈rankl, ·〉). Before dying, it releases death factors that will attractosteoblasts to the consumed part of bone, thus triggering theformation phase. In particular, a single “dead” osteoclast canbind with kOc = ⌊nOb/nOc⌋ osteoblasts, so fitting the ratiobetween active osteoclasts and active osteoblasts.

Osteoblast

Obdef= SOb[〈rankl,X〉.ρ(rankl,X).BOPG + ǫ(tPb).BOPG]

BOPGdef= 〈rankl,X〉.ρ(rankl,X).BOb + BOb

BObdef= δ(tOb , 〈deathF, X〉.form).Θ

An osteoblast initially behaves as a non differentiated cell:it produces RANKL, by exposing channel 〈rankl, ·〉. After theeffect of OPG-inhibition or after its differentiation time tPbhas elapsed, it starts behaving as a mature osteoblast whichproduces OPG. In particular, a mature osteoblast can inhibit asingle precursor, by binding on the channel 〈rankl, ·〉 (standingfor the OPG decoy receptor) (Fig. 3). Then, the formation phaselasts a time tOb, after which the cell undergoes apoptosis.

TABLE 2Shape Calculus specification and biological

description of bone remodeling.

Shape Calculus-based agents have associated twofurther features: stochasticity, that enriches agents’ ac-tions with stochastic rates modeling the propensity ofthe action itself; and perception, that is the capability tocommunicate at distance and sense the neighborhood.

Stochasticity. We follow the so-called rated out-put/passive input approach [28], where agent actions areequipped with stochastic rates and weights. In parti-cular, output actions are annotated with a rate λ ∈ R

+,while input actions have associated a weight w ∈ N+,

Param Value Description

nOc 20 [26], [27] Expected numberof Oc

nOb 2000 [26], [27] Expected numberof Ob

tOc 10 days [27] Oc lifetime

tOb 15 days [27] Mature Ob life-time

tPb 5 days [27] Ob differentiationtime

sizeBMU 2.4×1.6×0.01 mm3 [27] Size of the BMU

∆t 1 day Time step

kOc 2.5 days−1 Resorption rate

kOb 0.25 days−1 Formation rate

kRANKL ∈ [1, 2] RANKL factor

kaging ∈ [1, 2] Aging factor

dOc−Oy 0.06 mm Oc perception ra-dius to Oy

dPb−Oc 0.01 mm Pb perception toOc death factor

dOb−Pb 0.01 mm Ob/OPGperception toPb/RANKL

DOc 10−2 mm2/day Oc diffusion coef-ficent

DOb 2.15 × 10−2 mm2/day Ob diffusion coef-ficent

TABLE 3Model parameters. Size of the BMU, lifetime and

expected concentration of bone cells have been takenfrom literature. Other parameters have been estimated

from the model (see Appendix C).

Oydef= SOy[(〈can,X1〉+ 〈can,X2〉)

kOy + 〈oySig,X3〉.Θ],

SOy = 〈VOy, mOy,pOy,vOy〉

↓

Channel c1 = new InChannel("can", X_1, prob);

Channel c2 = new OutChannel("can", X_2, rate1, d1);

Channel c3 = new OutChannel("oySig", X_3, rate2, d2);

Behaviour b1, b2, b3, b4;

b1 = new Choice(new Bind(c1), new Bind(c2));

b2 = new Iteration(b1, k_Oy);

b3 = new Sequence(new Bind(c3), new Thanatos());

b4 = new Choice(b2,b3);

Shape oy = new BasicShape(m_Oy, p_Oy, v_Oy, b4);

TABLE 4Example of translation from a Shape Calculus term

describing an osteocyte into agent code.

representing the probability that the specific input isselected when a compatible output action is fired.Consider the system in Figure 4, where two agentsA and B are illustrated. Agent A can perform anoutput action a with rate λ; while B can performthe input actions (a, w1) and (a, w2). The relative

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 8

A B

a w1 a w2a λ, , ,

Fig. 4. Agents synchronizing on compatible channelswith stochastic rate λ and probability wi.

selection probability of action (a, wi), called p(a, wi), iscalculated as the ratio between wi and the total weightof a, that is the sum of the weights of all the enableda-actions; in this example,

p(a, wi) =wi

w1 + w2, i = 1, 2.

Then, the synchronization rate between the actions(a, wi) and (a, λ) is given by λ · p(a, ωi).Perception.Differently from the theory of the Shape

Calculus where binding between two 3D processescan exclusively occur when they share a common sur-face of contact, we allow compatible agents to bind atdistance, according to a given perception radius. Moreprecisely, output bind actions are annotated with aparameter d ∈ R+ called sensibility distance, meaningthat if an agent A can perform an action (a, λ)d, itcan bind with any agent B which can perform acompatible action (a, w) and which is distant from Aat most d. Figure 5 depicts a situation where an agentA is able to execute an output bind action (a, λ)d. Evenif agents C and D expose complementary channels,they are out of the perception radius of A, whichcan consequently bind only with agent B. The actualdistance between two agents affects their synchroniza-tion rate, as expected, since longer distances shouldlead to longer action durations and consequently tolower rates. Therefore, we define the binding rate be-tween two compatible bind actions (a, λ)d and (a, w)exposed by agents A and B respectively, by the ratioλ · p(a, ω)d(A,B) , where d(A,B) is the distance between A

and B.The simulator is based on the Repast Symphony

Suite [29], an agent-based modeling and simulation(ABMS) toolkit in Java 1. Figure 6 shows a screen-shottaken during a simulation. The simulator allows us tovisualize at runtime the location of bone cells in theBMU, the bone density and micro-structure, and thespatial diffusion of RANKL. Furthermore it supportsseveral runtime plots including bone mineral density,resorption and formation activities, RANKL concen-tration and number of active/total bone cells.Repast models are based on the concept of context,

which is a structured container serving as the agents’

1. A prototype version is available at http://www.cs.unicam.it/merelli/bone/

Ba,w1−−→

Ca,w2−−→

D

Fig. 5. Agents exposing binding capabilities. Agents Cand D is out of the perception radius of A (the bluecircle), determined by the sensibility distance d. Onthe other hand, A can detect and bind with agent Bbecause its center is within the perception radius of A.

environment. A context may include one or moreprojections, that represent the spatial domains whereagents are located (for example, continuous and dis-crete spaces). A projection in turn can have associ-ated value layers, i.e. spatial data structures holdinga numerical value for each position, and that can beaccessed by agents. Simulations in Repast are basedon a discrete-event execution model, where time pro-ceeds by discrete steps, called ticks and actions arescheduled for the execution at a specific tick. In thefollowing we provide some details on the variouscomponents of the agent-based model.

Computation model. Repast implements the discrete-event computation model through a scheduler hold-ing a list of pending actions as (action, tick) couples.At each tick t all the actions (a, t) are executed. Thisexecution model adapts well to the simulation algo-rithm defined in the theory of the Shape Calculus [5],[6], where the time-line is divided in time-steps ofduration ∆t. After each ∆t the velocities and thepositions of all the shapes are updated, according totheir particular motion law. However in the stochasticsettings we must consider actions with exponentiallydistributed durations that are determined by the rateof the action itself. Moreover the race condition ap-plies so that shorter actions should be executed earlier.Appendix D shows the pseudo-code implementingthe stochastic scheduler. In few words, if an action hasa duration less than or equal to ∆t, it is performed at∆t; otherwise, it is postponed to the subsequent timestep. Given that a single remodeling cycle lasts aboutone year and the resorption and formation processescan take months, in this model ∆t is set to 1 day.

Projections. We consider two different spatial do-mains: a continuous space, and a discrete space (i.e agrid). Space is approximated in two-dimensions, andhas a size of 240×160×10−4 mm2 such that each cellin the grid models a portion of size 0.01× 0.01 mm2.According to the parameter sizeBMU in Table 3, wecan ignore the depth of the BMU (0.01 mm) which ismuch smaller than its width (2.4 mm) and its height

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 9

Fig. 6. Screen-shot of the agent-based simulator forbone remodeling. The center panel shows the locationin the BMU of bone cells (blue: osteocytes, green: os-teoblasts, red: osteoclasts). On the left side the graphsfor BMD and RANKL concentration are displayed. Theright panels depict the bone micro-structure and theRANKL diffusion in the BMU (brightest zones: highestconcentrations).

(1.6 mm).Bone tissue. It is modeled as a Repast value layer

that is, a real-valued matrix Bone. The value of acell Boneij represents the percentage bone densityat the position (i, j) of the BMU grid, and variesfrom 0 (void) to 100 (fully mineralized). The simulatorsupports both cortical and trabecular bone.Molecular signals. While at the specification level

molecular signals like RANKL are modeled as com-munication channels, at the implementation level ex-posed output channels are assumed to release anamount of molecules diffusing in the space. The spa-tial concentration of such molecules is updated at each∆t according to a diffusion Cellular Automata updaterule [30], which corresponds to a discretization ofthe diffusion equation. The model considers both theRANKL signaling that affects osteoclasts’ motion, sothat osteoclasts are directed towards higher RANKLconcentrations (attraction), and the death factors pro-duced by osteoclasts that attract osteoblasts towardsthe portions of bone previously absorbed.Cell motion. In the absence of attraction factors, an

agent moves according to a random walk. At time

t its position→p (t) is updated by the following law

(adapted from [31]):

→p (t+∆t) =

→p (t) +

→

ξ√2D∆t

where ξ is a uniformly distributed random vectorranging in [−1, 1], and D is the diffusion coefficientof the agent.In the case that a cell is also affected by attraction

factors, like RANKL-attracted osteoclasts, its motion isdetermined by a biased random walk, i.e. a randomwalkaltered by the concentration gradient of an attractant

molecule u. Assuming that→p (t) = (x, y), let u⌊x⌋,⌊y⌋(t)

denote the concentration of u at the discrete position

(⌊x⌋, ⌊y⌋) and at time t. Then, the position of the agentis updated by:

→p (t+∆t) =

→p (t) +

→

ξ±√2D∆t

where now the vector→κ± models the “molecular

bias”, by taking into account the concentration gra-

dient of u. The x coordinate of→

ξ± is calculated asfollows:

(ξ±)x =

U [0, 1]1 + ux∗,⌊y⌋(t)

1 + u⌊x⌋,⌊y⌋(t), if x∗ ≥ x

U [−1, 0]1 + u⌊x⌋,⌊y⌋(t)

1 + ux∗,⌊y⌋(t), if x∗ < x

where U [a, b] is a uniformly distributed random valuebetween a and b; x∗ is the x component of the discreteposition with the highest RANKL concentration inthe Von Neumann neighborhood of (⌊x⌋, ⌊y⌋). Theequivalent equation holds for (ξ±)y .

Osteocytes. We assume that osteocytes produceRANKL, since osteocytes’ signaling has the effectof attracting osteoclasts to the source of the signal,similarly to RANKL signaling by pre-osteoblasts. Os-teocytes’ initial positioning in the bone matrix is notpre-determined, and the outcome of different shapesfor osteocytes’ positioning is discussed in Section 5.2.

Osteoclasts. In normal conditions, the number ofosteoclasts in the BMU is subject to little variationsfrom the average nOc = 20. Higher perceived densitiesof RANKL in the environment, which are causedby an inflammation or a signaling defect, lead to ahigher production of osteoclasts. Their motion lawis a biased random walk affected by the RANKLconcentration gradient, with a diffusion coefficientDOc = 10−2mm2/day. The expected lifetime tOc is 10days, but they can undergo apoptosis before tOc ifa high concentration of osteoblasts is detected in theneighborhood. Having reached the portion of bone toconsume, an osteoclast reduces the percentage densityin that part of bone by kOc = 2.5 each day.

Osteoblasts. The number of osteoblasts in the systemvaries randomly around the average nOb = 2000. Theymove according to a biased random walk affectedby the osteoclasts’ death factors, with a diffusioncoefficient DOb = 2.15 × 10−2mm2/day. Moreover,osteoblast precursors emit RANKL which contributesto osteoclasts’ production and stimulation. Matureosteoblasts inhibits RANKL signaling by binding withthe OPG channel. The lifetime of an osteoblast is about20 days: 5 days as a precursor, 15 days as a maturecell active in the formation process. When active, itincreases the bone density of a factor kOb = 0.25 eachday. It can die before its lifetime has passed, if thenumber of active osteoclasts in the system is higherthan expected.

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 10

5 RESULTS AND DISCUSSION

The aim of this work is to show how apparentlyinsignificant changes in RANKL signaling lead to dis-ease conditions, especially when aging factors are in-volved. In order to describe these two factors, we havedefined two parameters, kRANKL and kaging, modelingrespectively the effectiveness of the RANKL signalingand the aging factor in terms of a reduced cellularactivity. In particular, we defined two configurationsobtained by varying the parameters kRANKL and kaging

(see Appendix C for details on parameter estimation):

• healthy conf. (kRANKL = 1, kaging = 1), whereRANKL production and cellular activity is nor-mal;

• osteoporotic conf. (kRANKL = 2, kaging = 2), withan overproduction of RANKL and a reducedcellular activity.

When running the osteoporotic configuration, thehigher RANKL effectiveness leads to a higher produc-tion of osteoclasts, hence to a higher resorption activ-ity. On the other hand, osteoblasts are not effectiveenough to completely repair the consumed bone, dueto the aging factor lowering the cellular activity. Thiscauses a lower total bone density, weaker trabeculaeand consequently more frequent and more consistentmicro-fractures in osteoporotic patients. It is worthnoting that when the aging factor is absent, like in ayoung patient, an overproduction of RANKL does notdetermine a disease situation. Indeed, an unexpectedhigh absorbing activity can be successfully balancedby recruiting a higher number of osteoblasts, whichwould not be possible in an old patient characterizedby a lower cellular activity. Conversely if we consideran old patient with a regular RANKL signaling, lesseffective osteoclasts are balanced by less effectiveosteoblasts and we do not observe relevant negativeremodeling phenomena.Figure 7 groups the key snapshots taken during a

simulation of one trabecular remodeling cycle, com-paring the healthy and the osteoporotic configura-tions. We analyze the location of the cells in the BMU,the bone micro-structure, resorption and formationactivities, and the RANKL concentration at time:

• t = 150: the first osteoblasts (green spheres)are being recruited and osteoclasts (red spheres)have almost concluded the resorption phase. Os-teoblasts starts to fill the cavities excavated by os-teoclasts, following them in a highly coordinatedmanner. At this stage, osteocytes (blue spheres)have been absorbed almost entirely. While thereare no significant differences in the bone mass, weobserve that RANKL concentration is higher inthe osteoporotic case, as expected. This causes afaster resorption, leading in turn to a faster BMUactivity, evidenced by the fact that osteoblasts arerecruited earlier than in the healthy case.

• t = 400: the remodeling cycle is in its final

days. Differences in the bone density are quiteprominent: in Figure 7 (f) the cavities have al-most been repaired; while Figure 7 (h) showshow the reduced osteoblastic activity generatesseveral holes in the trabecular structure. Thiscan be observed also in the cumulative plots ofbone resorption and formation in Fig. 7 (j),(l). Inhealthy conditions formation (solid line) compen-sate resorption (dashed line), while in the osteo-porotic case formation does not counterbalanceresorption, even if the resorption activity is notgreater than in the healthy case. In addition, wenotice a lower RANKL concentration in the sec-ond half of the remodeling cycle, due to the factthat signaling osteocytes have been completelyabsorbed and RANKL-producing pre-osteoblastshave matured into bone-forming osteoblasts.

Figure 8 compares the bone mineral density inthe healthy and in the osteoporotic case, averagedover forty runs for each configuration of a singleremodeling cycle. Initial values of BMD are takenfrom statistical data [32] and represent the hip densityof a Caucasian woman aged 25 (see Section 5.1 for abroader discussion). In normal conditions we observethat at the end of remodeling the initial bone densityis reestablished, and that variations from the meandensity are limited. In case of osteoporosis, the mini-mal values of percentage density are reached duringresorption, and the final density is lower than the ini-tial one, since the osteoblastic activity is not effectiveenough to counterbalance resorption. In addition, thepathological case is characterized by a higher stan-dard deviation. However the sum of the mean densityand its standard deviation at the end of remodeling isstill below the optimal density, meaning that in morethan the 84% of cases the outcome of bone remodelingis negative in the osteoporotic configuration.

5.1 Model Validation

In this part the outputs of the agent-based model willbe validated against statistical datasets. In particular,we refer to the National Health and Nutrition Exami-nation Survey (NHANES III, 1988-94) [32], and to hipdensity values taken from [33]. Bone Mineral Density(BMD) is an indicator of the quality of the bone andit is measured with clinical techniques, the most usedof which is DEXA (Dual Energy Xray Absorptiome-try). Based on BMD measurement and according tothe patient’s gender and ethnicity, it is possible todetermine the presence of bone pathologies accordingto the following definitions [34]:

• Osteopenia, when the BMD is between 1 standarddeviation (SD) and 2.5 SD below the mean of theyoung reference group.

• Osteoporosis, when the BMD is 2.5 SD or morebelow the mean of the young reference group.

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 11

0 50 100 150

050

100

150

0 100 200 300 400

050

100

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200

0 50 100 150

050

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0 100 200 300 400

050

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150

200

Res Form

0 50 100 150

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0 100 200 300 400

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20

0 50 100 150

05

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0 100 200 300 400

05

10

15

20

(p)(o)

(m) (n)

(k) (l)

(i) (j)

(g) (h)

(e) (f)

(c) (d)Oc Ob Oy

(a) (b)

BM

UB

on

e m

ass

Res

- Form

[m

g/c

m2]

RA

NK

L [

pm

ol/

l]

0 1

H

O

H

O

H

O

H

O

150 days 400 days

Fig. 7. Simulation of healthy (H) and osteoporotic (O)configurations. The first two rows display the positionof bone cells in the BMU. The third and the fourth rowsshow the density of the trabecula. Graphs in the fifthand sixth row depict the cumulative values of boneresorption and formation. The last two rows show howthe RANKL concentration varies during remodeling.

HealthyOsteoporoticStdev HStdev O

time [days]

BM

D [

mg

/cm

2]

Fig. 8. Changes in BMD during one remodeling cy-cle. Continuous curves represent the mean density.Filled areas span the interval mean ± standard devi-ation. In the healthy configuration (blue curve), mineralhomeostasis is maintained. On the other hand, in thepathological configuration (red curve) the final densityis lower than the initial one (negative remodeling).

Therefore bone diseases are assessed by comparingthe BMD measurements with statistical data obtainedfrom population-wide surveys. Figure 9 shows thereference values for Caucasian women at about 25years (young reference group) a), 45 years b) and 65years c).Simulation results show that after just one remod-

eling cycle both the healthy and the osteoporotic con-figurations return density values within the normaldensity range µ± σ = 955± 123, which is determinedby the BMD distribution of the young reference group(see Fig. 9). After 40 simulation runs, the followingresults have been obtained at the end of a singleremodeling cycle:

• Healthy configuration: 955.01± 14.41 ⊂ µ± σ• Osteoporotic configuration: 921.86±19.36⊂ µ±σ

Although in the pathological configuration we ob-serve a lower mean density, in both cases no particularbone pathologies are diagnosed, as one can expect.Indeed it is very unlikely that pathological conditionscan occur after one single remodeling cycle (corre-sponding to about 400 days) and starting from anoptimal bone density.As a matter of fact, osteopenia and osteoporosis

are diseases resulting after several years of negativeremodeling. Figure 10 plots the results of a singlesimulation involving seven consecutive remodelingcycles, that correspond roughly to seven years. In thiscase we report that when running the osteoporoticconfiguration, bone mineral density keeps decreasingthroughout the years. At about t = 700 days, densityreaches the range [µ − 2.5σ, µ − σ], meaning thatosteopenia would be diagnosed after less than twoyears. Moreover at about t = 2100, bone densitygoes below the level µ − 2.5σ. Therefore running theosteoporotic configuration, osteoporosis is diagnosedafter less than six years of simulation.

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 12

a) Caucasian women, 25 yo. � = 955 mg/cm2; � = 123.

b) Caucasian women, 45 yo.

� = 920 mg/cm2; � = 136.

c) Caucasian women, 65 yo.

� = 809 mg/cm2; � = 140.

Fig. 9. Hip BMD distribution in Caucasian women fromstatistical data in [32]. a) refers to the young referencegroup (25 years) with mean BMD µ and standarddeviation σ. The filled areas below the curve and thedifferent shades of red identify patients with normalbone, osteopenia and osteoporosis. b) and c) showthe BMD distributions at 45 and 65 years respectively,where an increasing incidence of osteopenia and os-teoporosis can be observed.

Fig. 10. Changes in BMD during seven remodelingcycles. As in Fig. 9, different shades of red determinethe density ranges for normal bone, osteopenia andosteoporosis. In the healthy configuration (blue curve),the initial density is maintained after each remodelingcycle. In the osteoporotic configuration (red curve),density keeps decreasing throughout the years, goingbelow the threshold (µ− 2.5σ) under which osteoporo-sis is diagnosed.

H

O

H

O

H

O

1st cycle 2nd cycle 3rd cycle

Fig. 11. The effects of different osteocytes’ position-ing (straight, zig-zag, and random) in the density andmicro-structure of a portion of compact bone. In thehealthy (H) case, mineral and structural homeostasisis maintained. In the osteoporotic (O) case, the densitykeeps decreasing and the structure keeps weakening.

5.2 Osteocytes’ positioning affects bone micro-structure

We evaluate how the positioning of active (signaling)osteocytes affects in the long run the bone density andmicro-structure. Keeping in mind that we model onlysignaling osteocytes, their initial positions are mainlydetermined by factors at tissue level. Osteocytes actas mechanosensors giving a molecular response atthe cellular level to the mechanical stresses at tissuelevel, and they also activate along micro-fracturesin the bone matrix. Considering that osteocytes areresponsible for the activation of bone remodeling, thiskind of analysis could explain the multiscale looplinking

1) the origination of mechanical stimuli at the tis-sue level, driven by the optimization of the bonemicro-structure;

2) the activation of a certain number of osteocytesat a certain position in the bone matrix, deter-mined by the entity of the mechanical stimuli;

3) depending on osteocytes’ positioning, the out-come of bone remodeling in terms of updatedbone density and micro-structure, which in turntriggers a new mechanical adaptation process.

Figure 11 summarizes the results obtained afterthree remodeling cycles, comparing healthy and os-teoporotic patients and three different positioningpatterns: straight, zig-zag, and random. Firstly we

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 13

observe that mineral homeostasis and bone micro-structure is successfully maintained in the healthyconfiguration, regardless of the initial positioning ofosteocytes. There are just few parts characterized by alower density, which can be ascribed to the stochasticfluctuations in the agent-based model. Indeed, wenotice that osteoblasts are able to fill any hole left inthe previous remodeling cycles, but their stochasticbehavior may allow some parts not to be completelyrepaired. On the other hand we observe a progressivemicro-structural weakening in the osteoporotic con-figuration, so reproducing the expected defective dy-namics. This qualitative analysis could be useful alsoin the assessment of the fracture risk in a particularosteoporotic patient, providing a more detailed andpersonalized scenario with respect to current medicalpractice that estimates the fracture risk only on astatistical basis.

6 CONCLUSION

In this work we have investigated the complex mul-tiscale dynamics that connects disorders in RANKLsignaling at the molecular level, to bone diseases likeosteoporosis that are characterized by a lower bonemass and disruptions at tissue level. In particularour results evidence that defective bone diseases arisewhen a higher effectiveness of RANKL signaling iscombined with aging factors reducing the activity ofosteoblasts and osteoclasts. The parameters modelingthe RANKL effectiveness and the aging factor havebeen estimated over an ODE model. By means of sen-sitivity analysis procedures, we have determined twoconfigurations of parameters corresponding respec-tively to a healthy patient and to an osteoporotic pa-tient. Simulation results have been validated over sta-tistical data available from population-wide surveys.In the healthy configuration, the simulator is ableto reproduce the homeostatic process of remodeling,where resorption is counterbalanced by formation. Onthe other hand, when the simulator is executed in theosteoporotic configuration, osteoporosis is diagnosedafter less six years, even if starting from an optimalbone density. Furthermore we have looked into theconnection between the spatial location of signalingosteocytes and the bone micro-structure in the longrun.From the methodological point of view, we have

defined a multi-level computational framework thatcan be successfully employed also in other areas ofbiology and biomedicine. At the specification level,we describe the process of bone remodeling througha process-algebraic language, the Shape Calculus,which proved to be particularly suitable for express-ing the spatial and multiscale nature of bone remodel-ing. At the implementation level, we have developeda stochastic agent-based simulator supporting theShape Calculus as the model specification language

(by means of translation functions); and featuring avariety of analysis and runtime monitors. We reportthat the stochastic agent-based simulation is time-consistent with the real biological process and that re-sults agree with those obtained from well-establishedmathematical models like [26], [27].Besides being a biomedical problem of growing

importance, osteoporosis and bone remodeling arecharacterized by complex multiscale mechanisms notentirely explained yet, thus representing a testbed fornovel modeling paradigms and techniques. The com-bination of formal methods with agent-based tech-nologies has provided us with a deeper understand-ing of this challenging field of research and we believethat our approach could be possibly employed asa general-purpose toolkit in Computational SystemsBiology.

ACKNOWLEDGMENTS

Pietro Lio would like to thank RECOGNITION: Rele-vance and cognition for self-awareness in a content-centricInternet (257756), funded by the European Commis-sion within the 7th Framework Programme (FP7).Nicola Paoletti would like to thank the Rizzoli Or-thopaedic Institute in Bologna (IT) for funding hisPhD scolarship.

REFERENCES

[1] S. Jabbar, J. Drury, J. Fordham, H. Datta, R. Francis, andS. Tuck, “Osteoprotegerin, RANKL and bone turnover inpostmenopausal osteoporosis,” Journal of Clinical Pathology,vol. 64, no. 4, p. 354, 2011.

[2] N. e. a. Morabito, “Osteoprotegerin and RANKL in the Patho-genesis of Thalassemia-Induced Osteoporosis: New Pieces ofthe Puzzle,” Journal of Bone and Mineral Research, vol. 19, no. 5,pp. 722–727, 2004.

[3] P. Lio, E. Merelli, N. Paoletti, and M. Viceconti, “A combinedprocess algebraic and stochastic approach to bone remodel-ing,” Electronic Notes in Theoretical Computer Science, vol. 277C,pp. 41–52, 2011.

[4] N. Paoletti, P. Lio, E. Merelli, and M. Viceconti, “Osteoporosis:a multiscale modeling viewpoint,” in Proceedings of the 9thInternational Conference on Computational Methods in SystemsBiology, ser. CMSB ’11. ACM, 2011, pp. 183–193.

[5] E. Bartocci, D. Cacciagrano, M. Di Berardini, E. Merelli, andL. Tesei, “Timed Operational Semantics and Well-Formednessof Shape Calculus,” Scientific Annals of Computer Science,vol. 20, 2010.

[6] E. Bartocci, F. Corradini, M. Di Berardini, E. Merelli, andL. Tesei, “Shape Calculus. A Spatial Mobile Calculus for 3DShapes,” Scientific Annals of Computer Science, vol. 20, 2010.

[7] L. Milanesi, P. Romano, G. Castellani, D. Remondini, andP. Lio, “Trends in modeling biomedical complex systems,”BMC bioinformatics, vol. 10, no. Suppl 12, 2009.

[8] L. Geris, J. Vander Sloten, and H. Van Oosterwyck, “In silicobiology of bone modelling and remodelling: regeneration,”Philosophical Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences, vol. 367, no. 1895, p. 2031,2009.

[9] F. Gerhard, D. Webster, G. van Lenthe, and R. Muller, “Insilico biology of bone modelling and remodelling: adaptation,”Philosophical Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences, vol. 367, no. 1895, p. 2011,2009.

[10] P. Pivonka and S. Komarova, “Mathematical modeling in bonebiology: From intracellular signaling to tissue mechanics,”Bone, vol. 47, no. 2, pp. 181–189, 2010.

IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 14

[11] M. Viceconti, L. Bellingeri, L. Cristofolini, and A. Toni, “Acomparative study on different methods of automatic meshgeneration of human femurs,” Medical engineering & physics,vol. 20, no. 1, pp. 1–10, 1998.

[12] J. Fisher and T. Henzinger, “Executable cell biology,” Naturebiotechnology, vol. 25, no. 11, pp. 1239–1249, 2007.

[13] A. Tovar, “Bone remodeling as a hybrid cellular automatonoptimization process,” Ph.D. dissertation, University of NotreDame, 2004.

[14] L. Li and H. Yokota, “Application of Petri Nets in BoneRemodeling,” Gene Regulation and Systems Biology, vol. 3, p.105, 2009.

[15] D. Cacciagrano, F. Corradini, E. Merelli, and L. Tesei, “Multi-scale Bone Remodelling with Spatial P Systems,” in ProceedingsCompendium of the Fourth Workshop on Membrane Computing andBiologically Inspired Process Calculi, 2010.

[16] P. Lio, E. Merelli, and N. Paoletti, “Multiple verification incomputational modeling of bone pathologies,” in CompMod,2011, pp. 82–96.

[17] A. T. Bittig and A. M. Uhrmacher, “Spatial Modeling in CellBiology at Multiple Levels,” in Proceedings of the 2010 WinterSimulation Conference. IEEE, 2010, pp. 608–619.

[18] F. Ciocchetta and M. L. Guerriero, “Modelling Biological Com-partments in Bio-PEPA,” Electronic Notes Theoretical ComputerScience, vol. 227, pp. 77–95, January 2009.

[19] A. Regev, E. M. Panina, W. Silverman, L. Cardelli, andE. Shapiro, “Bioambients: an abstraction for biological com-partments,” Theoretical Computer Science, vol. 325, no. 1, pp.141 – 167, 2004.

[20] C. Priami and P. Quaglia, “Beta Binders for Biological Interac-tions,” in Computational Methods in Systems Biology, ser. LectureNotes in Computer Science, 2005, vol. 3082, pp. 20–33.

[21] L. Cardelli and P. Gardner, “Processes in Space,” in Programs,Proofs, Processes, ser. Lecture Notes in Computer Science, 2010,vol. 6158, pp. 78–87.

[22] M. John, C. Lhoussaine, J. Niehren, and A. Uhrmacher, “Theattributed π-calculus with priorities,” Transactions on Compu-tational Systems Biology XII, pp. 13–76, 2010.

[23] A. Stefanek, M. Vigliotti, and J. T. Bradley, “Spatial extensionof stochastic π calculus,” in 8th Workshop on Process Algebraand Stochastically Timed Activities, August 2009, pp. 109–117.

[24] R. Milner, A calculus of communicating systems. Springer-Verlag, 1980, vol. 92.

[25] D. Epari, G. Duda, and M. Thompson, “Mechanobiology ofbone healing and regeneration: in vivo models,” Proceedingsof the Institution of Mechanical Engineers, Part H: Journal ofEngineering in Medicine, vol. 224, no. 12, pp. 1543–1553, 2010.

[26] S. Komarova, R. Smith, S. Dixon, S. Sims, and L. Wahl, “Math-ematical model predicts a critical role for osteoclast autocrineregulation in the control of bone remodeling,” Bone, vol. 33,no. 2, pp. 206–215, 2003.

[27] M. Ryser, N. Nigam, and S. Komarova, “Mathematical Mod-eling of Spatio-Temporal Dynamics of a Single Bone Multicel-lular Unit,” Journal of bone and mineral research, vol. 24, no. 5,pp. 860–870, 2009.

[28] J. Hillston, A compositional approach to performance modelling.Cambridge Univ Pr, 1996, no. 12.

[29] M. North, T. Howe, N. Collier, and J. Vos, “A DeclarativeModel Assembly Infrastructure for Verification and Valida-tion,” in Advancing Social Simulation: The First World Congress.Springer Japan, 2007, pp. 129–140.

[30] X. Yang and Y. Young, Handbook of Bioinspired Algorithms andApplications. Chapman & Hall/CRC Computer and Informa-tion Science, 2005, ch. Cellular Automata, PDEs, and PatternFormation.

[31] A. Edelstein and N. Agmon, “Brownian simulation of many-particle binding to a reversible receptor array,” Journal ofComputational Physics, vol. 132, no. 2, pp. 260–275, 1997.

[32] A. Looker, H. Wahner, W. Dunn, M. Calvo, T. Harris, S. Heyse,C. Johnston Jr, and R. Lindsay, “Updated data on proximalfemur bone mineral levels of us adults,” Osteoporosis Interna-tional, vol. 8, no. 5, pp. 468–490, 1998.

[33] “T and Z scores,” http://courses.washington.edu/bonephys/opbmdtz.html, Accessed: 10/11/2011.

[34] WHO Study Group on Assessment of Fracture Risk and itsApplication to Screening for Postmenopausal Osteoporosis,

“Assessment of fracture risk and its application to screeningfor postmenopausal osteoporosis,” Tech. Rep. 843, 1994.

Nicola Paoletti received the B.S. and theM.S. degree in Computer Science from theUniversity of Camerino in 2008 and 2010,resp. Since 2011 he is a PhD student at theSchool of Information Sciences and ComplexSystems at the University of Camerino. Hisresearch interests include formal methodsfor the specification and verification of com-plex systems; computational systems biol-ogy; modeling of multiscale systems.

Pietro Li o is a Senior Lecturer and a mem-ber of the Artificial Intelligence Division ofThe Computer Laboratory (the departmentof Computer Science), University of Cam-bridge. He has an interdisciplinary approachto research and teaching due to the factthat he holds a Ph.D. in Complex Systemsand Non Linear Dynamics (School of Infor-matics, dept of Engineering of the Univer-sity of Firenze, Italy) and a Ph.D. in (Theo-retical) Genetics (University of Pavia, Italy).

Main research interests: Modeling multiscale biological systems;emerging properties from microscale to macroscale in biology;Modeling methodologies, parameter estimation techniques, reverseengineering of biological circuits; Bioinspired technology. He haspublished more than 200 papers and organized several workshopsand schools.

Emanuela Merelli is Associate Professor ofComputer Science at University of Camerino,Italy, where she currently leads the ComplexSystems Research Group. She is the coordi-nator of B.Sc and M.Sc Degree and DoctoralProgram in Computer Science. Her researchinterests include Formal Methods for Mod-eling and Verification of Complex Systems,Formal Methods applied to Systems Biologyand Biomedicine, Agent-based Modeling forMulti-level and Multi-scale systems.

Marco Viceconti is full Professor of Biome-chanics at the Department of MechanicalEngineering at the University of Sheffield andScientific Director of the Insigneo/SheffieldResearch Institute. Before this he was theTechnical Director of the Medical TechnologyLab at the Rizzoli Orthopaedic Institute inBologna, Italy. Prof. Viceconti has a Mechan-ical Engineering degree from the Universityof Bologna and a PhD from the Universityof Firenze. His main research interests are

related to the development and validation of medical technology,especially that involving simulation, and primarily in relation tomusculoskeletal diseases. He has published over 200 papers andserves as reviewer for many international funding agencies and peer-reviewed journals.