Computer Laboratory, Cambridge University, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK

School of Science and Technology, Computer Science Division, University of Camerino, Via Madonna delle Carceri 9, Camerino (MC) 62019, Italy

Department of Mechanical Engineering, University of Sheffield, Sir Frederick Mappin Building, Mappin Street, Sheffield S1 3JD, UK

Abstract

Background

This work focuses on the computational modelling of osteomyelitis, a bone pathology caused by bacteria infection (mostly

Results

Since both osteoporosis and osteomyelitis cause loss of bone mass, we focused on comparing the dynamics of these diseases by means of computational models. Firstly, we performed meta-analysis on a gene expression data of normal, osteoporotic and osteomyelitis bone conditions. We mainly focused on RANKL/OPG signalling, the TNF and TNF receptor superfamilies and the NF-

Conclusions

We present a modeling framework able to reproduce aspects of the different bone remodeling defective dynamics of osteomyelitis and osteoporosis. We report that the verification-based estimators are meaningful in the light of a feed forward between computational medicine and clinical bioinformatics.

Background

There are two main types of bone tissues:

The

Key steps in bone remodelling

**Key steps in bone remodelling**. 1) Osteocytes send signals to the fluid part, activating pre-osteoblasts (

1. **Origination**. During normal turnover or after a micro-crack, or as a response to mechanical stress, the osteocytes in the bone matrix produce biochemical signals showing sufferance towards the lining cells, i.e. the surface cells around the bone. The lining cells pull away from the bone matrix, forming a canopy which merges with the blood vessels.

2. **Osteoclast recruitment**. Stromal cells divide and differentiate into osteoblasts precursors. Pre-osteoblasts start to express RANKL, inducing the differentiation of and attracting pre-osteoclasts, which have RANK receptors on their surfaces. RANKL is a homotrimeric molecule displayed on the membrane of osteoblasts that stimulates differentiation in osteoclasts and is a key induction molecule involved in bone resorption leading to bone destruction.

3. **Resorption**. The pre-osteoclasts enlarge and fuse into mature osteoclasts. In cortical BMUs, osteoclasts excavate cylindrical tunnels in the predominant loading direction of the bone, while in trabecular bone they act at the bone surface, digging a trench rather than a tunnel. After the resorption process has terminated, osteoclasts undergo apoptosis.

4. **Osteoblast recruitment**. Pre-osteoblasts mature into osteoblasts and start producing osteoprotegerin (OPG). OPG inhibits the osteoclastic activity by binding to RANKL and preventing it from binding to RANK. When RANKL expression is high, osteoprotegerin levels are low and vice versa.

5. **Mineralization**. Osteoblasts fill the cavity by secreting layers of osteoids. Once the complete mineralization of the renewed tissue is reached, some osteoblasts can go apoptosis, other can turn into lining cells, while other can remain trapped in the bone matrix and become osteocytes.

6. **Resting**. Once the cavity has been filled by osteoblasts, the initial situation is re-established.

The bone remodelling undergoes a pathological process, generally related to ageing, termed osteopenia and with more severity, osteoporosis, during which an unbalance of the RANKL/OPG signalling equilibrium is typically observed. The osteoporosis is a skeletal disease characterized by low

The aim of this work is to provide a computational modelling framework able to reproduce and compare the defective dynamics of osteoporosis and osteomyelitis. We believe that this framework could easily be adapted to model also other bone diseases like multiple myelomas or Paget's disease, and that it could help in better understanding the disruptions of cellular and signalling mechanisms that underlie such bone pathologies.

Osteomyelitis

Osteomyelitis is a bone infection mainly caused by the aggressive pathogen

Although effective treatment of this disease is very difficult, one of most used drug is the fusidic acid that acts as a bacterial protein synthesis inhibitor by preventing the turnover of elongation factor G (EF-G) from the ribosome. Fusidic acid inhibits bacterial replication and does not kill the bacteria, and is therefore termed "bacteriostatic". Many strains of methicillin-resistant

We believe that a model of the infection could provide a framework for a better diagnosis and understanding the antibiotic intervention. Here we develop a hybrid modelling framework for combining and untangling the relationships of physiological and molecular data. We then apply the methodology to determine disease related abnormalities of the key osteogenesis molecular network. The universality of the approach is demonstrated by an integration of the modelling and diagnosis which resembles medical visits with blood testing for infection progress and bone mineralisation measurements along a period of time. Our perspective is that this approach would inch towards an automatized methodology for improving disease classification and diagnosis.

Results and discussion

Meta analysis of gene expression data

Important parameter values of bone remodelling models are based on various authors (see

Box plot representation of the gene expression of 82 genes corresponding to a) 48 osteomyelitis infected patients and b) 27 healthy controls)

**Box plot representation of the gene expression of 82 genes corresponding to a) 48 osteomyelitis infected patients and b) 27 healthy controls)**.

Box plot representation of the gene expression of 31 genes corresponding to a) 43 osteomyelitis infected patients and b) 17 healthy controls)

**Box plot representation of the gene expression of 31 genes corresponding to a) 43 osteomyelitis infected patients and b) 17 healthy controls)**.

Comparative representation of gene expression level for osteomyelitis and osteoporosis.

**Regulation for Osteomyelitis (GPL96)**

**Gene ID**

**Regulation for Osteomyelitis (GPL97)**

**Gene ID**

**Regulation for Osteoporosis (GPL96)**

**Gene ID**

Up regulated

NFKB2_1

Up regulated

NFKB2_1

Up regulated

NFKB1

NFKB2_2

NFKBIZ_1

NFKB2_1

NFKBIA

NFKBIZ_2

NFKB2_2

NFKBIE

RELL1

REL_2

REL_2

RELT

RELA_1

RELB

TNFSF13B_1

RELA_2

TNFRSF10B_2

TNFSF13B_2

RELB

TNFRSF10C_2

TRAF7_1

TNFRSF17

TNFRSF10C _3

TRAF7_3

TNFSF10_2

TNFRSF10C_4

TRAFD1_2

TRAF3_1

TNFRSF1A

Down regulated

TNFRSF10A

Down regulated

TNFRSF10B_2

TNFRSF1B

TNFRSF18_2

TNFRSF25_2

TNFSF10_1

TRAF1

TNFSF10_3

TNFSF10_2

TRAF3IP1

TRAF3IP3_1

TNFSF10_3

TRAF3IP3_1

TRAF3IP3_3

TNFSF12_3

TRAF3IP3_2

TRAF5

TNFSF12_4

TNFSF12_2

TNFSF13

TRAF3IP3_2

TRAF3IP3_3

TRAFD1_2

Down regulated

IKBKG 2

NFKB1

RELA_1

TNFRSF14

TNFRSF25_1

TNFRSF25_2

TNFRSF25_3

TNFRSF25_4

TNFRSF25_6

TRAF1

TRAF3IP2_2

TRAF3IP3_1

TRAF5

Interestingly we found that, despite a very small increase of RANKL gene expression in osteoporosis and a larger increase in osteomyelitis, OPG gene expression become more deregulated in both osteomyelitis and osteoporosis. There is the increased expression of different isoforms of OPG which are known to have different binding capability with RANKL and seem to be linked, from mice experiments, to hypocalcemia

A computational framework for bone dynamics

In this work we present a combined computational framework for the modelling, simulation and verification of the bone remodelling process, and of bone pathologies like osteomyelitis and osteoporosis. Based on the methods developed in

Mathematical model

We develop a differential equation model for describing the dynamics of bone remodelling and of bone-related pathologies at a multicellular level. The model describes the continuous changes of, and the interactions between populations of osteoclasts and osteoblast (including bacteria in the osteomyelitis model). Bone density is calculated as the difference between the formation activity which is proportional to osteoblasts concentration, and the resorption activity which is proportional to osteoclasts concentration. In the last twenty years, a variety of mathematical and computational models has been proposed in order to better understand the dynamics of bone remodelling (reviewed in

Model verification

We define a stochastic model for bone remodelling from the ODE specification, that allows us to analyse the random fluctuations and the discrete changes of bone density and bone cells. Given that randomness is an inherent feature of biological systems, whose components are naturally discrete, the stochastic approach could give useful insights on the bone remodelling process. Indeed, stochasticity plays a key role in bone remodelling, e.g. the fluctuations in molecular concentrations of RANKL and OPG produce changes in the chemotaxis (the process by which cells move toward attractant molecules) of osteoclasts and osteoblasts. This may affect for example the cell differentiation, number and arrival time, and consequently the whole remodelling process. Besides achieving a good fitting between the ODE model and the stochastic one, we employ

Modelling bone remodelling pathologies

The ODE model for bone remodelling is mainly inspired from the work by Komarova et al

The model describes the autocrine and paracrine relationships between osteoclasts and osteoblasts. Autocrine signalling usually occurs by a secreted chemical interacting with receptors on the surface of the same cell. In the paracrine process a chemical signals that diffuse outside the emitting cell and interacts with receptors on nearby cells. Here the parameters _{ij }_{11 }describes the osteoclast autocrine regulation, _{22 }the osteoblast autocrine regulation, _{21 }is the osteoblast-derived paracrine regulation, and _{12 }is the osteoclast paracrine regulation. The nonlinearities of these equations are approximations for the interactions of the osteoclast and osteoblast populations in the proliferation terms of the equations. The autocrine signalling has a positive feedback on osteoclast production (_{11 }_{21 }_{22 }_{12 }

Overall the regulatory circuit should lead to a positive mineralisation balance (_{1 }and _{2 }are the resorption and formation rates, respectively. More precisely, the bone density is determined by the difference between the actual resorption and formation activity when osteoclasts and osteoblasts exceed their steady levels. Therefore bone density is calculated as follows:

where _{c }_{b}

In order to reproduce the defective dynamics (i.e. bone negative balance) characterizing osteoporosis, we assumed an increased death rate for osteoclasts and osteoblasts, motivated by the fact that the occurrence of defective bone pathologies in elderly patients is partly attributable to the reduced cellular activity typical of those patients. Therefore we introduced the parameter g_{ageing }as a factor multiplying the death rates β_{i}

On the other hand, we modified the regulation factors in order to model an increased RANKL expression by osteoblasts, which results both from the analysis performed on gene expression data and from experimental evidences _{21 }is the result of all the factors produced by osteoblasts that activates osteoclasts and as explained in _{21 }= _{por }has been included as a factor incrementing _{21}, in order to incorporate the changes in the system RANKL, OPG associated to osteoporosis. The resulting equations for osteoclasts and osteoblasts are:

Osteomyelitis effects on bone remodelling

Starting from the above model of bone remodelling, we consider the progressing of osteomyelitis induced by the

where γ_{B }_{ij }_{ij}

This model has been inspired from Ayati's work on multiple myeloma bone disease _{ij}B/s _{11}, _{12}, _{21}, _{22 }are all nonnegative. The

• reducing osteoblasts' growth rate: in fact, the paracrine promotion of osteoblasts is reduced

• increasing RANKL and decreasing OPG expression: as previously stated, the paracrine inhibition of osteoclasts is a negative exponent resulting from the difference between the effectiveness of OPG signalling and that of RANKL signalling. Since

In addition the infection increases the autocrine promotion of osteoclasts (since _{11 }_{B }_{B }_{B}_{B}

Parameters for the three different models (control, osteoporosis and osteomyelitis) are given in Table

Model parameters.

**Parameter**

**Description**

**Value**

(α_{1}, α_{2})

_{c }_{b }

(3, 4) ^{-1}

(β_{1}, β_{1})

_{c }_{b }

(0.2, 0.02) ^{-1}

(_{11}, _{12}, _{22}, _{21})

Effectiveness of autocrine/paracrine regulation

(1.1, 1, 0, -0.5)

(_{1}, _{2})

Resorption and formation rates

(0.0748, 0.0006395) ^{-1}

g_{ageing}

Ageing factor

2

g_{por}

RANKL factor

0.1

γ_{B}

0.005 ^{-1}

100

Effectiveness of antibiotic treatment

(0.005, 0.007) ^{-1}

_{treat}

Dosage time

(200, 400, 600)

(_{11}, _{12}, _{22}, _{21})

Effect of infection on regulation factors

(0.005, 0, 0.2, 0.005) ^{-1}

Steady levels of _{c }_{b}

Control: (1.16, 231.72)

Osteoporosis: (1.78, 177.91)

Osteomyelitis: (5, 316)

(_{c0}, _{b0}, _{0})

Initial states

Control: (11.16, 231.72, 1)

Osteoporosis: (11.78, 177.91, 1)

Osteomyelitis: (15, 316, 1)

Values have been adapted from literature (mainly _{ageing }and g_{por }are relative to the osteoporosis model, while parameters γ_{B}, s, V, t_{treat }_{11}, _{12}, _{22}, _{21}) are specific to the osteomyelitis model.

Simulation results of the ODE model

**Simulation results of the ODE model**. Bone density (first row), number of osteoblasts (second row), and number of osteoclasts (third row) compared between control and osteoporotic (first column); control and osteomyelitis (second column); osteoporotic and osteomyelitis (third column). Red lines mark the steady states for the variables considered. Results show a negative remodelling balance in the osteoporotic case and much more critical in the osteomyelitis case. While we observe a higher (but constant) remodelling rate in the osteoporotic configuration, in the osteomyelitis scenario the remodelling period is unstable and longer.

Furthermore we simulate the dosage of a bacteriostatic treatment (_{B}_{B}_{treat }_{treat }

Simulation of a bacteriostatic (_{B}_{B}

**Simulation of a bacteriostatic ( V = 0.005 = γ**. Dots on the plots mark the points when treatment is given. As regards the bacteriostatic drug, the bone density is not subject to critical drops if the treatment is administered at

Stochastic model for the verification of bone pathologies

Following and extending the work in

We follow a

where

where _{max }_{min }

Table _{2}_{b}_{1}_{c}

Transitions in the stochastic model for bone remodelling.

(a) Osteoclasts

[]

: _{c }_{c }+

[]

_{c }

g_{ageing}β_{1}_{c}

: _{c }_{c }-

[

_{c }

_{1}
_{c}

:

(b) Osteoblasts

[]

: _{b }_{b }+

[]

_{b }

g_{ageing }β_{2}_{b}

: _{b }_{b }-

[

_{b }

_{2}
_{b}

:

(c) Bacteria

[]

0 <_{max }

:

[]

:

[]

0 <_{max }_{B }

:

[]

_{B }

:

(d) Bone resorbed reward

(e) Bone formed reward

[

[

We consider the model with bacterial infection, being equivalent to the model with no infection when _{ij }_{treat}_{B}_{B}_{1}_{c }_{2}_{b }

Potentialities in clinical bioinformatics and conclusions

Osteomyelitis and osteoporosis are assessed through the verification of quantitative properties over the defined stochastic model.

Let assume that the simulation of the PRISM implementation of the model is run in parallel with the determination of clinical parameters during the periodic medical visits of a patient. These medical visits provide a mean of fine tuning a personalised model of the disease and a measure of how a therapy is effective. Different diseases, when monitored in a continuous way, may produce different alterations in local mineral density. We could extend the statistical estimators of a disease to: 1) the BMD (measured as z-score, the number of standard deviations above or below the mean for the patients age, sex and ethnicity; or as t-score, i.e. the number of standard deviations above or below the mean for a healthy 30 year old adult of the same sex and ethnicity as the patient); 2) The rate of change of BMD. This estimator tells us the emergence of defects of the bone metabolism in terms of signaling networks of RANK/RANKL and decrease of pre-osteoblast number; 3) The variance, skewness and curtosis of the the local small scale intermittency of the signal. For example osteomyelitis and osteoporosis show slightly confounding pattern of BMD decrease; we could also think at the confounding patterns of IRIS in HAART therapy, co-morbidity of osteopetrosis and osteoporosis, multiple myelomas, breast cancer, diabetes and metabolic syndromes, etc. The variance could perhaps help in discriminating among bone-related diseases. From a technical viewpoint, properties to verify have been formulated in

• **Bone density estimator**. It is calculated as the difference between the cumulative _{{"..."}}) for bone formation and bone resorption, with the formula

• **Density change rate**. It allows to assess rapid negative and positive changes in bone density. This estimator could be particularly helpful in detecting the insurgence of osteomyelitis before critical values of bone density are reached, since osteomyelitis is typically characterized by a higher negative change rate than osteoporosis. In particular the estimator is defined as the difference quotient of BMD over a time interval of months, e.g. 50 days. The formula obtained is

• **Density variance**. While the first estimator computes the expected value of bone density, here we calculate the variance of BMD taking into account the whole state space and the actual bone density at each state.

Figure

Bone mineral density function and its standard deviation for the control (left, a), osteoporosis (middle, b) and osteomyelitis (right, c) simulations

**Bone mineral density function and its standard deviation for the control (left, a), osteoporosis (middle, b) and osteomyelitis (right, c) simulations**.

Rate of change of bone mineral density function for the control (left, a), osteoporosis (middle, b) and osteomyelitis (right, c) simulations

**Rate of change of bone mineral density function for the control (left, a), osteoporosis (middle, b) and osteomyelitis (right, c) simulations**.

Here we report that the genetic complexity and the gene expression data meta analysis shows that the underlying "mystery" of bone remodelling is much greater than handled by the current mathematical models. In other words we are not able to use all our gene expression results in a full model of BR diseases. Although our model of osteomyelitis and the comparison with the osteoporosis is not able to consider all this complexity, nevertheless it makes a partial use of the results of the analysis of the experimental data and produces a realistic description of the pathology. From a methodological point of view the combination of mathematical and formal method approach has led to the proposal of considering additional estimators (first derivatives and variance) of the bone pathologies as diagnostic tool. That could also inspire the ideal situation in which a personalised model is generated from (personalised) data and the comparison between clinical data obtained during periodic medical check-up is compared with the computer predictions.

Methods

Data analysis

We found that there are no comprehensive analysis on osteomyelitis; most studies focus on specific conditions. We have collected a large ensemble of gene expression data related to osteomyelitis and osteoporosis. For this reason, we have considered 6 microarray data sets of the same platform GPL96 from the Gene Expression Omnibus (

For more evidence about osteomyelitis, we have considered more gene expression data related to osteomyelitis on different platform GPL97. For this reason, we have considered additional 3 microarray data sets from the Gene Expression Omnibus (

ODE and probabilistic model checking models

We have implemented the ODE model based on Komarova et al

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

LP and NP conceived and designed the models, MM carried out data analysis. All authors contributed writing, reading and approving the final manuscript.

Acknowledgements

We thank Bruce P. Ayati (Iowa University) and Glenn Webb (Vanderbilt University) for suggestions and help in computation.

This article has been published as part of