Table 3

Transitions in the stochastic model for bone remodelling.

(a) Osteoclasts


[]

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M19">View MathML</a>

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M20">View MathML</a>

: Oc = Oc + 1

[]

Oc > 0 →

gageingβ1Oc

: Oc = Oc - 1

[resorb]

Oc > 0 →

k1Oc

: true


(b) Osteoblasts


[]

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M21">View MathML</a>

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M22">View MathML</a>

: Ob = Ob + 1

[]

Ob > 0 →

gageing β2Ob

: Ob = Ob - 1

[form]

Ob > 0 →

k2Ob

: true


(c) Bacteria


[]

0 <B <Bmax treat = 0 →

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M23">View MathML</a>

: B = B + 1

[]

treat = 0 →

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M24">View MathML</a>

: treat = 1

[]

0 <B <Bmax treat = 1 ∧ V < γB

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M25">View MathML</a>

: B = B + 1

[]

B > 0 ∧ treat = 1 ∧ V > γB

<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/S14/S12/mathml/M26">View MathML</a>

: B = B - 1


(d) Bone resorbed reward

(e) Bone formed reward



[resorb] true: 1

[form] true: 1


We consider the model with bacterial infection, being equivalent to the model with no infection when fij = 0. Guard predicates are set in order to avoid out-of-range updates and 0-valued transition rates. Maximum values for state variables have been estimated from the continuous model. The variable treat is used as a switch for the beginning of treatment firing with rate 1/ttreat, therefore with an exponentially distributed delay having mean treatTime. Bacteriocide (V >γB) and non-bacteriocide (V <γB) dynamics is considered separately. Bone density is calculated by subtracting the bone resorbed reward (d) from the bone formed reward (e). Resorption and formation rates in the ODE model, i.e. k1Oc and k2Ob respectively, become the stochastic rates of transitions incrementing the bone resorbed/formed reward.

Liò et al. BMC Bioinformatics 2012 13(Suppl 14):S12   doi:10.1186/1471-2105-13-S14-S12

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