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Biological background Computational modeling of bone pathologies Results Formal Analysis of Bone Clinical Pathologies Nicola Paoletti nicola.paoletti@unicam.it School of Science and Technology, Computer Science Division, University of

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Biological background Computational modeling of bone pathologies Results

Formal Analysis of Bone Clinical Pathologies

Nicola Paoletti

nicola.paoletti@unicam.it School of Science and Technology, Computer Science Division, University of Camerino, Italy.

joint work with Pietro Li`

  • , Emanuela Merelli

NETTAB 2011, Pavia, 13 October 2011

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Biological background Computational modeling of bone pathologies Results

Outline

1

Biological background

2

Computational modeling of bone pathologies

3

Results

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Biological background Computational modeling of bone pathologies Results

Outline

1

Biological background

2

Computational modeling of bone pathologies

3

Results

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Biological background Computational modeling of bone pathologies Results

Bone remodeling

Bone remodeling (BR) is the process by which aged bone is continuously renewed in a balanced alternation of bone resorption and formation BR is driven by osteoclasts (the diggers) and osteoblasts (the fillers), forming Basic Multi-cellular Units (BMUs) Imbalances between resorption and formation lead to bone pathologies (e.g. in osteoporosis resorption > formation)

Normal Osteoporotic

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Biological background Computational modeling of bone pathologies Results

Key events in BR (1/2)

Osteocytes(1) send signals to the fluid part, activating Pre-

  • steoblasts(2) (Pb) and Pre-osteoclasts(3) (Pc)

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Biological background Computational modeling of bone pathologies Results

Key events in BR (1/2)

Pbs express RANKL(4). Pcs express RANK(5) receptor.

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Biological background Computational modeling of bone pathologies Results

Key events in BR (1/2)

RANK/RANKL binding(6) induces Pcs’ proliferation(7). Pcs en- large and fuse, forming mature Osteoclasts(8) which start bone Resorption(9). Mature osteoblasts express the decoy receptor OPG(10).

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Biological background Computational modeling of bone pathologies Results

Key events in BR (2/2)

Osteoblasts start the bone Formation(11) process. RANKL/OPG binding(12) inhibits RANKL, protecting bone from excessive resorp- tion.

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Biological background Computational modeling of bone pathologies Results

Key events in BR (2/2)

During the Mineralization(13) process, osteoids secreted by os- teoblasts calcify.

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Biological background Computational modeling of bone pathologies Results

Key events in BR (2/2)

Resting(14): the initial situation is re-established

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Biological background Computational modeling of bone pathologies Results

Outline

1

Biological background

2

Computational modeling of bone pathologies

3

Results

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Biological background Computational modeling of bone pathologies Results

ODE model

˙ x1 =α1x1g11x2g21 − β1x1 ˙ x2 =α2x1g12x2g22 − β2x2 ˙ z = − k1x1 + k2x2 Osteoclasts Osteoblasts Bone mass

100 200 300 400 Cell populations Time [days] 5 10 15 20 25 30 Osteoclasts Osteoblasts 100 200 300 400 Cell populations 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 100 200 300 400 −10 −8 −6 −4 −2 Bone mass Time [days] Density

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Biological background Computational modeling of bone pathologies Results

ODE model

˙ x1 =α1x1g11x2g21 − β1x1 ˙ x2 =α2x1g12x2g22 − β2x2 ˙ z = − k1x1 + k2x2

  • 1. Modeling a portion of BMU

Parameter sensitivity and identifiability New parameters after model fitting

  • perations

100 200 300 400 200 400 600 800 1000 1200 1400 Osteoblast fit time [days] fitted

  • riginal

100 200 300 400 −30 −25 −20 −15 −10 −5 Bone Density fit time [days] fitted

  • riginal

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Biological background Computational modeling of bone pathologies Results

ODE model

˙ x1 =α1x1g11x2g21 − β1x1 ˙ x2 =α2x1g12x2g22 − β2x2 ˙ z = − k1x1 + k2x2

  • 2. RANKL and aging paramaters

RANKL strongly affects the resorption phase Aging expressed as reduced cellular activity

100 200 300 400 10 20 30 40

x1

time 100 200 300 400 200 400 600 800

x2

time 100 200 300 400 −60 −50 −40 −30 −20 −10

Bone

time q05−q95 q25−q75

Sensitivity to Rankl Paoletti, Merelli, Li`

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Biological background Computational modeling of bone pathologies Results

ODE model

˙ x1 =α1x1g11x2g21 − β1x1 ˙ x2 =α2x1g12x2g22 − β2x2 ˙ z = − k1x1 + k2x2

New model

˙ x1 = α1x1g11x2g21/rankl − β1x1 ˙ x2 = α2x1g12x2g22 − β2x2 ˙ z = −agk1x1 + agk2x2

100 200 300 400 10 20 30 40

x1

time 100 200 300 400 200 400 600 800

x2

time 100 200 300 400 −60 −50 −40 −30 −20 −10

Bone

time q05−q95 q25−q75

Sensitivity to Rankl Paoletti, Merelli, Li`

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Biological background Computational modeling of bone pathologies Results

Stochastic model

From the modified ODE, we derive a CTMC model for the PRISM model checker. Original BMU portion States 21,021 5,616 (-73.28%) Transitions 103,060 27,345 (-73.47%)

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Biological background Computational modeling of bone pathologies Results

Stochastic model

From the modified ODE, we derive a CTMC model for the PRISM model checker. Original BMU portion States 21,021 5,616 (-73.28%) Transitions 103,060 27,345 (-73.47%)

Modified ODE model

˙ x1 = α1xg11

1 xg21/rankl 2

− β1x1 ˙ x2 = α2xg12

1 xg22 2

− β2x2 ˙ z = −ag · k1x1 + ag · k2x2

Stochastic model

[] x1 > 0 − → β1x1 : x1 = x1 − 1 [] x1 < maxx1 − → α1xg11

1 xg21/rankl 2

: x1 = x1+1 [resorb] x1 > 0 − → ag · k1x1 : true [] x2 > 0 − → β2x2 : x2 = x2 − 1 [] x2 < maxx2 − → α2xg12

1 xg22 2

: x2 = x2 + 1 [form] x2 > 0 − → ag · k2x2 : true

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Biological background Computational modeling of bone pathologies Results

Model checking bone pathologies

Diagnostic estimators:

1 Bone density monitor 2 Rapidity of density changes

Comparison of two configurations over a 4 years-time: healthy conf: rankl = 1 and ag = 1 pathological conf: rankl = 1.2 and ag = 2

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Biological background Computational modeling of bone pathologies Results

Outline

1

Biological background

2

Computational modeling of bone pathologies

3

Results

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Biological background Computational modeling of bone pathologies Results

Bone mineral density

f+(t) : R{′′boneFormed′′} =?[C ≤ t], f−(t) : R{′′boneResorbed′′} =?[C ≤ t], fBD(t) : f+(t) − f−(t), t = 0, 10, . . . , 1460.

healthy pathological

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Biological background Computational modeling of bone pathologies Results

Density change rate

fBD(t + ∆t) − fBD(t) ∆t , t = 0, 50, . . . , 1450.

Healthy Pathological

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Biological background Computational modeling of bone pathologies Results

Conclusions

Statistical analysis of ODE model for reducing the state space and incorporating RANKL and aging parameters Derivation of a stochastic model in PRISM Comparison of healthy and pathological configurations Probabilistic verification of bone pathologies with clinical estimators

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Biological background Computational modeling of bone pathologies Results

Conclusions

Statistical analysis of ODE model for reducing the state space and incorporating RANKL and aging parameters Derivation of a stochastic model in PRISM Comparison of healthy and pathological configurations Probabilistic verification of bone pathologies with clinical estimators

Paoletti, Merelli, Li`

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Biological background Computational modeling of bone pathologies Results

Conclusions

Statistical analysis of ODE model for reducing the state space and incorporating RANKL and aging parameters Derivation of a stochastic model in PRISM Comparison of healthy and pathological configurations Probabilistic verification of bone pathologies with clinical estimators

Paoletti, Merelli, Li`

  • . Formal Analysis of Bone Clinical Pathologies

14/14

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Biological background Computational modeling of bone pathologies Results

Conclusions

Statistical analysis of ODE model for reducing the state space and incorporating RANKL and aging parameters Derivation of a stochastic model in PRISM Comparison of healthy and pathological configurations Probabilistic verification of bone pathologies with clinical estimators

Paoletti, Merelli, Li`

  • . Formal Analysis of Bone Clinical Pathologies

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