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` o, Emanuela Merelli NETTAB 2011 , Pavia, 13 October 2011 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 1/14
Biological background Computational modeling of bone pathologies Results Outline Biological background 1 Computational modeling of bone pathologies 2 Results 3 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 2/14
Biological background Computational modeling of bone pathologies Results Outline Biological background 1 Computational modeling of bone pathologies 2 Results 3 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 3/14
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 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 4/14
Biological background Computational modeling of bone pathologies Results Key events in BR (1/2) Osteocytes (1) send signals to the fluid part, activating Pre- osteoblasts (2) ( Pb ) and Pre-osteoclasts (3) ( Pc ) Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 5/14
Biological background Computational modeling of bone pathologies Results Key events in BR (1/2) Pb s express RANKL (4) . Pc s express RANK (5) receptor. Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 5/14
Biological background Computational modeling of bone pathologies Results Key events in BR (1/2) RANK/RANKL binding (6) induces Pcs’ proliferation (7) . Pc s en- large and fuse, forming mature Osteoclasts (8) which start bone Resorption (9) . Mature osteoblasts express the decoy receptor OPG (10) . Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 5/14
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. Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 6/14
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. Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 6/14
Biological background Computational modeling of bone pathologies Results Key events in BR (2/2) Resting (14) : the initial situation is re-established Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 6/14
Biological background Computational modeling of bone pathologies Results Outline Biological background 1 Computational modeling of bone pathologies 2 Results 3 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 7/14
Biological background Computational modeling of bone pathologies Results ODE model Osteoclasts x 1 = α 1 x 1 g 11 x 2 g 21 − β 1 x 1 ˙ Osteoblasts x 2 = α 2 x 1 g 12 x 2 g 22 − β 2 x 2 ˙ Bone mass z = − k 1 x 1 + k 2 x 2 ˙ Cell populations Cell populations Bone mass 30 1400 Osteoclasts 0 Osteoblasts 1300 1200 25 −2 1100 1000 20 900 −4 800 Density 15 700 600 −6 500 10 400 −8 300 5 200 100 −10 0 0 0 100 200 300 400 0 0 100 100 200 200 300 300 400 400 Time [days] Time [days] Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 8/14
Biological background Computational modeling of bone pathologies Results ODE model 1. Modeling a portion of BMU Parameter sensitivity and x 1 = α 1 x 1 g 11 x 2 g 21 − β 1 x 1 ˙ identifiability x 2 = α 2 x 1 g 12 x 2 g 22 − β 2 x 2 ˙ New parameters after model fitting operations z = − k 1 x 1 + k 2 x 2 ˙ Osteoblast fit Bone Density fit 1400 0 1200 −5 1000 −10 800 −15 600 −20 400 −25 200 −30 fitted fitted 0 original original 0 100 200 300 400 0 100 200 300 400 time [days] time [days] Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 8/14
Biological background Computational modeling of bone pathologies Results ODE model 2. RANKL and aging paramaters RANKL strongly affects the x 1 = α 1 x 1 g 11 x 2 g 21 − β 1 x 1 ˙ resorption phase x 2 = α 2 x 1 g 12 x 2 g 22 − β 2 x 2 ˙ Aging expressed as reduced cellular z = − k 1 x 1 + k 2 x 2 ˙ activity Sensitivity to Rankl x1 x2 Bone 40 800 0 −10 30 600 −20 −30 20 400 −40 200 10 −50 q05−q95 q25−q75 0 0 −60 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 time time time Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 8/14
Biological background Computational modeling of bone pathologies Results ODE model New model x 1 = α 1 x 1 g 11 x 2 g 21 − β 1 x 1 x 1 = α 1 x 1 g 11 x 2 g 21 / rankl − β 1 x 1 ˙ ˙ x 2 = α 2 x 1 g 12 x 2 g 22 − β 2 x 2 x 2 = α 2 x 1 g 12 x 2 g 22 − β 2 x 2 ˙ ˙ z = − ag k 1 x 1 + ag k 2 x 2 ˙ z = − k 1 x 1 + k 2 x 2 ˙ Sensitivity to Rankl x1 x2 Bone 40 800 0 −10 30 600 −20 −30 20 400 −40 200 10 −50 q05−q95 q25−q75 0 0 −60 0 100 200 300 400 0 100 200 300 400 0 100 200 300 400 time time time Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 8/14
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%) Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 9/14
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 Stochastic model 1 x g 21 / rankl [] x 1 > 0 − → β 1 x 1 : x 1 = x 1 − 1 x 1 = α 1 x g 11 ˙ − β 1 x 1 2 1 x g 21 / rankl → α 1 x g 11 x 2 = α 2 x g 12 1 x g 22 [] x 1 < max x 1 − : x 1 = x 1 +1 ˙ − β 2 x 2 2 2 [ resorb ] x 1 > 0 − → ag · k 1 x 1 : true z = − ag · k 1 x 1 + ag · k 2 x 2 ˙ [] x 2 > 0 − → β 2 x 2 : x 2 = x 2 − 1 → α 2 x g 12 1 x g 22 [] x 2 < max x 2 − : x 2 = x 2 + 1 2 [ form ] x 2 > 0 − → ag · k 2 x 2 : true Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 9/14
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 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 10/14
Biological background Computational modeling of bone pathologies Results Outline Biological background 1 Computational modeling of bone pathologies 2 Results 3 Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 11/14
Biological background Computational modeling of bone pathologies Results Bone mineral density f + ( t ) : R { ′′ boneFormed ′′ } =?[ C ≤ t ] , f − ( t ) : R { ′′ boneResorbed ′′ } =?[ C ≤ t ] , f BD ( t ) : f + ( t ) − f − ( t ) , t = 0 , 10 , . . . , 1460 . healthy pathological Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 12/14
Biological background Computational modeling of bone pathologies Results Density change rate f BD ( t + ∆ t ) − f BD ( t ) t = 0 , 50 , . . . , 1450 . , ∆ t Healthy Pathological Paoletti, Merelli, Li` o . Formal Analysis of Bone Clinical Pathologies 13/14
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` o . Formal Analysis of Bone Clinical Pathologies 14/14
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` o . Formal Analysis of Bone Clinical Pathologies 14/14
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