Model Mathematics

\frac{d}{d time} (OB_p) = (D_OB_u pi_TGF_beta_act - (D_OB_p pi_TGF_beta_rep OB_p)) + pd_OB_p

\frac{d}{d time} (OB_a) = (D_OB_p pi_TGF_beta_rep OB_p - (A_OB_a OB_a)) + pd_OB_a

\frac{d}{d time} (OC_a) = (D_OC_p pi_RANKL_act - (A_OC_a pi_TGF_beta_act OC_a)) + pd_OC_a

D_OC_p = \{\begin{matrix} 2.100 e -2 if time > 0.0 \land time ≦ 100.0 \\ 2.100 e -3 otherwise \end{matrix}

\frac{d}{d time} (BV) = k_form (OB_a - OB_a_initial) - (k_res (OC_a - OC_a_initial))

pi_TGF_beta_act = \frac{alpha OC_a}{KD_TGF_activate + alpha OC_a}

pi_TGF_beta_rep = \frac{1}{1 + \frac{alpha OC_a}{KD_TGF_repress}}

pi_PTH_act = \frac{PTH_tot}{PTH_tot + KD_PTH_activate}

pi_PTH_rep = \frac{1}{1 + \frac{PTH_tot}{KD_PTH_repress}}

PTH_tot = \frac{beta_PTH + P_PTH_d}{Deg_PTH}

OPG = \frac{P_OPG_d + OPG_eff}{\frac{OPG_eff}{OPG_max} + Deg_OPG}

OPG_eff = (beta1_OB_p_OPG beta_OPG OB_p + beta2_OB_a_OPG beta_OPG OB_a) pi_PTH_rep

prod_OPG = OPG_eff (1 - (\frac{OPG}{OPG_max})) + P_OPG_d

Degr_OPG = Deg_OPG OPG

RANKL = \frac{beta_RANKL + P_RANKL_d}{RANKL_tot (\frac{beta_RANKL}{RANKL_eff} + Deg_RANKL)}

RANKL_eff = (K_RANKL_OB_p OB_p + K_RANKL_OB_a OB_a) pi_PTH_act

K_RANKL_OB_p = R1_OB_p_RANKL R_RANKL

K_RANKL_OB_a = R2_OB_a_RANKL R_RANKL

RANKL_tot = 1 + K_A2_RANKL RANK + K_A1_RANKL OPG

pd_RANKL = (beta_RANKL (1 - (RANKL \frac{RANKL_tot}{RANKL_eff})) - (Deg_RANKL RANKL RANKL_tot)) + P_RANKL_d

pi_RANKL_act = \frac{RANKL_RANK}{KD_RANKL_activate + RANKL_RANK}

OPG_RANKL = K_A1_RANKL OPG RANKL

RANKL_RANK = K_A2_RANKL RANKL RANK