Model Mathematics

\frac{d}{d time} (OB_p) = D_OB_u \frac{TGF_beta}{K_D1_TGF_beta + TGF_beta} - (D_OB_p OB_p \frac{1}{1 + \frac{TGF_beta}{K_D2_TGF_beta}})

\frac{d}{d time} (OB_a) = D_OB_p OB_p \frac{1}{1 + \frac{TGF_beta}{K_D2_TGF_beta}} - (A_OB_a OB_a)

\frac{d}{d time} (OC_a) = D_OC_p OC_p pi_RANKL_act_OC_p - (A_OC_a OC_a \frac{TGF_beta}{K_D3_TGF_beta + TGF_beta})

\frac{d}{d time} (BV) = k_form OB_a_tilde - (k_res OC_a_tilde)

OC_a_tilde = OC_a - OC_a_initial

OB_a_tilde = OB_a - OB_a_initial

TGF_beta = \frac{alpha k_res OC_a + S_TGF_beta}{D_TGF_beta_tilde}

PTH = \frac{beta_PTH + P_PTH_d}{D_PTH_tilde}

pi_PTH_act_OB_p = \frac{PTH}{K_D4_PTH + PTH}

pi_PTH_act_OB_a = \frac{PTH}{K_D5_PTH + PTH}

pi_PTH_rep_OB_p = \frac{1}{1 + \frac{PTH}{K_D6_PTH}}

pi_PTH_rep_OB_a = \frac{1}{1 + \frac{PTH}{K_D7_PTH}}

OPG = \frac{(beta1_OPG OB_p + beta2_OPG OB_a) pi_PTH_rep_OB + P_OPG_d}{\frac{(beta1_OPG OB_p + beta2_OPG OB_a) pi_PTH_rep_OB}{OPG_max} + D_OPG_tilde}

P_OPG_e = (beta1_OPG OB_p + beta2_OPG OB_a) pi_PTH_rep_OB (1 - (\frac{OPG}{OPG_max}))

pi_PTH_rep_OB = pi_PTH_rep_OB_p

RANKL = \frac{RANKL_eff}{1 + K_A1_RANKL OPG + K_A2_RANKL RANK} \frac{beta_RANKL + P_RANKL_d}{beta_RANKL + D_RANKL_tilde RANKL_eff}

RANKL_eff = (R1_RANKL OB_p + R2_RANKL OB_a) pi_PTH_act_OB

RANKL_tot = RANKL (1 + K_A1_RANKL OPG + K_A2_RANKL RANK)

pi_RANKL_act_OC_p = \frac{RANKL}{K_D8_RANKL + RANKL}

P_RANKL = P_RANKL_e + P_RANKL_d

P_RANKL_e = beta_RANKL (1 - (\frac{RANKL_tot}{RANKL_eff}))

D_RANKL = (- D_RANKL_tilde) RANKL_tot

pi_PTH_act_OB = pi_PTH_act_OB_p