Model Mathematics

DifferOBpNormal = \frac{D_OBp_t0 1}{1 + \frac{TGFbNormal}{KD_TGF_beta_repress}} ProlifOBpNormal = \frac{frac_prolifOBp_vs_differOBu DifferOBpNormal}{1 - (\frac{OBpNormal}{OBp_sat})} DifferOBu = (1 - frac_prolifOBp_vs_differOBu) D_OBu_t0 Pi_TGFbeta_OBu_act_t0 ProlifOBp = ProlifOBpNormal (1 - (\frac{OBp}{OBp_sat})) OBp_in = DifferOBu OBu_t0 + \frac{ProlifOBp OBp Pi_WNT}{Pi_WNT_0} + pd_OBp_t0 OBp_out = D_OBp_t0 Pi_TGFbeta_OBp_rep OBp \frac{d}{d time} (OBp) = OBp_in - OBp_out

OBa_in = OBp_out + pd_OBa_t0 OBa_out = (D_OBa_t0 + A_OBa_t0) OBa \frac{d}{d time} (OBa) = OBa_in - OBa_out

OCp_in = D_OCu_t0 OCu_t0 Pi_RANKL_act_OCu + pd_OCp_t0 OCp_out = D_OCp_t0 Pi_RANKL_act_OCp OCp \frac{d}{d time} (OCp) = OCp_in - OCp_out

OCa_in = OCp_out + pd_OCa_t0 OCa_out = A_OCa_t0 Pi_TGFbeta_OCa_act OCa \frac{d}{d time} (OCa) = OCa_in - OCa_out

dfbmdt = K_form OBa - (K_res OCa) \frac{d}{d time} (fbm) = dfbmdt

OCY_act = fact_0 fbm OCY_act_0 = fact_0 20 \frac{d}{d time} (OCY) = fact_0 dfbmdt

\frac{d}{d time} (vm) = XKAPPA (vmmax - vm) - (\frac{vm K_form OBa}{fbm})

PTH_tot = \frac{Beta_PTH + P_PTH_d}{Deg_PTH} Pi_PTH_act = \frac{PTH_tot}{PTH_tot + KD_PTH_act} Pi_PTH_rep = \frac{1}{1 + \frac{PTH_tot}{KD_PTH_rep}}

RANK = \frac{N_RANK_OCp OCp}{1 + \frac{1}{K_RANK_RANKL} RANKL} OPG = \frac{Beta_OPG Pi_PTH_rep OBa}{\frac{Beta_OPG Pi_PTH_rep OBa}{OPG_max} + Deg_OPG + \frac{Deg_OPG_RANKL}{K_OPG_RANKL} RANKL} RANKL_tot = RANKL (1 + \frac{RANK}{K_RANK_RANKL} + \frac{OPG}{K_OPG_RANKL}) RANKL_max = N_RANKL_OBp_max OBp_t0 P_RANKL1 = Beta_RANKL_OCY (1 - (\frac{RANKL_tot}{RANKL_max})) OCY_act P_RANKL2 = Beta_RANKL_OBp Pi_NO_PTH_act_rep (1 - (\frac{RANKL_tot}{RANKL_max})) OBp P_RANKL = P_RANKL1 + P_RANKL2 RANKL = \frac{P_RANKL + P_RANKL_d}{Deg_RANKL + \frac{Deg_RANK_RANKL}{K_RANK_RANKL} RANK + \frac{Deg_OPG_RANKL}{K_OPG_RANKL} OPG} Pi_RANKL_act_OCp = \frac{RANKL}{KD_RANKL_act_OCp + RANKL} Pi_RANKL_act_OCu = \frac{RANKL}{KD_RANKL_act_OCu + RANKL}

TGF_beta_t0 = Alpha OCa_t0 Pi_TGFbeta_OBu_act_t0 = \frac{TGF_beta_t0}{KD_TGF_beta_activate + TGF_beta_t0} TGF_beta = Alpha OCa Pi_TGFbeta_OBu_act = \frac{TGF_beta}{KD_TGF_beta_activate + TGF_beta} Pi_TGFbeta_OCa_act = \frac{TGF_beta}{KD_TGF_beta_activate + TGF_beta} Pi_TGFbeta_OBp_rep = \frac{1}{1 + \frac{TGF_beta}{KD_TGF_beta_repress}}

Beta_Scl = Beta_Scl_0 OCYprod = Beta_Scl OCY_act Pi_eps_rep P_Scl_b = OCYprod (1 - (\frac{Scl}{Scl_max})) LRP5tot = LRP5perCell OBp A = \frac{1}{KD_SclLRP5} (Deg_Scl + \frac{OCYprod}{Scl_max}) B = A KD_SclLRP5 (1 + \frac{Wnt}{KD_WntLRP5}) + \frac{Deg_SclLRP5 LRP5tot}{KD_SclLRP5} - (\frac{P_Scl_d + OCYprod}{KD_SclLRP5}) C = (- (P_Scl_d + OCYprod)) (1 + \frac{Wnt}{KD_WntLRP5}) Scl = \frac{(- B) + \sqrt{B^{2} - (4 A C)}}{2 A} OCYprod_0 = Beta_Scl_0 OCY_act_0 Pi_eps_rep_stst P_Scl_b_0 = OCYprod_0 (1 - (\frac{Scl_0}{Scl_max})) LRP5tot_0 = LRP5perCell OBp_t0 A_0 = \frac{1}{KD_SclLRP5} (Deg_Scl + \frac{OCYprod_0}{Scl_max}) B_0 = A_0 KD_SclLRP5 (1 + \frac{Wnt_0}{KD_WntLRP5}) + \frac{Deg_SclLRP5 LRP5tot_0}{KD_SclLRP5} - (\frac{P_Scl_d + OCYprod_0}{KD_SclLRP5}) C_0 = (- (P_Scl_d + OCYprod_0)) (1 + \frac{Wnt_0}{KD_WntLRP5}) Scl_0 = \frac{(- B_0) + \sqrt{{B_0}^{2} - (4 A_0 C_0)}}{2 A_0}

Wnt = Wnt_0 Pi_WNT = \frac{Wnt}{KD_WntLRP5 (1 + \frac{Wnt}{KD_WntLRP5} + \frac{Scl}{KD_SclLRP5})} Pi_WNT_0 = \frac{Wnt_0}{KD_WntLRP5 (1 + \frac{Wnt_0}{KD_WntLRP5} + \frac{Scl_0}{KD_SclLRP5})}

PTH_tot = \frac{Beta_PTH + P_PTH_d}{Deg_PTH} Beta_NO = Beta_NO_0 NO_tot = \frac{Beta_NO Pi_eps_act OCY_act + P_NO_d}{Deg_NO + \frac{Beta_NO Pi_eps_act OCY_act}{NO_max}} NO_eq = \frac{Beta_NO_0 Pi_eps_act_stst OCY_act_0}{Deg_NO + \frac{Beta_NO_0 Pi_eps_act_stst OCY_act_0}{NO_max}} KD_rep_NO = \frac{NO_eq}{aa} Pi_PTH_act = \frac{PTH_tot}{PTH_tot + KD_PTH_act} Pi_NO_rep = \frac{1}{1 + \frac{NO_tot}{KD_rep_NO}} Pi_NO_PTH_act_rep = lambda_s (Pi_PTH_act + Pi_NO_rep) + lambda_c Pi_PTH_act Pi_NO_rep

sig_1 = 0.0 sig_2 = 0.0 sig_4 = 0.0 sig_5 = 0.0 sig_6 = 0.0 sig_macro = sig_macro_t0 + de_sig_macro sig_3 = sig_macro dens_tis = 1 + (rho_m - 1) vm + (rho_o - 1) vo fvas = 1 - (\frac{fbm}{100}) dens = \frac{dens_tis fbm}{100} dens_1 = \frac{dens}{1} E_mod = \{\begin{matrix} 2014 {dens_1}^{2.5} if dens_1 < 1.2 \\ 1763 {dens_1}^{3.2} otherwise \end{matrix} S_11 = \frac{1}{E_mod} S_12 = \frac{- nu}{E_mod} S_44 = \frac{2 + 2 nu}{E_mod} eps_1 = S_11 sig_1 + S_12 sig_2 + S_12 sig_3 eps_2 = S_12 sig_1 + S_11 sig_2 + S_12 sig_3 eps_3 = S_12 sig_1 + S_12 sig_2 + S_11 sig_3 eps_4 = S_44 sig_4 eps_5 = S_44 sig_5 eps_6 = S_44 sig_6 SED = 0.5 (sig_1 eps_1 + sig_2 eps_2 + sig_3 eps_3 + sig_4 eps_4 + sig_5 eps_5 + sig_6 eps_6) SED_bm = \frac{SED}{{(1 - fvas)}^{2}} valeq = omega deltaAct = taueq {(\frac{alphaAct - valeq}{valeq - rhoAct})}^{\frac{1}{gammaAct}} Pi_eps_act = rhoAct + \frac{(alphaAct - rhoAct) {SED_bm}^{gammaAct}}{{deltaAct}^{gammaAct} + {SED_bm}^{gammaAct}} Pi_eps_act_stst = rhoAct + \frac{(alphaAct - rhoAct) {taueq}^{gammaAct}}{{deltaAct}^{gammaAct} + {taueq}^{gammaAct}} deltaRep = taueq {(\frac{valeq - rhoRep}{alphaRep - valeq})}^{\frac{1}{gammaRep}} Pi_eps_rep = alphaRep - (\frac{(alphaRep - rhoRep) {SED_bm}^{gammaRep}}{{deltaRep}^{gammaRep} + {SED_bm}^{gammaRep}}) Pi_eps_rep_stst = alphaRep - (\frac{(alphaRep - rhoRep) {taueq}^{gammaRep}}{{deltaRep}^{gammaRep} + {taueq}^{gammaRep}})