Model Mathematics

sovr_ze = \{\begin{matrix} \frac{len_thick}{2} if \frac{len_thick}{2} < \frac{SL}{2} \\ \frac{SL}{2} otherwise \end{matrix} sovr_cle = \{\begin{matrix} \frac{SL}{2} - (SL - len_thin) if \frac{SL}{2} - (SL - len_thin) > \frac{len_hbare}{2} \\ \frac{len_hbare}{2} otherwise \end{matrix} len_sovr = sovr_ze - sovr_cle SOVFThick = \frac{len_sovr 2}{len_thick - len_hbare} SOVFThin = \frac{len_sovr}{len_thin}

konT = kon {Qkon}^{\frac{TmpC - 37}{10}} koffLT = koffL koffmod {Qkoff}^{\frac{TmpC - 37}{10}} koffHT = koffH koffmod {Qkoff}^{\frac{TmpC - 37}{10}} kn_pT = kn_p permtot {Qkn_p}^{\frac{TmpC - 37}{10}} kp_nT = kp_n inprmt {Qkp_n}^{\frac{TmpC - 37}{10}} permtot = \sqrt{| \frac{1}{1 + {(\frac{perm50}{Tropreg})}^{nperm}} |} inprmt = \{\begin{matrix} \frac{1}{permtot} if \frac{1}{permtot} < 100 \\ 100 otherwise \end{matrix} \frac{d}{d time} (TRPNCaL) = dTRPNCaL dTRPNCaL = konT Cai (1 - TRPNCaL) - (koffLT TRPNCaL) \frac{d}{d time} (TRPNCaH) = dTRPNCaH dTRPNCaH = konT Cai (1 - TRPNCaH) - (koffHT TRPNCaH) Tropreg = (1 - SOVFThin) TRPNCaL + SOVFThin TRPNCaH

fappT = fapp xbmodsp {Qfapp}^{\frac{TmpC - 37}{10}} gapslmd = 1 + (1 - SOVFThick) gslmod gappT = gapp gapslmd xbmodsp {Qgapp}^{\frac{TmpC - 37}{10}} hfmd = ⅇ^{\frac{- xXBprer}{| xXBprer |} hfmdc {(\frac{xXBprer}{x_0})}^{2}} hbmd = ⅇ^{\frac{xXBpostr - x_0}{| xXBpostr - x_0 |} hbmdc {(\frac{xXBpostr - x_0}{x_0})}^{2}} hfT = hf hfmd xbmodsp {Qhf}^{\frac{TmpC - 37}{10}} hbT = hb hbmd xbmodsp {Qhb}^{\frac{TmpC - 37}{10}} gxbmd = \{\begin{matrix} ⅇ^{sigmap {(\frac{x_0 - xXBpostr}{x_0})}^{2}} if xXBpostr < x_0 \\ ⅇ^{sigman {(\frac{xXBpostr - x_0}{x_0})}^{2}} otherwise \end{matrix} gxbT = gxb gxbmd xbmodsp {Qgxb}^{\frac{TmpC - 37}{10}}

\frac{d}{d time} (N_NoXB) = kp_nT P_NoXB - (kn_pT N_NoXB) \frac{d}{d time} (P_NoXB) = kn_pT N_NoXB - (kp_nT P_NoXB) \frac{d}{d time} (N) = kp_nT P - (kn_pT N) \frac{d}{d time} (XBprer) = dXBprer dXBprer = fappT P + hbT XBpostr - (gappT XBprer + hfT XBprer) \frac{d}{d time} (XBpostr) = dXBpostr dXBpostr = hfT XBprer - (hbT XBpostr + gxbT XBpostr) P = 1 - N - XBprer - XBpostr

\frac{d}{d time} (xXBprer) = dxXBprer dxXBprer = \frac{dSL}{2} + \frac{xPsi}{dutyprer} (fappT (- xXBprer) + hbT (xXBpostr - (x_0 + xXBprer))) \frac{d}{d time} (xXBpostr) = dxXBpostr dxXBpostr = \frac{dSL}{2} + \frac{xPsi}{dutypostr} hfT (xXBprer + x_0 - xXBpostr) dutyprer = \frac{hbT fappT + gxbT fappT}{fappT hfT + gxbT hfT + gxbT gappT + hbT fappT + hbT gappT + gxbT fappT} dutypostr = \frac{fappT hfT}{fappT hfT + gxbT hfT + gxbT gappT + hbT fappT + hbT gappT + gxbT fappT}

SSXBprer = \frac{hb fapp + gxb fapp}{fapp hf + gxb hf + gxb gapp + hb fapp + hb gapp + gxb fapp} SSXBpostr = \frac{fapp hf}{fapp hf + gxb hf + gxb gapp + hb fapp + hb gapp + gxb fapp} Fnordv = kxb x_0 SSXBpostr force = kxb SOVFThick (xXBpostr XBpostr + xXBprer XBprer) active = \frac{1 force}{Fnordv} ppforce_t = \frac{SL - SLrest}{| SL - SLrest |} PCon_t (ⅇ^{PExp_t | SL - SLrest |} - 1) ppforce_c = \{\begin{matrix} PCon_c (ⅇ^{PExp_c | SL - SL_c |} - 1) if SL > SL_c \\ 0 otherwise \end{matrix} ppforce = ppforce_t + ppforce_c preload = \frac{| SLset - SLrest |}{SLset - SLrest} PCon_t (ⅇ^{PExp_t | SLset - SLrest |} - 1) afterload = \{\begin{matrix} KSE (SLset - SL) if SEon = 1 \\ 0 otherwise \end{matrix} \frac{d}{d time} (SL) = dSL dSL = \{\begin{matrix} \frac{intf + (SLset - SL) visc}{massf} if SL ≦ SLmax \land SL > SLmin \\ 0 otherwise \end{matrix} \frac{d}{d time} (intf) = preload + afterload - (ppforce + active)

FrSBXB = \frac{XBpostr + XBprer}{SSXBpostr + SSXBprer} dFrSBXB = \frac{dXBpostr + dXBprer}{SSXBpostr + SSXBprer} dsovr_ze = \{\begin{matrix} (- 0.5) dSL if SL < len_thick \\ 0 otherwise \end{matrix} dsovr_cle = \{\begin{matrix} (- 0.5) dSL if 2 len_thin - SL > len_hbare \\ 0 otherwise \end{matrix} dlen_sovr = dsovr_ze - dsovr_cle dSOVFThin = \frac{dlen_sovr}{len_thin} dSOVFThick = \frac{2 dlen_sovr}{len_thick - len_hbare} TropTot = Trop_conc ((1 - SOVFThin) TRPNCaL + SOVFThin (FrSBXB TRPNCaH + (1 - FrSBXB) TRPNCaL)) dTropTot = Trop_conc ((- dSOVFThin) TRPNCaL + (1 - SOVFThin) dTRPNCaL + dSOVFThin (FrSBXB TRPNCaH + (1 - FrSBXB) TRPNCaL) + SOVFThin (dFrSBXB TRPNCaH + FrSBXB dTRPNCaH + (1 - FrSBXB) dTRPNCaL - (dFrSBXB TRPNCaL))) dforce = kxb dSOVFThick (xXBpostr XBpostr + xXBprer XBprer) + kxb SOVFThick (dxXBpostr XBpostr + xXBpostr dXBpostr + dxXBprer XBprer + xXBprer dXBprer)

beta = {(\frac{tau1}{tau2})}^{\frac{- 1}{\frac{tau1}{tau2} - 1}} - ({(\frac{tau1}{tau2})}^{\frac{- 1}{1 - (\frac{tau2}{tau1})}}) Cai = \{\begin{matrix} \frac{Ca_amplitude - Ca_diastolic}{beta} (ⅇ^{\frac{- (time - start_time)}{tau1}} - (ⅇ^{\frac{- (time - start_time)}{tau2}})) + Ca_diastolic if time > start_time \\ Ca_diastolic otherwise \end{matrix}