Model Mathematics

\frac{d}{d time} (wi) = 1 Lp (Nai + Ki + Cli + HCO3i + \frac{Xi}{wi} - (Nao + Ko + Clo + HCO3o))

wo = w_tot - wi

\frac{d}{d time} (N_Nai) = dN_Nai_dt

dN_Nai_dt = theta_NKCC1 J_NKCC1 + theta_NBC J_NBC - (1e10 (\frac{gNa}{F} (Vm - ENa) + 3 J_NaKATPase))

\frac{d}{d time} (N_Ki) = dN_Ki_dt

dN_Ki_dt = theta_NKCC1 J_NKCC1 + theta_KCC1 J_KCC1 + 1e10 2 J_NaKATPase - (\frac{1e10 gK}{F} (Vm - EK))

\frac{d}{d time} (N_HCO3i) = dN_HCO3i_dt

dN_HCO3i_dt = 2 theta_NBC J_NBC

\frac{d}{d time} (N_Cli) = dN_Ki_dt + dN_Nai_dt + dN_HCO3i_dt

\frac{d}{d time} (N_Nao) = 1e10 (\frac{gNa}{F} (Vm - ENa) + 3 J_NaKATPase) - (theta_NKCC1 J_NKCC1 + theta_NBC J_NBC + kc y)

\frac{d}{d time} (N_Ko) = \frac{1e10 gK}{F} (Vm - EK) + kc y - (theta_NKCC1 J_NKCC1 + theta_KCC1 J_KCC1 + 1e10 2 J_NaKATPase)

\frac{d}{d time} (N_HCO3o) = (- 2) theta_NBC J_NBC

N_Clo = N_Nao + N_Ko - N_HCO3o

ENa = \frac{R T}{1 F} \ln (\frac{Nao}{Nai}) EK = \frac{R T}{1 F} \ln (\frac{Ko}{Ki}) ECl = \frac{R T}{(- 1) F} \ln (\frac{Clo}{Cli})

J_KCC1 = \frac{\frac{1e10 gKCC1 ft}{F} R T}{F} \ln (\frac{Ko Clo}{Ki Cli}) ft = \{\begin{matrix} 1 if time \geq 10 \land time < 20 \\ 0 otherwise \end{matrix}

J_NBC = \frac{1e10 gNBC}{F} (Vm - ENBC)

J_NKCC1 = \frac{\frac{1e10 gNKCC1 ft}{F} R T}{F} \ln (\frac{\frac{Nao}{Nai} Ko}{Ki} {(\frac{Clo}{Cli})}^{2}) ft = \{\begin{matrix} 1 if time \geq 10 \land time < 20 \\ 0 otherwise \end{matrix}

ENBC = \frac{R T}{zNBC F} \ln (\frac{\frac{Nao}{Nai} {HCO3o}^{2}}{{HCO3i}^{2}})

Vm = \frac{gNa ENa + gK EK + gCl ECl + theta_NBC gNBC ENBC - (J_NaKATPase F)}{gNa + gK + gCl + theta_NBC gNBC}

J_NaKATPase = \frac{\frac{Pmax {Nai}^{1.5}}{{Nai}^{1.5} + {K_Nai}^{1.5}} Ko}{Ko + K_Ko}

I_K = \frac{1e10 2 J_NaKATPase F}{elementary_charge nA} - (\frac{1e10 gK}{elementary_charge nA} (Vm - EK))

I_Cl = \frac{1e10 gCl}{elementary_charge nA} (Vm - ECl)

I_Na = J_NBC - (\frac{1e10 gNa}{elementary_charge nA} (Vm - ENa) + \frac{1e10 3 J_NaKATPase F}{elementary_charge nA})

y = \{\begin{matrix} \frac{gamma_alpha_beta}{gamma_alpha gamma_beta} {(1 - (\frac{time - t0}{deltaT}))}^{beta - 1} {(\frac{time - t0}{deltaT})}^{alpha - 1} if time \geq t0 \land time ≦ t1 \\ -1 if time > t1 \land time < t2 \\ 0 otherwise \end{matrix} deltaT = t1 - t0 gamma_alpha_beta = (alpha + beta - 1)! gamma_alpha = (alpha - 1)! gamma_beta = (beta - 1)! Ki = \{\begin{matrix} \frac{N_Ki}{wi} if \frac{N_Ki}{wi} > 0 \\ 1e-180 otherwise \end{matrix} Ko = \{\begin{matrix} \frac{N_Ko}{wo} if \frac{N_Ko}{wo} > 0 \\ 1e-180 otherwise \end{matrix} Nai = \{\begin{matrix} \frac{N_Nai}{wi} if \frac{N_Nai}{wi} > 0 \\ 1e-180 otherwise \end{matrix} Nao = \{\begin{matrix} \frac{N_Nao}{wo} if \frac{N_Nao}{wo} > 0 \\ 1e-180 otherwise \end{matrix} Cli = \{\begin{matrix} \frac{N_Cli}{wi} if \frac{N_Cli}{wi} > 0 \\ 1e-180 otherwise \end{matrix} Clo = \{\begin{matrix} \frac{N_Clo}{wo} if \frac{N_Clo}{wo} > 0 \\ 1e-180 otherwise \end{matrix} HCO3i = \{\begin{matrix} \frac{N_HCO3i}{wi} if \frac{N_HCO3i}{wi} > 0 \\ 1e-180 otherwise \end{matrix} HCO3o = \{\begin{matrix} \frac{N_HCO3o}{wo} if \frac{N_HCO3o}{wo} > 0 \\ 1e-180 otherwise \end{matrix}