Model Mathematics

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v_in_W = \{\begin{matrix} water_intake if t > water_intake_start \land t < water_intake_start + water_intake_duration \\ 0 otherwise \end{matrix} v_out1_W = water_gi_excretion v_in_W_base v_out2_W = water_kidney_excretion v_in_W_base v_in_Na = \{\begin{matrix} v_in_Na_base + Na_intake if t > Na_intake_start \land t < Na_intake_start + Na_intake_duration \\ v_in_Na_base otherwise \end{matrix} v_out1_Na = Na_gi_excretion v_in_Na_base v_out2_Na = Na_kidney_excretion v_in_Na_base \frac{d}{d t} (q_gi_W) = v_in_W - (RT k_gi_W (\frac{q_vc_Na}{q_vc_W} - (\frac{q_gi_Na}{q_gi_W}))) - v_out1_W \frac{d}{d t} (q_pt_W) = k_gl_W (\frac{E_ac (q_ac_W - U_ac_W)}{{(L_ac_W - q_ac_W)}^{2}} - (\frac{E_pt (q_pt_W - U_pt_W)}{{(L_pt_W - q_pt_W)}^{2}})) - (RT k_pt_W (\frac{q_vc_Na}{q_vc_W} - (\frac{q_pt_Na}{q_pt_W}))) - v_out2_W \frac{d}{d t} (q_vc_W) = k_cc_W (\frac{E_ac (q_ac_W - U_ac_W)}{{(L_ac_W - q_ac_W)}^{2}} - (\frac{E_vc (q_vc_W - U_vc_W)}{{(L_vc_W - q_vc_W)}^{2}})) + RT k_gi_W (\frac{q_vc_Na}{q_vc_W} - (\frac{q_gi_Na}{q_gi_W})) + RT k_pt_W (\frac{q_vc_Na}{q_vc_W} - (\frac{q_pt_Na}{q_pt_W})) - v_lv_W \frac{d}{d t} (q_ac_W) = (v_lv_W + \frac{k_cc_W E_vc (q_vc_W - U_vc_W)}{{(L_vc_W - q_vc_W)}^{2}} - (\frac{(k_gl_W + k_cc_W) E_ac (q_ac_W - U_ac_W)}{{(L_ac_W - q_ac_W)}^{2}})) + \frac{k_gl_W E_pt (q_pt_W - U_pt_W)}{{(L_pt_W - q_pt_W)}^{2}} \frac{d}{d t} (q_gi_Na) = v_in_Na - (kK_gi_Na (q_gi_Na - q_giEpi_Na)) - v_out1_Na \frac{d}{d t} (q_giEpi_Na) = kK_gi_Na (q_gi_Na - q_giEpi_Na) - (\frac{3 k_giEpi_NKE ({(\frac{K_Na q_giEpi_Na}{q_giEpi_W})}^{3} {(\frac{K_K q_vc_K}{q_vc_W})}^{2} - ({(\frac{K_Na q_vc_Na}{q_vc_W})}^{3} {(\frac{K_K q_giEpi_K}{q_giEpi_W})}^{2} ⅇ^{\frac{2 F u_giEpi_e}{RT}}))}{(1 + {(\frac{K_Na q_giEpi_Na}{q_giEpi_W})}^{3}) (1 + {(\frac{K_Na q_vc_Na}{q_vc_W})}^{3}) (1 + {(\frac{K_K q_vc_K}{q_vc_W})}^{2}) (1 + {(\frac{K_K q_giEpi_K}{q_giEpi_W})}^{2})}) \frac{d}{d t} (q_pt_Na) = kK_gl_Na (q_vc_Na - q_pt_Na) - (kK_pt_Na (q_pt_Na - q_ptEpi_Na)) - v_out2_Na \frac{d}{d t} (q_ptEpi_Na) = kK_pt_Na (q_pt_Na - q_ptEpi_Na) - (\frac{3 k_ptEpi_NKE ({(\frac{K_Na q_ptEpi_Na}{q_ptEpi_W})}^{3} {(\frac{K_K q_vc_K}{q_vc_W})}^{2} - ({(\frac{K_Na q_vc_Na}{q_vc_W})}^{3} {(\frac{K_K q_ptEpi_K}{q_ptEpi_W})}^{2} ⅇ^{\frac{2 F u_ptEpi_e}{RT}}))}{(1 + {(\frac{K_Na q_ptEpi_Na}{q_ptEpi_W})}^{3}) (1 + {(\frac{K_Na q_vc_Na}{q_vc_W})}^{3}) (1 + {(\frac{K_K q_vc_K}{q_vc_W})}^{2}) (1 + {(\frac{K_K q_ptEpi_K}{q_ptEpi_W})}^{2})}) \frac{d}{d t} (q_vc_Na) = \frac{3 k_giEpi_NKE ({(\frac{K_Na q_giEpi_Na}{q_giEpi_W})}^{3} {(\frac{K_K q_vc_K}{q_vc_W})}^{2} - ({(\frac{K_Na q_vc_Na}{q_vc_W})}^{3} {(\frac{K_K q_giEpi_K}{q_giEpi_W})}^{2} ⅇ^{\frac{2 F u_giEpi_e}{RT}}))}{(1 + {(\frac{K_Na q_giEpi_Na}{q_giEpi_W})}^{3}) (1 + {(\frac{K_Na q_vc_Na}{q_vc_W})}^{3}) (1 + {(\frac{K_K q_vc_K}{q_vc_W})}^{2}) (1 + {(\frac{K_K q_giEpi_K}{q_giEpi_W})}^{2})} + \frac{3 k_ptEpi_NKE ({(\frac{K_Na q_ptEpi_Na}{q_ptEpi_W})}^{3} {(\frac{K_K q_vc_K}{q_vc_W})}^{2} - ({(\frac{K_Na q_vc_Na}{q_vc_W})}^{3} {(\frac{K_K q_ptEpi_K}{q_ptEpi_W})}^{2} ⅇ^{\frac{2 F u_ptEpi_e}{RT}}))}{(1 + {(\frac{K_Na q_ptEpi_Na}{q_ptEpi_W})}^{3}) (1 + {(\frac{K_Na q_vc_Na}{q_vc_W})}^{3}) (1 + {(\frac{K_K q_vc_K}{q_vc_W})}^{2}) (1 + {(\frac{K_K q_ptEpi_K}{q_ptEpi_W})}^{2})} - (kK_gl_Na (q_vc_Na - q_pt_Na)) u_vc_W = \frac{E_vc (q_vc_W - U_vc_W)}{{(L_vc_W - q_vc_W)}^{2}} u_ac_W = \frac{E_ac (q_ac_W - U_ac_W)}{{(L_ac_W - q_ac_W)}^{2}} u_pt_W = \frac{E_pt (q_pt_W - U_pt_W)}{{(L_pt_W - q_pt_W)}^{2}} c_vc_Na = \frac{q_vc_Na}{q_vc_W} c_gi_Na = \frac{q_gi_Na}{q_gi_W} c_pt_Na = \frac{q_pt_Na}{q_pt_W} c_giEpi_Na = \frac{q_giEpi_Na}{q_giEpi_W} c_ptEpi_Na = \frac{q_ptEpi_Na}{q_ptEpi_W} C_vc_Na = 1 e 3 c_vc_Na C_gi_Na = 1 e 3 c_gi_Na C_pt_Na = 1 e 3 c_pt_Na C_giEpi_Na = 1 e 3 c_giEpi_Na C_ptEpi_Na = 1 e 3 c_ptEpi_Na u_gi_osmotic = RT c_gi_Na u_vc_osmotic = RT c_vc_Na u_pt_osmotic = RT c_pt_Na v_gi_W = k_gi_W (u_vc_osmotic - u_gi_osmotic) v_pt_W = k_pt_W (u_vc_osmotic - u_pt_osmotic) v_gl_W = k_gl_W (u_ac_W - u_pt_W) v_cc_W = k_cc_W (u_ac_W - u_vc_W) v_gl_Na = kK_gl_Na (q_vc_Na - q_pt_Na) v_gi_Na = kK_gi_Na (q_gi_Na - q_giEpi_Na) v_pt_Na = kK_pt_Na (q_pt_Na - q_ptEpi_Na) v_giEpi_NKE = \frac{3 k_giEpi_NKE ({(\frac{K_Na q_giEpi_Na}{q_giEpi_W})}^{3} {(\frac{K_K q_vc_K}{q_vc_W})}^{2} - ({(\frac{K_Na q_vc_Na}{q_vc_W})}^{3} {(\frac{K_K q_giEpi_K}{q_giEpi_W})}^{2} ⅇ^{\frac{2 F u_giEpi_e}{RT}}))}{(1 + {(\frac{K_Na q_giEpi_Na}{q_giEpi_W})}^{3}) (1 + {(\frac{K_Na q_vc_Na}{q_vc_W})}^{3}) (1 + {(\frac{K_K q_vc_K}{q_vc_W})}^{2}) (1 + {(\frac{K_K q_giEpi_K}{q_giEpi_W})}^{2})} v_ptEpi_NKE = \frac{3 k_ptEpi_NKE ({(\frac{K_Na q_ptEpi_Na}{q_ptEpi_W})}^{3} {(\frac{K_K q_vc_K}{q_vc_W})}^{2} - ({(\frac{K_Na q_vc_Na}{q_vc_W})}^{3} {(\frac{K_K q_ptEpi_K}{q_ptEpi_W})}^{2} ⅇ^{\frac{2 F u_ptEpi_e}{RT}}))}{(1 + {(\frac{K_Na q_ptEpi_Na}{q_ptEpi_W})}^{3}) (1 + {(\frac{K_Na q_vc_Na}{q_vc_W})}^{3}) (1 + {(\frac{K_K q_vc_K}{q_vc_W})}^{2}) (1 + {(\frac{K_K q_ptEpi_K}{q_ptEpi_W})}^{2})} q_tot_W = q_ac_W + q_vc_W + q_gi_W + q_pt_W + q_giEpi_W + q_ptEpi_W q_tot_Na = q_vc_Na + q_gi_Na + q_pt_Na + q_giEpi_Na + q_ptEpi_Na