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All model variables

The set of all quantities whose evolution is determined by the model is long. These quantities divide into those whose evolution is determined by differential equations, and those whose evolution is not (algebraic variables). Clicking on a given variable name takes the reader to the equation for its evolution. We should note however that for a given algebraic relation involving several variables, which variable is said to be ``determined'' by this relation is a somewhat arbitrary choice.

The careful reader might notice that many of the algebraic variables are not strictly necessary to the model evolution - rather they are simply variables of convenience capturing intermediate steps in some calculation. For example the variables $gCa_1$ and $gCa_2$ - conductivities of calcium channels in vascular smooth muscle serve only as intermediates to determine $\ensuremath{\mathrm{k_{Cain, 1}}}$ and $\ensuremath{\mathrm{k_{Cain, 2}}}$, the rates of calcium entry into smooth muscle cells. However, in keeping with our philosophy of conceptual simplicity we have included such variables. Eliminating some of these non-crucial model variables, adds little to simulation speed, while making the model harder to understand and alter.


A list of model variables

variable (click for evolution) code version brief description default initial value other comments
         
$C_{vi}$ C_vi venous compliance 4.5930e-01 (fcn) see (1)
$G_1$ G_1 conductance of the proximal arterial segment 4.3619e-01 (fcn) see (1)
$G_2$ G_2 conductance of the distal arterial segment 2.6932e-01 (fcn) see (1)
$G_{vs}$ G_vs conductance of the terminal veins 1.5740e+00 (fcn) see (1)
$M_1$ M_1 level of muscle activation in proximal arterial segment 1.0000e+00 (fcn) normal value = 1
$M_2$ M_2 level of muscle activation in distal arterial segment 1.0000e+00 (fcn) normal value = 1
$PD_K$ PD_K membrane potential for potassium ions in both VSM segments -1.1129e+02 (fcn) assumed to be the same in each segment since it is dependent primarily on the concentration of extracellular potassium
$PD_{Ca, 2}$ PD_Ca2 membrane potential for calcium ions in the distal VSM segment 1.5387e+02 (fcn)  
$PD_{Ca,1}$ PD_Ca1 membrane potential for calcium ions in the proximal VSM segment 1.5387e+02 (fcn)  
$PD_{mem,1}$ PD_mem1 membrane potential of cells in the proximal VSM segment -5.0000e+01 (fcn)  
$PD_{mem,2}$ PD_mem2 membrane potential of cells in the distal VSM segment -5.3000e+01 (fcn)  
$\ensuremath{\mathrm{PMF}}$ PMF the proton motive force across the mitochondrial inner membrane 1.8693e+02 (fcn)  
$P_1$ P_1 blood pressure at the midpoint of the proximal arterial segment 8.5671e+01 (fcn) see (1)
$P_2$ P_2 blood pressure at the midpoint of the distal arterial segment 4.8136e+01 (fcn) see (1)
$P_c$ P_c capillary blood pressure 2.4929e+01 (fcn) see (1)
$P_v$ P_v cerebral venous pressure 1.3937e+01 (fcn) see (1)
$P_{ic}$ P_ic intracranial pressure 9.4265e+00 (fcn) see (1)
$P_{vs}$ P_vs venous sinus pressure 5.9997e+00 (fcn) see (1)
$Pbuf$ Pbuf Sites on cellular proteins capable of binding protons 1.0000e+01 (fcn)  
$PbufH$ PbufH Sites on cellular proteins bound to protons 2.0000e+00 (fcn)  
$T_{e,1}$ T_e1 elastic tension in the vessel walls of the proximal arterial segment -4.3856e-02 (fcn) see (1)
$T_{e,2}$ T_e2 elastic tension in the vessel walls of the distal arterial segment -1.1205e-01 (fcn) see (1)
$T_{m,1}$ T_m1 active tension in the vessel walls of the proximal arterial segment 1.8087e+00 (fcn) see (1)
$T_{m,2}$ T_m2 active tension in the vessel walls of the distal arterial segment 3.6079e-01 (fcn) see (1)
$T_{max, 1}$ T_max1 maximum active muscle tension in the proximal arterial segment 2.1600e+00 (fcn)  
$T_{max, 2}$ T_max2 maximum active muscle tension in the distal arterial segment 1.5000e+00 (fcn)  
$T_{v,1}$ T_v1 viscous tension in the vessel walls of the proximal arterial segment 0.0000e+00 (fcn) see (1)
$T_{v,2}$ T_v2 viscous tension in the vessel walls of the distal arterial segment 0.0000e+00 (fcn) see (1)
$V_1$ V_1 blood volume in the proximal arterial segment 2.5626e+00 (fcn) see (1)
$V_2$ V_2 blood volume in the distal arterial segment 7.9286e+00 (fcn) see (1)
$V_{mca}$ V_mca approximate velocity of the blood in the middle cerebral artery 5.8946e+01 (fcn) see (1)
$\ensuremath{\mathrm{Vmax_{glucN}}}$ Vmax_glucN Vmax for glycolysis 1.2973e-01 (fcn) a function of ATP and AMP concentrations
$Vol_{art}$ Vol_art proportion of total brain volume which is arterial blood 1.0491e+01 (fcn) see (1)
$Vol_{ven}$ Vol_ven proportion of total brain volume which is venous blood 3.0834e+01 (fcn) see (1)
$\ensuremath{\mathrm{[ADP_{cyt}]}}$ _ADP ADP concentration in cytoplasm 3.5000e-01 (fcn)  
$\ensuremath{\mathrm{[ADP_{im}]}}$ _mADP ADP concentration in mitochondria 8.2677e-01 (fcn)  
$\ensuremath{\mathrm{[AMP_{cyt}]}}$ _AMP AMP concentration in cytoplasm 1.1367e-01 (fcn)  
$\ensuremath{\mathrm{[ATP_{cyt}]}}$ _ATP ATP concentration in cytoplasm 3.0000e+00 (fcn)  
$\ensuremath{\mathrm{[ATP_{im}]}}$ _mATP ATP concentration in mitochondria 1.4173e+01 (fcn)  
$\ensuremath{\mathrm{[Ad_{cyt}]}}$ _Ad adenosine concentration in cytoplasm 1.0000e-05 (fcn)  
$\ensuremath{\mathrm{[Ad_{ec}]}}$ _eAd adenosine concentration in extracellular space 1.0000e-05 (fcn)  
$\ensuremath{\mathrm{[CO_{2, cyt}]}}$ _CO2 carbon dioxide concentration in cytoplasm 2.0520e+00 (fcn)  
$\ensuremath{\mathrm{[CO_{2, c}]}}$ CO2 carbon dioxide concentration in capillary blood 1.5157e+00 (fcn)  
$\ensuremath{\mathrm{[CO_{2, ec}]}}$ _eCO2 carbon dioxide concentration in extracellular space 1.7838e+00 (fcn)  
$\ensuremath{\mathrm{[CO_{2, im}]}}$ _mCO2 carbon dioxide concentration in mitochondria 2.4624e+00 (fcn)  
$\ensuremath{\mathrm{[CO_{2, v}]}}$ CO2_v carbon dioxide concentration in venous blood 2.0200e+00  
$\ensuremath{\mathrm{[Ca^{2+}_{i,1}]}}$ Ca_i1 intracellular calcium ion concentration in proximal VSM segment 1.0000e-04 (fcn)  
$\ensuremath{\mathrm{[Ca^{2+}_{i,2}]}}$ Ca_i2 intracellular calcium ion concentration in distal VSM segment 1.0000e-04 (fcn)  
$\ensuremath{\mathrm{[Cr_{cyt}]}}$ _Cr creatine concentration in cytoplasm 3.9000e+00 (fcn)  
$\ensuremath{\mathrm{[FADH_{2, im}]}}$ _mFADH $\ensuremath{\mathrm{FADH_2}}$ concentration in mitochondria 1.2200e+00 (fcn)  
$\ensuremath{\mathrm{[FAD_{im}]}}$ _mFAD FAD concentration in mitochondria 1.0000e-01 (fcn)  
$\ensuremath{\mathrm{[HCO^-_{3, cyt}]}}$ _BiC bicarbonate ion concentration in cytoplasm 1.9046e+01 (fcn)  
$\ensuremath{\mathrm{[HCO^-_{3, c}]}}$ BiC bicarbonate ion concentration in capillary blood 2.5760e+01 (fcn)  
$\ensuremath{\mathrm{[HCO^-_{3, ec}]}}$ _eBiC bicarbonate ion concentration in extracellular space 2.5060e+01 (fcn)  
$\ensuremath{\mathrm{[HCO^-_{3, im}]}}$ _mBiC bicarbonate ion concentration in mitochondria 9.0988e+01 (fcn)  
$\ensuremath{\mathrm{[HCO^-_{3, v}]}}$ BiC_v bicarbonate ion concentration in venous blood 2.5550e+01  
$\ensuremath{\mathrm{[H^+_{cyt}]}}$ _Hy hydrogen ion concentration in cytoplasm 8.5114e-05 (fcn)  
$\ensuremath{\mathrm{[H^+_{c}]}}$ Hy hydrogen ion concentration in capillary blood 4.4157e-05 (fcn)  
$\ensuremath{\mathrm{[H^+_{ec}]}}$ _eHy hydrogen ion concentration in extracellular space 5.6234e-05 (fcn)  
$\ensuremath{\mathrm{[H^+_{im}]}}$ _mHy hydrogen ion concentration in mitochondria 2.1380e-05 (fcn)  
$\ensuremath{\mathrm{[H^+_{v}]}}$ Hy_v hydrogen ion concentration in venous blood 6.2500e-05  
$\ensuremath{\mathrm{[Hb(H^+)_{n_H, c}]}}$ X_H concentration of capillary haemoglobin combined with protons 3.1960e-01 (fcn)  
$\ensuremath{\mathrm{[Hb(H^+)_{n_H,v}]}}$ X_H_v concentration of venous haemoglobin combined with protons 5.4000e-01  
$\ensuremath{\mathrm{[Hb(O_2)_4(H^+)_{n_H, c}]}}$ X_O4_H concentration of capillary haemoglobin combined with oxygen and protons 3.3572e-01 (fcn)  
$\ensuremath{\mathrm{[Hb(O_2)_4(H^+)_{n_H, v}]}}$ X_O4_H_v concentration of venous haemoglobin combined with oxygen and protons 8.3500e-02  
$\ensuremath{\mathrm{[Hb(O_2)_{4, c}]}}$ X_O4 concentration of capillary haemoglobin combined with four oxygen molecules 1.5206e+00 (fcn)  
$\ensuremath{\mathrm{[Hb(O_2)_{4,v}]}}$ X_O4_v concentration of venous haemoglobin combined with four oxygen molecules 1.5300e+00  
$\ensuremath{\mathrm{[HbC]}}$ CY concentration of $\ensuremath{\mathrm{CO_2}}$ bound to capillary haemoglobin 3.0181e+00 (fcn)  
$\ensuremath{\mathrm{[HbC_v]}}$ CY_v concentration of $\ensuremath{\mathrm{CO_2}}$ bound to venous haemoglobin 3.3500e+00  
$\ensuremath{\mathrm{[Hb_c]}}$ X concentration of capillary haemoglobin (not combined with oxygen or protons) 9.4093e-02 (fcn)  
$\ensuremath{\mathrm{[Hb_o]}}$ Y concentration of empty $\ensuremath{\mathrm{CO_2}}$ binding sites in capillary haemoglobin 6.0619e+00 (fcn)  
$\ensuremath{\mathrm{[Hb_v]}}$ X_v concentration of venous haemoglobin (not combined with oxygen or protons) 1.2000e-01  
$\ensuremath{\mathrm{[Hb_{o, v}]}}$ Y_v concentration of empty $\ensuremath{\mathrm{CO_2}}$ binding sites in venous haemoglobin 5.7300e+00  
$\ensuremath{\mathrm{[K^+_{cyt}]}}$ _K potassium ion concentration in cytoplasm 1.8000e+02 (fcn)  
$\ensuremath{\mathrm{[K^+_{c}]}}$ K potassium ion concentration in capillary blood 4.0700e+00 (fcn)  
$\ensuremath{\mathrm{[K^+_{ec}]}}$ _eK potassium ion concentration in extracellular space 2.8000e+00 (fcn)  
$\ensuremath{\mathrm{[K^+_{v}]}}$ K_v potassium ion concentration in venous blood 0.0000e+00  
$\ensuremath{\mathrm{[MLC_{1}]}}$ MLC1 concentration of unphosphorylated crossbridge heads of myosin in the proximal VSM segment 4.4800e-02 (fcn)  
$\ensuremath{\mathrm{[MLC_{2}]}}$ MLC2 concentration of unphosphorylated crossbridge heads of myosin in the distal VSM segment 4.7600e-02 (fcn)  
$\ensuremath{\mathrm{[MLC_{p,1}]}}$ MLCp1 concentration of phosphorylated crossbridge heads of myosin in the proximal VSM segment 1.1200e-02 (fcn)  
$\ensuremath{\mathrm{[MLC_{p,2}]}}$ MLCp2 concentration of phosphorylated crossbridge heads of myosin in the distal VSM segment 8.4000e-03 (fcn)  
$\ensuremath{\mathrm{[NADH_{cyt}]}}$ _NADH NADH concentration in cytoplasm 2.9703e-03 (fcn)  
$\ensuremath{\mathrm{[NADH_{im}]}}$ _mNADH NADH concentration in mitochondria 9.0909e-01 (fcn)  
$\ensuremath{\mathrm{[NAD_{cyt}]}}$ _NAD NAD concentration in cytoplasm 2.9703e-01 (fcn)  
$\ensuremath{\mathrm{[NAD_{im}]}}$ _mNAD NAD concentration in mitochondria 9.0909e+00 (fcn)  
$\ensuremath{\mathrm{[NO_1]}}$ NO1 nitric oxide concentration in proximal VSM segment 2.0000e-06 (fcn)  
$\ensuremath{\mathrm{[NO_2]}}$ NO2 nitric oxide concentration in distal VSM segment 2.0000e-06 (fcn)  
$\ensuremath{\mathrm{[Na^+_{cyt}]}}$ _Na sodium ion concentration in cytoplasm 1.5000e+01 (fcn)  
$\ensuremath{\mathrm{[Na^+_{c}]}}$ Na sodium ion concentration in capillary blood 1.3800e+02 (fcn)  
$\ensuremath{\mathrm{[Na^+_{ec}]}}$ _eNa sodium ion concentration in extracellular space 1.3800e+02 (fcn)  
$\ensuremath{\mathrm{[Na^+_{v}]}}$ Na_v sodium ion concentration in venous blood 0.0000e+00  
$\ensuremath{\mathrm{[O_{2, cyt}]}}$ _O2 oxygen concentration in cytoplasm 4.5500e-02 (fcn)  
$\ensuremath{\mathrm{[O_{2, c}]}}$ O2 dissolved oxygen concentration in capillary blood 6.8000e-02 (fcn)  
$\ensuremath{\mathrm{[O_{2, im}]}}$ _mO2 oxygen concentration in mitochondria 2.2750e-02 (fcn)  
$\ensuremath{\mathrm{[O_{2, v}]}}$ O2_v dissolved oxygen concentration in venous blood 6.2500e-02  
$\ensuremath{\mathrm{[O_{2,ec}]}}$ _eO2 oxygen concentration in extracellular space 5.0000e-02 (fcn)  
$\ensuremath{\mathrm{[Ox]}}$ _Ox oxaloacetate concentration in mitochondria 5.4000e-01 (fcn)  
$\ensuremath{\mathrm{[PCr_{cyt}]}}$ _PCr phosphocreatine concentration in cytoplasm 5.8500e+00 (fcn)  
$\ensuremath{\mathrm{[P_{i, cyt}]}}$ _Phos inorganic phosphate concentration in cytoplasm 1.5000e+00 (fcn)  
$\ensuremath{\mathrm{[P_{i, im}]}}$ _mPhos inorganic phosphate concentration in mitochondria 3.2000e+00 (fcn)  
$\ensuremath{\mathrm{[\alpha KG]}}$ _AKG alpha-ketoglutarate concentration in mitochondria 1.0000e-01 (fcn)  
$\ensuremath{\mathrm{[gluc_{cyt}]}}$ _gluc glucose concentration in cytoplasm 1.4000e+00 (fcn)  
$\ensuremath{\mathrm{[gluc_{c}]}}$ gluc glucose concentration in capillary blood 4.8396e+00 (fcn)  
$\ensuremath{\mathrm{[gluc_{ec}]}}$ _egluc glucose concentration in extracellular space 2.0879e+00 (fcn)  
$\ensuremath{\mathrm{[gluc_{v}]}}$ gluc_v glucose concentration in venous blood 0.0000e+00  
$\ensuremath{\mathrm{[lac^-_{cyt}]}}$ _L0 lactate concentration in cytoplasm 2.0000e+00 (fcn)  
$\ensuremath{\mathrm{[lac^-_{c}]}}$ L0 lactate ion concentration in capillary blood 1.0200e+00 (fcn)  
$\ensuremath{\mathrm{[lac^-_{ec}]}}$ _eL0 lactate ion concentration in extracellular space 1.9020e+00 (fcn)  
$\ensuremath{\mathrm{[lac^-_{v}]}}$ L0_v lactate ion concentration in venous blood 1.0300e+00  
$\ensuremath{\mathrm{[lac_{cyt}]}}$ _lac lactic acid concentration in cytoplasm 1.2335e-03 (fcn)  
$\ensuremath{\mathrm{[lac_{c}]}}$ L lactic acid concentration in capillary blood 3.2638e-04 (fcn)  
$\ensuremath{\mathrm{[lac_{ec}]}}$ _eL lactic acid concentration in extracellular space 7.7505e-04 (fcn)  
$\ensuremath{\mathrm{[lac_{v}]}}$ L_v lactic acid concentration in venous blood 4.6000e-04  
$\ensuremath{\mathrm{[pyr^-_{cyt}]}}$ _Py0 pyruvate concentration in cytoplasm 1.2000e-01 (fcn)  
$\ensuremath{\mathrm{[pyr^-_{ec}]}}$ _ePy0 pyruvate concentration in extracellular space 1.2000e-01 (fcn)  
$\ensuremath{\mathrm{[pyr^-_{im}]}}$ _mPy0 pyruvate concentration in mitochondria 3.8218e-01 (fcn)  
$\ensuremath{\mathrm{[suc]}}$ _suc succinate concentration in mitochondria 1.0000e-01 (fcn)  
$\Delta\Psi_m$ PD_mit membrane potential for the mitochondrial inner membrane (multiplied by membrane capacitance) 1.5000e+02 (fcn)  
$\sigma_{e,1}$ sigma_e1 elastic stress in the vessel walls of the proximal arterial segment -2.1716e+01 (fcn) see (1)
$\sigma_{e,2}$ sigma_e2 elastic stress in the vessel walls of the distal arterial segment -4.3339e+01 (fcn) see (1)
$\sigma_{v,1}$ sigma_v1 viscous stress in the vessel walls of the proximal arterial segment 0.0000e+00 (fcn) see (1)
$\sigma_{v,2}$ sigma_v2 viscous stress in the vessel walls of the distal arterial segment 0.0000e+00 (fcn) see (1)
$gCa_1$ gCa1 conductivity of calcium channels in the proximal VSM segment 6.7999e-08 (fcn)  
$gCa_2$ gCa2 conductivity of calcium channels in the distal VSM segment 6.7999e-08 (fcn)  
$gK_1$ gK1 total conductivity of all potassium channels in the proximal VSM segment 3.9161e-02 (fcn)  
$gK_2$ gK2 total conductivity of all potassium channels in the distal VSM segment 4.3235e-02 (fcn)  
$gK_{ATP, 1}$ gKATP1 conductivity of the ATP and pH sensitive potassium channels in the proximal VSM segment 7.8321e-03 (fcn)  
$gK_{ATP, 2}$ gKATP2 conductivity of the ATP and pH sensitive potassium channels in the distal VSM segment 1.2971e-02 (fcn)  
$gK_{Ca, 1}$ gKpres1 conductivity of the pressure sensitive potassium channels in the proximal VSM segment 1.9580e-02 (fcn)  
$gK_{Ca, 2}$ gKpres2 conductivity of the pressure sensitive potassium channels in the distal VSM segment 1.7294e-02 (fcn)  
$gK_{IR, 1}$ gKpot1 conductivity of the potassium sensitive potassium channels in the proximal VSM segment 7.8321e-03 (fcn)  
$gK_{IR, 2}$ gKpot2 conductivity of the potassium sensitive potassium channels in the distal VSM segment 8.6470e-03 (fcn)  
$gK_{V, 1}$ gKV1 conductivity of the voltage gated potassium channels in the proximal VSM segment 3.9161e-03 (fcn)  
$gK_{V, 2}$ gKV2 conductivity of the voltage gated potassium channels in the distal VSM segment 4.3235e-03 (fcn)  
$h_1$ h_1 vessel wall thickness in the proximal arterial segment 2.0195e-03 (fcn) see (1)
$h_2$ h_2 vessel wall thickness in the distal arterial segment 2.5854e-03 (fcn) see (1)
$\ensuremath{\mathrm{k_{CC}}}$ k_CC rate constant for the diffusion of carbon dioxide between blood and extracellular space 1.4713e+02 (fcn) perhaps dependent on transmural pressure
$\ensuremath{\mathrm{k_{Cain, 1}}}$ k_Cain1 rate constant for the inflow of calcium into the proximal VSM segment 1.3863e-05 (fcn) dependent on conductance of calcium channels
$\ensuremath{\mathrm{k_{Cain, 2}}}$ k_Cain2 rate constant for the inflow of calcium into the distal VSM segment 1.3863e-05 (fcn) dependent on conductance of calcium channels
$\ensuremath{\mathrm{k_{MLCac, 1}}}$ k_MLC1ac rate of phosphorylation of dephosphorylated myosin heads in the proximal VSM segment 1.3863e-01 (fcn)  
$\ensuremath{\mathrm{k_{MLCac, 2}}}$ k_MLC2ac rate of phosphorylation of dephosphorylated myosin heads in the distal VSM segment 1.0397e-01 (fcn)  
$\ensuremath{\mathrm{k_{MLCinac, 1}}}$ k_MLC1inac rate of dephosphorylation of phosphorylated myosin heads in the proximal VSM segment 5.5452e-01 (fcn)  
$\ensuremath{\mathrm{k_{MLCinac, 2}}}$ k_MLC2inac rate of dephosphorylation of phosphorylated myosin heads in the distal VSM segment 5.8918e-01 (fcn)  
$\ensuremath{\mathrm{k_{NOprod, 1}}}$ k_NOprod1 rate constant for the inflow of nitric oxide into the proximal VSM segment 1.3863e-08 (fcn) dependent on shear stress and pH
$\ensuremath{\mathrm{k_{NOprod, 2}}}$ k_NOprod2 rate constant for the inflow of nitric oxide into the distal VSM segment 1.3863e-08 (fcn) dependent on shear stress and pH
$\ensuremath{\mathrm{k_{OO}}}$ k_OO rate constant for the diffusion of oxygen between blood and extracellular space 2.4112e+03 (fcn) perhaps dependent on transmural pressure, though currently no effects are included.
$q$ q cerebral blood flow 1.2500e+01 (fcn) see (1)
$r_1$ r_1 radius of the larger (proximal) cerebral arteries 2.3501e-02 (fcn) see (1)
$r_2$ r_2 radius of the smaller (distal) cerebral arteries 7.1683e-03 (fcn) see (1)
$r_{mca}$ r_mca approximate radius of the middle cerebral artery 1.5000e-01 (fcn) see (1)
 
variable table ends

Bibliography

1
M. Ursino, C. A. Lodi, Interaction among autoregulation, $\ensuremath{\mathrm{CO_2}}$ reactivity, and intracranial pressure: a mathematical model, Am J Physiol Heart Circ Physiol 274 (5) (1998) H1715-H1728.


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Up: Index of documentation
Murad Banaji
2004-07-08