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.