Cerebral glucose delivery,transport and metabolism: Theory and modeling using four,three and two tissue compartments

Abstract

Flux equations describing brain D-glucose uptake are presented for up to four tissue compartments: blood, endothelial intracellular space in the blood-brain barrier (BBB), extravascular-extracellular space (EES), and intracellular space. Transport rates are described by Michaelis-Menten kinetics, including half-saturation constants ( $K_T$ ) and maximum rates for transport ${(T}_{\mathit{max}})$ over the BBB and the cell membrane (CMB). These transport parameters and the maximum rate for hexokinase-catalyzed metabolism ( $V_{\mathit{max}}^{HK}$ ) were determined by numerical fitting of the models to both steady-state and dynamic D-glucose uptake data in human gray matter from MRS. Two-, three-, and four-compartment results are compared, including effects of incorporating an endothelial compartment with unequal ratios ( $R_{A/L}$ ) of GLUT1 receptors on abluminal and luminal membranes. Four-compartment fitting with $R_{A/L}=2.0$ resulted in $T_{\mathit{max}}^{\mathit{BBB}}=0.804\pm 0.131$ µmol/g/min, $K_T^{\mathit{BBB}}=6.20\pm 1.53\hspace{0.17em}$ mM, $T_{\mathit{max}}^{\mathit{CMB}}=1.04\pm 0.25\hspace{0.17em}$ µmol/g/min, $K_T^{\mathit{CMB}}=3.10\pm 0.70\hspace{0.17em}$ mM and $V_{\mathit{max}}^{HK}=0.260\pm 0.039\hspace{0.17em}$ µmol/g/min, comparing well with the simpler models. A model with at least three tissue compartments (blood, EES, cell) is essential for quantification and interpretation of dynamic glucose-enhanced (DGE) MRI data in brain tumors, where signal intensities depend on compartmental pH in addition to concentration, and where the signal contribution from the EES is dominant. It should also be relevant to PET and MR(S) studies of pathologies where the BBB is compromised.

Introduction

D-glucose is the main source of energy for brain metabolism. Its uptake, which includes information on delivery, transport and metabolism, has been studied using several imaging technologies. Of these, autoradiography ^1,2 and positron emission tomography (PET) ^{3 –8} employed radioactively labeled glucose-derivatives, while magnetic resonance spectroscopy (MRS) studies used both magnetically labeled (¹³C or ²H) ^{9 –11} and natural D-glucose. ^12,13 Recently, dynamic glucose enhanced (DGE) MRI studies have been designed in which sugar uptake is detected via the water signal using the exchange interaction between hydroxyl group protons in sugar and water protons. ¹⁴ This has been accomplished using chemical exchange saturation transfer (CEST), ^{15 –18} chemical exchange spin lock (CESL), ¹⁹ and T₂-relaxation based MRI. ^20,21 DGE MRI shows promise to specifically diagnose malignant brain tumors, ^{14,17,20,22 –28} with the majority of signal originating from the extravascular-extracellular space (EES), due to the reduced pH affecting the proton exchange rate favorably for detection.

When modeling glucose utilization, the most common kinetic model used for transport and metabolism in vivo is a two-tissue-compartment model (2CM), consisting of a blood compartment for delivery and a parenchymal compartment for metabolism, separated by a blood-brain barrier (BBB) that facilitates transport of sugars. ²⁹ This single parenchymal compartment uses the assumption of rapid mixing of substrate between EES and intracellular space relative to the slow rate-determining step of glucose phosphorylation. This approach, introduced in the autoradiography ^1,30,31 and PET literature, ³² was continued in the ¹³C, ²H and ¹H MRS literature ^{9,11 –13,33} and recently in DGE MRI. ³⁴ The 2CM description is practical, because all detection approaches measure a single signal intensity that cannot distinguish glucose in blood, intracellular space, and EES. Unfortunately, a 2CM is unlikely to be truly representative for several pathologies. The reasons are two-fold: (i) in situations where the BBB is disrupted (e.g. in certain brain tumors or MS lesions), the glucose concentration in EES equilibrates with the blood concentration, but remains low in the cell; (ii) for DGE MRI, an additional issue arises in that the pH in EES is reduced in malignant tumors, leading to a reduced exchange rate for the hydroxyl protons in sugars and an increased dominant CEST signal contribution from this compartment. The CESL relaxation rate change is also strongly dependent on proton exchange rate, but pH reduction may increase or decrease the DGE effect depending on the chemical shift difference between the OH groups and water protons. ¹⁹ Issues (i, ii) necessitate the availability of uptake models treating EES and intracellular space as separate compartments, i.e. at least three compartments (blood, EES, cell).

A three-tissue-compartment model (3CM), using five rate constants for describing transport and metabolism of radioactively labeled PET-substrates, has been explored in skeletal muscle, ³⁵ brain, ³⁶ and multiple organs. ³⁷ Huang et al. extended this concept to include Michaelis-Menten (MM) kinetics with maximum influx rates ( $T_{\mathit{max}},V_{\mathit{max}}$ ) and half-saturation constants ( $K_T,K_M$ ) for glucose transport and metabolism, respectively. ³⁸ Contrary to PET, MR studies of D-glucose infusion use non-tracer concentrations (millimolar increases leading to hyperglycemia), causing the rate constants for transport and metabolism to become time dependent and requiring the use of MM kinetics with tissue-specific concentration-independent parameters. Also, contrary to FDG, D-glucose is metabolized completely, requiring an additional metabolic efflux rate constant. Another issue is that most existing models assume a single BBB membrane, while the occupancy of GLUT1 transporters has been reported to differ between the luminal and abluminal membranes of the endothelial cells forming the BBB. ^{39 –44} Therefore, we add a fourth (endothelial, EN) compartment, allowing the effect of different GLUT1 receptor densities on these membranes to be studied, as well as enabling the use of reversible GLUT1 transport kinetics ^11,12,45,46 for each membrane. While additional tissue compartments require more unknown parameters to characterize a single uptake curve, careful use of appropriate biophysical constraints and known tissue constants still allows assessment of transport and metabolic steps. A key constraint in addition to the MM constraints is that the D-glucose flux in steady-state is the same for EN, EES, and the intracellular compartment, namely the cerebral metabolic rate of glucose ( ${\mathit{CMR}}_{\mathit{Glc}}$ ), the magnitude of which is known from glucose extraction fraction (GEF) measurements at normoglycemia. ⁶ ${\mathit{CMR}}_{\mathit{Glc}}$ has been shown to be unaffected in acute hyperglycemia. ^7,47

In this study, expanded mass-balance differential equations for a four-tissue compartment model (4CM, called 3CM-EN) are derived. We also include correction terms for water versus tissue versus blood concentrations, based on well-known density and water-content constants from the literature, allowing the different concentrations of D-glucose in EES and cell to be accounted for. Using numerical simulations and D-glucose uptake data from the MRS literature, ^11,12 the MM constants for BBB transport, cell membrane (CMB) transport, and hexokinase-catalyzed phosphorylation are determined for gray matter. The model is compared to existing 2CM both with reversible and non-reversible BBB transport and further demonstrated for white matter and for malignant tumors with a disrupted BBB.

Methods

Theory

Three-tissue compartment reversible transport model

Tissues such as white matter (WM), gray matter (GM) and glioblastoma (GBM) are described using four compartments: blood (b), BBB endothelium (EN), EES (e), and a cellular compartment (c) in which sugar is phosphorylated in the hexokinase (HK) step (Figure 1). The fast efflux of phosphorylated D-glucose to pyruvate (pyr) is also accounted for. Blood has two physical compartments, plasma (p) and erythrocytes (ery), but, additionally, there are three microvascular compartments in tissue regions of interest, namely arteriolar/arterial (a), capillary, and venular/venous (v). For simplicity, a 30/70 volume ratio based arterial/venous distinction for microvasculature is used. ⁴⁸ The BBB is generally modeled as a single membrane, which has been shown not to affect determination of the 2CM glucose transport rates across the BBB under healthy conditions. ^49,50 However, histology studies indicate an unequal distribution of GLUT1 receptors on the luminal (L) and abluminal (A) membranes ^{39 –44} which may affect transport parameters. To evaluate the impact of including an endothelial compartment in the kinetic modeling of glucose transport, model versions including both single- and double-membrane BBB were introduced, as depicted in Figure 1.

Figure 1.

Four-, three-, and two-tissue compartment models (4CM, 3CM, 2CM) for D-glucose uptake in the brain. For practical purposes, to describe inclusion or exclusion of an endothelial compartment (EN), these are denoted as 3CM-EN, 3CM, 2CM-EN and 2CM. The EN-inclusive models account for reversible transport across GLUT1 in the luminal (L) and abluminal (A) membranes of the blood brain barrier (BBB). All membrane transport rates from plasma (p) to EN to EES (e) and cell (c) are described by Michaelis-Menten transport kinetics (Table 1). Separating the EES and cell compartments is important for comparing (i) glucose-based transport and metabolism with and without BBB disruption, for example, in tumors, or for (ii) CEST-MRI related signals when the pH differs between compartments, for example, in malignant tumors where the EES pH is lower than in the EES of healthy tissue. The formation of glucose-6-phosphate (G6P) is the rate determining step, followed by fast turnover of products during glycolysis to the end-product pyruvate (pyr), which is treated as a single step as the concentrations of these phosphorylated glucoses are low. Solid lines indicate tissue compartments and dashed lines indicate metabolic compartments. Rate constants $k_i$ are explained in Table 1.

For a 4CM, here denoted as a 3CM with a two-membrane endothelial step (3CM-EN), the mass-balance equations describing the glucose fluxes in µmol/g/min for a particular tissue are:

$$\frac{{f_b(1-\mathit{Hct})\rho }_p^{V/V}}{d_{\mathit{tot}}^{m/V}}\frac{d{C}_p^a\left(t\right)}{dt}=-k_{1L}\left(t\right)C_p^a\left(t\right)+k_{2L}\left(t\right)C_{EN}\left(t\right)$$

(1)

$$\frac{f_{EN}{\rho }_{EN}^{V/V}}{d_{\mathit{tot}}^{m/V}}\frac{d{C}_{EN}\left(t\right)}{dt}=k_{1L}\left(t\right)C_p^a\left(t\right)-{[k}_{2L}\left(t\right)+k_{1A}\left(t\right)\left]C_{EN}\right(t)+k_{2A}(t\left)C_e\right(t)$$

(2)

$$\frac{f_e{\rho }_e^{V/V}}{d_{\mathit{tot}}^{m/V}}\frac{d{C}_e\left(t\right)}{dt}=k_{1A}\left(t\right)C_{EN}\left(t\right)-{[k}_{2A}\left(t\right)+k_3\left(t\right)\left]C_e\right(t)+k_4(t\left)C_c\right(t)$$

(3)

$$\frac{f_c{\rho }_c^{V/V}}{d_{\mathit{tot}}^{m/V}}\frac{d{C}_c\left(t\right)}{dt}=k_3\left(t\right)C_e\left(t\right)-{[k}_4\left(t\right)+k_5\left(t\right)\left]C_c\right(t)$$

(4a)

$$\frac{f_c{\rho }_c^{V/V}}{d_{\mathit{tot}}^{m/V}}\frac{d{C}_{G6P}\left(t\right)}{dt}=k_5\left(t\right)C_c\left(t\right)-k_6\left(t\right)C_{G6P}\left(t\right)$$

(4b)

The mass-balance equations for the other models in Figure 1: three-tissue compartment with single-membrane endothelial step (3CM), two-tissue compartment with two-membrane endothelial step (2CM-EN), and two-tissue compartment with single-membrane endothelial step (2CM) are shown in Supplement 1. The compartmental glucose concentrations $C_i\left(t\right)(i=b, p, EN, e, c$ ) and the cellular glucose-6-phosphate concentration $C_{G6P}\left(t\right)$ are all in units of µmol/mL water (mM) within the compartment. The rate constants $k$ for all models follow MM kinetics as described in Table 1. Importantly, MM kinetics was originally developed to describe enzyme-catalyzed metabolism in water solution using substrate concentrations in mM. It is not straightforward to transfer this approach to tissue, where the rates reflect amount of glucose taken up or metabolized per volume or weight of tissue, and concentrations are generally given in µmol/mL tissue or µmol/g tissue. Such rates are tissue dependent because the MM maximum rates (in µmol/g/min) for membrane transport, $T_{\mathit{max}},$ and for biochemical reactions, $V_{\mathit{max}},$ depend on the number of available receptors and enzymes, respectively, in a particular tissue. For instance, GM has a larger $CM{R}_{\mathit{Glc}}$ than WM, with cerebral blood volume ( $\mathit{CBV})$ and cerebral blood flow (CBF) increased proportionally. This leads to a different permeable surface area for the capillaries and thus a different number of GLUT1 receptors for the BBB and a different rate of transfer on a per tissue weight/volume basis.

Table 1.

Michaelis-Menten description for compartmental transport and metabolism. ^*

3CM-EN and 3CM
Transport (influx/efflux) over luminal membrane of BBB (plasma - EN)	$k_{1L}\left(t\right)=k_{2L}\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{BBB}}}{C_p^a\left(t\right)+C_{EN}\left(t\right)+K_T^{\mathit{BBB}}}$
Transport (efflux/influx) over abluminal membrane of BBB (EN – EES)	$k_{1A}\left(t\right)=k_{2A}\left(t\right)=\frac{R_{A/L}{T}_{\mathit{max}}^{\mathit{BBB}}}{C_{EN}\left(t\right)+C_e\left(t\right)+K_T^{\mathit{BBB}}}$
Transport (influx/efflux) over cell membrane (EES – cell)	$k_3\left(t\right)=k_4\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{CMB}}}{C_e\left(t\right)+C_c\left(t\right)+K_T^{\mathit{CMB}}}$
Phosphorylation (inside cell) (D-Glc → G6P)	$k_5\left(t\right)=\frac{V_{\mathit{max}}^{HK}}{C_c\left(t\right)+K_M^{HK}}$
Residual glycolysis in cell G6P → pyr/lac	$k_6\left(t\right)=\frac{V_{\mathit{max}}^{\mathit{glyc}}}{C_{G6P}\left(t\right)+K_M^{\mathit{glyc}}}$
Reversible influx/efflux assuming a single membrane BBB (plasma – EES)	$k_1\left(t\right)=k_2\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{BBB}}}{C_p^a\left(t\right)+C_e\left(t\right)+K_T^{\mathit{BBB}}}$
2CM-EN, 2CM reversible and nonreversible
Transport (influx/efflux) over luminal membrane of the BBB	$k_{1L}\left(t\right)=k_{2L}\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{BBB}}}{C_p^a\left(t\right)+C_{EN}\left(t\right)+K_T^{\mathit{BBB}}}$
Transport (efflux/influx) over abluminal membrane of BBB (EN – [EES+cell])	$k_{1A}\left(t\right)=k_{2A}\left(t\right)=\frac{R_{A/L}{T}_{\mathit{max}}^{\mathit{BBB}}}{C_{EN}\left(t\right)+C_{e+c}\left(t\right)+K_T^{\mathit{BBB}}}$
Phosphorylation (D-Glc → G6P)	$k_5\left(t\right)=k_3^{2CM}\left(t\right)=\frac{V_{\mathit{max}}^{HK}}{C_{e+c}\left(t\right)+K_M^{HK}}$
Residual glycolysis: G6P → pyr /lac	$k_6\left(t\right)\hspace{0.17em}\mathrm{as}\hspace{0.17em}\mathrm{above}$ for 3CM
Reversible influx/efflux assuming a single membrane BBB (plasma – [EES+cell])	$k_1\left(t\right)=k_2\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{BBB}}}{C_p^a\left(t\right)+C_{e+c}\left(t\right)+K_T^{\mathit{BBB}}}$
Non-reversible influx/efflux for a single membrane BBB (plasma – [EES+cell])	$k_1\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{BBB}}}{C_p^a\left(t\right)+K_T^{\mathit{BBB}}}; k_2\left(t\right)=\frac{T_{\mathit{max}}^{\mathit{BBB}}}{C_{e+c}\left(t\right)+K_T^{\mathit{BBB}}}$

a: arterial; BBB: blood-brain barrier; c: intracellular space; CMB: cell membrane barrier; D-Glc: D-glucose; e: extravascular extracellular space (EES); EN: endothelial; G6P: glucose-6-phosphate; glyc: glycolysis; HK: hexokinase; $K_T$ and $K_M$ : Michaelis constants, i.e. the concentration in mM at which half the maximum rate is reached for catalysis by the transport receptor and the enzyme, respectively; p: plasma; $T_{\mathit{max}},\hspace{0.17em}{V}_{\mathit{max}}$ in µmol/g tis/min: Michaelis-Menten maximum rates for receptor-facilitated transport and enzyme-catalyzed metabolism, respectively; $R_{A/L}:$ abluminal to luminal receptor ratio.

*

Rate constants $k$ have units of (mL water/g tis/min).

The denominators of the rate constants in Table 1 include compartmental glucose concentrations and the Michaelis constant, $K_T^{\mathit{membrane}}$ or $K_M^{\mathit{enzyme}}$ , all in mM, leading to units of mL water/g tissue/min for $k$ . To balance the units for equations (1) to (4) to µmol/g/min, the left-hand side must be converted using the compartmental volume fraction ( $f_i$ ) in mL/mL tissue and water content ( ${\rho }_i^{V/V}$ ) in mL water/mL tissue, as well as tissue density $d^{m/V}$ in g tissue/mL tissue (Table 2). Notice that $f_b\mathrm{ }$ equals the $\mathrm{ }\mathit{CBV}\mathrm{ }$ for the particular tissue in mL blood/mL tissue, and $\mathrm{ }{f}_p=f_b(1-\mathit{Hct})\mathrm{ }$ the tissue plasma volume fraction in mL plasma/mL tissue.

Table 2.

Densities and water contents for different substances and brain tissues.

Tissue/substance	Density † (g tissue/mL tissue)	Water content (g water/g tissue)	Water content (g water/mL tissue)	Water content (mL water/mL tissue)	Water content (mL water/g tissue)
	$d^{m/V}$	${\rho }^{m/m}$	${\rho }^{m/V}$	${\rho }^{V/V}$	${\rho }^{V/m}$
Water, w	0.993a
Plasma, p	1.021b	0.912f	0.931¶	0.938§	0.919§§
Erythrocyte, ery	1.100c,d	0.662g	0.728¶	0.733§	0.666§§
Blood, b (average Hct 0.4)	1.052 *	0.808 ***	0.850 **	0.856§	0.813§§
Endothelial (EN) † †	1.050	0.765	0.803	0.809	0.770
EES$, e	1.021$	0.912$	0.931$	0.938$	0.919$
Gray matter, GM
- whole	1.050e	0.793&&&	0.833&&	0.839&	0.799§§
- cell	1.050$$	0.765g	0.803¶	0.809§	0.770§§
White matter, WM
- whole	1.050e	0.698&&&	0.733&&	0.738&	0.703§§
- cell	1.050$$	0.641g	0.673¶	0.678§	0.646§§
Whole brain	1.050#	0.770h,i	0.809¶	0.815§	0.776§§
Brain tumor, GBM
- whole	1.050$$	0.770j	0.809¶	0.815§	0.776§§
- cell	1.050$$	0.674&&&	0.708¶	0.674&	0.642§§

Notice that mass per volume (m/V) was used for tissue density, expressed in g/mL. Medical literature often uses weight, but in SI units, mass is measured in kilograms and weight is the force measured in Newton.

†

Calculated from relative density (tissue density at 37 °C relative to water density at 4 °C). ^††Assumed to be the same as GM cells. * from equation (8e); ** from equation (8c); *** from $\mathrm{ }{\rho }_b^{m/V}/d_b^{m/V}$ ; # based on equation (8d), with cell densities for GM and WM;^$ Assumed equal to plasma; ^$$Assumed equal to blood, GM and WM densities; ^¶from $\mathrm{ }{\rho }^{m/V}={\rho }^{m/m}·d^{m/V}$ ;^§ from $\mathrm{ }{\rho }^{V/V}={\rho }^{m/V}/d_w^{m/V}$ ;^§§ $\mathrm{ }{\rho }^{V/m}={\rho }^{V/V}/d^{m/V};$ ^&Calculated based on whole tissue and fractions of blood and EES using equation (8 b); Volume fractions: $f_{b,GM}$  = 0.038, $\mathrm{ }{f}_{b,WM}$  = 0.015, $\mathrm{ }{f}_{b,\mathit{GBM}}$  = 0.05; $\mathrm{ }{f}_{e,GM }$  = 0.22, $f_{e,WM}$  = 0.22; $f_{e,\mathit{GBM}}$  = 0.5; $\mathrm{ }{f}_c=1-f_e-f_b$ ;^&& from $\mathrm{ }{\rho }^{V/V}·d_w^{m/V}$ ;^&&& $\mathrm{ }{\rho }^{m/V}/d_w^{m/V}$ .

a) US Department of the interior, Bureau of reclamation ¹⁰³ ; b) Trudnowski RJ et al. ¹⁰⁴ ; c) Ashworth CT et al. ¹⁰⁵ ; d) van Slyke DD et al. ¹⁰⁶ ; e) Torack RM et al. ¹⁰⁷ ; f) Lijnema TH et al. ¹⁰⁸ ; g) Stewart-Wallace AM et al. ¹⁰⁹ ; h) Herscovitch P et al. ¹¹⁰ ; i); Dittmer DA et al. ¹¹¹ ; j) Takagi H et al. ¹¹² .

The rate constant equations in Table 1 contain tissue-based MM parameters that provide information about healthy or abnormal physiology, i.e. $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}},K_T^{\mathit{BBB}}, T_{\mathit{max}}^{\mathit{CMB}}, K_T^{\mathit{CMB}}, V_{\mathit{max}}^{HK}, K_M^{HK}, V_{\mathit{max}}^{\mathit{glyc}}, K_M^{\mathit{glyc}}$ . These equations differ from those typically used in the PET literature. MRS and DGE MRI use millimolar concentrations of glucose (of the order of magnitude of $\mathrm{ }{K}_T$ ) and molecules at each side of a membrane are competing for the same receptors. This can be accounted for using reversible transport kinetics, ^{11,12,45,46,51} where concentrations in both spaces are included in the rate constants. While this, at first, appears to complicate the description, it actually simplifies it, because the forward and reverse rate transport constants become equal, i.e., $\mathrm{ }{k}_{1L}\left(t\right)=k_{2L}\left(t\right),k_{1A}\left(t\right)=k_{2A}\left(t\right),$ and $k_3\left(t\right)=k_4\left(t\right)$ . A single $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ is defined to describe transport by GLUT1 for both endothelial membranes. To account for different rates over the luminal and abluminal membranes in case of different GLUT1 densities, a polarity ratio $\mathrm{ }{R}_{A/L}\mathrm{ }$ is added to the abluminal equation. Literature reports on this ratio are conflicting, ^{39 –44} with primates showing a higher luminal density, ^52,53 dogs approximately equal polarity, ⁵⁴ and small mammals a higher abluminal density. ^{52,55 –59} The effect of this polarity ratio on the MM constants is therefore investigated.

While the concentrations $\mathrm{ }{C}_e\left(t\right)\mathrm{ }$ and $\mathrm{ }{C}_c\left(t\right)\mathrm{ }$ are unknown, several steps can be taken to simplify the fitting and assess the values of the MM constants. Glucose phosphorylation, described by $\mathrm{ }{k}_5\left(t\right)C_c\left(t\right)=CM{R}_{\mathit{Glc}}$ , is the rate-determining step in the cascade of brain transport and metabolism. Its rate relates to the tissue-based GEF and the arteriovenous concentration difference through the Kety-Schmidt equation: ⁶⁰

$$\mathit{GEF}=\frac{C_p^a-C_p^v}{C_p^a}=\frac{C_p^{a,V_p}-C_p^{v,V_p}}{C_p^{a,V_p}}=\frac{\mathit{Glucose}\mathrm{ }\mathit{consumption}}{\mathit{Glucose}\mathrm{ }\mathit{delivery}}$$

(5a)

$$=\frac{{\mathit{CMR}}_{\mathit{Glc}}}{\mathit{CBF}·(1-\mathit{Hct})·C_p^a{·\rho }_p^{V/V}}$$

(5b)

$\mathit{CBF}\mathrm{ }$ is in units of mL/g/min and $\mathrm{ }{C}_p^a$ and $\mathrm{ }{C}_p^v\mathrm{ }$ in mM. These are related to the plasma glucose concentrations ( $C_p^{a/v,V_p}$ ) in µmol/mL plasma via the plasma water content: $\mathrm{ }{C}_p^{a/v,V_p}=C_p^{a/v}{·\rho }_p^{V/V}$ . Under conditions of constant arterial plasma concentration, one can assume steady-state for brain glucose uptake $\left(\frac{d{C}_{EN}\left(t\right)}{dt}=0;\frac{d{C}_e\left(t\right)}{dt}=0;\frac{d{C}_c\left(t\right)}{dt}=0;\frac{d{C}_{G6P}\left(t\right)}{dt}=0\right)$ . Applying several substitutions starting from equation (4a) back towards equation (2), it can be derived (Supplement 2) that:

$$k_5{C}_c\left(t\right)={\mathit{CMR}}_{\mathit{Glc}}=k_3{C}_e\left(t\right)-k_4{C}_c\left(t\right)=k_{1A}{C}_{EN}\left(t\right)-k_{2A}{C}_e\left(t\right)=k_{1L}{C}_p^a\left(t\right)-k_{2L}{C}_{EN}\left(t\right)$$

(6)

Thus, each individual flux step can be analyzed separately for its MM parameters. Using equation (6) and Table 1, we can derive (Supplement 2):

$$T_{\mathit{max}}^{\mathit{BBB}}={\mathit{CMR}}_{\mathit{Glc}}\frac{C_p^a\left(t\right)+C_{EN}\left(t\right)+K_T^{\mathit{BBB}}}{C_p^a\left(t\right)-C_{EN}\left(t\right)}$$

(7a)

$$T_{\mathit{max}}^{\mathit{BBB}}={\mathit{CMR}}_{\mathit{Glc}}\frac{C_{EN}\left(t\right)+C_e\left(t\right)+K_T^{\mathit{BBB}}}{R_{A/L}\left(C_{EN}\left(t\right)-C_e\left(t\right)\right)}$$

(7b)

$$T_{\mathit{max}}^{\mathit{CMB}}={\mathit{CMR}}_{\mathit{Glc}}\frac{C_e\left(t\right)+C_c\left(t\right)+K_T^{\mathit{CMB}}}{C_e\left(t\right)-C_c\left(t\right)}$$

(7c)

$$V_{\mathit{max}}^{HK}={\mathit{CMR}}_{\mathit{Glc}}\frac{C_c\left(t\right)+K_M^{HK}}{C_c}\left(t\right){\hspace{0.17em}=\mathit{CMR}}_{\mathit{Glc}}\left(1+\frac{K_M^{HK}}{C_c\left(t\right)}\right)$$

(7d)

Equations (7a) to (7d) for 3CM-EN and similar equations for 3CM/2CM/2CM-EN in Supplement 2 were used to determine the initial parameter ranges for data fitting as explained in the methods. The total glucose concentration in the tissue is the sum of all compartments, described in equation (8a) for the 3CM-EN and in Supplement 1 for the other models.

$$C_{\mathit{tot}}\left(t\right)={[f}_b{\rho }_b^{V/V}{C}_b\left(t\right)+f_{EN}{\rho }_{EN}^{V/V}{C}_{EN}\left(t\right)+f_e{\rho }_e^{V/V}{C}_e\left(t\right)$$

$$\hspace{1em}\hspace{1em}\hspace{1em}+{(f}_c-f_{EN}\left){\rho }_c^{V/V}\right(C_c\left(t\right)+C_{G6P}\left(t\right)\left)\right]/{\rho }_{\mathit{tot}}^{V/V}$$

(8a)

$${\rho }_{\mathit{tot}}^{V/V}=f_b{\rho }_b^{V/V}+f_{EN}{\rho }_{EN}^{V/V}+f_e{\rho }_e^{V/V}+{(f}_c-f_{EN}){\rho }_c^{V/V}$$

(8b)

$${\rho }_b^{V/V}=\left(1-\mathit{Hct}\right){\rho }_p^{V/V}+\mathit{Hct}·{\rho }_{\mathit{ery}}^{V/V}$$

(8c)

$$d_{\mathit{tot}}^{\hspace{0.17em}m/V}=f_b{d}_b^{\hspace{0.17em}m/V}+f_{EN}{d}_{EN}^{\hspace{0.17em}m/V}+f_e{d}_e^{\hspace{0.17em}m/V}+{(f}_c-f_{EN})d_c^{\hspace{0.17em}m/V}$$

(8d)

$${\rho }_b^{V/V}{C}_b\left(t\right)=(1-\mathit{Hct}){\rho }_p^{V/V}{[f_{a}C}_p^a\left(t\right)+f_v{C}_p^v\left(t\right)]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathit{Hct}{\rho }_{\mathit{ery}}^{V/V}\left[{f_{a}C}_{\mathit{ery}}^a\left(t\right)+f_v{C}_{\mathit{ery}}^v\left(t\right)\right]$$

(8e)

Notice that $f_{EN}$ is very small ⁶¹ ( $f_{EN,GM}=1.43·{10}^{-3}\mathrm{\hspace{0.17em}}$ mL EN/mL brain, Supplement 3), but included for accuracy of the equation. As its water content and density were assumed to be close to that of a GM cell, equations (8a), (8b) and (8d) complete the equation by subtracting $\mathrm{ }{f}_{EN}\mathrm{ }$ from the cell fraction.

Blood glucose concentrations

Calculation of the contribution of blood glucose to the total tissue glucose concentration (equations (8a) and (e)) requires knowledge of the relationships between arterial and venous glucose concentrations and between plasma and erythrocyte glucose levels.

Arterial versus venous glucose concentrations

Equation (1) includes the arterial plasma concentration in mM. However, most literature reports only venous plasma levels in µmol/mL plasma ( $C_p^{v,V_p}$ ). For steady-state situations over the range from normo- to hyperglycemia, the arteriovenous difference in glucose concentration is determined by tissue glucose consumption only. The GEF at normoglycemia is well known to be 0.108 ± 0.008 ^{6,62 –69} (Supplement 4), which can be used to determine $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ (equation (5)). For steady-state conditions at any glycemic level, the arterial plasma concentration can be determined from a measured venous plasma level using:

$$C_p^v\left(t\right)=C_p^a\left(t\right)+\frac{d{C}_p^a\left(t\right)}{dt}·\mathit{MTT}$$

(9a)

where $\mathrm{ }\mathit{MTT}\mathrm{ }$ is the mean transit time in minutes:

$$\mathit{MTT}=\frac{f_b}{d_{\mathit{tot}}^{m/V}·\mathit{CBF}}$$

(9b)

or

$$C_p^a\left(t\right)=C_p^v\left(t\right)+{\left(\frac{{\mathit{CMR}}_{\mathit{Glc}}}{\mathit{CBF}}\right)}^{\begin{array}{c}\mathit{normo}-\\ \mathit{glycemia}\end{array}}·\frac{1}{{\left(1-\mathit{Hct}\right)\rho }_p^{V/V}}$$

(9c)

or in terms of venous plasma glucose levels in µmol/mL plasma:

$$C_p^a\left(t\right)=\frac{C_p^{v,V_p}\left(t\right)}{{\rho }_p^{V/V}}+{\left(\mathit{GEF}·C_p^a\right)}^{\mathit{normoglycemia}}.$$

(9d)

Equations (9c) and (9d) are valid based on the results of Hasselbalch et al., ⁴⁷ who reported unchanged whole-brain $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ and $\mathrm{ }\mathit{CBF}\mathrm{ }$ between normoglycemia and acute hyperglycemia.

Plasma versus erythrocyte concentrations

The erythrocyte membrane (EMB) also has facilitated transport through GLUT1. Including metabolism of glucose, the mass-balance equations for the glucose fluxes in µmol/mL ery/min are:

$$\frac{{\mathrm{ }\rho }_p^{V/V}(1-\mathit{Hct})}{\mathit{Hct}}\frac{d{C}_p^{a/v}\left(t\right)}{dt}=-k_{in,\mathit{ery}}\left(t\right)C_p^{a/v}\left(t\right)+k_{\mathit{out},\mathit{ery}}\left(t\right)C_{\mathit{ery}}^{a/v}\left(t\right)$$

(10)

$${\rho }_{\mathit{ery}}^{V/V}\frac{d{C}_{\mathit{ery}}^{a/v}\left(t\right)}{dt}=k_{in,\mathit{ery}}\left(t\right)C_p^{a/v}\left(t\right)-k_{\mathit{out},\mathit{ery}}\left(t\right)C_{\mathit{ery}}^{a/v}\left(t\right)-{\mathit{EMR}}_{\mathit{Glc}}$$

(11)

in which the rate constants follow Michaelis-Menten kinetics:

$$k_{in,\mathit{ery}}\left(t\right)=\frac{T_{\mathit{max},in}^{\mathit{EMB}}}{C_p^{a/v}\left(t\right)+C_{\mathit{ery}}^{a/v}\left(t\right)+K_{T,in}^{\mathit{EMB}}}\mathrm{ }\mathrm{and}\mathrm{ }{k}_{\mathit{out},\mathit{ery}}\left(t\right)=\frac{T_{\mathit{max},\mathit{out}}^{\mathit{EMB}}}{C_p^{a/v}\left(t\right)+C_{\mathit{ery}}^{a/v}\left(t\right)+K_{T,\mathit{out}}^{\mathit{EMB}}}$$

(12)

The metabolic rate of D-glucose in erythrocytes ( ${\mathit{EMR}}_{\mathit{Glc}}=0.0265\pm 0.0017\hspace{0.17em}$ µmol/mL ery/min ⁷⁰ ) is negligible compared to the transport rates, which are three orders of magnitude larger. A first simple conclusion would thus be that at steady-state ( $\frac{d{C}_{\mathit{ery}}^{a/v}\left(t\right)}{dt}=0$ ), influx and efflux should be approximately equal, leading to equal concentrations inside and outside the cell. However, many erythrocyte glucose transport studies over the past five decades have shown that, while this applies to small mammals, it is not seen when studying human erythrocytes in vitro. ^71,72 Using a simple non-reversible transport MM model, without taking metabolism into account, several studies have shown a large difference between the maximum rates of influx ( $T_{in,\mathit{max}}^{\mathit{EMB}}$ ) and efflux ( $T_{\mathit{out},\mathit{max}}^{\mathit{EMB}}$ ) over the EMB as well as between the Michaelis constants for influx ( $K_{T,in}^{\mathit{EMB}})\mathrm{ }$ and efflux ( $K_{T,\mathit{out}}^{\mathit{EMB}}$ ). ^{73 –75} A recent re-review of data by Fuhrman, ⁷⁶ reflects the typical literature values: $\mathrm{ }{T}_{in,\mathit{max}}^{\mathit{EMB}}=33.1\hspace{0.17em}$ µmol/mL ery/min, $\mathrm{ }{T}_{\mathit{out},\mathit{max}}^{\mathit{EMB}}=132.0\hspace{0.17em}$ µmol/mL ery/min, $\mathrm{ }{K}_{T,in}^{\mathit{EMB}}=2.3\hspace{0.17em}\mathrm{mM}, K_{T,\mathit{out}}^{\mathit{EMB}}=11.3\hspace{0.17em}$ mM at 20 °C. However, all these dynamic transport studies were performed under non-physiological conditions, such as hundreds of mM D-glucose loaded into the erythrocyte or, vice versa, hundreds of mM outside the cell. Also, the erythrocyte solutions were many times diluted, and the measurements performed at lower temperature (typically 0–20 °C). Contrary to these data, several studies ^{77 –79} in the field of diabetic testing have shown a constant venous plasma (µmol/mL plasma) to venous blood (µmol/mL blood) concentration ratio over a wide range of concentrations, from normoglycemia to hyperglycemia, which, within experimental error, reflects their water content ratios (Table 2), i.e.:

$$C_p^{v,V_p}=\left(\frac{{\rho }_p^{V/V}}{{\rho }_b^{V/V}}\right)C_b^{v,V_b}=1.096C_b^{v,V_b}=\left(\frac{{\rho }_p^{V/V}}{{\rho }_{\mathit{ery}}^{V/V}}\right)C_{\mathit{ery}}^{v,V_{\mathit{ery}}}={1.280C}_{\mathit{ery}}^{v,V_{\mathit{ery}}}$$

(13a)

At steady-state, similar to the expectation above, this corresponds to approximately equal concentrations of D-glucose (in µmol/mL water) in plasma, erythrocytes, and thus blood:

$$C_b^a=C_{\mathit{ery}}^a=C_p^a$$

(13b)

$$C_b^{\mathrm{v}}=C_{\mathit{ery}}^v=C_p^v$$

(13c)

Another way to come to this conclusion is based on the logical expectation that $\mathrm{ }{K}_T\mathrm{ }$ for the GLUT1 transporter is the same for the BBB and EMB, with similar in- and outflux rates for erythrocytes at equal receptor densities. This corresponds to $\mathrm{ }{K}_{T,in}^{\mathit{EMB}}=K_{T,\mathit{out}}^{\mathit{EMB}}=K_T^{\mathit{BBB}}$ and $\mathrm{ }{T}_{in,\mathit{max}}^{\mathit{EMB}}=T_{\mathit{out},\mathit{max}}^{\mathit{EMB}}=T_{\mathit{max}}^{\mathit{EMB}}$ . At steady-state, using $EM{R}_{\mathit{Glc}}\ll K_T^{\mathit{BBB}}<T_{\mathit{max}}^{\mathit{EMB}}$ , it can then be derived that:

$$C_{\mathit{ery}}^{a/v}\left(t\right)=\frac{C_p^{a/v}\left(t\right)\left(T_{\mathit{max}}^{\mathit{EMB}}-EM{R}_{\mathit{Glc}}\right)-K_T^{\mathit{BBB}}EM{R}_{\mathit{Glc}}}{T_{\mathit{max}}^{\mathit{EMB}}+EM{R}_{\mathit{Glc}}}=C_p^{a/v}\left(t\right)-\frac{K_T^{\mathit{BBB}}}{T_{\mathit{max}}^{\mathit{EMB}}}EM{R}_{\mathit{Glc}}$$

(14)

again confirming that $\mathrm{ }{C}_{\mathit{ery}}^{a/v}\left(t\right)\hspace{0.17em}\approx \hspace{0.17em}{C}_p^{a/v}\left(t\right)$ . For our calculations, the only issue remaining is how fast glucose enters the erythrocytes when the arterial plasma concentration is increased, allowing for estimation of the erythrocyte contribution to total brain glucose. This requires knowledge of $\mathrm{ }{T}_{\mathit{max}}^{\mathit{EMB}}$ . Using the range of influx and efflux values from the literature (33–132 µmol/mL ery/min), ⁷⁶ simulations show that plasma and erythrocyte concentrations equilibrate quickly (Supplement 5) without much influence of the rate value. Therefore, an average $\mathrm{ }{T}_{\mathit{max}}^{\mathit{EMB}}=83\hspace{0.17em}$ µmol/mL ery/min was chosen to calculate erythrocyte D-glucose concentrations during infusion.

Modeling of tissue concentrations and Michaelis-Menten constants

To estimate the MM constants for transport and phosphorylation in healthy brain, all transport in healthy tissue is assumed to be facilitated by receptors, namely GLUT1 from blood into EN and EN into EES, GLUT1 into glial cells, and GLUT3 into neuronal cells. ⁴² Following the approach introduced by others, ^{11 –13} both steady-state and dynamic brain and plasma D-glucose levels from the MRS literature are used to estimate the unknown compartmental concentrations and MM constants through model fitting. For this fitting, appropriate initial ranges for these unknown parameters are set and the differential equations solved numerically using Matlab (R2024b) (MathWorks, Natick, MA, US). The approaches for determining constants and setting parameter ranges are explained in the steps below.

While it appears that there are many unknowns to be determined from a simple sugar uptake curve, several parameters can either be measured from blood ( $C_p^a\mathrm{ }$ or $\mathrm{ }{C}_p^v$ , Hct ⁸⁰ ) or are already reasonably well known from the literature. Here, this knowledge is used to derive the MM parameters for healthy tissue in a manner constrained by (i) plasma and brain D-glucose concentrations measured in MRS normoglycemia and hyperglycemia experiments, (ii) knowledge that during hyperglycemia the concentrations of brain products of glucose metabolism (e.g., glutamate) do not change significantly due to hexokinase being saturated as a consequence of its low $\mathrm{ }{K}_M^{HK}$ , which has been well studied in mammalian tissue, ^81,82 (iii) limits on ${\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ based on literature-based ranges for GEF and CBF (Supplements 4,6).

Step 1 – calculate $\mathrm{ }{\mathrm{CMR}}_{\mathrm{Glc}}\mathrm{ }$ range using normoglycemic GEF and CBF ranges and $\mathrm{ }{\mathrm{C}}_{\mathrm{p}}^{\mathrm{a}}$

Values from seven studies ^{6,64 –69} reporting $\mathrm{ }\mathit{GEF}\mathrm{ }$ or arteriovenous differences measured from blood samples obtained at rest in healthy volunteers were used to calculate a GEF range (Supplement 4), i.e., $\mathrm{ }\mathit{GEF}=0.108\pm 0.008$ . Using equation (5a), $\mathit{GEF}=0.108$ , and the average $\mathrm{ }{C}_p^{v,V_p}=5.14\hspace{0.17em}$ µmol/mL plasma from Shestov et al. ¹² (corresponding to $C_p^v=5.48\hspace{0.17em}$ mM), a representative normoglycemic $C_p^a=6.15\hspace{0.17em}$ mM was determined. In order to calculate a range of $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ values using equation (5b), a normoglycemic range of $\mathrm{ }\mathit{CBF}\mathrm{ }$ values is needed. Ranges of $\mathrm{ }\mathit{CBF}\mathrm{ }$ and $\mathrm{ }\mathit{CBV}\mathrm{ }$ of healthy volunteers from ten PET ¹⁵O studies ^{83 –92} show an average GM $\mathit{CBF}$ of $0.451\hspace{0.17em}\pm \hspace{0.17em}0.078\hspace{0.17em}$ mL/g/min and $\mathit{CBV}\mathrm{\hspace{0.17em}}$ of $\mathrm{ }0.038\hspace{0.17em}\pm \hspace{0.17em}0.007\hspace{0.17em}$ mL/mL (Supplement 6). The corresponding mean transit time is $\mathit{MTT}=4.85\hspace{0.17em}\pm \hspace{0.17em}0.86\mathrm{ }$ s (equation (9b)). Assuming a Hct of 0.4, the normoglycemic range for $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ in GM becomes: $0.131\le {\mathit{CMR}}_{\mathit{Glc}}\le 0.209\hspace{0.17em}$ µmol/g/min (Supplement 7).

Step 2 – calculate normoglycemic blood, plasma, and erythrocyte glucose concentrations

Using the reasoning with equations (13a) to (13c), the water-based normoglycemic glucose concentrations in erythrocytes and blood are the same as in plasma, namely $C_p^v=C_{\mathit{ery}}^v=C_b^v=5.48\hspace{0.17em}$ mM and $\mathrm{ }{C}_p^a=C_{\mathit{ery}}^a=C_b^a=6.15\hspace{0.17em}$ mM.

Step 3 – calculate $\mathrm{ }{\mathrm{V}}_{\max }^{\mathrm{HK}}\mathrm{ }$ value range

Determining a range of possible $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ values using equation (7d) requires knowledge of the ranges of $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}$ (Step 1), $\mathrm{ }{K}_M^{HK}$ , and $\mathrm{ }{C}_c$ . Hexokinase has been studied for mammalian cells and has a well-determined low $\mathrm{ }{K}_M^{HK}$ ( $\sim 0.05\hspace{0.17em}$ mM), ^81,82,93 indicating that at intracellular D-glucose concentrations of a few hundred µM, the saturation of hexokinase is already above 80% (Supplement 8). This corresponds well with data from MRS studies ^11,12 showing that the level of brain metabolic products of D-glucose (including glutamate) during steady-state hyperglycemia is not significantly different from that during normoglycemia, despite a large increase in brain D-glucose levels. As $\mathrm{ }{K}_M^{HK}\mathrm{ }$ is small compared to $\mathrm{ }{C}_c$ , it is valid to take 0.05 mM as the minimum for $\mathrm{ }{C}_c$ , which then determines the maximum $\mathrm{ }{V}_{\mathit{max}}^{HK}=2{\mathit{CMR}}_{\mathit{Glc}}.\mathrm{ }$ From the right-hand side of equation (7d), it can be concluded that increasing $\mathrm{ }{C}_c\mathrm{ }$ decreases the possible value of $\mathrm{ }{V}_{\mathit{max}}^{HK}$ , with a lower limit of $\mathrm{ }{V}_{\mathit{max}}^{HK}={\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ at infinite $\mathrm{ }{C}_c$ . Thus, $V_{\mathit{max}}^{HK}\mathrm{ }$ can only have a limited range:

$${\mathit{CMR}}_{\mathit{Glc}}<V_{\mathit{max}}^{HK}\le 2{\mathit{CMR}}_{\mathit{Glc}}$$

(15)

which corresponds to a $\mathrm{ }{V}_{\mathit{max}}^{HK}$ range of 0.131–0.418 µmol/g/min based on the $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ range for GM from Step 1. $V_{\mathit{max}}^{HK}\mathrm{ }$ has been measured as a function of ATP concentration in mammalian cells and was found to have a maximum at 0.30 µmol/g/min in non-tumor cells, ⁸¹ well in line with the above range. Notice that for the 2CMs, the concentration determining the rate for metabolism is $\mathrm{ }{C}_{c+e}\mathrm{ }$ (Supplement 1, Eq. S3e) for the mixed EES/intracellular compartment. A higher cytosolic concentration automatically forces the allowed range for $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ to be close to $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}\mathrm{ }$ or a few percent more (equation (7d), Supplementary Eq. S2d).

Step 4 – $\mathrm{d}$ etermine fitting ranges for the MM parameters

To determine sufficiently large and reasonable ranges for the MM maximum rates when fitting them from the experimental data, equations (7a) to (7c) for 3CM-EN and Supplementary Eqs. S4-6 for the other models were used. To accomplish this, input ranges of possible compartmental glucose concentrations and the MM constants $\mathrm{ }{K}_T^{\mathit{BBB}}$ and $K_T^{\mathit{CMB}}\mathrm{ }$ are required. Working from existing knowledge, the need for D-glucose to cross the BBB and later a cell membrane followed by phosphorylation and metabolism, will lead to tissue concentrations decreasing continuously from plasma to cell. To account for this, a minimum of 20% decrease between compartments and a maximum of 80% decrease was assumed. Using $\mathrm{ }{C}_p^a=6.15\hspace{0.17em}$ mM, this gives the following ranges:

$$\mathrm{ }1.23\mathrm{\hspace{0.17em}}\mathrm{mM}<C_{EN}<4.92\mathrm{\hspace{0.17em}}\mathrm{mM}\mathrm{ }0.25\mathrm{\hspace{0.17em}}\mathrm{mM}<C_e<3.94\mathrm{\hspace{0.17em}}\mathrm{mM}\mathrm{ }0.05\mathrm{\hspace{0.17em}}\mathrm{mM}<C_c<3.15\mathrm{\hspace{0.17em}}\mathrm{mM}\mathrm{ }$$

(16a)

$$\mathrm{ }1.23\mathrm{\hspace{0.17em}}\mathrm{mM}<C_e<4.92\mathrm{\hspace{0.17em}}\mathrm{mM}\mathrm{ }0.25\mathrm{\hspace{0.17em}}\mathrm{mM}<C_c<3.94\mathrm{\hspace{0.17em}}\mathrm{mM}$$

(16b)

$$\mathrm{ }1.23\mathrm{\hspace{0.17em}}\mathrm{mM}<C_{EN}<4.92\mathrm{\hspace{0.17em}}\mathrm{mM}\mathrm{\hspace{0.17em}}0.25\mathrm{\hspace{0.17em}}\mathrm{mM}<C_{e+c}<3.94\mathrm{\hspace{0.17em}}\mathrm{mM}$$

(16c)

$$1.23\hspace{0.17em}\mathrm{mM}<C_{e+c}<4.92\hspace{0.17em}\mathrm{mM}$$

(16d)

Interestingly, for 3CM-EN, the lower value of $\mathrm{ }{C}_c\mathrm{ }$ is equal to the MM constant for hexokinase, which is very reasonable at normoglycemia.

$K_T^{\mathit{BBB}}\mathrm{ }$ has been well analyzed in PET and MRS studies to be on the order of the baseline plasma D-glucose concentration, i.e., about 4–7 mM. $\mathrm{ }{K}_T^{\mathit{CMB}}$ , which for our purpose reflects a proportional average between the astrocyte GLUT1 and neuronal GLUT3 Michaelis constants, is expected to be lower than $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ due to the higher affinity of GLUT3. ^40,42 To be conservative, a range of $0.50<{K_T^{\mathit{CMB}},K}_T^{\mathit{BBB}}<15\hspace{0.17em}$ mM was chosen. Using equations 7(a) to (c), this range was used together with $\mathrm{ }{C}_p^a=6.15\hspace{0.17em}$ mM, the $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ range from Step 1, and the concentration ranges from equation (16a) to calculate the following ranges for the 3CM-EN fitting: $\mathrm{ }0.210\le T_{\mathit{max}}^{\mathit{BBB}}\le 4.43\hspace{0.17em}$ µmol/g/min and $\mathrm{ }0.529\le T_{\mathit{max}}^{\mathit{CMB}}\le 5.86\hspace{0.17em}\mu \mathrm{mol}/\mathrm{g}/\min$ . The corresponding $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ and $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}\mathrm{ }$ (when applicable) ranges for the other models were set the same as for the 3CM-EN, as derived from Supplementary Eqs. S4-6. Interestingly, the average range for $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}\mathrm{ }$ is higher than for $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}$ , which is not unexpected in view of reported faster transport for brain cells compared to endothelial cells. ^40,42

Assumed values of $\mathrm{ }{V}_{\mathit{max}}^{\mathit{glyc}-\mathit{brain}}=4.0\hspace{0.17em}$ µmol/g/min and $K_M^{\mathit{glyc}-\mathit{brain}}=1.5\hspace{0.17em}$ mM were used with the goal of having efficient removal of glycolytic products to form pyruvate.

Step 5 – further refinement of the MM parameters

The MM parameters $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}, K_T^{\mathit{BBB}}, T_{\mathit{max}}^{\mathit{CMB}}, K_T^{\mathit{CMB}}$ and $V_{\mathit{max}}^{HK}$ were further refined by numerical fitting of the model to the average (N = 5) of the dynamic D-glucose uptake curves from Shestov et al.. ¹² To construct a high-time-resolution arterial D-glucose input function, the time dependent $\mathrm{ }{C}_p^{v,\hspace{0.17em}{V}_p}\left(t\right)\mathrm{ }$ values from that study were averaged over all subjects (N  = 5) and interpolated linearly in steps of 0.001 min. A baseline arterial D-glucose concentration $\mathrm{ }{C}_p^{a,\hspace{0.17em}{V}_p}\mathrm{ }$ was calculated from $\mathrm{ }{C}_p^{v,\hspace{0.17em}{V}_p}\mathrm{ }$ using equation (5a) and $\mathrm{ }\mathit{GEF}=0.108$ , and $C_p^{a,\hspace{0.17em}{V}_p}\mathrm{ }$ was subsequently converted to $\mathrm{ }{C}_p^a$ using the water content. The original $\mathrm{ }{C}_p^{v,\hspace{0.17em}{V}_p}\left(t\right)\mathrm{ }$ curve consisted of 26 measurements between time 0 (baseline concentration) and 105 min. A baseline period of 30 min (i.e. repetition of the same concentration) was added to ensure that a steady-state baseline concentration would be reached before the onset of D-glucose infusion. The whole blood D-glucose concentration $\mathrm{ }{C}_b\left(t\right)\mathrm{ }$ was calculated using equation (8e), Table 2, and assuming $\mathrm{ }\mathit{Hct}=0.4$ . Hct can vary substantially with gender and vessel size. Also, in capillaries, where the glucose exchange occurs, Hct is about 85% of that in larger vessels, ⁹⁴ where $C_p^{v,\hspace{0.17em}{V}_p}\mathrm{ }$ is measured and which determines overall delivery of oxygen. However, because of the requirements of constant oxygen delivery to maintain metabolism and the availability of sufficient glucose in the blood, the determination of MM parameters for glucose transport is not affected by Hct, as explained in Supplement 9. The small changes in CBF and CBV ( $f_b$ ) resulting from a change in Hct are compensated by an appropriate change in GEF in equation (5b) and $\mathrm{ }\frac{d{C}_p^a\left(t\right)}{dt}\mathrm{ }$ in equation (1), respectively.

The model parameters (MM constants for transport and metabolism) were optimized by minimizing the root mean square error (RMSE) between the simulated total D-glucose concentration $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{sim}}\left(t\right)$ , and the experimental data $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{exp}}\left(t\right)\mathrm{ }$ from Shestov et al.. ¹² Optimization was performed using MATLAB’s Optimization Toolbox, version 24.2. The fmincon function was used with RMSE as the objective function and the sequential quadratic programming (SQP) algorithm, with two physiologically relevant constraints applied during the optimization process. First, $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ at steady-state normoglycemia was calculated for each $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ and $C_c\mathrm{ }$ using equation (7d). A penalty proportional to the deviation was applied if $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ exceeded the upper limit determined in Step 1. Second, the $\mathrm{ }{C}_v^{p,V_p}$ -to- $C_{\mathit{tot}}^{\mathit{sim}}\mathrm{ }$ ratio was calculated at normoglycemia and compared with experimental ratio ranges from the literature, ^11,12 and deviations outside ±2 SD were similarly penalized. A small time-increment dt of 0.001 min was used to ensure numerical stability. The optimization tolerances were set to a function value of $\mathrm{ }{10}^{-5}\mathrm{ }$ and a step tolerance of $\mathrm{ }{10}^{-5}$ . The lower and upper bounds for each fitted parameter were based on the $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}},T_{\mathit{max}}^{\mathit{CMB}}, \mathrm{and} K_T^{\mathit{BBB}}\mathrm{ }$ ranges determined in Step 4 and the $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ range from Step 3. To reduce the number of independent variables to be fitted, a physiological constraint was added through a ratio $R_{K_T}=K_T^{\mathit{BBB}}/K_T^{\mathit{CMB}}>1,$ based on the affinity considerations mentioned in Step 4. The initial guesses were $T_{\mathit{max}}^{\mathit{BBB}}=1.0\hspace{0.17em}$ µmol/g/min, $K_T^{\mathit{BBB}}=8.0\hspace{0.17em}\mathrm{mM},T_{\mathit{max}}^{\mathit{CMB}}=1.0\hspace{0.17em}$ µmol/g/min, and $\mathrm{ }{V}_{\mathit{max}}^{HK}=0.20\hspace{0.17em}$ µmol/g/min, based on physiological expectations from the literature. To check the robustness, the optimization was repeated twice with initial guesses set to the lower and upper bounds, respectively. Values for $\mathrm{ }{R}_{K_T}\mathrm{ }$ and the membrane polarization ratio $\mathrm{ }{R}_{A/L}$ were chosen (see Step 6 for optimization procedure).

For each set of parameters, compartmental D-glucose concentrations $C_{\mathit{ery}}^{a/v}\left(t\right), C_{EN}\left(t\right){, C}_e\left(t\right), C_c\left(t\right),$ and $C_{G6P}\left(t\right)\mathrm{ }$ were calculated stepwise by numerically solving the differential equations (equations (1) to (4), (10) and (11): The time-dependent rate constants $\mathrm{ }{k}_i\left(n\right)\mathrm{ }$ (Table 1, equation (12)) were calculated stepwise (n steps) using the concentration determined in the previous step $\mathrm{ }C\left(n-1\right);k_i\left(n\right)\mathrm{ }$ and $\mathrm{ }C\left(n-1\right)$ were then used to calculate $\mathrm{ }dC/dt\mathrm{ }$ using equations (1) to (4) and (11). The compartmental D-glucose concentration $\mathrm{ }C\left(n\right)\mathrm{ }$ was calculated as $\mathrm{ }C\left(n-1\right)+(dC/dt)·dt$ . The initial compartmental concentrations were all set to zero and their fitting ranges were not limited, except by the initial value of $\mathrm{ }{C}_p^a$ . $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{sim}}\left(t\right)\mathrm{ }$ was calculated for each combination of MM constants using equation (8a), $\mathrm{ }{C}^b\left(t\right)$ , the simulated $\mathrm{ }{C}_e\left(t\right), C_c\left(t\right), C_{G6P}\left(t\right)$ , as well as the constants for GM given in Table 2. Every $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{sim}}\left(t\right)\mathrm{ }$ curve was downsampled and compared to $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{exp}}\left(t\right)\mathrm{ }$ using the RMSE. To emphasize the steeper initial part of the curve, all data points between $\mathrm{ }t=0\mathrm{ }$ and $\mathrm{ }t\hspace{0.17em}=\hspace{0.17em}$ 10 min were included in the RMSE calculation. After that point in time, every second data point was included to avoid the flat part of the curve dominating the RMSE. Finally, the combination of MM constants resulting in the smallest RMSE was determined. To estimate the range of MM constants that produced a good fit, the best-fit MM constants were perturbed by introducing ±25% random Gaussian noise, and the corresponding glucose concentrations and RMSEs were calculated. The perturbation process was repeated 10000 times, and the standard deviation of each MM constant ( $T_{\mathit{max}}^{\mathit{BBB}},K_T^{\mathit{BBB}}, T_{\mathit{max}}^{\mathit{CMB}}$ , $K_T^{\mathit{CMB}}\mathrm{ }$ and $\mathrm{ }{V}_{\mathit{max}}^{HK}$ ) was calculated for all fits with an RMSE $\mathrm{ }\le 0.30\hspace{0.17em}$ mM.

To account for model complexity, the small-sample-corrected Akaike Information Criterion $\mathrm{ }\left(AI{C}_c\right)\mathrm{ }$ was calculated for all models. Practical identifiability of MM parameters was evaluated via Monte Carlo simulations, where both the input function $C_p^{v,V_p}$ and $C_{\mathit{tot}}^{\mathit{sim}}\left(t\right)\mathrm{ }$ were perturbed and refitted to get parameter distributions. Sensitivity analysis was performed by perturbing each best-fit parameter by ±50% while keeping the others fixed and assessing the resulting changes in $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{sim}}\left(t\right)$ .

Step 6 – evaluate the impact of $\mathrm{ }{\mathrm{R}}_{\mathrm{A}/\mathrm{L}}\mathrm{ }$ and $\mathrm{ }{\mathrm{R}}_{{\mathrm{K}}_{\mathrm{T}}}$

The impacts of the luminal-to-abluminal ratio $\mathrm{ }{R}_{A/L}\mathrm{ }$ and the Michaelis constant ratio $\mathrm{ }{R}_{K_T}$ on the fitting results were assessed by repeating the optimization process for ranges of $\mathrm{ }{R}_{K_T}\mathrm{ }$ and $\mathrm{ }{R}_{A/L}$ . For the 3CM-EN and 3CM, $\mathrm{ }{R}_{K_T}\mathrm{ }$ was varied over the range $\mathrm{ }1.0\le R_{K_T}\le 3.0\mathrm{ }$ based on the physiological expectation that $\mathrm{ }{K}_T^{\mathit{CMB}}<K_T^{\mathit{BBB}}.\mathrm{ }$ For the 3CM-EN and 2CM-EN, the abluminal-to-luminal ratio was varied over the range $\mathrm{ }0.5\le R_{A/L}\le 5.0\mathrm{ }$ (Supplement 10). Results were judged to be in the appropriate range for cases where $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}$ > $T_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ as based on literature knowledge of the relative magnitude of the transport rates for GLUT3 and GLUT1.

Step 7 – validation through simulation of steady-state GM concentrations

Steady-state compartmental GM D-glucose concentrations were simulated by creating 19 rectangular input functions $\mathrm{ }{C}_p^v\left(t\right)\mathrm{ }$ with a baseline value $\mathrm{ }{C}_p^v=5.48\hspace{0.17em}$ mM and ramped to a constant hyperglycemic value in the range 2–20 mM in steps of 1 mM. $\mathrm{ }{C}_{\mathit{tot}}^{\mathit{sim}}$ was then calculated as described in Step 5, using the MM constants from the best fit. This simulation was repeated for every combination of MM constants resulting in an RMSE $\le 0.30\hspace{0.17em}$ mM from Step 5 and the results were compared with experimental GM D-glucose concentrations at steady-state from Shestov et al. ¹²

Step 8 – determine MM constants and simulate compartmental concentrations in WM and GBM

The combination of MM constants that produced the best fit in GM was used to simulate D-glucose concentrations in WM, and GBM, using values of $\mathrm{ }{R}_{K_T}\mathrm{ }$ and $\mathrm{ }{R}_{A/L}\mathrm{ }$ determined in Step 6. A recent review by Ibaraki et al. ⁸³ reported GM-to-WM ratios of CBF and CBV of 2.75 and 2.15, respectively. Assuming equal density of GLUT1 receptors in the BBB for WM and GM, a reduction of CBV leads to a proportional reduction in maximum transport capability for WM relative to GM and $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ for WM was adjusted accordingly. Based on the coupling of flow and metabolism, a reduction of $\mathrm{ }{T}_{\mathit{max},}^{\mathit{CMB}}\mathrm{ }$ and $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ by a factor of 2.75 was assumed for WM relative to GM. The $\mathrm{ }{K}_T\mathrm{ }$ and the MM constants for phosphorylation were kept the same as in GM. Steady-state WM D-glucose concentrations were simulated using rectangular input functions as described in Step 7, and were compared to experimental MRS steady-state concentrations in GM and WM from de Graaf et al. ¹¹

For the tumor case, a glioblastoma with BBB breakdown was simulated by setting $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}=10\hspace{0.17em}$ µmol/g/min, leading to rapid equilibration of the D-glucose concentrations in plasma and EES. Nagamatsu et al. showed that, similar to astrocytes, brain tumor cells have mainly GLUT1 transporters, but with increased density to facilitate transport for accommodating faster glucose metabolism. ⁹⁵ The $\mathrm{ }{K}_T^{\mathit{CMB}}$ = $K_T^{\mathit{BBB}}\mathrm{ }$ from GM was therefore assumed for tumor, and increased transport was accounted for by setting $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}\mathrm{ }$ for GBM to double the GLUT1-determined $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ for GM. Additionally, Nagamatsu et al., ⁹⁵ Graham et al. ⁹⁶ and Muzi et al. ⁹⁷ showed that metabolism ( $V_{\mathit{max}}^{HK})\mathrm{ }$ in malignant cells is about three times faster than in healthy GM, which was used in our estimates. These investigators found a large $\mathrm{ }{K}_M^{HK}\mathrm{ }$ range, from which $\mathrm{ }{K}_M^{HK}=0.12\mathrm{ }$ was chosen. The $\mathrm{ }{C}_p^a\left(t\right)\mathrm{ }$ and $\mathrm{ }{C}_{\mathit{ery}}^{a/v}\left(t\right)$ curves simulated for GM were used in the simulation of WM and GBM.

Step 9 – determine $\mathrm{ }\mathrm{CM}{\mathrm{R}}_{\mathrm{Glc}}\mathrm{ }$ and $\mathrm{ }\mathrm{CBF}$

$CM{R}_{\mathit{Glc}}\mathrm{ }$ at normoglycemia ( $t=0$ ) was calculated for all tissues using equation (7d), $\mathrm{ }{V}_{\mathit{max}}^{HK}$ , and $\mathrm{ }{C}_c\left(t=0\right)$ . Subsequently, $\mathit{CBF}\mathrm{ }$ was calculated for GM and WM using equation (5b), $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}$ , and the normoxic GEF from Step 1.

Results

Figure 2(a) shows best fits of the 3CM-EN, 3CM, and 2CM-EN to the GM D-glucose uptake curve from the MRS data in Shestov et al. (average ± standard deviation from 5 subjects). ¹² The data were analyzed using an abluminal-to-luminal GLUT1 polarity ratio $R_{A/L}=2.0$ for models with an endothelial compartment, and $\mathrm{ }{R}_{K_T}=K_T^{\mathit{BBB}}/K_T^{\mathit{CMB}}=2.0\mathrm{ }$ for models with both BBB and CMB. Figure 2(b) shows $T_{\mathit{max}}^{\mathit{BBB}},K_T^{\mathit{BBB}}, T_{\mathit{max}}^{\mathit{CMB}}, K_T^{\mathit{CMB}}\mathrm{ }$ as a function of $\mathrm{ }{R}_{A/L}$ . Each data point represents a separate fit of the MRS curve in Figure 2(a), with a corresponding RMSE. Fits were performed using the ranges and $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ and normoglycemic $\mathrm{ }{C}_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ constraints described in Steps 1–5. The largest dependence on $\mathrm{ }{R}_{A/L}\mathrm{ }$ is observed for $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}$ , which reduces steeply with increasing $R_{A/L}$ for $\mathrm{ }{R}_{A/L}<1.0$ , leveling off in the range ${1.0<R}_{A/L}<2.0$ and becoming approximately constant within error limits for $\mathrm{ }{R}_{A/L}>2.0$ . At that point, values slowly approach those observed for the models with a single BBB membrane (solid symbols displayed at the borders to the right). The numbers displayed in Figure 2(b) are listed in Supplementary Table S3 in Supplement 10, where, over the $\mathrm{ }{R}_{K_T}\mathrm{ }$ range studied, all $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ values show the same pattern. $\mathrm{ }{K}_T^{\mathit{BBB}}$ is also affected by $\mathrm{ }{R}_{A/L}$ , but only over a limited value range from 5.85 to 7.12 for $\mathrm{ }{R}_{K_T}=2.0$ , and ranges of the same magnitude at other $\mathrm{ }{R}_{K_T}\mathrm{ }$ values. Interestingly, replacing the dual-membrane endothelial cell with a single membrane BBB leads to a larger $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ for 3CM-EN, while very limited change is apparent for 2CM-EN. For both models, the single membrane values appear to correspond to the extrapolation limit at high $\mathrm{ }{R}_{A/L}\mathrm{ }$ (see bottom of Supplementary Tables S3 for 3CM-EN and S5 for 2CM-EN). Interestingly, the RMSE for 3CM-EN is constant as a function of $\mathrm{ }{R}_{A/L}$ , while the 2CM-EN has a clear RMSE minimum around $R_{A/L}=1.0$ .

Figure 2.

Calculated RMSE and Michaelis Menten parameters for best fits of the models to the GM MRS data of Shestov et al. using the constraints from steps 1–5. (a) Results for 3CM-EN, 3CM, and 2CM-EN using the abluminal-to-luminal GLUT1 polarity ratio $\mathrm{ }{R}_{A/L}=2.0$ (3CM-EN, 2CM-EN) and $\mathrm{ }{R}_{K_T}=K_T^{\mathit{BBB}}/K_T^{\mathit{CMB}}=2.0\mathrm{ }$ (3CM-EN, 3CM). (b) Results for 3CM-EN (blue symbols) and 2CM-EN (orange symbols) as a function of $\mathrm{ }{R}_{A/L}\mathrm{ }$ ( $R_{K_T}=2.0\mathrm{ }$ for 3CM-EN). The filled symbols at the borders to the right represent the parameter values excluding the EN compartment, i.e. a single BBB membrane. Trends at $\mathrm{ }{R}_{K_T}=2.0\mathrm{ }$ were the same as for $\mathrm{ }{R}_{K_T}\mathrm{ }$ values from 1–3 (Supplement 10). (c) Results for the fitted parameters in the 3CM-EN (green symbols) and 3CM (blue symbols) as a function of $R_{K_T}\mathrm{ }$ ( $R_{A/L}=2\mathrm{ }$ for 3CM-EN). Trends were the same for $\mathrm{ }{R}_{A/L}\mathrm{ }$ values from 0.5–5 (Supplement 10). (d) 2-tissue-compartment results without the $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}\le 0.209\mathrm{ }$ constraint: 2CM-EN (left), 2CM (middle), 2CM non-reversible (right).

Figure 2(c) shows fitted MM parameters for the 3CM-EN and 3CM as a function of $\mathrm{ }{R}_{K_T}$ . Supplementary Table S3 and Supplementary Figure S6 (Supplement 10) show the 3CM-EN fitting results, and Supplementary Table S4 those for 3CM. The results indicate that excluding the EN compartment leads to smaller $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}$ , and larger $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}$ , resulting in a larger difference between these parameters for the 3CM (Supplement 10). Interestingly, at $\mathrm{ }{R}_{K_T}\ge 3.0$ , for the 3CM-EN and 3CM, the maximum rates are approximately equal for all membranes. Excluding the EN compartment also increases $\mathrm{ }{K}_T$ , but at higher $\mathrm{ }{R}_{K_T}\mathrm{ }$ values, the $\mathrm{ }{K}_T\mathrm{ }$ values for the two model versions approach each other, which occurs faster for $\mathrm{ }{K}_T^{\mathit{CMB}}\mathrm{ }$ than for $\mathrm{ }{K}_T^{\mathit{BBB}}$ . For 3CM-EN, the MM parameters for the CMB show a stronger dependence on $\mathrm{ }{R}_{K_T}\mathrm{ }$ than those for the BBB. While RMSE is barely affected, the fitting shows a clear tendency to keep $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ constant when increasing $\mathrm{ }{R}_{K_T}$ . This naturally leads to a reduction in $\mathrm{ }{K}_T^{\mathit{CMB}}$ , which is paired to a reduction in $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}$ . For 3CM-EN, the same trends as a function of $\mathrm{ }{R}_{K_T}\mathrm{ }$ were found at other $\mathrm{ }{R}_{A/L}\mathrm{ }$ values (Supplementary Figure S6). Interestingly, for 3CM, $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ reduces by about 13% over the $\mathrm{ }{R}_{K_T}\mathrm{ }$ range from 1.0–3.0, which is paired with a strong reduction in $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}$ .

To choose reasonable values for $\mathrm{ }{R}_{K_T}\mathrm{ }$ and $\mathrm{ }{R}_{A/L}$ , the parameter values in Supplementary Table S3 were assessed under the additional criterion of $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}<T_{\mathit{max}}^{\mathit{CMB}}$ , which applied to all results for $R_{A/L}\ge 2.0$ . $R_{K_T}=2.0\mathrm{ }$ and $\mathrm{ }{R}_{A/L}=2.0\mathrm{ }$ were chosen based on reported differences between $\mathrm{ }{K}_T\mathrm{ }$ values for GLUT1 and GLUT3 in the literature and the logic that $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ likely is close to the normoglycemic blood glucose concentration. These values were used to generate the results for 3CM-EN, 3CM, and 2CM-EN in Figure 2(a). The 3CM-EN and the 3CM have a reasonably low RMSE (0.15 mM), but the 2CM-EN fit showed higher RMSE (0.20 mM), which we attribute to the metabolic constraint not applying to the 2CM model. Figure 2(d) shows the fitting results for the 2CM-EN without applying fitting constraints. The RMSE for the 2CM-EN decreases from 0.20 to 0.14 mM and $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ remains well below the limit established in Step 3. The $\mathrm{ }{R}_{A/L}\mathrm{ }$ dependence for 2CM-EN, with and without the $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ constraint, is given in Supplement 11 (Supplementary Table S6). When removing the $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ constraint, RMSE becomes constant as a function of $\mathrm{ }{R}_{A/L}$ , indicating that the previously measured $\mathrm{ }{R}_{A/L}$ -dependence of RMSE was due to that constraint. Again the values at higher $R_{A/L}\mathrm{ }$ approach those without endothelial compartment, namely $\mathrm{ }{K}_T^{\mathit{BBB}}=4.48\hspace{0.17em}$ mM and $T_{\mathit{max}}^{\mathit{BBB}}=0.687\hspace{0.17em}$ µmol/g/min, which for $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ is comparable to that for the 3CM at $\mathrm{ }{R}_{K_T}=2.0$ , while $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ is about half.

Figure 2(d) also includes fitting for the unconstrained 2CM, both with reversible ( $k_1=k_2$ ) and non-reversible transport ( $k_1\ne k_2$ ). Table 3 summarizes the best fit MM parameters and compartmental concentrations for all models. The corresponding compartmental concentration curves and results for each model are shown in Supplement 12. Interestingly, the $C_{\mathit{tot}}\mathrm{ }$ and $\mathrm{ }{C}_p^{v{,V}_p}/C_{\mathit{tot}}\mathrm{ }$ values for 3CM-EN and 3CM at normoglycemia differ from those of 2CM-EN and 2CM when applying the $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ constraint, but become equal when removing this constraint for the 2-compartment models. Similarly, $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ values compare better when removing the $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ constraint for the 2CM. $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ values remain different, but two-compartment values are close to $\mathrm{ }{K}_T^{\mathit{CMB}}\mathrm{ }$ values for the 3CMs. Fitting the unconstrained non-reversible 2CM results in MM parameters similar to those for the 3CM.

Table 3.

Michaelis-Menten constants for transport and metabolism, normoglycemic $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}$ , CBF, tissue fractions and D-glucose concentrations at normoglycemia (n) and steady state hyperglycemia (h) ( $C_p^{v,V_p}=17\hspace{0.17em}$ mM) for three tissues and for all models ( $R_{A/L}$  = 2.0; $\mathrm{ }{R}_{K_T}$  = 2.0).

	Gray matter					White matter					Glioblastoma
	3CM-EN	3CM	2CM-EN#	2CM#	2CM#non-rev	3CM-EN	3CM	2CM-EN#	2CM#	2CM#non-rev	3CM-EN	3CM	2CM-EN#	2CM#	2CM#non-rev
$T_{\mathit{max}}^{\mathit{BBB}}$	0.80 ± 0.14 $	0.68 ± 0.11	0.72 ± 0.11	0.69 ± 0.07	0.65 ± 0.11	0.37€	0.32	0.34	0.32	0.30	10&	10	10	10	10
$K_T^{\mathit{BBB}}$	6.2 ± 1.5 $	8.5 ± 2.0	3.9 ± 0.9	4.5 ± 1.1	8.4 ± 1.7	6.2§	8.5	3.9	4.5	8.4	6.2§	8.5	3.4	4.5	8.4
$T_{\mathit{max}}^{\mathit{CMB}}$	1.0 ± 0.2 $	1.1 ± 0.2	N/A	N/A	N/A	0.38€€	0.39	N/A	N/A	N/A	1.6§§	1.4	N/A	N/A	N/A
$K_T^{\mathit{CMB}}$	3.1 ± 0.8¢	4.3 ± 1.0				3.1§	4.3				6.2§§	8.5
$V_{\mathit{max}}^{HK}$	0.26 ± 0.04 $	0.28 ± 0.04	0.33 ± 0.05	0.42 ± 0.04	0.28 ± 0.05	0.10€€	0.10	0.12	0.15	0.10	0.78§§	0.84	0.98	1.25	0.83
${\mathit{CMR}}_{\mathit{Glc}}$	0.21 ± 0.03†	0.21 ± 0.03	0.29 ± 0.04	0.36 ± 0.03	0.25 ± 0.03	0.09†	0.09	0.11	0.14	0.10	0.60†	0.49	0.95	1.22	0.82
CBF	0.56 ± 0.08£	0.56 ± 0.08	0.77 ± 0.12	0.97 ± 0.09	0.66 ± 0.09	0.23¢	0.23	0.30	0.38	0.26	N/A	N/A	N/A	N/A	N/A
f @	$f_b=0.038, f_{EN}=$ 1.4·10–3 $, f_e=0.22$					$f_b=0.018, { f}_{EN}=$  0.67·10–3 $, { f}_e=0.22$					$f_b=0.050, { f}_{EN}=$ 1.9·10–3 ${, f}_e=0.50$
$C_{EN}$ (n)	2.33	N/A	1.53	N/A	N/A	2.72	N/A	2.04	N/A	N/A	5.09	N/A	4.74	N/A	N/A
$C_e$ (n)	1.08	1.26	0.374 *	0.345*	0.396*	1.54	1.68	0.882*	0.914*	0.915*	4.61	5.17	4.13*	4.33*	4.36*
$C_c$ (n)	0.204	0.146				0.409	0.279				0.408	0.169
$C_{G6P}$ (n)	0.083	0.083	0.117	0.150	0.098	0.032	0.033	0.043	0.056	0.037	0.266	0.211	0.469	0.654	0.378
$C_{\mathit{tot}}$ (n)	0.692	0.694	0.692	0.695	0.695	0.855	0.803	1.02	1.07	1.05	3.20	3.41	4.66	5.02	4.79
$C_p^{v{,V}_p}/C_{\mathit{tot}}$ (n)	7.43	7.41	7.43	7.39	7.40	6.01	6.40	5.03	4.82	4.90	1.60	1.51	1.10	1.02	1.07
$C_{EN}$ (h)	8.13	N/A	5.92	N/A	N/A	9.08	N/A	7.08	N/A	N/A	15.7	N/A	15.1	N/A	N/A
$C_e$ (h)	5.02	5.40	3.03*	3.01*	3.09*	6.04	6.55	4.12*	4.45*	3.95*	14.4	15.5	13.5*	14.2*	4.33*
$C_c$ (h)	2.41	2.27				2.82	2.71				3.23	1.25
$C_{G6P}$ (h)	0.102	0.111	0.131	0.171	0.109	0.036	0.039	0.045	0.058	0.038	0.348	0.357	0.482	0.672	0.387
$C_{\mathit{tot}}$ (h)	3.74	3.74	3.75	3.77	3.78	4.07	4.13	4.45	4.80	4.28	10.6	10.5	14.2	15.0	13.7
$C_p^{v{,V}_p}/C_{\mathit{tot}}$ (h)	4.54	4.54	4.54	4.51	4.50	4.18	4.12	3.82	3.54	3.98	1.61	1.62	1.19	1.13	1.24

Units: $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}$ , $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}$ , $\mathrm{ }{V}_{\mathit{max}}^{HK}$ , $\mathrm{ }{\mathit{CMR}}_{\mathit{Glc}}$ : µmol/g/min; f mL/mL; $\mathrm{ }{K}_T^{\mathit{BBB}}, K_T^{\mathit{CMB}}$ , $\mathrm{ }{C}_i$ : mM; Compartmental concentrations were obtained from the simulated rectangular input function shown in Supplementary Figure S11 (Supplement 15) with $\mathrm{ }{C}_p^a=6.15\hspace{0.17em}$ mM at normoglycemia and 18.8 mM at hyperglycemia. $\mathrm{ }{K}_M^{HK}\mathrm{ }$ was assumed to be 0.050 mM for GM and WM ^81,82,93 , and 0.12 mM for GBM ^{95 – 97}. ^# Unconstrained 2CM-EN, 2CM and 2CM non-reversible. The following superscript symbols, shown after the 3CM-EN apply to all the models and are defined as:.

$

Determined from fitting the model to the GM D-glucose data of Shestov et al. ¹² ; ^¢ Assuming $\mathrm{ }{K}_T^{\mathit{BBB}}=R_{K_T}{K}_T^{\mathit{CMB}}$ , where $\mathrm{ }{R}_{K_T}=2.0\mathrm{ }$ while fitting the model; ^† Calculated using equation (7d) at normoglycemia, $\mathrm{ }{V}_{\mathit{max}}^{HK}\pm$ error and $\mathrm{ }{C}_c\mathrm{ }$ at baseline; ^£Calculated from equation (5 )b and ${\mathit{CMR}}_{\mathit{Glc}}\pm$ error; ^@ $f_b$ : described in Step 1 and Supplement 6, $\mathrm{ }{f}_{EN}$ : Calculated from Pardridge ⁶¹ , described in Supplement 3, $\mathrm{ }{f}_e\mathrm{ }$ from Zamecnik et al. ¹¹³ and Vargova et al. ¹¹⁴ ; ^€Assumed to be a factor of 2.15 lower than in GM; ^§ Assumed equal to GM; ^€€Assumed to be a factor of 2.75 lower than in GM; ^& Assumed to simulate BBB breakdown; ^§§ Related to GM as described in Step 9; * For the 2CM, this is $\mathrm{ }{C}_{e+c}$ .

The impact of the chosen endothelial volume fraction ( $f_{EN,GM}=1.43·{10}^{-3}$  mL EN/mL brain, Supplement 3) on the fitted parameters was also tested. The results in Supplement 13 show minimal effect on the fitted parameters for the $\mathrm{ }{f}_{EN,GM}\mathrm{ }$ range from $\mathrm{ }1.0·{10}^{-3} \mathrm{to}3.0·{10}^{-3}\mathrm{ }$ mL/mL brain.

Figure 3(a) shows the 3CM-EN $\mathrm{ }{(R}_{K_T}=2.0\mathrm{ }$ and $R_{A/L}=2.0)\mathrm{ }$ fit to the averaged $\mathrm{ }{C}_{\mathit{tot}}\left(t\right)$ curve in GM from Shestov et al, ¹² resulting in the smallest RMSE (0.15 mM). The MM constants corresponding to this best fit were $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}=0.804\hspace{0.17em}\pm \hspace{0.17em}0.138\hspace{0.17em}$ µmol/g/min, $K_T^{\mathit{BBB}}=6.20\hspace{0.17em}\pm \hspace{0.17em}1.52\hspace{0.17em}$ mM, $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}=1.04\hspace{0.17em}\pm \hspace{0.17em}0.24\hspace{0.17em}$ µmol/g/min, $K_T^{\mathit{CMB}}=3.10\hspace{0.17em}\pm \hspace{0.17em}0.75\hspace{0.17em}$ mM and $\mathrm{ }{V}_{\mathit{max}}^{HK}=0.260\hspace{0.17em}\pm \hspace{0.17em}0.038\hspace{0.17em}$ µmol/g/min. The error ranges (also shown in Table 3) are based on all fits resulting in an RMSE $\mathrm{ }\le 0.30\hspace{0.17em}$ mM (N = 1041), indicated in purple shading. Simulated GM D-glucose concentration curves and MM constants for the individual healthy subjects from Shestov et al. ¹² are given in Supplement 14. The fitting results obtained using initial guesses set to the lower and upper bounds were consistent with those obtained using the reported initial guesses in Step 5, but the model took longer to converge when starting from the boundary values. A validation (Step 7) using simulated steady-state $\mathrm{ }{C}_{\mathit{tot}}\mathrm{ }$ (MM constants for best fit and for all fits with RMSE $\mathrm{ }\le 0.30$ ) as a function of venous plasma concentrations together with steady-state values from Shestov et al. ¹² is shown in Figure 3(b). Figure 3(c) shows another validation, comparing simulated GM and WM steady-state concentrations with experimental data from de Graaf et al.. ¹¹ Figure 3(d) shows $C_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ ratios from the experimental data of both MRS studies. ^11,12 Two data points (shown in red) were omitted because they resulted in unreasonably large $\mathrm{ }{C}_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ ratios of 17 and 14. The normoglycemic $\mathrm{ }{C}_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ range is 3.8–7.4 mL water/mL plasma for $\mathrm{ }{C}_p^{v,V_p}=5.14\hspace{0.17em}$ µmol/mL plasma based on two standard deviations (dashed lines). Simulated steady-state GM $\mathrm{ }{C}_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ ratios are also shown, confirming a good fit. The ratio $\mathrm{ }{C}_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ was 7.43 mL water/mL plasma at normoglycemia ( $C_p^{v,V_p}=5.14\hspace{0.17em}$ µmol/mL plasma) and 4.54 mL water/mL plasma at steady-state hyperglycemia ( $C_p^{v,V_p}=17.0\hspace{0.17em}$ µmol/mL plasma).

Figure 3.

Use of MRS data to refine the Michaelis-Menten parameters in the proposed model. a) Experimental D-glucose concentration in gray matter (GM) (black and gray circles, averaged over 5 subjects with error bars indicating the standard deviation across all subjects) as a function of time together with the D-glucose concentration in GM simulated using the 3CM-EN ( $R_{K_T}=2.0\mathrm{ }$ and $\mathrm{ }{R}_{A/L}=2.0$ ). The purple line represents the fit with lowest RMSE. Gray values were not included in the fitting to avoid having the RMSE being overdetermined by the flat part of the curve. The purple shading represents all simulated fits adhering to RMSE $\mathrm{ }\le \mathrm{ }$ 0.3 mM (N  = 1041). b) Validation comparing simulated best fit (purple line) and all fits with RMSE $\mathrm{ }\le \mathrm{ }$ 0.3 mM (purple shading) to experimental (black circles) steady-state D-glucose concentrations in GM, as a function of venous plasma D-glucose concentration. c) Validation comparing simulated (lines) and experimental steady-state D-glucose concentrations in GM (triangles) and WM (squares) as a function of venous plasma D-glucose concentration. d) The experimental ratio of D-glucose concentration in venous plasma to total GM D-glucose concentration is shown as a function of venous plasma D-glucose concentration (black circles and triangles), along with a linear fit (black line) and two standard deviation bounds (dashed line). The corresponding simulated ratios are shown in purple. Two data points, shown in red, were excluded as they resulted in unreasonably large $\mathrm{ }{C}_p^{v,V_p}/C_{\mathit{tot}}\mathrm{ }$ ratios. Experimental MRS data from Shestov et al. ¹² (a, b, d) provided by Dr. Gulin Oz from the University of Minnesota; Experimental MRS data from de Graaf et al. ¹¹ (c, d) provided by Dr. Robin de Graaf from Yale University.

The simulated compartmental D-glucose concentrations in GM, WM and GBM determined in Step 8 are shown in Figure 4 for all model versions. The input for the numerical fitting was the arterial plasma concentration, $\mathrm{ }{C}_p^a\left(t\right)$ , also shown in each subfigure. The corresponding time dependent rate constants for transport and metabolism are shown in Supplement 15, together with the steady-state normoglycemic rate constants (Supplementary Table S8). In Supplementary Table S9, we compare the plasma transfer constant ( $K_1\mathrm{ }$ in mL plasma/g tissue/min) with the literature values for D-glucose deduced from PET FDG experiments, showing good agreement for 3CM and both reversible and non-reversible unconstrained 2CM. Table 3 summarizes parameters for GM, WM, and GBM, including MM constants, $\mathrm{ }CM{R}_{\mathit{Glc}}$ and $\mathit{CBF}$ , and the normo- and hyperglycemic steady-state compartmental D-glucose concentrations. The MM saturation curves for these tissues are given in Supplement 16.

Figure 4.

Compartmental and whole-brain D-glucose concentrations for gray matter (left column), white matter (center column) and glioblastoma (right column) for the 3CM-EN (first row), the 3CM (second row). The 2CM-EN (third row), and the 2CM (fourth row), both without $\mathrm{ }\mathit{CM}{\mathit{R}}_{\mathit{Glc}}\mathrm{ }$ constraint, are also shown. * Notice that the concentration graphs for 2CM are equivalent for all unconstrained 2CM models, reversible and non-reversible (data not shown). Compartmental concentrations $\mathrm{ }{C_{EN}\left(t\right),\mathrm{ }C}_e\left(t\right)$ , $\mathrm{ }{C}_c\left(t\right)\mathrm{ }$ and $\mathrm{ }{C}_{G6P}\left(t\right)\mathrm{ }$ were calculated together with rate constants $\mathrm{ }{k}_i\mathrm{ }$ ( $i\mathrm{ }$  = 1–6) using the MM constants from Table 3. $\mathrm{ }{C}_p^a\left(t\right)\mathrm{ }$ was calculated using equation (9d) and the averaged ( $N=5$ ) experimental venous plasma glucose concentration ( $C_p^v\left(t\right)=C_p^{v,V_p}\left(t\right)/{\rho }_p^{V/V}$ ) data from Shestov et al.. ¹² $\mathrm{ }{C}_{\mathit{tot}}\left(t\right)\mathrm{ }$ was calculated using equation (8a). Notice that the initial baseline is not flat in the graphs. The reason is that no constant baseline start value was used, but rather the model was applied using the MM constants, thus it takes time for the system to reach equilibrium. A baseline of 45 min was used to accomplish this. The G6P concentration is small compared to the D-glucose concentrations, and more detailed (magnified) plots are shown in Supplement 16.

PET studies or MR experiments with a glucose/insulin clamp may use faster input functions. Supplement 17 shows that the model can handle this well for rectangular and transient input functions. The 3CM-EN robustness to time resolution (data sparsity) is evaluated in Supplement 18. Despite removing up to one third of the time points, all parameters remain within their uncertainty boundaries, demonstrating that the fits are robust to data sparsity and that the time resolution is sufficient. A sensitivity analysis comparing 3CM and 2CM models is presented in Supplement 19, Supplementary Figure S18, showing good sensitivity for $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}},\mathrm{ }\mathrm{and}\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ for both model types, but limited sensitivity for $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}$ and $\mathrm{ }{K}_T^{\mathit{BBB}}$ , the latter in line with reports by others using 2CMs. Supplement 19 also includes a practical identifiability evaluation using Monte Carlo-based synthetic data simulation. The 3CMs (Supplementary Figure S20) and the three unconstrained 2CMs (Supplementary Figure S21) show reasonable maximum transport rate and $V_{\mathit{max}}^{HK}\mathrm{ }$ distributions at noise levels of 1% and 5% (in both plasma input and tissue response functions), except for the unconstrained reversible 2CM giving $V_{\mathit{max}}^{HK}\mathrm{ }$ equal to its upper limit. However, increasing the upper limit of $\mathrm{ }{V}_{\mathit{max}}^{HK}$ does not affect the best fit value for the reversible 2CM, indicating that this model is not optimal for fitting of $\mathrm{ }{V}_{\mathit{max}}^{HK}$ , most likely due to the unrealistically high D-glucose concentration of the mixed cell and EES compartments. This effect disappears for the unconstrained non-reversible 2CM. Identifiability for $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ was acceptable in all models, but became more limited in 3CM at higher noise levels, while this remained good for 2CMs. However, for the latter, the parameter value changed by a factor of two between reversible and non-reversible transport. Supplementary table S10 shows that the ground truth for the tested noise levels for these Monte Carlo simulations always fell within one standard deviation of the determined parameters, supporting robustness of all models.

Model comparison using the Akaike Information Criterion (AIC) is described in Supplement 20. The reversible 2CM and 2CM-EN have the lowest AICs, while the AIC for the non-reversible 2CM is next. However, the AICs for all models are of comparable magnitude (−84.5 to −91.6), with only a small delta-AIC, indicating that none of the models are strongly favored.

Discussion

A 4-tissue-compartment model (3CM-EN, consisting of blood, endothelium, EES and cell) was developed to describe sugar utilization kinetics in brain tissue and tumor and compared to two 3-tissue-compartment models (3CM and 2CM-EN) and the conventional 2-tissue-compartment model (2CM). The need for models with at least three compartments (blood, EES, and cell) was prompted by recent interest in DGE MRI for the study of brain tumors, highlighting several issues that cannot be accounted for with the classic 2CM used for PET and MRS. The first issue is the dependence of DGE MRI signal intensity on the exchange rate of sugar hydroxyl protons, which is pH dependent. For malignant tumors, the pH in EES is lower ^98,99 than in the cell and the vascular compartments. This improves detectability of the OH protons, and recent studies suggest that the majority of the DGE MRI signal originates from the EES. ¹⁴ The proposed model facilitates estimation of EES and cellular glucose concentrations in tumor, allowing one to account separately for DGE MRI signals in these compartments by using different exchange rates in different pH environments. Bloch-McConnell equation-based simulations of DGE MRI signal based on compartmental concentrations using this model will be explored in future work. The second issue is the existence of glucose concentration gradients between plasma and the EES, as well as between the EES and the cell, that need to be accounted for when compartmental signal properties differ. In the 2CM, glucose concentrations in EES and cell are averaged under the reasonable assumption that the phosphorylation of D-glucose is the rate determining step. However, in tumors there is a need to describe (partial) breakdown of the BBB, which is not possible when there is only a single extravascular compartment.

Using a normoglycemic $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ range constraint based on CBF and GEF data from multiple literature sources (Supplements 4,6,7), MM parameters for D-glucose transport and metabolism were derived (Table 3 and Supplements 10–12) for 3CM-EN by fitting dynamic GM D-glucose uptake data from MRS literature and using plausible tissue compartment fractions and water densities based on literature (Tables 2,3 and Supplements 3,13). Models with more compartments require fitting of more parameters and to avoid random results based on local minima, additional assumptions are needed. Following reports that transport for GLUT3 in neuronal cells is faster than for GLUT1, ^40,42 $T_{\mathit{max}}^{\mathit{CMB}}$ , reflecting an average for transport by GLUT1 into astrocytes and GLUT3 into neurons, is expected to be larger than $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}$ . For 3CM-EN and 2CM-EN, the dependence of these parameters on the abluminal/luminal GLUT1 polarity ratio $\mathrm{ }{R}_{A/L}\mathrm{ }$ was investigated and shown not to affect $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}.\mathrm{ }$ On the other hand, $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ strongly decreased with increasing $R_{A/L}\mathrm{ }$ for $\mathrm{ }{R}_{A/L}<2.0\mathrm{ }$ (Figure 2(b), Supplements 10,11), but converged to the limit without EN (i.e. 3CM) for values above that. Previous studies reported a higher affinity (lower $\mathrm{ }{K}_T$ ) for GLUT3 than GLUT1, ^40,42 so the affinity ratio for the BBB and CMB was varied through the ratio $R_{K_T}=K_T^{\mathit{BBB}}/K_T^{\mathit{CMB}}\mathrm{ }$ (Figure 2(c), Supplements 10,11). Interestingly, Figure 2(c) show that when increasing the affinity of CMB relative to BBB, their maximum transport rates converge. $\mathrm{ }{R}_{K_T}=2.0\mathrm{ }$ was chosen based on the reported higher affinity of the CMB.

The GM data for all models ( $R_{A/L}=2.0\mathrm{ }$ and $R_{K_T}=2.0$ ) are compared in Figures 2(a) and (d), Table 3 and Supplement 12. The first conclusion is that the 2CM models do not fit well for the imposed $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ range. When removing this metabolic constraint, the resulting $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ increases by 40% and 80% for 2CM-EN and 2CM, respectively, resulting in comparable increases in $\mathrm{ }CM{R}_{\mathit{Glc}}$ . Thus, using unconstrained reversible 2CM models results in a higher metabolic rate, which can be explained by the mixed EES and cellular compartment with a higher concentration (almost double) than an intracellular compartment. This is reflected in the 2CM literature, where the maximum phosphorylation rate for D-glucose in humans is reported to be around $\mathrm{ }{V}_{\mathit{max}}^{HK}\approx 0.5\hspace{0.17em}$ µmol/g/min, ^9,12,13 and the reported range for $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ is around 0.30–0.45 µmol/g/min, ^{11 –13} the same order of magnitude as our unconstrained reversible 2CM. However, a 2CM model with non-reversible transport ( $k_1\ne k_2$ ), compensates for this and a $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ similar to the 3CM cases is found. Our estimates $\mathrm{ }\mathrm{of}\mathrm{ }{V}_{\mathit{max}}^{HK}=0.260\pm 0.038\hspace{0.17em}$ µmol/g/min and $\mathrm{ }CM{R}_{\mathit{Glc}}=0.209\pm 0.031\hspace{0.17em}$ µmol/g/min are a consequence of the range determined from normoglycemic $\mathrm{ }\mathit{CBF}\mathrm{ }$ and GEF from a large number of experimental studies (Supplements 4,6,7). Using a higher $\mathit{CBF}$ increases the upper limit of the ${\mathit{CMR}}_{\mathit{Glc}}$ and $V_{\mathit{max}}^{HK}$ . The 2CM without constraints gives normoglycemic $\mathrm{ }{C}_p^{v{,V}_p}/C_{\mathit{tot}}\mathrm{ }$ ratios in line with 3CM and 3CM-EN (Table 3) and comparable $T_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ and $\mathrm{ }{C}_{\mathit{tot}}$ . Therefore, only the results for the unconstrained two-compartment models are compared below.

Interestingly, when comparing all models for the BBB GLUT1 polarity and BBB/CMB affinity ratios chosen, $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ for GM varies only from 0.65 to 0.80 µmol/g/min, all within fitting error limits. Inclusion of the endothelial compartment affects the BBB parameters by increasing the affinity and maximum transport rate (Table 3). The CMB affinity is similarly affected, but this is due only to the imposed $\mathrm{ }{R}_{K_T}$ . However, these changes are relatively small with respect to the parameter error margins and do not affect the normoglycemic EES and cellular concentrations by more than 10–30% and $\mathrm{ }{C}_{\mathit{tot}}\mathrm{ }$ not at all (Table 3). This suggests that eliminating the endothelial compartment may be a reasonable approach for 3CM. On the other hand, knowledge about abluminal/luminal GLUT1 polarity for the BBB endothelium has increased in recent decades and different diseases may have different polarity ratios. Use of reversible two-compartment models without metabolic constraint leads to lower transport affinity and a $\mathrm{ }CM{R}_{\mathit{Glc}}\mathrm{ }$ above the normal range established in Supplements 4,6 and 7. This is not the case for the unconstrained non-reversible 2CM, which gives transport and metabolic parameters comparable to the two reversible 3CMs, suggesting that reversible transport is a reasonable assumption for 3CM.

For additional validation of the 3CM-EN, results were compared with the concentration-dependent GM steady-state data of Shestov et al. (Figure 3(b)) and de Graaf et al. (Figure 3(c), purple curve) and reported steady-state ratios of $\mathrm{ }{C}_p^{v{,V}_p}/C_{\mathit{tot}}\mathrm{ }$ as a function of $\mathrm{ }{C}_p^{v{,V}_p}\mathrm{ }$ (Figure 3(d)), showing excellent agreement. While the simulated ratio at normoglycemia seems high, this is realistic due to higher plasma values for Shestov et al. ¹² compared to de Graaf et al., ¹¹ and consistent with our use of Shestov’s experimental data to simulate the purple curve. Further validation of the model comes from the simulated dynamic glucose concentration in WM (Figure 3(c)), which is comparable to the results by de Graaf et al., who reported steady-state D-glucose levels in GM and WM of about 0.8 and 1.0 mM at normoglycemia and 3.5 and 5.0 mM at hyperglycemia, respectively ¹¹ (Figure 3(c)). The fitting consistency was evaluated by repeating the optimization process with initial guesses set to the lower and upper bounds. In all cases, the optimization converged to the same set of MM constants, confirming the robustness of the parameter estimation process and indicating that the results were not sensitive to the choice of initial guesses within the defined bounds.

The sensitivity analysis (Supplementary Figure S18, Supplement 19) indicates that all models have limited sensitivity to $\mathrm{ }{K}_T^{\mathit{BBB}}$ , consistent with previous reports of limited $\mathrm{ }{K}_T\mathrm{ }$ identifiability, ^12,34 and that the 2CM has the best sensitivity to the remaining parameters. For the 3CM and 3CM-EN, $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}\mathrm{ }$ also showed low sensitivity, but the subsequent practical identifiability analysis in Supplement 19 (Fig. S20 and Table S10) demonstrated that $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}\mathrm{ }$ can be estimated reliably. However, using the physiologically reasonable assumption that $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ is on the order of the normoglycemic plasma concentration, and considering that $\mathrm{ }{C}_{\mathit{tot}}\left(t\right)\mathrm{ }$ is relatively insensitive to variations in $\mathrm{ }{K}_T^{\mathit{BBB}}$ , the parameter range for $\mathrm{ }{K}_T^{\mathit{BBB}}\mathrm{ }$ can be restricted, or the value can be fixed, and the remaining parameters can be estimated reliably even for noisier data sets. While all models had similar RMSEs, the $\mathrm{ }\Delta AI{C}_c\mathrm{ }$ favored the unconstrained 2CM (lowest $\mathrm{ }\Delta AI{C}_c$ ) and the 3CM had the highest. However, all models had $\mathrm{ }\Delta AI{C}_c$ <10 indicating acceptable support (Supplement 20). The motivation for advocating the more complex model, the 3CM-EN, is that it provides separate estimates of extracellular-extravascular and intracellular glucose concentrations, which is essential for DGE MRI in tumors where compartment-specific signal detection differs due to pH.

Finally, EES concentrations at normoglycemia (6.15 mM) of 1.1–1.3 mM were found, consistent in magnitude with Silver et al., ¹⁰⁰ who, at a blood concentration of 7.6 ± 0.3 mM, measured $C_e=2.4\hspace{0.17em}\pm 0.1\hspace{0.17em}$ mM in rats using microelectrodes. At steady-state hyperglycemia (18.8 mM), $\mathrm{ }{C}_e\mathrm{ }$ was 5.0 mM in our current study, comparable to Silver et al.’s $\mathrm{ }{C}_e\mathrm{ }$ of 4.5 ± 0.4 mM at 15.2 ± 2.4 mM blood glucose.

The MM parameters found by all models are comparable in order of magnitude to the rather broad range of literature values established with 2-compartment models. For transport, Gjedde performed an in-depth review of the extensive PET literature on human and rat GM data, ¹⁰¹ concluding that $\mathrm{ }{K}_T^{\mathit{BBB}}=4.0\hspace{0.17em}$ mM and $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}=1.0\hspace{0.17em}$ µmol/g/min in humans. The MRS literature reports a large range of values. When using a unidirectional uptake model in humans, Gruetter et al. deduced $\mathrm{ }{K}_T=6.2\hspace{0.17em}$ mM, $\mathrm{ }{T}_{\mathit{max}}=1.2\hspace{0.17em}$ µmol/g/min from ¹³C MR studies in GM ⁹ and $K_T=4.8\hspace{0.17em}$ mM, $T_{\mathit{max}}=0.80\hspace{0.17em}$ µmol/g/min from ¹H MR studies in GM. ¹³ In a re-assessment with reversible transport, ⁴⁶ this was adjusted to $\mathrm{ }{K}_T=0.6\hspace{0.17em}$ mM, $\mathrm{ }{T}_{\mathit{max}}=0.69\hspace{0.17em}$ µmol/g/min. De Graaf et al. reported $\mathrm{ }{K}_T=1.1\hspace{0.17em}$ mM and $\mathrm{ }{T}_{\mathit{max}}=0.65\hspace{0.17em}$ µmol/g/min for the reversible model and $\mathrm{ }{K}_T=6.2\hspace{0.17em}$ mM and $\mathrm{ }{T}_{\mathit{max}}=1.2\hspace{0.17em}$ µmol/g/min for the non-reversible model. ¹¹ Shestov et al., also using a reversible model and ¹H MRS in GM, found a large range of possible $\mathrm{ }{K}_T$ – $T_{\mathit{max}}\mathrm{ }$ combinations. ¹² When using $\mathrm{ }{K}_T=2.1\hspace{0.17em}$ mM, their corresponding $\mathrm{ }{T}_{\mathit{max}}\mathrm{ }$ was 0.97 µmol/g/min. However, it is not obvious that reversible transport applies to the previously used 2CMs, because the number of GLUT1 transporters on the abluminal and luminal sides of the BBB has been reported to differ. ^{39 –44} An endothelial compartment (3CM-EN and 2CM-EN) was therefore added to make the assumption of reversible transport more plausible. The fitted MM constants for the non-reversible 2CM were similar to those for the 3CM. This suggests further support for the reversible assumption when using three compartments, but that for the 2CM used for tracer concentrations in PET, non-reversible kinetics ( $k_1\ne k_2$ ) is required to account for the changed concentrations when mixing intra- and extracellular compartments and the tracer concentration differs between blood and extravascular space.

Another possible factor for some differences between this study and the literature is related to the use of different units throughout the literature (e.g., per mL plasma, blood, or water for blood glucose). Hct was also accounted for, which varies with vessel size and gender and may also vary between studies. However, the magnitude of Hct does not substantially affect the MM constants and concentrations (Supplement 9).

The results in Figure 4 and Table 3 illustrate similarities and differences in glucose dynamics between models and compartments for GM, WM and GBM. The 3CM-EN and the 3CM yield comparable compartmental concentrations, except in GBM, where the 3CM gives a 50% lower intracellular concentration (Figure 4, Table 3). This is caused by the combined effects of a lower $\mathrm{ }{T}_{\mathit{max}}^{\mathit{CMB}}\mathrm{ }$ in the 3CM (based on the lower $\mathrm{ }{T}_{\mathit{max}}^{\mathit{BBB}}\mathrm{ }$ in GM for 3CM), and high $\mathrm{ }{K}_T$ , reducing glucose influx into the cell. Moreover, as $\mathrm{ }{V}_{\mathit{max}}^{HK}\mathrm{ }$ is slightly higher for 3CM while $\mathrm{ }{K}_M^{HK}\mathrm{ }$ remains the same, the lower $\mathrm{ }{C}_c\mathrm{ }$ in the 3CM results in a larger $\mathrm{ }{k}_5$ (Supplementary Figure S11). Models with an EN compartment produce $\mathrm{ }{C}_{\mathit{tot}}\mathrm{ }$ comparable to models without EN, but removing the EN compartment leads to a slight increase in $\mathrm{ }{C}_e\mathrm{ }$ or $\mathrm{ }{C}_{e+c}$ , which is expected when a transport barrier is removed. Interestingly, the 2CM-EN and 2CM result in 50% higher $\mathrm{ }{C}_{\mathit{tot}}\mathrm{ }$ in GBM than the 3CM-EN and 3CM, potentially overestimating the tumor glucose concentration due to the assumption of a single mixed extravascular compartment.

The expanded model still has limitations, particularly regarding model complexity. Introducing additional parameters could make it difficult to reliably estimate all kinetic parameters from the available data. However, by applying constraints based on physiological knowledge as discussed above, good fitting stability was achieved. Tissue volume fractions, densities and water contents (Tables 2 and 3) were assumed based on well-established literature values, and yielded results that compare well with steady-state data for both GM and WM (Figure 3(c)). Contrary to Shestov et al., ¹² CSF partial volume effects in GM were not accounted for. Finally, the glucose transport in tumors differs from that in healthy brain, and the expression of different transporters may change with malignancy status. ¹⁰² Future knowledge about the different MM properties of different transporters and their distribution in tumors would allow modification of the model to account for that. Currently, any change in transport will be seen as a change in MM constants and thus still provide information on being clinically normal or abnormal. The fitted MM parameters reflect tissue-specific kinetics determined by the number of transporters and available metabolic enzymes, which will not change with the shape of the input function (Supplement 17). An important fact to realize is that while PET employs tracer concentrations, the MM equations for tissue still contain mM concentrations because the receptors experience the actual tissue D-glucose concentrations (in mM). However, as the EES and tissue volumes are substantially larger than the blood volume, the chance of return of the tracer to blood is small. Thus, the MM constants determined from the rate constants from the PET signal intensity changes may not reflect actual MM kinetics of the tissue, which is reflected in the need for a lumped constant.

In conclusion, a 4-tissue-compartment model (blood, endothelium, EES and cell) for describing D-glucose delivery, transport, and metabolism in brain tissue and brain tumors was derived. Different glucose concentration units (plasma-based, water-based, and tissue-based) were accounted for in the equations. Using Michaelis-Menten constants for transport and phosphorylation, derived from fitting the model to dynamic MRS data from the literature, the model reasonably reproduced literature steady-state brain D-glucose concentrations. The model should be adaptable to different sugars such as 2-Deoxy-D-glucose (2-DG), 3-O-methyl-D-glucose (3-OMG), and L-glucose by adjusting the transport steps to reflect their specific kinetics. A compartment theory for at least 3 tissues (blood, EES, cell) is needed for DGE MRI of tumors and may be more applicable than a 2-tissue-compartment model for all metabolic substrate uptake imaging methods in situations of BBB breakdown, where cellular and blood concentrations of infused agents are unlikely to equilibrate. The 3CM-EN and 3CM are expected to be applicable also to isotope labeled PET and MRS studies of tumors. For instance, in FDG-PET, the total radioactive label will increase rapidly in the EES of tumor regions with a broken BBB followed by transport into the cell and slow conversion into phosphorylated glucose. This early component in the single convolved uptake curve or signal measured at a particular time point is determined by the extent of BBB breakdown and may complicate interpretation of the metabolic rate based on hexokinase activity. In MRS, the product (lactate) has a different signal, but the cellular concentration of D-glucose depends on the amount of BBB breakdown. Application of 3CM-EN and 3CM to radioactively labeled sugars in PET, magnetically labeled sugars in MR, and exchange-based contrast in DGE MRI will be the topic of future studies.

Supplemental Material

sj-pdf-1-jcb-10.1177_0271678X251366074 - Supplemental material for Cerebral glucose delivery, transport and metabolism: Theory and modeling using four, three, and two tissue compartments

Supplemental material, sj-pdf-1-jcb-10.1177_0271678X251366074 for Cerebral glucose delivery, transport and metabolism: Theory and modeling using four, three, and two tissue compartments by Anina Seidemo, Linda Knutsson, Nirbhay N Yadav, Pia C Sundgren, Ronnie Wirestam and Peter CM van Zijl in Journal of Cerebral Blood Flow & Metabolism