Quantitative Theory for the Transverse Relaxation Time of Blood Water

Abstract

An integrative model is proposed to describe the dependence of the transverse relaxation rate of blood water protons (R_2blood = 1/T_2blood) on hematocrit fraction (Hct) and oxygenation fraction (Y). This unified model takes into account (i) the diamagnetic effects of albumin, hemoglobin, and the cell membrane; (ii) the paramagnetic effect of hemoglobin; (iii) the effect of compartmental exchange between plasma and erythrocytes under both fast and slow exchange conditions that vary depending on field strength and compartmental relaxation rates; (iv) the effect of diffusion through field gradients near the erythrocyte membrane. To validate the model, whole blood and lysed blood R₂ data acquired previously using Car-Purcell-Meiboom-Gill (CPMG) measurements as a function of inter-echo spacing τ_CP at magnetic fields of 3.0T, 7.0T, 9.4T and 11.7T were fitted to determine the life times (field independent physiological constants) for water diffusion and exchange, as well as several physical constants, some of which are field independent (magnetic susceptibilities) and some are field dependent (relaxation rates for water protons in solutions of albumin and oxygenated and deoxygenated hemoglobin, i.e. blood plasma and erythrocytes, respectively). This combined exchange-diffusion model allowed excellent fitting of the curve of the τ_CP dependent relaxation rate dispersion at all four fields using a single average erythrocyte water life time, τ_ery = 9.1±1.4 ms and an averaged diffusional correlation time, τ_D = 3.15±0.43 ms. Using this model and the determined physiological time constants and relaxation parameters, blood T₂ values published by multiple groups based on measurements at magnetic field strengths of 1.5T and higher could be predicted correctly within error. Establishment of this theory is a fundamental step for quantitative modeling of the BOLD effect underlying functional MRI.

Introduction:

The transverse relaxation rate of blood water protons (R_2blood) is sensitive to the physiological conditions of the blood, especially the oxygenation fraction (Y) and the hematocrit fraction (Hct). In vivo, the capillary and venous oxygenation are determined by the perfusion and metabolic functioning of the tissue. For instance, an increase in local cerebral blood flow (CBF) during brain activation causes a reduction in the oxygen extraction fraction (OEF)¹. This in turn reduces the arteriovenous oxygenation fraction difference (Y_a-Y_v) and R_2blood in the capillaries and veins, a phenomenon known as the BOLD effect^2-6. Quantification of whole-brain or local OEF has been done by measuring R_2blood in draining venous compartments and comparing it with available calibration curves^7-19 that relate R_2blood to Hct and Y. Such data can be combined with a measured CBF to determine the cerebral metabolic rate of oxygen^20-25. The development of an exact theory for R_2blood is needed to provide a foundation for physiological theories for the BOLD effect^5,6,26-31 and for BOLD-based applications for determining physiological parameters. However, while many models have been proposed for the description of R_2blood^{7,8,10,12,15,17,32-39}, an inclusive and conclusive theory that allows its prediction as a function of Hct and Y at multiple magnetic field strengths B₀ has remained elusive. This is not surprising as, contrary to R_1blood ^40-49, the rate R_2blood depends not only on the concentration of blood proteins and their diamagnetic and paramagnetic relaxivities, but also on the correlation times describing the exchange of water protons between compartments with different magnetic susceptibility and their diffusion through magnetic field gradients within or around such compartments. The sensitivity of R_2blood to these correlation times becomes apparent when varying the inter-echo spacing (τ_cp) in a multi-echo Carr Purcell Meiboom Gill (CPMG) experiment, or even just the echo time (TE) in a single echo experiment, and this R_2blood dispersion effect has a strong field dependence. The goal of this work is to establish a universal theory that can be used to predict R_2blood as a function of physiological conditions (Y, Hct), MRI parameters (TE, τ_cp) and field strength B₀. This theory is evaluated using a recently published large experimental R_2blood data set.⁵⁰ Development of such a theory for blood is important, because even gradient echo based blood transverse relaxation changes contribute substantially to the total BOLD effect, especially at lower field. For instance, during visual activation, extravascular BOLD signal change fractions are approximately 45±13%, 70±11% and 91±11% at 1.5, 3.0 and 7.0 T, respectively.^51,52

Previous studies^{5,7,8,12,16,17,32,33,36-39,48,53-61} have already provided much insight into the relationship between R_2blood, Hct, Y, and τ_cp. The rate R_2blood is usually described as the sum of a τ_cp-independent term R_20,blood (the intercept of the R_2blood-τ_cp dispersion curve at infinitely short τ_cp) and a τ_cp-dependent relaxation enhancement due to either chemical exchange, as previously described by the Luz and Meiboom (LM)⁶² or the more general Allerhand and Gutowsky (AG) models⁶³, or diffusion though local field gradients models, as proposed by Jensen and Chandra (JC model)³² or by Ziener et al. (Ziener model)³³. Previously, when fitting R_2blood dispersion data versus the rate 1/τ_cp, the intercept R_20,blood has been considered either as a free fitting parameter without any quantitative relationship with Hct and Y, or just assumed to be the R_2blood value at the shortest available experimental τ_cp^{7,8,12,15,17,36-39}. However, both of these approaches could induce errors due to residual diffusion contributions, especially under deoxygenated conditions at high magnetic fields (Fig. 1), and make it difficult to predict R_2blood from the blood physiological parameters. Here, we will instead start from basic principles for plasma and erythrocyte water proton relaxation rates based on albumin and hemoglobin protein solutions, respectively. This is possible because of the recent availability of relaxation data for plasma and lysed blood,⁵⁰ in which the cell sequestration of hemoglobin is broken and the membranes removed. This lysed blood was mixed with plasma to perform hemoglobin concentration dependent studies, which also allowed determination of the “intracellular” water relaxation rate R_2ery. Studying R_2ery as a function of Y allowed estimation of the diamagnetic and paramagnetic water relaxivities for this hemoglobin solution.

Figure 1.

Comparison between the R_2blood contributions at 11.7 T from diffusion (Jensen-Chandra model) and exchange (Luz-Meiboom model) as a function of τ_cp for a Hct of 0.4 and a chemical shift difference between intracellular and extracellular water of 0.1 ppm. It can be seen that the estimated diffusion contribution is substantial even at τ_cp values below a few ms, which is the approximate length of the 180° pulses on typical human scanners.

When fitting the R_2blood dispersion as a function of echo spacing τ_cp, previous studies have been limited mainly to single-mechanism descriptions based on either water exchange between compartments^{7,8,10,12,15,17,36-39,57,58} or diffusion through field gradients within compartments^32,33,64-68, sometimes including arguments why one would be better than the other. However, it is important to realize that both mechanisms affect the blood transverse relaxation enhancement, each with a characteristic τ_cp-dependent dispersion^32,36 that depends on the correlation times representative for these distinct processes (Fig. 1). Evidence of problems with using one mechanism comes from the fact that while both approaches are generally able to provide excellent curve fits at each field strength, it is difficult to get consistent erythrocyte life times (τ_ery) and diffusional correlation times (τ_D) between fields¹⁷. For instance, a gradual decrease in τ_ery is found when going to higher fields using the fast exchange models and in τ_D using the diffusion models, while correlation times should in principle not be field dependent parameters. We propose here that a further improvement can be made by combining exchange and diffusion models to characterize the blood transverse relaxation rate and finding appropriate starting values for model fitting by separating out data from experimental conditions where either exchange or diffusion contributions dominate. For instance, the effect of diffusion becomes apparent already at very short τ_cp (< 2ms) under conditions of large field gradients around the cell membrane, such as for venous oxygenations at high magnetic fields, while, due to the relatively long lifetime of water in the erythrocyte (τ_ery ~ 10 ms), compartmental exchange contributions are still minimal under such conditions (Fig. 1). A further improvement that we will implement is the use of the general exchange model^69,70 that not only covers all exchange regimes (fast, slow and intermediate), but can also be adjusted to include compartmental relaxation rates for the erythrocyte and plasma water (R_2ery and R_2plas, respectively). This is crucial, because the exchange regime changes with field strength and oxygenation fraction. Finally, with regards to diffusion, water protons in the plasma and erythrocyte may experience different shape effects of the erythrocyte membrane. If so, they will contribute in different proportions, the magnitude of which we will assess. The resulting analytical expression thus combines the intrinsic relaxation of protein solutions with the relaxation enhancements due to exchange between and diffusion around compartments with different magnetic susceptibilities. This expression, together with the field-dependent inherent relaxivities determined from our fits allows one to predict R_2blood at different Hct, Y, TE and τ_cp as a function of magnetic field strength.

Theory

Blood is composed predominantly of plasma and erythrocytes. The plasma consists mainly of buffered water and albumin, while the intracellular volume of the erythrocyte consists mainly of buffered water and a high concentration of hemoglobin. As illustrated in Fig. 2, R_2blood is affected by these protein-water interactions both at the molecular level and the cell level⁷¹. At the molecular level, a protein molecule (albumin or hemoglobin) enhances the overall relaxation of water protons because of fast chemical exchange between exchangeable protons in proteins and water protons in bulk water⁷². At the cell level, a magnetic susceptibility difference is induced by the diamagnetic hemoglobin and paramagnetic deoxyhemoglobin molecules being confined to the erythrocyte with its bi-concaved shape. This will generate a B₀ field gradient and a chemical shift difference between water protons in the erythrocyte and in plasma. When water molecules diffuse in the field gradient or exchange across the erythrocyte membrane, the field gradient/chemical shift difference will result in a loss of phase coherence for the transverse proton magnetization. This R_2blood enhancement can be mediated by the interplay between τ_cp and the correlation time of exchange (τ_ex) or diffusion (τ_D). A CPMG sequence with a τ_cp that is short compared to τ_ex or τ_D can promptly refocus the local chemical shift evolution of water protons before the molecule moves to a position with different spatial frequency, and thus minimizes the relaxation enhancement. When increasing τ_cp, more and more water protons change their positions either between the two compartments or over a larger range of field variance due to the presence of a gradient, and experience a larger relaxation enhancement. When τ_cp is much longer than τ_ex and τ_D, the water molecules have experienced most possible compartmental positions and thus the full range of field variance. At this point the relaxation will not enhance anymore when further increasing the τ_cp. However, this situation of full mixing (theoretically analogous to fast exchange) is generally not reached for the case of blood over the typical range of echo times TE used for in vivo experiments. Therefore, in addition to blood parameters such as the hemoglobin concentration in the erythrocyte (c_Hb), albumin concentration in plasma (c_Alb), the volume ratio taken by the erythrocyte (Hct), and the oxygenation fraction (Y), the experimentally measured R_2blood depends on the chosen τ_cp. If τ_cp is not kept constant while the echo time TE is varied, such as in a TE-dependent single spin-echo relaxation measurement, R_2,blood even varies with TE.

Figure 2.

Two-compartment model used to describe the transverse relaxation rate of blood water protons, R_2blood. The magnetic field line (gray line) simulated based on the biconcave shape of erythrocyte shows the smaller gradient inside the erythrocyte.

To characterize R_2blood, we propose the model shown in Fig. 2. The water in the blood is divided into two pools: (i) the erythrocyte water proton pool with relaxation rate R_2ery, having contributions from the hemoglobin solution relaxation rate R_2Hb and a diffusion relaxation enhancement R_2D,ery (ii) the plasma pool with the rate R_2plas with contributions from the albumin solution relaxation rate R_2Alb and a diffusion relaxation enhancement R_2D,plas. Then the whole-blood relaxation rate R_2blood is calculated by including the water exchange effect across the erythrocyte membrane described using a previously established general exchange model^69,70, erythrocyte shape factors (β), and a cell membrane component accounting for relaxation enhancement due to the presence of the erythrocyte membrane.

Transverse Relaxation in Hemoglobin and Albumin Solutions

As shown in a previous study⁷², the transverse relaxation of water in a protein solution measured by CPMG experiments can be well approximated in terms of chemical exchange between free water protons and exchangeable protein protons. To quantitatively describe the transverse relaxation consequences of this chemical exchange process, the general exchange model developed by Carver and Richards⁶⁹ and modified by Davis et al.⁷⁰ (Appendix 1) has to be used. The reason is that the transverse relaxation rate of exchangeable protein protons without exchange (R_2b,prot) is much faster than that of bulk water protons in buffered solution (R_2buf) and the regular expressions of Luz and Meiboom⁶² or Allerhand and Gutowsky⁶³ are not applicable⁷³ (See Appendix 2 for requirements). Therefore, the transverse relaxation rates of the water protons in hemoglobin solution and albumin solution (plasma) are expressed as⁶⁹:

R_2prot =

$$R_{2prot}=\frac{1}{2}\left\{R_{2b,prot}+R_{2buf}+k_{prot}+k_{buf}-\frac{1}{{\tau }_{cp}}cos{h}^{-1}[D_{prot}^{+}\cosh (2{\xi }_{prot})-D_{prot}^{-}\cos (2{\phi }_{prot})]\right\}$$ [1a]

$$D_{prot}^{\pm }=\frac{1}{2}[\pm 1+({\psi }_{prot}+2\Delta {\omega }_{prot}^2)∕({\psi }_{prot}^2+{\zeta }_{prot}^2{)}^{1∕2}]$$ [1b]

$$k_{buf}=k_{prot}\times P_{prot}∕(1-P_{prot})$$ [1c]

$${\xi }_{prot}=\left(\frac{{\tau }_{cp}}{\sqrt{8}}\right){\left[+{\psi }_{prot}+({\psi }_{prot}^2+{\zeta }_{prot}^2{)}^{1∕2}\right]}^{1∕2}$$ [1d]

$${\phi }_{prot}=(\frac{{\tau }_{cp}}{\sqrt{8}}){\left[-{\psi }_{prot}+({\psi }_{prot}^2+{\zeta }_{prot}^2{)}^{1∕2}\right]}^{1∕2}$$ [1e]

$${\psi }_{prot}=(R_{2b,prot}-R_{2buf}+k_{prot}-k_{buf}{)}^2-\Delta {\omega }_{prot}{}^2+4k_{prot}{k}_{buf}$$ [1f]

$${\zeta }_{prot}=2\Delta {\omega }_{prot}(R_{2b,prot}-R_{2buf}+k_{prot}-k_{buf})$$ [1g]

in which k_prot is the dissociation rate for protein-bound exchangeable protons and k_buf is the association rate onto the protein of protons from the free solution pool. While R_2buf and k_buf are orders of magnitude smaller than R_2b,prot and k_prot, respectively, and could in principle be neglected, we include them as R_2buf is available from direct measurements and k_buf can be calculated (Eq. 1c). In this paper we use cgs units for chemical shifts and magnetic susceptibilities. The chemical shift difference (rad/s) between exchangeable protein protons and solution water protons, Δω_prot, can be calculated for each field strength using the averaged chemical shift difference (Δδ_prot, ppm) between the protein protons and the water protons:

$$\Delta {\omega }_{prot}=\gamma B_0\times \Delta {\delta }_{prot}$$ [2]

P_prot is the proton fraction of the number of exchangeable protein protons ($N_{prot}^{ex}$) relative to the number of bulk water protons ($N_{buf}^{ex}$) and can be estimated as

$$P_{prot}=\frac{N_{prot}^{ex}}{N_{buf}^{ex}+N_{prot}^{ex}}=\frac{n_{prot}^{ex}\times c_{prot}}{2\times (55.6mol∕L\times f_{V,water})}$$ [3]

where $n_{prot}^{ex}$ is the number of exchangeable protein protons per molecule and c_prot the molar concentration (mol/L) of protein. The bulk water volume fraction, f_V,water of a protein solution depends on the protein concentration and is [1 - 0.3/(332g/L) × c_Hb(g/L)] for Hb inside erythrocyte and [1 - 0.05/(50g/L) × c_Alb(g/L)] for Alb in plasma. For normal blood, where c_Hb is 332g/L inside the erythrocyte and c_Alb is 50g/L in the plasma, f_V,water is a constant of 0.7⁷⁴ and 0.95⁷⁵ for Hb and Alb respectively. However, for the hemoglobin solutions of different concentration, the number has to be calculated appropriately. The water concentration in solution is 55.6 M and, because there are two protons for each water molecule, a pre-factor 2 is used to calculate the bulk water proton number. The term $N_{prot}^{ex}$ in the denominator of Eq. [3] was neglected when deriving the final expression on the right hand side.

The hemoglobin solution is a mixture of deoxygenated and oxygenated hemoglobin. Assuming minimal conformational changes from the oxygenated to the deoxygenated state, we use the same proton dissociation rate (k_prot) and the same number of exchangeable protons ($n_{prot}^{ex}$). However, the exchangeable protons in the deoxygenated and oxygenated hemoglobins have different chemical shifts (Δδ_prot) and relaxation rates (R_2b,prot) because the iron ion in oxygenated hemoglobin is diamagnetic, while it is paramagnetic in deoxygenated hemoglobin. Therefore, the overall relaxation rate (R_2Hb) of water protons in the hemoglobin solution is expressed as^17,36:

$$R_{2Hb}=Y\times R_{2OxyHb}+(1-Y)\times R_{2DeoxyHb}$$ [4]

where the rates of the oxygenated (R_2OxyHb) and deoxygenated (R_2DeoxyHb) hemoglobin contributions can be calculated based on Eqs. 1-3 using their particular chemical shift differences Δδ_OxyHb and Δδ_DeoxyHb for their exchangeable protons.

Membrane Relaxation Enhancement

The lipid bilayers of the erythrocyte may also enhance the transverse relaxation of blood water^76,77 as for instance demonstrated by the short T₂ of myelin water⁷⁸. Following our previous paper⁵⁰, we assume the transverse relaxation enhancement of the membrane (R_2mem) to be the same inside and outside the cell. However, this enters in different proportions in the equations for the plasma and erythrocyte contributions, because the intracellular contribution is always the same (concentration of exchangeable protons in the membrane inside the erythrocyte does not change), while the extracellular concentration of exchangeable membrane protons increases proportional to the Hct. In addition, the total effect for blood water has to include the appropriate water fractions for the erythrocyte (f_ery) and plasma, f_plas = (1 – f_ery), which depend on the protein concentrations in these compartments. Thus:

R_2blood total enhancement from the cells:

$$R_{2\mathrm{blood}}\text{total enhancement from the cells:}{f}_{ery}{R}_{2mem}$$ [5a]

R_2blood total enhancement from plasma:

$$R_{2\mathrm{blood}}\text{total enhancement from plasma:}(1-f_{ery})Hct\cdot R_{2mem}$$ [5b]

where R_2mem is a field dependent relaxation rate. The relationship between Hct and f_ery is defined in the whole-blood relaxation section below (Eq. [17]). Notice that Eq. [5b] correctly reflects that there will be no additional membrane water relaxation contribution to whole blood from plasma, both for pure plasma (Hct = 0 and f_ery = 0) and pure cells (Hct = 1 and f_ery = 1).

Oxygenation Dependence of the Magnetic Susceptibility

The volume magnetic susceptibilities of the hemoglobin solution inside the erythrocyte and the albumin solution (plasma) are given by⁷⁹:

$${\chi }_{ery}={\chi }_{H_{2}O}+c_{Hb}\left[{\chi }_{Hb}^M+4\times (1-Y)\times {\chi }_{heme}^M-v_{Hb}^M\times {\chi }_{H_{2}O}\right]$$ [6a]

$${\chi }_{plas}={\chi }_{H_{2}O}+c_{Alb}\left[{\chi }_{Alb}^M-v_{Alb}^M\times {\chi }_{H_{2}O}\right]$$ [6b]

in which χ and χ^M indicate volume and molar magnetic susceptibilities, respectively; ${\chi }_{Hb}^M=\text{-}3.82\times {10}^{\text{-}2}$ mL/mole is the molar magnetic susceptibility of hemoglobin in its quaternary form⁸⁰ and ${\chi }_{heme}^M=1.229\times {10}^{\text{-}2}$ mL/mole is the molar paramagnetic contribution due to a single deoxygenated heme⁸¹. This value corresponds to a magnetic moment of 5.435 Bohr magneton (B.M.), which agrees well with Pauling’s measurement 5.46 B.M⁸². χ_H2O is the volume magnetic susceptibility of water (−0.719×10⁻⁶), $v_{Hb}^M$ is the molar volume of the quaternary hemoglobin (M/ρ = 64.5×10³ [g/mole]/1.335 [g/mL])⁸³, and c_Hb is the total concentration of the quaternary hemoglobin inside the erythrocyte. For albumin, which has a molecular weight of 66.5×10³ g/mol⁴², we assume the same molar susceptibility and density as for quaternary hemoglobin, while c_Alb is 0.752 mM in the plasma. Substitution of these values gives:

$${\chi }_{ery}={\chi }_{H_{2}O}+0.253\times (0.930-Y)\times {10}^{-6}$$ [6c]

$${\chi }_{plas}={\chi }_{H_{2}O}-0.00179\times {10}^{-6}$$ [6d]

$$\Delta {\chi }_{ery-plas}={\chi }_{ery}-{\chi }_{plas}=0.253\times (0.937-Y)\times {10}^{-6}$$ [6e]

In previous papers, this susceptibility difference was written as

$$\Delta {\chi }_{ery-plas}=\Delta {\chi }_{deoxy}(1-Y)$$ [7a]

A Δχ_deoxy of 0.253 ppm is in the typical range found in recent measurements^38,84, but interestingly the oxygenation-dependent term derived in Eq. [6e] is not (1-Y) but correctly reflects the fact that the magnetic susceptibility difference transitions from being negative (diamagnetic) in the arteries and arterioles to positive (paramagnetic) when oxygenation decreases. The theoretical prediction indicates this transition happens at Y = 0.937, but for a more general description we define an oxygenation value for this diamagnetic-to-paramagnetic susceptibility difference transition (Y^off):

$$\Delta {\chi }_{ery-plas}=\Delta {\chi }_{deoxy}(Y^{off}-Y)=0.253(Y^{off}-Y)ppm$$ [7b]

Meanwhile, it is important to note that the water proton chemical shift difference between the plasma and the hemoglobin solution inside the erythrocyte is not naturally equivalent to the bulk susceptibility difference between inside and outside the erythrocyte. This chemical shift difference mainly comes from two parts: 1. The average proton chemical shift of water protons will change due to the fast exchange with the exchangeable protons of proteins in solution. As discussed above in the transverse relaxation in hemoglobin and albumin solutions, this chemical shift difference contribution can be expressed as

$$\Delta {\delta }_{ery-plasma}=\left\{P_{ery}\times \left[Y\times \Delta {\delta }_{OxyHb}+(1-Y)\times \Delta {\delta }_{DeoxyHb}\right]-P_{plas}\times \Delta {\delta }_{Alb}\right\}$$ [8]

in which P_ery and P_plas are the fractions of the number of exchangeable protein protons relative to the number of bulk water protons; 2. the chemical shift difference generated by the bulk susceptibility difference between the erythrocyte and plasma⁸⁵. This is very similar to the bulk susceptibility corrections for the chemical shift measurement when an external standard solution was used in a sealed capillary that is coaxially inserted in the sample⁸⁶. Because this chemical shift difference is proportional to the bulk susceptibility difference and depends on the erythrocyte’s shape and its orientation relative to the main magnetic field, a shape factor β_Ex will be introduced. Therefore, combing Eq. 7b and 8, the average chemical shift difference experienced between water protons inside and outside the erythrocyte becomes:

$$\Delta {\omega }_{ery-plas}=\gamma B_0\times [\Delta {\delta }_{ery-plasma}+{\beta }_{Ex}\Delta {\chi }_{ery-plas}]=\gamma B_0\times \left\{P_{ery}\times \left[Y\times \Delta {\delta }_{OxyHb}+(1-Y)\Delta {\delta }_{DeoxyHb}\right]-P_{plas}\times \Delta {\delta }_{Alb}+{\beta }_{Ex}\times 0.253(Y^{off}-Y)ppm\right\}$$ [9]

in which all Δδ values are also on the order of 10⁻⁶. In the fitting of the whole blood values we will start with using 0.937 (Eq. 6e) as the starting value for Y^off.

Diffusion-based Relaxation Enhancement due to the Presence of a Cell Membrane

In whole blood, the susceptibility difference between erythrocytes and plasma generates an inhomogeneous magnetic field both inside and outside the erythrocytes. Jensen and Chandra.³² derived an analytical solution for the diffusion relaxation enhancement based on a weak field approximation:

$$R_{2D}=G_0\frac{{\gamma }^2{\tau }_D}{2}F\left(\frac{4{\tau }_{cp}}{{\tau }_D}\right)$$ [10]

where

$$F\left(\frac{4{\tau }_{cp}}{{\tau }_D}\right)=\frac{1}{\sqrt{\pi }}{\int }_{y=0}^{\infty }\frac{e^{-y}}{\sqrt{y}}\left[1-\frac{{\tau }_D}{4{\tau }_{cp}y}tanh\left(\frac{4{\tau }_{cp}}{{\tau }_D}y\right)\right]dy$$ [11]

The diffusional correlation time ${\tau }_D=r_c^2∕D$ is defined to describe the process of water proton diffusion through a gradient that they experience,^32,33,87 in which D is the translational diffusion constant of water, and r_c is the length scale of the inhomogeneities (i.e. expected to be approximately on the order of the erythrocyte size). G₀ can be considered as a square of local gradient, which is obtained by averaging the square of the local field inhomogeneity (Eq. 27 in Ref. 32). If we assume the erythrocyte to be spherical from the view of a molecule close to the membrane, the value of G₀ can be approximated as ^17,32

$$G_0=\frac{64}{45}{\pi }^2{B}_0^2\eta (\Delta {\chi }_{ery-plas}{)}^2$$ [12a]

However, the deviation of the erythrocyte’s bi-concaved shape from spherical leads to a different field pattern and different diffusion-based relaxation enhancements. We therefore need to add a shape factor β.

$$G_0=\frac{64}{45}{\pi }^2{B}_0^2\eta {\beta }^2(\Delta {\chi }_{ery-plas}{)}^2$$ [12b]

The term η is the volume that experiences the gradient. In the plasma, we approximate this by the volume occupied by the erythrocyte (η = Hct). Therefore, the diffusion relaxation enhancement in the plasma can then be calculated as

$$R_{2D,plas}=\frac{32{\pi }^2}{45}Hct(\gamma B_0{)}^2\cdot [\Delta {\chi }_{ery-plas}\cdot {\beta }_{plas}{]}^2\cdot {\tau }_{D,plas}\cdot F(\frac{4{\tau }_{cp}}{{\tau }_{D,plas}})$$ [13]

In this study, the water diffusion contribution inside the erythrocyte is neglected due to the magnetic field homogeneity inside the erythrocyte. In fact, if we assume the erythrocyte to be an oblate ellipsoid of rotation, the magnetic field inside the erythrocyte will be fully homogeneous⁸⁸. Under this uniform magnetic field, the spin dephasing does not depend on position and not contribute to R₂ measured by CPMG experiments. Certainly, the erythrocyte’s shape is not perfectly ellipsoid, but as shown by the simulation based on the real erythrocyte’s shape (Figure 2), the magnetic field inside the erythrocyte is very homogeneous compared to the magnetic field outside the erythrocyte. Therefore, the water diffusion contribution inside the erythrocyte is neglected.

Whole-blood Relaxation Including Exchange-based Relaxation Enhancement

To describe transverse relaxation in the two-compartment model for blood (Fig. 2), the contributions from the individual compartments now can be written out using equations 1-4 for the hemoglobin and albumin solutions (R_2Hb and R_2Alb) and equation 13 for the diffusion enhancement in the plasma. The water proton transverse relaxation rates for the water protons in the plasma (R_2plas) and erythrocyte (R_2ery) then are:

$$\begin{gathered} R_{2plas}=R_{2Alb}\left(B_0,{\tau }_{cp},c_{Alb},R_{2b,Alb},k_{Alb},n_{Alb}^{ex},\Delta {\delta }_{Alb}\right) \\ +R_{2D,plas}(B_0,Y,{\tau }_{cp},Hct,{\beta }_{plas},{\tau }_{D,plas},Y^{off})+Hct\cdot R_{2mem} \end{gathered}$$ [14a]

$$R_{2ery}=R_{2Hb}(B_0,Y,{\tau }_{cp},c_{Hb},R_{2b,OxyHb},R_{2b,DeoxyHb},k_{Hb},n_{Hb}^{ex},\Delta {\delta }_{OxyHb},\Delta {\delta }_{DeoxyHb})+R_{2mem}$$ [14b]

The permeability of the erythrocyte membrane limits the life time of the water molecules in the erythrocyte to about 10 ms.⁸⁹ As a consequence, water molecules experience the different precession frequencies in the erythrocyte and plasma, leading to dephasing and a relaxation enhancement. Previously, this exchange-based enhancement has been described using the Luz-Meiboom (LM) Model^62,90, but this model implicitly assumes⁹¹ that: (i) the R₂ rates in the erythrocyte and in the plasma are of the same order of magnitude, which need not be always valid because the intrinsic relaxation rates and diffusion relaxation enhancements are very different inside and outside the erythrocyte, especially at lower oxygenations; (ii) the chemical shift difference between erythrocyte and plasma Δω is much smaller than the exchange rate, i.e. fast exchange on the NMR time scale, which is not fulfilled at high magnetic field¹⁷. Therefore, again the general exchange model (Appendix 1)^69,70 needs to be used to calculate the blood transverse relaxation rate:

$$\begin{gathered} R_{2blood}=\frac{1}{2}\left\{R_{2ery}+R_{2plas}+k_{ery}+k_{plas}-\frac{1}{{\tau }_{cp}}cos{h}^{-1}[D_{blood}^{+}\times \cosh (2{\xi }_{blood})-D_{blood}^{-}\times \\ \cos (2{\phi }_{blood})]\right\} \end{gathered}$$ [15a]

$$D_{blood}^{\pm }=1∕2[\pm 1+({\psi }_{blood}+2\Delta {\omega }_{ery-plas}^2)∕({\psi }_{blood}^2+{\zeta }_{blood}^2{)}^{1∕2}]$$ [15b]

$$k_{plas}=k_{ery}\times f_{ery}∕(1-f_{ery})$$ [15c]

$${\xi }_{blood}=\left(\frac{{\tau }_{cp}}{\sqrt{8}}\right){\left[+{\psi }_{blood}+{\left({\psi }_{blood}^2+{\zeta }_{blood}^2\right)}^{1∕2}\right]}^{1∕2}$$ [15d]

$${\phi }_{blood}=(\frac{{\tau }_{cp}}{\sqrt{8}}){\left[-{\psi }_{blood}+{\left({\psi }_{blood}^2+{\zeta }_{blood}^2\right)}^{1∕2}\right]}^{1∕2}$$ [15e]

$${\psi }_{blood}=(R_{2ery}-R_{2plas}+k_{ery}-k_{plas}{)}^2-\Delta {\omega }_{ery-plas}^2+4k_{ery}{k}_{plas}$$ [15f]

$${\zeta }_{blood}=2\Delta {\omega }_{ery-plas}(R_{2ery}-R_{2plas}+k_{ery}-k_{plas})$$ [15g]

in which k_ery and k_plas are the water exchange rate constants from the erythrocyte to the plasma and from the plasma to the erythrocyte, respectively. They can be related to water life time in the erythrocyte τ_ery through the equations:

$$k_{ery}=\frac{1}{{\tau }_{ery}}$$ [16a]

$$k_{plas}=\frac{f_{ery}}{1-f_{ery}}\times \frac{1}{{\tau }_{ery}}$$ [16b]

in which the water fraction in the erythrocyte f_ery is⁴²

$$f_{ery}=\frac{0.7Hct}{0.7Hct+(1-Hct)\ast 0.95}$$ [17]

The exchange life time for the two-compartment system is defined by

$$\frac{1}{{\tau }_{ex}}=\frac{1}{{\tau }_{ery}}+\frac{1}{{\tau }_{plas}}or{\tau }_{ex}={\tau }_{ery}(1-f_{ery})$$ [18]

Equations 15[a-g] may appear to be different from the widely used Luz-Meiboom (LM) fast exchange model⁶², where the intrinsic relaxation rate contributions of protein solutions in erythrocyte and plasma were appropriately proportioned through multiplication with the appropriate water fractions¹⁷. It is important to realize that, while not intuitive, the effect of these fractions is included appropriately in the current equations too. To show that the fractions are in there, one can take the limit of fast exchange and retrieve the equations used in previous studies where fast exchange was assumed (Supplementary S4).

Methods

To verify the model, we used the isolated plasma, lysed blood and whole blood R₂ data measured by Grgac et al.⁵⁰ as a function of Hct, Y, and τ_cp at 3.0 T, 7.0 T, 9.4 T and 11.7 T. The fitting approach is described in Figure 3. As discussed above, the whole blood R₂ includes contributions arising from the hemoglobin and albumin solutions (R_2Hb and R_2Alb), the erythrocyte membrane, and relaxation enhancements from water exchanging between inside and outside of the erythrocyte and diffusing around the erythrocyte. Therefore, we first fitted the isolated plasma and lysed blood R₂ data to obtain the parameters for calculating R_2Alb and R_2Hb, respectively. These were subsequently used for the fitting of whole blood R₂ values⁵⁰ for obtaining the parameters describing the membrane, diffusion and exchange relaxation enhancements. In the latter fitting, we started with baseline Y^off = 0.937 (Eq. 6e) and obtained further initial parameters by focusing on data acquired at short τ_cp and high Y, where the diffusional contributions dominate, before performing the final whole-blood fitting.

Figure 3:

Flow chart illustrating the fitting process.

Data Used in the Model Fitting

We used the isolated plasma, lysed blood and whole blood R₂ data measured by Grgac et al.⁵⁰, but several corrections were made. First, the length of refocusing pulse in the CPMG pulse sequence needs to be accounted for to have a correct τ_cp, using the equation⁹²:

$${\tau }_{cp,corr}={\tau }_{cp}-\frac{pw}{2}$$ [19]

in which pw is the length of refocusing pulse. This was previously done for the long pulses used at 3.0 T and 7.0 T, but due to the short pw (microsecond) on the high resolution vertical bore NMR scanners, the data at 9.4 T and 11.7 T in previous paper⁵⁰ were not corrected. The now corrected data for 9.4 T and 11.7 T are listed in Supplementary Tables S1, S2 and S3. Notice that the use of a corrected (shorter) τ_cp for refitting the TE-dependence of the experimental curves will lead to slightly larger relaxation rates than the original ones in reference⁵⁰ (Supplementary S5). Second, the previous paper contained multiple measurements of plasma at 9.4 T, with some larger differences for some of the inter-echo spacings (Table 2c in previous paper⁵⁰). We therefore re-measured the τ_cp dependence of R_2plas at 9.4 T (Supplementary Table S1) using the same approach as for the previous paper⁵⁰, and the new data only were used in the current fitting. Third, we acquired several new data at 3.0 T using the same method as previous paper⁵⁰, and found that the previous data with Hct of 0.45 and 0.56, which were measured at the same day, had significantly higher R_2blood at long τ_cp than the new data. This could indicate some experimental problems in the blood preparation. Therefore, we used only new data for this Hct range (new Hct values 0.46 and 0.56) at 3.0 T (Supplementary Table S3a). Fourth, we only included the data with Hct smaller than 0.58. This is because the irregular bi-concave shape of the erythrocyte can not effectively fill the space and Hct can not be over 0.58 without changing the shape and thus magnetic field gradient properties of the erythrocyte⁹³. This should not be problematic for use of the model, because the normal range of Hct in vivo is 0.36-0.53⁷⁵ which is covered in our fitting. Fifth, we only include data with Y higher than 0.55. Low oxygenation fractions greatly increase the susceptibility of the erythrocyte and thus the signal decay rate. As a consequence, we have limited data for such oxygenation fractions at high field and for consistency removed these Y-values also at lower fields. Since Y of venous blood is about 0.60-0.65, we are basically limiting the fitting of our data to the physiological range. All the data used in the fitting are listed in Supplementary Tables S1-S3.

Transverse Relaxation of Hemoglobin and Albumin Solution

To determine the parameters causing the exchange relaxation enhancement of water protons in albumin solution, the isolated plasma data were described using R_2plas = R_2Alb(B₀,τ_cp,R_2b,Alb,c_Alb,k_Alb,$n_{Alb}^{ex}$,Δδ_Alb). These data measured as a function of τ_cp at 3.0, 7.0, 9.4, and 11.7 T (supplementary Table S1) were then fitted for R_2b,Alb, k_Alb, $n_{Alb}^{ex}$, and Δδ_Alb using Eqs. 1-3 through minimizing the relative error ∣R_2.exp-R_2,fitted∣/R_2,exp, A value of 0.36 s⁻¹ was used for the bulk water relaxation rate (R_2buf = R_2saline) across all magnetic fields⁵⁰; the albumin concentration was 0.752 mM⁴² and the bulk water volume fraction f_V,water was 0.95, based on the fact that the albumin takes 5% volume in the solution⁷⁵. To improve the accuracy of the fitting, the ranges of the fitted parameters were limited based on existing knowledge. The typical chemical shift ranges for exchangeable NH protons and OH protons are from 6.6 to 8.8 ppm⁹⁴ and from 5.4 to 6.2 ppm⁹⁵, respectively, while the proton chemical shift of bulk water is about 4.8 ppm. Therefore, Δχ_Alb was limited from 0.6 ppm to 4 ppm in the fitting. The upper limit for the number of exchangeable protons in albumin ($n_{Alb}^{ex}$) was taken from the deuterium exchange experiments⁹⁶ by Benson et al. which showed a maximum $n_{Alb}^{ex}$ of about 910 ⁹⁶. The lower limit of $n_{Alb}^{ex}$ was estimated as 350 by directly counting the number of NH and OH groups in the accessible side chains (such as the side chains rich in lysine, arginine and threonine) of the albumin sequence⁷². The lower limit of the proton dissociation rate (k_Alb) was assumed to be 1000 s⁻¹, based on the experimental finding of the lack of large R_2plas changes with τ_cp shortening in our measured τ_cp range (0.5 ms – 20 ms). In the fitting, the plasma data at 9.4 T and 11.7 T were used first to estimate the field-independent parameters (k_Alb, $n_{Alb}^{ex}$ and Δδ_Alb) and the field-specific R_2b,Alb, because the τ_cp dependence of R_2plas at 3.0 T and 7.0 T was too small compared to the experimental noise (Table S1). The k_b,Alb, $n_{Alb}^{ex}$ and Δδ_Alb estimated this way were subsequently used to fit the 3.0 T and 7.0 T data for the R_2b,Alb values at these fields.

The parameters causing the exchange relaxation enhancement of water protons in hemoglobin solution were determined from fitting the lysed blood data available for mixtures of packed erythrocytes and plasma (Supplementary Tables S2a-d). To account for the addition of albumin, Eq. 4 was modified as:

$$R_{2lysed}=(1-Hct)\cdot R_{2Alb}+Hct\cdot [Y\cdot R_{2OxyHb}+(1-Y)\cdot R_{2DeoxyHb}]$$ [20]

where R_2Alb was calculated based on the fitted parameters above, and the Hct was estimated from the ratio of the hemoglobin concentration in the lysed blood sample and in the erythrocyte (332g/L = 5.15 mM): ^97,98

$$Hct=\frac{c_{Hb}(\frac{mol}{L})}{5.15\times {10}^{-3}mol∕L}$$ [21]

Because the hemoglobins occupy 30% of the volume in solution⁷⁴, P_ery in Eq. 3 can be calculated from

$$P_{ery}=\frac{n_{Hb}^{ex}\times 5.15\times {10}^{-3}mol∕L}{2\times [55.6mol∕L\times (1-0.3)]}$$ [22]

It is worthy to note that the hemoglobin concentrations for several of the lysed packed erythrocyte samples were higher than 5.15 mM, probably because the erythrocytes shrink during the centrifugation in the preparation of packed erythrocytes. For these cases, the Hct was set to one, and the P_ery modified to:

$$P_{ery}=\frac{n_{Hb}^{ex}\times c_{Hb}(\frac{mol}{L})}{2\times \left[55.6(\frac{mol}{L})\times (1-0.3\times \frac{c_{Hb}(\frac{mol}{L})}{5.15\times {10}^{-3}(\frac{mol}{L})})\right]}$$ [23]

assuming that the volume occupied by the hemoglobins is proportional to the hemoglobin concentration.

Similar to the fitting of isolated plasma data, the lysed blood data at 9.4 T and 11.7 T were fitted first to obtain the global parameters (k_Hb, $n_{Hb}^{ex}$ Δδ_DeoxyHb and Δδ_OxyHb) and R_2b,OxyHb and R_2b,DeoxyHb for each field. The fitting ranges of the global parameters were set the same as for albumin because both are globular proteins with comparable molecular weight (66.5 vs 64.5 kDa, respectively). The parameters k_Hb, $n_{Hb}^{ex}$, Δδ_DeoxyHb and Δδ_OxyHb from the fitting of 9.4 T and 11.7 T data were subsequently used as the constants in Eqs. 1-3 to fit the 3.0 T and 7.0 T data for determining the field-specific R_2b,Oxy and R_2b,Deoxy values.

Transverse Relaxation of Whole Blood

In the parameter fitting of the whole blood, R_2blood data obtained for various Hct and Y values at 3.0 T, 7.0 T, 9.4 T, 11.7 T (Supplementary Tables S3a-d) were fitted using Eqs. 14 and 15 through minimizing the relative error ∣R_2.exp-R_2,fitted∣/R_2,exp. The transverse relaxation rates of albumin solution in plasma (R_2Alb) and hemoglobin solution in the cytoplasm of erythrocytes (R_2Hb) were calculated based on Eq. 1 using an albumin concentration of 0.752 mM⁴², a hemoglobin concentration of 5.15 mM in the erythrocyte^97,98, R_2buf = 0.36 s⁻¹ at all fields⁵⁰, and the values of k_Alb, $n_{Alb}^{ex}$, Δχ_Alb k_Hb, $n_{Hb}^{ex}$, Δδ_DeoxyHb, Δδ_OxyHb, and the protein specific R_2b values as given by the field dependency equation (Eq. 31) derived from the previous fits of the isolated plasma and lysed blood data. Having the R_2Hb and R_2Alb available, the remaining parameters to be fitted from the whole blood data were τ_ery, β_plas, τ_D,plas, β_Ex, Y^off, and R_2mem, among which τ_ery, β_plas, τ_D,plas, β_Ex, and Y^off are expected to be magnetic field independent and thus set to be the same across all magnetic fields. R_2mem was fitted separately for each field.

To minimize the uncertainty of multi-parameter fitting (many possible fitting minima and co-dependent parameter changes for the diffusion and exchange life time), we first fit only the short τ_cp (< 2ms) data at high fields (9.4 T and 11.7 T), assuming contributions only from the albumin and hemoglobin solutions and the diffusion effects in plasma (Eq. 13). This is reasonable because for the exchange contribution, the water life time in the erythrocyte (~10ms) is much longer than the τ_cp (< 2ms) of these fitted data, thus the water exchange has little contribution compared with the diffusion contribution in the plasma which has the diffusional correlation time (~3ms estimated based on the erythrocyte size) close to τ_cp of these data (Fig. 1). Therefore, to fit the data acquired at short τ_cp (< 2ms), only the diffusion-based relaxation, the intrinsic protein solution relaxation, and the membrane relaxation contributions were used:

$$R_{2blood,short,{\tau }_{cp}}=f_{ery}\times (R_{2Hb}+R_{2mem})+(1-f_{ery})\times (R_{2Alb}+R_{2D,plas}+Hct\times R_{2mem})$$ [24]

where f_ery was calculated from the Hct using Eq. 17; R_2Hb and R_2Alb were pre-calculated using Eqs. 1-4 based on the parameters (Tables 1 and 2) obtained from the fits of lysed blood and plasma. Starting with this equation, the values of the parameters and accuracy of the fits were judged and the equations further adjusted by removing superfluous parameters (negligible contributions) and adding additional parameters, the logic of which is described in the results. As similar approach was used when subsequently fitting the whole blood values. From these fits it became clear that there was a need to have separate Y^off parameters for diffusion and exchange ($Y_D^{off}$ and $Y_{Ex}^{off}$), which are about 5% bigger and smaller, respectively, than the theoretically derived value of 0.937 (Eq. 6e). Thus:

$$\Delta {\chi }_{ery-plas}^D=\Delta {\chi }_{deoxy}(Y_D^{off}-Y)=0.253(Y_D^{off}-Y)ppm$$ [7c]

$$\Delta {\chi }_{ery-plas}^{Ex}=\Delta {\chi }_{deoxy}(Y_{Ex}^{off}-Y)=0.253(Y_{Ex}^{off}-Y)ppm$$ [7d]

Table 1.

Protein proton dissociation rates, number of exchangeable protons, and chemical shift differences relative to the bulk water protons.^*

		kprot (s−1)	$n_{prot}^{ex}$	∣Δδprot∣ (ppm)
Albumin		(11.6±5.8)×103	806±418	1.21±0.76
Hemoglobin	Oxy	(14.6±6.4)×103	499±206	0.614±0.304
	Deoxy			1.11±0.51

^*fitted from the data for plasma and lysed blood/plasma mixtures using Eqs. 1-4 and 20 (Fig. 3).

The error indicates 95% confidence interval of fitting.

Table 2.

Transverse relaxation rates for the exchangeable protein protons (R_2b,prot)

	3.0 T*	7.0 T*	9.4 T*	11.7 T*	1.5 T#	2.35 T#	4.7 T#
R2b,Alb (s−1)	291±170	523±190	363±160	672±180	83.7	131	262
R2b,OxyHb (s−1)	252±70	612±110	906±210	(1.13±0.20)×103	142	223	445
R2b,DeoxyHb (s−1)	378±120	841±242	(1.17±0.22)×103	(1.76±0.32)×103	311	346	521

^*fitted from the data for plasma and lysed blood/plasma mixtures using Eqs. 1-4 and 20 (Fig. 3). The error indicates 95% confidence interval of fitting.

^#calculated from the B₀ dependence of R_2b,prot measured at 3.0T, 7.0T, 9.4T, and 11.7T, using Eq. 31a-c.

Therefore, in total 7 parameters, i.e. three unifield parameters (τ_ery, $Y_{Ex}^{off}$ and β_Ex) and four R_2mem (one R_2mem for each field) were fitted using 130 R_2blood values at 3T, 104 R_2blood values at 7T, 191 R_2blood values at 9.4T and 285 R_2blood values at 11.7T. In the fitting, the τ_ery range was limited from 8 ms to 20 ms based on previous literature^89,99-105. $Y_{Ex}^{off}$ was limited from 0.8 to 1.0. The chemical shift shape factor β_Ex was limited from −10 to 10. The nonlinear fitting function “fmincon” in matlab was used in the fitting. To alleviate the problem of local minimum, a series of 126 initial values was sampled using τ_ery values of [8.0, 10.0, 12.0, 14.0, 16.0, 18.0] ms, β_Ex values of [−4, −2, −1, 0, 1, 2, 4] and $Y_{Ex}^{off}$ values of [0.85, 0.9, 0.95].

To compare our model with previous models, the same data set was fitted with

1. The Luz-Meiboom (LM) fast exchange model ⁶²

   $$R_{2blood}=R_{20}+f_{ery}(1-f_{ery})\Delta {\omega }_{ery-plas}^2{\tau }_{ex}\left[1-\frac{2{\tau }_{ex}}{{\tau }_{cp}}tanh\left(\frac{{\tau }_{cp}}{2{\tau }_{ex}}\right)\right]$$ [25]

   where τ_ex defined in Eq. 18 and Δω_ery–plas defined in Eq. 9 were used for the fitting; The R₂₀ is a weighted average of the relaxation rates of the hemoglobin solution inside the erythrocyte (Eq. 1a) and the albumin solution outside the erythrocyte (Eq. 1a) combined with the membrane contribution:

   $$R_{20}=f_{ery}\times (R_{2Hb}+R_{2mem})+(1-f_{ery})\times (R_{2Alb}+Hct\times R_{2mem})$$ [26]

   Therefore, in total 7 parameters (3 unifield parameters β_Ex, $Y_{Ex}^{off}$, τ_ery and four R_2,membrane, one for each field) will be fitted.

2. The Allerhand-Gutowsky (AG) exchange model ⁶³

   $$R_{2blood}=R_{20}+\left[\frac{1}{2{\tau }_{ex}}-\frac{1}{{\tau }_{cp}}sin{h}^{-1}{K}_{ex}({\tau }_{ex},f_{ery},\Delta {\omega }_{ery-plas})\right]$$ [27a]

   $$K_{ex}={\left[D^{+}sin{h}^2\left(\frac{{\tau }_{cp}{S}_r}{2}\right)+D^{-}si{n}^2\left(\frac{{\tau }_{cp}{S}_i}{2}\right)\right]}^{1∕2}$$ [27b]

   $$S={\left[(\frac{1}{{\tau }_{ex}}{)}^2-(\Delta {\omega }_{ery-plas}{)}^2+2i\left(f_{ery}-(1-f_{ery})\right)\left(\Delta {\omega }_{ery-plas}∕{\tau }_{ex}\right)\right]}^{1∕2}$$ [27c]

   $$2D^{\overline{+}}=\overline{+}1+\left[(\frac{1}{{\tau }_{ex}}{)}^2+(\Delta {\omega }_{ery-plas}{)}^2\right]{\left(S_r^2+S_i^2\right)}^{-1}$$ [27d]

   where S_r and S_i are the real and imaginary parts of S respectively and R₂₀ is described by Eq. 26. Similar to the Luz-Meiboom (LM) fast exchange model, in total 7 parameters (3 unifield parameters β_Ex, $Y_{Ex}^{off}$, τ_ery and four R_2,mem, one for each field) will be fitted.
3. The Jensen-Chandra (JC) diffusion model^17,32:

   $$R_{2blood}=R_{20}+\frac{32{\pi }^2}{45}Hct{\left[\gamma B_0\Delta {\chi }_{ery-plas}{\beta }_{plas}\right]}^2{\tau }_{D,plas}F\left(\frac{4{\tau }_{cp}}{{\tau }_D}\right)$$ [28]

   where the F function can be found in Eq. 11, Δχ_ery–plas in Eq. 7c, and R₂₀ is described by Eq. 26. Therefore, in total 7 parameters (3 unifield parameters β_plas, $Y_D^{off}$, τ_D,plas and four R_2mem for each field) will be fitted.
4. The Ziener diffusion model³³:

   $$R_{2blood}=R_{20}+{\left[\gamma B_0\Delta {\chi }_{ery-plas}{\beta }_{plas}\right]}^2{\tau }_{D,plas}H\left(\frac{4{\tau }_{cp}}{{\tau }_D}\right)$$ [28a]

   $$H(x)=\frac{8}{{\pi }^2}{\sum }_{n=0}^{\infty }\frac{1}{(2m+1{)}^2}{\sum }_{n=0}^{\infty }\frac{(H_n(\eta ){\lambda }_n(\eta ){)}^2}{{\lambda }_n(\eta {)}^4+(\pi (2m+1)∕x{)}^2}$$ [28b]

   where η is the occupied volume by the erythrocyte in the plasma (η = Hct), F_n(η), λ_n(η) are the expansion coefficients related to η and can be calculated as shown in Supplementary S6, and R₂₀ is described by Eq. 26. Therefore, in total 7 parameters (3 unifield parameters β_plas, $Y_D^{off}$, τ_D,plas and four R_2mem, one for each field) will be fitted.
   The Akaike information criterion (AIC)^106-108, which penalizes the goodness of fit with the number of free parameters, was used to compare the fitting goodness of the different models. The AIC with a correction for finite sample sizes was calculated as ¹⁰⁹:

   $$AIC=n\times \text{ln}\left(\frac{RSS}{n}\right)+2k+\frac{2k(k+1)}{n-k-1}$$ [29]

   where RSS is residue sum of square, n is the number of data points and k is the number of fitting parameters plus one. To interpret the calculated AIC values as conditional probabilities, the Alkaike weights were calculated using ¹⁰⁹:

   $$w_i(AIC)=\frac{exp\{-0.5\times (AI{C}_i-AI{C}_{min})\}}{{\sum }_{k=1}^{N}exp\{-0.5\times (AI{C}_k-AI{C}_{min})\}}$$ [30]

   where w_i(AIC) and AIC_i are the ith model’s Alkaike weights and AIC value, and AIC_min is the smallest AIC value among all the models.

Results

The fits of the relaxation data of isolated plasma and lysed blood used to determine the relaxation contributions of albumin and hemoglobin molecules, respectively, are shown in Figs. 4 and 5. The fitted number of exchangeable protons, the dissociation rate, the chemical shift difference relative to bulk water, and the transverse relaxation rates without exchange (R_2b,prot) are listed in Tables 1 and 2. The field dependencies of these rates are plotted in Fig. 6, where linear (including point 0,0) and quadratic functions were used to fit the diamagnetic and paramagnetic rates, respectively:

$$R_{2b,Alb}=(55.8\pm 11.3)\times B_0$$ [31a]

$$R_{2b,OxyHb}=(94.7\pm 3.1)\times B_0$$ [31b]

$$R_{2b,DeoxyHb}=(10.6\pm 0.7)\times B_0^2+(287\pm 63)$$ [31c]

Figure 4:

Fits of R₂ of isolated plasma at multiple fields used to determine the unknown contributions to R_2Alb, namely k_Alb, $n_{Alb}^{ex}$, and Δδ_Alb as well as the R_2b,Alb values at the four different fields (Tables 1,2).

Figure 5.

Fits of R₂ of lysed blood samples at multiple fields ((a) 3.0T; (b) 7.0T; (c) 9.4T; (d) 11.7T) to determine the unknown contributions to R_2Hb, namely k_Hb, $n_{Hb}^{ex}$, and ΔΔ_Hb, as well as the R_2b,Hb values at these four different fields (Tables 1,2).

Figure 6.

The B₀ dependence of the transverse relaxation rates of the exchangeable protons in albumin(a), oxygenated hemoglobin (b) and deoxygenated hemoglobin (c).

Based on these field dependencies, we also predicted the relaxation times of albumin and hemoglobin at 1.5 T, 2.35T, and 4.7 T (Table 2) to be used below for prediction of blood relaxation data at these field strengths and comparison with measured values in the literature. Due to the low relaxation rates in plasma, the error in the measured R_2b,Alb is expected to be larger than for R_2b,OxyHb and R_2b,DeoxyHb, which becomes clear when comparing the fit in Fig. 6a with the fits in Figs. 6b and c. When predicting literature values, we will therefore use Eqns. 31a-c (now part of the model) and not the individual numbers listed in Table 2.

To simplify the whole blood R₂ fitting through the use of appropriate data-based starting values, we exploited the fact that the diffusion contribution dominates at short inter-echo spacing (see Fig. 1). We did this by first fitting only the R_2blood data measured at short τ_cp (< 2ms) at 9.4T and 11.7T and considering only the protein solution R₂ values, the membrane relaxation contribution and the diffusion effect in the plasma (Eq. 24). In the determination of the diffusion contribution the susceptibility difference between inside and outside the erythrocyte ((Δχ_ery–plas) is very important. Except for a few studies^38,84, Δχ_ery–plas has usually been described by Δχ_deoxy(1 – Y), while we now have a theoretical derivation based on magnetic protein properties, Δχ_ery–plas = 0.253×(0.937-Y) ppm (Eq. 6e), indicating that there is an oxygenation value Y^off unequal to unity to account for the transition from diamagnetic to paramagnetic cell susceptibility. To test for the correctness of the theoretical value, two fits were conducted, one using Eq. 6e (Model 1 in supplementary S7), and another introducing a diffusion-specific fitting parameter $Y_D^{off}$ into Eq. 7b, leading to $\Delta {\chi }_{ery-plas}=0.253\times (Y_D^{off}-Y)$ ppm, corresponding to Model 2 in supplementary S7. The results (Table 3 and Supplementary Figs. S7.1 and S7.2) show that the fitting with $Y_D^{off}$ decreased the relative error by about 30%. To assess whether it is proper to neglect the intracellular diffusion contribution, we also estimated the intracellular diffusion contribution R_2D,ery using a form similar to extracellular diffusion contribution above (Eq. 13) as

$$R_{2D,ery}=\frac{32{\pi }^2}{45}\cdot (\gamma B_0{)}^2\cdot {\left[\Delta {\chi }_{ery-plas}\cdot {\beta }_{ery}\right]}^2\cdot {\tau }_{D,ery}\cdot F(\frac{4{\tau }_{cp}}{{\tau }_{D,ery}})$$ [32]

in which Hct dependence in Eq. 13 was dropped because the gradient is felt throughout the cytoplasm in the small erythrocyte, leading to the assumption of no Hct dependence for R_2D,ery. We then incorporated R_2D,ery into Eq. 24 (model 3 in supplementary S7):

$$R_{2blood,short{\tau }_{cp}}=f_{ery}\times (R_{2Hb}+R_{2D,ery}+R_{2mem})+(1-f_{ery})\times (R_{2Alb}+R_{2D,plas}+Hct\times R_{2mem})$$ [33]

Table 3:

Fitted parameters for the short τ_cp (< 2ms) whole blood data at 9.4 T and 11.7 T using the three models and Equation. 24 (Model 1, 2) and Equation 33 (Model 3)

	Model 1	Model 2	Model 3
τD,plas (ms)	3.30±0.43	3.15±0.43	4.04±1.58
βplas	0.778±0.013	0.661±0.027	0.608±0.054
$Y_D^{off}$		0.985±0.015	0.984±0.015
τD,ery (ms)			0.376±1.80
βery			0.641±0.980
Relative Error	6.75%	4.58%	4.57%

When fitting this model, we found that the fitted diffusional correlation time outside the erythrocyte (τ_D,plas) was about ten times larger than inside (τ_D,ery) (Table 3 and Fig. 7), corresponding to a diffusion contribution in the plasma that is much larger than inside the erythrocyte. Moreover, the results in Table 3 show that this new model had a fitting error (4.57%) very similar to the model (Eq. 24) without the intracellular diffusion contribution (4.58%), but even though there were two more fitting parameters (τ_D,ery and β_ery), the fitting error for the τ_D,plas was dramatically increased. This provided further support for neglecting the intracellular diffusion contribution and therefore Eq. 24 only was used to fit the data acquired at short τ_cp (< 2ms) (Table 3 and Supplementary Fig. S7.2) and all further fits were done without considering the diffusion contribution in the erythrocyte.

Figure 7.

The τ_cp dependence of R_2D,plas (red) and R_2D,ery (blue) for venous blood at Hct = 0.43 and Y=0.65 using the parameters fitted based on Model 3 in Supplementary 7.

In the analysis of the complete whole blood data set with all τ_cp values, the τ_D,plas (3.15 ms), β_plas (0.661) and $Y_D^{off}$ (0.985) fitted from short τ_cp data (Table 3) were used for the diffusion contribution and not adjusted further. This $Y^{off}=Y_D^{off}$, was initially also used in the exchange contribution (Eq. 9). However, we found that this approach could not fit the whole blood R₂ at high oxygenation fraction values and high magnetic field (9.4 T and 11.7T) and therefore needed to introduce a parameter $Y_{Ex}^{off}$ in the exchange susceptibility equation. Fig. 8 shows the fitting results of the whole data set using our general model (including eqns. 31a-c). As a comparison, the same data set was fitted with the LM exchange model⁶² (Supplementary Fig. S8.1), the AG exchange model⁶³ (Supplementary Fig. S8.2), the JC diffusion model³² (Supplementary Fig. S8.3) and the Ziener diffusion model³³ (Supplementary Fig. S8.4). The values of fitted parameters for each model are presented in Tables 4 and 5. The determined water residence times in the erythrocyte (τ_ery) fitted using the LM and AG exchange models are much shorter than τ_ery fitted in our general model, while the exchange-based shape factors (β_Ex) fitted in the LM and AG exchange models are much larger than those fitted in our general model. Notice the large deviations at long τ_cp, especially at low field for the fits in Supplementary Figs. S8.1 and S8.2, indicating that the fit of the exchange contributions is not very good using these two models. The diffusion correlation time (τ_D,plas) fitted in our general model is similar to the JC results and about 50% larger than the Ziener diffusion model. The diffusion shape factor β_plas fitted by our general model is a bit larger than the results fitted by the JC model but much smaller than Ziener model. Most importantly, the fits in the Supplementary Figs. S8.1 - S8.4 show that none of the single-mechanism models can consistently fit the correct curve shapes at all fields, while our general model can. Table 6 presents the average relative error (R₂-R_2,fit)/R₂ along with AIC analysis to compare the performance of different models. In line with the visual inspection of the curve fits, the average relative error in our general model is the best, and the calculated AIC values, which penalize the goodness of fit with the number of fitting parameters, also show the same trend as the relative error. Meanwhile, the Akaike weights calculated based Eq. 30, which represent the model likelihood in the comparison of the given models, shows that our model has a dominant advantage to fit the data.

Figure 8:

Results of multi-variant fits of whole blood R₂ at multiple fields with different Hct, Y and τ_cp using our general model (Eq. 15a-g)

Table 4.

Fitted parameters for the whole blood R₂ model

	This paper	LM	AG	JC	Ziener
τD,plas (ms)	3.15±0.43			3.42±0.22	2.26±0.09
βplas	0.661±0.027			0.490±0.012	1.02±0.03
${\mathit{Y}}_{\mathit{D}}^{\mathit{off}}$	0.985±0.015			1.00±0.01	1.00±0.01
τery (ms)	9.13±1.43	0.792±0.032	0.790±0.030
βEx	−1.20±0.20	3.22±0.090	3.29±0.088
${\mathit{Y}}_{\mathit{Ex}}^{\mathit{off}}$	0.889±0.019	0.979±0.010	0.972±0.009

The error indicates 95% confidence interval of fitting.

Table 5.

Membrane relaxation rate R_2mem (s⁻¹) as a function of magnetic field strength for whole blood obtained from fits with five different models

	3.0 T	7.0 T	9.4 T	11.7 T
This paper	2.65±0.87	2.27±1.01	3.33±0.75	1.29±0.63
Luz-Meiboom	2.79±1.06	2.60±1.19	2.54±1.02	0.00±0.97
Allerhand-Gutowskey	2.78±1.04	2.64±1.16	2.92±0.98	0.00±0.92
Jensen-Chandra	2.81±0.97	2.15±1.13	2.28±0.97	0.00±0.94
Ziener	2.26±0.09	2.38±1.21	2.58±1.04	0.00±1.00

The error indicates 95% confidence interval of fitting.

Table 6.

Summary of goodness of fit for the different models.

Model	Relative Error	No. of Parameters	AIC	w(AIC)
This paper	4.2%	10 *	1563	100%
Luz-Meiboom	6.5%	7	1876	0%
Allerhand-Gutowskey	6.5%	7	1844	0%
Jensen-Chandra	6.2%	7	1756	0%
Ziener	7.0%	7	1874	0%

^*Of these 10, only 7 were varied in the fitting of all data as β_plas, τ_D,_plas and τ_D,plas and $Y_D^{off}$ were kept constant.

Finally, our general model was used to predict the CPMG-based T₂ value of human blood at normal Hct for commonly used human field strengths (1.5T, 3T, and 7T). These results are compared with the literature values for CPMG measurements in Table 7 and Fig. 9a. We then did a similar prediction for all field strengths at which either T₂ of human or bovine blood samples was measured in vitro and the results are shown in Table 8 and Fig. 9b. Both comparisons show outstanding agreement between the model prediction and actual experimental determinations.

Table 7.

Comparison between literature values of human blood T₂ measured in vivo at different common field strengths and blood T₂ predicted^# by our model.

	B0	τcp (ms)	Hct	Y	Measured T2 (ms)	Predicted T2 (ms) #
Adult (Ref. 12)	1.5T	6	0.43*	0.67	168	172
Adult (Ref. 12)	1.5T	25	0.43*	0.62±0.04	128±9	131±14
Adult (Ref. 12)	1.5T	25	0.43*	0.82±0.04	174±18	209±14
Adult (Ref. 11)	1.5T	6	0.43*	0.45±0.05	113±6	114±12
Adult (Ref. 11)	1.5T	6	0.43*	0.67±0.05	166±15	172±15
Adult (Ref. 7)	1.5T	24	0.43*	0.97	230	242
Adult (Ref. 7)	1.5T	24	0.43*	0.77±0.05	152±34	190±19
Adult (Ref. 59)	1.5T	6	0.43*	1.0	224.5±5.3	253
Male (Ref. 14)	3T	10	0.45*	0.61±0.03	61.4±5.3	61.9±6.4
Female (Ref. 14)	3T	10	0.40*	0.62±0.04	63.4±7.0	68.4±9.8
Adult (Ref. 14)	3T	10	0.43*	1.0	155±10	155
Adult (Ref. 10)	3T	10	0.35-0.42	1.0	140-180	158-180
Adult IVJ (Ref. 19)	3T	10	0.43*	0.65±0.04	70±7	72.4±10.4
Adult SSS (Ref. 19)	3T	10	0.43*	0.63±0.03	67±7	67.7±7.1
Adult SS (Ref. 19)	3T	10	0.43*	0.68±0.04	77±9	80.1±11.5
Male Normoxia (Ref. 124)	3T	10	0.45*	0.605±0.036	60.3±5.9	60.9±7.6
Male Hyperoxia (Ref. 124)	3T	10	0.45*	0.689±0.037	77.2±8.3	80.3±10.5
Male Normoxia (Ref. 124)	7T	5	0.45*	0.600±0.056	20.5±4.3	19.5±4.4
Male Hyperoxia (Ref. 124)	7T	5	0.45*	0.690±0.061	29.7±6.9	27.4±8.2

^*The reference didn’t mention Hct, so Hct was assumed as the average value based on published of 0.45+/−0.05 for males, 0.40+/−0.04 for females and 0.43+/−0.08 for adult humans¹²⁵

^#For calculations, all the R_2b,Alb R_2b,OxyHb and R_2b,DeoxyHb values were calculated based on their B₀ dependence (Eq. 31a-c)

Figure 9.

Comparison of predicted CPMG-based R_2blood values with literature values for (a) in vivo human data (Table 7) and (b) in vitro human and bovine blood data (Table 8).

Table 8.

Comparison of predicted T₂ with literature values for human and bovine blood.

	B0	τcp (ms)	Hct	Y	Literature T2 (ms)	Predicted T2 (ms)
Human blood (Ref. 16)	1.5T	10#	0.512	0.93	181#	212
Human blood (Ref. 16)	1.5T	10#	0.512	0.87	156#	201
Human blood (Ref. 16)	1.5T	10#	0.512	0.72	129#	159
Human blood (Ref. 16)	1.5T	10#	0.512	0.66	111#	142
Human blood (Ref. 16)	1.5T	10#	0.512	0.62	102#	131
Human blood (Ref. 16)	1.5T	10#	0.512	0.48	80.0#	98.1
Human blood (Ref. 16)	1.5T	10#	0.512	0.43	68.0#	88.6
Human blood (Ref. 16)	1.5T	10#	0.512	0.42	67.1#	86.9
Bovine blood (Ref. 12)	1.5T	10	0.43#	0.65#	149#	156
Bovine blood (Ref. 12)	1.5T	20	0.43#	0.65#	136#	145
Human blood (Ref. 35)	1.5T	15	0.43*	0.719±0.026	181±20	174±10
Human blood (Ref. 35)	1.5T	15	0.43*	0.967±0.016	254±24	245±1
Human blood (Ref. 17)	2.35T	10	0.43#	0.65#	128#	98.9
Bovine blood (Ref. 10)	3T	5	0.43#	0.65#	82.6#	84.0
Bovine blood (Ref. 10)	3T	10	0.43#	0.65#	69.4#	72.4
Bovine blood (Ref. 10)	3T	15	0.43#	0.65#	68.0#	66.7
Bovine blood (Ref. 10)	3T	20	0.43#	0.65#	64.9#	63.2
Human blood (Ref. 34)	3T	10	0.43#	0.65#	78.1#	72.4
Human blood (Ref. 34)	3T	20	0.43#	0.65#	55.9#	63.2
Bovine blood (Ref. 48)	4.7T	10#	0.40	0.254	12.6#	12.2
Bovine blood (Ref. 48)	4.7T	10#	0.40	1.00	105#	110
Bovine blood (Ref. 48)	4.7T	10#	0.40	0.718	39.7#	51.5
Bovine blood (Ref. 48)	4.7T	10#	0.40	0.014	9.26#	9.1
Human blood (Ref. 17)	7T	10#	0.43#	0.65#	22.0#	19.4
Human blood (Ref. 9)	7T	5	0.34	0.65#	32.2#	27.0
Human blood (Ref. 9)	7T	5	0.42	0.65#	24.3#	24.1
Human blood (Ref. 9)	7T	5	0.54	0.65#	21.1#	22.0

^#For the references in which only fitted R_2blood models were given and the measured R_2blood values were not listed, the R_2blood values were calculated using their fitted model. In the calculation, if τ_cp was not specified in their model, 10ms was used, if Hct was not specified, the averaged adult Hct 0.43 was used and if Y was not specified, a typical venous oxygenation 0.65 was used.

^*The reference didn’t mention Hct, so Hct was assumed as the average value based on published of 0.45+/−0.05 for males, 0.40+/−0.04 for females and 0.43+/−0.08 for adult

Discussion

We designed a general mechanistic model for transverse relaxation of blood water that takes into account the effects of τ_cp, Hct, Y, and hemoglobin and albumin concentration, and validated it on R_2blood data measured previously using a CPMG sequence⁵⁰. Our approach differs from previous studies^{7,8,12,15,17,36-39}, in which the R₂ contribution of proteins (albumin and hemoglobin) was assumed to be a τ_cp independent term R₂₀ that was determined from whole blood R₂ fitting either by extrapolating to infinitely small τ_cp or by using the whole blood R₂ values measured at the shortest attainable τ_cp. However, the lysed blood and isolated plasma data⁵⁰ (Supplementary Tables S1 and S2) clearly have a τ_cp dependence, which could result from either diffusion or exchange effects. However, as reviewed by Kiselev and Novikov⁷¹, the protein molecule has a relatively small spatial scale, and the diffusional correlation time around the gradient induced by the protein molecule is much shorter than that cause by for instance the erythrocyte membrane. Because the diffusion contribution to water R₂ is roughly proportional to the correlation time, the diffusion contribution for proteins in solution is very small. Therefore, the fast exchange contribution dominates the τ_cp dependence of R₂ in the protein solutions, which is different from the whole blood case, where between-compartment water exchange is on a much slower time scale than direct exchange between protein and water. This latter exchange R₂ contribution can be interpreted in terms of either (i) the sub-millisecond exchange between exchangeable protein protons and bulk water protons⁷², (ii) the millisecond exchange between water in hydration shell of protein and bulk water¹¹⁰, or (iii) the cross relaxation between proteins protons and bulk water protons¹¹¹. Based on previous studies showing that the R₂ of water in protein solution is enhanced at characteristic pH values corresponding to the pK_a of NH or OH protons of the protein¹¹², we concluded that the fast exchange between exchangeable protein protons and bulk water protons is the main source of relaxation enhancement for proteins solutions and modeled this contribution following the work of Hills et al.⁷². Certainly, the exchange rate for exchangeable protons often is indeed very fast and will not induce a noticeable τ_cp dependence of R₂ of whole blood for typical longer TEs (longer τ_cp) used in vivo. However, as measured by previous proton-deuterium exchange experiments⁹⁶, large proteins like albumin and hemoglobin have hundreds of exchangeable protons with exchange correlation times in the millisecond range. Therefore, the albumin and hemoglobin solutions will have a high microsecond to low millisecond range τ_cp dependence. This fast exchange contribution to the intercept at long 1/τ_cp in a plot of whole-blood relaxation versus 1/τ_cp is relevant for a complete theory. Due to the fact that τ_cp cannot be very short because of machine-based limitations in the refocusing pulse length, we did not observe the characteristic R₂ jump with increasing 1/τ_cp when 1/τ_cp is comparable to the proton exchange rate, but the fitting results with small error (3.1%) at the available τ_cp range provided a reasonable estimate of the protein relaxation contributions as a function of Hct, Y and B₀, which were subsequently used in the fitting of the whole blood R₂ data. The B₀ dependencies of these protein solution relaxation rates due to the presence of exchangeable protons (R_2b,Alb R_2b,OxyHb and R_2b,DeoxyHb) were determined in Fig. 6 (Eqs. 31a-c). Following suggestions in the work of Gardener et al.¹⁷, the R_2b values for the diamagnetic proteins (R_2b,Alb and R_2b,OxyHb) were fitted using a linear model, while those of the paramagnetic protein (R_2b,DeoxyHb) were fitted using a quadratic model. Due to the range of relaxation rates, the fitted R_2b,Alb had a large fluctuation, but considering its small contribution of plasma R₂ to the whole blood relaxation, this is not detrimental and the fitted linear curve provides a good estimate. These field dependencies can now be used to predict the hemoglobin solution (cytoplasm) and albumin solution (plasma) R_2b values beyond the magnetic fields investigated here.

Once the albumin solution parameters R_2b,Alb, k_Alb, $n_{Alb}^{ex}$, Δδ_Alb and the hemoglobin solution parameters R_2b,oxyHb, R_2b,DeoxyHb, k_Hb, $n_{Hb}^{ex}$, Δδ_OxyHb, Δδ_DeoxyHb are available (Tables 1,2 and Eqs. 31a-c) from the data fits for isolated plasma and lysed blood, the number of unknown parameters to be determined by fitting the whole blood R₂ to Eqs. 14 and 15 is reduced to the exchange-based, diffusion-based, and membrane-based relaxation enhancements, with the latter very small. Thus, with the magnetic susceptibility differences calculated from basic principles and related to oxygenation through Equation 7b, the task is much simplified, with the only unknowns remaining being the field-independent parameters τ_D,plas, τ_ery, $Y_D^{off}$, $Y_{Ex}^{off}$, β_plas, , β_Ex and the field-dependent parameter R_2mem. We then exploited the dominant contribution of diffusion through field gradients in the plasma at high field and short τ_cp to determine the values for τ_D,plas, $Y_D^{off}$ and β_plas and thus further reduce the number of the unknown parameters. In this process, we neglected the intracellular diffusion contribution as explained in Theory section and supported by our fitting estimates (Fig. 7) showing that the intracellular diffusion contribution was one order of magnitude smaller than that in plasma (Fig. 7), in agreement with recent simulation results⁸⁸. It is important to realize that the assumption of a negligible relaxation enhancement from intracellular diffusion does not conflict with a previous observation at 7 Tesla that the width of the water proton peak in blood NMR spectra is the same at low Hct (0.42) and high Hct (0.69) for the same oxygenation (Y = 0.6) ¹¹³. In fact, using our theory, the blood T₂ (τ_cp=10ms) values are predicted as 16.1 ms and 15.8 ms for these two samples, respectively. At first this seems counter-intuitive to cytoplasm having a small diffusion contribution, because one would then expect R_2blood (linewidth) to decrease with an increasing fraction of cytoplasm (Hct). However, R_2blood at low oxygenation has a parabolic Hct dependence with the highest value at Hct of about 0.55, as shown by Thulborn et al.³⁶. This phenomenon can be explained by the fact that although the extracellular diffusion contribution is proportional to Hct (Eq. 13a), this contribution will be counteracted by the plasma’s water fraction (about 1-Hct, more precisely 1 – f_ery) when accounting for total blood water. Therefore, the previous proton linewidth equivalence finding is not proof for equivalence of intracellular and extracellular diffusion contributions^39,113.

Most interesting in the correlation time dependent fit limited to diffusion was the finding that $Y^{off}(Y_D^{off}=0.985,Y_{Ex}^{off}=0.889)$ differed from the one predicted from theory (Y^off = 0.937). When fitting the whole-blood data with the predicted Y^off, we noticed that we could not satisfactorily match the curve shapes at all fields unless we assumed individual values for diffusion ($Y_D^{off}$) and exchange ($Y_{Ex}^{off}$). While this appeared illogical at first, we later concluded that this is not unreasonable based on the known physiological phenomenon that the erythrocyte membrane has a much higher affinity to the deoxygenated Hb^114-116 and that this binding to the erythrocyte membrane coincides with a strong shift of the hemoglobin’s Hb-O₂ saturation curve to the right^115,117. This means that the membrane will bind a certain amount deoxygenated hemoglobin even if the blood oxygenation is high. These bound deoxygenated hemoglobins will largely determine the magnitude of the magnetic field gradient near the membrane and it is thus not surprising to find that Y^off for diffusion differs from that in the theoretical calculation (Eq. 6e) for which the distribution of deoxygenated hemoglobin is assumed to be homogeneous. Thus, the $Y_{Ex}^{off}$ reflects the susceptibility difference that water protons experience for the average compartments, while $Y_D^{off}$ accounts for the field gradients being affected by phenomena changing close to the membrane.

In the fitting results related to the exchange process, the fitted water residue time in the erythrocyte τ_ery was found to be 9.13 ± 1.43 ms, which is in the range of 6.0 to 9.6 ms measured by others at 37 °C ^{89,100,103,105}. Using the fitted τ_D,plasma of 3.15 ± 0.43 ms and the water diffusion coefficient at 37 °C (2.9 μm²/ms) ¹¹⁸, the effective radius of the erythrocyte r_c can be estimated to be 3.02 μm through the equation ${\tau }_{D,plas}=r_c^2∕D$ ^32,33,87. While this is on the correct order of magnitude for the erythrocyte, for which the short and long radii are about 1 μm and 4 μm, respectively, it appears to be on the high side. On the other hand, this may be explained by the diffusion constant of the viscous plasma water being smaller than that of neat water¹¹⁹.

The shape factors β_plas and β_Ex were introduced to account for the effect of the erythrocyte’s bi-concaved donut shape on the magnetic field gradient experienced in the plasma close to the membrane and the average field inside the cell, respectively. As pointed out in Theory Section, the need to include a shape effect in the diffusion contribution results from the fact that the erythrocyte shape is too complicated to find an analytical result for the pre-coefficient in the JC model. The fitted β_plas (0.661) is close to the theoretical derivation by Kiselev and Novikov⁶⁴ (~0.8 from their Fig. 2) which assumes the erythrocyte as a disk with height-to-radius radio of 0.5. This also agrees with Sukstanskii and Yablonskiy’s work⁶⁷ (~0.8 from their Fig. 3) if the erythrocyte is assumed as an ellipsoid with a long-to-short axis ratio of 4. In the exchange contribution, the shape factor is needed because the susceptibility difference between inside and outside the erythrocyte will induce magnetism on the external boundary of the erythrocyte and generate a demagnetization field that depends on the shape of the cell. In our fitting, the theoretically calculated susceptibility difference between deoxygenated and oxygenated hemoglobin (Δχ_deoxy) was used as the value in Eq. 6e and the introduction of shape factors for diffusion and exchange will correct the fitting uncertainty of using Δχ_deoxy, which is just based on solution susceptibility values. The fitted β_Ex (−1.20) value is also close to the theoretically predicted value (−1.62, Supplementary S9). The small deviation could result from the discrepancy between our assumed cylinder shape and the erythrocyte’s bi-concaved shape. The membrane contributions fitted here are smaller than found in our previous paper⁵⁰. Interestingly, we did not find a B₀ dependence for the membrane relaxivity. Therefore, the average of R_2mem of all four fields in Table 4 (2.39 s⁻¹) was used to predict blood R₂.

Using the fitted parameters, we can determine the contributions from different relaxation mechanisms, which are shown in Fig. 10. These calculations show that the relaxation rates of hemoglobin and albumin solutions (black curves), which have a weak proton-exchange based dependence on τ_cp, have a large contribution at low field (3T) that even dominates at very short τ_cp. However, at higher fields, the compartmental exchange and diffusion relaxation enhancements, which result from the sequestration of the hemoglobin solution by the erythrocyte membrane, increase with a quadratic dependence on B₀. Interestingly, the compartmental exchange contribution does not become much larger than that of the individual protein solutions, but the compartmental diffusion contribution dominates, being a factor of about five larger than the compartmental exchange contribution at every field. The data show that with increasing field, the chemical shift difference inside and outside the erythrocyte is changing from 264 rad/s⁻¹ (3T) to 1017 rad/s⁻¹ (11.7T). Given that the water exchange rate (k_ery + k_plas in Eq. 15) can be calculated to be 154 s⁻¹ at a Hct of 0.43, the exchange goes from the intermediate regime at 3T to the slow exchange regime at 11.7T. Therefore, the LM and AG models, based on fast exchange, are not able to fit the data with a consistent erythrocyte lifetime at all fields. Meanwhile, while the diffusion relaxation enhancement dominates at short τ_cp, and the JC and Ziener models should be able to fit such data with consistent correlation times between fields, the exchange contribution cannot be neglected at longer τ_cp, where these two models will therefore fail to give consistent life times.

Figure 10.

The τ_cp dependence of intrinsic relaxation contribution R₂₀ (black), exchange relaxation enhancement R_2ex (blue) and diffusion relaxation enhancement in the plasma R_2D,plas (red) for venous blood at Hct = 0.43 and Y=0.65.

For all our fittings, we minimized the relative error (i.e. (R_2,exp-R_2,fit)/R_2.exp) rather than the absolute error ∣R_2,exp-R_2,fit∣. This is because the experimental errors induced by the fluctuation of samples conditions such as oxygenation will be larger at the higher magnetic field and lower oxygenation due to the quadratic dependence of R₂ on the B₀ and susceptibility. The minimization of relative error can alleviate the effect of these experimental errors. This also may explain that we had larger deviations for some low oxygenation fittings at 11.7 T (Fig. 8). As shown in Supplementary S10, these deviations can possibly be explained by <5% oxygenation error. Certainly, other factors such as the aggregation and alignment of erythrocytes¹²⁰ and imperfection of the model also could introduce these deviations.

We also compared our model with pure exchange models (LM model and AG model) and diffusion models (JC model and Ziener model). Our model performed best as judged from the AIC statistics for current R₂ data set. Actually, as already commented upon by Thulborn et al.³⁶, both exchange and diffusion are expected to contribute to the τ_cp dependent relaxation enhancement. As measured and analyzed by previous studies, the correlation times for these two kinds of water displacements are very different (~10ms vs ~3ms), and their τ_cp dependent correlation function is different³² because exchange is water jumping between two environments with different chemical shifts while diffusion is water moving randomly through a continuously changing field. Therefore, it is very difficult to just use a single model with a single correlation time to fully characterize the τ_cp dependence of R_2blood. As shown in Fig. 10, the diffusion relaxation enhancement in the plasma dominates the R_2blood measured by a CPMG experiment. This is the reason that the exchange correlation time (Eq. 18 τ_ex = τ_ery/(1-f_ery)) fitted from LM and AG model is closer to the water diffusion correlation time and much shorter than the water residue time measured by membrane permeability experiments. However, the exchange contribution can not be ignored for an accurate estimation of blood R₂. For example, the experimental R_2blood is 22.7 s⁻¹ for Hct=0.44, Y = 0.99, τ_cp = 16.05 ms at 9.4T (Table S3c). Using our model including exchange, the predicted value is 22.4 s⁻¹, while using the JC model, the predicted value is 17.6 s⁻¹, indicating a ~20% error. Even more important than just accuracy (which represents a summary of both short and long inter-echo spacings) is the curve shape for the JC model versus our universal model at lower fields, physiological Hct, arterial oxygenation and small 1/τ_cp (longer τ_cp), the range important for typical fMRI experiments. This is clear in Figs. 8 and S8.3 and amplified in Supplementary Fig. S11 for convenience. The JC model cannot faithfully represent the curve and instead shows a straight line. Meanwhile, we found the fitted β_plas and τ_D,plas (Table 4) by JC model to be closer to our model than the Ziener model. This probably results from the Ziener model assuming erythrocytes to be impermeable solid spheres with water diffusion outside the erythrocyte being restricted, while the JC model assumes unrestricted water diffusion. However, because the diffusion R₂ contribution is positively correlated with τ_D,plas and β_plas, the shorter τ_D,plas and larger β_plas obtained from the Ziener model will not cause significant deviation of the fitted R_2blood results. Therefore, our combined model that considers both the exchange and diffusion effects and both the water inside and outside the erythrocyte can better depict the τ_cp dependence of blood R₂.

Our model also supplies the constants and equations to predict R_2blood at arbitrary Hct, Y and τ_cp at multiple fields. Figs. 9a-b (Tables 7, 8) show that the CPMG-based T_2blood values predicted by our model agree well with both the in vitro and in vivo CPMG blood T₂ measured at 1.5T, 2.35T, 3T, 4.7T and 7T by other groups. Since the model has many constants and terms and since the predicted and measured relaxation times will also depend on the length of the 180° RF pulses in the CPMG experiment, we have made available a website (Blood T1 T2 Calculation at http://godzilla.kennedykrieger.org/) that can be used for these predictions.

As a comparison, the JC model was also applied to predict the blood CPMG-based T₂ values, which are compared with literature numbers in Supplementary Tables S12a,b and Figs. S12a,b. Compared to the general model, we found that the JC model had a significantly larger averaged relative error for both in vivo measurements (0.67 s⁻¹ (6.4%) vs 1.21 s⁻¹ (13.2%)) and in vitro measurements (1.8 s⁻¹ (13.4%) vs 2.7 s⁻¹ (17.7%)). This further confirms that our model has better performance.

We also attempted to predict single-echo based blood T₂ measurements. As clear from Jensen and Chandra’s work³² and the exchange theories and as observed by Golay et al.¹² and in line with the model, T₂ values measured by CPMG experiments are always longer than T₂ values measured by single spin echo experiment. To predict T₂ values for single spin echo experiments, we use our model to calculate the magnetization intensity at each TE by setting TE = τ_cp for a range of 10 – 200 ms, followed by fitting the single-echo based T₂ from these intensities. As shown in Fig. 11 (Supplementary Table S13), however, these predicted R_2blood values are still smaller (T_2blood longer) by a systematic offset of a few Hz than the values reported previously in the literature^12,15,18,38. Several phenomena could explain such a deviation. One possibility is that the spin dephasing by background field gradients (either due to unidirectional flow or diffusion effects) cannot be effectively refocused in the single spin echo and induce an extra relaxation contribution¹²¹. We also performed a comparison with the JC model for these single echo-based measurements, also shown in Supplementary Table S13. Again our general model showed a better averaged relative error (6.3 s⁻¹ (24%) vs 9.4 s⁻¹ (35%)), confirming the CPMG-based comparison above.

Figure 11.

Comparison of our predicted single-echo (SE) based R_2blood values with the literature values measured for human and bovine blood (Table S13 in Supplementary)

The work here should also be considered in the context of several limitations. First, the model we built considers the compartmental exchange and diffusion as two separated processes based on the assumption that water diffusion is much faster than water exchange between inside and outside the erythrocyte. However, the diffusional correlation time is not short enough (as shown in Table 3) to clearly separate it from the exchange process. We also could approximate the water motion in the whole blood as a free diffusion process which implies that the membrane is fully permeable for water diffusion, and use current diffusion theories such as the JC diffusion model to calculate blood R₂. However, the capacity of the erythrocyte membrane to regulate water transport could cause the larger deviation for the JC model found in our comparisons above. A recent study shows that such a restricted diffusion by the permeable membrane can be described by the disorder-averaged diffusion propagator using a scattering approach¹²². However, further study is still needed to derive an analytical equation to describe the R₂ relaxation contribution from this restricted diffusion. Meanwhile, the derivation of the analytical equation (Eq. 41 in Ref 32) used to calculate whole blood R₂ in the JC model inherently implies that the erythrocyte is impermeable (shown in Supplementary S14). Therefore, we used this equation to calculate the diffusion contribution in the plasma. Furthermore, in Reference 32, after fitting the JC model to red blood cells, the data were interpreted invoking the “random sphere” assumption (Eq. 41 in Ref 32), which inherently implies that the whole blood R₂ is proportional to Hct, in conflict with the experimental observation³⁶. Several studies^64,65 have justified this by adding a multiplication factor (1-Hct), but further investigation is still needed to establish this justification as shown in supplementary S15. Therefore, although the diffusion-based theory^{32,33,64-68,122} to describe R₂ in biological tissue has undergone significant developments in recent years, a perfect analytical model is still missing and we here addressed the need to build a working theory for the blood R_2. In this theory, the continuous water diffusion and jump-like water exchange were considered to be non-interfering processes based on the fact that the exchange correlational time is three times longer than the diffusion correlational time. The isolated diffusion contribution was then calculated based on the classic JC model³² and incorporated into the general exchange model^69,70 to calculate the whole blood R₂.

A second limitation of our general model is that the JC model terms we used for diffusion are based on a weak field approximation and as a consequence, the accumulated phase by water spins at the time scale of diffusion should be much smaller than 1. If we assume the frequency induced by the erythrocyte to be at the scale of $\frac{4\pi }{3}\gamma \Delta {\chi }_{ery-plas}{B}_0$, the accumulated phase^64,66 α during the diffusion correlation time τ_c is $\frac{4\pi }{3}{\tau }_c\gamma \Delta {\chi }_{ery-plas}{B}_0$. As shown in Supplementary Fig. S16, all our fitting samples have α values ranging from 0 – 0.6, and thus there is higher probability of large fitting errors when increasing α. Third, in our fitting, we aimed to minimize the total relative error ∣R_2.exp-R_2,fitted∣/R_2,exp of all the samples using the nonlinear fitting function from MATLAB, which has the risk to accidentally end in a local minimum. As shown in the Methods, we used 126 initial values sampled in the physiological ranges of fitting parameters to alleviate this process. Fourth, the oxygenation of all our fitting samples was higher than 0.55. Further work should be done for lower oxygenation cases.

Conclusion

We derived a comprehensive general model for the effects of hemoglobin concentration, hematocrit fraction, oxygenation fraction and albumin concentration on the water proton transverse relaxation time of blood, and determined the relaxation constants and relevant diffusional and exchange correlation times in the equations using previously published whole blood R₂ data at multiple Hct and Y values at 3T, 7T, 9.4T and 11.7T. The fitted parameters presented here provided a good calibration for calculating CPMG based human blood T₂ as a function of field strength, as demonstrated by excellent agreement with human blood T₂ measured in vivo and in vitro in other studies. A website was made available to use this theory for calculating blood T₂ (Blood T1 T2 Calculation at http://godzilla.kennedykrieger.org/).

Abbreviations:

Hct

hematocrit fraction

Y

oxygenation fraction

τ_cp

inter-echo spacing in a multi-echo Carr Purcell Meiboom Gill (CPMG) experiment

τ_ery

water life times in erythrocyte

τ_D

water diffusional correlation times

c_Hb

hemoglobin concentration in the erythrocyte

c_Alb

albumin concentration in plasma

$n_{Alb}^{ex}$

the number of exchangeable protons in albumin

Δδ_Alb

the averaged chemical shift difference between bulk water protons and exchangeable protons in albumin

k_Alb

the averaged dissociation rate of exchangeable protons in albumin

$n_{Hb}^{ex}$

the number of exchangeable protons in hemoglobin

k_H

the averaged dissociation rate of exchangeable protons in hemoglobin

Δδ_DeoxyHb

the averaged chemical shift difference between bulk water protons and exchangeable protons in the deoxygenated hemoglobin

Δδ_OxyHb

the averaged chemical shift difference between bulk water protons and exchangeable protons in the oxygenated hemoglobin

f_v,water

the bulk water volume fraction

f_ery

the erythrocyte water fraction in whole blood

f_plas = (1 – f_ery)

the plasma water fraction in whole blood

Δχ_deoxy

volume magnetic susceptibility difference between erythrocyte and plasma at fully deoxygenated state

Δχ_ery-plas

volume magnetic susceptibility difference between erythrocyte and plasma

Y^off

the oxygenation value at which the volume magnetic susceptibility difference between erythrocyte and plasma changes from diamagnetic to paramagnetic, i.e. at which the volume magnetic susceptibility of plasma and erythrocyte is equal.

Δω_ery-plas

the chemical shift difference between inside and outside the erythrocyte

P_ery

the fractions of the number of exchangeable protein protons relative to the number of bulk water protons in the erythrocyte

P_plas

the fractions of the number of exchangeable protein protons relative to the number of bulk water protons in the plasma

β_plas

the shape factor that corrects the difference between the field gradient generated by the sphere we assumed in the model and the field gradient generated by the real shape of the erythrocyte

β_Ex

the shape factor that corrects the difference between bulk susceptibility difference and the real chemical shift effect generated by the erythrocyte

R_2blood

whole blood transverse relaxation rate of water

R_2b,Alb

the averaged relaxation rates of exchangeable protons in albumin without exchange

R_2b,OxyHb

the averaged relaxation rates of exchangeable protons in oxygenated hemoglobin without exchange

R_2b,DeoxyHb

the averaged relaxation rates of exchangeable protons in deoxygenated hemoglobin without exchange

R_2mem

the transverse relaxation enhancement of water due to presence of the cell membrane

R_2Hb

the transverse relaxation rate of water in the hemoglobin solution in the cytoplasm of erythrocytes

R_2Alb

the transverse relaxation rate of water in the albumin solution in plasma

R_2D,plas

diffusion-based relaxation enhancement for plasma water

R_2D,ery

diffusion-based relaxation enhancement for erythrocyte water

R_2ery

intracellular water relaxation rate for erythrocytes in blood

R_2plas

extracellular water relaxation rate for plasma in blood

APPENDICES

Appendix 1: General expressions for R₂ based on Carver-Richards chemical exchange model

The general form of the τ_cp dependence of R₂ for exchange between two sites can be obtained from the Bloch equation derived by Woessner¹²³ and Allerhand and Gutowsky⁹¹

$$\frac{d{u}_b}{dt}=-{\omega }_b{v}_b-R_{2b}{u}_b-k_b{u}_b+k_f{u}_f$$ [A1a]

$$\frac{d{u}_f}{dt}=-{\omega }_f{v}_f-R_{2f}{u}_f-k_f{u}_f+k_b{u}_b$$ [A1b]

$$\frac{d{v}_b}{dt}={\omega }_b{u}_b-R_{2b}{u}_b-k_b{u}_b+k_f{u}_f$$ [A1c]

$$\frac{d{v}_f}{dt}={\omega }_f{u}_b-R_{2f}{u}_f-k_f{u}_f+k_b{u}_b$$ [A1d]

in which “b” refers to bound, which we will use here for the exchangeable protein protons in protein solution and erythrocyte cytoplasm water protons in blood. On the other hand, “f“ refers to free, which we will use for water protons in solution in protein solutions and plasma water in blood. Following the derivation by Carver and Richards⁶⁹ and Davis et al.⁷⁰, this Bloch equation can be solved analytically and obtain

$$R_2=\frac{1}{2}\left\{R_{2b}+R_{2f}+k_b+k_f-\frac{1}{{\tau }_{cp}}cos{h}^{-1}[D^{+}\cosh (2\xi )-D^{-}\cos (2\phi )]\right\}$$ [A2a]

$$D^{\pm }=\frac{1}{2}[\pm 1+(\psi +2\Delta {\omega }^2)∕({\psi }^2+{\zeta }^2{)}^{1∕2}]$$ [A2b]

$$k_f=k_b\times P_b∕(1-P_b)$$ [A2c]

$$\xi =\left(\frac{{\tau }_{cp}}{\sqrt{8}}\right){\left[+\psi +({\psi }^2+{\zeta }^2{)}^{\frac{1}{2}}\right]}^{\frac{1}{2}}$$ [A2d]

$$\phi =(\frac{{\tau }_{cp}}{\sqrt{8}}){\left[-\psi +({\psi }^2+{\zeta }^2{)}^{1∕2}\right]}^{1∕2}$$ [A2e]

$$\psi =(R_{2b}-R_{2f}+k_b-k_f{)}^2-\Delta {\omega }^2+4k_b{k}_f$$ [A2f]

$$\zeta =2\Delta \omega (R_{2b}-R_{2f}+k_b-k_f)$$ [A2g]

Notice that k_b and k_f have the relationship that k_b × P_b = k_f × P_f and that P_b + P_f = 1 where P is the proton fraction for the protons of interest in protein solution. We replace this by water fraction (f_i, with i = ery or plas) in the blood. The term Δω is chemical shift difference between the sites in the two compartments.

Appendix 2: Fast exchange versus general exchange requirements

The derivation of Luz-Meiboom model^62,91 assumes that the exchange rate must be much faster than the chemical shift difference between two exchange sites; also the relaxation times at two exchange sites in the absent of the exchange must be of the same order of magnitude.

The modeled developed by Allerhand and Gutowsky⁶³ has expanded the suitable exchange rate range to the range that is comparable or slower than the chemical shift difference between two exchange sites, but it still requires the relaxation times at two exchange sites in the absence of the exchange must be of the same order of magnitude.