Quantitative theory for the longitudinal relaxation time of blood water

Abstract

Purpose

To propose and evaluate a model for the blood water T₁ that takes into account the effects of hematocrit fraction, oxygenation fraction, erythrocyte hemoglobin concentration, methemoglobin fraction and plasma albumin concentration.

Methods

Whole blood and lysed blood T₁ data were acquired at magnetic fields of 3T, 7T, 9.4T and 11.7T using inversion-recovery measurements and a home-built blood circulation system for maintaining physiological conditions. A quantitative model was derived based on multi-variable fitting of this data.

Results

Fitting of the model to the data allowed determination of the different parameters describing the blood water T₁ such as those for the diamagnetic and paramagnetic effects of albumin and hemoglobin, and the contribution of methemoglobin. The model correctly predicts blood T₁ at multiple fields, as verified by comparison with existing literature.

Conclusion

The model provides physical and physiological parameters describing the effects of hematocrit fraction, oxygenation, hemoglobin concentration, methemoglobin fraction and albumin concentration on blood water T₁. It can be used to predict blood T₁ at multiple fields.

Introduction

The longitudinal relaxation rate (R₁ = 1/T₁) of water proton in blood plays an important role in various quantitative MRI applications, such as measuring cerebral blood flow using arterial spin labeling (1) as well as determining the appropriate nulling point for black blood imaging (2) and vascular space occupancy (VASO) MRI, a cerebral blood volume weighted functional MRI approach (3). It is also a valuable diagnostic parameter for staging hematoma (4,5).

Under physiological conditions, blood is composed mainly of plasma and erythrocytes with the erythrocyte volume fraction referred as hematocrit fraction (Hct). The intracellular volume of erythrocyte consists mainly of a high concentration (~5mM tetramer) hemoglobin solution. Regular hemoglobin with Fe²⁺ in the heme is a diamagnetic protein when binding O₂ (HbO₂), but paramagnetic when not coordinated to O₂ (deoxyHb). Methemoglobin (MetHb) with Fe³⁺ in the heme, which is a trace amount in physiological blood but has high concentration in methemoglobinemia (6,7) and aged blood such as hematoma (4), is also paramagnetic with high relaxivity. In terms of water ¹H magnetic properties at a certain magnetic field, blood can be considered as consisting of two water compartments separated by the erythrocyte membrane: (i) the plasma, characterized by a constant water T₁ depending on the albumin concentration, and (ii) the erythrocyte cytoplasm characterized by a water T₁ depending on the hemoglobin concentration in the erythrocyte (mean corpuscular hemoglobin concentration, MCHC), methemoglobin fraction (f_MetHb) and oxygenation fraction (Y). The hemoglobin concentration in the erythrocyte and the albumin concentration in the plasma determine the water fractions in these two compartments.

To date, the longitudinal relaxation rate of water in blood has been measured in vivo and in vitro at multiple magnetic fields, showing a linear dependence of longitudinal relaxation rate R₁ = 1/T₁ on Hct, and Y (8–29). Several calibration equations (14,17,19,20,23,26–29) have been derived as a function of hematocrit fraction (Hct) and oxygenation fraction (Y). In addition to these two parameters, R₁ depends on the hemoglobin concentration in erythrocytes (MCHC), which is generally constant under physiological conditions (30), but may vary under pathological situations. The methemoglobin fraction (f_MetHb) may also greatly affect R₁, but this effect was previously measured only at magnetic fields below 3T (13,31–37). Therefore, we measured a series of lysed and whole blood water T₁ data and derived and evaluated a general theory that describes the effects of Hct, Y, MCHC and f_MetHb on blood water R₁ at higher magnetic fields (≥3T).

METHODS

Sample Preparation

Bovine blood, an analogue of human blood with similar hemoglobin structure (38), erythrocyte permeability (39), MCHC (40) and viscosity (40), was obtained from a local slaughterhouse. Fresh blood (3.5L) was mixed with 300mL 83mM sodium citrate solution to prevent coagulation. The samples were stored at 4°C, and used within 8 days. Samples with different hematocrit fractions were prepared by centrifuging blood samples and separating them into erythrocytes and plasma at 10,000G for 15min at 4°C, and recombining them in appropriate volume ratios. The lysed blood samples with different hemoglobin concentrations were prepared through lysing the whole blood with target hemoglobin concentration by sonication (Branson Ultrasonics, Danbury, CT) on ice for 6min with 45% amplitude and 10s “on”, 5s “off” working cycles. The cell lysate after centrifugation (10,000G, 15min at 4°C) was immediately used for T₁ measurements. For the experiments to determine methemoglobin’s relaxivity, concentrated sodium nitrite solution (23mM) was added into fully oxygenated lysed blood and equilibrated at room temperature for 2 hours to prepare samples with different methemoglobin concentrations (1 mole sodium nitrite can roughly transfer 1 mole hemoglobin to methemoglobin). The phosphate buffered saline (PBS) solution was prepared by diluting 10 times in PBS liquid (Fisher) with deionizing water.

A home-built blood circulation system (14,17,41) was used to prevent sedimentation of erythrocytes, and to control temperature and oxygenation fraction. Temperature was maintained at 37+/−0.3°C for all experiments. Oxygenation was controlled by blowing N₂/O₂ mixture into the mixing chamber in the blood circulation system and measured in the flow system using a fiber-optic pO₂ sensor (Oxford Optronix) placed outside the coil. Whole or lysed blood samples were circulated with a flowing speed of ~30 mL/min (~0.5 cm/s). During experiments, we used a flow-stop-flow pattern, i.e. a flowing blood sample for the first 5–10 seconds, after which the flow was stopped 5–10 seconds before MR signal acquisition and resumed after acquisition. This strategy prevented erythrocyte precipitation and “washout” effects during T₁ measurement. A cylinder sample tube (inner diameter 11 mm) was connected with the circulation system at both ends, and positioned vertically inside the RF coil. Experiments for determination of methemoglobin’s relaxivity and water T₁ in saline solution were performed in a tube not connected to the perfusion system. Samples were equilibrated in the 37. 5°C water bath for 20min, and then quickly put into the magnet for T₁ experiments (5–10min). Sample temperatures after experiments were measured as 36.5–37.0°C.

During the experiment, the O₂ needs to be carefully controlled because the free O₂ dissolved in the blood can decrease T₁ as reported before (14,28,42). In venous blood samples, the pO₂ is tightly related to Y following the oxygen-hemoglobin dissociation curve, but for arterial blood more free O₂ will be present at higher pO₂, leading to a reduction in T₁. This extra relaxation enhancement by free O₂ has been studied in arterial blood, plasma and CSF (14,28,42) and found to depend linearly on R₁ as described by: R1=0.00027xpO2 (mmHg) + 0.2263 in CSF at 1.5T (42), R1 = 0.00041xpO2 (mmHg) + 0.56 in arterial blood at 3T (28) and R1 = 0.00020xpO2 (mmHg) + 0.31 in plasma at 7T (14).. The slopes of these equations represent the pO₂ dependence of R₁. Therefore, a variation of arterial blood pO₂ over the typical physiological range of 95+/−13 mmHg (43) can induce R₁ differences of 0.002–0.004 s⁻¹, which are within the experimental error range. Most of our whole blood data were in this range, but some of the fully oxygenized lysed blood data used for calculating r_1dia,Hb over this range were not. To correct for the pO2 effect, the R₁ of these lysed blood data with pO₂ outside the mentioned range were calibrated to the condition of pO₂=95 mmHg using the equation R_1,cal = R_1,exp – 0.00020 (pO₂ – 95). The pO₂ of the fully lysed blood samples for calculating r_1para,MetHb were not measured, because the samples with different MetHb concentrations for the same magnetic field were prepared by pouring 50mL lysed blood from the same batch and ~1mL sodium nitrite into 50mL tubes and tightly sealed. The experiments at the same magnetic field were finished at nearly the same time, so pO₂ across different samples were nearly the same. Fortunately, the lack of this value will not affect the calculation of r_1para,MetHb, which is the slope of R₁ change with the concentration of MetHb.

A blood analyzer (Radiometer, ABL700) was used to monitor hematocrit fraction (Hct), oxygen partial pressure (pO₂), oxygenation saturation (sO₂), total hemoglobin concentration (ctHb) and methemoglobin fraction (f_MetHb) immediately before and after every experiment. Note that Hct value reported by the blood analyzer is calculated based on total hemoglobin concentration in the blood (ctHb) according to an empirical relationship Hct = 0.00301 × ctHb (g/L) (reference manual of the ABL700 blood analyzer). This corresponds to a hemoglobin concentration in erythrocyte (mean corpuscular hemoglobin concentration, MCHC) of 332 g/L (44). The oxygen saturation (sO₂) is the ratio between oxygenated hemoglobin concentration and total hemoglobin concentration (the sum of oxygenated and deoxygenated hemoglobin concentration). The oxygenation fraction (Y) is defined as sO₂/100%.

MR Methods

The MR experiments were conducted on 3T and 7T whole body scanners (Philips Medical Systems, Best, The Netherlands) as well as on 9.4T, 11.7T spectrometers (Bruker Biospin, Billerica, USA). A conventional inversion recovery sequence was used to measure T₁, with recovery time of 15s after each acquisition to relax the system back to equilibrium and have a consistent initial magnetization. The total TR thus increases with the inversion time (TI), but the initial magnetization is always the same. TI-values used were 0.1s, 0.4s, 0.7s, 1s, 1.4s, 1.9s, 2.5s, 4s, 6s, 9s for 3T and 7T scanners, and 0.1s, 0.2s, 0.4s, 0.8s, 1s, 1.2s, 1.5s, 1.8s, 2.2s, 2.5s, 3s, 4s, 5s, 8s, 12s, 15s, 20s for 9.4T and 11.7T spectrometers. The experiments performed at 3T used a quadrature head coil and the following imaging parameters: FOV=32×32mm², matrix=32×32, single-short EPI, 10ms no-slice-selection hyperbolic secant inversion pulse. A single slice (10mm thick) in the center region of the tube was imaged. A small dephasing gradient (0.6 mT/m) was applied continuously during the inversion delay time to minimize possible radiation damping effects (17,45,46). A similar protocol was applied on the 7T human scanner, as detailed in a previous paper (14). The experiments performed on spectrometers used quadrature volume coils with a 60μs 180° inversion pulse and a small dephasing gradient (0.10 G/cm) during the inversion delay time.

Blood T₁ Model

Here we build up the theory from basic principles to combine the individual components that are needed to describe the contributions of water in the erythrocyte (affected by the diamagnetic and paramagnetic relaxation of hemoglobin and the paramagnetic relaxation of methemoglobin) and water in plasma (affected by albumin and free O₂). The hemoglobin relaxation is affected by the oxygenation fraction Y and the iron oxidative state (Fe²⁺ versus Fe³⁺), while the relative contribution of the two compartments depends on the water fraction (f_water) in each compartment, often approximated using the Hct but more accurately calculated from the hemoglobin and albumin concentrations for erythrocytes and plasma, respectively. The addition of the components depends on the exchange regime for the MRI measurement. A two-compartment “fast exchange” model (47) has been applied to describe blood T₁ measured by inversion recovery experiments because the water exchange rate between plasma and cytoplasm is much faster than difference between the longitudinal relaxation rates between these two compartments (14,48). The baseline equation for the resulting single-compartment rate then is:

$$R_{1,\mathit{blood}}=f_{\mathit{ery},\mathit{water}}\times R_{1,\mathit{ery}}+(1-f_{\mathit{ery},\mathit{water}})\times R_{1,\mathit{plasma}}$$ [1]

with f_ery,water the water fraction in the erythrocyte, which can be derived to be (14,49,50)

$$f_{\mathit{ery},\mathit{water}}=\frac{0.70\times \mathit{Hct}}{0.70\times \mathit{Hct}+0.95\times (1-\mathit{Hct})}$$ [2]

based on the fact that hemoglobin occupies about 30% of the volume (51) in the erythrocyte and water volume fraction in the plasma is about 95% (49,50) mainly due to the high concentration of albumin. R_1,plasma is the longitudinal relaxation rate of water in plasma, and R_1,ery is the water longitudinal relaxation rate of water in the erythrocyte.

Diamagnetic relaxation enhancement

In a protein solution like plasma (albumin solution) and cytoplasm in the erythrocyte (hemoglobin solution), the large protein will slow down the water tumbling and increase the water longitudinal relaxation rate. This diamagnetic effect is a short range effect limited to a thin “hydration shell” (52), where water tumbling is slow and the cross relaxation between water and protein is strong (53–55). Due to the fast exchange between water in this hydration shell and bulk water, the diamagnetic relaxation enhancement (relative to plasma) ΔR_1,dia can be expressed as

$$\mathrm{\Delta }{R}_{1\mathit{dia}}=\mathrm{\Delta }{R}_{1\mathit{dia},\mathit{shell}}\frac{V_{\mathit{shell}}}{V_{\mathit{bulk}}}$$ [3]

ΔR_1dia,shell is the relaxation enhancement (relative to saline solution) of water in the hydration shell. V_shell and V_bulk are the water volumes in the hydration shell and in protein solution. Because the hydration shell is thin and its volume proportional to the protein volume, ΔR_1,dia can be rewritten as

$$\begin{array}{l}\mathrm{\Delta }{R}_{1\mathit{dia}}=\mathrm{\Delta }{R}_{1\mathit{dia},\mathit{shell}}\frac{a{V}_{\mathit{Alb}}}{V_{\mathit{bulk}}}=\mathrm{\Delta }{R}_{1\mathit{dia},\mathit{shell}}\frac{a\times m_{\mathit{protein}}/{\rho }_{\mathit{protein}}}{m_{\mathit{water}}/{\rho }_{\mathit{water}}}\\ =\left[\mathrm{\Delta }{R}_{1\mathit{dia},\mathit{shell}}\times a\times \frac{{\rho }_{\mathit{water}}}{{\rho }_{\mathit{protein}}}\times M_{\mathit{protein}}\right]\frac{n_{\mathit{protein}}}{m_{\mathit{water}}}\end{array}$$ [4]

where a is the volume ratio between the hydration shell and protein molecule, and m_protein, V_protein, ρ_protein, n_protein, M_protein are the mass, volume, density, number of moles and molecular weight of the protein, m_water is the water mass in the protein solution. Therefore, the diamagnetic relaxation enhancement of protein can be expressed as

$$\mathrm{\Delta }{R}_{1\mathit{dia}}=r_{1\mathit{dia}}\times b_{\mathit{protein}}$$ [5]

where $r_{1\mathit{dia}}=\mathrm{\Delta }{R}_{1\mathit{dia},\mathit{shell}}\times a\times \frac{{\rho }_{\mathit{water}}}{{\rho }_{\mathit{protein}}}\times M_{\mathit{protein}}$ and b_protein = n_protein/m _protein are the diamagnetic relaxivity and molality of the protein, respectively. The molality can be related to the mass concentration of protein (c_protein g/L) via:

$$b_{\mathit{protein}}=\frac{n_{\mathit{protein}}}{m_{\mathit{water}}}=\frac{m_{\mathit{protein}}/M_{\mathit{protein}}}{m_{\mathit{water}}}=\frac{m_{\mathit{protein}}/V}{(m-m_{\mathit{protein}})/V}\times \frac{1}{M_{\mathit{protein}}}=\frac{c_{\mathit{protein}}}{\rho -c_{\mathit{protein}}}\times \frac{1}{M_{\mathit{protein}}}$$ [6]

with the solution volume V, the solution mass m and the solution density ρ. The protein in plasma is mainly albumin (56), therefore the longitudinal relaxation of water in plasma can be written as

$$R_{1\mathit{plasma}}=R_{1\mathit{saline}}+r_{1\mathit{dia},\mathit{Alb}}\times b_{\mathit{Alb}}$$ [7]

in which R_1saline is the water longitudinal relaxation rate in saline solution and r_1dia,Alb is the diamagnetic relaxivity of water due to albumin. According to Eq. [6], the molality concentration of albumin b_Alb can be calculated via:

$$b_{\mathit{Alb}}=\frac{c_{\mathit{Alb}}(g/L)}{[(0.501\times c_{\mathit{Alb}}+1000)-c_{\mathit{Alb}}](g/L)}\times \frac{1}{66.5kg/\mathit{mol}}$$ [8]

The coefficient of 0.501 in the plasma density ρ_plasma was obtained from linear extrapolation between the density with the protein (1025g/L) (57) and the water density (i.e. zero protein concentration). This can be done because the protein solution density has a linear dependence on the protein concentration (58) and the albumin concentration is known: c_Alb=0.75mM×66.5kg/mol=49.9g/L (14).

The diamagnetic relaxation enhancement (relative to saline water) in the erythrocyte is mainly attributed to the hemoglobin, and can be expressed as

$$\mathrm{\Delta }{R}_{1\mathit{dia},Hb}=r_{1\mathit{dia},Hb}\times b_{Hb}$$ [9]

r_1dia,Hb is the diamagnetic relaxivity of hemoglobin. The molality b_Hb can be related to the concentration of hemoglobin (c_Hb) via

$$b_{Hb}=\frac{c_{Hb}(g/L)}{[(0.377\times c_{Hb}+1000)-c_{Hb}](g/L)}\times \frac{1}{64.5kg/\mathit{mol}}$$ [10]

The factor 0.377 in the cytoplasm density ρ_cytoplasm was determined from linear extrapolation between solution and pure water, similar to albumin above but now using the erythrocyte cytoplasm density of 1125g/L (44,59) and the erythrocyte hemoglobin concentration of 332g/L. Notice that the overall structures of oxyhemoglobin, deoxyhemoglobin and methemoglobin are comparable under the same physiological conditions (temperature, pH, etc), causing these three types of hemoglobin to share the same diamagnetic relaxivity r_1dia,Hb.

Paramagnetic relaxation enhancement

Similar to the diamagnetic effect, the paramagnetic relaxation enhancements of methemoglobin and deoxyhemoglobin, which also depend linearly on their concentration, can be expressed as

$$\mathrm{\Delta }{R}_{1\mathit{para},\mathit{MetHb}}=r_{1\mathit{para},\mathit{MetHb}}\times b_{\mathit{MetHb}}$$ [11a]

$$\mathrm{\Delta }{R}_{1\mathit{para},\mathit{deoxyHb}}=r_{1\mathit{para},\mathit{deoxyHb}}\times b_{\mathit{deoxyHb}}$$ [11b]

where r_1para,MetHb and r_{1para,deoxyHb} are the paramagnetic relaxivities of methemoglobin and deoxyhemoglobin respectively. b_MetHb and b_deoxyHb are the molalities of methemoglobin and deoxyhemoglobin in the erythrocyte, that can be related to the total molality of hemoglobin b_Hb through

$$b_{\mathit{MetHb}}=b_{Hb}\times f_{\mathit{MetHb}}$$ [12a]

$$b_{\mathit{deoxyHb}}=b_{Hb}\times [1-(1-f_{\mathit{MetHb}})\times Y-f_{\mathit{MetHb}}]=b_{Hb}\times (1-f_{\mathit{MetHb}})\times (1-Y)$$ [12b]

Under physiological conditions, the hemoglobin protein is a tetramer (60). While multiple possible tetramers can exist (e.g. binding one, two, three or four O₂), it can be calculated from the binding constants that hemoglobin exists predominantly as a mixture of the fully oxygenated state (all four O₂ binding sites occupied) and the fully deoxygenated state in the normal oxygenation range (61–64). Thus, thanks to this well-known cooperative binding process, one relaxivity parameter (r_{1para,deoxyHb}) is sufficient to approximate the paramagnetic effect of deoxy-hemoglobin. Therefore the water R₁ in erythrocyte can be expressed as

$$R_{1,\mathit{ery}}=b_{Hb}\times [r_{1\mathit{dia},Hb}+r_{1\mathit{para},\mathit{deoxyHb}}\times (1-f_{\mathit{MetHb}})\times (1-Y)+r_{1\mathit{para},\mathit{MetHb}}\times f_{\mathit{MetHb}}]+R_{1\mathit{saline}}$$ [13]

and the blood water R₁ can be written as

$$\begin{array}{l}{R}_{1,\mathit{blood}}=\\ f_{\mathit{ery},\mathit{water}}\times \{b_{Hb}\times [r_{1\mathit{dia},Hb}+r_{1\mathit{para},\mathit{deoxyHb}}\times (1-f_{\mathit{MetHb}})\times (1-Y)+r_{1\mathit{para},\mathit{MetHb}}\times \\ f_{\mathit{MetHb}}]+R_{1\mathit{saline}}\}+(1-f_{\mathit{ery},\mathit{water}})\times [R_{1\mathit{saline}}+r_{1\mathit{dia},\mathit{Alb}}\times b_{\mathit{Alb}}]\end{array}$$ [14]

Data Fitting

Averages of the magnitude intensities at the center of the slice profiles (scanners) and the integrals of the water peaks (spectrometers) were fitted as a function of inversion time (TI) using the following model valid under conditions of TR > 5T₁:

$$S(TI)=\mid S_0(1-C{e}^{-TI/T_1})\mid$$ [15]

The equilibrium signal S₀, the inversion efficiency C and blood ¹H₂O R₁ (1/T₁) were fitted using a nonlinear-least squares algorithm. The errors of fitted R₁ were calculated as 95% confidence interval in the fitting.

In whole blood, multiple T₁ data at 3T, 7T, 9.4T, 11.7T, at various Y (0.48 – 1.00), Hct (0.31 – 0.75) and f_MetHb (0.016–0.028) were fitted using Eq. [14]. The blood sample Hct, Y and f_MetHb were read directly from the blood analyzer. Albumin’s concentration was set as 0.75 mM (14) or 0.77 mmol/kg (Eq. [6]). The hemoglobin molalities b_Hb were calculated to be 6.49 mmol/kg (Eq. [10]) assuming that the hemoglobin concentration (c_Hb) in erythrocytes was the normal MCHC of 332 g/L (34,35). The water fractions in erythrocyte f_ery,water were calculated from Hct according to Eq. [2]. The r_1dia,Alb, r_1dia,Hb and r_1para,MetHb obtained from the linear fits of plasma and lysed blood samples were directly used in the fitting. A Matlab (Mathworks, Natick, MA) function fmincon was used to minimize the relative error |R_1,exp – R_1,fitted|/R_1,exp.

Results

The paramagnetic effect of methemoglobin was studied by measuring R₁ of fully oxygenated lysed blood samples with different methemoglobin fractions and the same total hemoglobin concentration (Figure 1). Because the diamagnetic effects from normal hemoglobin and methemoglobin are the same, r_1para,MetHb could be directly derived from the slope as listed in Table 1.

Figure 1.

a) The dependence of lysed blood R₁ values on the methemoglobin’s concentration. b) Linear fitting of lysed blood R₁ values as a function of methemoglobin’s concentration under fully oxygenated conditions.

Table 1.

Fitted paramagnetic relaxivities of methemoglobin and deoxyhemoglobin, diamagnetic relaxivities of albumin and hemoglobin, and relaxation rate of saline, with error indicating 95% confidence interval of fitting.

B0 (T)	3.0	7.0	9.4	11.7
r1para,MetHb/103 (mol/kg×s) −1	1.26+/−0.13	1.08+/−0.30	0.854+/−0.13	0.672+/−0.040
r1dia,Alb (mol/kg×s) −1	172+/−9	114+/−13	114+/−8	96.2+/−5.2
r1dia,Hb (mol/kg×s) −1	62.9+/−2.4	45.1+/−9.1	47.2+/−5.4	43.3+/−9.9
r1para,deoxyHb (mol/kg×s) −1	65.6+/−8.8	56.2+/−8.1	64.9+/−8.9	58.2+/−5.0
R1,saline (s−1)	0.287+/−0.009	0.263+/−0.007	0.232+/−0.007	0.235+/−0.004

Combining water R₁ data in plasma and saline solution at 37 °C (Table 2), the diamagnetic relaxivity of albumin at c_Alb = 0.75 mM (14) was calculated as

Table 2.

Measured longitudinal relaxation rates of plasma and saline solution at 37°C with error indicating 95% confidence interval in the fitting of inversion-recovery curves.

B0 (T)	3.0	7.0	9.4	11.7
R1plasma (s−1)	0.383+/−0.003	0.328+/−0.005	0.318+/−0.003	0.296+/−0.003
R1saline (s−1)	0.251+/−0.004	0.240+/−0.005	0.229+/−0.003	0.222+/−0.001

$$r_{1\mathit{dia},\mathit{Alb}}=\frac{R_{1\mathit{plasma}}-R_{1\mathit{saline}}}{b_{\mathit{Alb}}}$$ [16]

and listed in Table 1.

The diamagnetic effect of hemoglobin r_1dia,Hb was studied by measuring R₁ of fully oxygenated lysed blood samples with different hemoglobin concentrations (Figure 2). Because the lysed blood samples with targeted hemoglobin concentrations were prepared by mixing plasma, albumin concentrations in different samples were different. Moreover, the methemoglobin concentrations in different samples were different. Therefore, the measured R₁ should be corrected by removing contributions of albumin’s diamagnetic relaxation and methemoglobin’s paramagnetic relaxation following Eq. [17].

Figure 2.

a) The dependence of lysed blood R₁ values on the hemoglobin’s concentration. b) Linear fitting of the corrected lysed blood relaxation rate ΔR₁ as a function of hemoglobin’s molality concentration at fully oxygenated conditions.

$$\mathrm{\Delta }{R}_1=R_1-r_{1\mathit{para},\mathit{MetHb}}\times f_{\mathit{MetHb}}\times b_{Hb}-r_{1\mathit{dia},\mathit{Alb}}\times b_{\mathit{Alb}}$$ [17]

Fig. 2b shows the linear fitting of this corrected lysed blood ΔR₁ (Table 3) as a function of hemoglobin concentration. The r_1dia,Hb was derived from the slope (Table 4).

Table 3.

Measured (R₁) and corrected (ΔR₁) lysed blood relaxation rates at fully oxygenated conditions.

B0 (T)	bHb (mM/kg)	fMetHb	pO2(mmHg)	bAlb (mM/kg)	R1 (s−1)	ΔR1 (s−1)
3.0	6.66	0.024	111	0.00	0.873	0.672
	4.20	0.022	97	0.27	0.680	0.517
	2.03	0.028	144	0.53	0.547	0.385
	0.00	0	95	0.77	0.383	0.251
	6.44	0.026		0.01	0.827
	6.44	0.041		0.01	0.932
	6.44	0.057		0.01	1.056
	6.44	0.070		0.01	1.170
	6.44	0.082		0.01	1.282
7.0	4.58	0.022	350	0.23	0.617	0.482
	2.59	0.021	288	0.46	0.481	0.369
	1.26	0.021	268	0.62	0.407	0.307
	0.00	0	167	0.77	0.328	0.240
	6.36	0.040		0.01	0.747
	6.36	0.053		0.01	0.815
	6.36	0.066		0.01	0.912
	6.36	0.077		0.01	1.000
9.4	6.72	0.016	102	0.00	0.638	0.546
	4.54	0.015	105	0.23	0.514	0.429
	2.70	0.020	104	0.45	0.437	0.339
	1.23	0.020	103	0.62	0.371	0.279
	0.00	0	106	0.77	0.317	0.229
	5.58	0.027		0.10	0.638
	5.58	0.065		0.10	0.799
	5.58	0.102		0.10	1.003
	5.58	0.137		0.10	1.153
11.7	4.18	0.021	113	0.27	0.487	0.402
	2.53	0.022	114	0.47	0.418	0.335
	1.02	0.022	139	0.65	0.340	0.263
	0.00	0	98	0.77	0.296	0.222
	5.84	0.029		0.07	0.567
	5.84	0.045		0.07	0.627
	5.84	0.063		0.07	0.700
	5.84	0.081		0.07	0.765
	5.84	0.096		0.07	0.832

Table 4.

Measured relaxation rates of blood water at different fields, Hct, oxygenation fraction, metHb fraction and pO2

3T					7T					9.4T					11.7T
Hct	Y	fMetHb	pO2	R1(s)	Hct	Y	fMetHb	pO2	R1(s)	Hct	Y	fMetHb	pO2	R1(s)	Hct	Y	fMetHb	pO2	R1(s)
0.32	0.80	0.015	47	0.525	0.31	0.88	0.019	58	0.429	0.30	0.98	0.017	85	0.400	0.31	1.00	0.018	108	0.385
	0.68	0.015	44	0.531		0.74	0.019	42	0.436		0.89	0.017	57	0.408		0.88	0.017	50	0.393
	0.59	0.013	37	0.550		0.62	0.014	31	0.447		0.77	0.016	40	0.411		0.88	0.017	50	0.391
	0.59	0.013	40	0.549		0.51	0.015	29	0.458		0.61	0.016	35	0.423		0.75	0.019	88	0.408
0.45	0.98	0.019	109	0.589	0.45	0.79	0.017	45	0.487		0.53	0.016	31	0.440		0.59	0.018	34	0.434
	0.96	0.018	86	0.590		0.75	0.018	45	0.501	0.39	0.99	0.019	90	0.426	0.38	1.00	0.018	99	0.395
	0.81	0.017	74	0.615		0.61	0.014	31	0.509		0.80	0.016	43	0.440		0.83	0.016	48	0.405
	0.77	0.016	48	0.618		0.50	0.013	29	0.515		0.75	0.015	40	0.444		0.72	0.014	40	0.420
	0.72	0.014	45	0.615	0.56	0.80	0.017	47	0.545		0.66	0.014	34	0.453	0.39	1.00	0.018	106	0.403
	0.72	0.014	42	0.617		0.68	0.015	35	0.561		0.57	0.013	31	0.461		0.99	0.018	80	0.399
	0.63	0.014	42	0.639		0.58	0.014	30	0.577	0.44	0.99	0.017	88	0.443		0.90	0.017	53	0.409
	0.52	0.012	38	0.648		0.45	0.013	23	0.597		0.84	0.017	49	0.454		0.82	0.013	44	0.415
0.56	0.98	0.019	34	0.627	0.63	0.87	0.018	51	0.569		0.82	0.017	50	0.455		0.72	0.015	40	0.438
	0.81	0.018	93	0.661		0.76	0.017	38	0.579		0.75	0.016	41	0.470		0.61	0.014	36	0.451
	0.8	0.017	52	0.668		0.63	0.016	31	0.599		0.65	0.015	39	0.489		0.50	0.013	33	0.461
	0.71	0.016	49	0.679		0.50	0.015	28	0.607		0.55	0.013	32	0.504		0.51	0.013	32	0.452
	0.62	0.014	43	0.685	0.75	0.86	0.019	55	0.618	0.53	0.98	0.018	83	0.469	0.44	0.99	0.017	98	0.414
	0.54	0.013	38	0.693		0.71	0.017	37	0.651		0.93	0.018	63	0.476		0.98	0.017	94	0.416
0.67	0.97	0.019	33	0.685		0.63	0.016	31	0.673		0.80	0.016	44	0.497		0.95	0.017	91	0.421
	0.9	0.019	84	0.699		0.50	0.014	27	0.694		0.70	0.014	39	0.505		0.86	0.016	57	0.424
	0.76	0.016	68	0.700							0.63	0.014	38	0.517		0.74	0.015	50	0.440
	0.71	0.016	52	0.737							0.56	0.013	35	0.530		0.69	0.014	47	0.450
	0.62	0.014	49	0.740						0.64	0.98	0.017	81	0.518		0.64	0.014	44	0.456
	0.53	0.014	41	0.784							0.97	0.016	74	0.519		0.55	0.013	38	0.464
	0.49	0.013	39	0.763							0.82	0.016	49	0.554		0.54	0.013	38	0.455
											0.71	0.014	41	0.574	0.53	1.00	0.018	94	0.433
											0.63	0.014	38	0.602		0.98	0.018	93	0.436
											0.51	0.012	34	0.639		0.86	0.016	56	0.454
										0.75	0.98	0.017	87	0.54		0.81	0.015	44	0.466
											0.98	0.017	87	0.541		0.69	0.014	38	0.487
											0.79	0.016	52	0.598		0.67	0.013	36	0.505
											0.74	0.016	50	0.609		0.60	0.014	36	0.499
											0.65	0.015	41	0.628		0.53	0.012	32	0.506
																0.51	0.012	31	0.511
															0.64	0.99	0.017	89	0.468
																0.79	0.016	43	0.506
																0.69	0.013	36	0.531
																0.53	0.012	29	0.538
															0.71	0.98	0.017	78	0.483
																0.85	0.016	50	0.513
																0.78	0.017	46	0.532
																0.67	0.015	39	0.563

To verify our model for whole blood, multiple whole blood R₁ data (Table 4) at 3T, 7T, 9.4T, 11.7T, at various Y (0.48 – 1.00), Hct (0.31 – 0.75) and f_MetHb (0.016–0.028) were fitted using Eq. [14]. Figure 3 shows a summary of the fitting accuracy, indicating that our model fitted the experimental data well, with average relative error (R₁-R_1,fit)/R₁ of 1.4%. The fitted parameters r_{1para,deoxyHb} and R_1saline are listed in Table 1, and the fitted R_1saline agreed with the R_1saline measured separately in experiments (Table 2).

Figure 3.

Histogram summarizing the relative errors between the experimental whole blood R₁ and fitted R₁.

Fig. 4 shows the magnetic field dependence of the diamagnetic relaxivity of hemoglobin (Fig. 4a), diamagnetic relaxivity of albumin (Fig. 4b), paramagnetic relaxivity of deoxyhemoglobin (Fig. 4c) and paramagnetic relaxivity of methemoglobin (Fig. 4d), and the water R₁ of saline solution (Fig. 4e). The diamagnetic relaxivities of albumin and hemoglobin were fitted (Figs 4a,b) using the following empirical equation proposed by Lindstrom and Koenig (16).

Figure 4.

The magnetic field dependence of r_1dia,Hb (a), r_1dia,Alb (b), r_{1para,deoxyHb} (c) r_1para,MetHb (d) and R_1saline (e), and numerical fitting using Freed’s equation (65,66) (paramagnetic relaxivity, solid line) and Lindstrom and Koenig’s Equation (16) (diamagnetic relaxivity and R_1saline, solid line)

$$r_{1,\mathit{dia}}=P_1\times \frac{P_2}{1+{({\gamma }_H{B}_0\times P_2)}^{P_3}}+P_4$$ [18]

with the ¹H gyromagnetic ratio γ_H and empirical constants P₁ = 9.45×10⁹ s⁻²×kg/mol, P₂ = 2.88×10⁻⁴ s, P₃ = 0.822 and P₄ = 64.9 (s×mol/kg) ⁻¹ for r_1dia,Alb and P₁ = 4.22×10¹⁵ (s⁻²×kg/mol), P₂ = 4.55×10⁻⁴ s, P₃ = 1.97 and P₄ = 42.9 (s×mol/kg) ⁻¹ for r_1dia,Hb. The water T₁ in saline solution R_1saline also can be predicted using Eq. [18] with P₁ = 3.88×10⁹ s⁻², P₂ = 6.68×10⁻¹¹ s, P₃ = 1.11 and P₄ = 2.41×10⁻³ s⁻¹. The paramagnetic relaxivities of deoxyhemoglobin and methemoglobin were fitted (Figs 4c,d) using the outer sphere model of Freed (65,66) which described the water molecules diffusing around the paramagnetic complex:

$$\begin{array}{l}{r}_{1,\mathit{para}}=P_1\times \{3j({\omega }_H)+7j({\omega }_S)\}\\ j(\omega )=Re\left\{\frac{1+\frac{1}{4}{(i\omega \tau +\tau /{\tau }_S)}^{1/2}}{1+{(i\omega \tau +\tau /{\tau }_S)}^{1/2}+\frac{4}{9}{(i\omega \tau +\tau /{\tau }_S)}^{1/2}+\frac{1}{9}{(i\omega \tau +\tau /{\tau }_S)}^{1/2}}\right\}\end{array}$$ [19]

with the proton and electron Larmor frequencies ω_H and ω_S, and empirical constants P₁ = 6.40×10⁶ (s×mol/kg) ⁻¹, τ = 9.72×10⁻⁶ s, τ_S = 3.07×10⁻¹⁰ s for r_1para,MetHb, and P₁ = 8.55×10³ (s×mol/kg) ⁻¹, τ = 8.68×10⁻⁹ s, τ_S = 8.97×10⁻¹² s for r_{1para,deoxyHb}.

Discussion

Lysed and whole blood experimental R₁ data were used to validate a simple longitudinal relaxation model for blood water and to evaluate the paramagnetic and diamagnetic effects of albumin, hemoglobin and methemoglobin at multiple fields. For the diamagnetic relaxivities of hemoglobin, previous studies (8,10,14,17,20,21,23,26–28) have quantitatively described the diamagnetic relaxation enhancement of hemoglobin as Hct dependence using the empirical equation R₁=A×Hct+B. In our model, this Hct dependent coefficient A can be approximated by the product of the diamagnetic relaxivity r_1dia,Hb and the hemoglobin concentration in the erythrocyte (b_Hb = 6.49mmol/kg), which is 0.41 s⁻¹ at 3T and 0.29 s⁻¹ at 7T. This is comparable to the reported A value as ~0.5 s⁻¹ at 3T (17,20) and 0.34 s⁻¹ at 7T (19) in the previous human or bovine blood studies. As shown in Fig 4, the diamagnetic longitudinal relaxivities of albumin and hemoglobin decrease as magnetic field increases, which follows the trend observed by Lindstrom and Koenig (16) in the CO bound hemoglobin solution. Therefore, the equation from Lindstrom and Koenig (16) (Eq. [18]) was used to fit r_1dia,Alb and r_1dia,Hb in the Fig 4. Note that we have only limited number of field data for this multiple parameter fit, but can be used to numerically extrapolate and estimate diamagnetic relaxivity values at unknown magnetic fields around our measured magnetic field range.

Our paramagnetic relaxivities of deoxyhemoglobin r_{1para,deoxyHb} (per hemoglobin complex) at 3T (65.6 (mol/kg×s) ⁻¹) and 7T (56.2 (mol/kg×s) ⁻¹) agree reasonably with r_{1para,deoxyHb} reported by Blockley et al. (10) as 14.6×4 (mol/L×s) ⁻¹ per hemoglobin at 3T and 10.6×4 (mol/L×s) ⁻¹ per hemoglobin at 7T. Our paramagnetic relaxivities of methemoglobin r_1para,MetHb (per hemoglobin complex) are also comparable to the r_1para,MetHb of 175×4 (M×s) ⁻¹ per hemoglobin measured at 220MHz (67). At the same time, the paramagnetic relaxivity of methemoglobin r_1para,MetHb decreases with increasing magnetic field following the NMR relaxation dispersion curve measured by Koenig et al. (68), while the paramagnetic relaxivities of deoxyhemoglobin r_{1para,deoxyHb} have weak magnetic field dependence which was also observed by Blockley et al. (10). Although initial studies about the paramagnetic relaxation in methemoglobin attributed the relaxation enhancement to the “inner sphere” (67,69) mechanism, i.e. only water ligated with heme relaxes fast, and then exchange with bulk water. However, an extensive investigation of proton relaxation dependence on temperature (70) had shown that the exchange rate between ligated water and bulk water was not fast enough to induce the fast relaxation, and Koenig et al. (36) measured the relaxation dispersion curve at multiple magnetic fields and proved the “outer sphere” mechanism, i.e. the fluctuation of the dipolar coupling between water and paramagnetic protein due to water diffusion, dominated the paramagnetic relaxation. Therefore, Freed’s diffusion model (65,66) which described water molecules diffusing around the paramagnetic complex was used to fit the field dependence of the paramagnetic relaxivities of methemoglobin and deoxyhemoglobin in Fig 4. Similar to the fitting of diamagnetic relaxivity dependence on the magnetic field, the fit of Eq. [19] should be considered as a numerical extrapolation due to the limited number of experimental data.

The good fitting of our model in terms of the difference between the experimental and predicted T₁ values over a large number of blood data (Fig. 3) supports our model and thus should allow the fitted relaxivities at multiple fields (3.0 T, 7.0 T, 9.4 T and 11.7 T) to be used to calculate the blood T₁ for quantitative MRI experiments. Table 5 compares the blood T₁ calculated from our model and the human blood T₁ measured by other groups. They agree well except for a venous blood T₁ measured at 7T by Rooney et al. (29). Therefore, the blood T₁ calculated from our model (Table 6) can be a reference for studies on 9.4T and 11.7T animal scanners as well as on ultra high human field scanners that are now becoming available. In addition to adult blood, the results in Table 5 show that our model is also able to predict blood T₁ in neonates. In neonate blood, the fetal erythrocyte’s cell volume (MCV) is 20% larger than adult erythrocyte (71), and the most of hemoglobin is hemoglobin F, which replaces two beta subunits with two gamma subunits, but has similar overall structure as the adult hemoglobin. This shows the versatility of our model to predict abnormal blood T₁.

Table 5.

Comparison between human blood T₁ measured in vivo and blood T₁ calculated by our model.

		B0	Hct	Exp. T1 (ms)	Cal. T1 (ms)
Adult (20)	Arterial	1.5T	0.47+/−0.03	1499+/−74	1439
Adult Male(25)	Venous	1.5T		1429+/−21	1229–1387
Adult Female(25)	Venous	1.5T		1531+/−29	1317–1462
Adult (20)	Arterial	3T	0.47+/−0.03	1779+/−80	1849
Adult (20)	Venous	3T	0.47+/−0.03	1694+/−77	1668
Adult Male (24)	Venous	3T		1677+/−120	1597–1743
Adult Female (24)	Venous	3T		1769+/−100	1680–1809
Adult Male (25)	Venous	3T		1618+/−30	1597–1743
Adult Female (25)	Venous	3T		1681+/−87	1680–1809
Adult (18)	Venous	3T		1825+/−104	1597–1809
Neonate (22)	Venous	3T	0.423	1757	1726
Sickle Cell Patient	Venous	3T	0.233+/−0.039	1910+/−89	1990
Female (18)
Sickle Cell Patient	Venous	3T	0.235+/−0.029	1938+/−82	1987
Male (18)
Adult Male (25)	Venous	7T		2012+/−119	1945–2116
Adult Female (25)	Venous	7T		2163+/−94	2043–2193
Adult (29)	Venous	7T		2587+−283	1945–2193

In the calculation, all relaxivities and R_1saline were obtained from Table 1. f_MetHb was set as the average value 0.004 (73) in normal adult blood. The oxygenation fraction Y was 1.0 for arterial blood and 0.65 for venous blood, based on published range of 0.55 – 0.75 (74,75). If Hct is not specified, T₁ range was calculated according to the physiological range of 0.41 – 0.53 for male and 0.36 – 0.46 for female (71).

Table 6.

The predicted arterial and venous blood T₁ in the physiological range of health human. The calculation parameters are the same as the calculation in Table 5.

	Hct	Y	T1 (ms)
			3.0 T	7.0 T	9.4 T	11.7 T
Male	0.41–0.53	1.00	1789–1910	2191–2329	2472–2691	2585–2810
Female	0.36–0.46	1.00	1859–1963	2271–2388	2597–2789	2714–2910
Male	0.41–0.53	0.65	1597–1743	1945–2116	2122–2373	2239–2497
Female	0.36–0.46	0.65	1680–1809	2043–2193	2264–2490	2385–2616
Male	0.47	0.55–0.75	1623–1716	1971–2090	2163–2330	2283–2449
Female	0.41	0.55–0.75	1700–1788	2063–2173	2296–2456	2420–2579

Besides predicting blood T₁ values at 3T, 7T, 9.4T, 11.7T, the blood T₁ at other fields can also be predicted from the curves in Fig 4. For example, using Eq. [18], Eq. [19] and our fitted parameters, we can extrapolate the relaxivities at 4.7T as r_1dia,Hb of 51.1 (mol/kg×s) ⁻¹, r_1dia,Alb of 138 (mol/kg×s) ⁻¹, r_{1para,deoxyHb} of 61.7 (mol/kg×s) ⁻¹, r_1para,MetHb of 1.19×10³ (mol/kg×s) ⁻¹ and R_1saline of 0.246 s⁻¹. Meyer et al. (27) measured at 4.7T the oxygenation dependence of bovine blood (Hct=0.4) as R₁ = A × (1-Y) + B, where A=0.12+/−0.03 s⁻¹ was defined as the paramagnetic coefficient of deoxyhemoglobin. According to Eq. [14], this paramagnetic coefficient A can be expressed as f_ery,water×b_Hb×r_{1para,deoxyHb}, which can be calculated as 0.13 s⁻¹ based on our fitted parameters r_{1para,deoxyHb} (61.7 (mol/kg×s) ⁻¹) and agrees well with the result from Meyer et al. (27). At the same time, Dobre et al. (11) measured bovine blood T₁ as 1833+/−49 ms at Hct=0.43+/−0.03 and Y=0.81+/−0.13. Assuming a f_MetHb=2% for the regular bovine blood, we calculated blood T₁ as 2004 ms under the same conditions based on our fitted relaxivities, which is similar to the result from Dobre et al. (11). Extrapolating to relaxation times at 1.5T is less accurate due to the nonlinear steeper field dependence of the diamagnetic albumin and diamagnetic and paramagnetic hemoglobin relaxivities (Figs. 4a-c). When extrapolating relaxivities at 1.5 T, we found r_1dia,Hb = 121 (mol/kg×s) ⁻¹, r_1dia,Alb = 253 (mol/kg×s) ⁻¹, r_{1para,,deoxyHb} = 80.3 (mol/kg×s) ⁻¹, r_1para,MetHb = 1.34×10³ (mol/kg×s) ⁻¹ and R_1saline = 0.257 s⁻¹. Using these numbers to estimate the arterial blood T₁ at Hct = 0.36–0.53 at 1.5 T, we found it to be 1367–1585 ms for a methemoglobin fraction of 0.004. This is comparable with a previous in vivo human study (1499+/−74ms, (20)), in vitro human blood study (1435 ms (9), measured at 23°C) and in vitro bovine blood study (1400 ms, (72)). The venous blood T₁ measured in sagittal sinus (25) is also comparable to our prediction (Table 2).

The blood T₁ predicted from our model can also be used to evaluate the MRI experiments related to blood T₁ such as the VASO experiment to determine the relative cerebral blood volume (CBV) change during activation (3) and the ASL experiment to measure cerebral blood flow (CBF) (77). The quantification in both experiments depends on the accuracy of the blood T₁ value used because the VASO experiment uses blood T₁ to calculate an inversion time (TI) to null the blood signal while blood T₁ is an important parameter in ASL experiment to calculate the label decay. When performing these experiments, the blood T₁ is generally considered to be a constant at certain magnetic field, but as we discussed, even in a healthy human, the blood T₁ could vary in a range due to different Hct, Y (venous blood) and f_MetHb. For example, in healthy people, Hct has a ~10% variation (71), f_MetHb has a ~1% variation (73) while Y in venous blood has a range of 0.55–0.75 (74,75). The blood T₁ could change 100–150ms at 3T for these blood parameters’ variation. Using the typical experiment conditions (76,77), the CBV change during the activation could have a 1%–2% difference based on the equations in (76) while the CBF could have ~20% difference for PCASL and ~10% difference for QUIPSS II PASL based on the simple model in ASL “white paper” (77).

There are several assumptions in our model. The key assumption is that a single R₁ dominates the inversion-recovery process of the blood despite the erythrocyte and plasma compartments having different R₁ and that this single blood R₁ can be simply expressed as the weighted sum of R₁ in plasma and erythrocyte (Eq. 1). According to chemical exchange theory (48), this assumption is valid when

$$k_{\mathit{exch}}\gg \mid R_{1,\mathit{ery}}-R_{1,\mathit{plasma}}\mid$$ [20]

in which k_exch is the water exchange rate between plasma and erythrocyte. Using a lifetime of about 8 ms for water inside the erythrocyte (78), the exchange life time (1/τ_exch= 1/τ_ery + 1/τ_plasma) for this system would be 6.8 ms for hematocrit fraction (Hct) = 0.3 and 2.5 ms for Hct = 0.75, corresponding to k_exch (= 1/τ_exch) values of 147 and 400 Hz, respectively. This exchange rate is much faster than the R₁ difference in erythrocyte and in plasma as measured in this study and many other studies. Note that this fast-exchange condition for R₁ is different from the exchange-narrowing condition for R₂ which requires water exchange rate between plasma and erythrocyte to be much faster than the water chemical shift difference in plasma and in erythrocyte. Even if the peaks for water in plasma and in erythrocyte are separated, they still share the same averaged longitudinal relaxation time as shown in Eq. 1. (48).

Another assumption in our model is that all hemoglobins are homogenous. This assumption is valid for the normal adult blood. But for sickle cell disease, the hemoglobin is mutated as HbS and forms fibers in its deoxy-state (79,80). This different hemoglobin, especially the fibers formed by polymerization could change the water-protein interaction and thus the blood T₁. According to a previous in vivo study, the blood T₁ of patients with sickle cell disease is comparable to normal volunteers and thus the values found with our model (Table 5), but caution needs to be taken when using our model for sickle cell disease until more measurements of blood R1 under this condition confirm the above result.

Conclusion

We derived a comprehensive model for the effects of hemoglobin concentration, hematocrit fraction, oxygenation fraction, methemoglobin fraction and albumin concentration on the water ¹H T₁ in blood, and confirmed the equations experimentally in whole blood samples at 3T, 7T, 9.4T and 11.7T. The fitted parameters presented here provide a good calibration for calculating human blood T₁, as demonstrated by excellent agreement with human blood T_{1 values} measured in vivo by other groups. Our model should be useful to predict blood T₁ for arbitrary hemoglobin concentration, hematocrit fraction, oxygenation, methemoglobin fraction and albumin concentration at multiple fields.