A general model to calculate the spin-lattice (T₁) relaxation time of blood, accounting for haematocrit, oxygen saturation and magnetic field strength

Abstract

Many MRI techniques require prior knowledge of the T1-relaxation time of blood (T_1bl). An assumed/fixed value is often used; however, T_1bl is sensitive to magnetic field (B₀), haematocrit (Hct), and oxygen saturation (Y). We aimed to combine data from previous in vitro measurements into a mathematical model, to estimate T_1bl as a function of B₀, Hct, and Y. The model was shown to predict T_1bl from in vivo studies with a good accuracy (±87 ms). This model allows for improved estimation of T_1bl between 1.5–7.0 T while accounting for variations in Hct and Y, leading to improved accuracy of MRI-derived perfusion measurements.

Introduction

The spin-lattice relaxation time of blood (T_1bl) plays a critical role in a range of MRI applications. In particular, the accuracy of blood flow measurements made using arterial spin labelling (ASL)¹ depends directly on accurate knowledge of T_1bl in a given subject. Black blood angiography, vascular space occupancy imaging, dynamic contrast-enhanced magnetic resonance imaging, cardiac MRI applications, and MR-based temperature monitoring also rely on accurate estimation of this parameter.

A number of factors influence spin-lattice relaxation time (T₁). T₁ increases with Larmor frequency, and therefore magnetic field strength (B₀), which in single phase homogeneous substances can be described using the Bloembergen-Purcell-Pound theory.² In human tissues and blood, the relationship is more complex, as macromolecules such as proteins provide relaxation pathways which speed up T₁ relaxation. One such protein is haemoglobin, which is an oxygen transport protein that resides in erythrocytes (red blood cells), and hence haematocrit (Hct), which describes the volume fraction of erythrocytes in whole blood, will influence T_1bl. Proteins residing in blood plasma, such as albumin and globulin, will similarly effect T_1bl. In addition, paramagnetic materials provide a further T₁ relaxation pathway, and, as deoxyhaemoglobin is weakly paramagnetic, the oxygen saturation fraction of blood (Y) will also influence T₁ relaxation. Lastly, the T₁ relaxation time is temperature-dependent, with longer relaxation times found at higher temperatures.

In general, an assumed value of T_1bl will be used in the MRI techniques described above, particularly in the clinic, where technical and scan time constraints prevent measurement of T_1bl on a patient-by-patient basis. Although many previous studies have determined values of T_1bl empirically (see Table 1), these have generally been performed under differing physiological/experimental conditions (magnetic field strength, blood oxygenation, haematocrit, etc.), often without direct measurement of all the influencing factors. The aim of this study was to bring together the results of previous studies into a general mathematical model for predicting T_1bl, as a function of B₀, Hct, and Y. This will provide estimates of T_1bl values in arterial and venous blood, over the range of magnetic field strengths used in clinical practice (1.5–7.0 T). This should lead to subsequent improvements in the accuracy of ASL-derived blood flow measurements, particularly in pathologies where the constitution of the blood is known to be outside the normal range.

Table 1.

Sources of literature values of R_1bl.

Ref.	B0 (T)	No. data points	Blood source	Age range (years)	Gender	Hct	Y
Stefanovic and Pike6	1.5	8	Human (22℃)	–	–	Measured (0.51 ± 0.004)	Measured (0.42–0.93)
Lu et al.5	3.0	9	Bovine (37℃)	–	–	Measured (0.38–0.46)	Measured (0.69–0.99)
Rane and Gore7	7.0	10	Human (37℃)	–	–	Measured (0–1)	Measured (0.66–0.97)
Grgac et al.3	7.0	13	Bovine (37℃)	–	–	Measured (0–1)	Measured (0.4–1.0)
Dobre et al.8	4.7,7.0	2	Bovine (37℃)	–	–	Measured (0.43)	Measured (0.79–0.81)
Zhang et al.4	1.5, 3.0, 7.0	18	Human in vivo	24–38	M, F	Estimated	Estimated (venous)
Wu et al.9	3.0	8	Human in vivo	7–39	M, F	Estimated	Estimated (venous)
Varela et al.10	3.0	19	Human in vivo	0.4–37	M, F	Measured	Estimated (venous)
De Vis et al.11	3.0	3	Human in vivo	0.05–0.24	M,F	Measured	Estimated (venous)

Note: The range of Hct and Y are shown if these parameters were measured, if not estimated values were used, adjusted for the age and gender of the subject (for Hct), and whether the blood was arterial or venous (for Y).

Materials and methods

Theory

A two-compartment blood model, consisting of erythrocytes and plasma in fast exchange, was previously described in Grgac et al.³ In this model, the longitudinal relaxation rate of whole blood (R_1bl, equivalent to T_1bl⁻¹) is given by

$$R_{1\mathrm{bl}}(\mathrm{Hct},Y)=f_e·R_{1e}(Y)+(1-f_e)·R_{1p}$$ (1)

where Hct is the haematocrit (0–1), Y is the oxygen saturation fraction (0–1), f_e is the fraction of water in whole blood that resides in erythrocytes (0–1), R_1e is the longitudinal relaxation rate of erythrocytes (s⁻¹), and R_1p is the longitudinal relaxation rate of plasma (s⁻¹). Under normal physiological conditions, water volume fraction in erythrocytes is approximately 70%, and water volume fraction in plasma is 94–95%,³ which allows f_e to be expressed as a function of Hct as follows³

$$f_e=\frac{0.70·\mathrm{Hct}}{0.70·\mathrm{Hct}+0.95·(1-\mathrm{Hct})}$$ (2)

In addition, because deoxyhaemoglobin acts as a weak paramagnetic contrast agent, we can express R_1e as a function of Y

$$R_{1e}(Y)=R_{1\mathrm{eox}}+r_{1\mathrm{deoxyHb}}·[\mathrm{Hb}]·(1-Y)$$ (3)

where R_1eox is the longitudinal relaxation rate of erythrocytes when Y = 1 (100% oxygen saturation), [Hb] is the mean corpuscular haemoglobin concentration (5.15 mmol Hb tetramer/L plasma³), and r_1deoxyHb is the molar relaxivity of deoxyhaemoglobin (s⁻¹ L plasma in erythrocyte/mmol Hb tetramer).

When analysing data collected over a range of magnetic field strengths, the possible dependence of R_1eox, R_1p and r_1deoxyHb on B₀ must also be accounted for. The relationship between whole blood R₁ and B₀ between 1.5 and 7.0 T has been shown to be linear,⁴ and as such linear regression terms for the above parameters were substituted into equations (1) and (3) as follows (β₀ represents the ‘intercept’ term, β₁ represents the ‘gradient’ with respect to B₀)

$$R_{1\mathrm{eox}}={\beta }_{0,R1\mathrm{eox}}+{\beta }_{1,R1\mathrm{eox}}·B_0$$ (4)

$$r_{1\mathrm{deoxyHb}}={\beta }_{0,r1\mathrm{deoxyHb}}+{\beta }_{1,r1\mathrm{deoxyHb}}·B_0$$ (5)

$$R_{1p}={\beta }_{0,R1p}+{\beta }_{1,R1p}·B_0$$ (6)

By combining equations (1)–(6), a general model was constructed, with R_1bl as the dependent variable, B₀, Hct and Y as independent variables, and the following as fitted parameters: β_0,R1eox, β_1,R1eox, β_0,r1deoxyHb, β_1,r1deoxyHb, β_0,R1p, β_1,R1p.

Data analysis

All data analysis was performed using Matlab R2014a (MathWorks Inc., Natick, MA, USA), and Matlab’s fminsearch algorithm was used for model fitting. Forty-two literature values of T_1bl,^3,5–8 acquired in vitro between 1.5 and 7.0 T in conjunction with empirically controlled variations in Hct and Y (see Table 1), were fit simultaneously to the model. The T_1bl values in Stefanovic and Pike⁶ were increased by 12% to account for the shift in T_1bl between 22℃ and 37℃.^5,6 Following initial model fitting, a Monte Carlo simulation was performed to determine the uncertainty on the fitted parameters. Here, the pool of N = 42 raw data points were randomly sampled N times (with replacement), and the model was fit to this synthetic raw data set. This process was iterated 1000 times, and the median and 95% confidence interval (CI) of the distribution of fitted values for each parameter were used as the final parameter estimate and its uncertainty, respectively. Parameters for which the 95% CI crossed zero were classified as non-significant, and the process was repeated with these terms set to zero.

Following this, the fitted parameter values were substituted into equations (1) to (6), and the ability of the model to predict values of T_1bl taken from further literature sources^4,9–11 (Table 1), in which measurements were made in vivo, was tested. These comprised 48 values of T_1bl, taken from human studies performed between 1.5 and 7.0 T. In these studies, measurements were made in blood in the sagittal sinus, so normal values of Y = 0.68 (for venous blood) were assumed,¹² and, where not measured, a value of Hct, corrected for age and gender, was estimated based on literature values.^13,14

Results

The Monte Carlo simulation showed that the 95% CI around the fitted value of β_1,r1deoxyHb crossed zero, and as such this parameter was non-significant (i.e. there was no evidence to suggest r_1deoxyHb changes with B₀). With this term set to zero, the following values were obtained for the remaining fitted parameters (95% lower and upper CIs shown in brackets):

β_0,R1eox: 1.10 (0.97, 1.28) s⁻¹, β_1,R1eox: −0.058 (−0.085, −0.038) s⁻¹T⁻¹, β_0,r1deoxyHb: 0.033 (1.8 × 10⁻⁹, 0.056) s⁻¹ L plasma in erythrocyte/mmol Hb tetramer, and β_0,R1p: 0.49 (0.40, 0.57) s⁻¹, β_1,R1p: −0.023 (−0.035, −0.0079) s⁻¹ T⁻¹.

With the above values substituted into equations (1) to (6), the difference between predicted and measured T_1bl values (ΔT_1bl), taken from the in vivo literature sources,^4,9,10 is illustrated in Figure 1(a). Predicted T_1bl values were slightly underestimated compared to in vivo literature values, with a mean ΔT_1bl of −108 ms (predicted minus literature values, mean 6% under-estimation), and a standard deviation in ΔT_1bl of 89 ms. As such, an offset of +108 ms should be added to the model when predicting in vivo T_1b values (see Discussion). The final model, valid between 1.5 and 7.0 T, is therefore

$$R_{1\mathrm{bl}}\text{'}(\mathrm{Hct},Y,B_0)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=f_e·[1.099-(0.057·B_0)+((0.033·[\mathrm{Hb}])·(1-Y))]\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+[(1-f_e)·(0.496-0.023·B_0)]$$

where

$$f_e=(\frac{0.70·\mathrm{Hct}}{0.70·\mathrm{Hct}+0.95·(1-\mathrm{Hct})})$$ (7)

and

$$T_{1\mathrm{bl}}(\mathrm{Hct},Y,B_0)=\frac{1}{R_{1\mathrm{bl}}\text{'}(\mathrm{Hct},Y,B_0)}+\Delta T_{1\mathrm{bl}}$$ (8)

where ΔT_1bl = 0 and 108 ms for in vitro and in vivo studies respectively. Note that if a full blood analysis has been performed, a measured value for [Hb] (mean corpuscular haemoglobin concentration) can be used in equation (7), rather than the assumed value of 5.15 mmol Hb tetramer/L plasma.

Figure 1.

(a) Bland-Altman plot illustrating the difference between modelled and measured values of T_1bl (ΔT_1bl), between 1.5 and 7.0 T. The horizontal dashed lines show the mean value of ΔT_1bl (short dashes) and the limits of agreement (long dashes, mean(ΔT_1bl) ± (1.96xSD(ΔT_1bl)). (b) Variation in modelled values of T_1bl (black line, with grey shading indicating limits of agreement), with literature values overlaid (legend indicates literature source in (a) and (b)). All data in (b) represent in vivo T_1bl values, and as such the ΔT_1bl = 180 ms offset has been added to the modelled values (equation (8)).

Figure 1(b) illustrates the change in both modelled and measured in vivo T_1bl values as a function of Hct, shown here at 3.0 T.

Discussion

After correcting for the ΔT_1bl offset, our model was able to predict literature measurements of in vivo T_1bl with a good degree of accuracy (root mean square error = 87 ms). For example, in Zhang et al.,⁴ the measured T_1bl values in venous blood in the sagittal sinus of healthy subjects (mean age 31 years) at 1.5 T, 3.0 T, and 7.0 T were 1480 ms, 1650 ms and 2088 ms, respectively (males and females combined). The equivalent values predicted using our model were 1565 ms, 1688 ms, and 2147 ms, respectively (data in Zhang et al.⁴ were not used to ‘train’ the model). Our model fitting indicated that both the T₁ of plasma and fully oxygenated erythrocytes increase with B₀, and our results agree with previous findings that T_1bl is highly sensitive to haematocrit, but only weakly dependent on oxygenation.¹⁵ Also, our fitted value for r_1deoxyHb of 0.033 s⁻¹ L plasma in erythrocyte/mmol Hb tetramer lies between the mean fitted values of 0.052 in Grgac et al.³ and 0.012 in Blockley et al.¹⁵ (s⁻¹ L plasma in erythrocyte/mmol Hb tetramer).

The offset between predicted and literature T_1bl values arose from the fact that our model was ‘trained’ using data from in vitro studies (necessary to control the Hct and Y levels in the blood), then used to predict values from in vivo studies. Trisodium citrate was added to the blood in the in vitro studies to prevent coagulation, and higher osmolarity sodium concentrations will draw water out of erythrocytes via osmosis, shortening T₁³. In Grgac et al.,³ the addition of anti-coagulant was estimated to reduce T_1bl by 7%, which agrees very well with the average underestimation of 6% when our model was used to predict literature values acquired in vivo.

Implications for arterial spin labelling

When processing ASL data, an assumed value of T_1bl is used in the conversion of raw signal into cerebral blood flow (CBF) values (see Alsop et al.¹⁶ for details). Certain pathologies may result in a patient having a haematocrit outside the normal range expected for their age/gender, and assuming a normal Hct value in these patients will result in an incorrect estimation of T_1bl, which in turn will lead to an incorrect calculation of CBF. For instance, sickle cell anaemia is a genetic condition in which patients have atypical haemoglobin molecules, resulting in a low number of red blood cells (anaemia). In a severely anaemic patient (say Hct = 0.20), the calculated value of T_1bl in arterial blood (Y = 0.97) using our model would be 2.09 s at 3.0 T. If we assumed a normal Hct value in this subject (say Hct = 0.47 for a 30 year old male), the calculated value of T_1bl drops to 1.70 s. This 18% underestimation of T_1bl, resulting from an assumed normal value for T_1bl rather than a Hct-corrected value, would lead to an overestimation in the calculated CBF of approximately 30%, based on a typical pseudo-continuous ASL acquisition (see equation 1 in Alsop et al.¹⁶).

Conclusions

MRI applications such as ASL are dependent on accurate knowledge of T_1bl, which is rarely measured on a patient-by-patient basis. We have presented a mathematical model which brings together previous empirical measurements of T_1bl, and provides a general tool for estimation of this parameter in individual subjects, adjusted for magnetic field strength, haematocrit and oxygenation of the blood. In healthy subjects, this model can be used with assumed normal values of Hct (corrected for age/gender) and Y, whereas in patients measured Hct and Y values may be required, which will require blood sampling and non-trivial steps for accurate assessment. Following this, the model presented here will account for the influence of a patient’s atypical Hct and Y values on the estimated value of T_1bl. This in turn will have a significant influence on the quantification of CBF using ASL, and considerable errors in CBF quantification will occur if normal values of T_1bl are assumed.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by Great Ormond Street Hospital Children’s Charity.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Authors’ contributions

PWH designed the research, analysed data and wrote the manuscript; CAC, FJK edited the manuscript and provided clinical input. All authors approved the final manuscript.