T1-refBlochi: high resolution 3D post-contrast T1 myocardial mapping based on a single 3D late gadolinium enhancement volume, Bloch equations, and a reference T1

The LGE signal is composed of weightings due to coil sensitivity, proton density (PD) and T2*, and T1-weighting (Fig. 1a). We have developed an analytic expression for the signal from 3D LGE. The LGE sequence is an inversion recovery prepared FLASH sequence. During each RR interval, the sequence applies a number of small flip angle pulses (α pulses) for data acquisition following the inversion pulse and subsequent inversion time (TI). For a simple IR GRE sequence, which consists of repeated blocks of an IR pulse, a TI time, a 90° excitation pulse, and another regrowth period (RR-TI), there is a well-known formula for the steady state longitudinal magnetization M_zss [35]:

where M_zss is the magnetization just prior to the 90° excitation pulse, TI is the inversion time, RR is the RR-interval. M ₀ is a complex scaling factor that includes impacts from proton density, T2* weighting, and coil sensitivity, and is represented by.

where A is a spatially invariant constant, C is coil sensitivity, PD is proton density, and the last term is the T2* weighting. Eq. 1 has often been used to approximate the signal in the LGE sequence [36, 37], but the model ignores the perturbation of the magnetization by the α pulse train. To improve the model accuracy, we derived a more complex expression for M_zss (just prior to 1st α pulse) that includes the perturbation from the α pulse train:

where N is the number of α pulses in the train (the views per segment), α is the flip angle, TR is the time between two α pulses, and

is referred to as the perturbation coefficient (p ≥ 1). The derivation of the Eq. 3 is in the Appendix A. If α = π/2, N = 1, then p = 1, and Eq. 3 reduces to Eq. 1; otherwise, M_zss increases due to use of the α pulse train. Fig. 1b shows the difference between the two models, across a spectrum of T1 s when N increases from 1 to 54.

Most T1 mapping methods are based on multiple measurements of the image at different TIs. However, multiple TIs may be nearly impractical for clinical usage in high resolution 3D T1-mapping because of exceedingly long acquisition time. In this section, we demonstrate that with a single measurement it is also possible to generate a T1 map, given a known reference T1 for a single tissue, and some reasonable assumptions about proton density and T2* uniformity in the heart. Fig. 1 shows the basic idea of T1-refBlochi. The prerequisite for this method is a one-to-one relationship between the LGE signal and the underlying T1, which is true as long as the signal-T1 curve, defined in Eq. 3, is monotonic in the post-contrast T1 range. This condition is guaranteed if the TI is sufficiently large so that the entire post-contrast T1 range is within the descending part of the LGE signal-TI curve (cf. Fig. 1). Once the one-to-one relationship between signal and T1 is established, the only obstacle in solving Eq. 3 for T1 is M ₀, which is usually unknown and requires multiple TIs to jointly estimate the two variables via voxel-by-voxel curve fitting. However, the requirement of multiple TIs can be relieved if M ₀ is spatially uniform in the heart region. In this case, since M ₀ is only a scalar, it can be estimated by using a reference signal, with a known T1, which is essentially a single-point calibration in the signal-T1 curve (cf. Fig. 1b). In practice, this can be achieved by using a T1 standard, placed around the patient’s body, or using blood (Fig. 1c), whose T1 can be obtained by a 2D breath-hold scan (TI-scout) or a MOLLI sequence, for example. The blood pool is a reasonable choice for an endogenous reference. Firstly, it has a homogeneous T1—so is easily measured using ROIs. Secondly, its T1 is a value intermediate between myocardium and scar. Dividing the reference signal by the model signal yields the scaling factor M ₀ of Eq. 3. Then for each signal intensity in each slice, a T1 value can be determined based on Eq. 3. A grid search over a practical T1 range was used to generate the T1 value at each voxel. In our implementation, 100 ms–800 ms was used as the T1 range and 2.5 ms was used as the grid search density.

The spatial variation of M₀ in the region of heart is assumed to be minor, after removing the effects of coil sensitivity. This assumption permits the single-TI T1 mapping based on a single-point calibration. Below, literature data justifying this assumption is shown and the detrending method to remove the coil sensitivity in this paper is introduced. In results, we directly measure M₀ and demonstrate that this assumption is reasonable in a study of eight swine.

T2* differences in myocardium and blood provide only minor weightings. It’s known that the native T2* of blood is about 200 ms [38], while the T2* of myocardium is 50 ms [39], although lower in iron overload disease [40], and with hemorrhage [41]. Contrast agent will reduce these T2*s [38], differently in whole blood vs. tissue with and without infarction. Very recent data show that 5 to 10 min after injection of contrast, myocardial T2* decreases by 4 ms in normal myocardium, and 7 ms in regions of chronic infarct, compared to pre-contrast values [42]. An estimate of how T2* variation will effect M₀ (Eq. 2) for T2*s of myocardium ranging from 50 to 10 ms, and T2*s of blood from 200 to 10 ms yields a worst case effect of 10% for a TE of 2 ms.

Secondly, the proton density, or its analogue, water content, of blood and myocardium are similar, although data is limited. Myocardium has a water content of 80% [43]. The plasma which is (1-hematocrit) of blood contains 93% water [43]. The erythrocytes make up the rest of blood, and are 64% water by weight [44]. This yields, for a HCT of 45%, a water content of about 80% for blood. A study of normal, ischemic and occluded and reperfused (i.e. highly edematous) myocardium showed that the water content was 80, 81 and 84% respectively [45]. Acutely infarcted tissue was measured by Higgins et al. to have an elevated water content of 79% vs. 76% for normal myocardium [46]. In chronic fibrotic myocardium, the T2 is similar to normal myocardium, suggesting that water content returns to normal values [47]. This literature data suggests that proton density weighting is a minor source of error (up to 4%).

Finally, the signal intensity variation caused by coil sensitivity is removed by a custom post-processing algorithm before the actual T1 mapping (see Fig. 1a). A number of post-processing methods for removal of the slowly-varying coil sensitivity has been proposed, mainly in brain or musculoskeletal imaging [48, 49] and cardiac perfusion imaging [50]. Similarly to these methods, our method uses a non-contiguous volume of interest (VOI) placed in the (presumably homogeneous) blood pool to extract a 3D coil sensitivity map. A cubic polynomial fit was performed to extract the slowly-varying coil profile from the noisy signal in the VOI. The 3D trend was then extrapolated based on the cubic polynomial parameters to cover the heart. Finally, the 3D trend was divided out of the volume to remove the coil sensitivity effect.

The accuracy of T1-refBlochi is influenced by several factors, including inaccurate reference T1 estimate, M₀ heterogeneity, inaccurate RR duration estimate, and imperfect inversion. Other minor factors may also exist, such as off-resonance, heart rate, and flow; these we briefly address in Discussion.

The reference T1 estimated by a separate 2D scan, such as TI-scout or MOLLI, can be biased for example due to T1 drift. The inaccurate T1 estimate of the reference tissue leads to a biased M₀ estimate, which later leads to a biased estimate of T1 in every tissue. M₀ heterogeneity due to proton density and T2* spatial variation also causes a biased M₀ estimate. However, this effect should be small (≤10%) based on the previous literature and our observation, as described above. In practice, we used the average RR recorded in the DICOM header when evaluating Eq. 3. However, the RR duration varies in vivo, and the RR value present during collection of central k-space data determines the image contrast. Therefore, some T1 error could result from inaccurate LGE modeling due to the inaccurate RR estimate. Imperfect inversion is generally present in any inversion-based T1 mapping methods [51]. Like the inaccurate RR estimate, imperfect inversion generates errors in the LGE signal modeling (Eq. 3) that assumes a perfect inversion.

The impact of all these four major factors can be broadly divided into two categories. The first category has bias in the estimate of M₀, which later causes error in the T1 estimate. The second category uses an inaccurate modeling of the LGE signal, which then causes biased T1 estimate. The T1 error in both categories can be mathematically derived based on Eq. 3. Here, the formulae are given while the specific derivation is shown in Appendix B. In the simulation studies, we verified the accuracy of these formulae by comparing them to direct simulation of the bias. To begin the analysis, let.

represent the T1-weighting, hence Eq. 3 can be rewritten as

In Category 1, the estimate of M₀ from the single-point calibration is wrong, presumably by a factor of δ, i.e., \( {\widehat{M}}_0=\delta {M}_0 \), where \( {\widehat{M}}_0 \) represents the estimated M₀. The resultant bias of T1, ΔT ₁, is then approximately given by (see Appendix B).

where f ^′(·) is the first-order derivative of f(·) with respect to T1. For a fixed LGE protocol, both f(T ₁) and f ^′(T ₁) for each T1 is known, so the bias of T1 estimate can be approximately evaluated given the bias in the estimate of M ₀.

In Category 2, not only is M₀ incorrectly estimated, but the function f(·) is also slightly biased compared to the true function, here denoted by f ^True (·). In this case, the resultant bias of T1, ΔT ₁, is given by.

Hence, if f ^True (·) is given, the bias of T1 can also be predicted by this equation.

For MOLLI or other multi-TI T1 mapping approaches, the precision depends on a number of factors, including SNR of the raw images, number of TIs, and number of parameters in the curve fitting [14]. For T1-refBlochi, there is only one parameter, i.e. T1, and only one measurement, i.e. the LGE signal, in the curve fitting at each voxel. The precision thus depends on SNR of the LGE image and the first-order derivative of the signal-versus-T1 function, i.e. the absolute value of f ^′(·) at the underlying T1. SNR of blood for 3D LGE is typically low, in the range of 10–20. ∣f ^′(·)∣ is a monotonically decreasing function, making precision of the T1 estimate better for smaller T1 s and worse for larger T1 s. The precision for blood SNRs of 10 and 15 was explored in the simulations.

Simulations were performed to verify Eq. 7 and Eq. 8 and to show the accuracy and precision of T1-refBlochi under various sources of error. To simulate T1-refBlochi, LGE signal of T1 ranging from 200 ms to 600 ms was generated based on Eq. 3. The parameters of the LGE sequence in the simulations were TR/ α/TI/RR/N = 5 ms/15°/300 ms/900 ms/33. The reference tissue has a T1 of 280 ms, and M₀ of 1. This reference T1 was chosen to simulate the blood T1, which was used in the in vivo studies as the reference.

We performed simulations to study the impact of the four major sources of error in the accuracy of T1 estimate. To simulate the impact of biased reference T1 estimate, two biased reference T1 s, 250 ms (true reference T1 – 30 ms) and 310 ms (true reference T1 + 30 ms), were used. The single-point calibration generated a biased M₀ estimate, which was used to perform T1 mapping. To simulate spatial variation of M₀, we used M₀ of 1.1 or 0.9 (±10% change from the reference M₀) for all non-reference tissue. To simulate a biased RR duration, we used a nominal RR of 930 ms and 870 ms (true RR ± 30 ms) in T1-refBlochi. Finally, to simulate the imperfect inversion, we used an inversion factor of 0.92 to simulate the true LGE signal. This inversion factor is the worst-case scenario for the hyperbolic secant RF pulse we used, according to the investigation in [51]. For the imperfect inversion case, the LGE signal was generated based on the following formula:

where β represents the inversion factor. The derivation of the equation is similar to that of Eq. 3. β = 0.92 in the simulation.

For all the simulations, we calculated the bias of T1 estimate relative to the true T1 in the range of T1 from 200 ms to 600 ms, covering the typical post-contrast T1 range. To verify Eq. 7 and Eq. 8, we also calculated the bias of T1 directly from these equations.

The precision of T1-refBlochi was studied using Monte-Carlo simulations [14]. Specifically, 10,000 Monte-Carlo simulations of LGE signal were employed and T1-refBlochi was applied to each of them for T1 ranging from 200 ms to 500 ms. The parameters of the LGE protocol was TR/ α/TI/RR/N = 5 ms/15°/300 ms/950 ms or 750 ms/33. The reference T1 was 280 ms. White Gaussian noise that represented a SNR of 10 and 15 was added to each simulated LGE signal. Here SNR is defined as the blood signal (blood T1 is 280 ms) divided by the standard deviation of the Gaussian noise. Two RR values, 750 ms and 950 ms, were used in the simulations to study the performance of the two methods under different heart rates. The standard deviation of the T1 estimates were calculated over the 10,000 simulations to study the precision of the method.

All phantom, animal and human studies were performed on a 1.5 T Siemens scanner (Siemens Healthcare, Erlangen, Germany).

T1-refBlochi was validated in phantoms (Gd-doped water) consisting of 6 T1 s from 200 to 560 ms, which spanned the relevant range for post-contrast T1 values. The method was compared to a gold standard T1-mapping (spin echo inversion recovery, SE IR) method in a uniform head coil, with 6 TIs, and a TR = 3000 ms. The LGE protocol used for T1-refBlochi was TR/α/TI/RR/N = 3.8 ms/15°/300 ms/800 ms/37, 1 RR between inversions. The protocol was modified to explore sensitivity of T1-refBlochi to the protocol. Modifications in flip angle (changed to 10°), RR (changed from 900 to 700 ms), N (changed to 51), TR (changed to 5.4 ms) and underlying SNR (reduced by a factor of 4) were studied. The reference T1 for refBlochi was the shortest (194 ms).

We compared T1-refBlochi with a 3D multi-TI T1 mapping method, which is based on multiple LGE imaging, each with a different TI. The 3D multi-TI T1 mapping also provides an estimate of the combined proton density and T2* weighting, which was used to investigate its heterogeneity in vivo and its impact on T1-refBlochi accuracy. Four LGE measurements (with the same parameters as for refBlochi) were used in the multi-TI method, with three of them having different TIs (100,200 and 300 ms), and the last measurement having inversion disabled to mimic an infinite TI [11, 52]. For the three LGEs with different TI values, the steady-state longitudinal signal can be accurately modeled by Eq. 3. For the volume with no inversion pulse, however, the model is slightly different due to the absence of inversion pulse:

A separate fitting of Eqs. 3 and 5 is needed for the inversion and non-inversion data, resulting in the following optimization problem:

where S _1 − 3 are the signals corresponding to the three TIs and S ₀ is the signal for the non-inversion LGE, respectively. The M ₀ and T1 maps were then generated voxel-by-voxel, using the Levenberg-Marquardt algorithm, implemented with MATLAB (MathWorks, MA, USA) Curve Fitting toolbox.

Animal studies were performed in accordance with our Institutional Animal Care and Use Committee. All animals received humane care in compliance with the “Guide for the Care and Use of Laboratory Animals” published by the National Institutes of Health, the Animal Welfare Act, and Animal Welfare Regulations. Eight Yorkshire pigs were studied, including 3 control animals, and 5 animals imaged one to two weeks after infarction (average weight 36 ± 9 kg, average heart rate of 94 ± 16 bpm), with infarctions performed using a surgical left circumflex coronary artery occlusion or a 90 min balloon occlusion of the left anterior descending coronary artery.

Prior to CMR, animals were anesthetized, and ventilated. T1-mapping data was acquired starting from 20 to 30 min post-injection of 0.2 mmol/kg Gadobutrol (Bayer Healthcare). Three-dimensional LGE with 4 TIs (no inversion, 100,200, and 300 ms) were sequentially acquired, targeting both the LV and LA. 3D LGE used a fat-saturated, navigator-gated high resolution 3D inversion recovery spoiled gradient echo sequence, with ECG-triggering. Mid-diastole was chosen based on the RR interval. Spatial resolution was 1.2 × 1.2 × 3 mm³ before zero-filling, with FOV of 300 mm. TR/TE/α/N = 3.9 ms/1.9 ms/15°/37. Bandwidth was 490 Hz/pixel. The number of phase-encodings and slice-encodings was typically 256 and 18, respectively. No acceleration was used. One or 2 RR intervals between inversion pulses were used; two RR intervals were chosen for studies if the heart-rate was very high. The average interval between inversion pulses was 1010 ± 270 ms. Cartesian k-space sampling was used with centric reordering in the phase-encoding direction and sequential ordering in the slice-encoding direction. Navigator-gating used a ± 3 mm acceptance window, with no tracking. Data was acquired over an average of 14 ± 5 min (for four TI values). The data was processed with the multi-TI method to generate T1 maps and maps representing combined coil-sensitivity, T2* and proton density. These combined maps of coil-sensitivity, T2* and proton density, were analyzed to evaluate the variation of M₀. Spatial registration between the four LGE images was not performed, but the quality of the generated T1 maps was visually good.

T1-refBlochi was performed with the LGE dataset acquired at TI = 300 ms. Blood was chosen for the single-point calibration and its T1 from the multi-TI method was used as the reference T1. The T1 maps generated from T1-refBlochi was compared to those from the multi-TI method, using visual inspection, and direct subtraction. For blinded comparison, ROIs were placed in blood, normal myocardium, and scar, on T1 maps from the multi-TI method, and automatically copied to the corresponding T1-refBlochi T1 map. The mean and standard deviation of T1 s were measured in each ROI, and compared.

Five patients (3 males, age 53 ± 8) were imaged with 3D LGE covering the whole heart, about 20 min post 0.2 mmol/kg injection of gadolinium contrast agent, and a 2D 4-chamber ShMOLLI was acquired 5 ± 2 min later, on a Siemens 1.5 T scanner (Siemens Aera, Erlangen, Germany), using our clinical protocol: 360 mm × 270 mm FOV, 1.4 × 2.8 × 6 mm³, with 92 phase-encodings, and TR/TE/ α =2.6 ms/1.3 ms/35°. 3D LGE scan parameters were: FOV 380 mm × 498 mm, spatial resolution 1.5 × 1.5 × 3.6 mm³ before zero-filling, TR/TE/ α =3.9 ms/1.7 ms/15°, N = 27–53, RR = 904 ± 57 ms, TI = 287 ± 12 ms, and receiver bandwidth 360 Hz/pixel. The number of phase-encodings and slice-encodings was typically 336 and 44, respectively. Parallel imaging was used with GRAPPA factor of 2. The 3D k-space was sampled with centric reordering in the phase-encoding direction and sequential ordering in the slice-encoding direction. Navigator-gating was used with a ± 3 mm acceptance window, placed after the acquisition window in the mid-diastole. The 3D LGE image was transformed into a T1 map using refBlochi, and compared to ShMOLLI. Blood with T1 provided by ShMOLLI was used as the reference for refBlochi. The patient study was approved by our institutions IRB and all patients provided informed written consent. The patients were imaged for assessments of atrial fibrillation (n = 4) and hypertrophic cardiomyopathy (n = 1). ROIs were placed in the blood and myocardium, with mean T1 values compared between the two methods.

Using the left atrial post-contrast T1 maps in the patient, the extracellular volume fraction was approximated (ECVa) as:

With an assumed T1_m0 = 1150 ms, and a T1_b0 = 1500 ms, and HCT = 0.45. T1_b0 and HCT can be measured in each patient, but left atrial T1_m0 is highly challenging to measure, due to e.g. registration error, additional scan time, and blood-myocardial partial voluming. Therefore, an approximation of ECV with presumed native T1, defined in Eq. 12, was used, and the sensitivity of ECV to native myocardial T1 was explored to estimate the approximation error. A plot of ECV for three post contrast blood T1 s and a post-contrast myocardial T1 of 300 ms shows the weak dependence on T1_m0 (Additional file 1). A highly realistic scenario in which the native T1 in regions of edema are elevated by 100 ms [53, 54], from 1150 to 1250, results in an absolute deviation of ECV by +0.02 (+2%), when ECV is measured with T1_b = 350 ms, for any myocardial post-contrast T1. This absolute increase in ECV by 2% cannot be measured if T1_m0 is assumed constant at 1150 ms. Therefore, ECVa is a reasonable and practical approximation of true 3D ECV.

Figure 2 shows the T1-refBlochi bias caused by the four major sources of error. The error calculated based on the mathematical formulae (dash lines) excellently matched the simulated error (solid lines) in all cases, demonstrating the accuracy of the formulae. Error in the M₀ estimate (Category 1) caused by either an inaccurate T1 reference or M₀ heterogeneity, results in a bias that decreases with increasing T1 (Fig. 2a & b). For biased reference T1 estimate (±30 ms, Fig. 2a), the error for all T1 s longer than the reference T1 is less than the error in the reference T1. Therefore, choosing a short reference T1 reduces the bias present in the large T1 species. Fig. 2b shows the bias caused by a ± 10% change of M₀ relative to the reference M₀. The bias is within 20 ms, showing that T1-refBlochi is very robust against a mild spatial variation of M₀. Error in the LGE signal modeling (Category 2) causes bias which increases with longer T1 s (Fig. 2c & d). Fig. 2c shows the bias of T1 estimate with an incorrect nominal RR (true RR ± 30 ms). The resultant bias is within 30 ms. Fig. 2d shows the bias of T1 estimate under worst-case imperfect inversion (inversion factor = 0.92). The bias is within 50 ms.

The result of the Monte-Carlo simulations is shown in Fig. 3. The standard deviation of the T1 estimates increases as T1 increases, consistent with the theoretical analysis. The largest standard deviation at T1 = 500 ms was 35 ms and 45 ms at SNR = 15 and 58 ms and 76 ms at SNR = 10 for RR of 950 ms and 750 ms, respectively. A longer RR is favorable but the difference due to variation of RR is rather small.

In phantoms, T1–refBlochi showed excellent agreement with the gold standard (SE IR) over multiple protocols (Fig. 4). Over all protocols and T1 ranges, the mean bias and standard deviation was -3 ms ± 9 ms and the correlation coefficient between T1-refBlochi and SE IR was R² = 0.99. For the multi-TI comparison method, the mean bias and standard deviation was -8 ms ± 13 ms, and R² = 0.99, compared to SE IR.