Improving CPMG Liver Iron Estimates with a T₁-Corrected Proton Density Estimator

Abstract

Purpose:

CPMG spin echo acquisitions are attractive for diagnosing and monitoring liver iron concentration in iron overload disorders due to their time efficiency and potential to reveal unique information about tissue iron distribution. Clinical adoption remains low due to the insensitivity of CPMG-based R₂ estimates to liver iron concentration (LIC) when common fitting techniques are applied. In this work, we demonstrate that the inclusion of a proton density estimator (PDE) derived from the CPMG acquisition increase the sensitivity of CPMG R₂ estimates to LIC in both simulated and in-vivo human data.

Theory and Methods:

CPMG R₂ acquisitions from 50 clinically-indicated MRI studies in patients with iron overload were analyzed with and without PDE constraints. Liver regions of interest were fit to monoexpontial and nonexponential signal decay equations. LIC by ${\text{R}}_2^{*}$ served as the reference standard. The observed calibration between CPMG R₂ values and LIC were compared to results predicted from a previously validated Monte Carlo model.

Results:

The sensitivity of CPMG-derived R₂ triples when a proton density constraint is applied. When compared with ${\text{R}}_2^{*}\text{ -LIC }$ estimates, both monoexponential and nonexponential models were unbiased but demonstrated broad 95% confidence intervals particularly for LIC values below 12^mg⁄_g. Absolute error did not increase with LIC.

Conclusion:

A proton density constraint can increase the sensitivity of CPMG-based models to iron. CPMG acquisitions are time-efficient and could potentially improve the dynamic range of single spin echo techniques as well as providing insight into tissue iron distribution.

1. Introduction

Chronic anemias such as thalassemia and sickle cell disease represent the most common genetic disorders in the world. Transfusion therapy in these patients produces severe iron deposition in the liver and other organs, leading to cardiac and endocrine dysfunction and liver cirrhosis. Unlike in hereditary hemochromatosis, iron overload in transfusional siderosis cannot be treated with phlebotomy. Instead, patients receive iron chelators that bind and remove iron, with dosing specific to their tissue iron content. Determining safe chelation dosing and assessing the efficacy of a given chelator regimen requires reliable iron quantitation techniques. Before 2005, needle biopsies were required to quantify liver iron, with their attendant risks¹ and sampling error^2,3,4. Since then, liver iron concentration (LIC) estimation using non-invasive magnetic resonance imaging has become a key tool for diagnosing and monitoring iron overload in both transfusional and nontransfusional siderosis. Both transverse relaxivity and magnetic susceptibility measurements have been used to estimate tissue iron concentrations in the liver, heart, endocrine glands, and brain. MRI-based heart and liver iron quantitation at 1.5 Tesla(T) is now considered standard of care⁵.

One common approach for measuring LIC is to estimate the transverse relaxation rate R₂ (1/T₂) from images formed with a single spin echo (SSE) pulse sequence collected using separate acquisitions for each echo time. This approach yields curvilinear calibration curves relating SSE-R₂ to LIC which demonstrate high sensitivity at low iron loads and reduced sensitivity in high iron loads^6,7. The excellent sensitivity observed for low LIC^8,9 is useful for precisely estimating iron loads below 20^mg⁄_g. SSE series are also attractive due to the straightforward acquisition and wide availability across scanners. However, scan time grows linearly with the number of echoes desired, dynamic range and accuracy are limited at higher LIC values, and the analysis is non-trivial, often requiring a third-party vendor.

Less commonly, R₂ estimates are made using Carr-Purcell-Meiboom-Gill multiecho spin echo sequences (CPMG)^{10,11,12,13,14}. Such sequences require only one excitation to form images at a variety of TEs, reducing the scan duration to that of a single SSE image series while still acquiring multiple echoes. Compared to SSE, signal loss is reduced in later echoes due to the dependence of each echo on preceding echoes, leading to improved SNR in high-iron subjects. This dependence may also provide information about cellular iron distribution that cannot be assessed with SSE images¹⁴. Despite the potential benefits of CPMG, constraints on both the minimum TE and the interecho spacing, τ, limit CPMG’s sampling to a greater extent than SSE. These timing limitations lead to undersampling of initial signal decay, presenting a significant curve-fitting challenge as the early decay information is lost in moderate-to-high iron loads.

When performing curve fitting, unconstrained fits are attractive due to their simplicity and lack of initial assumptions but fail in the presence of rapid signal decay. One approach for overcoming the loss of early signal decay is to use an estimate of the signal intensity prior to any signal decay occuring. The inclusion of a signal intensity estimate at t=0 essentially restores the lost early decay information. Figure 1 illustrates the echo undersampling problem for two ideal monoexponential signals, having initial amplitudes of 1000 and T₂ decay values of 6 ms (moderate iron load) and 3 ms (high iron load), respectively. The red triangles represent typical echo sampling intervals and capture the true signal decay perfectly. Using these points, two exponential fits are performed. The unconstrained fit only uses the sampled data, while the constrained fit includes the signal value at time zero. For moderate siderosis, the unconstrained fit (dashed line) is only 7% lower than the constrained fit and true value, which is an acceptable level of bias comparable to the known coefficient of variation of LIC measurements⁶. Most of the error comes from the unconstrained technique vastly underestimating the early signal decay. However, the unconstrained fit fails for severe siderosis, with high iron loads demonstrating over 50% disagreement between constrained and unconstrained R₂ estimates. By anchoring the fit with an estimate of the intersection point between the y-axis and the decay curve, the R₂ estimate captures the decay effectively. A similar approach has been successfully used in conjuction with an empirical signal model for SSE¹⁵. In the present work, we investigate using a muscle-based S₀ constraint to improve the sensitivity of CPMG R₂ to LIC, thereby improving its potential clinical utility.

Figure 1:

Example of mono-exponential fits for different R₂ species with and without the use of a proton density estimate (PDE) constraint. When the decay rate is comparable to the first echo time (left), the proton density is underestimated by 8% and T₂ is overestimated by 10%. The same trend is seen to a more extreme degree in the right pane where the T₂ is shorter; proton density is underestimated by 51% and T₂ is overestimated by 57%.

2. Theory

The choice of signal model can have significant impact on quantitative results from MRI. In this work, we applied two signal equations to the acquired data: the Yamada CPMG signal equation and the Jensen model for non-exponential decay in the liver. The Yamada equation and similar mono-exponentials are the most common signal models used in MRI while the Jensen model specifically describes CPMG signal behavior in iron loaded tissue.

The Yamada (monoexponential) CPMG model accounts for the R₂ envelope that is traditionally seen in spin-echo sequences as well as the T₁ weighting that is present¹⁶. The signal over time, S(t), is given by:

$$S(t)=S_0\times W_{T_1}\times W_{T_2}$$ (1)

where S₀ represents proton density as a product of a constant k and the initial longitudinal magnetization M0, R₂ represents transverse relaxation rate, t represents echo formation time, and $W_{T_1}$ and $W_{T_2}$ represent the respective effects of T₁ and T₂ on the signal as follows:

$$S_0=k\times M_0$$ (2)

$$W_{T_1}=1-e^{\frac{-TR}{T_1}}$$ (3)

$$W_{T_2}=e^{-R_{2}t}$$ (4)

In the case of a fixed TR, the T₁ effects are constant for all acquisitions and the signal model can be simplified by including the constant effects in the proton density term $S_0^{\prime }$:

$$S(t)=S_0^{\prime }\times e^{-R_{2}t}$$ (5)

$$S_0^{\prime }=S_0\times W_{T_1}$$ (6)

Although the monoexponential equation is commonly used for R₂ relaxometry, it does not differentiate between the decay processes in SSE and CPMG sequences. Spins that are stationary with respect to magnetic inhomogeneities will demonstrate identical SSE and CPMG R₂ estimates. When spins diffuse their echoes are weighted by the magnetic heterogeneity experienced between any two RF pulses, leading to a cohort of spins whose apparent decay is echo spacing-dependent. Although the echo times produced by SSE and CPMG sequences may match, the distance between the excitation and inversion RF pulses increases with each TE for the SSE series while the spacing between the RF pulses in the CPMG series remains fixed. In cases where proton motion distance and tissue mesostructure scale are similar, as with iron, the diffusion and sampling scheme will lead to different R₂ estimates from the two sequences. The Jensen (nonexponential) model¹⁴ accounts for this by providing two decay parameters rather than one - T₁ effects are omitted for brevity:

$$S(t)=S_0\times e^{-R{R}_{2}t}\times exp\left(-{\alpha }^{\frac{3}{4}}{(\Delta t)}^{\frac{3}{4}}{\left(t-t_s\right)}^{\frac{3}{8}}\right)$$ (7)

where RR₂ represents “reduced R₂,” α represents a nonlinear aggregation parameter, t represents time, and t_s is a time shift given by:

$$t_s=2\tau \left[1-{\left(\frac{\tau }{\Delta t}\right)}^2\right]$$ (8)

where τ represents the time of the first RF pulse and Δt represent the time between successive RF pulses, such that echoes form at TE_n = 2τ + 2(n − 1)Δt (or 2nΔt when τ = Δt). The resulting signal in the presence of iron stores is therefore a function of properties that are intrinsic to the tissue, such as diffusion and iron aggregation, as well as extrinsic acquisition parameters such as echo spacing.

3. Methods

3.1. Imaging

Patients with β-thalassemia, sickle cell disease, and other rare anemias underwent MRI assessment of their liver iron burden at Boston Children’s Hospital for clinical care purposes. A total of 100 studies were deidentified and reviewed as part of a retrospective study jointly approved by institutional review boards at Children’s Hospital of Los Angeles and Boston Children’s Hospital (IRB#CHLA-15–00010). Fifty imaging series were sampled from the larger dataset to yield a wide and relatively uniform range of LIC burdens. All scans were completed on a Philips Achieva 1.5T magnet (Philips HealthTech, Best, Netherlands) on software revisions 2.6.1, 2.6.3, or 3.2.2. Breath holding was performed for all acquisitions. Three 8-echo gradient echo sequences with minimum TEs of 1.16 ms and maximum TEs of 8.6, 11.66, and 16.56 ms were used to estimate ${\text{R}}_2^{*}$ for use as a standard LIC estimate; the appropriate scan for each patient was selected based on subjective assessment of signal intensity in the liver of each series and analyzed by a clinician with 12 years of experience quantifying liver iron. An 8-echo CPMG sequence with equally spaced TEs from 6.5–52 ms was used to capture R₂. Expanded imaging parameters for both protocols are shown in Table 1.

3.2. Simulation

An internally developed, previously validated Monte Carlo simulation framework was used to generate synthetic signals matching the echo times used in the clinical scans for an LIC range of 1–50^mg⁄_g [iron/dry tissue weight]¹⁷. Briefly, liver tissue was modeled as 80 μm cubes containing 64 cuboidal hepatocytes and 18 cylindrical sinusoid regions¹⁸. Iron overload was modeled by distributing spheres of iron in the hepatocytes using gamma distributions to statistically describe their size and spacing. From the iron distributions, magnetic field disturbances were calculated based on the field strength and magnetic susceptibility of iron. Five thousand spin cohorts were tracked using a random walk simulation through iron distributions that obeyed cell and obstacle boundaries. The simulation framework, which was updated to include a 3D Bloch equation simulator to allow for the inclusion of T₁ effects, calculated the magnetization at each timestep. RF excitations and inversions were modeled as 90° and 180° instantaneous flips; the CPMG phase cycling scheme was applied. Iron-free relaxation rates T₁ and T₂ were set to 576 ms and 42 ms, respectively¹⁹. Iron-mediated T₁ enhancement was not modeled in the simulations because each spin cohort was initialized with the same magnetization. This is essentially equivalent using an infinitely long TR that would remove any T₁-weighting.

3.3. Analysis

In the unconstrained case, both acquired and simulated signal decay were processed identically. A region of interest (ROI) of the whole liver was selected from a single mid-hepatic slice in each imaging series by a cardiologist with 15 years experience and a graduate student with 5 years experience. CPMG mono-exponential parameters (R₂,$S_0^{\prime }$) were estimated using equation 5 with a constant added for noise bias. CPMG nonexponential decay parameters (RR₂,$S_0^{\prime }$,α) were estimated using equation 7 with a constant term added. A pseudopixel-wise Levenberg-Marquadt algorithm²⁰ was used for all fits with non-negativity constraints on $S_0^{\prime }$, RR₂, α and c.

Both the exponential and nonexponential fits were subsequently repeated after constraining subsequently repeated after constraining $S_0^{\prime }$ to be within ±20% of the PDE. However, for the patient scans, PDE was derived from a region of interest drawn bilaterally in the erector spinae muscles in the same slice as the liver ROI. The bounds for the PDE were chosen empirically to account for numerous uncorrected sources of intensity variation. Most notably, spatial intensity variation resulting from ${\text{B}}_1^{+}$ heterogeneity can cause significant intensity variations that will manifest in this context as a spatial variation in S₀. Further, the wide bounds prevent thermal noise and partial voluming in the relatively small paraspinal muscle ROI from having an outsided effect on the fit. For the Monte Carlo simulations, the PDE was known exactly. Skeletal muscle was chosen as a surrogate for proton density because its resistance to iron uptake provided a reliable reference tissue in all patients. Based on the reported densities of 1.06^kg⁄_L and 1.02^kg⁄_L for muscle²¹ and liver²², respectively, we assumed that skeletal muscle and liver would have approximately the same S₀.

Since liver and muscle have different T₁ values, it was necessary to correct the $S_0^{\prime }$ of muscle for T₁ weighting (equation 2) as follows:

$$S_{0,\mathit{\text{liver}}}^{\prime }=S_{0,\mathit{\text{muscle}}}^{\prime }\times \frac{W_{T_1,\mathit{\text{liver}}}}{W_{T_1,\mathit{\text{muscle}}}}$$ (9)

Muscle T₁ was assumed to be 1008 ms¹⁹ for all subjects, while liver T₁ had to be adjusted for iron load²³. This was initially performed using LIC values predicted by the reference standard $\left({\text{R}}_2^{*}\right)$ to establish the initial calibration curves. However, we subsequently determined that it was possible to estimate liver T₁ iteratively from LIC values predicted by CPMG R₂ with convergence over three iterations, i.e. T₁ is estimated first from an initial R₂ estimate and the fit is performed three times, each time recalculating T₁ based on the newest R₂ estimate. The liver T₁-R₂ relationship is as follows:

$$T_{1,\mathit{\text{liver}}}=\frac{1}{0.0049\times R_2+3.0871}$$ (10)

The experimental models were compared using gradient-echo based reference LIC estimates in human subjects or specified simulation iron loads as the standard iron load metric. ${\text{R}}_2^{*}$ estimates were derived from magnitude gradient echo images using a previously validated pseudo-pixelwise technique with gradient echo images²⁰. Reference LIC estimates were derived from ${\text{R}}_2^{*}$ estimates using the Wood ${\text{R}}_2^{*}\text{ -LIC}$ calibration⁶ via equation 11:

$$F{E}_{R_2^{*}}=0.0254\times R_2^{*}+0.202$$ (11)

Models of LIC as a function of R₂ and as a function of RR₂ and α were generated by fitting second-order polynomials to the fit parameters from patient data. Each patient’s LIC was computed from each model based on their fit parameters (R₂ or RR₂ and α) and compared to the reference LIC via Bland-Altman analysis. Both absolute and relative differences were calculated using the following equations:

$$\begin{gathered} LI{C}_{\mathit{\text{mean}}}=\frac{LI{C}_a+LI{C}_b}{2} \\ LI{C}_{\mathit{\text{difference}}}=\{\begin{array}{ll}LI{C}_a-LI{C}_b\hfill & \left(\text{Absolute}\text{Difference}\right)\hfill \\ \frac{LI{C}_a-LI{C}_b}{LI{C}_{\mathit{\text{mean}}}}\hfill & \left(\text{Relative}\text{Difference}\right)\hfill \end{array} \end{gathered}$$ (12)

where LIC_a is the experimental LIC estimate and LIC_B is the standard LIC estimate (based on ${\text{R}}_2^{*}$ for human subjects and known apriori for simulations). Additionally, the standard deviation and confidence intervals were calculated.

4. Results

The cohort included 24 thalassemia patients, 6 sickle cell disease patients, 5 Blackfan Diamond syndrome patients, and 15 patients with other rare anemia syndromes. The patients were mostly adolescents and young adults (ages 16.4±9.4 years) and were balanced for sex (24M, 26F). By design, LIC was evenly distributed between approximately 0.8 and 32.5 ^mg⁄_g (11.9±9.6). Figure 2 demonstrates the unconstrained R₂ estimates versus ${\text{R}}_2^{*}\text{ -derived }$ LIC (circles). Linear regression with 95% confidence intervals (shown as solid and dashed lines, respectively) demonstrates the flat relationship between LIC and R₂. A previously published patient dataset²⁴ comparing LIC and R₂ is included for comparison (points shown as triangles).

Figure 2:

Unconstrained CPMG R₂ estimates from our patient cohort are plotted against LIC. Linear regression of the data (shown as solid line with 95% confidence intervals shown as dashed lines) demonstrates the relatively flat relationship of unconstrained, CPMG-based R₂ estimates to liver iron. Open triangles represent previously published CPMG data²⁴ obtained by screen capture and transformed to LIC units using an appropriate calibration²⁷. Some positive bias is observed, particularly at very low iron concentrations, but the slope of R₂ versus iron is quite similar across the two studies.

Figure 3 compares the monoexponential fits from observed data (open symbols) to the calibration curves predicted by the Monte Carlo model (solid line). The dashed lines represent the 95% confidence intervals (CIs) of the model. The top panel represents unconstrained R₂ estimates while the bottom panel incorporates the PDE. For the unconstrained fits, there is almost perfect agreement between the simulated and observed data. However, note that the sensitivity of R₂ to iron concentration is minimal. An increase in LIC from 5 to 40 ^mg⁄_g (800% increase) results in less than a 200% increase in R₂.

Figure 3:

Comparison of fitting techniques. Top panel demonstrated the weak relationship between R₂ and LIC in the unconstrained fits. Lower panel shows the approximate tripling of the calibration curve sensitivity with the addition of the PDE constraint. The agreement between model predictions and observed data are excellent for both constrained and unconstrained fits. Human subject reference LIC was estimated via ${\text{R}}_2^{*}$ while the simulation LIC was known.

Addition of the S₀ constraint improves the sensitivity. Figure 3 (bottom panel) demonstrates observed R₂ and Monte Carlo-based calibration after including S₀ constraints in the fitting algorithm. R₂ sensitivity to iron concentration is markedly improved: an increase in LIC from 5 to 40 ^mg⁄_g now results in a predicted R₂ increase of 241 s⁻¹ compared to 39.5 s⁻¹ for the unconstrained fits over the same LIC range, making constrained R₂ estimates approximately 6 times more sensitive to iron than unconstrained estimates. The agreement between the observed and predicted R₂ is strong across the entire range of iron loads.

Figure 4 demonstrates the reduced R₂ (RR₂) (top panel) and iron aggregation (α) parameter (bottom panel) from equation 8 as a function of ${\text{R}}_2^{*}\text{ -LIC}$. Unconstrained patient fits (blue circles) demonstrate a relatively flat relationship comparable to the unconstrained R₂ estimated from the simulations (demonstrated in figure 3, top panel). Likewise, unconstrained estimates of α show a shallow relationship with LIC with wide confidence intervals that appears to plateau as iron increases. When a S₀ constraint is applied (patient data shown as red triangles), RR₂ estimates appear to rise similarly to unconstrained estimates until an LIC of approximately 5 ^mg⁄_g but then plateau and remain essentially flat over the remaining iron range. Constrained estimates of α demonstrate increased sensitivity with iron, rising about twice as fast as unconstrained estimates and demonstrating no plateau. Fitting the nonexponential model to the simulation data produces similar results. The RR₂ of the constrained fits (red dotted line) are lower than those of the unconstrained fits (blue dashed line) while the constrained α parameter is nearly 50% more sensitive to iron than the unconstrained estimates.

Figure 4:

Jensen model parameters, RR₂ and α, are plotted as a function of LIC for both constrained (red triangles) and unconstrained (blue circles) fits. Corresponding predictions from the Monte Carlo simulation data are demonstrated for constrained (red dotted lines) and unconstrained (blue dashed lines) fits. The observed RR₂ data are systematically larger than the predicted data for both constrained and unconstrained fits. For the unconstrained fits, the observed A values are unbiased with respect to the simulation data until LIC values exceed 15 ^mg⁄_g where there is catastrophic loss of sensitivity. Addition of the S₀ constraint improves the agreement between observed and predicted fits. Human subject reference LIC was estimated via ${\text{R}}_2^{*}$ while the simulation LIC was known.

Table 3 demonstrates Bland-Altman statistics comparing LIC estimates from the monoexponential and nonexponential models to the ${\text{R}}_2^{*}\text{ -based}$ reference LIC; plots are shown in supporting information figures S1, S2 and S3. All models were unbiased. The LIC estimates from both unconstrained and constrained monoexponential models showed standard deviations of 5.1^mg⁄_g. The unconstrained nonexponential model demonstrated standard deviation of 4.4^mg⁄_g while the constrained model showed a standard deviation of 3.8^mg⁄_g. Relative confidence intervals are noted in the table. Polynomial fit plots and fit parameters are available in supporting information figure S4 and supporting information tables S1 and S2

Figure 5 demonstrates fits for the monoexponential and nonexponential models in two manually selected patients demonstrating moderate and high iron loads. The unconstrained monoexponential model performs relatively well in moderate iron patients (top left pane). The decay rate and S₀ estimate is similar between the unconstrained (blue dashed line) and constrained (red solid line) fits, and both fits agree with the patient data (blue dots). However, for high iron patients, the unconstrained monoexponential model (lower left pane) vastly underestimates proton density, leading to a significantly reduced decay rate when compared with the constrained fit. In contrast, unconstrained fits of the nonexponential model for both low and high iron patients (right column, top and bottom panes) demonstrate strong agreement with the with the patient data and only slightly underestimate proton density compared to the constrained fits but fail to capture the aggregation parameter while the constrained fits show a recovery of the aggregation parameter.

Figure 5:

Examples of monoexponential and nonexponential model fits for a low and a high iron patient. Blue dots represent signal intensity values while red and blue lines represent the constrained and unconstrained fits respectively. Fit parameters and mean squared error (MSE) for each model are included. For the monoexponential model, the proton density (PDE) constraint systematically increases S₀ and R₂ estimates but the MSE also increases, demonstrating a degraded quality of fit with the observed data points. In contrast, the proton density constraint increases the RR₂ and α values for the nonexponential fits without any degradation of fit quality.

5. Discussion

CPMG sequences have been proposed as rapid alternatives to SSE acquisitions for LIC determination but remain impractical due to the insensitivity of R₂ to higher, clinically relevant LIC range at practically achievable echo times. We demonstrate, in-silico and in-vivo, that sensitivity can be improved by including a proton density constraint to capture rapid initial signal decay. The signal from iron loaded tissue acquired with a CPMG sequence is incompletely described by a monoexponential, requiring either biexponential or nonexponential models^13,14. The nonexponential model consists of a non-exponential rapid decay component (α) governed by “near-field” iron stores and an exponential slow decay component called “reduced R₂” that accounts for diffuse iron storage. In general, α reflects lysosomal hemosiderin deposits and ferritin aggregates and RR₂ represents cytosolic, soluble, ferritin. Jensen noted that capturing the initial decay is particularly important in high-iron tissues and his model permits a first echo time that is shorter than the CPMG interecho spacing for this reason. In fact, when the condition α ≲ (Δt)^−3/2 is not met, the rapid-decay signal cannot be detected at all¹⁴, limiting prior applications of this model to the heart.

Most clinical imaging magnets cannot achieve sufficiently short TEs to capture liver α for patients with moderate LIC or higher. The shortest first echo time is limited to approximately 4 ms for a slice-selective spin echo acquisition and was 6.2 ms in this study. Jensen tried to compensate for limited first echo times by varying the interecho spacing and first echo time over multiple acquisitions to improve fitting robustness, but this reduces any potential time savings of using CPMG.

In addition to the acquisition challenges, the problem of separating the fast and slow decay components with a single CPMG series is ill-posed because there are three free fit parameters (S₀,α,RR₂) in the nonexponential model. It is therefore attractive to fit with simpler models like a mono-exponential which has only two free parameters (S₀, R₂). Mono-exponential fitting leads to a weighted average of the fast and slow decay components with the proportion of fast decay signal increasing with LIC.¹³ Due to TE limitations, unconstrained mono-exponential fitting mostly ignores the fast decay and almost exclusively fits the slow-decay component of the non-exponential model. This is apparent in figure 4, where unconstrained R₂ tracks the relatively iron-insensitive RR₂ from both constrained and unconstrained fits of a single CPMG series while the α parameter increases with liver iron. The agreement between the constrained and unconstrained estimates of RR₂ and the unconstrained mono-exponential R₂ estimates suggests that the lack of a proton density constraint leads to the fitting of only the slow decay component, which saturates at LIC values between 2.5 and 8^mg⁄_g. Previous work by Ghugre¹³ demonstrated using a benchtop NMR relaxometer with interecho spacings as short as 0.1 ms that increasing echo spacing leads to a decrease in the fast-decay component of a bi-exponential fit rather than a change in the decay rate of either exponential. This is similar to our finding that the unconstrained mono-exponential fits primarily capture the slow-decay component.

Due to the complete loss of fast decay species at clinically achievable first echo times, unconstrained mono-exponential fits will underestimate proton density. Unconstrained exponential fitting of simulation data underestimated proton density by up to 70% of its known value. Similar behavior was noted in-vivo when comparing to a muscle-based proton density estimate. Such a precipitous drop in S₀ cannot be accounted for by the mass of the iron alone, which we estimate will not exceed 1.5% of the tissue mass in even the most highly iron loaded patients. Without a physical basis for the decrease in S₀, applying a constraint can reduce the sampling bias that favors the slow decay component. This approach is similar to that used to constrain bi-component exponential fits of SSE data in FerriScan(®, Resonance Health, Australia).²⁵ By using an S₀ constraint derived from tissue whose iron load is independent of disease status, we can effectively increase the weight of the fast decay component and increase the LIC sensitivity of mono-exponential R₂ estimates. Signal decay in tissue with mild iron overload is primarily governed by diffuse iron and exhibits slow decay that can be adequately fit with either a constrained or unconstrained exponential (figure 1, left panel). In high-iron tissues, the fast decay component is more prevalent and the signal demonstrates primarily rapid decay that cannot be sampled with long echo times. The decay is recovered by the application of the proton density constraint (figure 1, right panel). It is worth noting that applying a proton density constraint to the monoexponential model produces a less-than-ideal fit (figure 5, left panels). Because the signal decay is fundamentally nonexponential in the the presence of iron, the use of proton density constraint enables monoexponential models to demonstrate sensitivity to a wider range of iron at the expense of the goodness of fit to the acquired data.

Although CPMG sensitivity was improved with the PDE constraints, the relative CIs demonstrated in the Bland-Altman analyses are wide relative to comparisons between SSE and ${\text{R}}_2^{*}\text{ -LIC }$ values. Constrained monoexponential and nonexponential models demonstrated relative CIs of −110.1% to 183.7% and −157.2% to 235.9%, respectively. Prior studies comparing LIC via R₂ and ${\text{R}}_2^{*}$ have demonstrated relative CIs of −64.8% to 190.4%⁵ and −66% to 143%⁶. This suggests that none of the experimental models in this study are more accurate for LIC assessment across the entire LIC range than existing R₂ techniques. However, the largest relative uncertainty was observed at LIC values below 12^mg⁄_g. Split popluation Bland-Altman analysis demonstrates that LIC values of 12^mg⁄_g or greater demonstrate relative CIs of 57.4% to 164.1% for constrained monoexponential fits and 50.2% to 162.1% for constrained nonexponential fits, as seen in figure S2. Unlike other techniques, the absolute error demonstrated by the experimental techniques did not grow with LIC. Thus, at LIC values greater than 12^mg⁄_g, these models may demonstrate improved performance compared to other approaches. This is especially true for the constrained nonexponential model, which demonstrated absolute CIs of −7.5 to 7.5^mg⁄_g dry weight.

There are several known sources of error that contribute to the observed variability within the human data as well as between the human and experimental fitting results. The echo time limitations in the patient data weight the PDE at high LIC values. As a result, R₂ and α parameters are particularly sensitive to PDE errors. Both nonuniform RF excitation (${\text{B}}_1^{+}$ inhomogeneity) or spatial variation of coil sensitivity (${\text{B}}_1^{-}$ inhomogeneity) confound the PDE estimate. While the Philips image reconstruction corrects image intensity for coil sensitivity functions, errors in coil sensitivity maps will introduce PDE uncertainty. ${\text{B}}_1^{+}$ errors cannot presently be estimated from the image alone. In retrospect, ${\text{B}}_1^{+}$ mapping might have been used to improve PDE estimates. However, by only loosely (±20%) constraining the model PDE to measured PDE, we added some model resilience to PDE errors. Other signal fit confounders include liver fat and Rician noise bias. Fat is known to be present in concentrations of about 0–20% in the liver (unless there is severe metabolic syndrome), but contributions from fat have been ignored in this study. The CPMG protocol’s relatively short TR may lead to amplification of fat brightness in later echoes, which would have the effect of artificially decreasing the apparent decay rate. Rician noise effects were also not addressed in this study, other than including a fixed offset which is known to be suboptimal. Applying a Rician noise model or fitting signal decay in the complex domain would better describe noise behavior than a bias constant²⁶. Taken together, these effects may contribute to the observed difference in sensitivity to RR₂ and α noted between the human and simulation data and underscore the challenges present in trying to fit a higher-order model such as the nonexponential model to data that is inadequately sampled. A complete exploration of these differences and mitigation strategies will require additional research.

Despite of the potential sources of measurement error, it is also critical to realize that approximately 50% of the observed uncertainty in CPMG measurements and LIC arise from physiological variability in tissue iron distribution (see Figure 3). The Monte Carlo model provides noiseless measurements of proton density, signal decay, and LIC values, but the simulated CI are still half as large as for the real data. These measurement uncertainties arise because no two patients distribute iron exactly the same in the liver. The Monte-Carlo model captures intersubject variability in tissue iron distribution. As a result, 95% CI from the Monte Carlo model represents the lower limit of accuracy for the CPMG method with respect to a true LIC value. The real patient data in Figure 3, contains uncertainty not only from the imperfect CMPG measurements but uncertainty introduced by using liver ${\text{R}}_2^{*}$ as a reference standard.

6. Conclusion

Muscle-based proton density estimators derived from within existing imaging datasets provide a robust way to increase the sensitivity of CPMG-based R₂ estimates, imporving the potential clinical utility of such a sequence. The effect was demonstrated in both human and simulated data. CPMG underperformed relative to SSE R2 estimates of liver iron at low LIC values but has potential for superior performance at high LIC. Improved selection of scan timing parameters, signal models, and fitting techniques may improve the accuracy of estimates that rely on a proton density estimator.

Table 1:

Relevant scan parameters for patient exam.

Spin Echo
Type	CPMG Multi echo sequence
TE	6.5, 13, 19.5, 26, 32.5, 39, 45.5, 52 ms
TR	247 ms
NSA	1
Voxel Size	2.5×2.5mm
Slice Thickness	8 mm
Matrix	64×64
Bandwidth	4.1 kHz/pixel
Gradient Echo
Type	Multi-echo GRE Sequence
Echo Time	1.16, 2.22, 3.28, 4.35, 5.41, 6.47, 7.54, 8.60 ms (short ${\text{T}}_2^{*}$) 1.16, 2.66, 4.16, 5.66, 7.16, 8.66, 10.16, 11.66 ms (medium ${\text{T}}_2^{*}$) 1.16, 3.36, 5.56, 7.76, 9.96, 12.16, 14.36, 16.56 ms (long ${\text{T}}_2^{*}$)
TR	50 ms
NSA	1
Voxel Size	1.25×1.25mm
Slice thickness	8mm
Matrix	128×128
Bandwidth	1.5 kHz/pixel

Table 2:

Ranges of demographic and laboratory data for the participant population

Demographics	Quantity
Sex	24M,26F
	24 thalassemia
	6 sickle cell disease
Clinical diagnosis	5 Diamond-Blackfan syndrome
	15 other rare anemia

Measurement	Min	Max	Mean±StDev
Age [years]	2.0	43.1	16.4±9.4
Height [cm]	117	179.8	156.2±15.3
Weight [kg]	20.1	89.7	54.3±17.1
Body Surface Area [m2]	0.81	2.1	1.5±.3
Body Mass Index [$\raisebox{1ex}{$\text{kg}$}\!\left/ \!\raisebox{-1ex}{${\text{m}}^2$}\right.$]	14.7	34.3	21.9±5.1
Ferritin [ng/mL]	199	16300	4483±4890
ALT [U/L]	18	391	74.5±84.5

Table 3:

Bland-Altman comparison of LIC estimated by ${\text{R}}_2^{*}$ to constrained and unconstrained fitting models demonstrate that all models are unbiased. All mean values were statistically insignificant. Although the relative means appear large, this is due to division by near-zero LIC estimates for low-iron patients; the lack of bias is clearly seen in the absolute mean values. The simulation and human data confidence intervals are similar for the monoexponential model. However, the simulation data show much narrower intervals for the nonexponential model, demonstrating both the appropriateness of the model as well as the challenges posed when noise is present.

		Constrained Mean±StDev		Unconstrained Mean±StDev
Model		Human	Sim	Human	Sim
Monoexp	Relative %	−13.2±49.4	−17.3±65.7	−7.1±76.6	−13.6±42.1
	Absolute $\frac{mg}{g}$	0.0± 5.1	0.0± 2.9	0.0± 5.1	0.0±10.2
Nonexp	Relative %	1.6±44.2	−1.2±11.1	−1.8±52.5	−1.8±50.4
	Absolute $\frac{mg}{g}$	0.0± 3.8	0.0± 0.7	0.0± 4.4	0.0± 6.3

Supplementary Material

supinfo

Supporting Information Table S1 - Polynomial fit parameters for the monoexponential model.

Supporting Information Table S2 - Polynomial fit parameters for the non-exponential model.

Supporting Information Figure S1 - Bland Altman Plot for Patient Results. Bias is indicated by blue line; dashed blue line indicates non-bias (ie insignificant p value). Confidence intervals are specified by dashed red lines.

Supporting Information Figure S2 - Bland Altman analysis of Patient Results split at LIC=12^mg⁄_g. Open circles represent mean LIC < 12^mg⁄_g and filled circles represent mean LIC ≥ 12^mg⁄_g. Bias is indicated by blue line; dashed blue line indicates non-bias (ie insignificant p value). Confidence intervals are specified by dashed red lines.

Supporting Information Figure S3 - Bland Altman Plot for Patient Simulation Results. Bias is indicated by blue line; dashed blue line indicates non-bias (ie insignificant p value). Confidence intervals are specified by dashed red lines.

Supporting Information Figure S4 - Plots of polyfit equations relating LIC to R₂ and α.