Dynamic NMR effects in breast cancer dynamic-contrast-enhanced MRI

Abstract

The passage of a vascular-injected paramagnetic contrast reagent (CR) bolus through a region-of-interest affects tissue ¹H₂O relaxation and thus MR image intensity. For longitudinal relaxation [R₁ ≡ (T₁)⁻¹], the CR must have transient molecular interactions with water. Because the CR and water molecules are never uniformly distributed in the histological-scale tissue compartments, the kinetics of equilibrium water compartmental interchange are competitive. In particular, the condition of the equilibrium trans cytolemmal water exchange NMR system sorties through different domains as the interstitial CR concentration, [CR_o], waxes and wanes. Before CR, the system is in the fast-exchange-limit (FXL). Very soon after CR_o arrival, it enters the fast-exchange-regime (FXR). Near maximal [CR_o], the system could enter even the slow-exchange-regime (SXR). These conditions are defined herein, and a comprehensive description of how they affect quantitative pharmacokinetic analyses is presented. Data are analyzed from a population of 22 patients initially screened suspicious for breast cancer. After participating in our study, the subjects underwent biopsy/pathology procedures and only 7 (32%) were found to have malignancies. The transient departure from FXL to FXR (and apparently not SXR) is significant in only the malignant tumors, presumably because of angiogenic capillary leakiness. Thus, if accepted, this analysis would have prevented the 68% of the biopsies that proved benign.

Nuclear magnetic resonance (NMR) measurements carry information on the dynamics of the signal-producing molecules and/or others interacting with them. This is true whether the NMR sample is a liquid, a solid, or biological tissue. Thus, if correctly analyzed, magnetic resonance imaging (MRI) allows the mapping of molecular dynamic properties.

Quantitative MRI/Dynamic-Contrast-Enhanced MRI (DCE-MRI).

Quantitative MRI produces parametric maps of MR, pathophysiological, and/or pharmacokinetic biomarker properties. The DCE-MRI subcategory is particularly significant because it applies to a wide pathology range. In this technique, the T₁-weighted tissue ¹H₂O MRI signal intensity is acquired before, during, and after the (usually) bolus injection of a hydrophilic, paramagnetic contrast reagent (CR) (1). The CR passage through a tissue region-of-interest (ROI) can cause a transient increase of the longitudinal ¹H₂O relaxation rate constant [R₁ ≡ (T₁)⁻¹] with consequent elevated MR steady-state signal intensity.

Molecular Imaging of Water.

The CR is “detected” indirectly—via its effect on the water proton MR signal (2). Water is the most important biological molecule: Besides its solvent role, it fills the histological-scale tissue compartmental spaces. Water movement between these volumes is crucial in homeostasis and edemic abnormalities. Indeed, one can consider this equilibrium interchange an essential biological activity of water: It is regulated.

However, its very ubiquity makes water molecular MR imaging a challenge. Although the tissue water proton signal is the almost universal MR image source, the unambiguous discrimination of compartmental ¹H₂O resonances—subvoxel (volume element) in origin—is difficult. The use of a CR is almost always required. And, because the CR and H₂O compartmental distributions are always different, the equilibrium intercompartmental water molecule interchange kinetics may become influential and measurable.

Dynamic Nuclear Magnetic Resonance (DNMR).

Since almost its beginning, NMR has been recognized as enjoying a unique ability to detect and measure the kinetics of rapid equilibrium (exchange) molecular processes occurring within the sample. In the pre-MRI era, this aspect was commonly referred to as DNMR spectroscopy (3, 4). This terminology follows the deliberate oxymoron, “dynamic equilibrium,” of chemistry usage, and thus differs from the (time-dependence) meaning in DCE-MRI. In DNMR, however, the speed of an exchange process is manifest (in signal relaxation exponentiality) only as a heterodyne-like comparison of its intrinsic rate constant with its particular NMR shutter-speed [т⁻¹] (5) [т is a generalization of the NMR time-scale (3, 4)]. For the exchange between 2 molecular forms, the spectroscopic т⁻¹ for a nuclear spin is its (intensive) resonance frequency difference (Δω) in the two forms. For tissue ¹H₂O, however, ω is effectively compartment-independent (isochronous).

Just as with DNMR, the neglect of exchange kinetics considerations can lead to systematic errors in parameters extracted by quantitative DCE-MRI analyses. Examples here are the compartmental water mole fractions defining tissue spaces. Therefore, DCE-MRI is also a subcategory of in vivo MR molecular imaging—mapping the distribution and/or activity of molecules in living tissues.

Early in medical MRI development, it was realized that the DCE biomarkers contained considerable information, particularly about vascular properties. Major efforts have been mounted for the pharmacokinetic analyses of DCE time-course data (1). It was natural that mathematical models for these derivations were sought from mature algorithms in the nuclear medicine field. We refer to such paradigms as members of the standard model (SM) family. However, these formalisms were developed for tracer pharmacokinetics, with the intrinsic feature of direct radiotracer detection. Although the MRI CR plays the tracer role, the signal molecule remains water, which is differently distributed. The direct application of tracer pharmacokinetic models to MRI data (1) resulted in the inadvertent constraint that all intercompartmental equilibrium water exchange be treated as if infinitely fast. This corollary is not valid (2, 6), and its assumption can effectively short circuit MRI determination of CR compartmentalization (2, 7)— the pharmacokinetic essence. In a series of papers (2, 5, 6, 8–16), we have examined the significance of this implication. We refer to models incorporating equilibrium exchange effects in the pharmacokinetic derivation as belonging to the shutter-speed model (SSM) family.

Here, we survey the effects of intercompartmental water exchange kinetics in DCE-MRI breast cancer screening. They appear to facilitate discrimination of benign and malignant lesions, as we show here and in a companion paper (17).

Results

Pharmacokinetic Analyses.

Clinical DCE-MRI data are collected in Ernstian NMR steady-states [see Longitudinal Relaxation Pulse Sequences in supporting information (SI) Text]. An expression for such a tissue ¹H₂O steady-state is given in Eq. 1,

where S is the signal intensity and S₀ is that for the sample Boltzmann ¹H₂O magnetization (M₀). The symbols S are saturation (really unsaturation) factors (18). Eq. 1 recognizes that each tissue signal comprises potentially separable intracellular, interstitial, and blood compartmental ¹H₂O resonances; subscripts i, o (for outside), and b, respectively. (Plasma and intraerythrocyte blood ¹H₂O signals are averaged under almost all conditions.) The p′ coefficients represent the apparent water mole fractions (populations) in the physiological compartments of the signal-emanating volume (as small as an image voxel): p′_i + p′_o + p′_b = 1. Each S quantity comprises, in turn, exponential longitudinal and transverse relaxation factors (Theory in SI Text).

Eq. 1 recognizes that the tissue ¹H₂O signal can potentially exhibit triple-exponential longitudinal relaxation. This possibility is accommodated with a 3 × 3 exchange matrix accounting for equilibrium water exchange between blood and interstitium and between interstitium and cytoplasm—a 3-site 2-exchange (3S2X) system (2). But it is cumbersome to write nonmatrix algebraic expressions for the apparent compartmental quantity (p′_i, R′_{1 i}, p′_o, R′_{1 o}, p′_b, and R′_{1 b}) time-dependencies during CR pharmacokinetic passage in terms of the more fundamental intrinsic compartmental properties containing pathophysiological information.

Eliminating the often small blood term (i.e., let p′_bS_b → 0), on the right-hand side of Eq. 1 leaves a 2-site-exchange (2SX) system, for equilibrium water transcytolemmal interchange. But, nonmatrix algebraic expressions for the p′_i, R′_{1 i}, p′_o, and R′_{1 o} would still be cumbersome (19). However, there are simpler nonmatrix 2SX equations with phenomenological bases. We have presented (18) a version of Eq. 2,

where the notation is that of Eq. 1 except, instead of i and o subscripts, there are L and S subscripts (the primes are unnecessary), which distinguish 2 empirical exponential components with larger and smaller T₁ values, T_1L and T_1S, respectively (thus, R_1S > R_1L). [The saturation factors are defined: S_L,S {[sin α(1 − exp(−TR·R_1L,S))/(1 − exp(−TR·R_1L,S)cos α)]·exp(−TE·R*_2L,S)}, where α is the flip angle, TR and TE are the repetition and echo times, and R*₂ is the apparent transverse relaxation rate constant.] Ref. 5 provides R_1L, R_1S, and a_S (= 1 − a_L) expressions as functions of p_i, R_1i, p_o (= 1 − p_i), R_1o (the intrinsic values, in the absence of exchange), and τ_i, the mean intracellular water lifetime (reproduced in Theory in SI Text). [R_1o is expressed as (r_1o[CR_o] + R_1o0), where r_1o and [CR_o] are the interstitial CR relaxivity and concentration, respectively: R_1o0 is R_1o before CR_o arrival.] The p_i and p_o quantities (p_i + p_o = 1) are the true tissue properties of pathophysiological interest— they are measures of the compartmental volume fractions, v_i and v_e, respectively (e for extracellular, extravascular).

After [CR_o] is sufficient, compartmental identifications can be made (a_L = p′_i, R_1L = R′_{1 i}, a_S = p′_o, and R_1S = R′_{1 o}). Before [CR_o] becomes very large, it is possible that the S and L compartmental assignments are reversed, because it is possible that R_1i > R_1o0. However, it is not very long at all after CR_o arrival that an assignment switchover occurs (2) to yield the identifications given here, which remain for essentially the entire time CR is present in the tissue.

Before CR_o arrival, т⁻¹ [≡ |r_1o[CR_o] + R_1o0 − R_1i|] is effectively zero and sufficiently smaller than the exchange rate constant [τ_i⁻¹(1 + (p_i/p_o))] that the system is in the fast-exchange-limit (FXL) condition. This situation is tantamount to stating that S/S₀ is 〈p′S〉 or 〈aS〉 on the right-hand sides of Eqs. 1 and 2, respectively. As CR_o arrives and increases, the system departs the FXL for the fast-exchange-regime (FXR) condition, which we (5) defined to be the situation when the a_SS_S term on the right-hand side of Eq. 2 (vanished in the FXL) remains negligible. A significant contribution from the a_SS_S term defines the slow-exchange-regime (SXR) condition (5). As CR_o washes out of the tissue, this progression is reversed. Although cell suspension systems can be driven to the FXR, the SXR, and even the slow-exchange-limit (SXL) (Exchange Domains in SI Text), there has never been conclusive evidence for a tissue system reaching the SXR in a DCE-MRI study.

We conducted a number of different pharmacokinetic analyses here. The SM assumes that the equilibrium water intercompartmental exchange systems remain in the FXL: it is FXL-constrained (FXL-c). We also used the FXR- and SXR-allowed (FXR-a and SXR-a) versions of the first-generation Shutter-Speed Model (SSM1), BOLus Enhanced Relaxation Overview (BOLERO) (8, 11). In the SM analyses, the parameters varied are K^trans and v_e, while the τ_i value is (implicitly) zero (11, 13, 14). [K^trans measures the CR extra/intravasation rate (Theory in SI Text).] In some SSM analyses, τ_i is also varied. The complete 9-parameter set has been tabulated (14) and the constant values of the other parameters have been reported (13, 14).

MRI Data.

Data were obtained with consent from patients with positive mammographic and/or clinical MRI reports from standard institutional breast cancer work-ups and protocols. All had MRI contrast-enhanced lesions radiologically classified as Breast Imaging Reporting and Data System (BIRADS) four (B-4, suspicious) or five (B-5, highly suggestive of malignancy) (17). Emphasizing practicability and robustness, the data are of a rather routine clinical nature (they were obtained at 2 different institutions, with 2 different instruments, CRs, etc.): The 2 different data acquisitions were by no means optimized for quantitative DCE-MRI. For example, the pulse sequence parameters were prescribed by radiological spatial resolution and tissue volume coverage considerations because many images were to be used to guide immediately subsequent surgical interventions. The pharmacokinetic period is truncated, and although the spatial resolution is reasonable, the temporal resolution is not optimal. Also, the adipose tissue –¹H₂C– MR signal is suppressed in only one institution's acquisitions.

Fig. 1A shows the 3.3-min pharmacokinetic image of sagittal slice 20 (from lateral to medial) of a 57-year-old subject's left breast [patient 4 (17)]. The ROI marked represents 17 pixels (10 mm², 31 μL) containing the tiny lesion evident in this slice, subsequently found to contain both malignant invasive and in situ ductal carcinoma (IDC/DCIS). Fig. 1C shows the 3.6-min pharmacokinetic image of slice 12 of a 52-year-old subject's left breast [patient 8 (17)]. The ROI there represents 164 pixels (81 mm², 243 μL) containing the lesion evident in this slice, subsequently found to comprise benign lobular carcinoma in situ and stromal fibrosis (LCIS/SF). These images were acquired with adipose –¹H₂C– suppression (institutional protocol required). In contrast to those (13, 14) obtained with no fat suppression, these darker images show glandular regions brighter than fatty tissues.

Fig. 1.

Sagittal, fat-suppressed breast DCE-MRI image planes containing a malignant invasive and in situ ductal carcinoma (IDC/DCIS) (A) and a benign lobular carcinoma in situ with stromal fibrosis (C) are shown. The relative ¹H₂O signal (S/S_pre) DCE-MRI time courses (circles) for ROIs containing the malignant and benign lesions are shown in B and D, respectively. The arterial input function (the plasma CR concentration variation) used in the analyses is shown in the Inset in D. There are 3 model-fitted (black) curves associated with each lesion data set. The dashed curves represent the best fittings with the SM (FXL-c). The solid curves represent the best fittings with the FXR- and the SXR-a SSM versions. [All are indistinguishable except for the dashed fitting to the IDC/DCIS data (B).] The model parametric values of these fittings are given in Table 1. The dotted curves represent the exchange-limiting expectations with the SSM parametric values. The upper curves delimit the FXL conditions, whereas the lower curves delimit the NXL conditions—red and blue for the SXR-a and FXR-a SSM analyses, respectively.

The ¹H₂O S/S_pre time courses for these 2 ROIs are shown in Fig. 1 B and D (circles). The CR injection timing is suggested by the rectangular function below the Fig. 1D abscissa. The shape of the IDC/DCIS ROI data (Fig. 1B) is that often reported as characteristic of such tumors: much faster and greater uptake (followed by washout) compared with a benign lesion such as that of Fig. 1D (13, 20–23). The mean arterial input function (AIF), determined from axillary artery ¹H₂O signal time courses in 3 patients, shows the plasma CR concentration, [CR_p], peaking near 5 mM at ≈0.75 min (Fig. 1D Inset).

There are many traces in Fig. 1 B and D. Three curves each are fitted to the IDC/DCIS ROI data and the LCIS/SF ROI data. The black dashed curve distinguishable in Fig. 1B represents the best FXL-c (SM) fitting of the data. The model has the constraints that τ_i is vanishingly small (the FXL) and that the transverse relaxation factor [exp(−TE·R*₂)] = 1. The dashed curve shows the characteristic undershoot at the beginning, followed by overshoot and then undershoot during the washout (11, 13, 14). The K^trans and v_e parameter values from this fitting are listed in Table 1. [Other parameter relationships and fixed values are given in the table legend. Acquisition pulse sequence parameter values (Materials and Methods) were also used.] The black solid curves represent the fittings from the FXR-a and SXR-a versions of SSM, which are equally better than the FXL fitting and are thus superimposed. These fittings employ the constraints that τ_i is held fixed at 0.4 s (the reason for this value will be seen below) and that (especially for SXR-a) exp(−TE·R*_2L,S) are each = 1. The latter are of course not necessarily true (see below). The FXR-a version has the additional constraint to only the Eq. 2 S_L factor (a_L → 1, a_S → 0). Except for the FXL-c (SM) analysis of IDC/DCIS, the random fitting errors (χ²σ², given and defined in Table 1) are not widely different. Of course the FXR-a and SXR-a fittings return K^trans and v_e values different from FXL-c (Table 1): The v_e values are increased, as is K^trans for the FXR-a IDC/DCIS fitting. All 3 fitted curves are essentially superimposed for the Fig. 1D LCIS/SF data.

Table 1.

Fitting results for Fig. 1

Lesion	FXL-c (SM)			FXR-a (SSM)			SXR-a (SSM)
	Ktrans, min−1	ve	τi, s	Ktrans, min−1	ve	τi, s	Ktrans, min−1	ve	τi, s
IDC/DCIS (Fig. 1A and B) patient 4	0.153 (± 0.014)	0.29 (± 0.03)	→ 0	0.211 (± 0.017)	0.64 (± 0.06)	0.40 (fixed)	0.155 (± 0.008)	0.38 (± 0.01)	0.40 (fixed)
χ2σ2		0.14			0.018			0.0085
LCIS/SF (Fig. 1C and D) patient 8	0.032 (± 0.003)	0.27 (± 0.07)	→ 0	0.033 (± 0.004)	0.63 (± 0.07)	0.40 (fixed)	0.026 (± 0.001)	0.42 (± 0.07)	0.40 (fixed)
χ2σ2		0.0037			0.0036			0.0017

Other parameter relationships and fixed values: r_1o (fixed at 4.1 s⁻¹·mM⁻¹); v_e = f_W(1 − p_i), where f_W is the volume fraction accessible to mobile aqueous solutes (fixed at 0.8); R_1o0 = (R₁₀ − p_iR_1i)/(1 − p_i), where R₁₀ is the tissue ¹H₂O R₁ before CR arrival (fixed at 0.55 s⁻¹); R_1i (fixed at 0.55 s⁻¹); [CR_b] = (1 − h_s)[CR_p], where h_s is the microvascular hematocrit (fixed at 0.3) (17). Fitting errors: χ² is a weighted sum of squared residuals, and σ is the standard deviation at each time point [Yankeelov TE,et al. (2005) Evidence for shutter-speed variation in CR bolus-tracking studies of human pathology. NMR Biomed 18:173–185].

Normal-appearing glandular (NAG) or adipose tissue ROI signal time courses show almost no enhancement (see figures 1b and d of ref. 13 and 1c of ref. 14). Fittings of NAG data typically yield very small parameter values; K^trans = 0.0004 min⁻¹ and v_e = 0.03.

There are 4 additional dotted curves (blue and red pairs) each in Fig. 1 B and D. The top member of each pair represents the FXL (τ_i → 0) expectation for the FXR-a (blue) and for the SXR-a (red) Table 1 parameter values. The bottom members represent the corresponding no-exchange-limit (NXL) (τ_i → ∞) expectations. Thus, the vertical dynamic range between the top and bottom members of a pair is a measure of the model exchange sensitivity for this acquisition of these data. This property is not the same as the shutter-speed. The т⁻¹ variation during the CR bolus passage, in comparison with the constant exchange rate constant [τ_i⁻¹(1 + (p_i/p_o))], determines the position of the data within the exchange sensitivity range.

We have reported other examples of the top blue curves; see figures 1d of ref. 14 and 1b and d of ref. 13. Note the significantly greater difference between the top blue curve and the data for the malignant IDC/DCIS (Fig. 1B) than for the benign LCIS/SF (Fig. 1D) tumor. This mismatch was also seen in the comparison of an IDC with a benign fibroadenoma (figure 1b and d of ref. 13). This observation is a preview of the FXR-a analysis ability to discriminate malignant from benign breast tumors. The only way the FXL-c (SM) model can bring the blue dotted curve down to the data circles is by reducing parameter values below those for the FXR-a fitting. Although the SM comes close to matching the data for the benign lesion, it fails much more significantly for the malignant tumor—just when needed.

The NXL expectation for the FXR-a parameters (bottom blue dotted line) shows no S/S_pre change during the CR bolus passage. For the NXL, a_S → (1 − p_i), R_1S → r_1o[CR_o] + R_1o0, a_L → p_i, and R_1L → R_1i, as they must (R_1S > R_1L). Thus, in this limit, R_1L = R_1i, which is [CR_o]-independent, and because there is only S_L for the FXR-a Eq. 2 version, S is constant. The exchange sensitivity for analysis by the FXR-a version (blue curves) is necessarily greater than that for analysis of the same data by the SXR-a version (red curves). One way to understand this maxim is by inspecting theoretical putative calibration curves implicit in the analyses [Calibration Curves in SI Text]. Before CR_o arrival, a_S is zero and R_1L is p_iR_1i + p_oR_1o: the FXL. When CR_o first arrives, a_S grows only very slowly: the FXR condition obtains until its contribution becomes significant (5, 18).

However, it is very interesting that the Fig. 1B IDC/DCIS data are essentially superimposed on the no-exchange limiting (bottom red) curve for the SXR-a model. [We have obtained a similar result for transendothelial water exchange (16).] Consistently, if these data are analyzed with the SXR-a model floating τ_i, it returns the algorithm's upper bound value. We stopped when τ_i exceeded the unrealistically large 35 s. This behavior is true for 6 of the 7 malignant tumors in this population. In contrast, the Fig. 1D benign LCIS/SF data are clearly inside the smaller red dotted curve exchange range. Correspondingly, the SXR-a fitting with floating τ_i returned a value of 110 ms. However, most of the benign lesions (11 of the 15 total) exhibited the same upper bound indeterminacy as the malignant tumor in this regard.

The FXR-a calibration curves (SI Text) imply that there will be greater parameter value differences between FXL-c (SM) and FXR-a (SSM) analyses. This result is seen in Table 1. For the IDC/DCIS data, FXR-a analysis increases v_e by more than 2-fold and K^trans by 38%. If τ_i is let float, FXR-a analysis returns a value of 360 (± 58) ms (SD) (with no K^trans change and an 8% v_e decrease), which is very reasonable (as we will see below).

Ostensibly, the SXR-a model includes the FXR-a model. However, the results suggest that the SXR-a model is less compatible with the data, an indication of systematic error. We will discuss possible reasons for this below. We now present the results for analyses of the data from all of the lesions by using the FXL-c SM and the FXR-a SSM. As we will see, these breast screening DCE-MRI data seem quite compatible with FXR-a analyses.

Comparison of Pharmacokinetic Model and Gold Standard Pathology Analyses.

For each of the 22 patients, 2 investigators independently analyzed ROI data from 1 sagittal image slice (of 16 to 40 per breast) exhibiting a lesion biopsied only post-DCE. They were each blinded to the biopsy pathology reports. An ROI boundary was independently manually drawn around the entire lesion in a pharmacokinetic image showing near maximal enhancement. The results of 2 different pharmacokinetic analyses (FXL-c and FXR-a) of the DCE-MRI data from the 22 lesion ROIs by each investigator were then averaged lesion-by-lesion, model-by-model, and parameter-by-parameter. The FXL-c analyses varied K^trans and v_e, whereas the FXR-a analyses also varied τ_i. Fig. 2 presents 1D scatter plots in each of 5 different single parametric space dimensions: K^trans (A), v_e (B), τ_i (C), K^trans/v_e (D), and (1 − v_e)/τ_i (E). The latter two are pair-wise linear combinations from the first 3 traditional adjustable model parameters; their definitions are elaborated below. In each case, the 7 proven malignant lesion ROI results are shown as black circles, with the 15 solely benign lesion ROI results as red triangles. Subsequent biopsy and pathology analyses proved 15 (13 of the 17 B-4 and 2 of the 5 B-5) lesions to be of several different solely benign types. The (only) 7 (4 B-4 and 3 B-5) malignant lesions comprised 3 IDCs, 1 DCIS, and 2 IDC/DCIS and 1 IDC/LCIS mixtures. The companion paper (17) provides details.

Fig. 2.

Scatter plots show the DCE-MRI pharmacokinetic parameter values for 22 subject lesion ROIs. Black circles represent those proven malignant by subsequent biopsy/pathology, whereas red triangles indicate those found solely benign. A–C give the 3 traditional independent parameter (K^trans, v_e, and τ_i) values, respectively measuring the CR extravasation rate, the interstitial volume fraction, and the mean intracellular water lifetime. Precise definitions are given in the text. D and E give the analogous plots of the linear combination K^trans/v_e and (1 − v_e)/τ_i parameters, which respectively measure k_ep (the CR intravasation rate constant) and the cytolemmal water permeability coefficient surface area product (P_WS′). For A–D, the left column gives the SM (FXL-c) results, and the right column gives the SSM (FXR-a) results. Connector lines join individual ROI values. Because the SM considers P_WS′ infinitely large, E shows only the SSM results. The group mean values are given as squares on the left and right sides for the SM and SSM columns, respectively. The error bars represent (±SD) values within each category. One malignant lesion FXR-a column outlier is plotted in the Insets in A and D and is excluded from the SD calculations for both the FXL-c and FXR-a results in those panels.

For each parametric dimension [except (1 − v_e)/τ_i] (Fig. 2E), the left and right columns represent the results of the SM (FXL-c) and SSM (FXR-a) analyses, respectively. The SM/SSM pair representing each ROI has a connector line. The group mean values for each parameter are given as squares on the left and right sides for the SM and SSM analyses, respectively. The error bars represent SD values within each category. There is one malignant (B-5) outlier so elevated that it is plotted in Fig. 2 A and D FXR-a column insets and is excluded from the Fig. 2 A and D SD calculations.

It is obvious that none of the SM parameters allows complete separation of the circles (malignant) and triangles (benign). None achieve anywhere near 100% positive predictive value (PPV) (the best is 54%, for SM K^trans). Compared with the SSM, the SM generally underestimates the parameter value [albeit with some precision benefit from reduced parametrization (11, 14)], except for k_ep (Fig. 2D) and (1 − v_e)/τ_i (Fig. 2E). The latter is effectively infinite in the SM. Discussion arguments suggest that these underestimations are in the absolute sense. Importantly, in the K^trans and K^trans/v_e (Fig. 2 A and D) cases, it is the SM misestimation that prevents better separation of circles and triangles achieved by the SSM (70% PPV for each). For the K^trans parameter (Fig. 2A), the SM underestimation has occurred in only malignant tumor ROIs (circles). [These include the Fig. 1 malignant/benign ROI pair (Table 1), and the only other such pair reported (13).] One black circle is comingled with the red triangles in the Fig. 2A SSM column only because of partial volume dilution (17). For K^trans/v_e (Fig. 2D), it is the SM overestimation of particularly benign ROI values that prevents their almost complete separation achieved by the SSM. [The ratio K^trans/v_e is equal to k_ep, the unidirectional CR intravasation rate constant. For passive CR transport, k_ep is equal to (v_b/v_e)k_pe (1); v_b is the tissue blood volume fraction, and k_pe is the rate constant for CR extravasation.] These K^trans and k_ep parameter behaviors are not observed for the SXR-a analyses.

There are several other items of note in Fig. 2. The SSM v_e values are increased in both benign and malignant lesions, and they rise to be quite large in some cases, all but one of them benign (Fig. 2B). The large v_e malignant tumor is mostly benign (17). Although the malignant and benign τ_i values become finite [and quite reasonable (see below)] with the SSM analyses (Fig. 2C), they overlap considerably (thus, this behavior does not contribute to discrimination). It is interesting that the benign values tend to be somewhat larger, although less certain. Although τ_i⁻¹ is proportional to the mean individual cellular claustrophobic ratio [surface area (A) to volume (V)] (6), it also increases with the transcytolemmal water permeability coefficient [τ_i⁻¹ = P_W(A/V)]. The P_W magnitude can reflect the extent of cytolemmal aquaporin expression and/or the cellular metabolic status. The ratio (1 − v_e)/τ_i ≈ (1 − v_e − v_b)/τ_i = v_i/τ_i = P_WS′, where v_i is the tissue intracellular volume fraction and S′ is the total cell surface area per voxel volume. P_WS′ is considered infinitely large by the SM. When P_WS′ is calculated by the SSM, the benign tumor values cluster considerably around 1.1 s⁻¹ and the malignant tumors around 1.3 s⁻¹ (Fig. 2E). Even more remarkable clustering is seen for the SSM benign k_ep values, near 0.15 min⁻¹ (Fig. 2D). This finding reinforces the common notion of considerably greater heterogeneity in malignant tumors. It also means that, although the SSM v_e values show considerable scatter (and no discriminatory power) (Fig. 2B), they are well correlated with the SSM K^trans values; see the companion paper (17).

Discussion

The Fig. 2A results are particularly interesting. For the 15 benign lesions, there is essentially no change in K^trans between the SSM (FXR-a) and SM (FXL-c) analyses [in most cases, there is absolutely no change (within error)]. However, in every one of the 7 malignant tumors, there is a K^trans decrease upon going to the SM analysis. Because malignant breast tumors are angiogenic (21, 24), it is sensible that malignant lesions have larger K^trans values. This inference suggests that the greater overlap of the FXL-c (SM) benign/malignant K^trans values represents underestimations in the malignant cases: The K^trans values are actually different. The mean ΔK^trans [≡ K^trans(FXR-a) − K^trans(FXL-c)] is 0.06 min⁻¹ for 6 malignant tumors and 0.006 min⁻¹ for the 15 benign lesions (17). This biomarker allows their complete discrimination (100% PPV) (17). This result and other Fig. 2 entries suggest that the FXR-a analyses are yielding reasonable results in the absolute sense. Although with varying precision, the parameter values determined agree quite well with independent measures (Literature Comparison in SI Text).

It is particularly the τ_i aspects of applying the SXR-a SSM version to the malignant tumor data that suggest SXR-a incompatibility. As stated above, for 86% of the malignant tumor and 73% of the benign lesion ROIs, SXR-a analyses floating τ_i increased its value until the program incrementation limit was reached. In most cases, the upper bound allowed was 40 s. Fig. 1B shows that the IDC/DCIS data points are on the (red) NXL curve for the SXR-a model. As far as SXR-a analyses are concerned, τ_i → ∞ for these breast tumors. This extreme is unreasonable and cannot be attributed to only τ_i uncertainty.

Thus, the cumulative weight of the literature results cited and our own results suggests that the FXR-a analyses yield reasonable pharmacokinetic parameter values when the SXR-a analyses do not. Why might that be?

One possibility is that none of the many diverse biological systems studied in the literature proceed sufficiently to depart the FXR for the SXR. However, if so, SXR-a analyses would give the same results as FXR-a analyses, because the former differs only if the SXR condition is reached.

A second possibility lies in the way the FXR-a analyses are performed. At each pharmacokinetic time point, the (extensive) signal intensity is used to estimate the (intensive) R_1L value—because there is only the S_L factor on the right-hand side of Eq. 2. Then, it is the R_1L time-dependence that is fitted. This operation may effectively unweight any contribution from an actual a_SS_S term. Because there is only a single recovery time point (at TR), this seems a particularly parsimonious approach.

A third possibility is that the SXR-a analyses overestimate the S_S factor contribution, which could be disproportionately quenched by transverse relaxation. The a_SS_S term always begins at zero and grows only slowly with increasing [CR_o] (Calibration Curves in SI Text and Fig. S1). The fractional a_SS_S contribution transiently maximizes (at [CR_o]_max) at ≈0.05 and ≈0.29 for benign and malignant tumors, respectively, even with exp(−TE·R*_2S) = 1. It is much smaller for most of the time course. Surely, however, the exp(−TE·R*_2L,S) factors are not actually unity for real acquisitions. Although still mostly intravascular, paramagnetic CR causes significant magnetic field inhomogeneities in perivascular spaces. As [CR_o] rises from zero, the first populations of interstitial water molecules whose magnetization distinguishes itself as a_S are those furthest from cells; some is pericapillary water. Then, as [CR_o] increases further, the increasing interstitial/cytoplasmic magnetic susceptibility difference leads to increasing field inhomogeneities in the entire interstitium. These aspects cause tissue R*_2S to become significantly greater than R*_2L. Even with the small TE of 3.5 ms, exp(−TE·R*_2S) would be reduced to 0.57 if R*_2S increased to only 160 s⁻¹. For a homogeneous Lorentzian spectral line, this rate constant corresponds to a width at half maximum, Δν_½ [≡ R*_2S/π] of 51 Hz, which at 1.5 T is only 0.8 ppm. This linewidth is easily possible (Interstitial Field Inhomogeneities in SI Text). Thus, it may very well be that a_SS_S{≡ a_S[sin α(1 − exp(−TR·R_1S))/(1 − exp(−TR·R_1S)cos α)]·exp(−TE·R*_2S)} is always significantly quenched [<0.17 (= 0.29·0.57)] below the SXR-a expectation. At any rate, the FXR range is likely extended to a larger [CR_o] value. The data behave as if they have been edited for S_L: the dominant intensive R_1L property may be somewhat immune to variations of the extensive a_L and a_S properties.

Conclusions

The DCE-MRI acquisitions for these data, prescribed for radiological considerations, were not particularly exchange sensitive. Even so, exchange effects seem to facilitate very high discrimination of malignant from benign breast tumors (17). In recent years, there has been considerable effort devoted to decreasing DCE-MRI pulse sequence exchange sensitivity (25). Our results suggest that there could be significant profit, not the least for cancer screening, in working to increase exchange sensitivity, by adjusting not only α and TR, but also TE (a small increase may improve specificity). The considerations expressed here could guide such endeavors.

Materials and Methods

Details on some of the MRI data acquisitions and analyses have been published (13, 14); more particulars and differences are presented here.

Patients.

Twenty-two patients were recruited from clinical breast cancer populations that had undergone initial screening (mammographic and/or clinical breast MRI protocols) at 2 different institutions: Stony Brook (SB) and Sloan-Kettering (SK). All were placed in the B-4 (n = 17) or B-5 (n = 5) classifications, which were to lead to biopsy procedures. Numerous details on the subjects are given in the companion paper (17).

DCE-MRI Data Acquisition.

The study was conducted using 2 different 1.5-T MR systems [Edge (Philips/Marconi), SB; Excite (General Electric), SK] with body radiofrequency transmit coils, and 4- (SB and SK) and 7-channel (SK) phased-array, prone patient, bilateral breast radiofrequency receive coils having compression plates. A 3D SPGR pulse sequence was used to acquire 12–20 serial sagittal image volume sets continually, covering the entire breast with the suspicious lesion to be biopsied. Other parameters included 10° (SK) and 30° flip angles, 3- to 4-ms TE, 6- to 9-ms TR, 3-mm section thickness, 18- to 24-cm field of view, and 256 × 128 and 256 × 64 (SB) matrix sizes, zero-filled to 256 × 256 during image reconstruction. The SK acquisitions suppressed the fat –¹H₂C– signal with narrow-band saturation radiofrequency pulses centered on its ω value. The phase-encoding direction was superior/inferior. Depending on the breast size, each volume set contained 16–32 image sections, resulting in 13- to 28-s temporal resolutions. At the start of the second volume set acquisition, the CR [gadolinium diethylenetriamine pentaacetate bismethylamide (GdDTPA-BMA) (Omniscan; Nycomed), SB; GdDTPA²⁻ (Magnevist; Berlex), SK] was delivered via an antecubital vein (0.1 mmol/kg at 2 mL/s) and followed by a saline flush, using programmable power injectors (Spectris; Medrad).

DCE-MRI Pharmacokinetic Analyses.

An ROI was manually drawn within an axillary artery visible in 3D volume image slices. This region was used for AIF determination. For each of the 6 SB patients, the representative AIF from one of them was used. This trace is shown in figures 1 of refs. 13 and 14. Individual AIF time courses were measured for 3 different SK patients. A mean AIF time course (Fig. 1D Inset) was generated by simply averaging these trajectories and used for each of the 16 SK subjects. These AIF time-course data were interpolated by using a 7-parameter empirical expression (11). For each patient, an ROI was manually drawn to circumscribe the entire enhanced lesion in a DCE-MRI image exhibiting near maximal enhancement. The mean tumor tissue ¹H₂O ROI DCE signal intensity/AIF time-course pairs were then subjected to both SM (FXL-c) and SSM (FXR-a) pharmacokinetic analyses to extract K^trans, v_e, and τ_i (SSM only) parameter values. Many details of the SM and SSM pharmacokinetic modeling of time-course data are described above, and some images from the 6 SB subjects (3 malignant and 3 benign tumors) have been reported (13, 14).