Intratumor mapping of intracellular water lifetime: metabolic images of breast cancer?

Abstract

Shutter-speed pharmacokinetic analysis of dynamic-contrast-enhanced (DCE)-MRI data allows evaluation of equilibrium inter-compartmental water interchange kinetics. The process measured here – transcytolemmal water exchange – is characterized by the mean intracellular water molecule lifetime (τ_i). The τ_i biomarker is a true intensive property not accessible by any formulation of the tracer pharmacokinetic paradigm, which inherently assumes it is effectively zero when applied to DCE-MRI. We present population-averaged in vivo human breast whole tumor τ_i changes induced by therapy, along with those of other pharmacokinetic parameters. In responding patients, the DCE parameters change significantly after only one neoadjuvant chemotherapy cycle: while K^trans (measuring mostly contrast agent (CA) extravasation) and k_ep (CA intravasation rate constant) decrease, τ_i increases. However, high-resolution, (1 mm)², parametric maps exhibit significant intratumor heterogeneity, which is lost by averaging. A typical 400 ms τ_i value means a trans-membrane water cycling flux of 10¹³ H₂O molecules s⁻¹/cell for a 12 µm diameter cell. Analyses of intratumor variations (and therapy-induced changes) of τ_i in combination with concomitant changes of v_e (extracellular volume fraction) indicate that the former are dominated by alterations of the equilibrium cell membrane water permeability coefficient, P_W, not of cell size. These can be interpreted in light of literature results showing that τ_i changes are dominated by a P_W(active) component that reciprocally reflects the membrane driving P-type ATPase ion pump turnover. For mammalian cells, this is the Na⁺,K⁺-ATPase pump. These results promise the potential to discriminate metabolic and microenvironmental states of regions within tumors in vivo, and their changes with therapy.

INTRODUCTION

Tumor heterogeneity

Cell type and metabolic heterogeneity are crucial tissue characteristics. For example, there is much current interest in intratumor phylogenetic diversity. Its metabolic consequences are likely of great significance in therapy resistance or tolerance (1–6). This heterogeneity presents a challenge to blood tests, point biopsies, and ex vivo tissue homogenization, and puts a premium on in vivo assessment with the highest possible spatial resolution and on an individual-by-individual basis (7). The most feasible current methods of human metabolic imaging, ¹⁸ F positron emission tomography (8) and hyperpolarized ¹³C magnetic resonance spectroscopic imaging (9), are extremely informative, but generally have insufficient resolution (typically 5 and 7 mm, respectively) to clinically assay intratumor heterogeneity, and are costly. In contrast, water hydrogen proton (¹H₂O) MRI affords millimeter or sub-millimeter spatial resolution in human studies, uses no ionizing radiation, is minimally invasive, and is relatively inexpensive. Though millimeter or sub-millimeter does not match the resolution of histopathology microscopy, it detects considerable intratumor heterogeneity. Conventionally, however, ¹H₂O MRI is thought to provide only anatomical/vascular (and sometimes tissue functional) information. Fortunately, a new opportunity is presenting itself. It arises from the ability of MR to measure intercompartmental water molecule exchange kinetics. This study investigates the implications of the heterogeneity of such trans-cytolemmal kinetics within malignant human breast tumors before and after neoadjuvant chemotherapy (NACT).

BACKGROUND

Mean intracellular water molecule lifetime (τ_i)

The kinetics of the equilibrium (steady-state) exchange of water molecules across cell membranes have been studied by isotope labeling techniques for almost 60 years (reviewed in (10)) and by NMR methods for over 40 years (reviewed in (11–13). For a ‘well-mixed’ cytoplasm, the kinetics can be expressed in Equation [1]:

1

where τ_i is the mean lifetime of a water molecule inside the cell, P_W the cell membrane water permeability coefficient, and (A/V) the claustrophobia ratio (A is the individual cell surface area and V the individual cell volume) (14). The NMR community tends to report τ_i values (11–13) while the isotope community gives P_W values: P_W is often labeled P_d, the diffusive permeability (10). (We note that, even though they have the same dimensions (distance/time), P_W [≡ P_d] is not the same as P_f, the water permeability coefficient in the presence of a transmural osmotic gradient, which causes net trans-membrane water transport (‘flow’) simultaneous with the much faster exchange (15–17).) Using Equation [1], one finds published P_W and τ_i values to be in general agreement (13). The τ_i reciprocal, τ_i⁻¹, is the unidirectional first-order rate constant, k_io, for equilibrium water efflux from the cell – there is no net

2

transport (18). Values of τ_i⁻¹ span four orders of magnitude – the extremes range from 10⁻² s⁻¹ for Xenopus oocytes to 10² s⁻¹ for erythrocytes (11–13,19). For virtually all other cells, however, τ_i values are hundreds of milliseconds and the ‘well-mixed’ approximation is quite good (20). For only the very largest, e.g. the Xenopus oocyte (1 mm diameter), does this condition just begin to fail (19).

Equation [1] can be approximated with Equation [3]:

3

where C is a constant shape factor and d the cell diameter. If the cell is spherically shaped, C = 6 (if the cell shape is well approximated as a right cylinder, C = 4). The quantity d is a one-dimensional (1D) measure of cell size.

There has been some recent concern (21) that interpretation of literature data on the blood wash-out following bolus injections of isotopically labeled water implies cerebral cortical τ_i ≥ 30 s. This is almost two orders of magnitude greater than values generally found (10–13). Taking a typical value of P_W = 1.4 × 10⁻⁴ cm s⁻¹ (10), a τ_i of 30 s in Equation [3] yields a d value of 250 µm: clearly much larger than the average cell (6). A simple tracer intravasation interpretation of the same blood wash-out data yields (22) a mean lifetime for extravascular water (τ_exv) of 21 s – in good agreement with the blood capillary water permeability coefficient from isotopically labeled water experiments. Seemingly, τ_exv has been mis-assigned as τ_i in (21).

Active trans-membrane water cycling

The equilibrium water exchange process has been conceived as resulting from passive molecular mechanisms: simple diffusion across the lipid bilayer, passage through aquaporin membrane protein channels, leakage through membrane transporters, etc. (23–25). Changes in d (or C) – cellular swelling or shrinking (edema) – will alter τ_i⁻¹. However, the rate constant for d change is at least an order of magnitude smaller than τ_i⁻¹ (k_io) itself (15,17). Any τ_i⁻¹ changes not due to d (or C) alterations must be due to changes in P_W, which has always been thought of as a passive cell membrane property (P_W(passive)). However, NMR studies have recently revealed an active component (P_W(active)) much larger than the passive contribution (17). This is due to active trans-membrane water cycling that accompanies active trans-membrane osmolyte cycling, which is paced by the driving cell membrane P-type ATPase ion pump (17). For mammalian cells, this is the Na⁺,K⁺-ATPase, NKA (17,26–28). The existence of this water cycling is significant.

NMR principles

The classic NMR measurements are mostly of homogeneous cell suspensions. The presence of an extracellular paramagnetic contrast agent (CA) increases the intrinsic longitudinal relaxation rate constant R_1o (≡ (T_1o)⁻¹) of the outside water proton signal (¹H₂O_o). Though the term was introduced only in 1999 (29), this approach increases the longitudinal ‘shutter-speed,’ т₁⁻¹ (≡ | R_1o − R_1i | ), sufficiently that the water exchange NMR system is moved out of the fast-exchange-limit (FXL) condition (R_1i is the intrinsic intracellular relaxation rate constant). A sufficient outside CA concentration, [CA_o], allows the NMR system to reach the slow-exchange-regime (SXR) condition. This is characterized by non-mono-exponential longitudinal magnetization recovery, but is distinct from the slow-exchange-limit and no-exchange-limit conditions (13,17,22,29,30). It is quite common to achieve the SXR with cell suspensions (11–13,17,22). However, if the [CA_o] value is only modest, and the system reaches only the fast-exchange-regime (FXR) condition, then τ_i⁻¹ measurement requires т₁⁻¹ variation (by varying [CA_o]) (17,22,29–31). The FXR condition features apparent mono-exponential longitudinal recovery, i.e., the measured R₁ is single valued, but has a non-linear [CA_o] dependence. In the FXL condition, this dependence is linear (13,17,22,29,30). (Sometimes, transverse ¹H₂O relaxation is employed for cell suspensions (11,12)).

Dynamic-contrast-enhanced (DCE)-MRI

The spatial encoding of ¹H₂O in MRI offers the possibility of measuring, and mapping, τ_i values in human biological tissues in vivo. A stepped CA infusion producing incremental plasma and interstitial CA steady-state levels works well in animal models (29), but is impractical for human use. However, in the common clinical DCE-MRI approach, serial T₁-weighted ¹H₂O images are obtained before, during, and after a bolus CA injection. The time-course of the CA bolus passage through the field of view is measured. The shutter-speed concept can be incorporated into the pharmacokinetic analysis of the time-course. This differs from the tracer pharmacokinetic paradigm, which is based on the fact that compartmentalization is not encoded in the tracer signal: it does not distinguish compartments entered by tracer. In DCE-MRI, CA compartmentalization is intrinsic to the ¹H₂O signal, which continuously tracks the amounts of H₂O and CA in each compartment. Thus, imposition of a classic tracer analysis on DCE-MRI data joins these contradictory postulates about CA compartmentalization, and the only reconciliation is the inherent and incorrect assumption that the NMR system is always in the FXL condition, i.e., that τ_i is effectively zero. Thus, besides systematically distorting the values of other pharmacokinetic parameters, this renders τ_i⁻¹ inaccessible. For a recent detailed overview, see (30). Immediately after a bolus intravenous (IV) injection, in almost all tissues CA transiently extravasates and NMR systems transiently depart the FXL for the FXR condition. There is no evidence that, with CA doses approved for human subjects, any system ever reaches the SXR condition (13,30). However, the interstitial CA concentration, [CA_o] (and thus т₁⁻¹), value varies naturally – increasing and decreasing during the DCE time-course.

Using the shutter-speed pharmacokinetic paradigm (SSP), we (30,32–40) and others (41–45) have presented high-resolution, pixel-by-pixel, τ_i maps. Importantly, the τ_i magnitudes in the maps, and averaged over regions of interest (ROIs), are in general agreement with those from the many, precise NMR spectroscopic cell suspension measurements (11–13). This indicates that in vivo τ_i accuracy is reasonable.

Let us consider fundamental aspects of the DCE pharmacokinetic parameters as biomarkers. The K^trans quantity_, mainly a rate constant for capillary CA extravasation/tissue arrival, is elaborated in Equation [4]:

4

k_ep is the first-order rate constant for CA intravasation, v_e the CA distribution volume (mainly the extracellular, extravascular volume fraction), k_pe the rate constant for CA extravasation, v_p the blood plasma volume fraction, P_CA the capillary wall CA permeability coefficient, and S the total blood capillary surface area (13,18,30,34,35,46). The v_e, v_p, and S magnitudes depend on the extent of voxel or ROI cell density (for v_e) or capillary density (for v_p or S) magnitudes. Thus, though the K^trans, v_e, and v_p biomarkers are formally intensive quantities, their values reflect the numbers of cells or capillaries. The τ_i parameter, on the other hand, is a true intensive property. For spherical cells, Equation [3] yields τ_i = 0.17(d/P_W), where d and P_W represent, respectively, the mean voxel or ROI cell diameter and cytolemmal water permeability coefficient. The τ_i magnitude does not depend on the extent of the voxel or ROI cell density (ρ) value: τ_i is independent of v_e, or the intracellular volume fraction, v_i (≈(1 – v_e)). This is an important, and novel, characteristic for an imaging biomarker.

EXPERIMENTAL

Subjects

We report on a cohort of 11 consecutive women undergoing NACT for locally advanced breast cancer diagnosed by core needle biopsy. The six-cycle therapeutic course was TP₀–[NACT₁]–TP₁–[NACT₂]–[NACT₃]–TP₂–[NACT₄]–[NACT₅]–[NACT₆]–TP₃, where TP is therapy point. Each three-week cycle comprised a single IV infusion of a drug cocktail. As per standard care, surgeries (seven mastectomies, four lumpectomies) followed TP₃ – after 18 weeks of therapy. Pathology analyses of the surgical specimens are described below. The subjects underwent DCE-MRI studies at TP₀, TP₁, TP₂, and TP₃. In each case, DCE parameters were spatially averaged and/or computed pixel by pixel, and the results compared with pathology findings.

Two representative cases were selected for high-resolution parametric mapping and in-depth deductive analyses. In each of these, the tumor was a grade 2 invasive ductal carcinoma (IDC) and the patient was HER2 (human epidermal growth factor receptor 2) positive and BRCA1/BRCA2 mutation negative. One was determined a non-complete responder by pathology (non-pCR). (She is estrogen (ER) and progesterone (PR) receptor negative but has a family history of breast cancer.) Her NACT comprised a trastuzumab (monoclonal antibody interfering with HER2/neu receptor), docetaxel (anti-mitotic interfering with microtubules), and carboplatin (interferes with DNA repair) cocktail for each cycle. The other was determined a complete responder (pCR). (She is ER (100%) and PR (10%, focal) positive.) Her NACT comprised a trastuzumab and paclitaxel (anti-mitotic interfering with microtubules) cocktail for the first three cycles. Six weeks after our TP₁ study of the pCR subject (i.e. after three NACT cycles), her therapy was switched to a cyclophosphamide (alkylating DNA interference) and Adriamycin (DNA intercalating) cocktail for the last three cycles.

DCE-MRI

The protocol was approved by the local Institutional Review Board. With informed consent, the patients participated in this ongoing MRI research study, which played no role in their clinical management. We have reported extensive details on DCE-MRI data acquisition from the breast (13,18,45,46) and prostate (30,47). Here, axial bilateral DCE-MRI images with fat saturation and full breast coverage were acquired with a 3D gradient echo-based time-resolved angiography with stochastic trajectories sequence (45) using a Siemens 3 T instrument. DCE-MRI acquisition parameters included 10^o flip angle, 2.9/6.2 ms T_E/T_R, a parallel imaging acceleration factor of two, (30–34 cm)² FOV, (320)² matrix size, and nominal 1.4 mm slice thickness. The nominal in-plane resolution was (1.1 mm)². This yielded a nominal 1.7 μL voxel. The total acquisition time was ∼10 min for 32–34 image volume sets with 18–20 s temporal resolution. The CA Gd(HP-DO3A) (ProHance) IV injection – 0.1 mmol kg⁻¹ (the clinical single dose) at 2 mL s⁻¹ – was carried out following acquisition of two baseline image volumes. A population-averaged arterial input function was obtained from axillary arteries (13,18,46). We used the FXR-allowed SSP version – detailed in (30) – for analyses of spatially averaged and pixel-by-pixel DCE-MRI time-course data. These were subjected to both tracer and shutter-speed pharmacokinetic analyses to extract the K^trans, v_e, k_ep (= K^trans/v_e), and τ_i (SSP only) DCE parameters.

Our focus is mostly on high-resolution parametric maps. However, there is some spatial averaging. These tumors are sufficiently heterogeneous that the averaging method must be specified. The (a) ‘whole tumor’ mean parameter values were calculated as the weighted (by ROI pixel number) averages of the (b) ‘single image slice-averaged’ ROI values from the image planes covering the entire tumor. For image slice-averaged results, the DCE-MRI time-courses from all pixels in the chosen tumor image slice are averaged before pharmacokinetic analysis. For (c) ‘image slice pixel-averaged’ data, the individual pixel time-courses are analyzed and then the resulting parameters averaged.

Parameter precision from Monte Carlo simulations

For the pCR subject selected for in-depth analyses, the precision of fitted DCE parameters (τ_i and K^trans) was estimated using a Monte Carlo approach detailed previously (48). The DCE-MRI time-courses for each tumor pixel in the chosen image slice, obtained at both TP₀ and at TP₁, were analyzed. Six fittings of each voxel DCE-MRI time-course were made, starting with randomly different initial sets of parameter values spanning broad ranges. The standard deviations (SDs) of the returned parameter values were calculated.

Histology

The response to NACT for each patient was determined by pathology analysis of post-therapy surgical specimens (after TP₃) and comparison with pre-therapy biopsy specimens (before TP₀). This comprised the determination of the residual cancer burden (RCB) and the relative change in tumor cell density using published methods (49,50). These analyses revealed that three patients were pCR – no cancer cells found in resection specimens – while the other eight were non-pCR – reduced cancer cell density in resection specimens compared with biopsy specimens. For the non-pCR case selected for in-depth analysis, a whole-mount slide of a slice of a biopsy core obtained pre-therapy (before TP₀) was used to estimate the extent of necrosis and the cell densities in regions of the parametric tumor rim and tumor core. The biopsy core was obtained with a 14 gauge needle (1.6 mm ID), inserted from medial to lateral, and the hematoxylin and eosin (H&E)-stained slide was examined by microscopy at both low and high power (200×).

Statistical analyses

Tumor ROIs were drawn by experienced radiologists, who also measured the largest 1D tumor size according to the RECIST (Response Evaluation Criteria in Solid Tumors Group) (51) guideline. The results from pathology analyses were correlated with the MRI metrics using the ULR (univariate logistic regression) analysis in order to identify imaging biomarkers for early prediction of response and/or accurate assessment of residual disease following NACT.

RESULTS

Human breast cancer and therapy

We have presented many DCE time-courses of breast (13,18,46) and prostate (30,47) cancer data. Figure 1 displays whole breast axial DCE image slices obtained during CA passage for two subjects with IDC representative of the population here – one found a non-pCR (Fig. 1(a), (c); left breast) and one a pCR (Fig. 1(b), (d); right breast). The scale bars are 2 cm. The Figure 1(a), (b) images were obtained at TP₀ – before NACT; the Figure 1(c), (d) images were obtained at TP₁ – after one NACT cycle. Since three weeks separate the DCE-MRI acquisitions, the image slices cannot be perfectly registered. The image slice displayed for TP₁ represents a best estimate for equivalence to the image slice for TP₀. Enhancing tumor regions are clearly hyperintense in these T₁-weighted images. As examples, the red borders in Figure 1(a), (c), demarcate the radiologist-selected tumor ROIs in these image planes.

Figure 1.

Axial breast T₁-weighted images, obtained during CA bolus passage, from two representative subjects in the population. (a) The left breast, before therapy (TP₀), of one of eight subjects deemed a non-pCR after 18 weeks of NACT. (c) The same subject after 3 weeks of NACT (TP₁). (b), (d) Right breast images of one of three subjects declared pCR, obtained at TP₀ and TP₁, respectively. Each patient had grade 2 IDC breast cancer. The radiographic tumor outlines are marked in red in (a) and (c). The scale bars are 2 cm.

Figure 2 shows 12 zoomed pixel-by-pixel SSP parametric maps of the Figure 1 tumors, six from each subject. The upper six ((a)–(f)) are from the Figure 1(a), (c) non-pCR and the lower six ((g)–(l)) are from the Figure 1(b), (d) pCR case. The top row of maps ((a)–(c), (g)–(i)) for each patient was obtained at TP₀ – before NACT initiation. The bottom row ((d)–(f), (j)–(l)) was obtained at TP₁ – after one NACT cycle; three weeks. Besides the τ_i maps, the other pharmacokinetic parameters are K^trans and v_e. The color scales (K^trans in min⁻¹; τ_i in s) are given; v_e is dimensionless.

Figure 2.

Zoomed shutter-speed DCE-MRI parametric maps of the two tumors shown in Figure 1. (a)–(f) Subject non-pCR; (g)–(l) pCR. (a)–(c), (g)–(i) were obtained pre-therapy (TP₀), and (d)–(f), (j)–(l) 3 weeks into (TP₁) the 18 week NACT course. The biomarkers K^trans (a), (d), (g), (j), v_e (b), (e), (h), (k), and τ_i (c), (f), (i), (l) measure, respectively, CA extravasation kinetics, extracellular volume fraction, and mean intracellular water lifetime, and exhibit intratumor heterogeneity. Analyses of τ_i and v_e relationships for 21 pairs of seven different ROIs within these maps (two outlined in yellow in (c)) indicate that the τ_i maps reflect metabolic activity: the smaller τ_i, the greater the NKA turnover. Note the τ_i scale change.

We briefly divert to spatial- and population-averaged results. Figure 3 is a column graph showing the early therapeutic responses of whole tumor-averaged parameter values, further averaged over the pCR (black bars) and non-pCR (gray bars) sub-populations. The error bars reflect the considerable inter- and intratumor (Fig. 2) heterogeneity averaged. The parameters are RECIST, K^trans(tracer), K^trans(SSP), k_ep(tracer), k_ep(SSP), and τ_i. The vertical axis measures the percentage change in the biomarker after the first NACT cycle, i.e. in the three weeks between TP₀ and TP₁. The bar color (pCR versus non-pCR), however, is determined only after completion of the entire NACT therapeutic course (18 weeks), surgery, and pathology analyses. Though the mean tumor size (RECIST) decreased only slightly after three weeks, the DCE biomarkers showed larger changes. Interestingly, the SSP K^trans and k_ep parameters decreased for the pCR sub-population, while τ_i increased. Overall, the ULR analysis found that the percentage changes in tumor mean K^trans (tracer and SSP), k_ep (tracer and SSP), and pixel τ_i histogram median after the first NACT cycle (at TP₁ relative to TP₀) were excellent discriminators of pCRs from non-pCRs, each with the ULR c statistic value of 1.0 (meaning complete separation), while the early RECIST percentage change was a poor predictor with c = 0.60. (The v_e parameter has c values less than unity.) In addition, the absolute values of TP₁ tumor mean K^trans(SSP) and k_ep (tracer and SSP) were also effective (c = 1.0) early discriminators (not shown). After only one NACT cycle, changes in the tumor-averaged shutter-speed DCE biomarkers K^trans, k_ep, and τ_i are excellent predictors of the therapeutic outcome to be found after NACT completion. This is very encouraging. We have published a preliminary report with more results (40), and plan a full paper on these aspects. The tumor-averaged K^trans(SSP) and τ_i values at TP₃ significantly correlate with the RCB magnitude found by pathology analyses.

Figure 3.

Sub-population-averaged whole tumor biomarker values. The vertical axis measures the percentage change after the first three weeks of therapy (between time point TP₀ and time point TP₁). The bar colors represent the sub-populations after 18 weeks of therapy, discriminated by pathology analysis: black, pCR (n = 3); gray, non-pCR (n = 8). The RECIST parameter is a radiographic tumor size measure; the others are DCE-MRI kinetic parameters from tracer or SSP analyses: K^trans (mainly CA extravasation), k_ep (CA intravasation), and τ_i (mean intracellular water lifetime). The change in DCE-MRI biomarkers after three weeks, particularly those from SSP – including τ_i, are excellent predictors of therapeutic outcome after 18 weeks. For the responders, τ_i increases, signifying a therapeutically induced decrease in metabolic activity.

However, the significant intra- and intertumor heterogeneity (1–6) described above seriously calls for individualized tumor assessment (7). For the remainder of this paper, we return our focus to the intratumor parametric maps of the two representative Figure 1, 2 case studies. These allow in-depth deductive analyses of τ_i interpretation. In these, the subjects serve as their own controls. Figure 4 shows τ_i (ordinate), K^trans (abscissa) scatter plots of the 228 tumor pixels from Figures 2(g), (i) (0.38 mL ROI) (a) and the 142 tumor pixels from 2(j), (l) (0.24 mL ROI) (b). These are for the representative pCR patient. The τ_i values in NMR spectroscopic cell suspension studies and in ROIs can be determined with precisions better than 5% (17) and 10% (30), respectively. However, single voxel DCE-MRI data have greater relative noise, which could diminish pixel τ_i precision. Figure 4(a), (b) appraises this for the Figure 2(i), (l) pixels, respectively. Figure 4(a), (b) shows the mean parameter values (black circles) returned from fittings of each pixel time-course. The gray error bars show the SDs for the six fittings from randomly chosen initial parameter value sets (48). All error bars are present: many are smaller than the circles. In these cases, the τ_i precision seems comparable to, or better than, that of cell suspension studies. But some fitting uncertainties are larger. Inspection of Figure 4(a) shows, however, that these occur only in regions where K^trans is relatively small. In these cases, [CR_o] and thus the shutter speed (т₁⁻¹) does not become very large for very long, and the system does not depart the FXL for the FXR condition very extensively and/or for a very long duration (13,30,32). Interestingly, these greater uncertainties occur for both small and large τ_i values. This indicates that τ_i precision is not particularly dependent on τ_i magnitude (the error bar is relatively independent of the mean). It implies that the τ_i accuracy is relatively independent of τ_i precision. This is also found for ROI data (30).

Figure 4.

Pixel τ_i (ordinate), K^trans (abscissa) scatter plots of the Figure 1(b), (d) pCR tumor (a) pre- (Fig. 2(g), (i)) and (b) post- (Fig. 2(j), (l)) therapy. These show the mean τ_i, K^trans values (black points) returned from six fittings of single voxel DCE-MRI time-courses with randomly different initial parameter sets. The gray error bars show the SDs of the fittings, and thus reflect parameter precision. All error bars are present: many are smaller than the points. Poorer τ_i precision occurs for pixels in tumor regions with small K^trans values – in agreement with theory. There is a substantial K^trans decrease and an overall τ_i increase after therapy.

The situation in Figure 4(b) (pCR, TP₁; Figure 2(j)) is different from that in Figure 4(a). As shown in Figure 3, the large K^trans decrease is good news for the pCR patient. (The tumor image slice-averaged K^trans decreases from 0.19 min⁻¹ at TP₀ (Fig. 2(g)) to 0.04 min⁻¹ at TP₁ (Fig. 2(j)) – i.e. by 79%.) However, the greatly diminished K^trans provides a difficult scenario for precise τ_i determinations. The error bars reflect this. Nonetheless, one can discern that, on the whole, τ_i values are larger than in Figure 4(a). (The tumor image slice pixel-averaged τ_i goes from 0.32 s at TP₀ (Fig. 4(a)) to 0.39 s at TP₁ (Fig. 4(b)) – a 22% increase.) This result reinforces the notion that the τ_i magnitude itself does not decrease with K^trans: mostly, the precision of its determination becomes poorer. When clinically indicated, this precision can be considerably improved by moving up to the approved triple CA dose.

A 2D scatter plot is an effective way to take advantage of two responsive biomarkers (52). The slight negative spatial τ_i, K^trans correlation apparent to the eye in the Figure 2(g), (i) parametric maps can also be barely discerned in Figure 4(a). The Pearson correlation coefficient is −0.23. The substantial decrease in K^trans values after one NACT cycle (TP₀ (a), TP₁ (b)) is quite obvious. It is also clear from Figure 4(b) that the τ_i and K^trans magnitudes are independent: they are not numerically correlated by the analysis. As we have seen, the pCR tumor- and population-averaged τ_i increases after therapy (Fig. 3). From Figures 2(i), (l) and 4(a), (b), we can detect an overall increase in τ_i after NACT. (The whole tumor-averaged τ_i increases from 0.27 s to 0.41 s – i.e. by 52%.) Figure 2(l) suggests that this is localized mainly in the tumor core. From the τ_i perspective, we can see from Figure 3 (based on whole tumor averages) that the result for the patient of Figure 2(i), (l) represents the most conservative of the three pCR cases. Importantly, there are no significant v_e differences between core and rim after therapy (Fig. 2(k)). As we will see, the tumor core τ_i increase by therapy is due to a P_W decrease.

Parameter heterogeneity, relationships, and therapy responses

Though the tumors appear relatively homogeneous in Figure 1, the SSP parametric maps exhibit significant intratumor heterogeneity that appears anatomic in nature. For example, the TP₀ K^trans maps of each tumor display elevated values in the tumor rim relative to the core. In Figure 2(a), (g), the K^trans variation exceeds a factor of 10. This pattern is common. It is observed in malignant human tumors – breast (37,46), osteosarcoma (35), head and neck (43), and soft tissue sarcoma (53), in spontaneous murine breast tumors (39), and in implanted rodent cerebral gliosarcoma (32), RIF-1 (33) and prostate (38) tumors. However, it is by no means universal: tumors with bright K^trans cores (or multiple cores) have been reported for malignant human breast (18,34,35,41,42), head and neck (44), and prostate (47) tumors, and in implanted rat glioma (54). The observation of this diversity is very promising for individualized imaging.

Furthermore, there are often spatial correlations between the imaging biomarkers. In the Figure 2 K^trans/τ_i pairs (especially (a)/(c) and (j)/(l), less so for (d)/(f) and (g)/(i)), regions with relatively elevated K^trans values generally have relatively smaller τ_i values, and vice versa. (The non-pCR core τ_i is quite large, 1 s (Fig. 2(c)).) This is also often observed in other human malignant tumors – breast (34,35,37), osteosarcoma (35), head and neck (43,44), and prostate (36) – and in spontaneous murine breast tumors (39). However, this negative correlation is not always the case: there are counter-examples in the human breast (41,42) and in implanted rodent gliosarcoma (32,55), RIF-1 (33), and prostate (38) tumors. Thus, the sign of the biomarker spatial correlation is independent of the intratumor heterogeneity spatial pattern.

For each Figure 2 patient, a comparison of a TP₀ parametric map with that at TP₁ shows the effect of the first NACT cycle. Figure 2(g), (j) reveals a significant K^trans decrease after the first three weeks of therapy for the pCR subject, while there is little, if any, decrease for the non-pCR patient (Fig. 2(a), (d)). The K^trans decrease for the pCR tumor is consistent with the results reported for essentially every antivascular cancer drug tested (56). Interestingly, the τ_i values increase (44% for the tumor image slice-average) for the pCR patient (Fig. 2(i), (l); note the more sensitive color scale), but not for the non-pCR individual (Fig. 2(c), (f)). Thus, there is a negative τ_i, K^trans correlation in (therapy) time as well in space. Importantly, the τ_i increase and K^trans decrease after 3 weeks of NACT predict very well that no RCB will be surgically found after 15 more weeks of therapy. This response is representative of the pCR tumor- and population-averaged results, and occurs usually before significant tumor size decrease (Fig. 3). This is also true for the K^trans decrease caused by therapy on soft tissue sarcoma (53).

We also see τ_i increase/K^trans decrease after a different therapy on a spontaneous murine breast tumor (39). It is important to note that the rather large tumor τ_i value (0.56 s, image slice average) in Figure 2(l) is obtained while v_e is also very large, 0.87 (image slice average, Figure 2(k)). If v_e is large, then v_i (≈1 − v_e) is small. This result reinforces the fact that, contrary to what one might intuit, the τ_i magnitude does not decrease with v_i. This is because of its intensive nature.

Changes in τ_i reflect membrane permeability changes

For globular cells, Equation [3] gives τ_i⁻¹ ≈ 6P_W/d: P_W is the membrane water permeability, d the cell diameter. For a spherical cell with a conservatively large d value (15 µm (6)), τ_i⁻¹ = 4000P_W (τ_i in s, P_W in cm s⁻¹). For a spherical cell with a typical P_W value (1.4 × 10⁻⁴ cm s⁻¹ (10)), τ_i⁻¹ = 8.4/d (d in µm). Thus, τ_i⁻¹ (k_io) is linearly related to P_W and linearly related to d⁻¹, with different coefficients. Do observed τ_i variations reflect changes in P_W, in d, or in both? In this paper, we compare relative (%) changes in τ_i⁻¹ values within human breast tumors with the accompanying % d⁻¹ changes, to show that P_W dominates τ_i.

Let us inspect a τ_i variation in Figure 2. Consider the τ_i map of the non-pCR patient at TP₀ (Fig. 2(c)). We choose an ROI representative of the annular tumor rim: we designate it RN₀, for rim, non-pCR, at TP₀. The average τ_i value for RN₀ is 0.60 s. For a conservative ROI representative of the outer core, CN₀, the τ_i value is 0.81 s (the inner core has even larger τ_i values). These ROIs are outlined with yellow borders in Figure 2(c). RN₀ comprises 55 pixels (66 mm²; 94 μL), while CN₀ comprises 43 pixels (52 mm²; 73 μL). The ratio {(τ_i[R])⁻¹/(τ_i[C])⁻¹} is 1.7/1.2 = 1.4: there is a 40% increase in k_io for the rim over the core. Thus, the relationship {P_W[R]/P_W[C]}{(d_R)⁻¹/(d_C)⁻¹} = 1.4 must be satisfied. There is an infinite number of possibilities. If d_C = 0.9d_R, P_W[R] = 1.6P_W[C]; if d_C = d_R, P_W[R] = 1.4P_W[C] (the mean transcytolemmal water permeability is 40% larger in the rim); and if d_C = 1.4d_R (the mean cell diameter in the core is 40% larger than in the rim), P_W[R] = P_W[C].

If possible, it is extremely difficult (and invasive) to determine actual d values in vivo. However, one can evaluate their variations by combining DCE-MRI results with histology. We obtain and map v_e values (e.g. Fig. 2(b)) and (1 − v_e) ≈ v_i ≡ ρV, where v_i is the intracellular volume fraction, ρ the mean cell number density, and V the mean individual cell volume (Equation [1]). For spherical cells, the mean effective cell diameter d′ ∼ V^1/3 ≈ {(1 − v_e)/ρ}^1/3. For the Figure 2(b) RN₀ and CN₀ ROIs (those marked in Fig. 2(c)), the v_e[R] and v_e[C] values are 0.38 and 0.73, respectively. Thus, {(1 − v_e[R])/(1 − v_e[C])}^−1/3 = 2.3^−1/3 = 0.76. The relationship {(d′_R)⁻¹/(d′_C)⁻¹}{(ρ_C)^1/3/(ρ_R)^1/3} = 0.76 must be satisfied.

A τ_i ratio measurement gives an experimental relationship between d and P_W ratios. A v_e ratio measurement gives an experimental relationship between d (taking d = d′) and ρ ratios. In the appendix, the Figure A1 3D plot shows the trace of all points that simultaneously satisfy both experimental relationships for the non-pCR tumor at TP₀. If the ρ_C/ρ_R ratio is determined independently, one can evaluate whether τ_i changes are dominated by d changes or by P_W changes, or whether both contribute significantly.

To do this, we examined an H&E histology slide of a slice of a biopsy core obtained 24 days before the non-pCR tumor was imaged at TP₀. The whole mount slide was studied, and microscopic images made at two higher powers. We identified the biopsy core sections corresponding to the 8 mm thick tumor rim and the 5 mm radius tumor core seen in Figure 2(a)–(c). Five different tissue types were identified with predominantly (1) stromal cells, (2) adipocytes, (3) invasive carcinoma cells, (4) inflammation (usually lymphocytes), or (5) necrosis. These of course have different cell sizes (d) and densities (ρ) (6). We estimated the five tissue type area fractions in the tumor rim and tumor core. This tumor (before NACT) was not particularly necrotic – ∼15% in both the tumor rim and tumor core. The highest power (200×) H&E fields were used to determine cell densities representative of each tissue type. Thus, we calculated ρ_R as 416 × 10³ and ρ_C as 256 × 10³ cells mm⁻³. (These correspond to 706 × 10³ and 434 × 10³ cells/voxel in rim and core (1.7 μL voxels), respectively; 39 × 10⁶ cells in RN₀ and 19 × 10⁶ cells in CN₀.) This gives ρ_C/ρ_R = 0.6: the tumor core has 60% of the cell density of the tumor rim.

In Figure A1, the point with experimental ρ_C/ρ_R = 0.6 has coordinates d_C = 0.9d_R, and P_W[R] = 1.6P_W[C]. Even accounting for uncertainty in our cell density determination, Figure A1 shows that the mean d_C and d_R values are similar, and P_W[R] is 40–75% larger than P_W[C]: i.e., if anything, greater than the experimental τ_i⁻¹ ratio. The conclusion is strong that the 40% increase in τ_i⁻¹ measured for the rim over the core ROIs in non-pCR at TP₀ (Fig. 2(c)) is dominated by a P_W increase and not a d decrease. (This is one of the most conservative ROI pairs (very small τ_i⁻¹ ratio) we could have chosen.) One may normally expect that it is cells in a tumor core that have an increased P_W(passive) value (57), and this might be the case here as well. However, it is shown below that τ_i⁻¹ is dominated by P_W(active), and the P_W[R] > P_W[C] result for non-pCR at TP₀ may constitute additional evidence that P_W(active) > P_W(passive). The tumor core cells could have increased P_W(passive) but decreased P_W(active), with a net P_W decrease (or increased P_W(active), as long as the rim cells have an even greater P_W(active) increase).

What about τ_i variations in other ROIs? Besides RN₀ and CN₀ (Fig. 2(b), (c)), five other representative ROIs were chosen: RN₁ and CN₁ (Fig. 2(e), (f)), RC₀ and CC₀ (Fig. 2(h), (i)), and CC₁ (Fig. 2(k), (l)) (in the ROI labels, the first character is R (rim) or C (core), the second character is N (non-pCR) or C (pCR), and the subscript is 0 (TP₀) or 1 (TP₁)). Taken two at a time, these seven ROIs afford 21(= 7!/(5!2!)) ROI pairs and thus 21 τ_i⁻¹ and (1 − v_e)^−1/3 ratios. With analyses similar to the above, percentage τ_i⁻¹ and d changes were estimated. (We did not separate the ρ ratio and d ratio factors. Since such ROIs average over at least 20 million cells of different types, sizes, and densities, these ratios are unlikely to deviate far from unity.) The results for 21 ROI pairs average to a 74% τ_i⁻¹ increase accompanied by a 5% d decrease (not statistically different from zero). (Details are given in appendix Table A1.) This result is displayed in the Figure 5 column graph. It is very consistent with literature results also compiled and presented in Figure 5, which summarizes reports of experimentally induced τ_i changes that also have accompanying measurements allowing calculation or estimation of d changes. These span three model systems: two cell suspensions (17,31 and murine myocardium 58). (Details for these are also given in Table A1.) Cisplatin treatment induces an apoptotic state in acute myeloid leukemia (AML) cells and elevates τ_i⁻¹ by almost 400% (31). However, the mean d decreases by only 9%. Switching the bubbling gas from N₂ to O₂ increases yeast τ_i⁻¹ by 110%, but the mean d decreases by only 7% (17). For in vivo mouse myocardium, the normal τ_i⁻¹ is 130% greater than in chronic hypertension (58), and yet the mean ex vivo cylindrical cardiomyocyte d is decreased by only 23% (58). It is important to note that two of these studies (31,58) included microscopy before and after the perturbation.

Figure 5.

Precedents comparing transmural water exchange kinetics increases with cell size decreases. The green bars represent percentage increases in the rate constant k_io (τ_i⁻¹), the red bars percentage decreases in mean cell diameter. Exchange sped up when AML cells were incubated with cisplatin (31), when yeast cells were switched from bubbling with N₂ to O₂ (17), and when hearts in chronically (induced) hypertensive mice were compared with control mice (58). In this work, 21 pairs of ROIs (in two human breast tumors pre- and post-therapy, Figure 2(a)–(l)) were compared, always taking the ROI with larger k_io as the numerator, and the results averaged. (Details in Table 2.) The exchange kinetics increases are not dominated by cell size decreases.

The comparable nature of these model system results and our in vivo human breast tumor findings makes a compelling case that observed τ_i⁻¹ variations are dominated by P_W changes, not size changes. This is reinforced by the fact that, even though they are of similar (small) size, τ_i⁻¹ is ∼4000% greater for the human erythrocyte (59) than for the yeast cell (17). It is very fortunate that nature exhibits this phenomenon.

DISCUSSION

As described above, the analysis of DCE-MRI time-course data with any formulation of the tracer pharmacokinetic paradigm (most common by far) is incorrect. It neglects the finite kinetics of the inter-compartmental water exchange equilibria that precede the rate limiting step of CA extravasation. This causes systematic changes in the DCE-MRI pharmacokinetic parameters, K^trans and v_e. In particular, spatially averaged K^trans is disproportionately depressed in malignant tumors of the breast (13,18,34,35,46) and prostate (30,47). The SSP allows extremely high specificity in the detection of these cancers: specificity not possible with tracer analysis. This makes effective cancer detection (13,18,34,46,47) and therapy prediction (40,53) practical.

Here, however, we focus on the second benefit of the SSP, access to the water exchange kinetics themselves. This is not possible with the tracer paradigm, where water is not considered molecular but merely a continuum filling tissue compartmental spaces. The equilibrium trans-cytolemmal water exchange kinetics are measured by τ_i, the reciprocal of the unidirectional rate constant for water efflux, k_io.

Changes in τ_i reflect changes in the driving membrane P-type ATPase ion pump turnover

The results above indicate that the intratumor τ_i variations observed in human breast cancer are dominated by P_W differences. If they were dominated by d, the τ_i⁻¹ (k_io) percentage increases and d percentage decreases would be similar. Still, are the differences observed in P_W(passive), in P_W(active), or in both?

Almost all cells have a driving membrane P-type ATPase ion pump enzyme, which serves to generate transmural ion gradients and membrane potentials (26). For yeast cells, this is the H⁺-ATPase, Pma1 (17,26). For mammalian cells, it is NKA (26). The forward reaction catalyzed by NKA can be written as in Equation [5], where intracellular adenosine triphosphate (ATP_i) is hydrolyzed to ADP_i and P_i,

5

extracellular potassium (K_o⁺) is transported into the cell, and intracellular sodium (Na_i⁺) is expelled. Table 1 summarizes the literature on the dependence of equilibrium transcytolemmal water exchange kinetics (τ_i⁻¹) on driving membrane P-type ATPase ion pump gene dosage, substrate concentration, and specific inhibitor concentration in yeast suspension, perfused rat heart, and erythrocyte suspension studies. Without exception, τ_i⁻¹ increases with gene copy number and substrate concentration, and decreases with extracellular inhibitor concentration. (Ebselen and ouabain are specific inhibitors of Pma1 (17) and NKA (26), respectively.) This is strong evidence that τ_i⁻¹ (k_io) is dominated by P_W(active), which in turn is driven by P-type ATPase ion pump turnover. The greater the turnover, the faster the exchange. All of these model systems were homeostatic for the Table 1 entries.

Table 1.

τ_i^-1 (k_io) reflects turnover of driving membrane P-type ATPase ion pump

P-type ATPase ion pump	Yeast17	Cardiomyocyte28	Erythrocyte61
Gene dosage	↑↑
Substrate
ATPi	↑↑		↑↑
Ko+		↑↑
Specific inhibitor
Ebselen	↑↓
Ouabain		↑↓

↑↑ positively related; ↑↓ inversely related.

¹⁷Measured 17; ²⁸measured 28; ⁶¹inferred 61.

Figure 6 presents a schematic diagram of the general molecular mechanism we have proposed 17 for active trans-membrane water cycling. The passive water exchange (P_W(passive)) equilibrium is indicated at the top right. It involves simple water diffusion through the phospholipid bilayer, transport through aquaporin channels 60,61, and leakage through membrane protein transporters 25. Active trans-membrane water cycling (P_W(active)), which can have three times the P_W(passive) flux 17, almost certainly involves water co-transporting membrane symporters 24. In the diagram, active water efflux is pictured as passing through NKA, and influx through the sodium glucose co-transporter (SGLT 62), but this is only for the purpose of illustration. It is not yet known which symporter molecules dominate active water cycling: there are a number of candidates, SGLT certainly being one of them 24. By themselves, aquaporins catalyze only passive transmural water transport. However, to the extent to which they co-localize with substrate transporters, say K⁺ channels 60 or NKA 63, they may participate in active trans-membrane water cycling.

Figure 6.

Schematic depiction of active trans-membrane water cycling (P_W, cell membrane water permeability coefficient; C₆H₁₂O₆, glucose; SGLT, sodium glucose co-transporter; NKA, Na⁺/K⁺-ATPase; KcsA, K⁺ channel). Water transport through NKA and SGLT is shown only for illustration. The principal transporters are not yet known.

Because it maintains the trans-membrane ion gradients that drive much secondary active transport and produce the membrane potential, one can argue that NKA is the most important enzyme in mammalian biology. Because of the particular, dual (‘vectorial’ and ‘scalar’ 64) characteristics of the reaction catalyzed by NKA (Equation [5]), measurements of its activity have always been adapted to the nature of the sample. For solubilized, purified enzyme or tissue homogenate preparations, one cannot measure the kinetics of (vectorial) ion transport. Thus, spectrophotometric 65 or radiolabeled (³²P) assays 66 of the rate of ATP hydrolysis are used. On the other hand, for intact cells in culture or in tissue preparations, one cannot easily measure the kinetics of the (scalar) intracellular ATP hydrolysis caused by NKA activity. However for such samples, voltage clamp current, ion-selective (Na⁺/K⁺) microelectrode response, radioisotope (²²Na⁺/²⁴Na⁺/⁴² K⁺/⁸⁶Rb⁺) uptake/release 67–70, or ²³Na, ⁸⁷Rb MR spectroscopic 71,72 methods can be used to measure NKA-driven trans-membrane ion transport kinetics. This is how it was learned that, when the concentrations of the other reactants and products have typical values, the intracellular Na⁺ concentration, [Na_i⁺], is generally the rate-determining factor 71,73,74. It is also possible to measure [Na_i⁺] using a fluorescent indicator 75. A breakthrough found that phospholipid vesicles reconstituted with purified NKA facilitated measurement of both ATP hydrolysis and ion transport 66. This allowed confirmation of the NKA reaction stoichiometry 66.

It is obvious that each of these methods is best suited to macroscopically homogeneous samples. None are particularly appropriate for use with normally heterogeneous tissue. (NKA distributions can be mapped histologically 76.) Except for the microelectrode approaches, these do not involve spatial encoding; and one cannot insert electrodes in all of the cells of a tissue. Furthermore, many of these methods directly measure only net NKA activity, not homeostatic NKA turnover. (An analogous problem arises measuring net versus steady-state water transport kinetics 15.) A ²⁴Na⁺ study reveals the existence of an equilibrium transmural Na⁺ exchange process in the cardiomyocyte over an order of magnitude faster than net Na⁺ transport 69. However, the radioisotope approach has been generally abandoned for ∼20 years, and deemed too problematic for even tissue preparations 73. As far as we are aware, the NKA turnover has never been measured, let alone mapped, in a living animal or human subject. Therefore, τ_i⁻¹ measurement offers the possibility of quantifying perhaps the most crucial ongoing cellular metabolic turnover. For a spherical cell with d = 12 µm, τ_i = 400 ms signifies active cycling of 7 × 10¹³ H₂O molecules s⁻¹/cell 17. If the stoichiometric flux ratio flux_Na+ = (10⁻³ to 10⁻²)flux_H2O 24 pertains, the Na⁺ flux is 10¹¹–10¹² Na⁺ ions s⁻¹/cell. If τ_i maps are reciprocal NKA turnover maps, they represent high-resolution metabolic images.

We show that τ_i exhibits significant intratumor heterogeneity in human breast cancer in vivo. That the biomarker spatial correlations are never perfect and that counter-examples exist suggest that, when seen, these are not numerical co-variance artifacts resulting from the three parameter fittings of DCE-MRI data time-courses. They appear to be physiological correlations.

The synergism of two responsive imaging biomarkers can be very powerful. In this case, one reports on kinetic processes occurring outside cells: K^trans measures mostly the microvascular CA extravasation/tissue arrival rate. The other (τ_i) measures metabolic fluxes inside cells. Since CA employs a para(endothelial)cellular pathway 77,78, it serves also as a surrogate for paracellular extravasation of plasma solutes with similar molecular sizes. Nutrients, particularly glucose 79, represent a crucial sub-class of these. Though glucose has specific transcellular transporters, when K^trans is relatively large it is likely that additional paracellular glucose extravasation is also relatively large. When K^trans is relatively large but τ_i is relatively small, as in the rims of both Figure 2 tumors before therapy, it may be signaling that greater nutrient delivery enables faster cell metabolism. The pre-therapy biopsy core for non-pCR showed clearly that the parametric rim seen in Figure 2(a)–(c) corresponds to a band with the greatest density of invasive carcinoma cells and inflammatory lymphocytes. Figure 2 suggests that NACT on the pCR tumor causes rim tissue to move from larger K^trans/smaller τ_i to smaller K^trans/smaller τ_i, and core tissue from smaller K^trans/smaller τ_i to smaller K^trans/larger τ_i. We find very similar behavior with a different therapy (phosphatase 2A re-activation) on a spontaneous murine breast tumor 39. Perhaps this pathway is common for therapy-induced tumor regression. The most parsimonious explanation is that a K^trans decrease echoes a nutrient delivery decrease and then, subsequently, there is a decrease in metabolic activity signaled as a τ_i increase.

The correlation of the results we see at TP₁ with the pathology results after 15 additional weeks of therapy (Fig. 3) is very encouraging. This is crucial for early, personalized therapy evaluation and adjustment. The two cases reported here exemplify this. If we had sufficient statistical experience that our approach informed clinical decisions, the therapy of the non-pCR patient might have been altered after TP₁, and the therapy of the pCR patient might have been ended after TP₁, or not switched after TP₃.

The region of positive τ_i, K^trans correlation in the rat gliosarcoma rim 32,55 coincides with the region of the implanted rat glioma that stains positive with EF5 55,80, a marker for hypoxia. This correlation is also positive for the RIF-1 tumor 33, which is known to be highly hypoxic 81. New results in a rat model of head and neck cancer show extensive overlap of an elevated τ_i region with that of EF5 staining 55. Also, increased tumor τ_i strongly correlates with survival times of human head and neck cancer patients (H. Poptani, personal communication). When τ_i is relatively large, the NKA activity is relatively small. Thus, it makes sense that K^trans and τ_i are large and positively correlated in tissues where the cells may have entered a hypoxic state. Though the delivery of glucose is sufficient, it is not metabolized efficiently w.r.t. ATP synthesis 81,82.

The K^trans value can be sensitive to the degree of tissue vascularization. If vascularization is low, the apparent K^trans value decreases because, after extravasation, CA arrival in the tissue also requires diffusion 83. We do observe very small K^trans and relatively large τ_i in breast adipose tissue 34,35. Presumably, this is due to the low vascularization. Also, in necrotic areas, one would expect K^trans to be relatively small and τ_i to be relatively large, since cell metabolism should be slow.

Certainly, the spatial resolution of ¹H₂O MRI does not compare with that of the optical microscopy of pathology. A typical high-resolution MRI pixel covers 4 × 10⁶ high power ‘20 × ‘ pixels ((0.5 µm)²) from a modern digital pathology microscope 84. However, the results presented here introduce the potentially highest resolution in vivo metabolic imaging.

Glossary

AML

acute myeloid leukemia

CA

contrast agent

CC_X

core ROI/complete responder/therapy point X

CN_X

core ROI/non-complete responder/therapy point X

DCE

dynamic contrast enhanced

FXL

fast exchange limit

FXR

fast exchange regime

H&E

hematoxylin and eosin

IDC

invasive ductal carcinoma

IV

intravenous

k_ep

CA intravasation rate constant

K^trans

CA extravasation transfer constant

L-NAME

N_ω-nitro-L-arginine methyl ester

NACT

neoadjuvant chemotherapy

NKA

Na⁺,K⁺-ATPase

non-pCR

non-complete responder by pathology

pCR

complete responder by pathology

RCB

residual cancer burden

RC_X

rim ROI/complete responder/therapy point X

RN_X

rim ROI/non-complete responder/therapy point X

ROI

region of interest

SDs

standard deviations

SSP

shutter-speed paradigm

SXR

slow exchange regime

τ_i

mean intracellular water molecule lifetime

TP_X

therapy point X

ULR

univariate logistic regression

v_e

extracellular, extravascular volume fraction.

CHANGES IN τ_i REFLECT MEMBRANE PERMEABILITY CHANGES

Analyses of non-pCR subject pre-therapy RN₀ and CN₀ ROI (Fig. 2(b), (c)) parametric relationships

Consider the τ_i map of the non-pCR patient at TP₀ (Fig. 2(c)). We choose an ROI representative of the annular tumor rim: we designate it RN₀, for rim, non-pCR, at TP₀. The τ_i value for RN₀ is 0.60 s. For a conservative ROI representative of the outer core, CN₀, the τ_i value is 0.81 s (the inner core has even larger τ_i values). These ROIs are outlined with yellow borders in Figure 2(c). The ratio {(τ_i[R])⁻¹/(τ_i[C])⁻¹} is 1.7/1.2 = 1.4: there is a 40% increase in k_io for the rim over the core. Thus, the relationship {P_W[R]/P_W[C]}{(d_R)⁻¹/(d_C)⁻¹} = 1.4 must be satisfied. Now, consider the accompanying v_e map (Fig. 2(b)). By definition, (1 − v_e) ≈ v_i, where v_i is the intracellular volume fraction. Furthermore, v_i = (n/V_T)V, where (n/V_T) is the number of cells in the ROI (or voxel) volume (V_T), and V is the mean individual cell volume (Equation [1]). The quantity (n/V_T) is the ROI (or voxel) mean cell number density (ρ): a few hundred thousand cells/voxel. Thus, (1 – v_e) ≈ v_i = ρV. For spherical cells, the mean effective cell diameter d′ ∼ V^1/3 ≈ {(1 − v_e)/ρ}^1/3. Therefore, for an ROI pair labeled R and C, the ratio {(1 − v_e[R])/(1 – v_e[C])}^−1/3 ≈ {(d_R′)⁻¹/(d_C′)⁻¹}{ρ_C/ρ_R}^1/3. For the Figure 2 (b) RN₀ and CN₀ ROIs (those marked in Figure 2(c)), the v_e[R] and v_e[C] values are 0.38 and 0.73, respectively. For these values, the ratio (1 − v_e[R])/(1 − v_e[C]) = (1 − 0.38)/(1 – 0.73) = 2.3. Thus, {(1 − v_e[R])/(1 − v_e[C])}^−1/3 = 2.3^−1/3 = 0.76. The relationship {(d′_R)⁻¹/(d′_C)⁻¹}{(ρ_C)^1/3/(ρ_R)^1/3} = 0.76 must be satisfied.

The red curve in Figure 7 is the trace of all points, in a 3D space of tissue cellular properties, which simultaneously satisfy the experimental {P_W[R]/P_W[C]}{(d_R)⁻¹/(d_C)⁻¹} = 1.4 and {(d′_R)⁻¹/(d′_C)⁻¹}{(ρ_C)^1/3/(ρ_R)^1/3} = 0.76 relationships for the rim and core ROIs of Figure 2(c), (b), respectively (taking d_R/d_C = d′_R/d′_C). The axes are P_W[R]/P_W[C] (vertical), (d_R)⁻¹/(d_C)⁻¹ (inverse cell diameter ratio (= d_C/d_R)), and ρ_C/ρ_R (cell density ratio). This is very informative. A black circle marks the point on the red curve where ρ_C/ρ_R = 0.6, the value determined from histology analysis of the biopsy core (see text). Its other coordinates are d_R⁻¹/d_C⁻¹ (= d_C/d_R) = 0.9, and P_W[R]/P_W[C] = 1.6. If anything, the permeability ratio is larger than {(τ_i[R])⁻¹/(τ_i[C])⁻¹} = 1.4. To allow for uncertainty in the ρ determination, a conservatively generous range for ρ_C/ρ_R of 0.9 to 0.3 is shaded gray in the Figure A1 horizontal plane. The dot–dashed red projection shows that the allowed (d_R)⁻¹/(d_C)⁻¹ values vary from 0.8 to 1.1 in the gray shaded region. This indicates that, averaged over the many different cell types and sizes of these ROIs (RN₀, 39 million cells; CN₀, 19 million cells), the mean cell diameter in the rim is approximately the same as in the core. The P_W[R]/P_W[C] values (vertical) over the gray shaded area range from 1.8 to 1.3 (80–30% increases in rim P_W, relative to core). We conclude that P_W is substantially larger in the rim of the untreated non-pCR tumor than in the core: τ_i variation is not due to d variation.

Figure 7.

Biomarker inter-relationships for the non-pCR tumor before therapy (Fig. 2(b), (c)). Values for ROIs representative of the tumor rim (R) (39 × 10⁶ cells) and the tumor core (C) (19 × 10⁶ cells) are used (yellow in Figure 2(c)). Experimentally, (τ_i[R])⁻¹/(τ_i[C])⁻¹ = 1.4, and (1 − v_e[R])/(1 − v_e[C]) = 2.3, where τ_i is the mean intracellular water lifetime and v_e the interstitial volume fraction. The red curve is the trace of points that satisfy these relationships simultaneously, in a 3D space of tissue cellular properties. The vertical axis is P_W[R]/P_W[C], the mean cell membrane water permeability coefficient ratio. The horizontal axes are the inverse mean cell diameter ratio, d_R⁻¹/d_C⁻¹ (= d_C/d_R), and the mean cell density ratio, ρ_C/ρ_R. The black point is the position on the curve where ρ_C = 0.6ρ_R, determined from histology on a biopsy specimen. The other coordinates indicate that the d_C and d_R cell diameters are similar and P_W[R] = 1.6P_W[C]. The gray shading allows for some uncertainty in the ρ_C/ρ_R determination. The result is consistent only with τ_i being dominated by the P_W factor.

Other Figure 2 ROIs

Besides RN₀ and CN₀ (Fig. 2(b), (c)), we have chosen five other representative ROIs: RN₁ and CN₁ (Fig. 2(e), (f)), RC₀ and CC₀ (Fig. 2(h), (i)), and CC₁ (Fig. 2(k), (l)) (in the ROI labels, the first character is R (rim) or C (core), the second character is N (non-pCR) or C (pCR), and the subscript is 0 (TP₀) or 1 (TP₁)). Taken two at a time, these seven ROIs afford 21 (=7!/(5!2!)) ROI pairs and thus 21 τ_i⁻¹ and (1 − v_e)^−1/3 ratios. We take these ratios such that the larger τ_i⁻¹ (k_io) value (smaller τ_i, τ_i[S]) is the numerator: i.e., (τ_i[S])⁻¹/(τ_i[L])⁻¹ > 1. For the 21 ratios, (τ_i[S])⁻¹/(τ_i[L])⁻¹ ranges from 1.05 to 3.58, and averages 1.74, a 74% increase. The companion {((1 − v_e)[S])/((1 − v_e)[L])}^−1/3 ratios (≈{(d_S′)⁻¹/(d_L′)⁻¹}{ρ_L/ρ_S}^1/3) range from 0.67 to 1.50, and average 1.05, a 5% increase that is not statistically different from 0%. The details are given in Table A1. The inter-ROI pair SDs are also given. Since these represent inter-subject and/or inter-session comparisons, their relatively small SD sizes are quite encouraging for the pseudo-absolute nature of these imaging biomarkers.

A1.

Trans-membrane water exchange increases are disproportionately larger than cell size decreases

System	AML cells (n = 5)	Yeast cells (n = 6)	Murine myocardium in vivo/ex vivo (n = 17)	Human breast tumors in vivo (21 ROI pairs)
Smaller τi−1 (kio) (s−1) (larger τi, τi[L])	1.4 (±43%)a	1.5 (±6%)b	2.3 (±29%)c
Larger τi−1 (kio) (s−1) (smaller τi, τi[S])	6.8 (± 57%)d	3.1 (±3%)e	5.3 (±43%)f
τi−1 (kio) change	390% increase	110% increase	130% increase	74% increase (±37%)i
(dL’)−1 for smaller τi−1	1.0 (±10%)g	2.2 (±2%)g	0.038 (±6%)c,h (µm−1)
(dS’)−1 for larger τi−1	1.1 (±4%)g	2.4 (±2%)g	0.051 (±5%)f,h (µm−1)
(d’)−1 change	10% increase	9% increase	34% increase	5% increase (±22%)j
Reference	31	17	58	this work

^aControl cells.

^bN₂ gasification.

^cL-NAME-treated mice, in vivo.

^dCisplatin-treated cells.

^eO₂ gasification.

^fControl mice, in vivo.

^gd’ = (p_i)^1/3.

^hCylindrical diameter by microscopy on ex vivo fixed tissue.

ⁱAverage for 21 ROI pairs in Figure 2 (see text).

^jAverage {(d_S′)⁻¹/(d_L′)⁻¹}{ρ_L/ρ_S}^1/3 for 21 ROI pairs in Figure 2 (ρ is cell number density); not statistically different from 0%.

Literature results

Table A1 also summarizes literature reports of experimentally induced τ_i changes that also have accompanying measurements allowing calculation or estimation of d changes. These span three model systems: two cell suspensions 17,31 and murine myocardium 58. In the two cell suspension studies, the yeast 17 and AML 31 cell densities were not radically changed by O₂ gasification and cisplatin treatment, respectively. Since the cells are spherical, the ratio of (p_i)^−1/3 (where p_i is the measured intracellular water mole fraction, ‘population’) values approximates the (v_i)^−1/3 ratio and thus the (d′)⁻¹ ratio. Table A1 shows that the 9% and 10% (d′)⁻¹ increases caused by O₂ gasification of the yeast cells 17 and cisplatin treatment of the AML cells 31, respectively, are much smaller than the concomitant 110% and 390% τ_i⁻¹ increases. In the mouse heart, in vivo τ_i⁻¹ was observed to decrease after seven weeks treatment with an NO biosynthesis inhibitor, N_ω-nitro-L-arginine methyl ester (L-NAME) 58. The ex vivo fixed tissue (cylindrical) cell d values were microscopically determined for control and L-NAME treated mice. Systematic d errors from tissue fixing must be hard to avoid. As seen in Table A1, the 130% τ_i⁻¹ increase from L-NAME treated to control is accompanied by only a 34% increase in d⁻¹. The AML study 31 also included microscopy before and after the perturbation. The consistent nature of these model system results and our in vivo human breast tumor findings makes a compelling case that τ_i⁻¹ changes are dominated by P_W changes.