Kinetic Assessment of Breast Tumors Using High Spatial Resolution Signal Enhancement Ratio (SER) Imaging

Abstract

The goal of this study was to investigate the relationship between an empirical contrast kinetic parameter, the signal enhancement ratio (SER), for three-timepoint, high spatial resolution contrast-enhanced (CE) MRI, and a commonly analyzed pharmacokinetic parameter, k_ep, using dynamic high temporal resolution CE-MRI. Computer simulation was performed to investigate: 1) the relationship between the SER and the contrast agent concentration ratio (CACR) of two postcontrast time-points (t_p1 and t_p2); 2) the relationship between the CACR and the redistribution rate constant (k_ep) based on a two-compartment pharmacokinetic model; and 3) the sensitivity of the relationship between the SER and k_ep to native tissue T₁ relaxation time, T₁₀, and to errors in an assumed vascular input function. The relationship between SER and k_ep was verified experimentally using a mouse model of breast cancer. The results showed that a monotonic mathematical relationship between SER and k_ep could be established if the acquisition parameters and the two postinjection timepoints of SER, t_p1, t_p2, were appropriately chosen. The in vivo study demonstrated a close correlation between SER and k_ep on a pixel-by-pixel basis (Spearman rank correlation coefficient = 0.87 ± 0.03). The SER is easy to calculate and may have a unique role in breast tissue characterization.

Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) has become a method increasingly used for evaluating breast tumors. Signal enhancement on T₁-weighted DCE-MRI can be assessed in two ways by using either a semiquantitative method to estimate signal intensity changes or a pharmacokinetic model to quantify changes of tissue contrast agent (CA) concentration (1). Pharmacokinetic model-based analysis of breast DCE-MRI data (2–9) has the advantages of providing parameters related to the changes in perfusion and vessel permeability of the microcirculation and allowing cross-comparison between different sites. However, compromises have to be made trading imaging spatial resolution, which is critical for detecting small features of breast lesions, for high temporal resolution, which is necessary for performing pharmacokinetic analysis. In addition, the native tissue T₁ relaxation time (T₁₀) and vascular input function (VIF) are required for the calculation of perfusion and vessel permeability (10). Establishing robust methods to rapidly measure T₁₀ and incorporating local VIF into kinetic modeling remain as challenges in model-based breast DCE-MRI.

High spatial resolution imaging is advantageous for depicting the heterogeneous microvascular network in breast cancers using parametric methods (11–15). As the importance of tumor morphology for making a correct diagnosis is increasingly being recognized, images with high spatial resolution and high signal-to-noise ratio (SNR) are much more desired over those with high temporal resolution, but low spatial resolution and low SNR in breast MRI. Previous studies have utilized high spatial resolution three-dimensional (3D) DCE-MRI with relatively low temporal resolution, e.g., 60–90 sec/frame (12). Methods based on three-timepoint examination consisting of one pre- and two postcontrast scans are commonly used in clinical studies (11,16,17) to evaluate morphological changes of breast lesion, using high spatial resolution 3D imaging with a typical isotropic pixel size of 1 × 1 × 1 mm covering the entire symptomatic breast or both breasts. Furthermore, our group has utilized such a three-time-point acquisition strategy to calculate high spatial resolution maps of a semiquantitative parameter, the signal enhancement ratio (SER), for clinical studies of breast cancer (13,18). SER is defined as (S₁-S₀)/(S₂-S₀), where S₀, S₁, and S₂ represent the signal intensity (SI) of each voxel in the precontrast, first postcontrast (t_p1), and second postcontrast (t_p2) images, respectively.

The aim of this study was to investigate the relationship between SER and the more physiologically understood pharmacokinetic parameter, k_ep. We hypothesize that SER is a good approximation of the redistribution rate constant, k_ep, calculated from typical two-compartment pharmacokinetic models (10). In this study we first present a theoretical framework to illustrate that SER is related to the CA concentration ratio (CACR) in tissues between two postcontrast time points, if short TR (<10 ms), short TE (⪡T₂* of tissue), high flip angle (FA), and a low dose of CA are used. Second, we demonstrate that the monotonic mathematical relationship between the CACR (≡ C(t_p1)/C(t_p2)) and the k_ep value can be established if optimal t_p1 and t_p2 are selected based on a two-compartment pharmacokinetic model (19). Third, the effects of the variation in tissue T₁₀ and the errors in the estimated VIF on the relationship between SER and k_ep are estimated using computer simulation. Finally, the theoretical relationship between SER and k_ep is verified experimentally using a mouse model of breast cancer.

MATERIALS AND METHODS

Theory: Relationship between SER and k_ep

Computer Simulation of the Relationship between Tissue SER and Contrast Agent Concentration Ratio

Tissue SER calculated from the three-timepoint CE-MRI data, using a short TE (TE ⪡ T₂* of tissue), short TR (TR ⪡ T₁ of tissue) transverse-spoiled gradient-echo sequence with low dose Gd-DTPA administration, can be expressed as SER = A×CACR. Factor A can be expressed as (see the Appendix for derivation):

$$A=\frac{\mathrm{SER}}{\mathrm{CACR}}=\frac{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot (R_{10}+C(t_{p2})\cdot \Re 1)}{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot (R_{10}+C(t_{p1})\cdot \Re 1)}$$ [1]

where R₁₀ (≡1/T₁₀) is the precontrast longitudinal relaxation rate of breast tissue, α is the flip angle, and ℜ1 (= 4.52 s⁻¹ mM⁻¹ (20)) is the T₁ relaxivity of Gd-DTPA.

Figure 1 plots A vs. T₁₀ using two different flip angles (α = 20°, 40°), a short TR (8 ms), and two pairs of C(t_p1) and C(t_p2), representing washing in and washing out of CA, respectively. The plot shows weak dependence of A on T₁₀, which indicates that the relationship between SER and CACR is not affected by the variations in the tissue T₁₀. Furthermore, the value of A is close to unity for large flip angle (A = 1 if α = 90°), and thus SER approximates CACR of the two postcontrast timepoints.

FIG. 1.

Plots of A (≡SER/CACR) vs. T₁₀ with different FA, C(t_p1) and C(t_p2). A TR of 8 ms was used in the simulation.

Computer Simulation of the Relationship between CACR and k_ep

The computer simulation was based on a two-compartment pharmacokinetic model (19). The model assumes that the time course of blood plasma concentration of CA, C_p(t) (≡VIF), follows a bi-exponential decay (19):

$$C_p\left(t\right)=D\cdot \sum _{i=1}^2{a}_i\cdot \exp (-m_i\cdot t)$$ [2]

where D is the administered dose of Gd-DTPA (mmol/kg body weight) and a_i and m_i are the amplitudes and the rate constants of the decaying exponentials, respectively. The amplitudes a_i are normalized for unit dose.

The CA concentration in tissue at time t, C(t), is then expressed as (19):

$$C\left(t\right)=D\cdot K^{\mathrm{trans}}\sum _{i=1}^2{a}_i\frac{\exp (-k_{\mathrm{ep}}\cdot t)-\exp (-m_i\cdot t)}{m_i-k_{\mathrm{ep}}}$$ [3]

where K^trans represents the endothelial transfer constant for transport from plasma to the extracellular extravascular space (EES), and k_ep represents the rate constant for transport from the EES to plasma. k_ep is the ratio of the transfer constant to the fractional leakage space, ν_e (21):

$$k_{\mathrm{ep}}=K^{\mathrm{trans}}∕{\nu }_e$$ [4]

The contribution of intravascular CA to tissue concentration is neglected in Eq. [3] (21).

Based on Eq. [3], the CACR at t_p1 and t_p2 can be represented as:

$$\begin{array}{cc}\hfill \mathrm{CACR}=& \frac{C\left(t_{p1}\right)}{C\left(t_{p2}\right)}\hfill \\ \hfill =& \frac{\sum _{i=1}^2{a}_i\cdot \left(\exp \right(-k_{\mathrm{ep}}\cdot t_{p1})-\exp (-m_i\cdot t_{p1}\left)\right)∕(m_i-k_{\mathrm{ep}})}{\sum _{i=1}^2{a}_i\cdot \left(\exp \right(-k_{\mathrm{ep}}\cdot t_{p2})-\exp (-m_i\cdot t_{p2}\left)\right)∕(m_i-k_{\mathrm{ep}})}\hfill \end{array}$$ [5]

The effect of t_p1 and t_p2 selection on the relationship between CACR and k_ep was depicted by plotting the CACR vs. k_ep for different combinations of t_p1 and t_p2, based on an assumed bi-exponential VIF and Eq. [5].

Measurement of VIF

The assumed bi-exponential VIF employed in the computer simulations was a mean VIF measured from eight patients with breast cancer in a previous DCE-MRI study (7). The temporal resolution of the measurement in that study was 66 sec. A bolus of Gd-DTPA (0.1 mmol/kg) was given intravenously over a period of 4 sec after the acquisition of the first three dynamic volumes. A bolus of 35 mL of saline solution was given immediately after the CA bolus at the same rate. The plasma concentration–time curve, C_p(t), was calculated for each patient from the MR signal intensity changes of blood in the descending thoracic aorta, with the first three data frames used as a baseline. The bolus arrival time, t_0, was placed at the end of T_{1/2 bolus} + T_transit (see Fig. 2), where T_{1/2 bolus} was the middle of the injection duration, i.e., 2 sec after administration of CA bolus, and T_transit was the transit time of CA from the site of injection to descending aorta. A literature value of T_transit = 13 sec was used (22). A functional form of VIF was obtained by fitting the C_p(t) of aorta to a bi-exponential model (Eq. [2]) for each patient. The data frame acquired at the first timepoint after administration of contrast (the fourth point in the DCE series), t_{i = 4} = t_baseline + 33 sec, was excluded from the fit to allow the steady-state part of C_p(t), after the systemic recirculation of CA bolus, to dominate the fit, reducing the effect of peak fluctuation on the value of (a₁ + a₂) (23).

FIG. 2.

The fit of the steady-state part of C_p(t) (bold asterisk), measured from the descending thoracic aorta (arrow) of a patient with breast cancer, to a bi-exponential model (solid line). The arrival time of the bolus, i.e., the time zero for blood kinetic analysis, was estimated as: t₀ = 0.5*(duration of bolus injection = 4 sec) + (transit time to the vessel under observation = 13 sec) = 15 sec.

Error Estimation

The effects of variations in tissue T₁₀ on the relationship between SER and k_ep were investigated by simulations of the SER vs. k_ep curves at various tissue T₁₀ values. A range of “true” k_ep values of 0.1–2.0 min⁻¹ and “true” ν_e = 0.3 or 0.6 were assumed. Corresponding “true” K^trans values were calculated using Eq. [4] with each of the assumed values of k_ep and ν_e. Theoretical SER corresponding to each k_ep was generated based on Eq. [A5] using 1) TR = 8 ms; 2) α = 40° or 20°; 3) an array of R₁₀ (≡ 1/ T₁₀) ranged from 0.77–1.67 s⁻¹ (24); and 4) The R₁₁ and R₁₂, calculated using R₁₁ = C(t_p1) · ℜ1+ R₁₀ and R₁₂ = C(t_p2) · ℜ1+ R₁₀. The values of the C(t_p1) and C(t_p2) were calculated with Eq. [3] using D = 0.1 mmol/(kg body weight), [t_p1, t_p2] = [1 min, 9 min] or [2.5 min, 7.5 min], the assumed “true” K^trans and “true” k_ep values, and the bi-exponential VIF as described above. The curve of CACR vs. k_ep was also generated using the same bi-exponential VIF for comparison. For an assumed k_ep, ν_e, FA, and [t_p1, t_p2] combination, eight SERs were generated with the array of R₁₀ values. Values of mean and standard deviation (SD_ser) of the eight SERs were obtained. The mean value was used to assess the agreement between SER and CACR, and the SD_ser was used to estimate the variability of SER resulting from variation in tissue T₁₀. The contribution from the intravascular CA to tissue SI was not taken into account in the simulation (21).

In order to visually inspect the effects of variations in the assumed bi-exponential VIF on the relationship between SER (approximated by CACR) and k_ep, we replotted the curve of CACR vs. k_ep by changing each of the four VIF parameters (a₁, a₂, m₁, and m₂) in turn by ±20% of its nominal value. The size of variations used in the simulations is consistent with a recent report that the percent SD of C_p(t) was ≈25% in the washout period (25). We also estimated the sensitivity of the assumed CACR vs. k_ep curve to errors in each of the assumed VIF parameters using the error propagation method suggested by Tofts et al. (3): A representative curve of CACR vs. k_ep was obtained using Eq. [5] with a range of “true” k_ep values of 0.1–2.0 min⁻¹, [t_p1, t_p2] = [1 min, 9 min] or [2.5 min, 7.5 min], and the bi-exponential VIF described in the previous section. Each VIF parameter, a₁, a₂, m₁, and m₂, in turn was reduced to 99% of its nominal value. The curve of CACR vs. k_ep was recalculated with the reduced VIF parameter value using Eq. [5]. For each k_ep the fractional change in the corresponding CACR was obtained and divided by 1% to give the error propagation ratio (EPR), defined as (fractional change in the estimate of the CACR corresponding to a k_ep value)/(fractional change in a VIF parameter). The EPR values give quantitative guidance on which parameters are most critical (3).

Experimental Validation of the Relationship between SER and k_ep

We employed the Gd-DTPA-enhanced MRI data acquired in a previous study (26) to produce the k_ep and SER maps for verifying the relationship between the two parameters. Details regarding the tumor model, the imaging protocol, and the pharmacokinetic analysis of the data have been previously described (26). In brief, six nude mice implanted with the human breast cancer line BT474 underwent DCE-MRI before and after treatment with a VEGF-receptor tyrosine kinase inhibitor (n = 3) or vehicle (n = 3). MRI was performed on a 1.5 T whole-body MRI scanner (General Electric Medical Systems, Milwaukee, WI) using a wrist RF coil (Medical Advances, Milwaukee, WI). Mice were anesthetized by inhalation of 1.5% isoflurane. Axial images were obtained for tumor localization. Tissue T₁₀ was measured using a coronal 3D variable flip angle fast gradient echo technique (27). Dynamic contrast-enhanced imaging was performed using a coronal T₁-weighted 3D gradient echo sequence with TR = 9.3 ms, TE = 4.2 ms, FA = 20°, FOV = 10 × 10 cm, imaging matrix = 256 × 256 × 28, slice thickness = 1.0 mm, NEX = 1. The DCE-MRI series were of a time resolution of 66 sec over a period of 22 min. Five precontrast scans were acquired, and then mice were injected via the tail vein with 0.2 mmol/kg Gd-DTPA (Magnevist; Schering, Berlin, Germany) over a period of 4 sec. The postinjection scan started immediately after completion of the injection. The animal studies were performed with institutional approval according to institutional and national guidelines.

k_ep Mapping

Pharmacokinetic analysis was performed on a pixel-by-pixel basis for the regions of interest (ROIs) encompassing the entire tumor. The time course of the intravascular CA concentration measured from the abdominal aorta of the mouse was used to calculate an effective VIF for each examination by fitting the plasma CA concentration time course data to a bi-exponential decay curve. The estimated bolus arrival time was 6 sec after completion of the injection (28). K^trans and ν_e were estimated using the Tofts-Kerm (T-K) model (10). Maps of scaled fitting error (SFE) (26,29) were generated as an integral part of each fitting procedure to assess the discrepancy between the derived curve and the original data. Maps of k_ep were calculated using Eq. [4].

SER Mapping

SER was computed on a pixel-by-pixel basis. For each voxel within the tumor volume, S₀ was calculated as the mean SI of the five precontrast scans; S₁ was the SI of the first postcontrast scan (t_p1 = 0.55 min) or the third postcontrast scan (t_p1 = 2.75 min); S₂ was the SI of the ninth postcontrast scan (t_p2 = 9.35 min). We use SER_0.55 to denote the SER calculated with [t_p1, t_p2] = [0.55 min, 9.35 min], and SER_2.75 to denote the SER calculated with [t_p1, t_p2] = [2.75 min, 9.35 min].

Exclusion Criteria for Correlation Analysis

Voxels that fell into the following criteria were excluded from the correlation analysis: 1) voxels associated with poor fits (SFE >35%) in pharmacokinetic analysis (30); 2) voxels showing pseudo-K^trans due to contributions from large vessel (K^trans >1.2 min⁻¹) (3,30) or exceedingly low or high ν_e (ν_e <0.1 or ν_e >1.0) (3); 3) voxels with excessively small k_ep (<0.1 min⁻¹) (3).

Correlation Analysis

The relationship between the pixel-by-pixel calculations of SER and the pixel-by-pixel calculations of k_ep was evaluated from a scatterplot of the ordered pairs of the two parameters in a region of tumor for each mouse. The effect of t_p1 selection on the monotonic relationship between SER and k_ep was assessed by visual inspection of the scatterplots generated from the same DCE-MRI dataset but using different t_p1 values. To measure the monotonicity of the relationship between the SER and k_ep, Spearman’s rank order correlation was used. The Spearman rank correlation coefficient (Splus 7, Insightful, Seattle, WA) between the paired pixel values of SER_0.55 (or SER_2.75) and k_ep was calculated for each tumor xenograft, denoted as r_0.55 and r_2.75, respectively. To test the difference between the two overlapping (in the sense of having a variable in common) correlation coefficients, r_0.55 and r_2.75, measured on each of the tumor xenografts, a Z-test as described by Meng et al. (31) was used:

$$Z=(z_{r1}-z_{r2})\sqrt{\frac{N-3}{2(1-r_x)h}}$$ [6]

where N is the number of pixels in the tumor xenograft, Z_r1 and Z_r2 are the Fisher z-transformed r_0.55 and r_2.75, respectively, and r_x is the correlation between SER_0.55 and SER_2.75,

$$h=\frac{1-f{r}_{\mu }^2}{1-r_{\mu }^2}=1+\frac{r_{\mu }^2}{1-r_{\mu }^2}(1-f)$$ [7]

$$f=\frac{1-r_x}{2(1-r_{\mu }^2)}$$ [8]

and $r_{\mu }^2=(r_{0.55}^2+r_{2.75}^2)∕2$. An “R” program (R Foundation for Statistical Computing v. 2.0.1) for the calculation of the Z values in the Meng-Rosenthal-Rubin method is available online (Li K-L, Zhu XP. http://lib.stat.cmu.edu/R/CRAN/src/contrib/Descriptions/compOverlapCorr.html). A significance level of 0.05 was used for all tests.

For assessing the impact of using population VIF as opposed to individual VIFs on the relationship between SER and k_ep, we recalculated k_ep maps for each mouse using the group mean VIF, denoted as k_{ep, group VIF}. Spearman rank correlation coefficients between the paired pixel values of SER_0.55 and k_{ep, group VIF} were calculated for each tumor xenograft, and denoted as r_{0.55, group VIF}. A paired t-test was used to compare the mean value for r_{0.55, group VIF} and r_0.55.

RESULTS

Computer Simulation and Error Estimation

Figure 2 shows an example of VIF measured from blood in the descending thoracic aorta of a patient with breast cancer. A mean bi-exponential VIF was obtained from the group of patients (n = 8) with parameters a₁ = 11.3 ± 1.75 kg/L, a₂ = 8.13 ± 1.68 kg/L, m₁ = 0.46 ± 0.11 min⁻¹, and m₂ = 0.042 ± 0.012 min⁻¹. The mean bi-exponential VIF was then used for the computer simulations.

Figure 3 shows the effects of the selection of timepoints, t_p1 and t_p2, on the relationship between CACR and k_ep. With t_p1 = 1.0 min, the CACR increases monotonically with k_ep over a wide range of k_ep values (0 ≈ 3 min⁻¹). With t_p1 = 2.5 min, the CACR maintains the monotonic relationship with k_ep over the k_ep range of 0 ≈ 1 min⁻¹; it, however, exhibits a plateau where k_ep is higher than 1 min⁻¹. Prolonging t_p2 enlarges the dynamic range of CACR (i.e., the sensitivity); e.g., the CACR range increases from 0.2–2.15 for [t_p1, t_p2] of [1 min, 7.5 min] to 0.2–2.35 for [t_p1, t_p2] of [1 min, 9 min].

FIG. 3.

The effect of different combinations of [t_p1, t_p2] on the relationship between CACR and k_ep.

Figure 4 shows the effects of variations in the tissue T₁₀ on the value of SER and the agreement between SER and CACR for different FA and [t_p1, t_p2]. In general the values of SD_ser are small (<0.01) for all the different conditions, indicating that variation in tissue T₁₀ values has little effect on the SER values and the relationship between SER and k_ep. The value of SER and the SD_ser are also affected by variations in the tissue ν_e, especially when FA is smaller than 40°. Using a small FA makes the dynamic range of SER narrower than the corresponding CACR, showing larger negative deviation from CACR at higher k_ep. With FA ≥40°, the SER agrees better with the CACR. Moreover, Fig. 4 shows that the use of the same FA but different [t_p1, t_p2] combinations has little effect on either the SD_ser or the agreement between SER and CACR in the range of k_ep of 0.1–2 min⁻¹.

FIG. 4.

The effects of variation in tissue T₁₀ on the value of SER and the agreement between SER and CACR. Theoretical SER values were simulated for k_ep ranged from 0.1 min⁻¹ to 2.0 min⁻¹, and ν_e = [0.3, 0.6]. For each k_ep and ν_e, SER is presented as mean ± 4 SD of eight SERs generated with an array of T₁₀, [600, 700, 800 … 1300] ms. The curves of CACR vs. k_ep are also included in each graph in order to view the agreement between the SER and the CACR for different [t_p1, t_p2] and FA: (a) [t_p1, t_p2] = [1 min, 9 min], FA = 20°; (b) [t_p1, t_p2] = [1 min, 9 min], FA = 40°; (c) [t_p1, t_p2] = [2.5 min, 7.5 min], FA = 20°; (d) [t_p1, t_p2] = [2.5 min, 7.5 min], FA = 40°.

Table 1 lists the propagation of errors in assumed VIF parameter values to estimate of the CACR. The EPR value in the CACR for each one of the VIF parameters was calculated using an array of k_ep values (k_ep = [0.1, Δk_ep = 0.1, 2.0]) and is reported as mean ± SD of the 20 corresponding EPR values. The a₁ and a₂ have EPR values with similar magnitude, but different sign (positive values for a₁; negative values for a₂). The effects of m₂ are mild, while the effects of m₁ are minimal.

Table 1.

Propagation of Errors in Assumed VIF Parameter Values to Estimate of CACR^a

Parameter	Nominal value	EPR in CACRb
		[tp1, tp2] = [1, 9]	[tp1, tp2] = [2.5, 7.5]
a 1	11.3 kg/L	0.44 ± 0.05	0.29 ± 0.04
a 2	8.13 kg/L	−0.45 ± 0.05	−0.30 ± 0.04
m 1	0.46 min−1	0.06 ± 0.05	−0.03 ± 0.06
m 2	0.042 min−1	0.28 ± 0.05	0.19 ± 0.04

^aThe CACR were estimated using Eq. [5].

^bThe EPR is defined as (fractional change in the estimate of CACR)/(fractional change in a VIF parameter). An array of k_ep, [0.1, 0.2, 0.3 … 2.0] was used in the estimation. EPR value in CACR for each of the VIF parameters was reported as mean ± SD of the 20 EPRs generated with the array of k_ep values.

Figure 5 shows the effects of variation in the VIF parameter values on the simulated curve of CACR vs. k_ep. The general shape of the curve of the SER vs. k_ep is not significantly affected by variations in the VIF parameters; however, the corresponding values between SER and k_ep may be affected. For [t_p1, t_p2] of [1 min, 9 min], the monotonic relationship between SER and k_ep maintains over a wide range of k_ep values (k_ep = 0 ≈ 3 min⁻¹), regardless of the variation in the VIF parameters (Fig. 5a–d).

FIG. 5.

The effects of variations of the functional VIF on the curve of CACR vs. k_ep. The “true” CACR was simulated with k_ep ranged from 0.1 min⁻¹ to 2.0 min⁻¹, a fixed ν_e (0.58), [t_p1, t_p2] = [1 min, 9 min], and the assumed VIF. CACR with ± 20% variations in a₁ (a), a₂ (b), m₁ (c), and m₂ (d) were plotted together with the “true” CACR for visual inspection.

Experimental Verification of the Relationship between SER and k_ep

Figure 6 shows a pair of representative pixel-by-pixel scatterplots of SER vs. k_ep from a pretreatment DCE-MRI dataset of a tumor xenograft. For [t_p1, t_p2] of [0.55 min, 9.35 min] (Fig. 6a), the values of SER are closely correlated with k_ep over a broad range of k_ep (0 ≈ 3 min⁻¹). However, when a delayed t_p1 (=2.75 min) was used for the same dataset (Fig. 6b), the range of k_ep showing concordance with SER narrows to k_ep = 0 ≈ 1 min⁻¹; the correlation plot reaches a plateau and starts to disperse at k_ep ≈ 1 min⁻¹ (marked with arrows).

FIG. 6.

Representative pixel-by-pixel scatterplots of SER vs. k_ep of a breast cancer in a mouse model for [t_p1, t_p2] = [0.55 min, 9.35 min] (a) and [t_p1, t_p2] = [2.75 min, 9.35 min] (b). The correlation plot with longer t_p1 (b) reaches a plateau and starts to disperse at k_ep ≈ 1 min⁻¹ (arrows).

With use of the Meng-Rosenthal-Rubin Z-test (31) for differences between overlapping correlation coefficients, we found that the pixel values on the k_ep maps, generated with the pretreatment DCE-MRI datasets of tumor xenografts, were more significantly correlated with the pixel values on the SER_0.55 maps than those on the SER_2.75 maps (all Zs > 15.0, all Ps < 0.001) generated from the same dataset (the number of voxels for each correlation analysis was 1466 ± 911). The paired pixel values of SER_0.55 and k_ep were closely positively correlated (r_0.55 = 0.87 ± 0.03). In contrast, the correlation coefficients between the paired pixel values of SER_2.75 and k_ep showed significantly lower (r_2.75 = 0.48 ± 0.07, P = 0.0001). However, increased r_2.75 was observed in tumors, with most voxels showing lower k_ep (<1 min⁻¹) after the anti-angiogenic treatment (details of the effect of the treatment are described elsewhere (26)).

Analysis of the effect of using the group mean VIF vs. individual VIFs on the relationship between SER and k_ep yielded Spearman coefficients of r_0.55 = 0.87 ± 0.03 and r_{0.55, group VIF} = 0.86 ± 0.04. The correlation between SER and k_ep was not significantly affected by the use of different VIFs, i.e., the group mean VIF and the individual VIFs (paired t-test, P = 0.22, df = 5).

DISCUSSION

This article presents a shortcut of the two-compartment pharmacokinetic model (19) that can be used in breast DCE-MRI studies. Using computer simulations, we demonstrated that 1) the signal differences between each of two postcontrast scans and the reference precontrast scan can form a ratio, SER, that tracks the CACR for scans with short TR (⪡T₁₀) and a high FA (≥40°); and 2) in the two-compartment model, the ratio can be made to monotonically relate to the redistribution rate constant k_ep by choosing the appropriate delay times for the postcontrast measurements. The comparison of pixel values on maps of SER and k_ep determined from DCE-MRI data acquired in a mouse model of breast cancer confirmed such a relationship.

One of the advantages of the SER method is that T₁₀ calibration is not needed. This makes the method superior to another model-based three-timepoint method (14), which is not based on the signal enhancement ratio. The T₁₀ of the breast tumor tissue may have a wide range of values, which makes a pronounced difference to the enhancement curve and must be taken into account in the analysis of the curve, whatever model is used. T₁₀ should be measured as accurately as possible, and this may become a limiting factor in the use of the pharmacokinetic model method for tissue characterization of the tumors in clinical practice (3). The simulations in this work have shown that the effect of T₁₀ on SER is very small at short TR (<10 ms), short TE (⪡T₂*), and a low dose of Gd-DTPA administration. This agrees with the observation by Tofts et al. (3) that the effect of T₁₀ on K^trans/ν_e (≡ k_ep) is very small. The fact that SER imaging can provide information about k_ep without spending additional scan time to measure T₁₀ makes the method simple and robust, whereas the link to underlying biochemical mechanisms is preserved.

In addition, an analytical expression relating SER (approximated as CACR) and k_ep (Eq. [5]) was derived based on the two-compartment pharmacokinetic model (19). The formula (Eq. [5]) was subsequently used to optimize t_p1 and t_p2 for the monotonic mathematical relationship between SER and k_ep over a limited range of k_ep values. Reducing t_p1 broadened the dynamic range of concordance between k_ep and SER and increasing t_p2 broadens the dynamic range of CACR. Moreover, in considering the effects of the first few intravascular CA bolus passages on the CACR (27), a t_p1 ⪡1 min should be avoided. We suggest that, in general, a combination of t_p1 = 1–1.5 min and t_p2 = 7–9 min should be used for studies in human subjects. However, specifically optimized [t_p1, t_p2] can be obtained depending on the specific range of k_ep of the lesion under study.

Another pivotal aspect of the DCE method is the determination of the VIF. In the computer simulations for this work we employed the mean bi-exponential VIF derived from the DCE-MRI data acquired in a previous breast cancer study (7), instead of the commonly used (32,33) T-K standardized bi-exponential VIF (10). The T-K VIF, which was derived from a dataset comprising a time span of 120 min, used two assumptions: 1) an instantaneous mixing of CA throughout the central compartment after bolus injection (10), and 2) two dominant pharmacokinetic processes, i.e., the equilibration between the central compartment and the extravascular extracellular space (EES) and the elimination via the kidneys. Neither assumption is valid for a VIF observed in the short time span (≤10 min) typical of clinical breast DCE-MRI exams. The reality is that after contrast administration the initial distribution of CA bolus into the central compartment, i.e., a hypothetical volume including plasma and the rapidly perfused tissues (34), takes several passages. For a 10-min DCE-MRI analysis, the initial intra-central compartment distribution and the equilibration between the central compartment and EES become the two dominant pharmacokinetic processes. The effect of elimination process of contrast agents, such as Magnevist, which has a mean half-life of 96 min (http://www.drugs.com/magnevist.html), is negligible for the 10-min DCE-MRI analysis. As such, the new bi-exponential VIF derived from the dataset with a comparable time span is more appropriate because the aforementioned systematic errors resulting from the use of standardized VIF (25) can be avoided.

Simulations in the study show that SER is a close estimate of CACR and solely depends on k_ep at a high FA (>40°). In contrast, with low FAs the value of SER is not only dependent on k_ep, but also affected by ν_e (Fig. 4). This makes high FAs attractive if quantitative DCE-MRI is the main concern; however, at the expense of a lower contrast to noise ratio, given a short TR. Based on the computer simulation using the Ernst equation, the highest contrast between tumor (T₁₀ = 1.3 sec) and normal breast tissue (T₁₀ = 0.6 sec) occurs at an FA of 14°, with TR = 8 ms. Contrast drops to 55% of the maximum with FA = 40°. As such, the pre-contrast images may provide less optimal image quality for evaluation of tumor morphology.

One limitation of this study is that only limited data were available for validating the simulation results. The validation of the simulated relationship between SER and k_ep was performed using the DCE-MRI data from a previous study (26), which were of low temporal resolution (66 sec) and low FA (20°). In the SER analysis of the animal data, the [t_p1, t_p2] were selected as close as possible to those used in the simulations. They were [0.55 min, 9.35 min] and [2.75 min, 9.35 min], taken from the timepoints in the contrast enhancement curves which were t_i = [0.55, 1.65, 2.75, … 14.85] min. Because of the rapid heart rate in mice, a t_p1 value as early as 0.55 min is long enough to allow the first couple of CA bolus to pass and the t_p1 of 0.55 min can be used to maximize SER-k_ep concordance range. The scatterplots of the pixel-by-pixel comparison of SER and k_ep in Fig. 6 show the same trend as shown by the simulation, even though the VIFs used in the animal study were different from the human VIF used in the simulations. The relatively large dispersion was partially due to the relatively low SNR in the animal DCE-MRI data and the variation of ν_e in tumors. The effects of various ν_e on the relationship between SER and k_ep were more prominent at low FA (20°) as demonstrated in the simulation (Fig. 4). Prolonging t_p2 enlarges the dynamic range of SER, and may help in reducing such dispersion.

Finally, the results from this study suggest the importance of adequate quality control in the parametric mapping. The importance of quality control in model-based analysis using a fitting procedure has been previously addressed in the pharmacokinetic model-based DCE-MRI studies (26,27,35). However, such quality control has not yet become the common practice in semiquantitative parametric mapping. For pharmacokinetic parameters estimated using a fitting procedure, voxel elimination was performed using numerical criteria assessing the goodness-of-fit (29,36). Criteria for the voxel elimination were also set according to the physiological range of the pharmacokinetic parameters (35). Fitting error, defined as a measure of the difference between modeled and measured data, arises from both the modeling error and the random noise, which may be assessed separately as reported in a recent article (36). The fitting procedure, generally based on least-squares regression, renders the model-based parametric mapping more robust (insensitive) to noise than the semiquantitative parametric mapping. However, accidental modeling of some of the random noise and covariance error in multiparameter curve fitting (37) may occur, especially for data with low SNR. In contrast, for the semiquantitative parametric mapping we do not estimate fitting uncertainties. Possible bias in the measurements of the parameter due to physiological and random noise becomes major concern, although SER images, in general, are inherently of higher SNR than other high temporal resolution DCE-MRI data. It has been demonstrated that the voxel selection using empirical criteria (16) is effective in preventing errors resulting from noise. Figure 7 shows the k_ep and SER ([t_p1, t_p2] = [0.55 min, 14.85 min], FA = 20°) maps from slices covering the entire volume of a tumor xenograft. The k_ep maps in Fig. 7a were generated with the inclusion criteria of SFE <0.35, ν_e >0.1, and K^trans >0.0. The SER maps in Fig. 7b were generated by identifying voxels with significant enhancement, i.e., either S₁–S₀ or S₂–S₀ was greater than 4 SDs in baseline signal intensities (SD_blsi). Specifically, SER was calculated for voxels with S₁–S₀ >4 SD_blsi and (S₁–S₀)/S₀ >0.55, or voxels with S₂–S₀ > 4SD_blsi and (S₂–S₀)/S₀ >0.2. The k_ep maps in Fig. 7a and the SER maps in Fig. 7b, both generated by eliminating voxels with potentially meaningless results, show great similarity. In contrast, the SER maps generated without proper exclusion criteria (Fig. 7c) appear different from the k_ep maps.

FIG. 7.

Five-color coded parametric maps from a pretreatment DCE-MRI dataset of a tumor xenograft: (a) k_ep maps for each slice level; the inclusion criteria were SFE <0.35, ν_e >0.1, and K^trans >0.0; the k_ep value was set to zero for voxels beyond these inclusion criteria. (b) SER maps; SER were calculated for voxels with S₁–S₀ >4 SD_blsi and (S₁–S₀)/S₀ > 0.55, or voxels with S₂–S₀ >4 SD_blsi and (S₂–S₀)/S₀ >0.2; SER value was set to zero for voxels beyond these inclusion criteria. (c) SER maps generated without inclusion criteria.

In summary, this study provides a systematic analysis for a better understanding of the physiologic relevance of the simple parameter SER. The model-based SER method is superior to the standard, semiqualitative methods of analyzing DCE-MRI. With optimized acquisition parameters and selective postinjection timepoints (t_p1 and t_p2), SER reflects the redistribution rate constant, k_ep, which has been reported as a putative marker of tissue vascular endothelial growth factor (VEGF) expression in breast tumors (38), and has been successfully used in the differentiation between benign and malignant tumors with high sensitivity and specificity (2,5,6). The optimized SER is advanced by its model-based feature giving insight into the pathophysiology of disease, while maintaining advantages as a high spatial resolution 3D fat-suppressed breast MRI with large imaging volumes covering whole breasts. The model-based SER has promise for improved tissue characterization of breast tumors and may ultimately guide the development of novel therapeutic strategies.

APPENDIX

The theoretic prediction of the steady state signal, S, from a transverse-spoiled gradient-echo acquisition, assuming short echo time (TE ⪡ T₂*), is given by:

$$S=M\cdot \sin \alpha \cdot \frac{1-\exp (-\mathrm{TR}\cdot R_1)}{1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_1)}$$ [A1]

where M is the proton density, α is the flip angle, TR is the repetition time, and R₁ is the spin-lattice relaxation rate (R₁ ≡ 1/T₁).

Substituting R₁ in Eq. [A1] with R₁ at the three acquisition timepoints, R₁₀, R₁₁, and R₁₂, the signal intensity at the three timepoints can be calculated as:

$$S_0=M\cdot \sin \alpha \cdot \frac{1-\exp (-\mathrm{TR}\cdot R_{10})}{1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{10})}$$

$$S_1=M\cdot \sin \alpha \cdot \frac{1-\exp (-\mathrm{TR}\cdot R_{11})}{1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{11})}$$

and

$$S_2=M\cdot \sin \alpha \cdot \frac{1-\exp (-\mathrm{TR}\cdot R_{12})}{1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{12})}$$ [A2]

The signal enhancement at t_p1 is:

$$S_1-S_0=M\cdot \sin \alpha \times \frac{\left(\exp \right(-\mathrm{TR}\cdot R_{11})-\exp (-\mathrm{TR}\cdot R_{10}\left)\right)(\cos \alpha -1)}{(1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{11}\left)\right)(1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{10}\left)\right)}$$ [A3]

Similarly, the signal enhancement at t_p2 is:

$$S_2-S_0=M\cdot \sin \alpha \times \frac{\left(\exp \right(-\mathrm{TR}\cdot R_{12})-\exp (-\mathrm{TR}\cdot R_{10}\left)\right)(\cos \alpha -1)}{(1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{12}\left)\right)(1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{10}\left)\right)}$$ [A4]

The ratio of signal enhancement at the two points is:

$$\frac{S_1-S_0}{S_2-S_0}=\frac{\exp (-\mathrm{TR}\cdot R_{11})-\exp (-\mathrm{TR}\cdot R_{10})}{\exp (-\mathrm{TR}\cdot R_{12})-\exp (-\mathrm{TR}\cdot R_{10})}\cdot \frac{1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{12})}{1-\cos \alpha \cdot \exp (-\mathrm{TR}\cdot R_{11})}$$ [A5]

For a short TR typically used for rapid acquisition and a low dose (0.1 mmol/kg) of Gd-DTPA administration, it can be assumed that TR ⪡ T₁; therefore, exp(-TR · R₁) ≈ 1-TR · R₁. Eq. [A5] can be rewritten as:

$$\frac{S_1-S_0}{S_2-S_0}=\frac{R_{11}-R_{10}}{R_{12}-R_{10}}\cdot \frac{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot R_{12}}{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot R_{11}}$$ [A6]

Remembering that C(t_p1) = (R₁₁ − R₁₀)/ ℜ1 and C(t_p2) = (R₁₂ − R₁₀)/ ℜ1, where ℜ1 is T₁ relaxivity of Gd-DTPA, SER in tissues can therefore be expressed as:

$$\mathrm{SER}=\frac{S_1-S_0}{S_2-S_0}=\frac{C\left(t_{p1}\right)}{C\left(t_{p2}\right)}\cdot \frac{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot R_{12}}{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot R_{11}}$$ [A7]

By defining a factor, A, as SER/CACR, where CACR ≡ C(t_p1)/C(t_p2), we can rewrite Eq. [A7] as:

$$A=\frac{\mathrm{SER}}{\mathrm{CACR}}=\frac{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot (R_{10}+C(t_{p2})\cdot \Re 1)}{1-\cos \alpha +\cos \alpha \cdot \mathrm{TR}\cdot (R_{10}+C(t_{p1})\cdot \Re 1)}.$$ [A8]