Polyaspartic Acid-Coated Paramagnetic Gadolinium Oxide Nanoparticles as a Dual-Modal T₁ and T₂ Magnetic Resonance Imaging Contrast Agent

Featured Application

Dual-modal T1 and T2 magnetic resonance imaging contrast agent.

Abstract

Surface-coating polymers contribute to nanoparticle-based magnetic resonance imaging (MRI) contrast agents because they can affect the relaxometric properties of the nanoparticles. In this study, polyaspartic acid (PASA)-coated ultrasmall Gd₂O₃ nanoparticles with an average particle diameter of 2.0 nm were synthesized using the one-pot polyol method. The synthesized nanoparticles exhibited r₁ and r₂ of 19.1 and = 53.7 s⁻¹mM⁻¹, respectively, (r₁ and r₂ are longitudinal and transverse water–proton spin relaxivities, respectively) at 3.0 T MR field, approximately 5 and 10 times higher than those of commercial Gd-chelate contrast agents, respectively. The T₁ and T₂ MR images could be obtained due to an appreciable r₂/r₁ ratio of 2.80, indicating their potential as a dual-modal T₁ and T₂ MRI contrast agent.

1. Introduction

Biocompatible magnetic nanoparticles have drawn attention owing to their potential applications in biomedical theragnosis [1,2,3,4]. Among various biomedical applications, magnetic resonance imaging (MRI) contrast agents have been an issue because MR image sensitivity and resolution can be improved using contrast agents [5,6,7,8]. Thus, nanoparticles with advanced magnetics properties have been investigated to develop advanced MRI contrast agents [9,10,11].

The clinically available Gd-chelates act as positive (or T₁) MRI contrast agents due to their efficient induction of longitudinal (or T₁) water–proton spin relaxation with respect to transverse (or T₂) water–proton spin relaxation, providing r₂/r₁ ratios (r₁ and r₂ are longitudinal and transverse water–proton spin relaxivities, respectively), which are close to one [5,6]. This is due to a high pure spin magnetic moment (S = 7/2) of Gd³⁺ [5]. The r₁ values of Gd-chelates are low [5,6,7,8]. However, the r₁ values and as a result, imaging performance can be improved using ultrasmall Gd₂O₃ nanoparticles because of their high Gd-densities per nanoparticle and appreciable nanoparticle magnetic moments [11]. The ultrasmall Gd₂O₃ nanoparticles have shown r₁ values higher than those of commercial Gd-chelates [11,12,13,14].

To use ultrasmall Gd₂O₃ nanoparticles as T₁ MRI contrast agents, surface coating is necessary to endow them with hydrophilicity and biocompatibility [15]. In addition, surface-coating ligands can affect their r₁ and r₂ values [16,17] because these values depend on the water-attracting power of surface-coating ligands around Gd₂O₃ nanoparticles. According to the inner-sphere model [5], water molecules close to Gd₂O₃ nanoparticles can affect the r₁ value, whereas the remaining water molecules can contribute to the r₂ value according to the outer-sphere model [5,18]. Therefore, surface-coating ligands may allow us to vary the r₂/r₁ ratio and as a result, T₁ and T₂ imaging modes of ultrasmall Gd₂O₃ nanoparticles. In this study, polyaspartic acid (PASA), which is a biodegradable and hydrophilic amino acid polymer [19], was used as a surface-coating ligand. PASA-coated ultrasmall Gd₂O₃ nanoparticles exhibited an appreciable r₂/r₁ ratio, thus allowing us to obtain both T₁ and T₂ MR images.

In this study, PASA-coated ultrasmall Gd₂O₃ nanoparticles were synthesized using the one-pot polyol method. The synthesized nanoparticles were subject to various analyses to characterize their particle size, colloidal stability, surface coating, in vitro cellular toxicity, and magnetic and relaxometric properties. They were applied to in vivo MRI in mice to obtain both T₁ and T₂ MR images.

2. Materials and Methods

2.1. Chemicals

GdCl₃∙6H₂O (99.9%), NaOH (99.9%), triethylene glycol (TEG) (99.9%), and poly–(αβ)-DL-aspartic acid (PASA) sodium salt (M_w = ~9900 amu) were purchased from Sigma Aldrich, St. Louis, MO, USA. C₂H₅OH (99.99%) was purchased from Duksan, Anshan, South Korea, and used for the initial washing of nanoparticles. All chemicals were used as received. Triple-distilled water was used for the final washing of nanoparticles and preparation of nanoparticle solution (i.e., suspension) samples.

2.2. Synthesis

PASA-coated Gd₂O₃ nanoparticles were synthesized through a polyol method (Figure 1). An amount of 0.5 mmol of GdCl₃∙6H₂O and 0.1 mmol of PASA were added to 10 mL TEG in a three-necked round-bottom flask. The mixture was magnetically stirred at room temperature under atmospheric conditions until the precursor and PASA dissolved in TEG. In a separate beaker, NaOH solution was prepared by dissolving 2.0 mmol of NaOH in 10 mL TEG at room temperature using a magnetic stirrer. The NaOH solution was slowly added to the precursor solution using a pipette until the pH of the solution reached ~10. At constant pH, the reaction temperature increased from room temperature to 110 °C and was maintained at that temperature for 15 h with magnetic stirring. After the reaction completion, the product solution was cooled to room temperature and transferred to a 500 mL beaker. The product nanoparticles were washed thrice with 400 mL ethanol. For this step, the product solution was magnetically stirred for 10 min and then, stored at a cool place (4 °C) until the nanoparticles settled down. The supernatant was decanted and the remaining product solution was washed again using ethanol. To remove ethanol and concentrate the nanoparticle solution sample, 400 mL triple-distilled water was added to the product solution and rotary evaporation was used to reduce the solution volume to approximately 50 mL. This process was repeated thrice. The obtained concentrated nanoparticle solution sample was split into two equal volumes: one part was subject to air-drying to obtain a powder sample for various characterizations and the other part was diluted using triple-distilled water to obtain a nanoparticle solution sample (30–50 mM Gd) for cytotoxicity and MRI analyses.

2.3. Characterizations

The particle diameter (d) of the nanoparticles was measured using a high-resolution transmission electron microscope (HRTEM) (Titan G2 Chemi STEM CS Probe, FEI, Hillsboro, OR, USA) operated at 200 kV acceleration voltage. The copper grid (PELCO No.160, TED PELLA, Inc., Redding, CA, USA) covered with carbon film was used to support nanoparticles for imaging. The hydrodynamic diameter (a) and zeta potential (ξ) were measured using a dynamic light scattering (DLS) particle size and zeta potential analyzer (Zetasizer Nano ZS, Malvern, Malvern, UK) using a 0.01 mM Gd solution sample. The crystal structural analyses of the nanoparticles before and after thermogravimetric analysis (TGA) were conducted using a multi-purpose powder X-ray diffraction (XRD) spectrometer (X’PERT PRO MRD, Philips, The Netherlands) with unfiltered CuKα radiation (λ = 1.54184 Å). The Gd-concentration of the aqueous nanoparticle solution sample was measured using an inductively coupled plasma-atomic emission spectrometer (ICP-AES) (IRIS/AP, Thermo Jarrell Ash Co., Waltham, MA, USA). A Fourier transform-infrared (FT-IR) absorption spectrometer (Galaxy 7020A, Mattson Instrument, Inc., Madison, WI, USA) was used to study the surface coating of the nanoparticles. To record FT-IR absorption spectra, pellets of the powder samples in KBr were prepared and the spectra were recorded between 400 and 4000 cm⁻¹. A TGA instrument (SDT-Q600, TA Instrument, New Castle, DE, USA) was used to record the TGA curve of the powder sample between room temperature and 900 °C while air flowed over the sample. The average amount of PASA polymers coating a nanoparticle was quantified from the mass drop in the TGA curve after considering the initial mass drop from room temperature to ~110 °C due to water desorption. A vibrating sample magnetometer (VSM) (7407-S, Lake Shore Cryotronics Inc., Westerville, OH, USA) was used to obtain the magnetic properties of the powder sample by recording a magnetization (M) versus applied field (H) (or M−H) curve (−2.0 T ≤ H ≤ 2.0 T) at 300 K. The net M value of the sample (i.e., only the Gd₂O₃ nanoparticles without PASA coating) was estimated using the net mass of Gd₂O₃ nanoparticles extracted from the TGA curve.

2.4. In Vitro Cytotoxicity Tests

Cellular toxicity of an aqueous nanoparticle solution sample was assessed using a CellTiter-Glo Luminescent Cell Viability Assay (Promega, Madison, WI, USA). In this assay, a luminometer (Synergy HT, BioTek, Winooski, VT, USA) was used to quantify intracellular adenosine triphosphate. Human prostate cancer (DU145) and normal mouse hepatocyte (NCTC 1469) cell lines were used as test cells. Dulbecco’s Modified Eagle medium and Rosewell Park Memorial Institute-1640 culture media were used for NCTC1469 and DU145 cells, respectively. Both cells were seeded on a 24-well cell culture plate and incubated for 24 h (5 × 10⁴ cell densities, 500 μL cells per well, 5% CO₂, and 37 °C). Five test nanoparticle solution samples were prepared by diluting an original nanoparticle solution sample with a sterile phosphate buffer saline solution. Two microliters of each test solution sample were added to the cells to obtain 10, 50, 100, 200, and 500 μM Gd-concentration, and the treated cells were then incubated for 48 h. The viability of the treated cells was measured and normalized with respect to the control cells. The measurement was repeated thrice for all treated cells to obtain average cell viabilities.

2.5. Relaxivity Measurements

Both longitudinal (T₁) and transverse (T₂) relaxation times as well as longitudinal (R₁) and transverse (R₂) map images were measured using a 3.0 T MRI scanner (MAGNETOM Trio Tim, Siemens, Munchen, Bayern, Germany) at 20 °C. Various concentrations of aqueous nanoparticle solution samples (0.1, 0.05, 0.025, 0.0125, 0.00625, and 0.0 mM Gd) were prepared in a series by diluting an original aqueous nanoparticle solution sample using triple-distilled water. T₁ relaxation times were measured using an inversion recovery method. T₂ relaxation times were obtained using a multiple spin-echo method with a Carr–Purcell–Meiboom–Gill pulse sequence. Then, r₁ and r₂ values were estimated from the slopes in the plots of 1/T₁ and 1/T₂ versus Gd-concentration, respectively.

2.6. In Vivo T₁ and T₂ MR Image Measurements

In vivo animal imaging was conducted based on the rules and regulations and permission of the animal research committee of the Korea Institute of Radiological and Medical Sciences (KIRAMS). The same 3.0 T MRI scanner used for relaxivity measurements was used to obtain T₁ and T₂ MR images. Two Institute of Cancer Research mice weighing ~30 g (N_mice = 2) were used for each modal imaging. The mice were anesthetized using 1.5% isoflurane in oxygen. Measurements were made before and after injecting an aqueous nanoparticle solution sample into mice tail veins. The injection doses were approximately 0.013 and 0.1 mmol Gd/kg for T₁ and T₂ MR image measurements, respectively. During measurements, the body temperature of the mice was maintained at 37 °C using a warm water blanket. After measurements, the mice were revived from anesthesia and placed in a cage with free access to food and water. The spin-echo sequence was used for MR image measurements at 3.0 T. The typical measurement parameters were as follows: echo time = 10 (or 37) ms, repetition time = 385 (or 1620) ms, pixel bandwidth = 299 (or 197) Hz, number of acquisitions = 8 (or 4), echo train length = 3 (or 13), flip angle = 120^o (or 120^o), slice thickness = 1.0 (or 1.0) mm, and slice gap = 1.1 (or 1.1) mm for T₁ (or T₂) MR image measurements.

3. Results

3.1. Nanoparticle Size and Structure

The particle diameters were ultrasmall (Figure 2a) and the average particle diameter (d_avg) was estimated to be 2.0 nm by fitting an observed particle diameter distribution to a log-normal function (Figure 2b and

Table 1. Average particle diameter (d_avg), average hydrodynamic diameter (a_avg), surface-coating amount (P, σ, N_PAA), magnetic properties, and water-proton spin relaxivities (r₁, r₂) of PASA-coated Gd₂O₃ nanoparticles.

davg (nm)	aavg (nm)	Surface-Coating Amount			ξavg (mV)	Magnetic Properties		Water–Proton Spin Relaxivities at 3.0 T and 20 °C
		P 1 (wt.%)	σ 2 (1/nm2)	NPAA 3		Magnetism	M at 2.0 T and 300 K (emu/g)	r1 (s−1mM−1)	r2 (s−1mM−1)
2.0 ± 0.1	12.7 ± 0.2	63.2 ± 0.2	0.29 ± 0.05	4 ± 1	−28.0 ± 0.2	Paramagnetic	1.95 ± 0.05	19.1 ± 0.1	53.7 ± 0.1

¹ Average surface-coating amount in wt.% of the PASA polymers per nanoparticle. ² Average grafting density (i.e., average number of polymers coating a nanoparticle unit surface area). ³ Average number of PASA polymers coating a nanoparticle.

). The average hydrodynamic diameter (a_avg) of the nanoparticles dispersed in triple-distilled water (0.01 mM Gd) was estimated to be 12.7 nm by fitting an observed DLS pattern to a log-normal function (Figure 2c and Table 1). An aqueous nanoparticle solution sample is presented in Figure 3a. To confirm the colloidal dispersion of the nanoparticles in aqueous media, laser scattering (i.e., Tyndall effect) was conducted (Figure 3b). Laser scattering was observed only in the nanoparticle solution sample whereas no such scattering was observed in reference triple-distilled water, confirming the colloidal dispersion. In addition, the average zeta potential (ξ_avg) was estimated to be −28.0 mV in aqueous media (Figure 3c). This high value supported the observed good colloidal stability.

XRD patterns of the synthesized nanoparticles before and after TGA were recorded (Figure 4). The XRD pattern of the as-synthesized nanoparticles was broad and structureless, indicating that they were poorly crystallized (i.e., amorphous) due to their ultrasmall particle size [20]. However, sharp peaks were obtained after TGA, indicating crystallization and particle size growth of the nanoparticles after TGA, as observed previously [21]. The observed XRD pattern after TGA was in good agreement with that of cubic Gd₂O₃ (JCPDS card no: 43-1014) and the estimated lattice constant (10.81 Å) was comparable to the reported value (10.818 Å) [22].

3.2. Surface Coating

The coating of PASA polymers on the nanoparticle surface was demonstrated by recording an FT-IR absorption spectrum (bottom spectrum labeled III in Figure 5a). In addition, FT-IR absorption spectra of bare Gd₂O₃ nanoparticles and PASA were taken (top and middle spectra labeled I and II in Figure 5a, respectively) as reference. The observed absorption frequencies are provided in

Table 2. Observed FT-IR absorption frequencies in cm⁻¹.

Assignment	Bare Gd2O3	PASA	PASA-Coated Gd2O3 Nanoparticle	Ref. [23]	Ref. [24]
O−H Stretch	3441	~3440	~3440	–	3616
N−H (Amide A) Stretch	–	~3283	~3296	3500	3423
C−H Stretch	–	2925	2868	2945	–
O−H Bend	1635	1635	~1635	–	1645
C=O (Amide I) Stretch	–	1717	~1653	–	–
COO− Antisymmetric Stretch	–	1585	1591	1595	1586
COO− Symmetric Stretch	–	1395	1395	–	–
N−H (Amide II) Bend	–	1521	1521	–	–
C−O Stretch	–	1151	1113, 1062	–	1102

. Characteristic absorption peaks of PASA [23,24] were also observed in the FT-IR absorption spectrum of PASA-coated Gd₂O₃ nanoparticles, confirming the surface coating. The frequency difference of ~196 cm⁻¹ between symmetric and asymmetric COO⁻ stretches indicated that COO⁻ groups were electrostatically bonded to Gd³⁺ on a nanoparticle surface through a bridge bond [25], as drawn in Figure 5b. Considering numerous COO⁻ groups per PASA (i.e., n = ~72), numerous bridge bonds are expected per PASA with a nanoparticle although only one bond is presented in Figure 5b.

The coating amount of PASA polymers per nanoparticle was estimated in weight percentage (%) from a TGA curve (Figure 5c). The initial mass drop of 5.5% between room temperature and ~110 °C was due to water desorption, and the next mass drop of 63.2% was due to PASA burning through its reaction with flowing hot air. The final mass drop of 31.3% was due to the remaining Gd₂O₃ nanoparticles. To estimate the average number of PASA polymers coating a nanoparticle per unit surface area, a grafting density (σ) [26] was calculated to be 0.29 nm⁻² using a molecular mass of PASA (i.e., 9900 amu), the average particle diameter estimated from HRTEM (i.e., 2.0 nm), and the bulk density of 7.407 g/cm³ [27] for Gd₂O₃. By multiplying σ with the average nanoparticle surface area (= πd_avg²), the average number of PASA polymers coating a nanoparticle was estimated to be ~4. The results are provided in Table 1.

3.3. In Vitro Cell Viability

As free Gd³⁺ ions are toxic in vivo [28,29], in vitro cytotoxicity of PASA-coated Gd₂O₃ nanoparticles was measured in DU145 and NCTC1469 cells. As shown in Figure 6, cell viabilities were approximately 100 and 77% in DU145 and NCTC1469 cells at 500 μM Gd, respectively, indicating good biocompatibility of the nanoparticles.

3.4. Magnetic Properties

Magnetic properties of PASA-coated Gd₂O₃ nanoparticles were characterized by obtaining an M−H curve at T = 300 K. As shown in Figure 7, the M−H curve showed no hysteresis (i.e., zero value in coercivity and remanence), indicating paramagnetism of the nanoparticles, similar to bulk Gd₂O₃ [30,31]. In the plot, the net M value of the nanoparticles (i.e., only Gd₂O₃ nanoparticles without PASA) was used, which was estimated using the net mass of Gd₂O₃ nanoparticles extracted from a TGA curve. The net M at H = 2.0 T was estimated to be 1.95 emu/g (Table 1). This appreciable M value is due to a high-spin magnetic moment (S = 7/2) of 4f-electrons of Gd³⁺.

3.5. R₁ and R₂ Values and R₁ and R₂ Map Images

As shown in Figure 8a, r₁ and r₂ values were estimated to be 19.1 and 53.7 s⁻¹mM⁻¹ (r₂/r₁ = 2.80) (Table 1), respectively, from the slopes in the plots 1/T₁ and 1/T₂ versus Gd-concentration using an equation 1/T_i = r_iC + I (i = 1, 2) where C is the Gd-concentration and I is the intercept. The obtained r₁ and r₂ values were approximately 5 and 10 times higher than those of commercial Gd-chelates [5,6], respectively. Dose-dependent contrast enhancements were observed in both R₁ and R₂ map images (Figure 8b), demonstrating the ability of the nanoparticles to enhance both T₁ and T₂ MR contrasts in vitro.

3.6. In Vivo T₁ and T₂ MR Images: Dual-Modal Imaging

As shown in Figure 9a, positive (or brightened) and negative (or darkened) contrasts were observed in T₁ and T₂ MR images of mice livers, respectively, after intravenous injection of the nanoparticle solution sample into mice tails. Time-course contrast changes of signal-to-noise ratios (SNRs) of a region-of-interest (ROI) (labeled as dotted circles) are plotted as a function of time before and after injection in Figure 9b. The contrast increased initially and then, decreased with time in T₁ MR images whereas the contrast decreased initially and then, slowly increased with time in T₂ MR images. These results indicated that PASA-coated Gd₂O₃ nanoparticles should act as a dual-modal T₁ and T₂ MRI contrast agent. Ultrasmall nanoparticles (d <3.0 nm) can be excreted via the renal system [32,33]. Therefore, most of the nanoparticles were likely to be excreted via the renal system.

4. Discussion

In this study, PASA-coated Gd₂O₃ nanoparticles were synthesized via the one-pot polyol method and characterized with various techniques. Their relaxometric and MRI imaging properties were studied. Surface-coating ligands can affect r₁ and r₂ values of Gd₂O₃ nanoparticles [16,17]. The PASA has not been used as a surface-coating ligand and thus, was investigated here. The r₁ and r₂ values of PASA-coated Gd₂O₃ nanoparticles were 19.1 and 53.7 s⁻¹mM⁻¹, respectively, which are much higher than those [5,6,7] of commercial Gd-chelates. Interestingly, PASA-coated ultrasmall Gd₂O₃ nanoparticles exhibited an appreciable r₂/r₁ ratio of 2.80 and provided both T₁ and T₂ MR images. To explain this, we used the relationship between hydrophilicity of polymers and r₂/r₁ ratio of polymer-coated Gd₂O₃ nanoparticles using various polymer-coated Gd₂O₃ nanoparticles studied here and previously [34,35,36]. As provided in

Table 3. r₁ and r₂ values of various polymer-coated Gd₂O₃ nanoparticles.

davg (nm)	Polymer	CO2− Group Weight Percent	Temperature (20 °C)	Applied Field (T)	r1 (s−1mM−1)	r2/r1	Ref.
2.0	PASA (9900 Amu)	38.6	22	3.0	19.1	2.80	This Study
2.0	PAA (5100 Amu)	62.0	22	1.5	31.0	1.2	[34]
1.8	PAAMA (3000 Amu)	71.4	22	3.0	40.6	1.56	[35]
1.9	PMVEMA (80,000 Amu)	51.2	22	3.0	36.2	2.0	[36]

PAA = polyacrylic acid; PAAMA = polyacrylic acid-co-maleic acid; PMVEMA = polymethyl vinyl ether-alt-maleic acid.

, the r₂/r₁ ratio of PASA-coated Gd₂O₃ nanoparticles was higher than those [34,35,36] of the other hydrophilic polymer-coated Gd₂O₃ nanoparticles. To understand this, the r₂/r₁ ratio of various polymer-coated Gd₂O₃ nanoparticles [34,35,36] was plotted as a function of COO⁻ group weight percentage of polymers in Figure 10, showing that the r₂/r₁ ratio generally decreased with increasing COO⁻ group weight percentage of the polymers. In this plot, it was assumed that among hydrophilic groups of the polymers, the COO⁻ group could most strongly attract water molecules around the Gd₂O₃ nanoparticles through its negative charge and electronegative oxygens. Thus, other hydrophilic groups such as C−O, C=O, and NH were excluded from the hydrophilic group weight percentage of the polymers. This plot indicates that a more water-attracting (or more hydrophilic) polymer can provide the r₂/r₁ ratio which is closer to one. This is because T₁ water–proton spin relaxation can be more strongly induced if more water molecules are attracted closer to the Gd₂O₃ nanoparticles according to the inner-sphere model [5]. Under such conditions, the r₂/r₁ ratio will be close to one; thus, contrast agents can efficiently act as T₁ MRI contrast agents. However, PASA has the lowest COO⁻ group weight percentage among the polymers (Table 3 and Figure 10) and thus, the lowest water-attracting power around the Gd₂O₃ nanoparticles. Therefore, PASA-coated Gd₂O₃ nanoparticles will most weakly induce T₁ water–proton spin relaxation among polymer-coated Gd₂O₃ nanoparticles in Table 3. This explains the observed appreciable r₂/r₁ ratio of 2.80 in this study, which is higher than those of the other hydrophilic polymer-coated Gd₂O₃ nanoparticles in Table 3. Due to this, PASA-coated Gd₂O₃ nanoparticles showed T₂ MR images as well as T₁ MR images, thus acting as a dual-modal T₁ and T₂ MRI contrast agent. This result indicates that dual-modal T₁ and T₂ MRI is possible through a proper choice of hydrophilic polymers as surface-coating ligands of Gd₂O₃ nanoparticles with which the r₂/r₁ ratio is appreciable.

5. Conclusions

In summary, surface-coating ligand plays an important role in relaxometric and MRI properties of Gd₂O₃ nanoparticles. The PASA has not been used as a surface-coating ligand of Gd₂O₃ nanoparticles and thus, was investigated here. PASA-coated ultrasmall Gd₂O₃ nanoparticles were synthesized using the one-pot polyol method and their relaxometric and imaging properties in MRI were investigated. The PASA-coated Gd₂O₃ nanoparticles were ultrasmall with a d_avg of 2.0 ± 0.1 nm and exhibited good biocompatibility up to 500 μM Gd as determined from cellular cytotoxicity tests. The PASA-coated Gd₂O₃ nanoparticles were paramagnetic and exhibited r₁ = 19.1 ± 0.1 s⁻¹mM⁻¹ and r₂ = 53.7 ± 0.1 s⁻¹mM⁻¹, which are higher than those of commercial Gd-chelates. Due to an appreciable r₂/r₁ ratio of 2.80, the PASA-coated ultrasmall Gd₂O₃ nanoparticles exhibited both in vivo T₁ and T₂ MR images in mice. This result indicates that dual-modal T₁ and T₂ MRI is possible through a proper choice of hydrophilic polymers as surface-coating ligands of Gd₂O₃ nanoparticles which provide appreciable r₂/r₁ ratios.