Host-guest interactions between dapsone and β-cyclodextrin (Part II): Thermal analysis, spectroscopic characterization, and solubility studies

Article abstract of DOI:10.1007/s10847-011-0089-1

The complex of dapsone with β-cyclodextrin was prepared by the co-precipitation/freeze-drying method. The physical-chemical characteristics of the complex were investigated by different methods and compared with those of the physical mixture and of the isolated compounds. The methods used were infrared spectroscopy, X-ray diffractometry and differential scanning calorimetry. The stability constant was calculated from phase solubility diagram (Higuchi-Connors) and fluorescence spectroscopy. The stoichiometry of the complex was confirmed by Job's plot. Fluorescence measurements at different temperatures provided the thermodynamic parameters of the complexation. The infrared spectrum showed the disappearance of the SO₂ asymmetric stretching band of the drug at 1275 cm^-1 after complexation. The amorphization of the samples, as revealed by the X-ray diffraction patterns, was an indirect proof of the inclusion complex. The thermal analysis showed that the curves of the physical mixture are combination of the curves of both constituents (dapsone and β-cyclodextrin) while the absence of the melting peak of the drug in the DSC curve of the complex suggests the inclusion of the drug molecule in the host cavity as a 1:1 complex as indicated by Job's plot. There was a linear increase in its solubility with the increase of the cyclodextrin concentration and the complex was classified as an A _L-type. The value of the stability constant was 3,998 L mol ^-1 calculated by the Higuchi-Connors diagram and around 18,100 L mol^-1 from the fluorescence method indicating a strong interaction between the host and the guest. Complex formation was a spontaneous and enthalpy directed process.

Full text of DOI:10.1007/s10847-011-0089-1

J Incl Phenom Macrocycl Chem (2012) 74:109–116
DOI 10.1007/s10847-011-0089-1
ORIGINAL ARTICLE
Host–guest interactions between dapsone and b-cyclodextrin
(Part II): thermal analysis, spectroscopic characterization,
and solubility studies
•
M. H. Martins A. Calderini F. B. T. Pessine
•
Received: 25 August 2011 / Accepted: 15 November 2011 / Published online: 16 December 2011
Ó Springer Science+Business Media B.V. 2011
Abstract The complex of dapsone with b-cyclodextrin
was prepared by the co-precipitation/freeze–drying method.
The physical–chemical characteristics of the complex were
investigated by different methods and compared with those
of the physical mixture and of the isolated compounds. The
methods used were infrared spectroscopy, X-ray diffrac-
tometry and differential scanning calorimetry. The stability
constant was calculated from phase solubility diagram
(Higuchi–Connors) and fluorescence spectroscopy. The
stoichiometry of the complex was confirmed by Job’s plot.
Fluorescence measurements at different temperatures pro-
vided the thermodynamic parameters of the complexation.
The infrared spectrum showed the disappearance of the SO2
asymmetric stretching band of the drug at 1275 cm-1 after
complexation. The amorphization of the samples, as
revealed by the X-ray diffraction patterns, was an indirect
proof of the inclusion complex. The thermal analysis showed
that the curves of the physical mixture are combination of the
curves of both constituents (dapsone and b-cyclodextrin)
while the absence of the melting peak of the drug in the DSC
curve of the complex suggests the inclusion of the drug
molecule in the host cavity as a 1:1 complex as indicated by
Job’s plot. There was a linear increase in its solubility with
the increase of the cyclodextrin concentration and the com-
plex was classified as an AL-type. The value of the stability
constant was 3,998 L mol-1 calculated by the Higuchi–
Connors diagram and around 18,100 L mol-1 from the
fluorescence method indicating a strong interaction between
the host and the guest. Complex formation was a spontane-
ous and enthalpy directed process.
Keywords Dapsone ꢀ b-cyclodextrin ꢀ Thermal analysis ꢀ
Fluorescence ꢀ Stability constant
Abbreviations
CD
Cyclodextrin
DPS
DSC
FD
Dapsone
Differential scanning calorimetry
Freeze–drying
FT-IR Infrared spectroscopy
PM
L
Physical mixture
Lyophilized complex
X-ray diffraction
XRD
Introduction
Dapsone (DPS, Fig. 1) has been the main drug for treat-
ment of leprosy. As an anti-infective agent, it is also used
for treating malaria, pneumocystic carinii pneumonia in
AIDS patients, Kaposi’s sarcoma and dermatoses. In some
cases, therapy may be continued for 3–5 years. DPS ther-
apy may cause adverse effects such as methemoglobine-
mia, anorexia, hemolysis, anemia, dyspnea, nausea,
tachycardia, fatigue, anorexia, headache, dizziness, exan-
thematous eruption, Stevens-Johnson syndrome, toxic
epidermal necrolysis, and others [1, 2]. The encapsulation
of DPS in cyclodextrins could improve its water solubility,
bioavailability and minimize side effects. This work
describes the inclusion complex formation of this drug with
b-cyclodextrin (bCD).
M. H. Martins (&) ꢀ A. Calderini ꢀ F. B. T. Pessine (&)
Chemistry Institute, State University of Campinas,
P.O. Box 6154, Campinas-SP 13083-970, Brazil
e-mail: milene@iqm.unicamp.br
F. B. T. Pessine
e-mail: fpessine@iqm.unicamp.br
A. Calderini
e-mail: adrianacalderini@yahoo.com.br
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J Incl Phenom Macrocycl Chem (2012) 74:109–116
LabSynth Ltda. Products for Laboratories, freeze–dryer
FTS Systems, rotary evaporator RE111, water bath 461 and
vacuum pump Bu¨chi Labortechnik AG, UV/vis spectro-
photometer Hewllet Packard–HP8453, Phoenix AP22
homogeneizer, DSC 910 DuPont Instruments calorimeter,
Shimadzu-XDR-6000 X-ray diffractometer, Bomem MB
Series MB102 FTIR spectrometer, and Cary Eclipse
spectrofluorimeter fromVarian.
Fig. 1 Dapsone molecular structure
Cyclodextrins are cyclic oligomers of a-D-glucose
connected through glycosidic a-1,4 bonds. They are crys-
talline, non-hygroscopic, and torus-like macrocycles, pro-
duced by the action of certain enzymes on starch. Due to
their characteristics and large molecular cavity; they have
the ability to include guest molecules, thus altering the
physical, chemical, biological and pharmacological prop-
erties of the guest through the formation of inclusion
complexes [3, 4]. Cyclodextrins are the more extensively
studied host molecule in supramolecular chemistry,
because of their innumerous advantages as: biocompati-
bility and natural production in the enzymatic degradation
of starch, relatively inexpensive and non-toxic, allowing
applications in several industries [5]. CDs are often used to
increase the aqueous solubility, stability and bioavailability
of drugs [6]. The natural CDs are: aCD (six glucose units),
bCD (seven glucose units), and cCD (eight glucose units)
[7]. The size of the cavity of bCD is more appropriate to
interact with a great number of molecules and, therefore,
this CD was used in this study. Many forces are involved in
the complexation of drugs in the CD cavity, which include
electrostatic interactions, van der Waals forces, hydro-
phobic interactions, hydrogen bonding, release of confor-
mational strain, charge-transfer interactions and exclusion
of cavity-bound water [24].
Preparation of the inclusion complex (DPS:bCD)
The inclusion complex DPS:bCD was prepared by adap-
tation of two methods: co-precipitation and lyophilization.
bCD was dissolved in 70 mL of deionized water, followed
by addition of 20 mL of DPS in ethanolic solution at molar
ratio of 1:1 and 2:1. The mixtures were stirred at room
temperature for 24 h. After removal of ethanol in rotary
evaporator at 40 °C, the suspension was frozen in liquid
nitrogen and lyophilized for 48 h.
Preparation of PMs
The PMs were prepared using the molar ratios of 1:1 and
2:1 of bCD and DPS by simply mixing the two compounds
for 2 min.
Solubility studies
Excess amount of DPS were added to 10 mL of deionized
water containing a range of bCD concentration (from 0 to
10 mmol L-1). The tubes were shaken for 48 h at
25 0.5 °C. After, the samples were filtered through a
0.22 lm membrane and the absorbance at 291 nm was
measured using UV/vis spectrophotometry. Calibration
curve (7 to 35 lmol L-1) was used to determine the con-
centration of the dissolved DPS. All studies were carried
out in triplicate.
There are many techniques that can be used to study
interactions between hosts and cyclodextrins. In our pre-
vious work, the inclusion compound DPS:bCD was studied
by different NMR techniques, which showed a strong
interaction between host and guest. In the present work, we
characterize the inclusion complex of DPS and bCD by
thermal analysis and spectroscopy. The solid complex was
characterized by DSC, XRD and FTIR, and it was com-
pared with the isolated compounds and their physical
mixture (PM). The spectroscopic experiments were done to
obtain physical–chemical parameters such as association
constant, DG, DH, and DS.
Differential scanning calorimetry (DSC)
The DSC thermograms were recorded on a DSC 910
(DuPont Instruments). Samples were heated in hermeti-
cally sealed aluminum pans over the temperature range of
30–300 °C with a heating rate of 10 °C min-1 under an
argon purge.
Experimental
X-ray powder diffraction (XRD)
Materials and equipments
The powder X-ray diffraction patterns of the samples were
´
recorded over the interval 5–50°/2h using Cu-ja radiation
Dapsone was supplied by Ecofarma Farmacia Ltda., bCD
was a gift from ISP Technologies, Inc. and was used as
˚
at 1.5406 A, voltage of 40 kV and current of 30 mA, using
a Shimadzu-XRD-6000 X-ray diffractometer.
received, ethyl alcohol, 99.5% P.A., was purchased from
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J Incl Phenom Macrocycl Chem (2012) 74:109–116
Infrared absorption spectroscopy
111
suspension was freeze–dried to remove the remaining
water and obtain the final powder.
The infrared spectra of the samples were recorded in a
Bomem MB Series MB102 FTIR spectrometer using KBr
pellets as support.
The inclusion complexes (DPS:bCD molar ratios 1:1
and 1:2) obtained by co-precipitation/lyophilization modi-
fied method were compared with individual constituents
and the PM in both molar ratio. The results of all analysis
are discussed below.
Method of continuous variation (Job’s plot)
The stoichiometry of the inclusion complex was deter-
mined by varying the mole fraction (r = 0, 0.2, 0.33, 0.5,
0.67, 0.8, 1.0) of the compounds by mixing equimolar
solutions of DPS and bCD. The final concentration
remained constant (3 mmol L-1). After stirring the solu-
tions for 1 h, the absorption spectra in the UV/vis region
were recorded at 298 K. The difference in absorbance
(A) measured at 291 nm between solutions containing only
DPS and the DPS/bCD mixtures, multiplied by the molar
ratio (r) of DPS, was plotted as a function of the r of DPS.
The point where the derivative of the curve was zero cor-
responded to the stoichiometric ratio for the inclusion
complex. Each solution was measured in triplicate.
X-ray diffractometry
The DRX patterns of DPS, bCD, PMs and corresponding
inclusion complexes are in Fig. 2.
The formation of inclusion complexes usually results in
visible changes in the X-ray diffraction patterns of mole-
cules, such as amorphization (i.e. reducing the crystallinity
degree) of the diffraction pattern, the disappearance of
characteristic peaks of cyclodextrins and/or the appearance
of new peaks. This technique is an indirect proof of the
formation of complexes [7].
Upon analysis of the X-ray patterns in the Fig. 4, it is
observed that DPS and bCD present numerous sharp
reflections indicating a high degree of crystallinity. The
characteristic diffraction peaks of DPS and bCD crystals
were detected in the PMs M11 and M12, whereas they
considerably decreased or have disappeared in the corre-
sponding inclusion complexes (L11 and L12). The XRD
patterns of the complexes were more amorphous than the
individual components and PMs. This is indicative that the
inclusion compound was formed in the co-precipitation/
freeze–drying process, as seen by other authors [8].
Fluorescence studies
To test tubes containing solutions of DAP (5 lmol L-1) were
added increasing amounts of bCD (0, 0.03, 0.06, 0.1, 0.3, 0.6,
1, 3, 6 and 10 mmol L-1). After stirring for 1 h, the solutions
were thermostated at the working temperatures (298, 310 and
322 K) for at least 1 h. Fluorescence measurements were
performed using a Cary Eclipse (Varian) spectrofluorimeter
coupled with its temperature controller with a precision of
0.5 °C. The emission spectra were recorded with excitation
at 295 nm, range of the 315–570 nm and scan rate fixed to
1200 nm min-1. Slit widths were 5 nm for excitation and
20 nm for emission. The time of thermal stabilization of the
solution in the equipment before the reading was 5 min. Each
solution was measured in triplicate.
FT-IR spectroscopy
The infrared spectra of inclusion complexes showed
changes in intensities and frequencies of some bands in
Results and discussion
Studies on solids compounds: bCD, DPS, PMs
and inclusion compounds
DPS is a drug slightly soluble in water. It’s solubility at
298 K is 0.16 mg mL-1 [8]. Therefore, in order to obtain
the complex, it was first dissolved in ethanol to improve its
interaction with the cyclodextrin in an aqueous solution.
After evaporation of the ethanol, a white solid was
obtained due to decreased solubility of the bCD in the
aqueous solution. The elimination of ethanol favored
the direction of water molecules into the hydrophobic
cavity of cyclodextrin, promoting complex formation. The
Fig. 2 X-ray diffractograms of DPS, bCD, physical mixtures (M11
and M12) and inclusion coplexes (L11 and L12)
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J Incl Phenom Macrocycl Chem (2012) 74:109–116
relation to PMs and individual constituents (Fig. 3). For
example, in the spectra of the complexes L11 and L12 the
disappearance of the DPS band in 1275 cm-1 corre-
sponding to asymmetric stretching of the group SO2 was
observed. This is also observed in the PMs (M11 e M12).
The absence of this band may indicate the approach of the
SO2 group of the DPS molecule to the region of the edge of
the cyclodextrin, with possible formation of hydrogen
bonds with the hydroxyl groups of this edge. This is an
indication of interaction between DPS and CD in the
complex. The formation of hydrogen bonds of drug-bCD at
the edge was also proposed by Enoch et al. [9].
Fig. 4 DSC curves of DS, bCD, physical mixtures (M11 and M12)
and inclusion complexes (L11 and L12)
Differential scanning calorimetry
The DSC curve of DPS, bCD and the binary systems are in
Fig. 4. The DSC curve of DPS showed a sharp endothermic
peak at 181 °C, which is attributed to the melt temperature
of the drug. In the DSC curve of bCD, one wide endo-
thermic peak was observed in the range between
100–180 °C due to the loss of hydration water. The thermal
behavior of the inclusion complexes was distinct from that
found for the individual components. The DSC pattern of
the PMs M11 and M12 showed a presence of peaks of both
pure compounds. Finally, the absence of the melting peak
of the DPS in the DSC curves of the inclusion complexes
L11 and L21 suggests the inclusion of DPS molecules in
the bCD (absence of crystals of DPS). Also, the shift of the
bCD peak to lower temperatures, which indicates a
reduction of DH reduction and a peak broadening are all a
strong evidence of disorder in the water molecules inside
the cavity caused by the drug [10].
Solubility studies
The phase-solubility diagram corresponding to DPS:bCD
system is illustrated in Fig. 5. It was performed according
to the method reported by of Higuchi and Connors [11] in
order to determine the stability constant of the suggested
complex and evaluate its stoichiometry. Measurements
were made at 291 nm, which corresponds to the wave-
length of maximum absorption of DPS solutions.
One can observe that the aqueous solubility of DPS
increased linearly as a function of bCD concentration,
therefore water-soluble inclusion complex was obtained
using that concentration range of bCD. The phase solu-
bility diagram behavior can be classified as AL-type
according to Higuchi and Connors. The linear host–guest
correlation with slope less than 1 indicates the formation of
Fig. 5 Phase-solubility diagram of DPS in aqueous bCD solutions, at
298 K (mean SD; n = 3; adjusted parameters: R = 0.9993;
y = 0.71103 ? 0.73979x)
Fig. 3 FT-IR spectra of DPS, bCD, inclusion complex physical
mixtures (M11 and M12) and inclusion complexes (L11 and L12)
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J Incl Phenom Macrocycl Chem (2012) 74:109–116
113
a 1:1 (DPS:bCD) complex. Although the structural formula
of DPS suggests a possible 2:1 stoichiometry, possibly this
complex is not favorable due to steric hindrance or the
approximation of the two CD rings [12].
molecule of bCD for one molecule of DPS. These results
are in agreement with the stoichiometry suggested by the
phase solubility study.
The apparent association constant (K) of the drug-CD
complex, assuming a 1:1 stoichiometry, was calculated from
the linear curve of phase solubility diagram using Eq. 1.
Spectrofluorimetric studies
In organic solvents, DPS exhibits intense fluorescence
signals, but in aqueous solution DPS has a low fluorescence
quantum yield and almost does not fluoresce, which may be
due to the formation of exciplexes between the DPS
excited singlet states and water molecules. However, a
huge increase in fluorescence intensity is observed when
solutions of bCD were added to an aqueous solution of
DPS [12]. In this way, spectrofluorimetry was used in this
work to estimate some parameters as apparent association
constants, thermodynamic parameters, as well as the
stoichiometry.
slope
K ¼
ð1Þ
Soð1 ꢁ slopeÞ
where So is the apparent solubility value of DPS in the
absence of cyclodextrins and the value of the slope was
obtained from Fig. 6.
The calculated Ka value was 3,998 L mol-1 suggesting
that the bCD and DPS molecules have high affinity and the
inclusion complex is thermodynamically highly stable [13].
Usually the interaction constant between a drug and
cyclodextrins should be at least 50 L mol-1 and not larger
than 5,000 L mol-1, since within this range the complex is
sufficiently stable [14].
As observed by Ma et al. [12], a dramatic increase in the
fluorescence signal was observed when the bCD solution
was added to an aqueous solution of DPS and the signal
was intensified with increasing concentrations of bCD
(Fig. 7). This phenomenon can be explained by the
encapsulation of a part of DPS molecule in the CD cavity,
which preludes the deactivation of the excited singlet by
quenching or other nonradioactive decay process occurring
in the bulk aqueous solution.
The apparent solubility of DPS in water (in the absence
of cyclodextrin) was determined directly by the linear
coefficient (0.177 mg mL-1 or 0.713 mmol L-1). The
value of solubility is close to that obtained by Porchopin
et al. [15] 0.16 mg mL-1 in phosphate buffer pH 7.4. The
solubility showed a steep increase from 0.177 to 2.013
mg mL-1 or 8.109 mmol L-1 with bCD 10 mmol L-1, i.e.
nearly 12 times more solubility.
A blue-shift occurred in the fluorescence signal with
addition of the bCD solution, which can be explained by
partial shielding of the excitable chromophore of DPS
inside the bCD cavity [12]. Therefore, one can confirm the
formation of an inclusion compound between the drug and
bCD.
Analyzing the enhancement of fluorescence of DPS
upon bCD addition, it is possible to investigate the stoi-
chiometry of the suggested complex and to determine its
apparent association constant [18, 19]. Besides, knowing
this constant at three temperatures (298, 310 and 322 K),
Method of continuous variation (Job’s plot)
This method [16] was used to determine the stoichiometry
of the inclusion complex bCD:DPS. This method is one of
the experimental mixing techniques used in the study of
reaction stoichiometries [17]. Figure 6 corresponds to the
Job’s plot, which showed a maximum at r = 0.5, a highly
symmetric plot, indicating that the complex between the
drug and bCD has a 1:1 stoichiometric ratio, i.e. one
Fig. 6 Job plot determined by measurement in the region of UV/vis
region
Fig. 7 Effect of addition of increasing concentration of bCD on the
fluorescence emission spectra of DPS at 298 K
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J Incl Phenom Macrocycl Chem (2012) 74:109–116
one can determine thermodynamic parameters by applying
the van’t Hoff’s equation. Figure 8 shows the plot of the
maximum fluorescence emission intensity of DPS with
increasing concentrations of bCD. One can observe that the
saturation region of DPS fluorescence started at bCD
the Fig. 9 the fluorescence intensity and, therefore, the K
values decrease with increasing temperature.
Determination of thermodynamic parameters
around 2 mmol L-1
.
The thermodynamic parameters DH and DS for the DPS/
bCD complex were determined from the temperature
dependence of the constants using the van’t Hoff’s equa-
tion (Eq. 4). The values of DH and DS were calculated
from the slope and intercept of the plot ln K versus 1/T,
respectively.
Determination of the apparent association constants
by fluorescence spectroscopy
The relationship between changes in DPS fluorescence,
excited at 248 nm, and the concentration of bCD can
provide K through the Scatchard equation (Eq. 2) [13, 20].
ðꢁDHꢃÞ ðDSꢃÞ
ln K ¼
þ
ð4Þ
RT
R
F ꢁ Fo
where R is the gas constant and T is the absolute
temperature. The constant is related to the Gibbs free
energy (DG) as described in Eq. 5.
¼ ðF1 ꢁ FoÞK ꢁ ðF ꢁ FoÞK
ð2Þ
½CDꢂ
where F is the area under the fluorescence emission
spectrum observed at each point of the titration, Fo is the
area of DPS fluorescence in the absence of bCD, [CD] is
the concentration of bCD at each point of the titration, I?
is the area of fluorescence when all DPS molecules have
been complexed with bCD (i.e. a large excess of CD), and
K is the apparent association constant (L mol-1). Using
this equation, a plot of (F-Fo)/[CD] as function of (F-Fo)
can give the binding constant, which is calculated from the
intercept and the slope of the straight line. Eq. 2 can be
rearranged in the format developed by Benesi-Hildebrand
(Eq. 3) to calculate the K value [13, 21].
DGꢃ ¼ DHꢃ ꢁ TDSꢃ
ð5Þ
The thermodynamic parameters of the inclusion
compound were calculated based on K values obtained
1
1
1
¼
þ
ð3Þ
ðF ꢁ FoÞ ðF1 ꢁ FoÞK½CDꢂ ðF1 ꢁ FoÞ
Applying both equations to the data of Fig. 8, a line is
obtained in the plot of (F-F0)/[bCD] versus (F-F0) for
Scatchard equation and a line in the plot of 1/(F-F0)
versus 1/[bCD] for Benesi-Hildebrand equation, as
illustrated in Figs. 9, 10 for the data at 298 K. The K
values are summarized in Table 1. As one can be seen in
Fig. 9 Scatchard Plot for DPS at 298 K (adjusted equation: (F-F0)/
[bCD] = 1.5514.107-18143.76
K298 K = 18,144 m-1
(F-F0),
R = 0.99707)
and
Fig. 10 Benesi-Hildebrand Plot for DPS at 298 K (adjusted equation:
1/(F-F0) = 0.00117-6.46669.10-8 1/[bCD], R = 0.99707) and
K298K = 18,093 L mol-1
Fig. 8 Maximum fluorescence emission intensity of DPS with
increasing concentrations of bCD at 298, 310 and 322 K
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J Incl Phenom Macrocycl Chem (2012) 74:109–116
115
indicate formation of a complex between DPS and bCD.
The solubility of DPS was significantly increased in the
presence of bCD. This complex is thermodynamically
stable and exhibits 1:1 stoichiometry, consistent with the
molecular dimensions of DPS and those of the hydrophobic
cavity of bCD. Although there are differences in the values
of the apparent stability constant obtained by different
methodologies, all results shows a strong interaction
between the drug and cyclodextrin.
Table 1 Association constants and thermodynamic parameters
obtained through the Benesi-Hildebrand, Scatchard and van’t Hoff
equations of the inclusion compound DPS:bCD
Parameter
298 K
310 K
322 K
K (Benesi-Hildebrand) (m-1
)
18,093
18,144
-24.4
-44.7
-67.9
10,897
10,632
-23.6
–
4,692
4,715
-22.8
–
K (Scatchard) (m-1
DG (KJ mol-1 a
DHo (KJ mol-1 a
DSo (J (mol K)-1 a
)
)
)
)
–
–
a
Acknowledgments The authors gratefully acknowledge the finan-
cial support from CAPES, ISP Technologies, Inc. for supplying bCD
´
and Claudia Martelli for the assistance with the spectroscopic work.
K obtained by Benesi-Hildebrand equation
by the Benesi-Hildebrand method and they are summarized
in Table 1.
In agreement with Alvariza [22] and Jullian [23], a
negative DH° is typical of hydrophobic molecules where
the complexation process is favored by van der Waals
attractive interactions between host and guest and/or by the
presence of intermolecular hydrogen bridges between, for
example, the hydroxyls groups of the CD and the DPS
molecule. Negative values of DS° usually describe guests
that do not totally penetrate into the CD cavity or that have
their movement restricted due to strong interactions with
the rigid host structure, decreasing their degrees of
freedom. Therefore, the formation of the complex is an
enthalpically driven process. Besides, the negative value of
DG° shows a spontaneous process for host–guest inclusion
[24].
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for the Scatchard and the Benesi-Hildebrand methods, the
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