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Kinetics of dissolution of calcium hydroxyapatite
254
Journal of Crystal Growth 62 (1953) 254-264 North-Holland Publishing Company
KINETICS OF DISSOLUTION OF CALCIUM HYDROXYAPATITE VI. The effects of adsorption of methylene diphosphonate, stannous ions and partly-peptized collagen J. CHRISTOFFERSEN and M.R. CHRISTOFFERSEN Medicinsk - Kemisk Institut, Panurn Instituttet, University of Copenhagen, Blegdamst’ej 3, DK - 2200 Copenhagen N. Dentnark
S.B. CHRISTENSEN Calcified Tissue Research Laboratory, Department of Orthopaedic Surgery, Rigshospita/et. Blegdainscej 9, DK -2100 Copenhagen 0, Denmark
and G.H. NANCOLLAS Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 142/4, USA Received 27 January 1983
Kinetic effects of the rate of dissolution of calcium hydroxyapatite crystals (HAP) due to adsorption of methylene diphosphonate ions (MDP), stannous ions, partly peptized collagen and mixtures thereof are determined. Adsorption of MDP onto HAP can he described in terms of a modified Langmuir adsorption isotherm, in which each MDP molecule replaces or occupies two phosphate sites in the crystal surface. A model is presented relating adsorption constants determined from kinetic experiments to adsorption constants determined from equilibrium measurements. The model is based on the assumption that the rate-controlling process is a surface process involving a critical phenomenon such as surface nucleation. An expression for the kinetic effect on a crystal growth or dissolution surface process, due to adsorption of more than one inhibitor, is given. The concept of additivity of the effects of inhibitors is discussed.
1. Introduction
such as the available surface area of mineral, are responsible for the amount of uptake [3,4]. For
Although bone scintigraphy has become a routine method for investigating disorders of bone, the mechanisms of the adsorption processes are still being discussed. The bone-seeking agent, 99m technetium methylene diphosphonate, is pre9smTc~pertechnetatewith methylpared by mixing ene diphosphonate (MDP) and stannous chloride in a commercially available kit and is given intravenously. The stannous ion is needed to reduce pertechnetate to Tc(IV), in which form it is bound to the diphosphonate ion or to the tin—diphosphonate complex [1,2]. This bone-seeking agent labels bone surfaces, particularly areas of bone formation and bone resorption [3], and it is most likely that, besides bone perfusion, other factors,
simplicity, we use calcium hydroxyapatite (hereafter HAP) as a model for bone mineral. As collagen accounts for 90—95% of the organic bone matrix, there is considerable interest in the interaction between collagen and bone mineral [5]. Thus it has been proposed that collagen is adsorbed on the surface of mineral in mature bone [6]. For an understanding of the adsorption process of the bone-scanning agent, it is important to know how the adsorption of MDP or collagen on bone mineral is perturbed by the presence of the other. As collagen is insoluble in aqueous solution, we have investigated how adsorption of MDP on HAP is affected by the presence of partly-peptized collagen, “Polypep” (Sigma), which is slightly
0022-0248/83/0000--0000/$03.00 © 1983 North-Holland
J. Christoffersen et al.
/
Kinetics of dissolution of Ca,
soluble. Werness et al. [7] and White et al. [8] have recently studied the effect of inhibitor mixtures on urinary crystal growth, but otherwise little is known of the effect on adsorption of mixtures of inhibitors. The dissolution of HAP in the pH range 5—7.2 and in 30—90% undersaturated solution, has been shown to be controlled by a surface process [9]. The rate of diffusion of hydrated ions from the interface region to the bulk solution is rapid compared to the rate of the reaction by which ions leave the crystal surface. It has [10] been shown that hydrogen ions play an important role in the dissolution process, not only as a complexing agent for hydroxyl and phosphate ions in solution, causing the solubility to increase with decreasing pH, but also as a catalyst for the exchange of phosphate ions between the crystal surface and the solution. The dissolution of HAP can be described by a polynuclear dissolution mechanism, for which the dependence of the rate constant on pH can be explained by hydrogen ion catalysis [10]. For crystal growth or dissolution which is controlled by a surface process, the rate is expected to be strongly retarded if foreign ions or molecules are adsorbed on the crystal surface. The effects of adsorption of inhibitors on the dissolution of HAP can be described by comparing the rate of dissolution, ~L’ with inhibitor present to the rate, J0, in the absence of inhibitor. The ratio ~L/~o can only be J used to describe the effect of inhibitor if ~L and 0 are determined for the same values of the parameters, such as undersaturation, pH, extent of reaction and surface area and structure, determining the rate of dissolution of the crystals [11].The last two parameters have been shown [12] to be a function of rn/rn0, the fraction of the initial mass of crystals remaining at the time the rate is measured. In the present work, ~L/~o will Polypep be used on to 2~and aq describe thedissolution effect of MDP, Sn the rate of of HAP. If each adsorbate molecule adsorbed on the HAP surface occupies more than one adsorption site, i.e. the adsorption is “bidentate”, “tridentate”, etc., one would expect the adsorption isotherm to be complicated. Despite this, simple Langmuir adsorption isotherms are often found. Possible reasons for this are discussed in the next sections.
0(P04)6(OJ-f)2
255
2. Adsorption equilibria In the following, it is assumed that adsorbate units, e.g., MDP entering the HAP crystal surface, replace surface phosphate groups or enter vacant phosphate group sites, that the adsorbate units do not interact with each other and that activity coefficients of the surface components are constant. Adsorption of an anion into a vacant site for a lattice anion may cause lattice anions to leave the crystal surface or cause lattice cations to enter the crystal surface. These reactions are electrically coupled, otherwise the electrical potential difference between the solution and the crystal surface would become very large. In the present investigation adsorption of MDP causes a release of phosphates, but we have not been able to demonstrate any important influence of the concentration of phosphate in the solution on the adsorption equilibrium for MDP. In this paper, therefore, the reaction between HAP crystals and MDP will be described in terms of a simple adsorption equilibrium rather than an ion exchange phenomenon. At adsorption equilibrium the fluxes of adsorbate to and from the surface,j~andj~,respectively, are equal. When an adsorbate unit reacts with or substitutes only one surface phosphate group, adsorption equilibrium can be expressed by k~(l ~C k -“ X1 L —J 1 and k are rate constants for adsorpin which k tion and desorption, respectively, CL is the concentration of adsorbate in the liquid phase and x is the mole fraction of phosphate surface sites occupied by adsorbate, i.e., x n~/n~,where n~is the amount of adsorbate on the crystal surface and n~is the amount of phosphate surface sites. Eq. (1) can be rearranged to give the conventional Langmuir equations (2) and (3): L’ _i±/1—_ ~ -~-—
—
—
—
—
=
l~Lit
l/(1
—X/~1 ~X)~L,
/I~ —
x)
=
1
+
K C L
“3 L~
A plot of l/(1 x) or of x/(1 x) against CL should give a straight line with a slope equal to KL, the Langmuir equilibrium constant. If the adsorption is bidentate or tridentate, eqs. (1), (2) and (3) are no longer valid. For simplicity, —
—
256
J. C’hr:stoffersen ci al.
/
Kinetics of dissolution of Ca,
11(PO4)~/OH)~
we shall discuss only bidentate adsorption in de-
tail, but the method outlined can readily be extended to more complex adsorption. In the case of bidentate adsorption, x 2n~/n~.For small values of x, the probability any the two number sites to be 2. Withyforbeing of vacantone is particular (1 —x) site can be involved in adsorpways tion, the flux of adsorbate to the surface is given =
—
x)2yC~.
K1~ x/(1 or =
1. eq. (6) can be replaced by —
2x)C1
.
2x)CL.
2KL2x/(t
(9)
(10)
(4)
(5)
(2) and (10). If each adsorbate unit occupies h single adsorption sites, eq. (6) should be replaced by
and the flux away from the surface is j=kx.
<<
which is similar to eq. (2). Eq. (10) can also he derived, if we work in terms of x’, the degree of surface coverage. Substituting .v’ = 2n~/n~ for x in eq. (1) leads to an equation of the type of eqs.
by
J~ k~(l
For x
The rate constants k~ and k contain not only kinetic terms, but also possible geometrical factors
~
=
x/(l
—
x)”Ct~ (x small),
(11)
other than y. Adsorption equilibrium leads to j ~
=
or KL
=
k~y/k= x/(l
—
x)2CL.
(6)
For relatively large values of x, this equation is not expected to be valid, even when all activity coefficients for the surface components are constant. The reason for this is that for x 1 we may have a situation in which most of the vacant sites
occur in the form of pairs of vacant sites. In this case, eq. (4) should be replaced by
J~ k~(l
—
x)[l
+ 6(x,v)]CL.
(7)
where ~(x, y) is a small correction term representing the possibility of finding a suitable vacant site
close to a pair of vacant sites. The flux of adsorbate from the surface is still given by an expression of the type of eq. (5). Adsorption equilibrium can be described by
KL
=
k~y= —
(1
—
x (I
—
x)CL’
x x)CL[l
+
6(x, y)J (8)
with x = 2n~/n~and with ~ —s 0 for x —~ 1. For this type of adsorption, a plot of x/(l — x) against CL will give points which asymptotically approach a line through the origin with slope KL. A straight line through points for which tS * 0 will have a slope less than KL and a positive intercept with the
y-axis.
with x = hn~/n~.For high values of x. the power of bin eq. (11) will decrease and at the limit .v 1. eq. (11) should be replaced by eq. (8).
3. Effects of adsorption on growth or dissolution kinetics The kinetic effects of adsorption of inhibitors onto growing or dissolving crystals have been and are still being investigated in great detail [7,8,13-15]. However, relatively little progress has been made in quantifying the kinetic effects of adsorption since the work of Cabrera and Vermilyea [16]. In thts section we shall only discuss the effects of inhibitors, the adsorption of which can be described by various forms of Langmuir adsorption isotherms. In the following we shall assume the rate of crystal growth or dissolution to be controlled by a surface process and that exchange of substance between the crystalline phase and the solution does not take place through the area A~.covered by the inhibitor, whereas exchange of substance through the uncovered area, A — A[. is not affected by the presence of the inhibitor. As a first approximation we assume that the fraction of the area covered by inhibitor is equal to the mole fraction of adsorption sites occupied by the inhibitor. AL/A = x, and that we may express the rate ~L with inhibitor present, relative to the corre-
J. Christoffersen et a!.
/
Kinetics ofdissolution of Ca,
0(P04)6(OH),
sponding rate J0, with no inhibitor present, as
kinetic effects of inhibition is thus (12)
AL/A
KL
=
1 1 —x
(1 —x)CL
=
0
1 =
I—AL/A
—
Jo ~
AL/A)CL’
1 +Kk~flCL,
Jo/JL=l/(l—~x), ~l,
(13)
(14)
nucleus, which in turn increases as the solution composition approaches saturation. When r* is
small, ~ 1. For monodentate adsorption we obtain from eqs. (3) and (16) 1 1 1
l+KLCL
—
in which Kkfl is a Langmuir adsorption constant, which can be determined from kinetic experiments. Expressions of the type of eq. (14) have been used [11] to express the effects of inhibitor on the rate of dissolution of HAP. In the following we
shall extend the description of the effects of inhibitors on crystal growth and dissolution processes controlled by a surface nucleation mechanism, and show that an equation of the type of eq. (14) can be applied, not only in the case of monodentate adsorption, but also for bidentate adsorption. We shall also explain why the kinetic Langmuir constant studied.depends on the affinity of the reaction The effect of an inhibitor for surface nucleation may be described as prevention, or strong retardation, of the nucleation process in areas around the adsorbed inhibitor molecules or ions. Each inhibitor unit in the crystal surface is surrounded by a neighbouring ions, all having the same sign of charge. Due to interaction with the inhibitor, such ions are strongly attached to the crystal surface. These ions, including the inhibitor, occupy an area (~+ l)d2, in which d is the diameter of a mean ion. The area around an inhibitor unit in which nucleation is strongly retarded is of the order ~r(r 2, r* being the radius of the criti+0 r*) cal nucleus. Assuming the surface to have a stoichiometric composition, the mean area per adsorption site can be calculated. For HAP we obtain A/ne 3d2. The fraction of the area inhibited for nucleation is thus of the order AL
n~y(i~ + r*)2
—
A
np(A/np)
—
IT(tb +
3d2
r*)2
—
(15)
A better equation than eq. (14) to describe the
—
—
KLCL(~
—
(17)
—
For sufficiently low values of KLCL, eq. (17) can be approximated by
1 + KL4~CL 1 + KkIflCL. (18) A plot of Jo/JL against CL should give a straight line for low values of KLCL, which corresponds to low values of CL. For higher values of CL, the slope of the curve ~o/~L plotted against CL, obtamed from eq. (17) is
~o/~L
=
=
~(~o/~L) 8CL
=
[1
—
KL4 KLCL(q~
—
1)12
(19)
For KLCL approaching l/(4 1) the slope becomes infinite, which means that the crystal growth or dissolution process has been effectively blocked by inhibitors. Despite this, the amount of adsorbed inhibitor may still increase with increasing values of CL. For bidentate adsorption we obtain from eqs. (6) and (16) —
— _____ —
1
—
4x 2KLCL
=
2KLCL(l —~)—~+~(1 +4KLCL)~2
(20) which for low values of KLCL can be approximated by 1 + KL4CL 1 + KkIflCL by using the approximation,
(21)
(1
(22)
~o/~L
x~4x.
(16)
in which 4 increases with the size of the critical
Eqs. (2) and (3) can thus be expressed as _______
257
=
+ 4KLCL)V2
=
1
+ 2KLCL
—
2K~C~.
258
J. Christoffersen et a!.
/
Kinetics of dissolution of Ca,~(PO
4)~(OH),
From eq. (20) we obtain
For a mixture of two inhibitors, one of which
~(J \ 0//J L// ~8CL
has an adsorption tsotherm ofisotherm the typeofofthe eq.form (2), the other with an adsorption
2KL~{2KLCL + 1 —(1 + 4KLcL)1/2]
=
t~2 [2KLC,
(I
of eq. (9), we have
K1 =x1/(l —x)C~. K~=x
—~)—~
x{(I +4KLCL) ~212\
+~(l +4KLCL)
(30)
~~2’
from which we obtain
I
—
(23)
if
K~q
— —
(29)
~
2/(l—2x)C2,
K~C1+ K2C2
x=
l+K~C~+2K.~C2
If we assume that 4~
(1 +2KLCL—2K~C~)(l—KLCL4) —~KL4, KLCL<
(31)
.
2 (24)
Jo/fL
Eq. (23) shows that one can correlate the kinetic constant Kkfl with the corresponding equilibrium adsorption constants, K 1.
~2. we obtain
)/a( K C
=
~Elx/~(K2C,) 4. Additivity of the effects of inhibitors
=
(l+Kc+2K2c2)—(Kc~+K2c2) (1
same value of
=
I
—
x
=
I
—
x~
(25)
~ K,C, +
=
I
—
Kc,
>
1,
(32)
from which is seen that an increase in K1C~causes a larger increase in the combined inhibitory effect
__________
1
K~C1+ 2K2C-,) —2(K1C1 + K2C2)
~,
X,
~ K,C1
+
I + K5C,
For a mixture of inhibitors with adsorption isotherms of the type of eq. (2) and all having the
(26)
~K c,
than a similar increase in K2C2. For the same value of KLCL, monodentate inhibitor molecules are thus more effective than inhibitor molecules with bidentate or tridentate adsorption. .
.
.
In the case under discussion constant inhibitory effect is obtained when 1/(l 4x) is constant, i.e. —
Introducing q, we have
for constant value of
l+~K,C1(l—4) I
l+~K,C,
(27)
,
l—~x —
l+K1C1+2K2C2
1+K1C1+2K2C2—~(K1C~+K2C2) Constant inhibitory effect corresponds to a constant value of 1 +~KC f0 I 28 J~ I —~x I +~K,C1(l
=
1 ±K1C1 + 2K2C2 1 +K1C1(l —~)+K2C2(2—~)’
()
-
—
—
—
—~)‘
5.
which is constant when L,K1C, is constant. This sum can thus be used as a measure for the combined inhibitory effect,
Experimental
HAP crystals were prepared and analysed as described earlier [9]. The specific surface area (SSA)
J. Christoffersen et a!.
/
Kinetics of dissolution of Ca
2/g, was measured by a BET of the crystals, 33.7 m using a Quantasorb instrusingle point method ment. The MDP used and a preparation kit, containing a mixture of MDP and 6% SnCl 2 2H20, were supplied by the Atomic Energy Institute, Kjeller, Norway. The ssmTc..5n..MDp was to pre99mTcO~frommixture a generator a pared by adding Sn—MDP solution. The i4C..MDP specific activity 40 mCi/g was kindly donated by Dr. M.D. Francis and Dr. J.J. Benedict, of the Procter and Gamble Company. The partly-peptized collagen was low viscosity ~~Polypep®~~, Sigma Chemical Company. This protein is prepared by trypsin digestion and has a molecular weight of approximately l0~.All other chemicals were of analytical grade. Adsorption experiments were made by adding HAP crystals to an undersaturated solution 4C..MDP. The pH of thisHAP solution was containing i total calcium and phosphate con6.6 and the centrations were 6.0 X iO~ M and 3.6 X iO~M, respectively. Under these conditions, the rate of dissolution of HAP is practically zero. I ml samples of solution for radioactive counting were separated from the mixture by centrifugation, followed by filtration through a 0.22 ~emMillex®-GS filter unit. No significant time effects were detected. The remaining mixture was acidified to dissolve all crystals and 1 ml of this solution was used for radioactive counting. The mass of crystals was calculated from the phosphate concentration of the acidified solution. For kinetic experiments, crystals were added to water, the pH of which was kept constant by
10(P04)6(OH)2
259
absorption spectrometry using a Perkin Elmer 305A instrument. The total phosphate concentration was measured by the molybdenum blue method, using a Carl Zeiss PMQII spectrophotometer. 6. Analysis and results
6.1. Equilibrium measurements of the adsorption of 14C—MDP on HAP
From the specific surface area, SSA 33.7 m2/g, and the dimensions of the crystallographic unit cell, a b 9.42 A and c 6.88 A, the specific amount of surface phosphate sites, n~ 5, can be calculated to be 1.73 x l0~mol/g. In this calculation each unit cell in the surface is assumed to have tion. two phosphate groups exposed to the soluThe results of the adsorption of 14C—MDP on HAP are given in table 1 and in figs. 1, 2 and 3. In table 1, column 1 gives the total concentration of MDP, n L/ V; column 2 gives the solid to solution ratio, rncr/V; column 3 gives the mole fraction of =
=
=
=
.0
0.5
0 4
~-~r
D 1
means of a pH-stat, which controlled the addition of dilute nitric acid to the system. The instruments used were PHM64, ABU 13, TTT6O and recorders from Radiometer, Denmark. The rate of dissolution was determined from the rate of acid consumption. Inhibitors were added to the water prior to the addition of crystals. The system was kept free of carbon dioxide by passing nitrogen through the aqueous phase. The mass of crystals used in an experiment was determined by acidifying the final reaction mixture, causing complete dissolution of the HAP crystals. The total concentration of calcium in this solution was determined by atomic
0.5 0.3
0.2 .
01
00
_____________________
Oo
6
7CL/M Fig. 1. Adsorption IO of ‘4C—MDP on HAP. Langmuir adsorption isotherm assuming monodentate adsorption, eq. (2). The values of D 1 and CL are given in table I.
/
260
J. (‘hnstoffer.sen ci al.
20
-
/
Kinetics of dissolution of Ca,
0 8
4
/// D2 10-
/
/
0
08
03
/~
x
1/
X
,1/
2
•~‘
//
07
/
0
1/P044(OH)
0.6
7
.: 7CL/M 1.
2
6
00
0
0
0.4
~L__~.I
2 JO7CL/M 4
6
00
JO Fig. 2. Adsorption of ‘4C—MDP on HAP. Adsorption isotherm for bidentate adsorption. eq. (6). The values of D 2 and Ci are given in table I. The slope of the line is KL = 3.6>< l0~l/mol.
Fig. 3. Adsorption of ‘4C—MDP on HAP. Adsorption isotherm for bidentate adsorption, eq. (8). when the degree of coverage. .v, is large. The values of D~are given in table 1. The slope of the solid line is Ki
0.9 ~ l0~l/mol.
=
MDP adsorbed. From these experimental data, the
to bidentate adsorption. D1 in column 8 is the
equilibrium concentration of MDP in solution, CL, and the amount of MDP adsorbed per unit mass, n~/mcr, can be calculated~ these data are
mole fraction term in eq. (8), corresponding to bidentate adsorption at high values of x, x 2n~/n11. Fig. 1, a plot of D~against ~L’ shows
given in columns 4 and 5. Assuming monodentate adsorption, the mole fraction term in eq. (2) is given in column 6, i.e., D1 = x/(1 — x), with x = nj~/np.D2, given in column 7, is the mole fraction term in eq. (6), with x = 2n~/n~,i.e.. corresponds
that the adsorption of MDP to HAP cannot be described by monodentate Langmuir adsorption. In fig. 2, a plot of D2 against C~.a straight line is obtained for 0 < ~L < ~ X IO~ M. In this concentration interval, adsorption of MDP onto HAP
Table I 4C_MDP on HAP crystals, n~/m~ The adsorption of ‘ 1 = l.73x 10
106 n
~/
V
m~/V
(mol/l)
(mg/I)
0.801 1.076 1.324 1.444 1.618 1.741
18.6 17.7 16.8 17.7 18.5 18.4
=
mol/g
nt/ni
l0~CL (mol/l)
l0~n~/Fn~, (mol/g)
D1
D2
D~
0.943 0.882 0.802 0.785 0.768 0.738
0.46 1.27 2.62 3.10 3.75 4.56
4.06 5.36 6.32 6.40 6.72 6.98
0.31 0.45 0.58 0.59
1.67 4.28 10.1 10.9 15.6 21.7
((.88 1.63 2.71 2.84 3.48 4.18
0.64 0.68
The equilibrium distribution of inhibitor given in columns 4 and 5 is calculated from the experimental data in columns I —3. The distribution between crystal surface and solution will be independent of the amount of inhibitor. nL. the mass of crystals, mrr. and the volume of the system, V. D~.D2 and D1 are the mole fraction 2, x = functions 2n~/n in the Langmuir adsorption isotherms. eqs. (2). (6) and (8), respectively. D1 =x/(I— x), x
=
n~/np; D2 = x/(l— x)
6 D~ x/(l— x).x
=
2n~/n1,.
J. Christoffersen et a!.
/
Kinetics ofdissolution of Ca
can be described as bidentate adsorption, with KL, see eq. (6), equal to the slope of the line, 3.6 x I0~ 1/mo!. For higher values of CL, the slope of the curve in fig. 2 increases, corresponding to the expected break-down of the validity of eq. (6), which for high values of coverage, should be replaced by an equation of the type of eq. (8). In fig. 3 the mole fraction term in eq. (8), D3 x/(1 x), is plotted against CL. For high values of x, the points approach a line through the origin with slope KL 0.9 x l0~l/mol, about one quarter of the value of KL obtained in fig. 2 using eq. (6). This corresponds to a value of the geometrical factor y 4 in eq. (6). This appears to be a reasonable value to represent the number of nearest —
=
=
6.35 iü~
261
10(P04)6(OH)2
adsorption sites to any particular adsorption site in the surface, considering that we have not taken the symmetry of the crystals into account, except when calculating n~. 6.2. The effect of MDP on HAP dissolution rate
Kinetic effects of adsorption of MDP onto dissolving HAP crystals are shown in fig. 4. In this figure, JL/Jo and Jo/fL are plotted against CL. CL was calculated from the total concentration of MDP, assuming equilibrium with respect to the type of adsorption expressed by eq. (6) with KL 3.6 X l0~1/mo!. From the lower plot in fig. 4 is seen that ~L/~o for constant values of C/CS decreases with CL and that ~L/~o for constant value of CL decreases with increasing values of C/CS. From the upper plot is seen that the kinetic effects of adsorption of MDP onto HAP crystals can be empirically expressed by an equation of the type ~o/~L 1 + Kk~flCL, (34) =
=
,
cf. eqs. (14), (18) and (21). In fig. 5 the points from 4.4
the upper plot in fig. 4 are plotted again; the
curves drawn are theoretical values of
J0/k
calculated from eq. (20) with KL 3.6 X l0~1/mol and ~ equal to the values indicat in the plot. =
,
8CL/M
3
0
1O Fig. 4. Effect of MDP on the rate of dissolution of HAP at pH 7.15. ~L and] 0 are the rates with and without inhibitor present, respectively. CL was calculated from the total amount of inhibi tor, the mass of crystals and the volume of the suspension assuming bidentate adsorption, eq. (6), with KL = 3.6x iO~ 1/mol. (•) C/Cs = 0.25; (X) C/C1 = 0.35; (0) C/C1 = 0.45. The values of Kkfl, the slopes of the plots of ~o/~L against CL, are given in the figure.
0
-___________________________________ 8 10 15 10 CL/M Fig. 5. The points in this figure are the same as the upper plot in fig. 4. The curves are drawn through values calculated from eq. (20) with the values of I~ given in the figure.
262
J. Christoffersen et a!. / Kinetics of dissolution of Ca,
11(PO4),,(OH)~
From this plot is seen that the theoretical expression eq. (20) may explain the inhibitory effect of MDP for C/c,, 0.25 8)< and l0~ C1 < M. l0~ M C/C.,= and for For C/C,,=0.35 0.45, eq. (20) and does CL< not give a good agreement with the experimental data. This may be a result of an uneven distribution of the inhibitors on the surface,
I
=
0
05
.1 a
as
i
IS
20
28
‘ItO
in
6.3. The effects of stannous chloride, Polypep, and mixtures of these with MDP
Fig. 7. Effect of a partly-peptized collagen. ‘Polypep” (Sigma).
M
= 10 kg/mol. on the rate of dissolution of HAP: mcr 7 mg, V = 0.9 I. C/C, 0.25. Circles: pH = 6.76. Squares: pH =
, ,
The effect of the presence of stannous chloride on the rate of dissolution of HAP crystals is shown in fig. 6 in which JL /J U is plotted against the total amount of stannous chloride per unit mass of HAP crystals. In all experiments represented in this plot approximately the same mass of crystals, 6 ±1 mg and volume of solution, 0.93 ±0.03 1, was used and all results are given for C/p,, 0.25. The inserted broken line gives the effect of MDP under similar conditions. As the actual concentration of stannous chloride in the aqueous solution was not determined, the data given in fig. 6 cannot be used in a plot of JO/JL against CL. However, from fig. 6 can be seen that stannous chloride is an even stronger inhibitor for HAP dissolution than MDP and that the effect of stannous chloride is independent of small changes in pH around pH 7. =
=
8
105n/mcr
moilg Fig. 6. Effect of stannous chloride on the rate of dissolution of HAP, nL is the total amount of stannous chloride added; us,,, = 6 mg. V= 0.9 I. C/C,, = 0.25: (0) pH = 6.76; (0) pH = 7.16; (X) n~~y. 11= nMDP, (— — —) effect of MDP on rate of dissolution of HAP, taken from fig. 3 of ref. [Ill. Stannous chloride is more effective than MDP in retarding the rate of dissolution of HAP.
7.16.
Closed symbols: “Polypep” dialysed. Open symbols:
‘Polypep” non-dialysed. Low concentration, I mg/I. of
Poly-
pep” give reduces thefurther rate of dissolution by 60%: tions little reduction in the rate. higher concentra-
The effect of the presence of the partly-peptized collagen, Polypep, is shown in fig. 7, in which ~L/~o is plotted against ML/V, the total mass concentration of Polypep in the system. All data in the plot are for C/C,, 0.25, mass of crystals 7 ±2 mg and volume of system 0.92 1. The plot shows that Polypep has quite a large inhibitory effect for small values of mL/V, but the increase in inhibition with concentration of Polypep is very small for a concentration larger than 1 mg/I. The effect of Polypep is not affected by small changes in pH around pH 7. The inhibitory effect of a 6% mixture of stannous chloride with MDP, as delivered in the hone-scanning agent from Kjeller. is shown in fig. =
=
2
JL/JN~N~~
——
Fig. 8. Effect of MDP—Sn and MDP—”Polypep” mixtures on the rate of dissolution of HAP; pH = 6.76, C/C,, = 0.25. The line drawn is the best line through the results for MDP alone, i.e. the same as the broken line in fig. 6. (0) MDP+6% SnCI,: (X)nMDP = ~s~c2~ (~) mixtures of”Polypep” and MDP, see table 3; nL = ~MDp for all points in this plot. The retarding effects of the mixtures, MDP—SnCI2 and MDP—” Polypep” are smaller than expected from the sum of their individual effects.
J. Christoffersen et a!.
/
Kinetics of dissolution of Ca
Table 2 4CMDP (L = I) and Polypep (L The adsorption of mixtures of ‘ 106 ni/V m~~/V rn,/V iO~C~
=
2) on HAP crystals I0~n’~/m,,r
(mol/I)
(mg/I)
(mg/l)
(mol/l)
(mol/g)
1.1 1.1
17 17
25 25
1.3 1.8
5.2 5.0
8, in which JL/JO is plotted against the total amount of MDP present per unit mass of crystals for C/CS 0.25, mcr 5.5 ±0.5 mg, pH 6.8 and V 0.94 ±0.02 1. The curve drawn is for MDP alone as in fig. 6. The presence of the small amount of stannous chloride appears to have no significant effect. The point marked x gives the effect of a mixture of stannous chloride with an equal amount of MDP. The combined effect of these two inhibitors appears to be less than the effect expected for twice the amount of one of them. We have also determined the amount of ‘4C—MDP on the surface under these conditions, and found 60% of the amount expected in the absence of stannous chloride. It appears that mixtures of MDP and stannous chloride have “negative synergistic” effect on each other. Complex formation between MDP and stannous ions in solution may be partly responsible for this. The points in fig. 8 marked ~ correspond to the presence of mixtures of MDP and Polypep, see tables 2 and 3. From table 2 is seen that if Polypep is added to HAP crystals in equilibrium with MDP, a nearly normal amount of MDP is found on the surface. Polypep does apparently not displace MDP already on the surface to any great extent. However, that there is also some interaction of Polypep with the surface can be seen from table 3, which shows that the inhibitory effect of the mixture on the dissolution rate is greater than the effect of MDP alone. When Polypep is added =
=
‘—
=
Table 3 Effect of mixtures of MDP (L
=
I) and Polypep (L
=
263
10(P04)/OH),
Added first 3.8 3.2
2
to the crystals prior to MDP, less MDP than normal is found on the surface (table 2) and the combined inhibitory effect is greater than in the previous example (table 3 and fig. 8). As the equilibrium adsorption isotherm for Polypep adsorption onto HAP has not been studied yet, additivity of the effects of MDP and Polypep will not be discussed in any detail, but it is not surprising that the sequence of addition of small and large inhibitor molecules is important. If the surface contains adsorbed small inhibitor units, the statistical possibility for finding adsorption sites for large molecules is obviously reduced, whereas adsorption of large molecules prior to small ones should have a much smaller effect on the adsorption of the small molecules as long as the surface is not covered completely by the large molecules, which is also most unlikely to occur.
7. Conclusion A Langmuir adsorption isotherm with an inhibitor unit occupying two adsorption sites can explain the adsorption of MDP on HAP crystals. The adsorption can be represented by KL x/(l —x)2CL with KL 3.6x l0~1/mol, when up to 75% of the available sites are occupied, x 0.75. At higher values of x, the equation KL x/( 1 x)CL with KL 0.9 x l0~1/mol fits the results. The inhibitory effect of MDP on the rate of dis=
=
=
=
=
2) on the rate of dissolution of HAP crystals
106 ni/V
mcr/V
m J2/J0
J/J0
(mg/I)
2/V (mg/I)
J1/J0
(mol/I) 0.41 0.41
7.3 6.8
27 27
0.26 0.23
0.20 0.20
0.18 0.09
Added first 2
—
J. Christoffersen ci a!.
264
/
Kinetics ofdissolution of Ca,
solution of HAP can be fitted by the equation Jo/fL 1 + KkflCL, when the undersaturation, C/C5 is in the range 0.25—0.45, with the efficiency of the inhibitor increasing as the composition of the solution approaches equilibrium. This can be explained by the increase in the size of the critical nucleus as the solution composition approaches equilibrium, the dissolution process being controlled by a polynuclear mechanism. Stannous chloride has been shown qualitatively to be an even more effective inhibitor for the dissolution of HAP than MDP. A partly-peptized collagen also inhibits the dissolution of HAP at very low concentration. Studies of the adsorption of mixtures of MDP and partly-peptized collagen show that the adsorption of MDP is reduced if the large molecules (collagen) are adsorbed first, but nearly normal, if the MDP is adsorbed first. The adsorption of collagen is affected in both of these cases, but more when the MDP is adsorbed first. In regions with high bone turnover the apatite crystals may not be covered by collagen to the same degree as crystals in areas with low bone turnover. The results indicate that bone-scanning agents will then preferentially adsorb in regions of high bone turnover with the larger free surface area. =
11(PO4),,(OH)~
given to J.C. and G.H.N. by the Danish Natural Science Research Council (81-3393) and by the Danish Medical Research Council (12-3339), to J.C. by Novo’s Foundation, and to G.H.N. and J.C. by NATO (RG 143.80). We are very grateful to Nobuko Christiansen, Iben Junghans and Lene Høyer for laboratory assistance.
References [I] G. Subramanian, R.J. Blair, E.A. Kallfelz, F.D. Thomas and J.G. McAfee. J. NucI. Med. 14 (1973) 638. [2] PH. Cox, Brii. J. Radiol. 47 (1974) 845. [3] SB. Christensen and OW. Kf’ogsgaard. J. NucI. Med. 22 (1981) 237. [4] M.D. Francis, A.J. Tofe, J.J. Benedict and J.A. Bevan, in: Radiopharmaceuticals
II, Proc. 2nd Intern. Symp. on
Radiopharmaceuticals, Seattle, WA, 1979 Nuclear Medicine, New York, 1979) p. 603.
(Society of
[5] J.L. Matthews, J.J. Reynolds, W.G. Roberison and K. Simkiss, in: Biological Mineralization and Demineralizalion. Rept. Dahlem Workshop on Biological Mineralizalion and Demineralization, Berlin, 1981 (Springer, Berlin, 1982) P. 327. [6] J.M. Holmes, R.A. Beebe, AS. Posner and R.A. Harper. Proc. Soc. Exptl. Biol. Med. 133 (1970) 1250. [7] P.Cj Werness, J H Bergert and K.E Lee. Clin Sci 61 (1981) 487. [81 D.J. White, J. (‘hristoffersen, T. Herman. AC. Lanzalaco
Acknowledgements
and G.H. Nancollas, J. Urol., in press. [9] J. Christoffersen, MR. Christoffersen and N. Kjiergaard, i. Crystal Growth 43 (1978) 501. [10] J. Christoffersen and MR. Christoffersen. J. (‘rystal Growth 57 (1982) 21.
J.C. and M.R.C. acknowledge the following grants for equipment and laboratory assistance: the Danish Natural Science Research Council (511-1042, 511-1590, 511-8181), the Danish Medical Research Council(12-3617, 512-15458, 12-0323. 12-2215), Novo’s Foundation, and the Carlsberg Foundation (1981/82, 35/IV). S.B.C. acknowledges a research grant from the Danish Medical Research Council (12-1829). G.H.N. acknowledges grants from the National Institutes of Health (No. 03223, No. RO1AM19O48). Travel support has been
[ill
J. Christoffersen
and
MR.
Christoffersen.
J. Crystal
Growth 53(1981) 42. [12] Christoffersen, J. Crystal Growth (1980) [13] J. MC. van der Leeden. J. Reedijk and49 G.M. van29. Rosmalen. unpublished. [14] E.C. Moreno, M. Kresak and DI. Hay. J. Biol. Chem. 257 (1982) 2981. [15] E.C. Moreno, K. Varughese and Dl. Hay. Calcified Tissue Res. 28 (1979) 7. [16] N. Cabrera and D.A. Vermilyea, in: Growth and Perfection of Crystals, Eds. RH. Doremus, B.W. Roberts and D. Turnbull (Wiley. New York. and Chapman and Hall, London, 1958) p. 393.
Journal of Crystal Growth 62 (1953) 254-264 North-Holland Publishing Company
KINETICS OF DISSOLUTION OF CALCIUM HYDROXYAPATITE VI. The effects of adsorption of methylene diphosphonate, stannous ions and partly-peptized collagen J. CHRISTOFFERSEN and M.R. CHRISTOFFERSEN Medicinsk - Kemisk Institut, Panurn Instituttet, University of Copenhagen, Blegdamst’ej 3, DK - 2200 Copenhagen N. Dentnark
S.B. CHRISTENSEN Calcified Tissue Research Laboratory, Department of Orthopaedic Surgery, Rigshospita/et. Blegdainscej 9, DK -2100 Copenhagen 0, Denmark
and G.H. NANCOLLAS Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 142/4, USA Received 27 January 1983
Kinetic effects of the rate of dissolution of calcium hydroxyapatite crystals (HAP) due to adsorption of methylene diphosphonate ions (MDP), stannous ions, partly peptized collagen and mixtures thereof are determined. Adsorption of MDP onto HAP can he described in terms of a modified Langmuir adsorption isotherm, in which each MDP molecule replaces or occupies two phosphate sites in the crystal surface. A model is presented relating adsorption constants determined from kinetic experiments to adsorption constants determined from equilibrium measurements. The model is based on the assumption that the rate-controlling process is a surface process involving a critical phenomenon such as surface nucleation. An expression for the kinetic effect on a crystal growth or dissolution surface process, due to adsorption of more than one inhibitor, is given. The concept of additivity of the effects of inhibitors is discussed.
1. Introduction
such as the available surface area of mineral, are responsible for the amount of uptake [3,4]. For
Although bone scintigraphy has become a routine method for investigating disorders of bone, the mechanisms of the adsorption processes are still being discussed. The bone-seeking agent, 99m technetium methylene diphosphonate, is pre9smTc~pertechnetatewith methylpared by mixing ene diphosphonate (MDP) and stannous chloride in a commercially available kit and is given intravenously. The stannous ion is needed to reduce pertechnetate to Tc(IV), in which form it is bound to the diphosphonate ion or to the tin—diphosphonate complex [1,2]. This bone-seeking agent labels bone surfaces, particularly areas of bone formation and bone resorption [3], and it is most likely that, besides bone perfusion, other factors,
simplicity, we use calcium hydroxyapatite (hereafter HAP) as a model for bone mineral. As collagen accounts for 90—95% of the organic bone matrix, there is considerable interest in the interaction between collagen and bone mineral [5]. Thus it has been proposed that collagen is adsorbed on the surface of mineral in mature bone [6]. For an understanding of the adsorption process of the bone-scanning agent, it is important to know how the adsorption of MDP or collagen on bone mineral is perturbed by the presence of the other. As collagen is insoluble in aqueous solution, we have investigated how adsorption of MDP on HAP is affected by the presence of partly-peptized collagen, “Polypep” (Sigma), which is slightly
0022-0248/83/0000--0000/$03.00 © 1983 North-Holland
J. Christoffersen et al.
/
Kinetics of dissolution of Ca,
soluble. Werness et al. [7] and White et al. [8] have recently studied the effect of inhibitor mixtures on urinary crystal growth, but otherwise little is known of the effect on adsorption of mixtures of inhibitors. The dissolution of HAP in the pH range 5—7.2 and in 30—90% undersaturated solution, has been shown to be controlled by a surface process [9]. The rate of diffusion of hydrated ions from the interface region to the bulk solution is rapid compared to the rate of the reaction by which ions leave the crystal surface. It has [10] been shown that hydrogen ions play an important role in the dissolution process, not only as a complexing agent for hydroxyl and phosphate ions in solution, causing the solubility to increase with decreasing pH, but also as a catalyst for the exchange of phosphate ions between the crystal surface and the solution. The dissolution of HAP can be described by a polynuclear dissolution mechanism, for which the dependence of the rate constant on pH can be explained by hydrogen ion catalysis [10]. For crystal growth or dissolution which is controlled by a surface process, the rate is expected to be strongly retarded if foreign ions or molecules are adsorbed on the crystal surface. The effects of adsorption of inhibitors on the dissolution of HAP can be described by comparing the rate of dissolution, ~L’ with inhibitor present to the rate, J0, in the absence of inhibitor. The ratio ~L/~o can only be J used to describe the effect of inhibitor if ~L and 0 are determined for the same values of the parameters, such as undersaturation, pH, extent of reaction and surface area and structure, determining the rate of dissolution of the crystals [11].The last two parameters have been shown [12] to be a function of rn/rn0, the fraction of the initial mass of crystals remaining at the time the rate is measured. In the present work, ~L/~o will Polypep be used on to 2~and aq describe thedissolution effect of MDP, Sn the rate of of HAP. If each adsorbate molecule adsorbed on the HAP surface occupies more than one adsorption site, i.e. the adsorption is “bidentate”, “tridentate”, etc., one would expect the adsorption isotherm to be complicated. Despite this, simple Langmuir adsorption isotherms are often found. Possible reasons for this are discussed in the next sections.
0(P04)6(OJ-f)2
255
2. Adsorption equilibria In the following, it is assumed that adsorbate units, e.g., MDP entering the HAP crystal surface, replace surface phosphate groups or enter vacant phosphate group sites, that the adsorbate units do not interact with each other and that activity coefficients of the surface components are constant. Adsorption of an anion into a vacant site for a lattice anion may cause lattice anions to leave the crystal surface or cause lattice cations to enter the crystal surface. These reactions are electrically coupled, otherwise the electrical potential difference between the solution and the crystal surface would become very large. In the present investigation adsorption of MDP causes a release of phosphates, but we have not been able to demonstrate any important influence of the concentration of phosphate in the solution on the adsorption equilibrium for MDP. In this paper, therefore, the reaction between HAP crystals and MDP will be described in terms of a simple adsorption equilibrium rather than an ion exchange phenomenon. At adsorption equilibrium the fluxes of adsorbate to and from the surface,j~andj~,respectively, are equal. When an adsorbate unit reacts with or substitutes only one surface phosphate group, adsorption equilibrium can be expressed by k~(l ~C k -“ X1 L —J 1 and k are rate constants for adsorpin which k tion and desorption, respectively, CL is the concentration of adsorbate in the liquid phase and x is the mole fraction of phosphate surface sites occupied by adsorbate, i.e., x n~/n~,where n~is the amount of adsorbate on the crystal surface and n~is the amount of phosphate surface sites. Eq. (1) can be rearranged to give the conventional Langmuir equations (2) and (3): L’ _i±/1—_ ~ -~-—
—
—
—
—
=
l~Lit
l/(1
—X/~1 ~X)~L,
/I~ —
x)
=
1
+
K C L
“3 L~
A plot of l/(1 x) or of x/(1 x) against CL should give a straight line with a slope equal to KL, the Langmuir equilibrium constant. If the adsorption is bidentate or tridentate, eqs. (1), (2) and (3) are no longer valid. For simplicity, —
—
256
J. C’hr:stoffersen ci al.
/
Kinetics of dissolution of Ca,
11(PO4)~/OH)~
we shall discuss only bidentate adsorption in de-
tail, but the method outlined can readily be extended to more complex adsorption. In the case of bidentate adsorption, x 2n~/n~.For small values of x, the probability any the two number sites to be 2. Withyforbeing of vacantone is particular (1 —x) site can be involved in adsorpways tion, the flux of adsorbate to the surface is given =
—
x)2yC~.
K1~ x/(1 or =
1. eq. (6) can be replaced by —
2x)C1
.
2x)CL.
2KL2x/(t
(9)
(10)
(4)
(5)
(2) and (10). If each adsorbate unit occupies h single adsorption sites, eq. (6) should be replaced by
and the flux away from the surface is j=kx.
<<
which is similar to eq. (2). Eq. (10) can also he derived, if we work in terms of x’, the degree of surface coverage. Substituting .v’ = 2n~/n~ for x in eq. (1) leads to an equation of the type of eqs.
by
J~ k~(l
For x
The rate constants k~ and k contain not only kinetic terms, but also possible geometrical factors
~
=
x/(l
—
x)”Ct~ (x small),
(11)
other than y. Adsorption equilibrium leads to j ~
=
or KL
=
k~y/k= x/(l
—
x)2CL.
(6)
For relatively large values of x, this equation is not expected to be valid, even when all activity coefficients for the surface components are constant. The reason for this is that for x 1 we may have a situation in which most of the vacant sites
occur in the form of pairs of vacant sites. In this case, eq. (4) should be replaced by
J~ k~(l
—
x)[l
+ 6(x,v)]CL.
(7)
where ~(x, y) is a small correction term representing the possibility of finding a suitable vacant site
close to a pair of vacant sites. The flux of adsorbate from the surface is still given by an expression of the type of eq. (5). Adsorption equilibrium can be described by
KL
=
k~y= —
(1
—
x (I
—
x)CL’
x x)CL[l
+
6(x, y)J (8)
with x = 2n~/n~and with ~ —s 0 for x —~ 1. For this type of adsorption, a plot of x/(l — x) against CL will give points which asymptotically approach a line through the origin with slope KL. A straight line through points for which tS * 0 will have a slope less than KL and a positive intercept with the
y-axis.
with x = hn~/n~.For high values of x. the power of bin eq. (11) will decrease and at the limit .v 1. eq. (11) should be replaced by eq. (8).
3. Effects of adsorption on growth or dissolution kinetics The kinetic effects of adsorption of inhibitors onto growing or dissolving crystals have been and are still being investigated in great detail [7,8,13-15]. However, relatively little progress has been made in quantifying the kinetic effects of adsorption since the work of Cabrera and Vermilyea [16]. In thts section we shall only discuss the effects of inhibitors, the adsorption of which can be described by various forms of Langmuir adsorption isotherms. In the following we shall assume the rate of crystal growth or dissolution to be controlled by a surface process and that exchange of substance between the crystalline phase and the solution does not take place through the area A~.covered by the inhibitor, whereas exchange of substance through the uncovered area, A — A[. is not affected by the presence of the inhibitor. As a first approximation we assume that the fraction of the area covered by inhibitor is equal to the mole fraction of adsorption sites occupied by the inhibitor. AL/A = x, and that we may express the rate ~L with inhibitor present, relative to the corre-
J. Christoffersen et a!.
/
Kinetics ofdissolution of Ca,
0(P04)6(OH),
sponding rate J0, with no inhibitor present, as
kinetic effects of inhibition is thus (12)
AL/A
KL
=
1 1 —x
(1 —x)CL
=
0
1 =
I—AL/A
—
Jo ~
AL/A)CL’
1 +Kk~flCL,
Jo/JL=l/(l—~x), ~l,
(13)
(14)
nucleus, which in turn increases as the solution composition approaches saturation. When r* is
small, ~ 1. For monodentate adsorption we obtain from eqs. (3) and (16) 1 1 1
l+KLCL
—
in which Kkfl is a Langmuir adsorption constant, which can be determined from kinetic experiments. Expressions of the type of eq. (14) have been used [11] to express the effects of inhibitor on the rate of dissolution of HAP. In the following we
shall extend the description of the effects of inhibitors on crystal growth and dissolution processes controlled by a surface nucleation mechanism, and show that an equation of the type of eq. (14) can be applied, not only in the case of monodentate adsorption, but also for bidentate adsorption. We shall also explain why the kinetic Langmuir constant studied.depends on the affinity of the reaction The effect of an inhibitor for surface nucleation may be described as prevention, or strong retardation, of the nucleation process in areas around the adsorbed inhibitor molecules or ions. Each inhibitor unit in the crystal surface is surrounded by a neighbouring ions, all having the same sign of charge. Due to interaction with the inhibitor, such ions are strongly attached to the crystal surface. These ions, including the inhibitor, occupy an area (~+ l)d2, in which d is the diameter of a mean ion. The area around an inhibitor unit in which nucleation is strongly retarded is of the order ~r(r 2, r* being the radius of the criti+0 r*) cal nucleus. Assuming the surface to have a stoichiometric composition, the mean area per adsorption site can be calculated. For HAP we obtain A/ne 3d2. The fraction of the area inhibited for nucleation is thus of the order AL
n~y(i~ + r*)2
—
A
np(A/np)
—
IT(tb +
3d2
r*)2
—
(15)
A better equation than eq. (14) to describe the
—
—
KLCL(~
—
(17)
—
For sufficiently low values of KLCL, eq. (17) can be approximated by
1 + KL4~CL 1 + KkIflCL. (18) A plot of Jo/JL against CL should give a straight line for low values of KLCL, which corresponds to low values of CL. For higher values of CL, the slope of the curve ~o/~L plotted against CL, obtamed from eq. (17) is
~o/~L
=
=
~(~o/~L) 8CL
=
[1
—
KL4 KLCL(q~
—
1)12
(19)
For KLCL approaching l/(4 1) the slope becomes infinite, which means that the crystal growth or dissolution process has been effectively blocked by inhibitors. Despite this, the amount of adsorbed inhibitor may still increase with increasing values of CL. For bidentate adsorption we obtain from eqs. (6) and (16) —
— _____ —
1
—
4x 2KLCL
=
2KLCL(l —~)—~+~(1 +4KLCL)~2
(20) which for low values of KLCL can be approximated by 1 + KL4CL 1 + KkIflCL by using the approximation,
(21)
(1
(22)
~o/~L
x~4x.
(16)
in which 4 increases with the size of the critical
Eqs. (2) and (3) can thus be expressed as _______
257
=
+ 4KLCL)V2
=
1
+ 2KLCL
—
2K~C~.
258
J. Christoffersen et a!.
/
Kinetics of dissolution of Ca,~(PO
4)~(OH),
From eq. (20) we obtain
For a mixture of two inhibitors, one of which
~(J \ 0//J L// ~8CL
has an adsorption tsotherm ofisotherm the typeofofthe eq.form (2), the other with an adsorption
2KL~{2KLCL + 1 —(1 + 4KLcL)1/2]
=
t~2 [2KLC,
(I
of eq. (9), we have
K1 =x1/(l —x)C~. K~=x
—~)—~
x{(I +4KLCL) ~212\
+~(l +4KLCL)
(30)
~~2’
from which we obtain
I
—
(23)
if
K~q
— —
(29)
~
2/(l—2x)C2,
K~C1+ K2C2
x=
l+K~C~+2K.~C2
If we assume that 4~
(1 +2KLCL—2K~C~)(l—KLCL4) —~KL4, KLCL<
(31)
.
2 (24)
Jo/fL
Eq. (23) shows that one can correlate the kinetic constant Kkfl with the corresponding equilibrium adsorption constants, K 1.
~2. we obtain
)/a( K C
=
~Elx/~(K2C,) 4. Additivity of the effects of inhibitors
=
(l+Kc+2K2c2)—(Kc~+K2c2) (1
same value of
=
I
—
x
=
I
—
x~
(25)
~ K,C, +
=
I
—
Kc,
>
1,
(32)
from which is seen that an increase in K1C~causes a larger increase in the combined inhibitory effect
__________
1
K~C1+ 2K2C-,) —2(K1C1 + K2C2)
~,
X,
~ K,C1
+
I + K5C,
For a mixture of inhibitors with adsorption isotherms of the type of eq. (2) and all having the
(26)
~K c,
than a similar increase in K2C2. For the same value of KLCL, monodentate inhibitor molecules are thus more effective than inhibitor molecules with bidentate or tridentate adsorption. .
.
.
In the case under discussion constant inhibitory effect is obtained when 1/(l 4x) is constant, i.e. —
Introducing q, we have
for constant value of
l+~K,C1(l—4) I
l+~K,C,
(27)
,
l—~x —
l+K1C1+2K2C2
1+K1C1+2K2C2—~(K1C~+K2C2) Constant inhibitory effect corresponds to a constant value of 1 +~KC f0 I 28 J~ I —~x I +~K,C1(l
=
1 ±K1C1 + 2K2C2 1 +K1C1(l —~)+K2C2(2—~)’
()
-
—
—
—
—~)‘
5.
which is constant when L,K1C, is constant. This sum can thus be used as a measure for the combined inhibitory effect,
Experimental
HAP crystals were prepared and analysed as described earlier [9]. The specific surface area (SSA)
J. Christoffersen et a!.
/
Kinetics of dissolution of Ca
2/g, was measured by a BET of the crystals, 33.7 m using a Quantasorb instrusingle point method ment. The MDP used and a preparation kit, containing a mixture of MDP and 6% SnCl 2 2H20, were supplied by the Atomic Energy Institute, Kjeller, Norway. The ssmTc..5n..MDp was to pre99mTcO~frommixture a generator a pared by adding Sn—MDP solution. The i4C..MDP specific activity 40 mCi/g was kindly donated by Dr. M.D. Francis and Dr. J.J. Benedict, of the Procter and Gamble Company. The partly-peptized collagen was low viscosity ~~Polypep®~~, Sigma Chemical Company. This protein is prepared by trypsin digestion and has a molecular weight of approximately l0~.All other chemicals were of analytical grade. Adsorption experiments were made by adding HAP crystals to an undersaturated solution 4C..MDP. The pH of thisHAP solution was containing i total calcium and phosphate con6.6 and the centrations were 6.0 X iO~ M and 3.6 X iO~M, respectively. Under these conditions, the rate of dissolution of HAP is practically zero. I ml samples of solution for radioactive counting were separated from the mixture by centrifugation, followed by filtration through a 0.22 ~emMillex®-GS filter unit. No significant time effects were detected. The remaining mixture was acidified to dissolve all crystals and 1 ml of this solution was used for radioactive counting. The mass of crystals was calculated from the phosphate concentration of the acidified solution. For kinetic experiments, crystals were added to water, the pH of which was kept constant by
10(P04)6(OH)2
259
absorption spectrometry using a Perkin Elmer 305A instrument. The total phosphate concentration was measured by the molybdenum blue method, using a Carl Zeiss PMQII spectrophotometer. 6. Analysis and results
6.1. Equilibrium measurements of the adsorption of 14C—MDP on HAP
From the specific surface area, SSA 33.7 m2/g, and the dimensions of the crystallographic unit cell, a b 9.42 A and c 6.88 A, the specific amount of surface phosphate sites, n~ 5, can be calculated to be 1.73 x l0~mol/g. In this calculation each unit cell in the surface is assumed to have tion. two phosphate groups exposed to the soluThe results of the adsorption of 14C—MDP on HAP are given in table 1 and in figs. 1, 2 and 3. In table 1, column 1 gives the total concentration of MDP, n L/ V; column 2 gives the solid to solution ratio, rncr/V; column 3 gives the mole fraction of =
=
=
=
.0
0.5
0 4
~-~r
D 1
means of a pH-stat, which controlled the addition of dilute nitric acid to the system. The instruments used were PHM64, ABU 13, TTT6O and recorders from Radiometer, Denmark. The rate of dissolution was determined from the rate of acid consumption. Inhibitors were added to the water prior to the addition of crystals. The system was kept free of carbon dioxide by passing nitrogen through the aqueous phase. The mass of crystals used in an experiment was determined by acidifying the final reaction mixture, causing complete dissolution of the HAP crystals. The total concentration of calcium in this solution was determined by atomic
0.5 0.3
0.2 .
01
00
_____________________
Oo
6
7CL/M Fig. 1. Adsorption IO of ‘4C—MDP on HAP. Langmuir adsorption isotherm assuming monodentate adsorption, eq. (2). The values of D 1 and CL are given in table I.
/
260
J. (‘hnstoffer.sen ci al.
20
-
/
Kinetics of dissolution of Ca,
0 8
4
/// D2 10-
/
/
0
08
03
/~
x
1/
X
,1/
2
•~‘
//
07
/
0
1/P044(OH)
0.6
7
.: 7CL/M 1.
2
6
00
0
0
0.4
~L__~.I
2 JO7CL/M 4
6
00
JO Fig. 2. Adsorption of ‘4C—MDP on HAP. Adsorption isotherm for bidentate adsorption. eq. (6). The values of D 2 and Ci are given in table I. The slope of the line is KL = 3.6>< l0~l/mol.
Fig. 3. Adsorption of ‘4C—MDP on HAP. Adsorption isotherm for bidentate adsorption, eq. (8). when the degree of coverage. .v, is large. The values of D~are given in table 1. The slope of the solid line is Ki
0.9 ~ l0~l/mol.
=
MDP adsorbed. From these experimental data, the
to bidentate adsorption. D1 in column 8 is the
equilibrium concentration of MDP in solution, CL, and the amount of MDP adsorbed per unit mass, n~/mcr, can be calculated~ these data are
mole fraction term in eq. (8), corresponding to bidentate adsorption at high values of x, x 2n~/n11. Fig. 1, a plot of D~against ~L’ shows
given in columns 4 and 5. Assuming monodentate adsorption, the mole fraction term in eq. (2) is given in column 6, i.e., D1 = x/(1 — x), with x = nj~/np.D2, given in column 7, is the mole fraction term in eq. (6), with x = 2n~/n~,i.e.. corresponds
that the adsorption of MDP to HAP cannot be described by monodentate Langmuir adsorption. In fig. 2, a plot of D2 against C~.a straight line is obtained for 0 < ~L < ~ X IO~ M. In this concentration interval, adsorption of MDP onto HAP
Table I 4C_MDP on HAP crystals, n~/m~ The adsorption of ‘ 1 = l.73x 10
106 n
~/
V
m~/V
(mol/l)
(mg/I)
0.801 1.076 1.324 1.444 1.618 1.741
18.6 17.7 16.8 17.7 18.5 18.4
=
mol/g
nt/ni
l0~CL (mol/l)
l0~n~/Fn~, (mol/g)
D1
D2
D~
0.943 0.882 0.802 0.785 0.768 0.738
0.46 1.27 2.62 3.10 3.75 4.56
4.06 5.36 6.32 6.40 6.72 6.98
0.31 0.45 0.58 0.59
1.67 4.28 10.1 10.9 15.6 21.7
((.88 1.63 2.71 2.84 3.48 4.18
0.64 0.68
The equilibrium distribution of inhibitor given in columns 4 and 5 is calculated from the experimental data in columns I —3. The distribution between crystal surface and solution will be independent of the amount of inhibitor. nL. the mass of crystals, mrr. and the volume of the system, V. D~.D2 and D1 are the mole fraction 2, x = functions 2n~/n in the Langmuir adsorption isotherms. eqs. (2). (6) and (8), respectively. D1 =x/(I— x), x
=
n~/np; D2 = x/(l— x)
6 D~ x/(l— x).x
=
2n~/n1,.
J. Christoffersen et a!.
/
Kinetics ofdissolution of Ca
can be described as bidentate adsorption, with KL, see eq. (6), equal to the slope of the line, 3.6 x I0~ 1/mo!. For higher values of CL, the slope of the curve in fig. 2 increases, corresponding to the expected break-down of the validity of eq. (6), which for high values of coverage, should be replaced by an equation of the type of eq. (8). In fig. 3 the mole fraction term in eq. (8), D3 x/(1 x), is plotted against CL. For high values of x, the points approach a line through the origin with slope KL 0.9 x l0~l/mol, about one quarter of the value of KL obtained in fig. 2 using eq. (6). This corresponds to a value of the geometrical factor y 4 in eq. (6). This appears to be a reasonable value to represent the number of nearest —
=
=
6.35 iü~
261
10(P04)6(OH)2
adsorption sites to any particular adsorption site in the surface, considering that we have not taken the symmetry of the crystals into account, except when calculating n~. 6.2. The effect of MDP on HAP dissolution rate
Kinetic effects of adsorption of MDP onto dissolving HAP crystals are shown in fig. 4. In this figure, JL/Jo and Jo/fL are plotted against CL. CL was calculated from the total concentration of MDP, assuming equilibrium with respect to the type of adsorption expressed by eq. (6) with KL 3.6 X l0~1/mo!. From the lower plot in fig. 4 is seen that ~L/~o for constant values of C/CS decreases with CL and that ~L/~o for constant value of CL decreases with increasing values of C/CS. From the upper plot is seen that the kinetic effects of adsorption of MDP onto HAP crystals can be empirically expressed by an equation of the type ~o/~L 1 + Kk~flCL, (34) =
=
,
cf. eqs. (14), (18) and (21). In fig. 5 the points from 4.4
the upper plot in fig. 4 are plotted again; the
curves drawn are theoretical values of
J0/k
calculated from eq. (20) with KL 3.6 X l0~1/mol and ~ equal to the values indicat in the plot. =
,
8CL/M
3
0
1O Fig. 4. Effect of MDP on the rate of dissolution of HAP at pH 7.15. ~L and] 0 are the rates with and without inhibitor present, respectively. CL was calculated from the total amount of inhibi tor, the mass of crystals and the volume of the suspension assuming bidentate adsorption, eq. (6), with KL = 3.6x iO~ 1/mol. (•) C/Cs = 0.25; (X) C/C1 = 0.35; (0) C/C1 = 0.45. The values of Kkfl, the slopes of the plots of ~o/~L against CL, are given in the figure.
0
-___________________________________ 8 10 15 10 CL/M Fig. 5. The points in this figure are the same as the upper plot in fig. 4. The curves are drawn through values calculated from eq. (20) with the values of I~ given in the figure.
262
J. Christoffersen et a!. / Kinetics of dissolution of Ca,
11(PO4),,(OH)~
From this plot is seen that the theoretical expression eq. (20) may explain the inhibitory effect of MDP for C/c,, 0.25 8)< and l0~ C1 < M. l0~ M C/C.,= and for For C/C,,=0.35 0.45, eq. (20) and does CL< not give a good agreement with the experimental data. This may be a result of an uneven distribution of the inhibitors on the surface,
I
=
0
05
.1 a
as
i
IS
20
28
‘ItO
in
6.3. The effects of stannous chloride, Polypep, and mixtures of these with MDP
Fig. 7. Effect of a partly-peptized collagen. ‘Polypep” (Sigma).
M
= 10 kg/mol. on the rate of dissolution of HAP: mcr 7 mg, V = 0.9 I. C/C, 0.25. Circles: pH = 6.76. Squares: pH =
, ,
The effect of the presence of stannous chloride on the rate of dissolution of HAP crystals is shown in fig. 6 in which JL /J U is plotted against the total amount of stannous chloride per unit mass of HAP crystals. In all experiments represented in this plot approximately the same mass of crystals, 6 ±1 mg and volume of solution, 0.93 ±0.03 1, was used and all results are given for C/p,, 0.25. The inserted broken line gives the effect of MDP under similar conditions. As the actual concentration of stannous chloride in the aqueous solution was not determined, the data given in fig. 6 cannot be used in a plot of JO/JL against CL. However, from fig. 6 can be seen that stannous chloride is an even stronger inhibitor for HAP dissolution than MDP and that the effect of stannous chloride is independent of small changes in pH around pH 7. =
=
8
105n/mcr
moilg Fig. 6. Effect of stannous chloride on the rate of dissolution of HAP, nL is the total amount of stannous chloride added; us,,, = 6 mg. V= 0.9 I. C/C,, = 0.25: (0) pH = 6.76; (0) pH = 7.16; (X) n~~y. 11= nMDP, (— — —) effect of MDP on rate of dissolution of HAP, taken from fig. 3 of ref. [Ill. Stannous chloride is more effective than MDP in retarding the rate of dissolution of HAP.
7.16.
Closed symbols: “Polypep” dialysed. Open symbols:
‘Polypep” non-dialysed. Low concentration, I mg/I. of
Poly-
pep” give reduces thefurther rate of dissolution by 60%: tions little reduction in the rate. higher concentra-
The effect of the presence of the partly-peptized collagen, Polypep, is shown in fig. 7, in which ~L/~o is plotted against ML/V, the total mass concentration of Polypep in the system. All data in the plot are for C/C,, 0.25, mass of crystals 7 ±2 mg and volume of system 0.92 1. The plot shows that Polypep has quite a large inhibitory effect for small values of mL/V, but the increase in inhibition with concentration of Polypep is very small for a concentration larger than 1 mg/I. The effect of Polypep is not affected by small changes in pH around pH 7. The inhibitory effect of a 6% mixture of stannous chloride with MDP, as delivered in the hone-scanning agent from Kjeller. is shown in fig. =
=
2
JL/JN~N~~
——
Fig. 8. Effect of MDP—Sn and MDP—”Polypep” mixtures on the rate of dissolution of HAP; pH = 6.76, C/C,, = 0.25. The line drawn is the best line through the results for MDP alone, i.e. the same as the broken line in fig. 6. (0) MDP+6% SnCI,: (X)nMDP = ~s~c2~ (~) mixtures of”Polypep” and MDP, see table 3; nL = ~MDp for all points in this plot. The retarding effects of the mixtures, MDP—SnCI2 and MDP—” Polypep” are smaller than expected from the sum of their individual effects.
J. Christoffersen et a!.
/
Kinetics of dissolution of Ca
Table 2 4CMDP (L = I) and Polypep (L The adsorption of mixtures of ‘ 106 ni/V m~~/V rn,/V iO~C~
=
2) on HAP crystals I0~n’~/m,,r
(mol/I)
(mg/I)
(mg/l)
(mol/l)
(mol/g)
1.1 1.1
17 17
25 25
1.3 1.8
5.2 5.0
8, in which JL/JO is plotted against the total amount of MDP present per unit mass of crystals for C/CS 0.25, mcr 5.5 ±0.5 mg, pH 6.8 and V 0.94 ±0.02 1. The curve drawn is for MDP alone as in fig. 6. The presence of the small amount of stannous chloride appears to have no significant effect. The point marked x gives the effect of a mixture of stannous chloride with an equal amount of MDP. The combined effect of these two inhibitors appears to be less than the effect expected for twice the amount of one of them. We have also determined the amount of ‘4C—MDP on the surface under these conditions, and found 60% of the amount expected in the absence of stannous chloride. It appears that mixtures of MDP and stannous chloride have “negative synergistic” effect on each other. Complex formation between MDP and stannous ions in solution may be partly responsible for this. The points in fig. 8 marked ~ correspond to the presence of mixtures of MDP and Polypep, see tables 2 and 3. From table 2 is seen that if Polypep is added to HAP crystals in equilibrium with MDP, a nearly normal amount of MDP is found on the surface. Polypep does apparently not displace MDP already on the surface to any great extent. However, that there is also some interaction of Polypep with the surface can be seen from table 3, which shows that the inhibitory effect of the mixture on the dissolution rate is greater than the effect of MDP alone. When Polypep is added =
=
‘—
=
Table 3 Effect of mixtures of MDP (L
=
I) and Polypep (L
=
263
10(P04)/OH),
Added first 3.8 3.2
2
to the crystals prior to MDP, less MDP than normal is found on the surface (table 2) and the combined inhibitory effect is greater than in the previous example (table 3 and fig. 8). As the equilibrium adsorption isotherm for Polypep adsorption onto HAP has not been studied yet, additivity of the effects of MDP and Polypep will not be discussed in any detail, but it is not surprising that the sequence of addition of small and large inhibitor molecules is important. If the surface contains adsorbed small inhibitor units, the statistical possibility for finding adsorption sites for large molecules is obviously reduced, whereas adsorption of large molecules prior to small ones should have a much smaller effect on the adsorption of the small molecules as long as the surface is not covered completely by the large molecules, which is also most unlikely to occur.
7. Conclusion A Langmuir adsorption isotherm with an inhibitor unit occupying two adsorption sites can explain the adsorption of MDP on HAP crystals. The adsorption can be represented by KL x/(l —x)2CL with KL 3.6x l0~1/mol, when up to 75% of the available sites are occupied, x 0.75. At higher values of x, the equation KL x/( 1 x)CL with KL 0.9 x l0~1/mol fits the results. The inhibitory effect of MDP on the rate of dis=
=
=
=
=
2) on the rate of dissolution of HAP crystals
106 ni/V
mcr/V
m J2/J0
J/J0
(mg/I)
2/V (mg/I)
J1/J0
(mol/I) 0.41 0.41
7.3 6.8
27 27
0.26 0.23
0.20 0.20
0.18 0.09
Added first 2
—
J. Christoffersen ci a!.
264
/
Kinetics ofdissolution of Ca,
solution of HAP can be fitted by the equation Jo/fL 1 + KkflCL, when the undersaturation, C/C5 is in the range 0.25—0.45, with the efficiency of the inhibitor increasing as the composition of the solution approaches equilibrium. This can be explained by the increase in the size of the critical nucleus as the solution composition approaches equilibrium, the dissolution process being controlled by a polynuclear mechanism. Stannous chloride has been shown qualitatively to be an even more effective inhibitor for the dissolution of HAP than MDP. A partly-peptized collagen also inhibits the dissolution of HAP at very low concentration. Studies of the adsorption of mixtures of MDP and partly-peptized collagen show that the adsorption of MDP is reduced if the large molecules (collagen) are adsorbed first, but nearly normal, if the MDP is adsorbed first. The adsorption of collagen is affected in both of these cases, but more when the MDP is adsorbed first. In regions with high bone turnover the apatite crystals may not be covered by collagen to the same degree as crystals in areas with low bone turnover. The results indicate that bone-scanning agents will then preferentially adsorb in regions of high bone turnover with the larger free surface area. =
11(PO4),,(OH)~
given to J.C. and G.H.N. by the Danish Natural Science Research Council (81-3393) and by the Danish Medical Research Council (12-3339), to J.C. by Novo’s Foundation, and to G.H.N. and J.C. by NATO (RG 143.80). We are very grateful to Nobuko Christiansen, Iben Junghans and Lene Høyer for laboratory assistance.
References [I] G. Subramanian, R.J. Blair, E.A. Kallfelz, F.D. Thomas and J.G. McAfee. J. NucI. Med. 14 (1973) 638. [2] PH. Cox, Brii. J. Radiol. 47 (1974) 845. [3] SB. Christensen and OW. Kf’ogsgaard. J. NucI. Med. 22 (1981) 237. [4] M.D. Francis, A.J. Tofe, J.J. Benedict and J.A. Bevan, in: Radiopharmaceuticals
II, Proc. 2nd Intern. Symp. on
Radiopharmaceuticals, Seattle, WA, 1979 Nuclear Medicine, New York, 1979) p. 603.
(Society of
[5] J.L. Matthews, J.J. Reynolds, W.G. Roberison and K. Simkiss, in: Biological Mineralization and Demineralizalion. Rept. Dahlem Workshop on Biological Mineralizalion and Demineralization, Berlin, 1981 (Springer, Berlin, 1982) P. 327. [6] J.M. Holmes, R.A. Beebe, AS. Posner and R.A. Harper. Proc. Soc. Exptl. Biol. Med. 133 (1970) 1250. [7] P.Cj Werness, J H Bergert and K.E Lee. Clin Sci 61 (1981) 487. [81 D.J. White, J. (‘hristoffersen, T. Herman. AC. Lanzalaco
Acknowledgements
and G.H. Nancollas, J. Urol., in press. [9] J. Christoffersen, MR. Christoffersen and N. Kjiergaard, i. Crystal Growth 43 (1978) 501. [10] J. Christoffersen and MR. Christoffersen. J. (‘rystal Growth 57 (1982) 21.
J.C. and M.R.C. acknowledge the following grants for equipment and laboratory assistance: the Danish Natural Science Research Council (511-1042, 511-1590, 511-8181), the Danish Medical Research Council(12-3617, 512-15458, 12-0323. 12-2215), Novo’s Foundation, and the Carlsberg Foundation (1981/82, 35/IV). S.B.C. acknowledges a research grant from the Danish Medical Research Council (12-1829). G.H.N. acknowledges grants from the National Institutes of Health (No. 03223, No. RO1AM19O48). Travel support has been
[ill
J. Christoffersen
and
MR.
Christoffersen.
J. Crystal
Growth 53(1981) 42. [12] Christoffersen, J. Crystal Growth (1980) [13] J. MC. van der Leeden. J. Reedijk and49 G.M. van29. Rosmalen. unpublished. [14] E.C. Moreno, M. Kresak and DI. Hay. J. Biol. Chem. 257 (1982) 2981. [15] E.C. Moreno, K. Varughese and Dl. Hay. Calcified Tissue Res. 28 (1979) 7. [16] N. Cabrera and D.A. Vermilyea, in: Growth and Perfection of Crystals, Eds. RH. Doremus, B.W. Roberts and D. Turnbull (Wiley. New York. and Chapman and Hall, London, 1958) p. 393.