kinetics of acid demineralization in his to logical technique

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DOI: 10.1177/22.6.4341974 22: 434J Histochem Cytochem

HENNING BIRKEDAL-HANSENKINETICS OF ACID DEMINERALIZATION IN HISTOLOGIC TECHNIQUE

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43 4

TH E JOURNAL OF H ISTO CH EM ISTRY AN I) CY TOC HEM ISTR Y

Copyr ight © 1974 by The H is lochem ica l oc ietv . Inc .

Vol . 22 , N o . 6 , pp. 4 :34-441 , 1 974

Pr,n ted in t’ .S .A .

K IN ET IC S O F A C ID D EM IN ERA L IZA T IO N IN H ISTO LO G IC TECHN IQ UE ’

H ENNING BIR KEDAL-HANSEN2

In stitu te for C ario lo gy and E ndo do n tic s, R oya l D en ta l C o lleg e , C openh s:ig en , D enm ark

Receiv ed for public ation August 21, 1973

T he ra te o f p rogress ion o f the dem in era liza tio n fron t d ur in g ac id dem in era liza t io n is in -

vestiga ted for s tan da rd ized e lep h an t ’s ivory den tine spec im en s. A theore tica l m od el for the

pen etra tion b ased up on a com b ined d iffu sion w ith rea c tio n p rocess is se t u p . It is dem on-

stra ted th a t the resu lts ar e in acco rdan ce w ith the prop osed m ode l. F or spec im en s o f cer -

ta in geom etr ica l shap es ( sem i-in fin ite or p la ne shee t) the d istan ce pen etra ted b y the fron t

is d irec tly p ropo rt io na l to the squ are roo t o f the tim e.

D em in eral iza tion is freq uen tly an inev ita b le

step in hard tissue h isto logy . M uch a tten tion

h as b een pa id to the m ore o r less de le te rious

e ffec t o f various decalc ifv ing agen ts (3 , 4 , 9 , 1 4 ,

21-2 5), w hereas on ly litt le in te res t has ga thered

aroun d th e k ine tic law s govern in g the p roc ess .

It h as been em ph as ized tha t d iffu sion p lays

an im por tan t ro le (3 ,3 , 30 ) and tha t the

pro gre ssio n of dec alc ific atio n is s ign ific an tly

in fluen ced by the rate of’ d iffusio n of d ec aicth ’-

in g so lu tio n in the specim en , bu t deca lcifica -

ti()n expe rim en ts h ave m ostly b een pe rfo rm ed

on nonsta nd ard ized spec im ens . w hereby unde r-

ly ing physicochem ic al p henom en a a re ob scu red

by uncon tro llab le variab les. X -ray techn iques

hav e revea led th at dem inera lization proceeds as

the fo rw ard m ovem ent o f ’ a sharp ly d e linea ted

fro n t and no t as a d if fuse loss o f’ m inera l sa lts. It

has fu rthe r been dem onstra ted tha t deca lc itlca -

tion proceeds w ith decreasing ra te , the cen tra l

pa rts of’ the spec im ens requ irin g m uch long er

tim e to ach ieve com plete dem inera liza tion th an

the periphera l p a rts . B rain (3 ) exp la ined th is as

a resu lt o f’ the d ifficu ltie s fo r hyd rogen ion s to

penetrate the specimens. These observations,

how ever, h ave n o t been exp ressed quan tita -

tiv ely . Th us th is study deals w ith the k in etic s o f

the progression o f’ the d em inera liza tion fron t.

THEOR ET I C A L

F rom a kine tic po in t o f’ v iew , ac id dem inera li-

zation of h is to lo g ic specim ens con sists o f’ th e

‘T h is p ro jec t wa s su pported by the D an ish D en ta lA ssoc ia tion ) ‘Fonden til s t#{2 46}tteo r v idenskabe lige o g

prak tisk e un dersg eise r inden fo r tand laegekuns-

ten”) .

2 Address requ es t for rep rin ts to :D r. H . B irk ed a l-

Hansen , Ins titu te o f C a rio logy an d Endodon tics ,

R oyal D ental C ollege , 4niversi tetsparken, DK 2100

Copenhagen 0. Denma rk .

foll owing p rocesses: (a ) diffusion of’ acid in the

ba th tow ard the sp ecim en surface : (b ) dif fusion

o f’ ac id in the organ ic m a trix of ’ the dec alc ifie d

part o f’ th e spec im en tow ard the d eca lc ifica tion

f ron t; (c ) re action w ith hyd ro xyapatite at the

de calcificat ion fron t; an d (d) outw ard d iffus ion

of rea ctio n p rod uc ts. T he ex act k inet ic treat-

m en t o f’ th is se ries of ’ ev en ts is com plicated an d

e lab ora te , b u t if’ c ertain a pproximations and

simplif v in g a ssum p tio ns a re m ad e, as sta ted

b elow , the treatm ent can be a ccom p lish ed w ith

a reasonab le resu lt.

T he reac tion be tw een ac id an d hydroxyapa-

tite is co nsid ered in stan tan eo us. A s th e d iffu -

sion of ’ a ci d in the d eca lcified org an ic m a trix is

m u ch slow e r than diffu sion of a cid in th e ba th

(5) , mat rix diffusion is conside red rate- deter-

m in ing . T he prob lem then is o ne of’ d iffu sio n o f

ac id in th e liqu id phase of an organ ic po lym er,

fu rn ish ed w ith a fin ite num ber of fix ed reac tive

s ites capab le o f re act ing w ith th e d iffu sin g

so lu tio n b y an in stan tan eo us reac tio n . R eact ive

s ites a re h ydroxy ap atite cry sta ls and ac id -b in d-

ing g roups in th e org an ic m a trix (m ain ly co lla-

gen). H ence we can cons ide r th e d iffu sion w ith

react ion proce ss from tw o s lig h tly d iffere n t

ang les (approach A and B as d iscussed b e-

low ), e ithe r (A ) as sim ple d iffusion ofc id in

the liqu id ph ase of ’ th e sp ec im en , go verned by

the d iffusion coe ffic ien t D0 and com b ined w ith

tw o d iffe ren t re act ion s-a cid b ind ing to co lla-

ge n (5 ) and ac id “d isso lu tion” o f hyd roxyap atite

-o r (B ) as d iffu sion of ac id in th e co llagenous

m atrix , go ve rned by the effect ive d iffus ion co-

ef f ic ien t , Dr, , com bined w ith on ly on e re act ion ,

hyd roxyap atite d isso lu tio n . Dr, . thus is th e ob -

se rv ed o ver-a ll d if fusion coeff ic ien t fo r the com -

b ined d iffusion an d ac id -b ind in g process . T he

re la tion be tw een D0 an d D dep en ds on the








1 0 2 0 4 0 6 0 8 0

a / b

z . io 2

20

16

12

F I G . 1. G raph of the function Z .expZ 2 ) .e rfZ

a/b .

(8)

and

(9)

V alues of Z can be obtained f’rom graphs of’ the

function (18) for any specified value of/b . The

function is plotted below (Fig. 1) w ithin the

range of’ values needed for the present study.

Thus if the actual constants (D 0 , D and a /b )

are determ ined, th e p ro gres sion o f the d em iner -

a liza tio n fro nt can b e calcula ted from th e above

(2) equations.

If a << b , expansion of the functions in

equation 3 gives a/b = 2Z 2 and hence equations

6 and 8 reduce to:

= ( 2M ,D, t (10)

\ M 5 ± M 0 1

or

= ( 2 (M , + M,,)D, , t H (11)

M 0 /

Practically this m eans that if’ the am ount of

acid needed for dem ineralization (M0 ) is m uch

greater than the am ount of free acid or acid

com bining w ith collagen (M , and M 5) the sim -

pler equations 10 and 11 can be used instead of

equations 6 and 7 and 8 and 9. T his situation

usually prevails during decalcification due to

the considerable m ineral content of bones and

teeth.nd

K I N E T IC S O F A C I D D E M I N E R A L I Z A T IO N 435

H ZA exp(ZA 2)erfZA = M, / (M5 + M ,) (7) It is seen from equations 2, 10 and 11 that the

kinetics of the acid-binding reaction, as it has

been discussed elsew here (5).

T he total am ount of acid used for com plete

decalcification of the specim en (M ) can be

expressed as

M=M,+M5# {2 4 7 }M0 (1)

w here M , is the equilibrium am ount of free,

diffusing acid contained in the specim en, M 0 is

the equilibrium am ount of acid bound to pro-

teins and Mh is the am ount of acid needed for

hydroxyapatite dissolution. If w e assum e that

the reaction associated w ith diffusion proceeds

to com pletion in the presence of sm all am ounts

of diffusing substance, as in the case of acid

dem ineralization, there w ill exist a front behind

w hich reaction has occurred at all reactive sites

and ahead of’ w hich no reaction has occurred at

all. A s all reactive sites have already reacted

behind the front, the penetration of acid occurs

as sim ple diffusion in the liquid phase of’ the

decalcified part of the specim en from an infinite

bath of’ concentration C0, but diffusing sub-

stance is continually los t a t the fron t by instan-

taneous reaction.

Treatm ents of analogous problem s have been

given by C rank (10), D ankw erts (11), H erm ans

(17) and H ill (19). D etails of the solution

procedure can be otained from these authors.

For the sem i-infinite m edium the solution ends

up w ith:

w here Z is giv en by

Xm = 2ZV”DT

7r- ’ Z e x p ( Z 2 ) e r f Z = a/ b

Xm is the m ovem ent of the front,D the diffusion

coefficient, t the tim e, exp the exponential

function, erf’ the error function. a/ b is the

fraction betw een concentrations of free and

reacted acid during decalcification and hence,

a c c o r d i n g t o a p p r o a c h A :

a/ b = M, / (M5 + M 5) (4)

and according to approach B :

a/ b = (M , + M /M 5 (5)

Inserting this in equations 2 and 3 gives for

approach A :

Xv , = 2ZA’. I ’ (6)

F or appro ach B correspondingly

X v, = 2Z0\/’D

Tr”ZR exp(Z52)erfZR = (M , +








1 2 )

0= 1 -- (13 )

r02

FIG. 2 . R ad iograph s o f decalc if y ing , cy lind rical iv o ry d en tine spec im en s w ith on ly rad ial ac id pen etration .

D ecalc if ic atio n , respectiv ely , in 1 .0 ’N HC1 f or 4, 8 , 2 4 , 48 , 72 and 79r. N o te sharp bou ndary b etw een

m ineraliz ed and dem ineraliz ed p arts o f th e sp ecim ens .

43 6 B IR K E D A L - H A N SE N

progre ssion of the d em ine raliz at ion f ron t is

d irectly proportio nal to the square root o f ’ the

tim e (parabo lic law ) .

S o f ar w e hav e d ealt on ly w ith the sem i-in f i-

n ite m ed ium . For theplane sheet, hounded by

tw o parallel p lan es and suspended in d if ’f ’using

s o l u t i o n t h e a b o v e e q u a t i o n s a r e a l s o v a l i d ,

p r o v i d e d t h a t t h e c o n c e n t r a t i o n o f ’ d i f f u s i n g

subs tance in th e center o f ’ the sheet is z e ro o r

n early z e ro (10 ). A s the reac tio n rate o f ’ ac id

d i s s o l u t i o n of ’ hy dro x y apatite is great and a/ b is

s ma l l t h i s wi l l h e f u l f ille d practically un til the

tw o d ecalci f icatio n f ro nts m ee t at th e cente r o f ’

th e sheet.

For the infinite cylinder, the treatm ent is

m o re com plicated and no sim ple re lat ion such

as the parabolic law can be fo und (10 , 17 . 1 9). In

conn ec tion w ith dem ineraliz ation , how ev er, th e

approx im ate treatm en t o f H ill (19 ) is in reason -

ab le acco rdance w ith the co nd itio n s p rev ailin g .

If it is as sum ed that the re is a steady s tate

concentration d istribu tio n o f ’ f ree acid beh ind

th e f ro n t, H ill g iv e s the f o llow ing so lu tion :

where

O - (1 - 0 ) ln = D t = 4aDt

1 - o r,2b r,2b

r0 is the rad ius o f ’ the cy linde r, r th e d is tan ce

f rom the cen te r to th e f ron t. W hen is k now n,

th e re lativ e d istance pen etrated b y the f ron t p

(r0 - r)/r0 can be calculated as

p 1 1 - 0 (14)

T he erro r com m itted by assum ing s tead y s tate

c o n d i t i o n s wi l l be sm all p rov ided a/ b i s s ma l l ,

so equations 12-1 4 are pp licab le to the dem in -

e raliz at ion pro cess. as a/ b ac tually is sm all.

M A T ER IA L S A N D M ET H OD S

Cy lindric al spec im ens o f ’ ele phan t’s iv ory d en tin e,

e i ther 3 , 0 m m in d iam e ter and 10 .0 m m of len gth

( 1 1 0 : 3 mg d r we i g h t ) ( g r o u p 1 ) o r 1 0 . 0 mm o f

diam eter and 4 .0 m m o f he igh t (500 m g d ry w eig ht)

(group 2) w ere m ade on a turn ing lathe .The curved

surf ac e o f group 1 specim en s and the end su rf aces of

group 2 spec im ens w ere dou b le lacq uered as prev i-

ou s ly desc ribed (5 ). T h e sp ec im en s w ere decalc if ie d in

1 .0 and 0 .1 N HC1 at ro om tem peratu re ( 22 #{ 1 76 } C)i n t he

presence of great ex cess o f decalci f )’ing f lu id , w h ich

w as ch ang ed daily or ev ery 2nd day to ensu re con stant

ex ternal con cen tration of h y d rog en io ns . D ecalcif ica-

tio n w as perf orm ed w ith and w ith out m agne tic s tir-

ring . W hen solu tion s w ere no t stirred , the spec im ens

w ere placed on gauz e to av o id contact w ith th e bottom

of the bath . T he p rog ressio n of the decalcif ication

f ro nt w as m easured on s tandardiz ed radiograph s

(S tructurix D4 , A g f ’a-G ev ae rt. B elg ium ) at a 28 tim es

m agn if ication (Fig . 2 ). X -rav s w ere tak en at regu lar








D IS TANC E 1 m m)

5 . 0

40

3. 0

2 0

1 0

/

/

/

//

/

//

D I S TANCE m m )

5 0

4 0

3 0

2 0

10

1 0 ’ ! . EDTA

DISTANCE m m )

8 1 6 2 4 3 2 4 0

S Q UAR E R O O T O F T IME HO UR S )

4 0

30

2 0

1 0

2 0 4 0 6 0 8 0

T I ME I HOURS I

K IN E T I C S O F A C I D D E M I N E R A L IZ A T I O N 4 3 7

2 4 6 8 1 0

S Q UAR E R O O T O F T IME HO UR S )

F I G . 3. Progressi on of decal ci f i cat ion f ront during

d e c a l c i f i c a t i o n i n 1 . 0 N HCI a s f un c t i o n of square root

of time. Rod l ike i vory dentine specimens wi th o n l y

ax ial acid penetration. Poi nts and verti cal l i nes de-note means and standard dev iations. T he slope of the

l i ne gives the value of the penetration coef f i cient

(2ZDH).

FIG. 4. Progression of decalcified f ront duri ng de-

calci f i cation in 0.1 N HC1 and 10% EDTA as f un ct ion

of square root of time. Rodl i ke iv ory dent ine speci -

mens w i th onl y ax ial acid penetration. T he slope of

the l ine gi ves the value of the penetration coef f i cient

(2ZDH),

i ntervals unti l decalci f i cation w as completed. A f ter

each exam inati on, the specimens were put back intothe decal ci f y ing solut ion. For comparison a f ew speci -

mens were decalci f ied in0 ethvlenediaminetetraa-

c e t i c a c i d ( EDTA) i n 0 . 2M Tr i s b u f f e r ( p H 7.0) at

r oom tem perature.

Further specimens w ere analyzed f or thei r organic

and m ineral content. T he constant w eight obtained

af ter dry i ng at 110#{ 176} Cs refer red to as “ dry w eight.”

The residue obtained af ter ashing in oven at 650#{ 176} C

unti l constant w eight i s referred tos “ mineral

content.” T he weight di f f erence betw een dry w eight

and m ineral content is supposed to be organi c content

and f i rm ly bound water (resista’ nt to dry ing at 110#{ 176} C).

From the relativ e amount of organic mater ial to

f i rm ly bound w ater reported f or other dentine samples

(14), the absolute contents of organi c materi al and

f i rm ly bound water w ere estimated. The w ater con-

tent of the decal ci f ied part of the specimens w as

determ ined in a prev ious investi gati on (5).

R E S U L T S

The progression of the decal ci f i cati on f r o n t

duri ng decal ci f i cati on in 1.0 and 0.1 N H C 1 is

shown i n Figures 3-5. I t i s seen that the di s-

tance penetrated by the f ront i s di rectl y pro-

porti onal to the square root of the t ime fo r

specimens w i th l i near di f f usion (decak i f i ca-

t i on through end surf aces onl y ), w hereas speci -

mens w i th radial di f f usi on dev iate cl earl y f rom

the p a r a b o l i c r e l a t i on s h i p ( F i g . 5 ) . Th e e f f e c t

of sti rr ing on the rate of dem ineral i zati on w as

negl i gi bl e (T able I I I ). A nal ysi s of organic and

m ineral content i s summari zed in T able I .

Cal culati on of a/b: A s before, i t i s assumed

that M, = qC, ,V (5) w here q is the f racti onal

pore volume of the decalci f i ed part of the speci -

men and C0 the acid concentrati on of the bul k

of the bath and V the volume of the specimen.

F I G . 5 . Progression of decalci f i cation f ront during

decal ci f i cat ion i n 1.0 N HC1 as f unction of t ime.

Cy l indrical i vory dentine specimens w i th onl y radi al

acid penetrati on. Points and verti cal l i nes denote

means and standard dev iati ons (-), theoretical

curve (equati ons 12-14) f or experimental l y deter-

m ined val ue of ’D0a/b . - - - - , theoreti cal curv e (equa-

ti ons 12-14) normal i zed to the points.








( 1 5 )

TABLE II

43 8 B I R K E D A L - H A N S E N

“V alues obtained from two diffe rent samples .

V alues of M , and M 6 have been determ ined pre-

v iously (5 ) and are summarized in Table II but

h er e con v en ien t ly ex p r essed p er u n it o f v o lu m e .

M h = pG V, w here p is the number of moles o f

ac id needed for complete decalc ification of Gmoles o f hydroxyapatite contained w ithin 1

unit o f vo lume of the spec imen; G is es timated

in Table I assuming pure hydroxyapatite in

the m ineral phase; andp is determ ined from the

sto ichiometry o f the hydroxyapatite disso lution

pro cess. The net reaction for demineralizatio n

o f hydroxyapatite in w eak ac ids (3 , 15 , 18) is :

TA BLE I

C ontent of Ivory D entin Specim ens per U nit

of V olum e”

Mm- Organic I Or-

Initial D ry era! I Material )+ Firm ly ganicB ound rial#{ 17 6}t W t Con- ‘Mate.

tent Water j

Hydroxy-

apatite

Equiva.lent

Amountof A c id

(HCI)#{176}

mg ig !

1 .74 1,56 0 . 9 9 0 . 5 6 0 . 4 8

gm ole/

0 . 9 8

gm ole

1 9 . 5 8

a A nalys is prov ided by B iochemical D epartment,

Royal D ental College , Copenhag en, D enmark.

C alculated by subtraction o f 5% w ater (Eastoe

( 1 3 ) ) .

Calculated acco rding to equations 15-17 .

Ca ,0 (PO4)0(OH)2 + 8H -,

lOCa5 + + 6HPO4 - + 2H,O

In a strong ly ac id medium , how ev er, the reac-

tion proceeds:

and

± 6H -. H2PO,, (16)

H2PO4- + 6H -, H,P04 ( pK 2 ) ( 1 7 )

Hence a to tal o f’ 2 0 moles o f’ acid is needed

during decalc ification o f 1 mole o f hydroxyapa-

tite . Calculated values of/b acco rding to

equations 4 and 5 are show n in Table II.

C h eck o f p r op osed m od e l: A ccor d in g t o

equation 2 the term Xm /t is c ons tant for plane

sheet spec imens and has the v alueZ D H . The

parameters and constants determ ined so far

allow the calculatio n of a theoretical value of2ZD #{189}ither by approach A o r appro ac h B as

and D 91 have been determ ined prev ious ly (5 )

and are summarized in Table II andA and Z5

can be read from Figure 1 for any spec ified value

of a/b. Calculated theoretical v alues o fZD H

are summarized in Table III, together w ith

experimental v alues . For radial diffus ion, calcu-

lated and experimental results are demon-

strated in Figure 5 by depic ting depth of pene-

P aram eters C alculated from Experim ental Findings

Ac i d ( ‘oncen- I 15,’ Z4 . I Z, . 10 2 IM ,/V M ,/V

f MR / V

N f nole/g1 I m m ’/hr

1 . 0 I 0 . 6 4 I 19 .58 I 0. 32 I 3 2 . 2 4 9 . 0 I 1 2 . 6 I 1 5 . 6 I I 2 . 2

0 . 1 0 . 0 6 4 1 9 . 5 8 0 . 3 2I 3 . 2 f 19 .6 I 3 . 6 I 9 . 8 3 , 9 0.67

“ Ca l c u l a t e d a s a/b = M ,/(M 0 + M according to approach A.

# { 1 7 6 }Ca lc ul ate d a s a/b = (M + M b)/M R according to approach B.

TA BLE III

C alculated Theoretical and Experim ental Values for Rate of D ecalcification

.

A cid (‘oncentrat ion

2 z 4 D 0 # { 1 7 6 } ’

Calculated

from

Equatio n 7

2 Z, , D, . , 1Calculated

from

Equation 8

Xv,/t I

Calculated I

I from

Equation_10],

X v,/t TCalculated

from

Equation 11

Xv,/t

Experimental

- Stirring”

Xv,/t”

Experimental

+ Stirring

N

1 . 0

0 . 1

0 . 4 5

0 . 1 4

0 . 4 6

0 . 1 3

m m/h

0 . 4 6

0 . 1 6

r’5

0 . 4 6

0 . 1 6

0 . 4 9

0 .47

0 . 1 5

0 . 1 6

0 .48

0 . 1 7








K IN E T IC S O F AC ID DEM INE R AL IZ AT IO N 43 9

tration a s a f u n c t i o n o f time. Calculatio ns us ing

equations 1 2-14 and inserting estimated values

of a/b and D 0 gives the upper curv e, predicting a

penetration rate somew hat faster than that

observed. The low er curve w as calculated from

e q u a t i o n s 1 2 - 14 but w as no rmalized w ith the

pos ition o f the po ints. It is seen, that the

experimental observ atio ns (points) fit the H ill

model, hut the actual values of either/b or

seem to be somew hat low er than expected.

DISCUSSION

The agreement betw een predic ted theoretical

and experimental values for the penetration

rate of the demineraliz atio n front supports the

validity of the proposed models despite the

approximations made in the course o f’ the calcu-

lations . The physical meaning o f the model is

that for spec imens generally o fhe form of a

plane sheet the depth o f penetration of’ the

demineralizatio n front is direc tly proportional

to the square roo t o f the time , w ith a penetra-

tio n co ef ficie nt, 2 Z D H , which is direc tly propor-

tional to the square root o f the diffus ion coeffi-

cient of’ acid in the decalcified part o f’ the

specimen and further is a function o f’ the

partition factor a/b betw een the amount of ac id

in the liquid phase of the decalcified specimens

and the amount of ac id needed for the deminer-

a l i z a t i o n r e a c t i o n . For the demineralizat io n

process. there is almost direc t proportionality

betw een the penetration co effic ient and the

square ro ot o f’ this fac tor, w hich means, that

w ithin certain lim its , the depth of penetration is

direc tly proportio nal to the square ro ot o f the

concentration of ac id in the bath (equations 10

and 11 ). From equation 2 it further appears ,

that the time needed for complete dem inerali-

zation of a “plane shee t” spec imen is direc tly

proportional to the square of the thickness o f’

the specimen.

A s i mi l a r f r o n t l i k e p e n e t r a t i on o b e y i n g the

so -called parabo lic law has been demonstrated

experimentally in a great number of investiga-

t i o n s ( 1 . 2 , 6 . 7, 1 2 , 2 0 , 2 4 , 2 7 . 2 9 ) f o r f i x a t i v e

penetratio n into tissue specimens. hutt has not

previously been demons trated for the decalcifi-

c atio n proce ss.

For cy lindrical spe cimens, the frontlike mode

o f penetratio n is still present, but no such

simple relatio nship as the parabolic law can be

found fo r the dependence o f depth o f penetra-

tio n on time . Earls ’ unpublished results o f

Bung enberg de Jong3 fo r frontlike penetratio n

in cy lindrical spec imens accord w ith results

obtained by Hermans (17) and Hill (19 ). w ho

propo sed an approx imate model for the pene-tration. H ill’s model is based upon the assump-

tion that the amount of’ free diffus ing substance

i s s ma l l c o mp a r e d t ohe amount o f’ subs tance

reacting w ith reactiv e s ites in the medium . This

assumption is fulfilled during decalcification,

and the results obtained g iv e a curve corre-

sponding to H ill’s model. although the observ ed

value of’ aD o/b is somew hat smaller than pre-

dicted (Fig . 5). The main difference betw een

penetration in the plane sheet and the cy linder

is that the rate of’ penetration is continually

decreasing for the plane sheet according to the

square root re lationship, but for the cy linder,

the rate o f pene tration decreases initially and

later inc reases again (Fig . 5). Hence simple

calculation of the decalc ification time , w hich is

eas ily carried through fo r the plane sheet (equa-

tion 2 ), cannot be made for the cy linder.

Prev ious inv estigation of the kinetics o f’ hard

tissue demineralization refer only to enamel

dis so lutio n (1 5 , 16 . 18). The models se t up in

the se publicatio ns diffe r considerably from the

present models , as itig ht be expected. D uring

ename l disso lution, the o rganic mate rial is neg -

l ig ible and g enerally disappears during decal-

c ifIcation, so that the mineral surface is alw ay s

in contac t w ith the bath. D uring dem ineraliza-

tion of bone, cementum and dentine , the or-

g anic matrix , w hich cons titutes 50-70 w /w o f’

the tissue, is retained, so that acid can reach

the m ineral surface only after dif fusion in the

decalc if ied o rganic matrix of the spec imen. In

the present model the reaction betw een ac id

a n d h y d r o x y a p a t i t e i s c o n s i d e r e d i n s t a n t an e -

o u s . A c c o r d i n g t o Gr a y ( 1 5 ) t he r a t e c o n s t an t f o r

e n a me l d i s s o l ut i o n i n HC1 is approx imately 50

> 10 g /cm2/sec . M easured in the terms used

in this inv estigation (m illime ter prog res sio n o f’

the front per square centimeter surf’ace area per

hour) this v alue corresponds to approximately

50 mm/hr. C ompared w ith the actual penetra-

ti( )n rate fo r ivo ry dentine , 0 .5m in the 1st hr.

i t appears reasonable to assume an instantane-

ous reac tio n, so that dif’ f’usio n in the o rganic

matrix is rate -determ ining . It also fo llow s f’rom

B ungenberg de Jong HG: Unpublished re sults,

1926. C i t e d by Hermans (1 7),








44 0 B I R KEDAL - HAN S EN

Gray’s investigatio n (15 ) that, for demineraliza-

tion in w eak acids , the rate constants are much

less (1 /500-1 /1 000) and hence the reactio n or

disso lution rate can no longer be considered

instantaneous , so the present model applies

only to decalc ification of’ dentine, cementum

and bone in strong , fully disso c iated ac ids .

The rate of’ penetration of the decalc if ication

front in the present experiment probably is

c lose to the greatest obtainable penetration rate

during ac id decalc ification. The parameters

invo lved in equations 2 and 3 can hardly be

changed in order to increase the rate o f decal-

c ification. Probably the diffusion coeffic ient for

the small H and CL ions is fas ter than fo r any

other ac id. Further the relative m ineral content

seems to be low er in elephant’s ivo ry dentine

than in dentine from most other spec ies (13 ).

A ccording to B rain (3 ) even increase of ac idconcentratio n w ill not s ignificantly speed up the

penetration rate beyond a concentration of 1 -2

N. Furthermore, at this concentration the pro-

teo ly tic action of the acid w ill become signifi-

cant. D emineralization, therefore, usually w ill

progress w ith a penetratio n co effic ient less than

and frequently much less than 0 .5 mm/hr in

histo log ic technique. This w ill be true fo r w eak

ac ids and especially for ED TA . Fo r comparison,

penetration of the decalc ification front during

decalc if ication in 10% ED TA pH 7 .0 has been

demons trated in Figure 4 . The kinetics of this

reac tion, o f course, is no t inc luded in the

present model, but it appears that for planesheet spec imens , the parabo lic re lationship be-

tw een depth o f penetratio n and time still ho lds .

W ith a few ex ceptions (22 , 2 6 , 30 ) prev ious

decalcificatio n experiments have been per-

formed on unstandardized spec imens , w hich

makes comparison w ith the present results

difficult. Further the above investigations w ere

perfo rmed on cortical bone w ith access o f ac id

to all surfaces of the spec imens, so that the rate

of penetration o f the decalcificatio n front dev i-

ates from the equations g iven in this s tudy .

The deve lopment o f the present models is

based upon the assumption of a homogeneous

medium . A lthough this is actually not presenti n i v o r y d e n t me d u e t o t h e t u b u l a r a n d l a me l l a r

nature of e lephant’s ivory (25) the uniform

thickness o f the decalc ified rim in cy lindrical

spec imens (Fig . 2 ) w ith radial diffusion show

that under the conditions of the present experi-

ment the ex is tence and direction of tubules and

lamellae do not interfere w ith the rate of dem in-

eralization in different direc tions . Thus the rate

of penetration o f the front is not greater in the

directio n of the tubules than perpendicular to

the tubules as it m ight be expected. This

probably is a result o f’ the concentration o f acid

used, as S ognnaes (28) using w eak ac ids ob-

served a lamellar decalc ification pattern in

e lephant’s ivo ry dentine . So it m ight be ex-

pected that although ivory dentine spec imens

are satisfacto ry as model substance in the

present investigation, they maye les s useful

w hen study ing demineralization in w eak ac ids .

In the pre sent inves tig atio n, the rate of’ de -

mineralization is only insignificantly influenced

by stirring . S imilar results w ere obtained by

B rain (3 ) and Clayden (8) and are cons is tent

w ith the obse rvation that the diffusion o f ac id in

the decalcified part o f the spec imen is not

influenced by stirring (5 ). It is , how ev er, at

variance w ith results obtained by Molenaar (27)

and V erdenius and A lma (30) . The lacking

effect of stirring w as explained by B rain (3 )y

the as sumption that even vigo rous stirring in

the bath w ill not affec t the liquid contained

ins ide the spec imen.

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kinetics of acid demineralization in his to logical technique

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