Hixson-Crowell Equations
& 　
A Modified Higuchi Equation


　 　　　

　Yorinobu Yonezawa, Ph.D
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　in preparation　　 Jan. '12

HOME of Yonezawa [JPN]，[ENG]
PAGE for MEMORANDUM


Hixson-Crowell Equations　　　- a reasonable derivation -

　　Monodispers system of spherical particles　&　　　　　　　　　　　　　　　Ref '89，Ref '94

Estimation of the effective surface area at time t for isotropic dissolution
　　initial 　　　Mo： the initial amount　　　　　　　　　　 = n πLo3ρ/6，
　　　　　　　　 So ： the initial effective surface area　 = n πLo2
　　　　　　　　 Lo ： the initial particle diameter，　　　n ： the initial particle number，　　　ρ：the density，

　　at time t　M ： the undissolbed remaining amount　at time t ( = Mo-m)　　　= n πL 3ρ/6， 　　m ： the dissolbed amount
　　　　　　　　 S ： the effective surface area　at time t 　　　　　　　　　　　　　　 = n πL 2
　　　　　　　　　L： the particle diameter at time t，
　　　　　　　　m ： the dissolbed amount ( = Mo - M = n πLo3ρ/6 - n πL 3ρ/6 )

　　　　　　　　M /Mo = (L /Lo)3　，L/Lo = (M /Mo)1/3，　　　S/So = (L /Lo)2 = (M /Mo)2/3


from Nernst-Noyes-Whitney Eq. to Hixson-Crowell Eqs.
　　　　dC/dt = (1/V ) k S ( Cs - C )
　　　　　k ： the intrinsic dissolution rate constant，　Cs： the solubility
　　　expression by using the amount of sample，　　　
　　　　dm/dt = (1/V ) k So (M /Mo)2/3 ( Ms - m ) 　　　　　Ms： the amount required to saturate the solution ( =VCs)
　　　the initial effective surface area can be expressed by using the specific surface area (Ssp= 6/ρlo ) as：　
　　　　　　 So = SspMo
　　　　dm/dt = (1/V ) k Ssp Mo (M /Mo)2/3 ( Ms - m )　　　 　　can be rewritten as
　　　　-dM/dt = (1/V ) k Ssp Mo1/3 M 2/3 {Ms - (Mo- M ) }

　1) sink condition ( Mo << Ms )　　　　　　　the cube root law equation　( '89)
　　　　-dM/dt = (1/V ) k Ssp Mo1/3 M.2/3 {Ms - (Mo - M ) }　　　　　　can be expressed as：
　　　　-dM/dt = (1/V ) k Ssp Mo1/3 M 2/3 Ms = k CsSsp Mo1/3 M 2/3
　　　　-M -2/3 dM = k CsSsp Mo1/3 dt
　　　by integrating and rearrangement
　　　　M 1/3 = Mo1/3 - (1/3)k Cs Ssp Mo1/3 t
　　　　( M /Mo )1/3 = 1 - (1/3) k Cs Ssp t 　　　

　2) a non-sink condition ( Mo = Ms )　　　the negative two-thirds law equation　( '89)
　　　　-dM/dt = (1/V) k Ssp Mo1/3 M 2/3 {Ms - (Mo - M ) }　　　　　　can be expressed as：
　　　　-dM/dt = (1/V) k Ssp Ms1/3 M 5/3 = k CsSsp Ms-2/3 M 5/3
　　　　-M -5/3 dM = k CsSsp Ms-2/3 dt
　　　by integrating and rearrangement
　　　　M -2/3 = Ms-2/3 + (2/3) k Cs Ssp Ms-2/3 t
　　　　( M /Mo )-2/3 = 1 + (2/3) k Cs Ssp t 　　　　　　　　　　　　

　3) non-sink condition ( 0 <Mo<Ms )　　( '94)
　　　General equation for dissolution with optional initial amount within the solubility was derived.
　　　Dissolution equations as approximate simple dissolution equations were
　 　　　the z-law equation，　( M /Mo )z = 1 - z k Cs Ssp t 　　　　　　z = 1/3 - Mo/Ms
　　 　　the Ln-z equation，　 ln( M /Mo ) = - k Cs Ssp t 　　　　　　　　when Mo = Ms/3
　　 were available for particles , non-disintegrating tablets　．．．．．．
　　 were available for dissolution with optional initial amounts within solubility
　　 were available for treatment, simulation and prediction of dissolution process for optional initial amounts.



The H-my Equation　　- a modified Higuchi equation -　　　　　　Ref '05
　　　The amount of drug in the wax matrix decrease with increasing the released amount,
　　　　and the release process graddually deviates from Higuchi equation.
　　 Higuchi equation was modified by using the released amount,　 　the H-my equation
　　 The appricability was confirmed by using wax matrix tablet prepared from physical mixtures.
　　 　The simulated value fitted well with the entire release process.
　　　　

　　　　　For the commom case of εCs << A in the initial state.
　　　Q(=m/So) = { P ( 2A - εCs) Cs t }1/2 was simplified as:
　　　Q(=m/So) = ( 2 P A Cs t )1/2　,　　　　dQ/dt = ( P A Cs / 2 t )1/2 ,　　
　　　dm/dt = So ( P A Cs / 2 t )1/2　　　　　
　　　　　released amount of drug should be taken into account as:
　　　dm/dt = So { P(Mo- m) Cs / 2Vmt }1/2　　　　　　　　　　　　　　　　　　 A = Mo/ Vm ,　M = Mo- m
　　　-dM/dt = So ( PMCs / 2Vmt )1/2　　　
　　　( M/Mo )1/2= 1 - So ( PCs t / 2VmMo )1/2 　　
　　　simulation　　m= Mo[ 1 - { 1 - So ( PCs t / 2VmMo )1/2 }2 ]


　　　HOME of Yonezawa [JPN]，[ENG]　　　 PAGE for MEMORANDUM