Sustained Release Dosage Forms
The PK of drugs presented in controlled or sustained release forms has not received adequate attention in the scientific literature. In this regard, although enteric dosage forms represent some kind of controlled release forms, they will not be subject to the discussion in this section. This is due to the fact that they only differ from the conventional dosage form by releasing its content of the drug substance after a certain prescribed period of time. In which case, a slight modification for the time element in the one-compartment oral model should be enough to describe it PK features. This amounts to simply subtracting the time at which the dosage form starts to release its content \(t_{lag}\). Hence, $$C_p=\frac{K_a F X_0}{V_d (K_a - K)} \left( e^{-K(t-t_{lag})} - e^{-K_a(t-t_{lag})} \right)$$ There are situations where there is a definitive need to instill a drug substance gradually into the systemic circulation. This is mainly encountered with drugs having relatively short half-life where desirable plasma levels could not be sustained. Also, a gradual input of the drug into the system is often preferred in drugs with rapid absorption property which may result in abrupt increase of the drugs level in the system; thus causing unwarranted adverse effect.
However, for other sustained release (SR) forms where the drug is gradually released at a predetermined rate, the situation is completely different from the enteric coated dosage forms. In this case, the native absorption characteristic of the drug will be masked or confounded by the release pattern of the drug from the dosage form. It is generally accepted that an ideal or optimal SR form must release it content in an approximate zero-order fashion. This situation may be exemplified by what has been previously discussed for the intravenous infusion mode of administration. Consequently, the release time of the drug substance from the dosage form could be conveniently considered as some kind of infusion time or period. Hence, if the release time is made available, the mathematical models introduced to account for the intravenous infusion, could be slightly modified so that the PK behavior of the SR forms is adequately catered for.
To start with the basic one-compartment intravenous infusion model may be re-written in terminology suited for the SR forms as follows: $$C_p=\frac{F X_{SR}}{T_r K V_d} ( 1- e^{-K t_r} ) e^{-K (t-T_r)}$$ where \(F X_{SR}/T_r\) is the release rate if the active drug from the SR matrix, \(t_r\) is the variable release time, \(T_R\) is a time constant representing the duration of release, \((t-T_R)\) is the post release time \(t_{pr}\) and \(K\) is the first order overall elimination rate constant.
This model is suited for drugs that confer upon the body the characteristic of single compartments. It could be also argued that it could be also suited for multiple compartment drugs since not only the absorption, but also the distribution, characteristics will be masked by the zero order release pattern from the SR form.
It must be noted that the conventional absorption rate constant \(K_a\) have been completely dropped from the model since the release pattern of the drug from the dosage from has become the rate-limiting step in the entire absorption process. The above model could be expressed in terms of two discrete entities, with one accounting for the release period and the other for the post-release time. Hence, $$C_p=\frac{F X_{SR}}{T_r K V_d} ( 1- e^{-K t_r} )\text{, for }t \leq T_r$$ $$+ \frac{F X_{SR}}{T_r K V_d} ( 1- e^{-K T_r} ) e^{-K (t-T_r)}\text{, for }t \geq T_r$$ This equation may be used to predict drugs plasma levels consequent to the administration of a single dose. In this case, information on the claim made about the release time must be known so that drug’s input rate is inserted into them. They could be also used for the characterization of the elimination properties where experimental plasma conc.-time data are provided. However, is must be emphasized that the slope of the terminal segment of such data should be estimated after the release time has elapsed by at least three absorption half-lives. Unfortunately, this is not how plasma data obtained consequent to the administration sustained release for is presently evaluated within the context of bioequivalence studies.
For drugs with relatively long biological half-lives, a steady state plasma level will be attained after a lengthy period of time. Hence, a loading dose has to be administered together with the initiation of therapy. For the estimation of such dose, either of the equations used to estimate the maximum, minimum or the average plasma level at steady state could be employed since all these equations will give the same value. It has been always assumed that a steady state level with no fluctuation is an ideal situation which can only be observed if the drug has been administered by intravenous infusion.
Multiple Dosing of SR Forms
Like other dosage forms, the SR forms are usually administered as multiple dosing regimens. Hence, plasma levels at any number of doses as well as at steady state become of direct concern. These include the maximum, minimum and the average levels throughout the dosing regimen. To account for these levels, the above model could be adopted to account for the repeated dosing of SR drug products in a similar way to that of the intermittent infusion. This requires the incorporation of the right multiple dosing factors for the release and post release phases. $$C_p=\frac{F X_{SR}}{T_r K V_d} ( 1- e^{-K t_r} ) + \frac{X_{SR}}{T_r K V_d} ( 1- e^{-K T_r} ) e^{-K (\tau-T_r)} e^{-K t_r} \left(\frac{1-e^{-(N-1)K\tau}}{1-e^{K\tau}}\right )\text{, for }t < T_r$$ $$+ \frac{F X_{SR}}{T_r K V_d} ( 1- e^{-K T_r} ) e^{-K (\tau-T_r)} \left(\frac{1-e^{-NK\tau}}{1-e^{K\tau}}\right ) \text{, for }t \geq T_r$$ Further simplification to the above model results in the following expression: $$C_{p(r)}^N=\frac{F X_{SR}}{T_r K V_d(1-e^{-K\tau})} \left( 1-e^{-K t_r}[1-e^{-K(\tau-T_r)}+e^{-K(N\tau-T_r)}-e^{-KN\tau}]-e^{-K\tau}\right)\text{ for }t \leq T_r$$ $$C_{p(pr)}^N=\frac{F X_{SR}}{T_r K V_d(1-e^{-K\tau})} (1-e^{-KT_r})(1-e^{-NK\tau})e^{-Kt_{pr}} \text{ for }t \geq T_r$$ Which could be readily modified to account for plasma level at SS condition, as provided hereunder: $$C_{p(r)}^N=\frac{F X_{SR}}{T_r K V_d(1-e^{-K\tau})} \left( 1-e^{-K t_r}[1-e^{-K(\tau-T_r)}+e^{-K(N\tau-T_r)}-e^{-KN\tau}]-e^{-K\tau}\right)\text{ for }t \leq T_r$$ $$C_p^{ss}=\frac{F X_{SR}}{T_r K V_d(1-e^{-K\tau})} \left( 1-e^{-Kt_r}[1-e^{-KT_{pr}}]-e^{-K\tau}\right) \text{ for }t \leq T_r$$ $$C_p^{ss}=\frac{F X_{SR}}{T_r K V_d(1-e^{-K\tau})} (1-e^{-KT_r})e^{-Kt_{pr}} \text{ for }t \geq T_r$$ Although the analogy between this model and that used for intermittent infusion is unmistakable, its output may vary with different dosing regimens of the SR release forms. Such variability is related to compliance of these forms with the compendial requirements as well as the compliance of patients with the intake schedule as specified by the dosing regimen (DR). Accordingly, three specific scenarios for the SS conditions could be contemplated. These are outlined as follows: In cases where the dosage form releases its contents in a period that exactly equals the claimed release time in the presence of strict compliance with the dosing schedule. This implies a continuous infusion condition, in which case the above equation reduces to the expression: $$C_p^{ss}=\frac{F X_{SR}}{T_r K V_d} \text{ for }\tau=T_r$$ $$C_p^{pr}=\frac{F X_{SR}}{T_r K V_d}e^{-Kt_{pr}} \text{ upon the completion of DR }$$ In another instance, doses are administered during the release period (i.e. \(\tau < T_r\)), which will result in positive values of the post-release time, i.e. \(\tau-T_r=T_{pr}\) will be more than zero. This implies that higher SS average plasma level than that where the dosing interval equals the release time since the exponent associated with this parameter will serve as an input function.
Finally, in cases where consequent doses are administered after the release period (i.e. \(\tau > T_r\)), the post-release time \(T_{pr}\) will be less than zero. Contrary to the above instance, the SS average plasma level will be lower than that when the dosing interval equals the release time.
These two instances are operationally equivalent to other situations in the presence of patient compliance and varying release rates inconsistent with the label claim. It is evident that the value of \(T_{pr}\) is the critical element that will eventually determine the magnitude of the SS plasma levels since its exponential term will be either an input or a decay function.
Average Plasma Levels for SR Forms
The above equation could be integrated from time zero to the end of the dosing interval to get an estimate for body exposure after and number of infusions throughout the dosing regimen. It could be verified that dividing such exposure by the dosing interval provides an estimate of the average plasma levels at any dosing interval: $$C_{av}^N=\frac{FX_{SR}/T_r}{\tau K V_d} \left( T_r+\frac{e^{-KT_r}}{K}-frac{1}{K}\right) - \frac{C_{min}^{N=1}F_r}{\tau K} \left( e^{-KT_r}-1\right)+\left( \frac{C_{max}^{N=1}-C_{min}^{N=1}}{\tau K}\right)F_{pr}$$ The above equations have been developed with the aim of estimating the average plasma levels during repeated dosing of the SR forms. The multiple dosing factors (\(F_I\) and \(F_{PI}\)), as well as the plasma levels at steady state contained in this equation are defined as: $$\text{Assume that }C_{av}^N=\frac{FX_{SR}/T_r}{\tau K V_d}\text{, }\frac{1-e^{-(N-1)K\tau}}{1-e^{-K\tau}}=F_r\text{ and }\frac{1-e^{-NK\tau}}{1-e^{-K\tau}}=F_{pr}$$ Although the above model could be suited for estimating drugs plasma levels at any time during multiple dosing, a special case could be considered whereby the dosing interval is set identical to the claimed release time of the drugs from the SR matrix. This could be conceptually thought of as being a case of continuous release pattern resulting in a zero-order input rate in the system. Under such condition, and with continued dosing, either of the two multiple dosing factors, that are cited above, will be reduced to the expression \(F_I=1/(1-e^{-K\tau})\).
Furthermore, considering that the post release term in Equation (\ref{ref11}) will be irrelevant during repeated dosing where the dosing interval is considered to be identical with the release time, this equation reduces to the following: $$C_{av}^N=С_{ss}\left( 1+ \frac{e^{-KT_r}}{\tau K}-\frac{1}{\tau K}\right)-\frac{C_{min}^{N=1}F_r}{\tau K}\left( e^{-KT_r}-1\right)\label{ref11}\tag{11}$$ Also, recalling that the minimum plasma level after the first dose \(C_{min}^{N=1}\) has been earlier defined as \(C_{ss}(1-e^{-KT_R})e^{-K(\tau-T_R)}\), hence substitution for this value and the multiple dosing factor \(F_I\) as defined above, in Equation (\ref{ref11}) the average concentration at any number of doses as: $$C_{av}^N=С_{ss}\left( 1+ \frac{e^{-KT_r}}{\tau K}-\frac{1}{\tau K}\right)-\frac{C_{ss}(1-e^{-KT_r})e^{-K(\tau-T_r)}}{\tau K}\left( \frac{1-e^{-(N-1)K\tau}}{1-e^{-K\tau}}\right)(e^{-KT_r-1})$$ However, for orally administered SR forms, the fluctuation of steady state plasma levels depends mainly on whether the release of the active drug material from dosage form is zero order or not. A zero order release rate could be hardly guaranteed specially during the first few hours after the administration of the dosage form. Consequently, the guidance has set rather relaxed limits (15 - 30%) for the percent release during the first hour.
In addition, it has been a common practice by too many drug manufacturers to incorporate an immediate release component in the SR form. It has been argued (Gibaldi) that it will take less time to attain the steady state levels in the presence of such component. Although this could be true only for drugs with relatively short biological half-lives. However, such component would seriously affect the peak to trough fluctuation at steady state. Since the latter property of the dosage form represents a significant quality aspect of the SR preparation, it has been stated as a quality measure by the international guidance.
It remains imperative that either impact of usual early release of the drug substance from the SR form, or the incorporation of an IR component be accurately assessed. This could be done by the employment one of the additive characteristic of all linear PK models. Hence, a third term could be added to the standard model presented in the equation provided hereunder to represent the absorption and disposition of the IR component. This will result in the following elaborate model that could be used to precisely predict drugs plasma levels attained consequent to the administration of the combined SR/IR forms. $$C_p^N=\frac{FX_{SR}/T_r}{KV_d}\left[ (1-e^{-Kt_r})+(e^{-K(\tau-T_r)}-e^{-K\tau})e^{-Kt_r}\left( \frac{1-e^{-(N-1)K\tau}}{1-e^{K\tau}}\right)\right]\text{ for }t<T_r$$ $$+\frac{FX_{SR}/T_r}{KV_d} (1-e^{-KT_r})e^{-Kt_{pr}}\left( \frac{1-e^{-NK\tau}}{1-e^{K\tau}}\right)\text{ for }t\geq T_r=t_{pr}$$ $$+\frac{K_aFX_{IR}}{V_d(K_a-K)} \left( \frac{1-e^{-NK\tau}}{1-e^{K\tau}}e^{-Kt} \frac{1-e^{-NK_a\tau}}{1-e^{K_a\tau}}e^{-K_at} \right)\text{ for the IR Component}$$ The utilization of the above model requires a prior knowledge of the IR component or the amount of the dosage form that would be released within the first two hours after the administration of the dose. Also, the release claim related to any specific dosage form must be stated so that the release (or the input) rate of the drug is determined.
The flexibility of this model is manifested by the fact that the dosing interval does not have to be necessarily set identical to the release time of the drug from the dosage form. On the contrary, the dosing interval could be set at any value so that a desired plasma profile or a predefined therapeutic window is achieved.
Plasma Levels for SR Dosage Forms
Maximum and minimum plasma levels at SS are generally employed as basis for targeting therapeutic ranges or windows. Hence, estimates of \(C_{min}\) and \(C_{max}\) at SS for SR formulations, in the presence of IR components, must be determined. These levels represent a summation of levels ensuing from IR and SR portions of such dosage forms at the time where maximum levels are likely to occur. For either IR or SR forms, the following equations describe the maximum and minimum plasma levels after a single dose based on the assumption that the release time from the dosage form \(T_r\) is equal to the dosing interval \(\tau\). It may be further assumed that maximum levels will only occur at \(T_r\). In addition, the concept of minimum and maximum concentration is irrelevant for the SR portion. Based on such assumptions, the following set of equation are devised:
For \(C_{min}\) and \(C_{max}^{N=1}\) $$C_{min}^{N=1}=\frac{K_a FX_{IR}}{V_d (K_a-K)}e^{-KT_r}\text{ from Oral Part}$$ $$C_{max}^{N=1}=\frac{FX_{IR}}{V_d}e^{-KT_{max}^{ss}}\text{ from Oral Part}$$ $$C_p^{N=1}=\frac{FX_{SR}}{T_rKV}(1-e^{-KT_r})\text{ from SR Part}$$ For \(C_{min}\) and \(C_{max}\) (\(N\) Doses) $$C_{max}^{N=1}=\frac{FX_{IR}}{V_d}e^{-KT_{max}}\left(\frac{1-e^{-NKT_r}}{1-e^{-KT_r}}\right)\text{ from Oral Part}$$ $$C_{min}^{N=1}=\frac{FX_{IR}}{V_d}e^{-KT_r}\left(\frac{1-e^{-NKT_r}}{1-e^{-KT_r}}\right)\text{ from Oral Part}$$ $$C_p^N=\frac{FX_{SR}}{T_r KV}\left(1-e^{-NKT_r}\right)\text{ from SR Part}$$ All the above equations become SS by setting all exponents containing \(N\) to zero. The same applies to all absorption exponents containing tau since such exponents will approximate zero.
Considering the above equation, it becomes evident that targeting plasma levels for SR in the presence of an IR component is not a straight forward matter. This further implies that targeting can never have mathematical basis if it is to be approached though a modified tau as is the case with other dosage forms. Hence, it could be only done by estimating the difference between the newly targeted \(C_{max}\) and \(C_{min}\). It should be born in mind that fluctuation in the plasma levels is strictly caused by the IR component. Accordingly the following mathematical treatment is applicable
For \(C_{max}\) & \(C_{min}\) (at SS)
Assuming that the dosing interval was set at exactly the release time \(T_r\), then there will not be a maximum and minimum levels ensuing from the SS portion of the dosage form. Rather we will have a steady state level only. Hence, the following applies: $$C_{min}^{ss}=\frac{FX_{IR}}{V}\left(\frac{e^{-KT_{max}^{ss}}}{1-e^{-KT_r}}\right)e^{-K(T_r-T_{max}^{ss})}\text{ from Oral Part}$$ $$C_{max}^{ss}=\frac{FX_{IR}}{V}\left(\frac{e^{-KT_{max}^{ss}}}{1-e^{-KT_r}}\right)\text{ from Oral Part}$$ $$C_p^{ss}=\frac{FX_{SR}}{T_r KV}\text{ from SR Part}$$ Accordingly, the overall maximum and minimum plasma levels may be given as follows: $$C_{max}^{ss}=\frac{FX_{SR}}{T_r KV}+\frac{FX_{IR}}{V} \left(\frac{e^{-KT_{max}^{ss}}}{1-e^{-KT_r}}\right)$$ $$C_{min}^{ss}=\frac{FX_{SR}}{T_r KV}+\frac{FX_{IR}}{V} \left(\frac{e^{-KT_r}}{1-e^{-KT_r}}\right)$$ This is a paradoxical expression since it appears to contradict those cited above for the estimation of maximum and minimum plasma levels. However, it is based on the realization that the contribution of the IR component to the ascending plasma levels to the SS condition is quite different from that once SS levels have been attained. It is evident that the SR portion has the overriding impact compared to the contribution of the IR component. This will be reversed once SS has been achieved. The result obtained from this equation will be either a fraction of a multiple of unity which will be readily transformed into concentration terms by multiplying it by the value of \((X_{IR} + X_{SR})/V_d\).
Targeting Therapeutic Ranges
For the purpose of targeting plasma ranges in case of SR dosage forms, the dosing interval should not be equated with the release time of drugs from such forms since the latter must be assumed as a constant, representing a definitive characteristic of the dosage form. Accordingly, the above expressions for \(C_{min}\) and \(C_{max}\) may be re-written as follows: $$C_{max}^{ss}=\frac{FX_{SR}}{T_r KV}+\frac{FX_{IR}}{V} \left(\frac{e^{-KT_{max}^{ss}}}{1-e^{-K\tau}}\right)$$ $$C_{min}^{ss}=\frac{FX_{SR}}{T_r KV}+\frac{FX_{IR}}{V} \left(\frac{e^{-K\tau}}{1-e^{-K\tau}}\right)$$ This equation may be re-written in terms of the maintenance dose with the IR portion represented as a fraction of such dose. $$C_{max}^{ss}=\frac{FX_0 (1-P)}{T_r KV}+\frac{FX_0}{V} \left(\frac{e^{-KT_{max}^{ss}}}{1-e^{-K\tau}}\right)$$ $$C_{min}^{ss}=\frac{FX_0 (1-P)}{T_r KV}+\frac{FX_0}{V} \left(\frac{e^{-K\tau}}{1-e^{-K\tau}}\right)$$ \(P=IR\) fraction $$C_{max}^{ss}=\frac{FX_0}{V}\left( \frac{1-P}{T_r K}+\frac{Pe^{-KT_{max}^{ss}}}{1-e^{-K\tau}}\right)$$ $$C_{min}^{ss}=\frac{FX_0}{V}\left( \frac{1-P}{T_r K}+\frac{Pe^{-K\tau}}{1-e^{-K\tau}}\right)$$ Divide \(C_max^ss\) to \(C_min^ss\) and we get $$\frac{C_{max}^{ss}}{C_{min}^{ss}}=\frac{\frac{1-P}{T_r K}+\frac{Pe^{-KT_{max}^{ss}}}{1-e^{-K\tau}}}{\frac{1-P}{T_r K}+\frac{Pe^{-K\tau}}{1-e^{-K\tau}}}=\frac{(1-P)(1-e^{-K\tau})+PT_rKe^{-KT_{max}^{ss}}}{(1-P)(1-e^{-K\tau})+PT_rKe^{-K\tau}}=\frac{(1-P)+PT_rKe^{-KT_{max}^{ss}}-(1-P)e^{-K\tau}}{(1-P)-(1-P-PT_rK)e^{-K\tau}}$$ Thus we get result expression for this fraction $$\frac{C_{max}^{ss}}{C_{min}^{ss}}=\frac{(1-P)+PT_rKe^{-KT_{max}^{ss}}-(1-P)e^{-K\tau}}{(1-P)-(1-P-PT_rK)e^{-K\tau}}$$ Then we express term \(e^{-K\tau}\) $$C_{max}^{ss}[(1-P)-(1-P-PT_r K)e^{-K\tau} ]=C_{min}^{ss} [(1-P)+PT_r Ke^{-KT_{max}^{ss}} -(1-P) e^{-K\tau} ]$$ $$C_{max}^{ss}(1-P)+C_{max}^{ss} (P+PT_r K-1) e^{-K\tau}=C_{min}^{ss} [(1-P)+PT_r Ke^{-KT_{max}^{ss}} ]-C_{min}^{ss} (1-P) e^{-K\tau}$$ $$[C_{max}^{ss} (P+PT_r K-1)+C_{min}^{ss} (1-P)] e^{-K\tau}=C_{min}^{ss} [(1-P)+PT_r Ke^{-KT_{max}^{ss}} ]-C_{max}^{ss} (1-P)$$ Finally we get expression for \(\tau\). $$\tau =-\frac{1}{K} \ln\left[\frac{C_{min}^{ss} [(1-P)+PT_r Ke^{-KT_{max}^{ss}} ]-C_{max}^{ss} (1-P)} {C_{max}^{ss} (P+PT_r K-1)+C_{min}^{ss} (1-P)} \right]$$ The above equation may be solved to obtain the value of the dosing interval associated with the pre-defined (or user-defined) therapeutic range. This new tau may inserted in either of the two equations provided above for \(C_{min}\) or \(C_{max}\) to obtained the new maintenance dose that will ensure that plasma levels are contained within the desired plasma range.