Intravenous Infusion Model

There are some situations that necessitate the administration of drugs by intravenous infusion. Drugs with very sort half-lives are often administered via this route. Also, for comatose patients, the intravenous route of administration remains the only way of administering drugs.

Unlike most other convoluted PK models with continuous flow of time, the PK models used to predict drug levels consequent to intravenous infusion consist of discrete terms that account for infusion and post-infusion phases. This implies that for any such model two mathematical expressions will be needed to describe the entire course of plasma levels during both phases. Hence, the following expression may be used to determine the impact of missed doses(s) during the infusion phase.

The development of a PK model suited for intravenous must take into account some unique features of this kinetic situation. Unlike many other kinetic models, the intravenous infusion model may be considered as mixed-rate model. This assumption is based on the realization that most linear PK models consist of one or more than one exponential driving force, representing different simultaneous kinetic processes. The totality of these processes amounts to some sort of a continuous function. This is not the case with intravenous infusion models since they contain a zero-order input rate which concurrently proceeds with a first-order decay rate. The abrupt cessation of infusion brings the value of the input function of the model to a zero, implying the termination of a phase and resulting in a discrete quantity. Consequently, the post infusion phase will represent another distinct phase with its discrete quantities. Such considerations have not been avidly expressed in the conventional mathematical models that have been developed for the intravenous infusion dosing.

A schematic presentation for a one compartment intravenous infusion model could be described as follows:

Figure 2. Schematic presentations for intravenous infusion of a drug to the systemic circulation and its consequent elimination.

A differential expression describing the input or delivery of the drug to the systemic circulation and the elimination from the body could be described as: $$\frac{dX}{dt}=k_0-KX$$ where $k_0$ is a zero-order input rate, $K$ is an overall elimination rate constant and $X$ is the total amount of the drug in the body.

Solving Equation (1.9) for $X$ gives the following expression which describes the build-up of drugs levels in the body during infusion: $$X=\frac{k_0}{K}(1-e^{-Kt_i})\label{ref2.1}\tag{2.1}$$ This equation describe the change (increase) of the drug quantity during infusion $$X=\frac{k_0}{K}(1-e^{-KT_i})e^{-K(t-T_i)}\label{ref2.1a}\tag{2.1a}$$ where $t_i$ is the variable infusion time, $T_i$ is a time constant representing the duration of infusion and $(t-T_i)$is the post infusion time $t_{pi}$. $$C_p=\frac{k_0}{KV}(1-e^{-KT_i})e^{-Kt_{pi}}\label{ref2.1b}\tag{2.1b}$$ This model could be re-written in terms of two discrete quantities as follows: $$C_p=\frac{k_0}{KV_d}(1-e^{-Kt_i})+\frac{k_0}{KV_d}(1-e^{-KT_i})e^{-Kt_{pi}}\label{ref2.1c}\tag{2.1c}$$ where $k_0$ is the infusion rate, $t_i$ is a variable infusion time and $T_i$ is a constant time value representing the length of the infusion period. Other parameters where defined earlier.

After collation of terms in Equation (\ref{ref2.1b}) could be also expressed as: $$C_p=\frac{k_0}{KV_d}(1-e^{-Kt_i})+\frac{k_0}{KV_d}(e^{-K(t-T_i)}-e^{-Kt})\label{ref2.1d}\tag{2.1d}$$ It must be noted that the infusion time $t_i$ varies during infusion, until it reaches a constant value $T_i$ which equals the entire infusion time.

Equations (1.9a) through to (1.9d) could be used to generate plasma conc.-time profiles during and post-infusion periods. Plasma levels during the infusion phase could be expressed by the first discrete term of Equation (\ref{ref2.1}) which given as: $$C_p^{t_i}=\frac{k_0}{KV_d}(1-e^{-Kt_i})\text{, and }C_p^{ss}=\frac{k_0}{KV_d}\label{ref2.1e}\tag{2.1e}$$

According to this expression, when the exponent within the bracketed term in this expression become insignificant as time increases, hence, the steady state level becomes equivalent to $k_0/KV_d$. It is obvious that the fraction of this level is determined by the value of the exponent $e^{-Kt_i}$in the equation that determines the plasma levels during infusion.

In order that further verification to assumption made above, the following table has been generated to demonstrate the relationship between the fraction of the $C_p^{ss}$, the biological half-life and the infusion time expressed in terms of the number of half-lives.

PK of Intermittent Infusion

A new approach for the development of a one-compartment multiple dosing model, for drugs administered by IVI, could be contemplated on the basis of Equation (\ref{ref2.1a}) provided above. It is evident that this equation represents a combination of two equations that are mathematically discrete. This is due to the fact that its first term is a mixed-order rate since it contains a zero-order input, as well as a first-order decay function; whereas second term represents a first-order decay term. Accordingly, these functions have to be treated differently from a temporal standpoint.

The second part of Equation (\ref{ref2.1a}) could be readily transformed into a multiple dosing expression that accounts for the increase in the plasma levels in the post-infusion phase upon repetitive dosing. This is done by multiplying it by the accumulation ratio $(1-e^{-NK\tau})/(1-e^{-K\tau})$. However, a different approach must be adopted for the transformation of the first term of this equation so that it accounts for the plasma profile during the infusion phase. It has been acknowledged that the ascending part of the plasma profile during intermittent infusion consists of a standard quantity produced by the term $k_0/KV(1-e^{-Kt_i})$in addition to the plasma levels emanating from minimum plasma levels of the previous infusion up to the cessation of the current infusion. This realization has been reflected in Gibaldi’s work as he and his co-workers have devised an intermittent infusion multiple-compartment model. However, since no such model for one-compartment has been given by them, the following model has been devised: $$C_p^N=\frac{k_0}{KV}(1-e^{-Kt_i})+\frac{k_0}{KV}(1-e^{-KT_i)}e^{-K(\tau-T_i)}e^{-Kt_i} \left( \frac{1-e^{-(N-1)K\tau}}{1-e^{-K\tau}}\right)\text{ for }t \leq T_i$$ $$C_p^N=\frac{k_0}{KV}(1-e^{-Kt_i})\left( \frac{1-e^{-NK\tau}}{1-e^{-K\tau}}\right)e^{-Kt_{pi}}\text{ for }t \geq T_i\label{ref2.2}\tag{2.2}$$ A simplified version of this model is also presented hereunder and will be used throughout the present study. $$C_p^N=\frac{k_0}{KV(1-e^{-K\tau})} \left[ 1-e^{-Kt_i} \left\{1-e^{-KT_{pi}}+e^{-NK\tau}(e^{KT_i}-1)\right\}-e^{-K\tau}\right]\text{ for }t \leq T_i$$ $$C_p^N=\frac{k_0(1-e^{-NK\tau})}{KV(1-e^{-K\tau})}(1-e^{-KT_i})e^{-Kt_{pi}}\text{ for }t \geq T_i\label{ref2.3}\tag{2.3}$$

The interesting characteristic of this model is its profiling capability during the entire infusion phase. This is expressed by its exponential driving force represented by the expression $1-e^{-Kt_i}$ which causes the increase in plasma levels until a SS level $k_0/KV(1-e^{-K\tau})$ has been reached. These ascending levels are determined by the interplay of this driving force and the rest of constant values of all other exponential terms contained in the square brackets in Equation (\ref{ref2.3}). It evident that this equation may be readily transformed into an expression which account for plasma levels under SS situations through the cancellation of the term associated with the expression $NK\tau$, hence, $$C_p^{ss}=\frac{k_0}{KV(1-e^{-K\tau})} \left[ 1-e^{-Kt_i} (1-e^{-KT_{pi}})-e^{-K\tau} \right] \text{ for }t \leq T_i$$ $$C_p^{ss}=\frac{k_0}{KV(1-e^{-K\tau})} (1-e^{-KT_i})e^{-Kt_{pi}}\text{ for }t \geq T_i\label{ref2.4}\tag{2.4}$$

Model Verification by Superposition

As explained in the PK of multiple dosing, the principle of superposition may be regarded as the most basic and rugged technique for testing the validity of profile generated by different mathematical models.

The above model has been verified against the most basic drug accumulation principle which is the superposition principle.

Mathematical Model Verification

Results obtained with this new model could be also demonstrated to be consistent with the continuous infusion model by setting the infusion time $T_i$ equivalent to the dosing interval $\tau$. In this case the second term in the above expression will be cancelled since there will be no post infusion phase. Hence, the above expression becomes as follows: $$C_p^{T_i=\tau}=\frac{k_0}{KV_d}(1-e^{-K\tau})+\frac{k_0}{KV_d}(1-e^{-K\tau})e^{-K(\tau-\tau)}\left( \frac{1-e^{-(N-1)K\tau}}{1-e^{-K\tau}} \right)e^{-K\tau}$$ Further simplification of the above equation gives the following: $$C_p=\frac{k_0}{KV_d}(1-e^{-K\tau}+e^{-K\tau}-e^{-NK\tau})=\frac{k_0}{KV_d}(1-e^{-NK\tau})$$ This is a similar expression to that given in (\ref{ref2.1a}) which will ultimately give the same steady state levels provided that the same PK metrics are used in either situation. However, the initial part of the plasma conc.-time profile will consist of discrete quantities until the steady level has been attained. The above expression may be re-written to account for the changes in the plasma levels during any dosing interval, hence, $$C_p=\frac{k_0}{KV_d}(1-e^{-NK_{t_{0-\tau}}})\text{, and }C_p=\frac{k_0}{KV_d}$$ However, it should be emphasized that this relationship is meant to show that there exists a term or an entity in the full model that can generate continuous plasma levels under intermittent infusion conditions.

Another approach has been attempted to mathematically verify the validity of this model. This approach is based upon the axiomatic realization related to area estimations. It is universally accepted that the area under the plasma conc.-time curve resulting after the administration of a single dose, from time zero to infinity, is equivalent to that measure between two consecutive doses at steady state, i.e. $$\int_0^\infty C_p^{N=1}\,dt=\int_0^\tau C_p^{ss}\,dt \Rightarrow \tau \geq T_i$$

Average Plasma Concentration ($C_{av}$)

The average plasma concentration at different dosing interval could be defined upon the integration of the first equation $C_{p1}$. Due to its therapeutic efficacy, the average plasma level is more important than just defining $C_{max}$ and $C_{min}$. $$AUC_{0-\tau}^{ss}=\frac{k_0}{K^2 V}\left[ -(1-e^{-KT_i})+1-(e^{-K(\tau-T_i)}-e^{-K\tau})\left( \frac{e^{-KT_i}-1}{1-e^{-K\tau}} \right) - (1-e^{-KT_i}) \left( \frac{e^{-K\tau}-e^{-KT_i}-1}{1-e^{-K\tau}} \right) \right]$$ $$AUC_{0-\tau}^{N}=\frac{k_0}{K^2 V_d}(KT_i+e^{-KT_i}-1)-(C_{min1}F_I(e^{-KT_i-1}))+((C_{max1}-C_{min1})F_{PI})$$ $$C_{av}=\frac{k_0}{\tau K V_d} \left( T_i+ \frac{e^{-KT_i}}{K}-frac{1}{K}\right)- \left( \frac{C_{min1}F_I}{\tau K} (e^{-KT_i}-1) \right)+\left( \frac{(C_{max1}-C_{min1})F_{PI}}{\tau K}\right)$$ where $C_{max}$ and $C_{min}$ are the respective maximum and minimum and plasma concentration attained at the end of the first infusion period and the end of the first dosing interval, $F_I$ and $F_{PI}$ at are the multiple dosing factor during the infusion and post infusion periods.

Targeting Therapeutic Plasma Levels

Provided that a definition for a therapeutic window is available, it is quite possible to determine a dosing rate and frequency that will generate the desired plasma concentration range.

Different approaches are provided hereunder:

Using $C_{max}^{ss}$ and $C_{min}^{ss}$

One can determine a dosing rate (infusion rate and a dosing interval) if a certain plasma concentration range is to be attained. This requires a prior definition of maximum and minimum plasma levels. The mathematical considerations for this objective are provided hereunder: $$\tau_{new}=T_i-\frac{1}{K} \ln \frac{C_{min}^{ss}}{C_{max}^{ss}}$$ If new maximum and minimum steady state plasma levels are to be targeted, while the infusion time is kept constant, a new dosing interval has to be estimated according as described hereunder.

The calculated dosing interval $\tau_{new}$ is then used to determine the infusion rate that will produce the desired targets by two ways: $$C_{max}^{ss}=\frac{k_{0L}}{VK}(1-e^{-KT_i})\left( \frac{1}{1-e^{-K\tau}}\right)$$ $$k_{0T}=\frac{C_{max}^{ss}V_d K (1-e^{-K\tau_{new}})}{1-e^{-KT_i}}$$ Alternatively, the similar values could be obtained for the infusion rate on the basis of the minimum plasma levels as per the following equalities: $$C_{min}^{ss}=\frac{k_{0L}}{Vk}(1-e^{-KT_i})\left( \frac{1}{1-e^{-K\tau}}\right)e^{-K(\tau_{new}-T_i)}$$ $$k_{0T}=\frac{C_{min}^{ss}V_d K (1-e^{-K\tau_{new}})}{e^{-K(\tau_{new}-T_i)}(1-e^{-KT_i})}$$ An infusion rate estimated according to either of the above equations will exactly give the same value.

Loading Dose for Intermittent Infusion

Loading dose consideration assumes critical significance for drug with relatively long biological half-life. This is related to the fact that the attainment of steady state levels for such drugs will take a long time. It is equally important in case of severe impairment of the drugs elimination processes. Estimates of the loading dose are generally based on drugs accumulation ratio $AR$ expressed as: $C_{0-\tau}^{ss}/C_{0-\tau}^{N=1}$. This is based on a gross generalization with regard the estimation of area after a single infusion and that are obtained at steady state between two consecutive doses.

These area related assumptions are true in cases of intravenous and oral bolus administrations of drugs. Hence, estimates for the loading dose such as $X_L = AR \times X_0$ where suggested and successfully employed. This approach has been demonstrated to be suitable for the intravenous infusion dosing and further extrapolated to the intermittent infusion. However, the administration of an intravenous bolus loading dose estimated as $X_L = C_p^{ss} V_d$ or $X_L = k_0/K$ as suggested by the entirety of the relevant PK literature.

As an alternative approach to that suggested by Gibaldi, the loading dose could be better defined as $X_L = C_{min}^{ss} V_d$, where $C_{min}^{ss}$ could be defined as the minimum steady state plasma level. This suggestion is based on the intuition that the administration of a bolus dose, estimated on $C_{max}^{ss}$ or even the $C_{av}^{ss}$ plasma levels at steady state, together with the initiation of infusion, higher levels that such steady state levels are bound to occur. It has been demonstrated that the simultaneous administration of an intravenous bolus dose with the predefined infusion regimen has produced steady state levels immediately with the initiation of infusion if the loading dose was determined as $C_{min}^{ss}V_d$.

Another intravenous infusion loading approach, similar to that attempted by Wagner, could be contemplated in situations where gradual instillation or infusion of the dose is favored. However, the direct implementation of Wagner’s approach, as detailed for the continuous infusion has resulted in initial plasma levels that were higher than the predicted steady state levels. A new method could be suggested for the estimation an intravenous infusion loading dose (or rate). This method is based on the theoretical estimation of the expected maximum steady state plasma level $C_{max}^{ss}$ for any infusion situation. This is generally expressed as: $$C_p^{ss}=\frac{k_0(1-e^{-KT_i})}{KV_d(1-e^{-K\tau})}$$ where $k_0$ is the regular infusion rate used for the post infusion loading phase.

This quantity could be then used to estimate the infusion loading rate $k_{0L}$ for any predefined infusion time $t_{iL}$ that will be enough to attain this plasma levels. Once such level has been reached, after the drug has been infused at this rate $k_{0L}$ for any infusion period $T_{iL}$, the will have been reached. $$k_{0L}=\frac{C_p^{ss}KV_d}{1-e^{-KT_{iL}}}$$ After that the infusion rate is switched to the regular rate $k_0$ which will be used throughout the intermittent dosing regimen after the cessation of the infusion loading.

The validity of this procedure could be verified for any dosing interval during the intermittent infusion. Noteworthy, it has been suggested (Milo Gibaldi) that a loading dose may be determined as: $$X_L=\frac{k_0 T_i}{1-e^{-K\tau}}$$

I.V. Bolus Loading and Simultaneous Infusion

As detailed above, in instances with drug substances have a relatively long biological half-life, the attainment of a steady state concentration will take considerable length of time. This may not of benefit where the therapeutic efficacy is associated with drugs steady state plasma level. Hence, a loading dosing should be considered. This dose must be of such magnitude so that it produced the steady state plasma level at the initiation of the dosing regimen. This can only be achieved if the loading dose is administered by an intravenous bolus mode together with the initiation of infusion.

For any defined infusion criterion, the value of the steady state level could be estimated as $k_0 / KV_d$. Consequently, the value of the intravenous loading dose could be obtained by multiplying this quantity by the volume of distribution of the drug, i.e. $k_0\times V_d / KV_d$ or simply $k0 / K$. Under these circumstances, it could be verified that the resulting plasma level will be identical to the steady state level right from the initiation of infusion and will be maintained at this level throughout the infusion period. This could be verified as follows:

One of the prime characteristics of the PK linear model is their additive nature. This implies that if more than one dose have been simultaneously administered via different routes, such as intravenous and oral, the resulting plasma levels from one model could be summed up to those ensuing from the other model.

I.V. Infusion Loading Followed by Maintenance Infusion

The loading approach explained above may not be feasible or has its limitations. This is due to the fact that the intravenous bolus administration may cause rapid highly plasma levels that may entail the occurrence of unwarranted adverse reactions. In such cases an alternative loading approach must be contemplated. It has been suggested that the dosing regimen consists of two consecutive infusions with different infusion rates $k_{01}$ and $k_{02}$. Wagner has suggested that once the maintenance infusion criteria has been defined, the steady state plasma levels could be determined as $k_{02}/KV_d$. The maintenance infusion rate $k_{02}$ is doubled and the drug is infused over a period of one biological half-life. Doubling the maintenance infusion rate is based on the realization that, for any drug and/or any infusion criteria, it usually takes one biological half-life to attain 50% of the steady state plasma levels. Hence, 100% of such level will be reached with the same if the infusion rate is doubled.

The above approach could be applied for drug with relatively short biological half-lives (less than 2 or 3 hours). However, for drug with much longer half-lives, or drugs with severely impaired elimination characteristics due to kidney failure or hepatic dysfunction, the Wagner’s approach becomes unattractive.

As an alternative to the above approaches, it is feasible to determine the rate of the first infusion according to any chosen infusion time. Once the infusion period has been selected, the first infusion will be automatically estimated. The estimation of the infusion is lift flexible to the practitioner to decide upon according to the case at hands. This approach is better illustrated by examining the following mathematical considerations. Rearrangement of Equation (1.9b) for the infusion phase to estimate the loading infusion rate gives the following expression: $$k_{01}=\frac{C_p^{ss}V_d K}{1-e^{-KT_{inf(01)}}}$$ where $C_p^{ss}$ is the target plasma level associated with the maintenance infusion rate and $T_{inf}$ is the length of the loading infusion time and $C_{p(01)}^{ss}=k_{01}/V_d K$ represents the steady state level had the drugs been administered with a continuous loading infusion at $k_{01}$ rate. Hence, plasma levels resulting from the loading could be estimated according to the following equation: $$C_{p(01)}=C_{p(01)}^{ss}(1-e^{-Kt_{inf(01)}})$$ This clearly signifies that the maximum plasma levels during the first infusion period will not exceed the planned or targeted steady state levels associated with the maintenance.

For PK-naïve practitioners, an infusion loading of 1400 mg/hours seems to be a huge rate if compared by the maintenance infusion rate of 40 mg/hour. However, as demonstrated in the above, the plasma levels under extremely varying scenarios remained within the controls specified for the particular dosing regimen.