A Multi-Compartment Model Capturing the Pharmacokinetics of the Calcimimetic Cinacalcet
Gudrun Schappacher-Tilpa Doris H. Fuertingerb Peter Kotankoc,d
aUniversity of Graz, Institute of Mathematics and Scientific Computing, Graz, Austria, bFresenius Medical Care Germany, Global Research and Development, Bad Homburg, Germany, cRenal Research Institute, New York, NY, USA, dIcahn School of Medicine at Mount Sinai, New York, NY, USA
Key Words
Cinacalcet • pharmacokinetics • mathematical model
Abstract
Background/Aims:Chronic kidney disease-mineral bone disorder is a major complication affecting the vast majority of chronic kidney disease patients. A hallmark of the disorder is an altered parathyroid gland biology resulting in secondary hyperparathyroidism. This condition is widely treated by calcimimetics like cinacalcet which act by allosteric activation of the calcium sensing receptor. Methods: Here, we present a linear multi-compartment model based on physiological principles such as first-pass metabolism and protein binding, which captures all relevant pharmacokinetic parameters of cinacalcet. Results: Due to the linear structure of the model, simulations are numerically stable and allow fast and accurate short or long-term predictions of cinacalcet concentrations in the body. Conclusion: The model compartments are physiological meaningful and can be easily adjusted to various conditions like impaired hepatic clearance or different drug administration regimens. Moreover, the model can be easily adapted to specific patient groups.
Introduction
Blood ionized calcium concentrations (Ca2+) have to be maintained within a very narrow range of around 2%;
a deviation from this range will result in serious health issues and if persistent in life threatening situations [1]. A key endocrine regulator for Ca2+ is the parathyroid hormone (PTH) [2-4].
PTH stimulates bone remodeling thereby ensuring a net release of calcium from the bone storage. Furthermore, it stimulates vitamin D conversion to its metabolic active form 1, 25-dihydroxyvitamin D3,
and it inhibits renal phosphate reabsorption while enhancing renal calcium reabsorption. The key regulator for the detection of changes in blood ionized calcium levels and the subsequent secretion of PTH
is the calcium-sensing receptor (CaSR) on the surface of parathyroid gland (PTG) cells [5-7]. If activated by calcium binding, the CaSR down-regulates PTH production and secretion as well as cell proliferation of PTG cells. In patients suffering from chronic kidney disease (CKD) the impairment of vitamin D metabolism as well as the impaired clearance of PTH and phosphate by the kidneys trigger a cascade of feedback loops resulting
eventually in chronic kidney disease-mineral bone disorder (CKD-MBD). CKD-MBD is a complex disorder affecting the vast majority of chronic kidney disease patients. It is associated with a variety of pathological
effects like renal osteodystrophy [8-10] or vascular calcification and an increased risk for cardiovascular events, the leading cause of mortality in CKD patients [11-14]. A hallmark of CKD-MBD is secondary
hyperparathyroidism, reduced expression of calcium-sensing receptor and parathyroid gland cell hyperplasia [15-18]. The 2017 KDIGO CKD-MBD Guideline Update suggests maintaining intact PTH levels in the range of approximately 2 to 9 times the upper normal limit for the assay [19].
One strategy to reach this goal is to target the CaSR. Calcimimetic drugs like cinacalcet or etelcalcetide enhance the interaction between the ionized calcium concentration (Ca2+) and the CaSR by allosteric activation [20].
The higher sensitivity of the CaSR to Ca2+ leads to an inverse relationship between plasma PTH and cinacalcet concentrations. PTH concentration declines after the administration of cinacalcet until it reaches a minimum
approximately 2-3 hours after dosing. Cinacalcet hydrochlorid, which was approved by the FDA in 2004, is widely used in hemodialysis patients [21] and has been listed under the 100 best selling pharmaceuticals world wide in the last years.
In addition to secondary hyperparathyroidism in patients with CKD on dialysis, cinacalcet is also approved for the treatment of hypercalcemia in patients with parathyroid carcinoma, and severe hypercalcemia
in patients with primary hyperparathyroidism who are unable to undergo parathyroidectomy. Its pharmacokinetics are well established in clinical studies [20, 22-26]. Following oral administration the plasma
concentration peaks within 2-6 hours. The absolute bio-availability is only between 20-25% while absorption is close to 100% indicating a high first-pass metabolism. Since almost 95% of the drug in the plasma
is protein bound, the affect of hemodialysis on the pharmacokinetics of the drug can be neglected [25]. The mechanisms leading to CKD-MBD are many and highly complex. Thus, a multitude of mathematical models were established in order to gain a better understanding of the multidimensional factors controlling the PTH household [27-29]
or PTH and bone remodeling [30]. For clinical application any proposed model should be able to predict the state of individual patients and ideally indicate an optimal treatment strategy.
Due to the vast clinical use of cinacalcet, a realistic model of CKD-MBD has to feature the use of cinacalcet and its effects. To be of clinical use the cinacalcet model should be readily
adaptable to various conditions, such as hepatic impairment enhancing cinacalcet exposure [31]. Moreover, it should be able to reflect different administration scenarios including patient
adherence which is known to be poor, due to gastrointestinal side effects and a general high pill load in this patient group [32]. Therefore, the aim of this study was to build a multi-compartment model based on physiological considerations capturing all major pharmacokinetics parameters of cinacalcet.
Moreover, the model should allow an intuitive individualization to various conditions or administration regimens and omit any numerical instabilities in order to be easily
combined with or incorporated to other physiological models.
Materials
and Methods Pharmacokinetics The low bio-availability of 20-25% indicates a high first pass metabolism. Hepatic impairment results in higher terminal half-life values
(40 - 70% for moderate to severe impairment) and higher total exposure (2.4 old to 4.2-fold higher in patients with moderate to severe impairment) [31]. Administration with
food results in higher bio-availability and consequently higher peak concentrations and total exposure values [32]. Around 95% of the drug is protein bound. Therefore, the
effect of hemodialysis on the pharmacokinetics is negligible [26]. The high volume of distribution at steady state of 1000 liters indicates a vast distribution outside the systemic circulation [33].
Experiments with radioactive labeled cinacalcet have shown that the drug peaks at different times in different tissues. Roughly, tissue can be lumped into two groups:
one group with short time to peak and one group with longer time to peak operating like a buffer for cinacalcet [33].
Mathematical Model The proposed model is depicted in Fig. 1. We use six physiological motivated compartments,
i.e. compartments for absorption, first pass metabolism, plasma free drug and protein bound as well as two tissue compartments. It is notable that the tissue compartment with the
longer time to peak can be omitted in most situations reducing the system to 5 compartments. The impact of the sixth compartment on the predictions based the parameter set
presented in this manuscript is marginal. However, the tissue compartment with longer time to peak includes adipose tissue and adrenals [33]. Therefore, the six compartment
model will be useful when addressing patients deviating from the norm regarding adipose tissue or adrenal gland mass. We use constant rate functions between the compartments.
The amount of drug in each compartment can be calculated by solving the following system of linear ordinary differential equations: where y1→ corresponds to the amount of drug in the absorption compartment, y2→ to the amount of drug in the first pass metabolism,
y3→ and y4→ to the amount of drug in the free and protein bound plasma compartments. y5→ and y6→
correspond to the amount of drug in the fast and slow tissue, respectively. The initial dose is delivered to the absorption compartment.
In cinacalcet naïve patients the initial condition is given by the vector y0→ = (Administered dose, 0,0,0,0,0)'.
In non-naïve patients the condition for a drug administration at time ta is yta→ = y(ta)→+y0→.
The coefficient matrix A can be written as This linear system has the distinct advantage that it can be solved by solving the corresponding eigenvalue problem,
i.e. if λ1,...,λ6 are the distinct eigenvalues of A and Φ = (v1→,...,v6→) is a matrix containing the corresponding eigenvectors,
then the solution of the differential equation with the last dose administration at time ta is Therefore, at any given time t we can calculate the drug amount by evaluating the amount of drug in all compartments only for the preceding administration times.
Long time simulations are therefore very fast and numerically stable. We use data from the literature to estimate the coefficient matrix A. The natural constriction is that all eigenvalues should be distinct and negative to ensure
the convergence to zero in all compartments in the absence of further drug administrations. We assume an average plasma volume of 3 liters and a single oral dose
of 75 mg cinacalcet if not stated otherwise. The target maximum plasma concentration Cmax should be close to 26.8 ng/mL, the time until the maximum plasma concentration is reached around 2.3 hours,
terminal half-life t1/2 30-40 hours, distribution half-life tD 6 hours [26, 33]. The apparent oral clearance rate CL/F which is defined as should be around 314 L/h [25]. The apparent volume of distribution at steady state VD which is defined as should be around 1000 L [25, 33]. Finally, the bio-availability Bio should be between 20 and 25% for fastening patients. We calculate the bio availability as where Civ is the plasma concentration of the intravenously administered drug. We calculate Civ by analyzing the model modified for intravenous bolus injection.
The drug is administered to the “Plasma free drug” compartment thereby omitting the absorption and first pass metabolism compartments. Coral is the plasma concentration of the orally administered drug [25].
Model parameters and sensitivity analysis Typical model parameters are given in Table 1. We set kf = 0.0526kb to ensure that 95% of the drug is
protein bound in the steady state.
For simulations regarding food intake we slightly enhance the absorption rate to ka = 0.06h-1 in order to accommodate a slightly lower tmax and we reduce
the first pass metabolism rate ke = 0.13h-1. We conducted a sensitivity analysis to study the impact of uncertainties in all parameters in the coefficient matrix A and to identify key parameters of inter-individual variability.
The base values of each of these parameters was either divided or multiplied by a factor of 2. We then calculated the respective pharmacokinetic parameters assuming a single dose administration of 60mg cinacalcet.
We compared these values to the corresponding base case values. The results are presented in Table 2. The analysis reveals that bio-availability is most sensitive to changes of ke and klp.
Accordingly, Cmax is most sensitive to changes of ke , klp. Moreover, Cmax is sensitive to changes of kPTf which determines the buffer function of the tissue. Due to the buffer function of the tissue compartments
the terminal half-life is most sensitive to changes of kPTf , kPTs and kPTs. The half-life of distribution as well as tmax are sensitive to changes
in kb and ke . Due to its definition,
the apparent oral clearance rate CL/F is only sensitive to changes in ke and klp. Adaptation to specific patient groups
The model can be personalized to individual patients by adjusting the various parameters based on the
physiological meaning of the compartments and the sensitivity analysis. As an example we aim to adjust the model to patients with hepatic impairment.
It is known that the area under the curve AUC∞ is 2.4 and 4.2 fold higher in patients with moderated and severe hepatic impairment, respectively,
than in healthy control patients. Terminal half-life t1/2 is 1.3 and 1.7 fold higher in patients with moderated and severe hepatic impairment,
respectively, than in healthy control patients. Cmax and tmax are not significantly altered [31]. The rates associated with liver function are the first pass
metabolism rates ke and klp as well as the elimination rate keP which is associated with metabolization of the drug by the liver.
Since Cmax and tmax are most sensitive to changes in ke and kpl, these parameters should not be adjusted. While target AUC and t1/2
levels could be reached by adjusting the protein binding rate or the transfer rate to the tissue compartments, the meaningful rate to adjust is keP.
We can use the linear structure of the model to calculate the area under the curve AUC∞:
where the subscripts 3 and 4 denote the third and fourth vector component, respectively. Furthermore, we can use the linear structure of the model to estimate terminal half-life
which is associated with the tissue compartments and the largest eigenvalues. Since the fast tissue compartment dominates, we can use the largest eigenvalue λk associated with the fast tissue compartment:
With the parameters set presented in Table 1 the difference between the exact value and the approximated value is less then 0.01%.
The reduction of klP by 53% enhances the terminal half-life 1.4 fold and the AUC∞ 2.2-fold,
the reduction of klP by 76% enhances the terminal half-life 1.8 fold and the AUC∞ 4.2-fold. Another example would be the administration with food. It has been shown that the administration of a single dose of 90mg cinacalcet with high fat and
low fat meal enhances the bio-availability significantly resulting in increase of AUC∞ of 68% (90% confidence interval: 48% to 89%) and 50% (90% confidence interval: 33% to 70%), respectively [25].
Mean tmax is higher in fastening subjects (6h) then in subjects receiving cinacalcet with a high fat diet (4h) and a low fat diet (3.5h).
Terminal half-life was comparable between the administration schemes. Since bio-availability is associated with absorption and first-pass metabolism, the only meaningful parameters to
adjust are ka, ke and klp. The sensitivity analysis suggests to decrease ke and/or increase klp.
Since an increase in ke would lead to an increase of tmax we adjust only klp. Results Main pharmacokinetics profiles
The simulated pharmacokinetics profile of a single oral administration based on the proposed six-compartment model
is shown in Fig. 2 and Fig. 3. The results are qualitatively and quantitatively in close agreement with published clinical data. The simulations of daily administration of cinacalcet predict a steady state within seven days (Fig. 4) which is similar to clinically observed behavior of the drug [33].
Different dosing regimens To assess the impact of dosing regimens on the total exposure we modeled the weekly average cinacalcet concentration
in the blood for three different dosing regimens: daily administration, administration three times a week, and administration with a
realistic medication refill rate of 64% [34]. For the realistic medication refill rate scenario we chose days without administration randomly (Fig. 5).
Discussion The linear multi-compartment model introduced in this article
is used to describe the complex pharmacokinetic behavior of cinacalcet in the human body. This approach allows the analysis of the effect of different dosing
regimens on the drug concentration in the body that could otherwise only be addressed by clinical studies. The analysis can help to evaluate dose titration and administration schemes.
Calcium homeostasis in the human body is achieved by a complex network of regulators, one of them being the kidney. If the kidney function starts to decline,
a cascade of physiological processes eventually result in secondary hyperparathyroidism. Due to the complexity of the system, a growing number of
mathematical models have been published with the aim to analyze the driving factors of calcium homeostasis, secondary hyperparathyroidism, and CKD-MBD [27, 29, 30, 35-38].
However these models did not include a detailed model of cinacalcet, a drug widely prescribed in patients with secondary hyperparathyroidism. Our pharmacokinetic
model can be easily incorporated in all other models by using an operational model of allosterism which transfers the actual Ca2+ concentration by the amount of free drug in the plasma [39-41]. It is worth noticing that many model approaches with more than one compartment are able to predict one or two pharmacokinetics parameter within a desired range.
The model introduced in this article is minimal in the sense that a linear mass-balance model with less than five compartments is not able to capture all seven key pharmacokinetic parameters
(i.e. Cmax, tmax, Bio, CL/F, t1/2, tD1/2, and VD) simultaneously. Conclusion In conclusion, we have developed a mathematical model incorporating key components of cinacalcet pharmacokinetics.
A great advantage of this model is its linear structure allowing long term predictions which are fast and numerically stable.
The model is a general compartment model which could be used to evaluate different drugs. An example would be etelcalcetide, another
calcimimetic which is administered intravenously [42]. In this case the model reduces to a four compartment model since the intestinal absorption and first-pass metabolism
compartments are obsolete. Due to the physiological meaningful compartments, adjustments to conditions like hepatic impairment are straight forward.
Combined with a model for the parathyroid gland and bone metabolism it provides a ready to use tool for clinical trial simulations to explore effects of relevant factors,
such as patient adherence, off-label administration regimens, or the effect of administration combined with food intake. Acknowledgements The funding for this study came from Renal Research Institute, a wholly owned subsidiary of Fresenius Medical Care.
Disclosure
Statement Peter Kotanko holds stock in Fresenius Medical Care. The other authors have no conflicts of interest to declare.
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