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Pharmacokinetic modeling of Gadolinium nanoparticles (GdNPs) with the sojourn time in vasa vasorum for the contrast enhanced MRI
Advances in Difference Equations volumeÂ 2020, ArticleÂ number:Â 662 (2020)
Abstract
Deposition of lipid in the artery wall called atherosclerosis is recognized as a major cause of cardiovascular disease that leads to death worldwide. A better understanding into factors that may influence the delivery of gadolinium nanoparticles (GdNPs) that enhances quality of magnetic resonance imaging in diagnosis may provide a vital key for atherosclerotic treatment. In this study, we propose a delay differential model for describing the dynamics of GdNPs in bloodstream, peripheral arteries, and vasa vasorum with two phenomena of GdNPs during a sojourn in vasa vasorum. We then investigate the dynamical behaviors of GdNPs and explore the effects of sojourn time and transfer rates of GdNPs on the concentration of GdNPs in vasa vasorum at the 12th hour after the administration of gadolinium chelates contrast media and also the maximum concentration of GdNPs in peripheral arteries and vasa vasorum. Our results suggest that the sojourn of GdNPs in vasa vasorum may lead to complex behaviors of GdNPs dynamics, and transfer rates of GdNPs may have a significant impact on the concentration of GdNPs.
1 Introduction
Atherosclerosis, which is a common cardiovascular disease involving inflammation and accumulation of lipid plaques within the arterial wall, is recognized as a major cause of death worldwide [1, 2]. There is recent evidence suggesting that vasa vasorum, which is a network of microvessels contained in tunicae adventitia that supply oxygen and nutrients to the arterial wall, is associated with the occurrence of atherosclerotic plaque progression [3, 4]. For normal people, the oxygen is usually delivered by vasa vasorum and lumen to respectively reach outer and inner layers of the arterial wall. However, for atherosclerotic patients, vasa vasorum has changed in shape due to angiogenic expansion resulting in it becoming an only supply source for the entire wall [5, 6]. Consequently, observation of vasa vasorum expansion can clinically help predict the pathogenesis of atherosclerosis [7].
Magnetic resonance imaging (MRI) is an important technique currently used to diagnose and detect the pathogenesis of early plaque. It is known that contrast agents are clinically employed to help enhance the quality of medical imaging modalities [8, 9]. Usually, the contrast agents are presented in forms of nanoparticles (NPs) or liposomes containing metal ions such as iron(II), iron(III), gadolinium(II), gadolinium(III), manganese(II), or manganese(III) [10â€“12]. Gadolinium(III) complexes (gadoliniumbased contrast agents (GBCAs) or gadolinium nanoparticles (GdNPs)) are the most widely clinically used agent through the intravenous route [13]. The working principle of GdNPs to assist the image acquisition is binding between the GdNPs and the specific biological environment such as molecular markers of disease. For atherosclerosis, there are \(\alpha _{\nu }\beta _{3}\)integrins acting as the receptors of inflammatory endothelial cells on vasa vasorum, and GdNPs are designed and synthesized in a variety of chemical methods with the aim of targeting the \(\alpha _{\nu }\beta _{3}\)integrin receptors [6, 14, 15].
Even though the integrin binding helps enhance MR images, there are concerns about the safety of the contrast agents. In order to investigate factors that may influence their delivery and to predict possible effects inside the organs, physiologicallybased pharmacokinetic (PBPK) models have often been chosen as important tools in many research studies. To investigate the multimodality imaging technologies, Bartlett et al. [16] developed a threecompartmental model for plasma, interstitial and intracellular fluids to investigate the biodistribution of transferrin (tf)targeted and nontargeted siRNA nanoparticles and the cellular uptake of a tumor. The study indicated that the tftargeted NPs were more effective in attaching to the cell target and can help reduce tumor activity by 50 percent. Barboriak et al. [17] constructed three PBPK models including wellmixed, delay, and dispersion models to study the concentrationtime curves (CTCs) of the gadolinium contrast agent at three position organs, which are cerebral cortex, superior sagittal sinus, and psoas muscle. Their results suggest that the best model to predict GdCTCs is the dispersion model. Neubauer et al. [18] examined the behavior of perfluorocarbon NPs while they were transported in blood, peripheral, and vasa vasorum by using a threecompartmental pharmacokinetic model. In their work, the NPs have the ability to target \(\alpha _{\nu }\beta _{3}\)integrin receptors of atherosclerotic tissue resulting in the enhancement of contrast MR images. The purpose of their study was to develop models that are compatible with the new drug carriers. Wenger et al. [19] proposed a mathematical model based on a mass balance equation to evaluate distribution, accumulation, and excretion of the synthesized two NPs, polyacrylamide (PAA) and polyethylene glycolcoated polyacrylamide (PAApeg). Taheri et al. [20] studied the pharmacokinetics of GdDTPA administered at a bolus dose to optimize an area under the curve (AUC) of GdDTPA and found that reduction of GdDTPA together with optimized acquisition of DCEMRI results in shorter acquisition time and less exposure of subjects.
In this present study, we incorporate the timedelay terms that reflect the sojourn of GdNPs in vasa vasorum into the mathematical model developed [18]. We also explore two types of functions describing circulation of GdNPs during the temporary stay in vasa vasorum: with and without natural clearance of GdNPs. Our goals are to (i) investigate the dynamical behaviors of GdNPs including periodic, asymptomatic, and oscillating solutions in bloodstream, peripheral arteries, and vasa vasorum; and (ii) explore the effects of certain parameters such as the sojourn time that GdNPs stay in vasa vasorum and transfer rates between the compartments on the concentration of GdNPs in vasa vasorum at the 12th hour after administration and the maximum concentration of GdNPs within peripheral arteries and vasa vasorum.
2 Pharmacokinetic model
Previously, a threecompartmental pharmacokinetic model was proposed to investigate the dynamical behaviors of GdNPs involved in the enhancement of MRI of early atherosclerosis [18]. Those three compartments represent bloodstream (\(y_{1}\)), peripheral arteries (\(y_{2}\)), and vasa vasorum \((y_{3})\). In this work, the previous model is further extended to incorporate a fixed length of the sojourn time \((\tau )\) that GdNPs remain trapped in the third compartment so that the MR image signal remains elevated in comparison to the blood signal. Transfer of GdNPs between the first two compartments occurs at rates \(k_{12}\) and \(k_{21}\). Transfer of GdNPs between the third compartment and the first compartment occurs at rates \(k_{13}\) and \(k_{31}\). In addition to the distribution phase of GdNPs, the model also includes the clearance phase since GdNPs are normally removed from the bloodstream by phagocytosis in the reticuloendothelial system. An elimination constant for the clearance phase is denoted by \(k_{e}\). As specific binding of \(\alpha _{v}\beta _{3}\)targeted GdNPs to \(\alpha _{v}\beta _{3}\)integrin can take place in vasa vasorum to enhance the MR signal, such transfer may include both passive and active transports of GdNPs. Consequently, the dynamics of GdNPs in those three compartments can be described by
where \(y_{1}(t)\), \(y_{2}(t)\), and \(y_{3}(t)\) represent the concentration of GdNPs at time t (Î¼M) in bloodstream, peripheral arteries, and vasa vasorum, respectively. In the model, f is a function of Ï„ where Ï„ represents the sojourn time in the vasa vasorum compartment. Two types of f are considered here to describe two different behaviors of GdNPs. The first case is \(f(\tau )=1\), which reflects the completely retained amount of GdNPs during their temporary stay in the vasa vasorum compartment. The second case is \(f(\tau )=e^{k_{31}\tau }\), which reflects the decreasing amount of GdNPs at rate \(k_{31}\) during the sojourn stay in the vasa vasorum compartment. To tackle the delay system of equations in (1) numerically, the corresponding nondelayed system is initially solved with the initial conditions
on the time interval \(t\in [0,\tau ]\), where D is the gadolinium dosage of 4.6 Î¼mol Gd/kg body weight injected to each test subject and \(v_{1}\) is the distribution volume of bloodstream. Such a model is solved until the time reaches \(t=\tau \) to obtain a new set of initial conditions for (1) on \([0, \tau ]\) as follows:
where \(Y_{i}^{0}(\theta )\) for \(i = 1, 2, 3\) denotes a solution of the corresponding nondelayed system at time Î¸. For \(t\in (\tau ,\infty )\), the delay system is solved with conditions in (2). Parameters used in the model are presented in Table 1, while the flow diagram for describing the dynamical behaviors of GdNPs is illustrated in Fig. 1.
Note that model (1) can be extended to incorporate diffusion and chemotaxis of GdNPs. Examples of preceding studies that take into account those terms are such as Li et al. [21] and Viglialoro and Woolley [22].
3 Analytic results
Some analytic results of the model including an equilibrium and certain stability conditions are demonstrated in this section. Some other works related to the analysis of oscillatory and periodic solutions in detail can be found in [23] and [24]. By setting each equation in (1) equaling zero, the GdNPsfree equilibrium is obtained, \(E^{0}=(y^{0}_{1}, y^{0}_{2}, y^{0}_{3})=(0, 0, 0)\).
3.1 \(\tau =0\) and \(f(\tau )=1\)
Theorem 1
The GdNPsfree equilibrium is locally asymptotically stable.
Proof
First, we compute the eigenvalues (Î») of the Jacobian matrix
from the equation, det \((J\lambda I) = 0\). The characteristic equation of J is given by
with
It is clear that \(a_{1}>0\), \(a_{3}>0\), and \(a_{1}a_{2}>a_{3}\). Hence, according to the Routhâ€“Hurwitz criteria [25], all eigenvalues of J have negative real parts. Consequently, \(E^{0}\) is locally asymptotically stable.â€ƒâ–¡
3.2 \(\tau \neq 0\) and \(f(\tau )=1\)
For this case, as an equilibrium solution does not depend on time, there exists the GdNPsfree equilibrium of (1), \(E^{0}\). To investigate the stability of \(E^{0}\), let us define
Then the linearized system of (1) in a matrix form can be described by
where \(A_{1}\) and \(A_{2}\) are given by
The characteristic equation of the linearized system is given by
Hence,
with
Since Ï„ is transcendental and the system has infinitely many eigenvalues, investigating the stability of \(E^{0}\) is much more difficult. Hence, we shall instead explore the distribution of roots of Eq. (3).
By Roucheâ€™s theorem and the continuity in Ï„, Eq. (3) has positive real parts if and only if it has purely imaginary roots. Then we study whether we can find purely imaginary roots. Assume that \(\lambda (\tau )=i\omega (\tau )\) with \(\omega >0 \). Substituting \(\lambda (\tau )\) into Eq. (3) gives
By separating real and imaginary parts and squaring their equations, we obtain
Next, let
Consequently, Eq. (5) can be rewritten as
Proposition 1
Because \(\gamma <0\), Eq. (6) has at least one positive root, denoted by \(Z_{0}\).
Proof
If \(Z=0\), it follows that \(h(0)=\gamma \). Since \(h(0)<0\) and \(\lim_{x \to \infty } Z^{3}+\alpha Z^{2}+\beta Z+\gamma = \infty \), Eq. (6) has at least one positive root. Consequently, let \(\lambda (\tau )=\mu (\tau )+i\omega (\tau )\) be the eigenvalue of Eq. (3) such that \(\mu (\tau _{0})=0\) and \(\omega (\tau _{0})=\omega _{0}\). According to the separated real and imaginary parts of Eq. (4), we obtain
â€ƒâ–¡
Theorem 2
Assume that
hold.

(i)
If \(\gamma \geq 0\) and \(\beta >0\), then \(E^{0}\) is locally asymptotically stable for any \(\tau \geq 0\).

(ii)
If either \(\gamma <0\) or \(\gamma \geq 0\) and \(\beta <0\), then

\(E^{0}\) is locally asymptotically stable when \(\tau < \tau _{0}\),

\(E^{0}\) undergoes a Hopf bifurcation when \(\tau =\tau _{0}\),

\(E^{0}\) is unstable when \(\tau >\tau _{0}\),
where
$$\begin{aligned} \tau _{0} = \frac{1}{\omega _{0}} \arccos \biggl( \frac{(b_{5}b_{1}b_{4})\omega ^{4}_{0}+(b_{4}b_{2}b_{3}b_{1})\omega ^{2}_{0}}{b^{2}_{5}\omega ^{4}_{0}+(2b_{3}b_{5}b^{2}_{4})\omega ^{2}_{0}b^{2}_{3}} \biggr). \end{aligned}$$ 
3.3 \(\tau \neq 0\) and \(f(\tau )=e^{k_{31}\tau }\)
Since there is still only a GdNPsfree equilibrium (\(E^{0}\)) for this case, we follow a similar approach in the previous part to explore stability of the equilibrium. Hence, the characteristic equation of the linearized system is given by
with \(b_{n}\) for \(n=1,\dots ,5\).
Substituting \(\lambda (\tau )=i\omega (\tau )\) with \(\omega >0 \) into Eq. (7), we have
By separating real and imaginary parts in Eq. (8), the real part satisfies the following equation:
while the imaginary part satisfies the equation
Next, Eq. (10) can be rewritten as
Then, substituting Eq. (11) into Eq. (9) yields
Notice that Eq. (12) is a transcendental equation and can be very difficult to solve for a closedform solution. Hence, we use a geometrical approach by considering intersections of two curves from the equation. Let us define \(f_{1}= \cos {\omega \tau }\) and \(f_{2}= \frac{(b_{1}b_{5}b_{4})\omega ^{4}+(b_{2}b_{4}b_{1}b_{3})\omega ^{2}}{b^{2}_{5}\omega ^{4}+(2b_{3}b_{5}b^{2}_{4})\omega ^{2}b^{2}_{3}} \cdot e^{k_{31}\tau }\). Based on parameter ranges of this study, there is no intersection between those two curves. Hence, we cannot find Ï‰ that satisfies Eq. (12) (see Fig. 2(a)â€“(d)). Consequently, there is no periodic solution of the system.
Theorem 3
If Eq. (12) has no Ï‰, then \(E^{0}\) is always locally asymptotically stable for all \(\tau >0\).
4 Numerical results
In this section, numerical results for dynamical behaviors of GdNPs in bloodstream, peripheral arteries, and vasa vasorum are illustrated to support theoretical results obtained from the previous section and to demonstrate the effects of transfer rates and the sojourn time that GdNPs remain trapped in the vasa vasorum compartment. Note that the numerical simulations are based on the Rungeâ€“Kutta method of order 4 incorporated with the initial conditions for solving delay differential equations [26].
In case \(f(\tau )=1\) for \(\tau >0\) or equivalently GdNPs are trapped in the vasa vasorum at a fixed length of time before escaping to other parts, dynamical behaviors of GdNPs according to different values of Ï„ (0.3, 1.7, 1.96, 2.5) are shown in Fig. 3. As a critical value of Ï„ (\(\tau _{0}\)) based on the parameter values in Table 1 is approximately 1.96 hours, GdNPs dynamics change around such a value. If \(\tau <1.96\), the system solution tends to the GdNPsfree equilibrium, and it starts oscillating when Ï„ becomes closer to \(\tau _{0}\) (see Fig. 3(a)â€“(b)). The results correspond to Theorem 2 that the GdNPsfree equilibrium of system (1) is locally asymptotically stable when \(\tau <1.96\). At \(\tau =1.96\), Fig. 3(c) shows that the solution fluctuates in a periodic pattern. Those dynamical behaviors are in agreement with Theorem 2 so that a Hopf bifurcation occurs at \(\tau =\tau _{0}=1.96\). In Fig. 3(d), when \(\tau >1.96\), the solution fluctuates with an increasing amplitude and becomes divergent. This is in agreement with the unstable behaviors around the equilibrium after \(\tau >\tau _{0}\) in Theorem 2. In Fig. 4, we demonstrate phase portraits at different values of Ï„ in correspondence with results in Fig. 3. In Fig. 4(a)â€“(b), with \(\tau =0.3\) and \(\tau =1.7\), solution trajectories approach the origin without and with fluctuation. The trajectory neither tends toward the origin nor infinity if \(\tau =1.96\) in Fig. 4(c). When \(\tau =2.5\), the solution trajectory diverges to infinity over time (see Fig. 4(d)).
Next, we investigate the effects of transfer rates on the concentration of GdNPs at the 12th hour after the initial injection according to different values of Ï„ (see Fig. 5). Results in Fig. 5(a) suggest that increasing the transfer rate of GdNPs from bloodstream to peripheral arteries may result in the decreased concentration of GdNPs in vasa vasorum. However, such a trend of results may slightly be affected by the increasing value of Ï„. In Fig. 5(b), the transfer rate from peripheral arteries to bloodstream has a significant impact on the concentration of GdNPs when it is quite small. When it becomes larger, the concentration gradually increases. The increasing value of Ï„ may alter the trend by delaying the increasing concentration of GdNPs. Results in Fig. 5(c) demonstrate the increasing concentration of GdNPs according to the transfer rate from bloodstream to vasa vasorum. Figure 5(d) suggests that increasing the transfer rate from vasa vasorum to bloodstream may result in the reduction of GdNPs in vasa vasorum. Similar to the previous results, the increasing value of Ï„ may affect the result trend. In Fig. 5(e), the higher elimination rate drastically reduces the concentration of GdNPs in vasa vasorum and higher Ï„ may affect the normal trend of results.
Moreover, we explore the effects of transfer rates on the maximum concentration of GdNPs inside peripheral arteries and vasa vasorum when GdNPs do not and do remain in vasa vasorum at the fixed length of time \(\tau =0, 1.7\). Note that based on the preceding study from the preceding work, GdNPs remain trapped in vasa vasorum for approximately \(\tau =1.7\) hours. In Fig. 6, the overall results suggest similar trends of results according to transfer rates for both \(\tau =0\) and \(\tau =1.7\), but the concentration of GdNPs for the \(\tau =0\) case is generally higher. Our results in Fig. 6(a) demonstrate that when the transfer rate from bloodstream to peripheral arteries increases, the concentration of GdNPs in peripheral arteries considerably increases, while the concentration of GdNPs in vasa vasorum moderately decreases. Figure 6(b) shows the significantly decreasing concentration of GdNPs in peripheral arteries and the slightly increasing concentration of GdNPs in vasa vasorum according to the increasing transfer rate from peripheral arteries to bloodstream. In Fig. 6(c), when the transfer rate from bloodstream to vasa vasorum increases, it may result in the increase of GdNPs concentration in vasa vasorum but in the moderate decrease of GdNPs concentration in peripheral arteries. The increasing transfer rate from vasa vasorum to bloodstream may lead to the decreasing concentration of GdNPs in vasa vasorum but the slightly increasing concentration of GdNPs in peripheral arteries for \(\tau =0\) and the slightly increasing then decreasing concentration of GdNPs in peripheral arteries for \(\tau =1.7\) (see Fig. 6(d)). In addition, when the sojourn time is present, it may also result in the increasing concentration after the reduction of GdNPs in vasa vasorum. Our results in Fig. 6(e) demonstrate that the concentration of GdNPs in vasa vasorum and peripheral arteries decreases if the elimination rate increases.
In the case \(f(\tau )=e^{k_{31}\tau }\) for \(\tau \neq 0\) or equivalently GdNPs are trapped in the vasa vasorum at a fixed length of time and eliminated during the stay at a rate of \(e^{k_{31}\tau }\) before escaping to other parts, timeseries plots and phase portraits that represent the dynamical behaviors near the GdNPsfree equilibrium for \(\tau =0.3\) and \(\tau =2.5\) are illustrated in Fig. 7. Figure 7(a)â€“(b) demonstrate the concentration of GdNPs according to time that GdNPs are eventually eliminated and a phase portrait that shows a solution approaching the origin for \(\tau =0.3\). In Fig. 7(c)â€“(d), our results show that the concentration of GdNPs approaches zero according to time and a corresponding phase portrait that shows a solution approaching the origin for \(\tau =2.5\).
We further investigate the effects of transfer rates on the concentration of GdNPs at the 12th hour after the initial injection when \(f(\tau )=e^{k_{31}\tau }\) for \(\tau \neq 0\). Overall, our results suggest that increasing Ï„ leads to the higher concentration of GdNPs. In Fig. 8(a), the transfer rate of GdNPs from bloodstream to peripheral arteries rarely affects the concentration of GdNPs. Figure 8(b) shows that when the transfer rate from peripheral arteries to bloodstream increases, the concentration of GdNPs increases at the beginning before becoming saturated according to the transfer rate. The increasing transfer rate of GdNPs from bloodstream to vasa vasorum significantly influences the concentration of GdNPs and results in higher concentration (see Fig. 8(c)). In contrast, when the transfer rate from vasa vasorum to bloodstream increases, the concentration of GdNPs significantly decreases as shown in Fig. 8(d). Figure 8(e) illustrates that the increasing clearance rate may reduce the concentration of GdNPs in vasa vasorum.
Figure 9(a)â€“(e) illustrates that the longer sojourn time that GdNPs remain trapped generally results in the higher concentration of GdNPs in vasa vasorum and the lower concentration of GdNPs in peripheral arteries. In Fig. 9(a), increasing the transfer rate of GdNPs from bloodstream to peripheral arteries may moderately increase the concentration of GdNPs in peripheral arteries and slightly reduce the concentration of GdNPs in vasa vasorum. Results in Fig. 9(b) suggest that increasing the transfer rate of GdNPs from peripheral arteries to bloodstream may result in the significant decrease of GdNPs concentration in peripheral arteries and the small increase of GdNPs concentration in vasa vasorum. As illustrated in Fig. 9(c), when the transfer rate of GdNPs from bloodstream to vasa vasorum increases, the concentration of GdNPs in peripheral arteries intermediately decreases, while the concentration of GdNPs in vasa vasorum considerably increases. In Fig. 9(d), increasing the transfer rate of GdNPs from vasa vasorum to bloodstream may result in the slightly decreasing concentration of GdNPs in peripheral arteries and the significantly increasing concentration of GdNPs in vasa vasorum when the sojourn of GdNPs in vasa vasorum is present. The increasing elimination rate of GdNPs generally leads to the decreasing concentration of GdNPs in both peripheral arteries and vasa vasorum (see Fig. 9(e)). When results in Figs. 6 and 9 are compared, the difference of \(f(\tau )\) may affect the concentration trends in vasa vasorum. When GdNPs remain trapped in vasa vasorum, the concentration of GdNPs is lower than when they do not for \(f(\tau )=1\), while it is higher for \(f(\tau )=e^{k_{31}\tau }\).
5 Conclusion
In this study, the threecompartmental pharmacokinetic model for describing the distribution of GdNPs used in the contrast enhanced MRI is extended to incorporate a sojourn period of particle circulation in vasa vasorum. The particle dynamics are then described by a system of delay differential equations. Two types of circulation are considered: with and without natural clearance during the particle sojourn.
When the sojourn of GdNPs in vasa vasorum is not included \((\tau =0)\), we show that the particlefree equilibrium point is locally asymptotically stable. When the fixed period of circulation is present \((\tau \neq 0)\), the delay system is analyzed to determine the conditions for asymptomatic stability and the critical time of a periodic solution. Our findings suggest that only when GdNPs are not eliminated during the sojourn in vasa vasorum, a periodic solution occurs under certain conditions. If GdNPs are eliminated by natural clearance during their stay in vasa vasorum, a system solution always tends to the particlefree equilibrium point.
In addition, we demonstrate the dynamics of GdNPs numerically according to the stability conditions. Our results are in agreement with the conditions and show asymptotic, periodic, and unstable solution behaviors of GdNPs when there is no clearance of GdNPs during their stay in vasa vasorum \((f(\tau )=1)\). Generally, a solution starts to oscillate when Ï„ approaches its critical value (\(\tau =1.96\)), shows a periodic behavior when \(\tau = 1.96\), and then shows an unstable behavior when \(\tau >1.96\). Consequently, to ensure that GdNPs are completely eliminated, GdNPs should be designed not to linger in vasa vasorum for too long. However, if GdNPs are naturally cleared during their stay in vasa vasorum \((f(\tau )=e^{k_{31}\tau })\), the particlefree equilibrium point is asymptotically stable. In such a case, a solution will tend to the equilibrium point or GdNPs will always be eventually eliminated from the body. As Ï„ is normally less than the critical value in reality, we further investigate how transfer rates of GdNPs affect their presence in blood stream, peripheral arteries, and vasa vasorum after twelve hours of injection, and also the maximum concentration of GdNPs. It is found that increasing transfer rates (\(k_{12}\), \(k_{31}\), and \(k_{e}\)) lowers the concentration of GdNPs, while increasing \(k_{21}\) and \(k_{13}\) leads to the higher concentration of GdNPs. For the maximum concentration, our results suggest that transfer rates affect the concentration of GdNPs in peripheral arteries and vasa vasorum. Note that results are similar for both types of GdNPs circulation. In addition, overall results suggest that increasing Ï„ may result in the higher concentration of GdNPs. Moreover, the maximum of GdNPs in vasa vasorum for \(f(\tau )=1\) is generally lower than that for \(f(\tau )=e^{k_{31}\tau }\).
Finally, we believe that this study may help gain a better understanding into the effects of the fixed period of particle circulation and transfer parameters, may also help design a therapy that enhances the efficacy of MRI, and may suggest possible administration strategies that could reduce toxicity from overdose.
Availability of data and materials
Not applicable.
Abbreviations
 GdNPs:

Gadolinium nanoparticles
 MRI:

Magnetic resonance imaging
 NPs:

Nanoparticles
 GBCAs:

Gadoliniumbased contrast agents
 PBPK:

Physiologicallybased pharmacokinetic
 tf:

Transferrin
 siRNA:

Small interfering RNA
 CTCs:

Concentrationtime curves
 GdCTCs:

Gadolinium contrast agent concentrationtime curves
 PAA:

Polyacrylamide
 PAApeg:

Polyethylene glycolcoated polycrylamide
 GdDTPA:

Gadoliniumdiethylenetriamine pentaacetic acid
 AUC:

Area under the curve
 DCEMRI:

Dynamic contrast enhancedmagnetic resonance imaging
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Acknowledgements
AP and FC would like to thank Assoc. Prof. Charin Modchang and Asst. Prof. Pairote Satiracoo for suggestive comments.
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AP was partially financially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand for her tuition fees. The funding body had no role in the study design, analysis, and interpretation of results nor in writing the manuscript.
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Prommarat, A., Chamchod, F. Pharmacokinetic modeling of Gadolinium nanoparticles (GdNPs) with the sojourn time in vasa vasorum for the contrast enhanced MRI. Adv Differ Equ 2020, 662 (2020). https://doi.org/10.1186/s1366202003114w
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DOI: https://doi.org/10.1186/s1366202003114w