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Pharmacokinetic models are mathematical models which provide insights into the interaction of chemicals with biological processes. During recent decades, these models have become central of attention in industry that caused to do a lot of efforts to make them more accurate. Current work studies the process of drug and nanoparticle (NPs) distribution throughout the body which consists of a system of ordinary differential equations. We use a tri-compartmental model to study the perfusion of NPs in tissues and a six-compartmental model to study drug distribution in different body organs. We have performed global sensitivity analysis by LHS Monte Carlo method using PRCC. We identify the key parameters that contribute most significantly to the absorption and distribution of drugs and NPs in different organs in body.

Nanotechnology is the study of materials, devices, and systems at the nanometer scale. Nanotechnology and nanoscience have been used widely in many areas of research and applications [

Mathematical and statistical modeling helps us to understand the interaction between the components of systems biology and prediction of the future of different biological models [

To check the accuracy of any mathematical model, we need to use different methods and because of existence of uncertainty in experimental data, it can be often complicated. Uncertainty and sensitivity analysis are useful techniques which help us to identify these uncertainties in data and then control them [

Current work studies the process of drug distribution throughout the body which consists of a system of ordinary differential equations. There are several biological parameters related to distribution of drug through different body organs. We start with a simple three compartmental model to demonstrate NP distribution from capillary to tissue. Globally sensitivity analysis LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC) has been performed to investigate the key parameters in model equations. Also, we study a six compartmental system for which we assume the specific drug has been distributed through different rout of drug administrations, such as intravenous injection, intramuscular injection, water and or feed. We have used the same global sensitivity analysis PRCC method to compare different physiological parameters. We have used the parameters variations based on different studies [

There are some efforts to develop physiologically based pharmacokinetic models for nanoparticles distribution through the body, which will be useful tools for predicting nanoparticle distribution in different organs to assist with extrapolation of responses from in vitro and in vivo [

As we can see in

For NPs that move from blood vessels into different tissues this tricompartmental model is needed to characterize NPs infusion in the body. For simplicity, we have supposed that there is no interaction between Endothelial cells and their surrounding cells.

As we can see in

If we apply the mass balance laws to this tri-compartmental model, we have:

{ d A C a p d t = k 12 A A r t − k 23 A C a p − k 25 A C a p + k 32 A E C d A E C d t = k 23 A C a p − k 34 A E C − k 32 A E C d A D T d t = k 34 A E C (2.1)

We have two possible routes for NPs in compartment two. First possibility, they can distribute to compartment 3 and or they leave compartment two via venous efflux. Here, also we have assumed that the uptake depends on NP concentration in compartment two and it does not depend on perforate blood flow [

J 25 = J 12 − k 23 A C a p + k 32 A E C (2.2)

where J 25 = k 25 A C a p and J 12 = Q C 12 ( k 25 is a variable rate function).

Physiologically speaking, we have considered that at starting time t = 0 NPs enter from artery to compartment 2 and then they leave capillary bed to shallow tissue. Before the time for the venous effluent, venous efflux of NPs is zero, and at this moment, t = τ , and then after that the sum of NPs fluxes to shallow tissue compartment and venous effluent should be equal to arterial flux or J 12 . By this assumption that τ is small at steady state, we can compute the initial mass M 2 for capillary bed compartment that would be M 2 = C 12 V 2 . Here, C 12 is

the concentration of infused NPs, and V 2 is the vascular volume. Also, we can calculate the capillary transit time by the following equality τ = V 2 / Q , where, Q is perfusate flow through skin flap and we consider it as a constant approximately equal 1 mL/min. The value of k 25 as a variable rate function after the time that flux k 32 A E C reaches to compartment two increased and we will prove it by some computations later [

A C a p = M 2 (2.3)

A E C = M 2 k 23 k 32 + k 34 ( 1 − e − ( k 32 + k 34 ) t ) + A E C ( 0 ) e − ( k 32 + k 34 ) ( t − Γ ) (2.4)

A D T = M 2 k 23 k 34 k 32 + k 34 ( t − 1 − e − ( k 32 + k 34 ) t k 32 + k 34 ) + k 34 A E C ( 0 ) k 32 + k 34 [ 1 − e − ( k 32 + k 34 ) t ( t − Γ ) ] + A D T ( 0 ) (2.5)

where we consider Γ as the beginning time of washout phase that is zero during dosing phase [

k 25 0 = J 12 + k 32 A E C ( 0 ) A C a p ( 0 ) − k 23 (2.6)

Because at t = 0 , A E C ( 0 ) = 0 and A C a p ( 0 ) = M 2 , so we have:

k 25 0 = J 12 M 2 − k 23 (2.7)

Therefore, k 25 at steady-state has the following value:

k 25 s s = J 12 + k 32 A E C ( s s ) A C a p ( s s ) − k 23 (2.8)

such that A C a p s s = M 2 and

A E C s s = l i m t → ∞ A E C ( t ) = k 23 M 2 k 32 + k 34 (2.9)

If we substitute the value of A C a p s s and A E C s s into k 25 s s , then we have:

k 25 s s = ( k 32 + k 34 ) J 12 + k 32 k 23 M 2 ( k 32 + k 34 ) M 2 − k 23 (2.10)

and because we have, J 12 M 2 = τ − 1 , so;

k 25 s s = τ − 1 + k 32 k 23 k 32 + k 34 − k 23 (2.11)

and when k 34 = 0 , we have k 25 s s = τ − 1 . Moreover, we can write k 25 0 = τ − 1 − k 23 . When we compare k 25 0 and k 25 s s , we see that k 25 0 is less than k 25 s s by the following result:

k 25 s s = k 25 0 + k 32 k 23 k 32 + k 34 (2.12)

During dosing phase A C a p = 0 . At t = Γ , A C a p changes from M 2 to 0, and also A E C ( 0 ) from 0 to A E C ( Γ ) . During decay phase J 12 is non-zero.

Physiological and pharmacokinetic models are useful to determine drug distribution into different target tissues, which helps for the evaluation of drug efficacy and drug safety. We study a six-compartmental pharmacokinetic model with application in food safety and we use the physiological parameters variations based on different studies [

Cardiac output and blood flows to tissues (L/h):

{ Q C = Q C C × B W , Cardiacoutput Q L = Q L C × Q C , Liver Q K = Q K C × Q C , Kidney Q L u = Q L u C × Q C , Lung Q F = Q F C × Q C , Fat Q M = Q M C × Q C , Muscle Q R = Q R C × Q C , Restofbody (3.1)

where, Q C C = 4.944 is cardiac output (L/h/kg), Q L C = 0.2725 is fraction of blood flow to the liver (unitless), Q K C = 0.12 is fraction of blood flow to the kidneys (unitless), Q F C = 0.1275 fraction of blood flow to the fat (unitless), Q M C = 0.251 is fraction of blood flow to the muscle (unitless), Q L u C = 1 is fraction of blood flow to the Lung (unitless), Q R C = 1 − Q L C − Q K C − Q F C − Q M C ; − Q L u C is fraction of blood flow to the rest of body and BW is body weight [

Tissue volumes (L):

{ V v e n = V v e n C × B W , V a r t = V a r t C × B W , V L = V L C × B W , Liver V K = V K C × B W , Kidney V L u = V L u C × B W , Lung V F = V F C × B W , Fat V M = V M C × B W , Muscle V R B = V R C × B W , Restofbody (3.2)

where, V L C = 0.0245 Fractional liver tissue (unitless), V K C = 0.004 fractional kidney tissue, V F C = 0.32 fractional fat tissue (unitless), V M C = 0.4 fractional muscle tissue (unitless), V L u C = 0.010 fractional Lung tissue (unitless), V v e n C = 0.044 venous blood volume, fraction of blood volume (unitless), V a r t C = 0.016 Arterial blood volume, fraction of blood volume (unitless) and V R C = 1 − V L C − V K C − V F C − V M C − V L u C − V v e n C − V a r t C fractional rest of body tissue (unitless).

Permeability surface area coefficients:

{ P A F = P A F C × V F , P A M = P A M C × V M , (3.3)

where, permeability constants (L/h/kg tissue) (Permeation area cross products) are: P A F C = 0.012 fat tissue permeability constant, P A M C = 0.225 muscle tissue permeability constant [

Volume of tissue as blood:

{ V F b = F V B F × V F , Fat compartment blood volume V F t = V F − V F b , Fat compartment tissue volume (3.4)

where, F V B F = 0.02 blood volume fraction of fat [

Muscle:

{ V M b = F V B M × V M , Muscle compartment bloodvolume V M t = V M − V M b , Muscle compartment tissue volume (3.5)

where, F V B M = 0.01 blood volume fraction of muscle [

Dosing:

{ D O S E o r a l = P D O S E o r a l × B W , ( mg ) Oraldose D O S E i v = P D O S E i v × B W , ( mg ) IVdose D O S E i m = P D O S E i m × B W , ( mg ) IMdose D O S E o r a l w = P D O S E o r a l w × B W , ( mg ) Oral through water dose D O S E o r a l f = P D O S E o r a l f × B W , ( mg ) Oral through feed dose (3.6)

where, P D O S E o r a l , P D O S E i v , P D O S E i m , P D O S E o r a l w and P D O S E o r a l f are parameters for exposure scenario.

Intramuscular (IM) injection equations:

{ R i m = K i m × Amtsite d d t ( Absorb ) = R i m Rsite = − R i m + K d i s s × Doseimremain d d t ( Amtsite ) = Rsite Rdoseimremain = − K d i s s × Doseimremain d d t ( Doseimremain ) = Rdoseimremain (3.7)

where, K i m = 0.15 or K i m = 0.3 I M IM absorption rate constant (/h), K d i s s = 0.02 IM absorption rate constant [

Intravascular (IV) injection to the venous equations:

{ I V R = D O S E i v / T i m e i v R i v = I V R × ( 1 − h e a v i s i d e ( T − T i m e i v ) ) d d t ( A i v ) = R i v (3.8)

where, T i m e i v is IV injection/infusion time (h).

Urinary elimination rate constant:

K u r i n e = K u r i n e C × B W (3.9)

Liver compartment:

{ R L = Q L × ( C A − C V L ) + R A O , d d t ( A L ) = R L , C L = A L / V L , C V L = A L / ( V L × P L ) , d d t ( A U C C L ) = C L , (3.10)

Blood compartment:

{ R V = ( Q L × C V L + Q K × C V K + Q F × C V F + Q M × C V M + Q R × C V R B + R i v + R i m ) − Q C × C V , ( mg ) RV the changing rate in the venous blood ( mg / h ) d d t ( A V ) = R V , AV the amount of the drug in the venous blood ( mg ) C V = A V / V v e n , CV drug concentration in the venous blood ( mg / L ) R A = Q C × ( C V L u − C A ) , RA the changing rate in th earterial blood ( mg / h ) d d t ( A A ) = R A , AV the amount of the drug in the venous blood ( mg ) C A = A A / V a r t d d t ( A U C C V ) = C V , AUCCVAUCofdrug A B l o o d = A A + A V (3.11)

Kidney compartment:

{ R K = Q K × ( C A − C V K ) − R u r i n e , d d t ( A K ) = R K , C K = A K / V K , C V K = A K / ( V K × P K ) , d d t ( A U C C K ) = C K , R u r i n e = K u r i n e × C V K , d d t ( A u r i n e ) = R u r i n e , (3.12)

Muscle compartment:

{ R M B = Q M × ( C A − C V M ) − P A M × C V M + P A M × C M t / P M , d d t ( A M B ) = R M B , C V M = A M B / V M B , R M t = P A M × C V M − P A M × C M t / P M , d d t ( A M t ) = R M t , C M t = A M t / V M t , A M t o t a l = A M t + A M B , C M = A M t o t a l / V M , d d t ( A U C C M ) = C M , (3.13)

Lung compartment:

{ R L u = Q L u × ( C V − C V L u ) , d d t ( A L u ) = R L u , C L u = A L u / V L u , C V L u = A L u / ( V L u × P L u ) , d d t ( A U C C L U ) = C L u , (3.14)

Fat compartment:

{ R F B = Q F × ( C A − C V F ) − P A F × C V F + P A F × C F t / P F , d d t ( A F B ) = R F B , C V F = A F B / V F B , R F t = P A F × C V F − P A F × C F t / P F , d d t ( A F t ) = R F t , C F t = A F t / V F t , A f t o t a l = A F t + A F B , C F = A f t o t a l / V F , (3.15)

Rest of body:

{ R R B = Q R × ( C A − C V R B ) , d d t ( A R ) = R R B , C R = A R / V R B , C V R B = A R / ( V R B × P R ) , d d t ( A U C C R ) = C R , (3.16)

Mass balance equation:

{ Q b a l = Q C − Q L − Q K − Q M − Q F − Q R , T m a s s = A b l o o d + A L + A K + A M t o t a l + A F t o t a l + A R + A u r i n e + A L u , B a l = A A O + A i v + A b s o r b − T m a s s , Permeability-limited model mass balance (3.17)

Global sensitivity analysis allows us to change all parameters simultaneously over the entire parameter interval. This is a way to evaluate the relative effects of each input parameter and also to identify the interactions between parameters to the model output. In global sensitivity analysis we determine that with variation of input parameters in a certain range, which parameters and interactions have the most influential impact on the overall behavior of our model [

There are several types of global sensitivity analyses, such as weighted average of local sensitivity analysis, partial rank correlation coefficient, multi parametric sensitivity analysis, Fourier amplitude sensitivity analysis (FAST) and Sobol’s method, which can be used for systems pharmacology models [

LHS method is a sampling method and requires fewer samples compare to simple random sampling to achieve the same accuracy [

In LHS method, sampling is independent for each parameter and can be done by randomly selecting values from each pdf. We may sample each interval once for each parameter without any replacement. The LHS matrix is consisting of N rows corresponding to the number of simulations or sample size and also it includes k columns corresponding to the number of varied parameters. Then, N model solutions may be simulated, using each combination of parameter values which they represent each row of the LHS matrix [

Here, a parameter sensitivity analysis has being conducted to identify the pharmacokinetic parameters that have the most significant effect on our model system by the LHS Monte Carlo method using PRCC with uniform distributions for the 95 percent confidence intervals. The global sensitivity results with p-values corresponding to capillary compartment, endothelial cell compartment and deep tissue compartment have been demonstrated in Figures 4(1)-(3) respectively.

According to LHS, we simulated the responses of the model for each organ by randomly selecting values for the parameter set from the 95 percent confidence intervals. These analyses were done by developing a LHS/PRCC method with uniform distributions for the 95 percent confidence intervals. We found that some parameters illustrate significant performance in terms of sensitivity of the output to the variations of these parameters in some organs while they do not have this effect for other organs. These results have been depicted in

Nowadays, nanoparticles have a growing use in industry specially medicine. There are some studies about applications of NPs in therapeutic areas, however, the number of these studies is not a lot. Increasing the importance of studies about tumors and concentration of drugs and NPs in tumors or other tissues has enhanced the role of in vitro models to simulate absorption process of drugs and NPs. Pharmacokinetic and physiological models are useful means to demonstrate the relationships between different drug administrations, and drug exposure or concentration.

An uncertainty analysis may be applied on the physiological and pharmaceutics models to investigate the uncertainty in system output that is generated from uncertainty in parameter inputs. Sensitivity analysis assesses how variations in model outputs can be apportioned, qualitatively or quantitatively, to different inputs.

In this research we reviewed two physiological systems which have been reported by different authors and we have used the presented physiological parameters from different published works. In the first case, we presented a three compartmental model which can be used to exhibit the distribution of drug and or NPs from capillary compartment to endothelial cells compartment and then tissue compartment. The objective of this study was to determine the key parameters in NPs infusion from blood vessels to target tissue in the ex vivo tissue perfusion system using sampling-based method (Partial Rank Correlation Coefficient-PRCC). As we have seen, some parameters have positively and some others negatively affected NPs infusion process.

We have presented another physiological model with six compartments, such as kidney, liver, lung, fat, muscle and plasma compartment. We identified the key parameters that contribute most significantly to the absorption and distribution of drugs in different organs in body using PRCC. Our findings imply that this identification is clearly dependent upon the dose and target tissues but not on the exposure route.

This work was supported by the Institute of Computational Comparative Medicine (ICCM) and department of Mathematics of Kansas State University. With a special thanks to Dr. Majid Jaberi-Douraki for his full support.

The authors declare no conflicts of interest regarding the publication of this paper.

Azizi, T. and Mugabi, R. (2020) Global Sensitivity Analysis in Physiological Systems. Applied Mathematics, 11, 119-136. https://doi.org/10.4236/am.2020.113011