Numerical Simulations of Stochastic Circulatory Models

Transcription

1 Bulletin of Mathematical Biology (1999) 61, Article No. bulm Available online at on Numerical Simulations of Stochastic Circulatory Models Gejza Wimmer Mathematical Institute, Slovak Academy of Sciences, SK Bratislava, Slovak Republic Ladislav DedÍk, Marek Michal, Andrea MudrÍkovÁ Department of Automation and Measurement, Faculty of Mechanical Engineering, Slovak echnical University, SK Bratislava, Slovak Republic MÁria ĎuriŠovÁ Institute of Experimental Pharmocology, Slovak Academy of Sciences, SK Bratislava, Slovak Republic Properties of two of the stochastic circulatory models theoretically introduced by Smith et al., 1997, Bull. Math. Biol. 59, 1 22 were investigated. he models assumed the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme. he reason for selecting these models was the fact that the probability density functions of the residence time of these models are formally similar to those of the Bateman and gamma-like function models, i.e., the two common deterministic models. Using published data, the analytical forms of the probability density functions of the residence time and the distributions of the simulated values of the residence time were determined on the basis of the deterministic models and the stochastic circulatory models, respectively. he Kolmogorov Smirnov test revealed that even for 1 xenobiotic particles, i.e., a relatively small number if the particles imply drug molecules, the probability density functions of the residence time based on the deterministic models closely matched the distributions of the simulated values of the residence time obtained on the basis of the stochastic circulatory models, provided that parameters of the latter models fulfilled selected conditions. c 1999 Society for Mathematical Biology 1. INRODUCION Stochastic circulatory models have been developed from the viewpoint of stochastic theory, taking into account the blood circulatory system (BCS). he observed o whom all correspondence should be addressed /99/ $3./ c 1999 Society for Mathematical Biology

2 366 G. Wimmer et al. time course of the blood concentration of a xenobiotic consists of a finite series of local time courses which are generated in each cycle passage through the BCS. he basic variables of the stochastic circulatory models are as follows: (1) number of cycles, i.e., the number of the passages of the xenobiotic particle through the BCS; (2) cycle time, i.e., the time necessary for the xenobiotic particle to return to the injection site; (3) residence time, i.e., the time necessary for the xenobiotic particle to leave the BCS. Mathematical formulations of the stochastic circulatory models, based on different continuous and/or discrete probability density functions of their basic variables, were described in the literature (Waterhouse and Keilson, 1972; Weiss, 1986; Mari, 1993, 1995; Lánský, 1996; Smith et al., 1997). he goals of our study were twofold: (1) to present the practical applicability of two of the stochastic circulatory models theoretically introduced by Smith et al. (1997), by employing measurements obtained in real world biomedical studies; (2) to determine the relationships between parameters of these models and parameters of two common deterministic models with the probability density functions of the residence time formally similar to those of the stochastic models. 2. HEORY he residence time of the xenobiotic particle in the BCS is a non-negative random variable with the probability density function h (t), where t is time. Under the assumption that the xenobiotic particles are identical (at least with respect to the distribution of the residence time), behave independently of one to another, and once eliminated they never return back into the BCS, it is commonly considered that the probability density function h (t) can be estimated according to equation (1): h (t) C(t) C(t)dt, (1) where C(t) is either a measured concentration-time profile after a single bolus administration of the xenobiotic, or its deterministic model. Equation (1) implies the following formulae: MR = th (t)dt tc(t) C(t)dt, (2) for the mean residence time (MR) of the xenobiotic particles in the BCS. Equation (3): = J j, (3) j=1

3 Stochastic Model Simulations 367 defines the stochastic circulatory model used through our study. he residence time is considered to be composed of a random number of the cycle times j s and J is a discrete positive random variable, representing the number of the cycles of the xenobiotic particle in the BCS. he cycle time j is a non-negative continuous random variable and the j s are assumed to be independent and identically distributed with a common cumulative distribution function. Equation (3) represents a form of the general stochastic circulatory model theoretically introduced by Smith et al. (1997), simplified by using assumptions that the xenobiotic particle is eliminated at the instant of completing the cycle J and that the first cycle is obligatory for all the injected xenobiotic particles. Below, the two stochastic circulatory models representing special cases of the model given by equation (3) are described Stochastic model description. In the model M 1, the cycle time j was assumed to follow the gamma distribution given by equation (4): h j (t) = t α 1 e t β, t, (4) β Ɣ(α) where α and β are the parameters, α, β>. he number of cycles J was assumed to follow the shifted geometric distribution with the probability mass function expressed by equation (5): h J ( j) = p(1 p) j 1, j = 1, 2,..., (5) where p is the parameter, < p < 1. In the model M 2, the same distribution of the cycle time j was assumed as in model M 1. he number of cycles J was assumed to follow the shifted Poisson distribution with the probability mass function expressed by equation (6): ν j 1 h J ( j) = e ν, j = 1, 2,..., (6) ( j 1)! where ν is the parameter, ν>. he models M 1 and M 2 were simplified in the following way: (1) in the model M 1, the parameter value α = 2 was employed in the probability density function given by equation (4); (2) in the model M 2, the conditions given by expressions (7), (8), and (9): α, (7) ν +, (8) α(ν + 1) ψ i.e., αν ψ, (9) where ψ was a real number, ψ>, were assumed to be valid for the parameter α of the probability density function given by equation (4) and the parameter ν

4 368 G. Wimmer et al. of the probability density function given by equation (6). he probability density functions h (1,s) (t) and h (2,s) (t) corresponding to the simplified form of the models M 1 and M 2 have the form of equations (1) and (11): h (1,s) (t) = h (2,s) p 2β 1 p (t) = t ψ 1 e t β β ψ Ɣ(ψ) ( e ( 1 1 p β ) t e ( 1+ 1 p β ) ) t, t, (1), t, (11) representing the exponential probability density function and the gamma probability density function, respectively, of the residence time. he MR in the form of equation (12) and equation (13), corresponding to the models M 1 and M 2, respectively, MR = 2β p, (12) MR = αβ(ν + 1), (13) can be determined using formulae (2), (1), and (11). Both equation (12) and equation (13) represent the products of the mean cycle time and the mean cycle number. In equation (12) the mean cycle time and the mean cycle number are given by 2β and 1/p, respectively. In equation (13) these variables are given by αβ, and ν + 1 (Smith et al., 1997) Stochastic and deterministic model relationships. Equations (14) and (15): C(t) = (e At e Bt ), t, (14) C(t) = ϱt ζ e λt, t, (15) are the two deterministic models commonly used in biomedicine,, A, B, ϱ, ζ, and λ are the parameters, B > A, and A, B,ϱ,ζ,λ > (Sheiner and Beal, 1985; Weiss, 1986). According to equation(1), the probability density functions h (1,d) (t) and h (2,d) (t) of the residence time corresponding to these models can be written in the form of equations (16) and (17): h (1,d) (t) = AB ( e At e Bt), t, (16) B A h (2,d) (t) = t ζ λ ζ +1 e λt, t. (17) Ɣ(ζ + 1)

5 Stochastic Model Simulations 369 he comparison of equations (1) and (16) yields expressions (18) and (19): p = 4AB (A + B) 2, (18) β = 2 A + B, (19) representing the relationships between the parameters p and β of the probability density function h (1,s) (t) of the stochastic model M 1 and the parameters A and B of the probability density function h (1,d) (t) corresponding to the deterministic model given by equation (14). Analogously, the comparison of equations (11) and (17) yields expressions (2) and (21): ψ = ζ + 1, (2) β = λ 1, (21) showing the relationships between the parameters ψ and β of the probability density function h (2,s) (t) of the stochastic model M 2 and the parameters ζ and λ of the probability density function h (2,d) (t) corresponding to the deterministic model given by equation (15). he MR in the form of equation (22) and equation (23) corresponding to the deterministic models given by equation (14) and (15), respectively MR = 1 A + 1 B, (22) MR = 1 + ζ λ, (23) can be determined using formulae (2), (16), and (17). Expressions (22) and (23) are analogous to expressions (12) and (13). 3. MAERIAL AND MEHODS Using a standard non-linear least squares method, deterministic models were obtained in the respective form of equations (14) and (15): (1) of the serum concentration-time profile of salicylic acid ingested by volunteer A in a single dose of 1 mg of the drug (rnavská and rnavský, 1983); (2) of the serum concentration-time profile of 14 C-cholesterol in guinea pigs after a single intracardial dose of 15 Bq of 14 C-cholesterol (Ginter, 1975). he probability density functions h (1,d) (t) and h (2,d) (t) corresponding to these models were determined according to equations (16) and (17). For both stochastic models M 1 and M 2, the simulated values of the residence time were generated as described below, for the same values of the total particle numbers N arbitrarily chosen from the interval N [1, 1 ].

6 37 G. Wimmer et al. For the model M 1, the parameters p and β of the probability density function h (1,s) (t) were estimated using equations (18) and (19). he value of the parameter α in the gamma distribution given by equation (4) was α = 2. Realizations of the simulated numbers of the cycles J n following the shifted geometric distribution given by equation (5), i.e., the random numbers of the cycles j n of the nth particle n [1, 2,...,N], were generated by using equation (24): ln u n j n = 1 + int ln(1 p), (24) where u n [, 1] were random numbers generated on the basis of the uniform distribution and the term int indicated an integer part (Kruskal, 1969). For the model M 2, the parameters ψ and β of the probability density function h (2,s) (t) were estimated according to equations (2) and (21). Realizations of the simulated numbers of the cycles J n following the shifted Poisson distribution given by equation (6), i.e., the random numbers of the cycles j n of the nth particle n [1, 2,...,N], were generated as: j n = K n, (25) where K n was the first number fulfilling condition (26): K n k n =1 u kn < e ν, (26) u kn [, 1] were random numbers generated on the basis of the uniform distribution, and ν was the parameter of the shifted Poisson distribution given by equation (6) (Kruskal, 1969). his parameter was calculated as ν = ψ/α, where α was the parameter of the gamma distribution given by equation (4). In order to meet the conditions given by expressions (7) (9), the following arbitrarily selected values of the parameter α were used: 1,.1,.1, and.1. he j n -fold self-convolution of the probability density function given by equation (4) was used to obtain the probability density function h n (t) of the residence time n of the nth particle performing j n cycles in the form of equation (27): h n (t) = t αjn 1 e t β, t. (27) β αj n Ɣ(αjn ) he realizations t n of the simulated residence time n were obtained according to equation (28): t n = H 1 n (u n ), (28)

7 Stochastic Model Simulations Salicylic acid (µg ml 1 ) ime (h) Figure 1. Serum concentration-time profile of salicylic acid in volunteer A after a single oral dose of 1 mg of the drug (rnavská and rnavský, 1983) (full circles); deterministic model in the form of the Bateman function given by equation (14) (full line); deterministic model in the form of the gamma-like function given by equation (15) (dashed line). where H n was the distribution function corresponding to the probability density function h n, H 1 n was the inverse function of H n, and u n [, 1] were random numbers generated on the basis of the uniform distribution (Kruskal, 1969). he simulated values t 1, t 2,...,t n corresponding to the stochastic models M 1 and M 2 were generated using the respective values of the parameters of these models and equations (27) and (28). Finally, the histograms of the simulated values t 1, t 2,...,t N were constructed for both stochastic models. he Kolmogorov Smirnov test (Sachs, 1978) was used to test the null hypothesis that the distributions of the simulated values of the residence time obtained on the basis of the stochastic models M 1 and M 2 conformed to the probability density functions derived on the basis of the deterministic models given by equation (16) and equation (17), respectively. 4. RESULS he estimation of the parameters, A, and B of the deterministic model in the form of equation (14), used to approximate the serum concentration-time profile of salicylic acid in volunteer A, yielded the values = 86 µg ml 1, A =.212 h 1, and B = 1.34 h 1. Both the data and model are illustrated in Fig. 1. he calculation of the parameters p and β according to equations (18) and (19) yielded the values p =.472 and β = 1.29 h. Using these results, equation (29): h (1) (t) =.252(e.212t e 1.34t ) (29) could be written. he function h (1) (t), representing the estimate of the probability density functions h (1,d) (t) and h (1,s) (t) given by equations (16) and (1), i.e., by the

8 372 G. Wimmer et al..18 Probability density ime (h) 2 25 Figure 2. Probability density function given by equation (29) of the residence time of salicylic acid in volunteer A after a single oral dose of 1 mg of the drug (rnavská and rnavský, 1983) (full line); histogram of the simulated values of the residence time of salicylic acid in the volunteer obtained on the basis of stochastic model M 1 with the parameters α = 2, β = 1.29 h, and p =.472, for 1 particles C-cholesterol (Bq µmol 1 ) ime (day) Figure 3. Serum concentration-time profile of 14 C-cholesterol in guinea pigs after a single intracardial dose of 15 Bq of 14 C-cholesterol (Ginter, 1975) (full circles); deterministic model in the form of the gamma-like function given by equation (15) (full line). deterministically and/or stochastically based formulae, is illustrated in Fig. 2. his figure simultaneously shows the histogram of the simulated values of the residence time obtained on the basis of the stochastic model M 1 with the parameters α = 2, β = 1.29 h, and p =.472, for 1 particles. he calculation of the value of the test statistic D of the Kolmogorov Smirnov test applied to the results shown in Fig. 2 yielded the value of.27. he critical value of the test statistic D is.43 for a significance level of.5 (Sachs, 1978). It follows then, that the distribution of the simulated values of the residence time shown in Fig. 2 can be said to conform to the probability density function h (1) (t), even for the lowest value of the particle number used in this study.

9 Stochastic Model Simulations Probability density ime (day) Figure 4. Probability density function given by equation (3) of the residence time of 14 C- cholesterol in guinea pigs after a single intracardial dose of 15 Bq of 14 C-cholesterol (Ginter, 1975) (full line); histogram of the simulated values of the residence time of 14 C- cholesterol obtained on the basis of stochastic model M 2 with the parameters α 1 = 1, β = 37.7 day, and ν 1 =.545, for 1 particles. he estimation of the parameters ϱ, ζ, and λ of the deterministic model in the form of equation (15), used to approximate the serum concentration-time profile of 14 C-cholesterol in guinea pigs, yielded the values ϱ = 39 Bq µmol 1 day.455, ζ =.455, and λ =.265 day 1. Both the data and model are shown in Fig. 3. he calculation of the parameter ψ and β according to equations (2) and (21) yielded the values ψ =.545 and β = 37.7 day. Using these results, equation (3): h (2) (t) =.848t.455 e.265t (3) could be written. he function h (2) (t), representing the estimate of the probability density functions h (2,d) (t) and h (2,s) (t) given by equations (17) and (11), i.e., by the deterministically and/or stochastically based formulae, is illustrated in Figs 4 and 5. hese figures simultaneously show the histograms of the simulated values of the residence time obtained for 1 particles on the basis of stochastic model M 2 with the parameters α = 1, β = 37.7 day, and ν =.545 or the parameters α =.1, β = 37.7 day, and ν = 545. he calculation of the values of the test statistic D of the Kolmogorov Smirnov test applied to the results presented in Figs 4 and 5 yielded the value of.299 and.137, respectively. It follows then, that the distribution of the simulated values of the residence time obtained for the model M 2 with the highest value of the parameter α did not conform to the probability density function h (2) (t). On the other hand, the results presented in Fig. 5 indicate that the distribution of the simulated values of the residence time for the same model as that in Fig. 4 but with the lowest value of the parameter α conformed to the probability density function h (2) (t).

10 374 G. Wimmer et al..2 Probability density ime (day) Figure 5. Probability density function given by equation (3) of the residence time of 14 C- cholesterol in guinea pigs after a single intracardial dose of 15 Bq of 14 C-cholesterol (Ginter, 1975) (full line); histogram of the simulated values of the residence time of 14 C- cholesterol obtained on the basis of stochastic model M 2 with the parameters α 4 =.1, β = 37.7 day, and ν 4 = 545, for 1 particles..18 Probability density ime (h) Figure 6. Probability density function based on equation (17) of the residence time of salicylic acid in volunteer A after a single oral dose of 1 mg of the drug (rnavská and rnavský, 1983) (full line); histogram of the simulated values of the residence time of salicylic acid obtained on the basis of stochastic model M 2 with the parameters α =.1, β = 2.95 h, and ν = 15.9, for 1 particles. Both stochastic models M 1 and M 2 produced the distributions of the simulated values of the residence time for the values 1 < N 1 similar to these for N = DISCUSSION In biomedicine, the deterministic models given by equation (14) and equation (15) are called, respectively, the Bateman function (Sheiner and Beal, 1985) and the gamma-like function (Weiss, 1983). he model in the form of equation (14) is very

11 Stochastic Model Simulations 375 frequently employed in practice to describe the drug concentration-time profiles after a single bolus extravascular administration of the drug (mostly oral administration), where = Dose Cl AB B A, (31) Dose stands for the given amount of the drug, and Cl, A, and B are interpreted as the drug clearance, elimination rate constant, and absorption rate constant, respectively. he gamma-like function given by equation (15), where ϱ = Dose Cl λ ζ +1 Ɣ(ζ + 1), (32) is mostly used in theoretical biomedical studies (Weiss, 1983). As presented in this study, the parameters of the stochastic models M 1 and M 2, assuming the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme, can be derived on the basis of the parameters of the deterministic models in the form of the Bateman function and the gamma-like function, respectively. It follows then that the probability density functions of the residence time given by equations (29) and (3) can be interpreted physiologically in the two different ways, i.e., in the terms of the deterministic or stochastic models. Eventually, it is possible to perform an interesting comparison. he Bateman function model of the drug profile shown in Fig. 1 yielded the value 35.1 of the Akaike information criterion (AIC) (Akaike, 1976). he same figure illustrates another successful approximation of this profile, i.e., the gamma-like function model with the parameters ϱ = 64.8 gµml 1 h.592, ζ =.592, and λ =.339 h 1, yielding the AIC value of Using the parameters of the latter model, the probability density function of the residence time of the drug can be determined according to equation (17) and the simulated values of the residence time can be generated using the stochastic model M 2 with the parameters α =.1, β = 2.95 h, and ν = 15.9 for 1 particles, as presented in Fig. 6. he value of the test statistic D of the Kolmogorov Smirnov test applied to the results shown in this figure was.186. Furthermore, the calculation of the MR for both deterministic models according to expressions (22) and (23) and for both stochastic models according to expressions (12) and (13) yielded the value 5 h, due to the analogy between these expressions. On the basis of these results, the observation can be made that, in this example, the two different deterministic models produced similar profiles and the two different stochastic models yielded similar distributions of the residence time with similar values of the MR. However, the mean cycle time and the mean number of the cycles corresponding to these two different stochastic models were different. he calculation of the mean cycle time and of the mean number of the cycles of the model M 1 yielded the values 2.57 h and 2 h, respectively. On the other hand, the analogous calculation for the model M 2 yielded the values.295 h

12 376 G. Wimmer et al. and 17 h. According to Smith et al. (1997), the stochastic model M 2 corresponds to a high number of very short cycles needed for the elimination of the xenobiotic particles from the BCS. On the other hand, the stochastic model M 1 corresponds to a situation when the difference between the numbers of short and long cycles need not be very marked. It follows then, that an a priori knowledge about elimination processes of the xenobiotic under study could be useful for discrimination between these models in their applications in real world biomedical studies. 6. CONCLUSIONS he present simulation study was designed to provide indicative rather than general results in the area of properties of the stochastic circulatory models. Yet, as recently pointed out by (Lánský, 1996) in the view of the practical importance of this new interesting class of models, before their application in routine biomedical analysis, a broad range of problems needs to be addressed. he examples of the stochastic circulatory models presented in our study gave results similar to those of the deterministic models even for small numbers of particles, i.e., 1 provided that the parameters of the stochastic circulatory models fulfilled selected conditions. his number is very small if the particles imply drug molecules. However, if the particles imply microspheres, for example those used in various diagnostic methods (Huang et al., 1998; Mizuno et al., 1998), the results presented indicate that in the evaluation of measurements obtained by these techniques the stochastic models might have a potential advantage. 7. ACKNOWLEDGEMENS his work was supported in part by Grants 2/4196/98 and 2/4249/98 from the Slovak Grant Agency. he authors are grateful to the journal reviewer for her or his very helpful comments and suggestions on an earlier version of this paper. REFERENCES }Akaike, H. (1976). Canonical correlation analysis of time series and the use of an information criterion, in System Identification. Advances and Case Studies, A. F. Mehra and D. G. Lainiotis (Eds), New York: Academic Press, pp }Ginter, E. (1975). he role of vitamin C in cholesterol catabolism and artherogenesis, Bratislava: Veda SAV, pp }Huang, Y.., Y. R. Cheng, H. C. Lin, S. M. Chen and C. Y. Hong (1998). Haemodynamic effects of chronic octreotide and tetrandrine administration in portal hypertensive rats. J. Gastroenterol. Hepatol. 13,

13 Stochastic Model Simulations 377 }Kruskal, J. B. (1969). Extremely portable pandom number generators. Commun. ACM. 12, }Lánský, P. (1996). A stochastic model for circulatory transport in pharmacokinetics. Math. Biosci. 132, }Mari, A. (1993). Circulatory models of intact-body kinetics and their relationship with compartmental and non-compartmental analysis. J. heor. Biol. 16, }Mari, A. (1995). Determination of the single-pass impulse response of the body tissues with circulatory models. IEEE rans. Biomed. Eng. 42, }Mizuno, N., Y. Kato, K. Shirota, Y. Izumi,. Irimura, H. Harashima, H. Kiwada, N. Motoji, A. Shigematsu and Y. Sugiyama (1998). Mechanism of initial distribution of blood-borne colon carcinoma cells in the liver. J. Hepatol. 28, }Sachs, L. (1978). Angewandte Statistik, Berlin: Springer-Verlag, pp }Sheiner, L. B. and S. L. Beal (1985). Pharmacokinetic parameter estimates from several least squares procedures: superiority of extended least squares. J. Pharmacokin. Biopharm. 13, }Smith, C. E., P. Lánský and. H. Lung (1997). Cycle-time and residence-time density approximations in a stochastic model for circulatory transport. Bull. Math. Biol. 59, }rnavská, Z. and K. rnavský (1983). Sex differences in the pharmacokinetics of salicylates. Eur. J. Clin. Pharmacol. 25, }Waterhouse, C. and J. Keilson (1972). ransfer time across the human body. Bull. Math. Biophys. 34, }Weiss, M. (1983). Use of gamma distributed residence times in pharmacokinetics. Eur. J. Clin. Pharmacol. 25, }Weiss, M. (1986). Generalizations in linear pharmacokinetics using properties of certain classes of residence time distributions. I. Log-convex drug disposition curves. J. Pharmacokin. Biopharm. 14, Received 2 August 1998 and accepted 15 December 1998