2006 Model of PRA After Nicardipine Treatment_Full

Journal of Information, Control and Management Systems, Vol. 4, (2006), No.2 203

MODEL OF A PLASMA RENIN ACTIVITY AFTER NICARDIPINE TREATMENT

Anna TOLEKOVA, Kaloyan YANKOV

Thracian University, Medical Faculty Stara Zagora, Bulgaria

e-mail: [email protected], [email protected]

Abstract

Mathematical model of plasma renin activity after nicardipine treatment is developed. System identification of the process is done applying the cyclic coordinate descent as optimization procedure. The model allows to predict the effects of different drug doses and allows the researcher to examine the behavior of the system under all conceivable conditions.

Keywords: system identification, mathematical model, cyclic coordinate descent, renin angeotensin system, plasma renin activity.

1 INTRODUCTION The values of blood pressure are an important indicator of human cardio-vascular

status. A considerable part of the long-term regulation is exerted by the renin-angiotensin system. The deviations of the blood pressure out of reference limits are treated with different groups of drugs, that influence various regulatory factors, including plasma renin activity (PRA). The later is an integral index of renin-angiotensin system state.

The changes of PRA and their regulatory kinetics have been studied after physiological stimulation [1] and after treatment with different drugs: inhibitors of calmodulin [2], blockers of prostaglandin syntase [3], angiotensin-converting enzyme inhibitors [4] and calcium channel blockers [5, 6] on male Wistar rats. Their dynamic characteristics are investigated employing statistics methods and numerical analysis [7, 8, 9, 10]. The received data are insufficient and it is incorrect to translate the results and the conclusions upon humans. In general resources and methods are necessary in order to predict the effects of certain treatment on humans and then to pass to the clinical research phase of that same treatment. An alternative approach, which proves its effectiveness for system evaluation and determination the quality of regulation at reasonable price, is the use of mathematical models [11]. The power of modeling lies in its abstraction, since a single family of models may describe many real systems. Mathematical modeling means to find out the relations that characterize the internal

Model of a Plasma Renin Activity 204

structure of the system and to describe the interdependencies between the input and output variables when the system is aftected by the environment in a particular way.

The purpose of this work is a mathematical model formulation of PRA dynamics under the influence of different doses of nicardipine. The model will be used for investigation of PRA from the point of view of system theory.

2 EXPERIMENTAL DESIGN The experiments were carried out on 145 male white Wistar rats. PRA was

assessed radioimmunologicaly (DiaSorin-Biomedica Ltd.) after perorally application of Nicardipine in doses 10, 20, 40 60 mg/kg body weight (b.w.). PRA was sampled in the intervals showed in a Table 1. Animals received humane care and the study complied with the Institution's guidelines of Thracian University and with the Guidelines for Breeding and Care of Laboratory Animals (Veterinary Public Health Reports, 1994).

Dose [mg/kg] b.w. Time

[hours] 10 20 40 60 0 7.58±0.8 7.58±0.8 7.58±0.8 7.58±0.8

0.5 27.13±3.1 31.68±3.1 32.13±2.9 33.80±6.8 1 35.18±4.5 40.92±9.4 44.44±1.7 51.22±11.2 3 40.07±6.4 45.91±3.4 48.56±2.6 49.61±8.7 5 23.21±4.3 27.55±4.5 28.84±3.9 31.18±5.4 7 11.17±2.8 12.47±1.5 14.58±1.5 17.24±3.8 9 7.58±0.8 7.83±0.9 8.85±0.9 9.80±2.1

11 7.56±0.6 7.50±0.7 7.80±0.6 7.58±0.4 Table 1. Plasma renin activity [ng/ml/h], presented as means and standard deviation.

3 DESIGN OF SYSTEM IDENTIFICATION PROCEDURE An identification experiment is performed by exciting the system using some sort

of input signals and observing a change in one of the system state variables as an output signal over a time interval. For modeling of the PRA production process, the experiment was designed to observe the following sequence: • Input signal U(t). A short perorally application of nicardipine is considered as

Dirac function. The signal amplitude is correlated to the drug dose. • Output response y(t). During the experiment a few values of the output signal

PRA are recorded. Let Φ(t) ⊂ y(t) be the observational vector, which is known from the experiment:

Φ(t) = [ϕ1, ϕ2,... ϕN,]Т, where N is the number of samples The vector Φ(t) is used during the identification process.


• Identification time tp. That is the time for process modeling. After this time the system response reaches the steady state level and then the system state variables are time independent. The maximal duration time was fixed to 11 hours.

• Sampling time. The first two samples were taken at 30 min and 1 hr and the subsequent were taken at every 2 hrs (see Table 1).

• Structure of the proces model: Consists of the mathematical equations which describe the behaviour of the process. The precision of the model depends on the appropriate choice of equations. Let y(t,q1,q2,…,qm) is the model for identification, t is the time, Q = [q1,q2,…,qm] is the identification vector make of the parametersin terms of which we want to describe the system behavior. Defining the values of those parameters is the object of the system identification. PRA follows an oscillation curve. The most appropriate model is the second order ordinary differential equation:

)()()(2)(0

22

2

tUKKtydt

tdydt

tyduϖϖζϖ =+++ (1)

where: U(t) – the input signal. In our case the applied dose of nicardipine. ζ(d) – the damping ratio. ω(d) - the undamped, natural frequency of the system. К0(d) – the base level. Ku(d) – the sensibility of the process to the input influence (proportionality

coefficient). The parameters above are unknown and they must be calculated in order to

identify the process. All of them are dose (d) dependent and they form the identification vector Q(d):

Q(d)=Q(ζ(d), ω(d), К0(d), Ku(d)) • Identification method. The mathematical model is determined using some

identification methods. For nonlinear models very few results have been obtained and no standard algorithm exists for testing global identifiability. Various approaches have been proposed, e.g., power series [12], differential algebra [13], stochastic approximation [14], similarity transformation methods [15]. As an identification method we use cyclic coordinate descent (CCD) [16] realized

by software KORELIA-DYNAMICS [17]. The values ϕi of the model y(t,Q) are known in m number of points ti :

y(ti, q1, q2,…,qm) = ϕi(ti), i=1,2,…,m (2) The identification vector Q must be defined from the equation system (2). But

this system is not well defined. For each point ϕi we define the residual di: di(Q) = ||yi-ϕi||

The aim is to minimize di for the identification time tp:


D(Q) = inf|| di(Q) || < ε, ε>0 (3) Thus the identification goal is translated into an optimization problem. The CCD

method is an iterative heuristic search technique that attempts to minimize D(Q) by varying one parameter at a time. It reduces the multi-criteria optimization to the single-criteria one. A number of iterations are made over the model equation to find the global minimum D for the system (3). The successive approximation of the identification parameters qi on the k-th iteration is evaluated using the procedure:

),...,,,...,(minarg 11121

−−+= k

Nkii

kkki qqqqDq θ

θ (4)

where argmin(…θ…) is the value of θ for which D(Q) has a minimum on the corresponding coordinate while the other coordinates are fixed. The implemented criterion to improve the estimation of the model parameters is the uniform fitting:

min||max)( →= idQD (5)

The criteria for termination of the identification process is the unequality: | D(Qк) - D(Qк-1) | < δ, where δ is an accuracy estimation (6)

CCD is simple for realization, relatively robust method. The main disadvantage is its locality and the fact that it may take a lot of calculations.

4 REZULTS The system identification is made using as a model the differential equation (1) at

initial conditions:

y(0) = 7.58 0)0(=

dtdy

(7)

The calculated values of the ζ(d), ω(d), К0(d) and Ku(d) using CCD are presented in Table 2.

Dose [mg/kg] b.w. parameter 10 20 40 60 ζ(d) 0.67 0.95 1.35 1.58 ω(d) 0.64 0. 64 0.64 0.64 К0(d) -5.97 -3.00 -1.62 0.42 Ku(d) 30.00 25.83 19.03 15.18 Table 2. Calculated values of the identification parameters

4.1 Damping ratio ζ(d). The dose-dependent values are graphically showed on Figure 1. The graph could

be described with the exponential growth model equation of the type :

constD

d

CeCd +−=∆+

−

∞ )1()(ζ (8)


Where the unknown parameters for identification are: C∞ = ζ(d→∞) D – dose-constant. ∆ – dose correction parameter; Cconst – free term

Using CCD, the calculated values are: C∞ =6.6; D=123.4; ∆=73.9; Cconst = -2.6 and the dose-dependent equation for the damping ratio is:

4.1239.73

4.1239.73

46.2)1(6.6)(+

−+

−−=−−=

dd

eedζ (9)

Figure 1. Damping ratio ζ(d)

4.2 Proportionality coefficient Ku(d).

Figure 2. Proportionality coefficient Ku(d)


This is an exponential decrease model of type:

constD

d

CeCd +=∆+

−

0U )(K (10)

Where the unknown parameters are: C0 = Ku(0) D – dose-constant. ∆ - dose correction parameter; Cconst – free term

Using CCD the calculated values are: C∞ =24.28; D=47.6; ∆=-457, and the obtained equation is:

6.47457

28.24)(−

−=

d

u edK (11)

4.3 Constant base level К0(d) – figure D. Some models were approved and the equation that gives minimal residual is:

K0 = C0*ln(d ) + Cconst (12) After optimization the calculated coefficients leads to the equation:

47.14)ln(64.3)(0 −= ddK (13)

Figure 3. Constant base level K0

Finally, the time and dose dependent plasma renin activity model is described by the system of equations:

4.1239.73

4)(+

−−=

d

edζ

6.47457

28.24)(−

−=

d

u edK 47.14)ln(64.3)(0 −= ddK

(14)


)()()(K)(406.0)()(27.1),(02

2

tUdKdtydt

tdyddt

dtydu=+++ ζ

With initial conditions:

y(0) = 7.58 0)0(=

dtdy

The graphics of the experimental data interpolated using cubic spline and generated models of PRA for dose of 10 mg.kg and 60 mg/kd are shown on Figure 4.

Figure 4. Experimental data and simulation curves of PRA after treatement with

doses 10 and 60 mg/kg b.w.

5 TRANSFER FUNCTION The transfer function summarizes a system in the Laplace domain. It is the ratio

of the output signal to the input one. A general second-order transfer function is:

22

2

2)(

ωζωω++

=ss

KsG U (15)

After substituting the corresponding coefficients from system (14) we obtain the dose-dependent transfer function:

4096.0)4.1239.73

6.6- .28(112

9.9451),(6.47

457

+

+−

+

=

−−

s

d

es

edsG

d

(16)


6 STATE-SPACE REPRESENTATION The state of a system is described by a set of first-order ordinary differential

equations (ODEs). The PRA model is described by second-order ODEs, therefore two variables x1(t,d) and x2(t,d), are necessary to represent the model in state-space. Upon substitution:

),(),(1 dtydtx =

dtdtdydtx ),(),(2 =

(17)

And the state space form is:

)()()()().()(),()()(2),(0

22 dUddKdKdtyddt

dtdydddt

dtdxu ϖϖϖζ +−−−=

(18)

For most practical purposes the analysis of the system is conducted by local linearization of the nonlinear system near a particular operating point XO(xO1, xO2) in the state space. This allows using well established tools from linear systems theory. The state space model (18) can be linearized around an operating point XO(xO1, xO2) using the Jacobi matrix J(t,d):

dX(t,d)/dt = J(t,d).(X(t,d) - XO) The state-space mathematical model for the kinetics of regulation of PRA can be

represented in matrix form:

)(.)()(

0),(),(

.)()(2)(

10),(

),(

22

112

2

1

dUddKxdtx

xdtxddd

dtdtdx

dtdtdx

uO

O⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡

−−

⎥⎦

⎤⎢⎣

⎡−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ϖωζω (19)

7 CONCLUSIONS Mathematical model is a powerful tool for better understanding and simulating of

processes. Modeling can be used to describe, interpret, predict or explain. Simulation results can be obtained at lower cost than real experimentation reducing the time and number of animals during the test.

Using a specialized software and aplying cyclic coordinate descent the mathematical model of PRA as a function of nicardipine dose is obtained. The model shows, that the natural frequency of PRA system is dose independent. The parameters ζ(d), К0(d) и Ku(d), have non-linear dependence of the nicardipine dose. The model is filled out with dose-dependent transfer function in Laplase domain and state space linearized model in matrix form. The proposed PRA model provides a means to study the role of various drugs and doses for regulation of renin-angiotensin system and human cardio-vascular status.


In future the efforts will be oriented to formulation of PRA models after treatement with different drugs [2,3,4,5,6], frequence ans stability analysis of obtained models and comparison of received models of studied drugs from the point of view of system theory.

REFERENCES

[1] TOLEKOVA, A., YANKOV, K.: System Analysis of Plasma Renin Activity upon Condition of Pharmacological and Physiological Stimulation. Proc Jubilee Scientific Conference, oct.18-20 2002, Stara Zagora, Bulgaria, Vol.1, Biomedical Sciences, pp. 61-65. (in Bulgarian).

[2] ILIEVA, G., TRIFONOVA, К., TOLEKOVA, A., LOGOFETOV, А.: Plasma renin activity in tachistin-stimulated hypercalcemia and under the effect of chlorazine. Eksp Med Morfol, 1994, XXXII (3-4): 1-7. [PubMed PMID: 8857027].

[3] TOLEKOVA, A., ILIEVA, G., TZANEVA, M., GANEVA, M.: Еffect of Acute and Chronic Acetylsalicylic Acid Administration on Renin-Angiotensin-Aldosterone System and Stomach Mucosa in Rats. Endocrine Regulations, Bratislava ,1995, 29(2): pp.115-120.

[4] TOLEKOVA, A. ILIEVA, G., PARVANOVA, A.: Comparison of the effects on RAAS of the combination of ACE inhibition with calcium channel blockade. IMAB Annual proceedings (scientific papers), 1996, V. 2(1), 209-210.

[5] ILIEVA, G., TOLEKOVA, A., SLAVOV, E., PARVANOVA, A., LOGOFETOV, A.: The change of Electrolyte Balance and Renin-Angiotensin-Aldosterone system after peroral and intraperitoneal treatment with Nifedipine. -In: Acid-Base and Electrolyte Balance, Molecular, Cellular, and Clinical aspects (eds. Natale Gaspare De Santo end Giovambatista Capasso), Papers from the Second G.A.Borelli Conference, presented July 8 and 9,1995, Istututo Italiano per gli Studi Filosofici,Naples, 1995, 335-340.

[6] YANKOV, К., TOLEKOVA, A., SPASOV, V.: Mathematical modelling of drug-induced changes in plasma renin activity, Pharmacia, 1998, Vol. 45 (1), 26-29.

[7] YANKOV, K.: Software Utilities for Investigation of Regulating Systems, Ninth Nat. Conf. "Modern Tendencies in the Development of Fundamental and Applied Sciences". June, 5-6, 1998, Stara Zagora, Bulgaria, pp.401-408.

[8] YANKOV, K.: Evaluation of Some Dynamics Characteristics of Transient Processes. 12-th Int.Conf. SAER'98. St.Konstantin resort, sept.19-20, 1998, Varna, Bulgaria,. pp.113-117.

[9] TOLEKOVA A., YANKOV, K.: System Analysis of Plasma Renin Activity upon condition of Pharmacological and Physiological Stimulation. Proc. Jubilee Scientific Conference, oct.18-20 2002, Stara Zagora, Bulgaria, Vol.1, Biomedical Sciences, pp. 61-65.

[10] TOLEKOVA, A.: Study of Plasma Renin Activity After some Physiological and Pharmacological Influences. Ph.D. Thesis, Sofia, 2003. (in Bulgarian).


[11] Bennett, B.: Simulation Fundamentals. Prentice Hall. 1995. ISBN 013-813262-3 [12] POHJANPALO H.: System Identifiability Based on the Power Series Expansion

of the Solution, Math. Biosci., vol. 41, pp. 21–33, 1978. [13] CARSON E., COBELLI, C., FINKELSTEIN, L.:The Mathematical Modeling of

Metabolic and Endocrine Systems. New York: Wiley, 1983. [14] PETROV N.: Stochastic Approximation for Estimation of Biological Models.

Trakia Journal of Sciences, vol.3, No2, pp 17-21, 2005. [15] Vajda S., K. Godfrey, and H. Rabitz, “Similarity Transformation Approach to

Identifiability Analysis of Nonlinear Compartmental Models,” Math. Biosci., vol. 93, 1989, pp. 217–248.

[16] BAZARAA M., SHETTY, C.: Nonlinear Programming. Theory and Algorithms. John Wiley and Sons. New York Chichester Brisbane Toronto, 1979.

[17] YANKOV, K.: System Identification оf Biological Processes. Proc. 20-th Int.Conf. "Systems for Automation of Engineering and Research (SAER-2006). St. Konstantin resort, sept.22-24, 2006, Varna, Bulgaria.

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2006 Model of PRA After Nicardipine Treatment_Full