System of Thermal Transfers Inside the Lungs Using Fractional Models Pierre Melchior*, Mathieu Pellet*, Youssef Abdelmoumen** Alain Oustaloup* *IMS UMR 528 CNRS Université de Bordeaux, IPB/ENSEIRB-MATMECA, 35 cours de la Libération, Bât. A4, F3345 TALENCE Cedex, France (first name.last name@ims-bordeaux.fr) **Hopital du Haut-Lévéque, CHU de Bordeaux, Avenue Magellan, 3364 PESSAC Cedex, France (youssef.abdelmoumen@chu-bordeaux.fr) Abstract: This paper is about fractional identification of thermal transfers in the lungs. Usually, during open-heart surgery, an extracorporeal circulation (ECC) is carried out on the patient. In order to plug the artificial heart/lung machine on the blood stream, the lungs are disconnected from the circulatory. No longer receiving blood, pulmonary cells die which, for the patient, may result in postoperative respiratory complications. A method to protect the lungs has been developed by surgeons anesthetists. It is called: bronchial hypothermia. The aim is to cool the organ in order to slow down its deterioration. Unfortunately the thermal properties of the lungs are not well-known yet. Mathematical models are useful needed in order to improve the knowledge of these organs. As proved by several previous works, fractional models are very appropriate to model thermal s (model compactness, accuracy) the dynamic of fractal s. Thus, this paper studies the comparison between two fractional models, a classical one another one using the Havriliak-Negami function. Keywords: System identification, Bioengineering, Fractional model, Lungs model, Non-integer differantiation.. INTRODUCTION Most of the time during open-heart surgery, it is necessary to stop the heart from beating to allow the surgeon to operate properly. This is possible thanks to artificial heart/lung device. This device is called extracorporeal circulation (ECC). In order to connect the machine with the circulatory, the heart lungs are excluded from the blood stream. The lack of blood in the lungs during ECC is recognized as a potential cause of postoperative pulmonary dysfunction. Thus, anesthetists are developing methods to protect the lungs. One of these methods is to cool down the lungs by insufflating fresh gas inside it. This method has shown results but is based on the assumption that the thermal insulation of the lungs increases during This assumption has not been verified yet. A solution to verify it, is to design a thermal model of the lungs during ECC during stard condition, to compare both models. This paper presents the first step of this study. Different models of thermal transfer inside the lungs are identified. Fractional transfer functions are especially appropriate to model thermal Victor et al.(29),battaglia et al.(2) Battaglia(22), Aoun et al.(23) biological s Sommacal et al.(27a, b), Sommacal et al.(28), Pellet et al.(2), Ionescu et al.(2a, b), Magin (2) they are useful in identification because they allow a few number of parameter (parametric compactness of the model) Oustaloup et al.(25), Gabano Poinot (2). Among the fractional functions, the Havriliak-Negami function is rarely used, though it showed interesting results in identification Sommacal(27). Thus this paper studies the comparison between two fractional models, a classical one another one using the Havriliak-Negami function. The results of these models are also compared with an integer model in order to emphasize the interest of fractional model with such application as biological /or thermal. This paper is composed of four parts. First, the approximations used for time simulation of fractional operator Havriliak-Negami function are recalled. The second part deals with the experimental protocol. The results of the identification comparison of the models are presented in the third part. Finally, a conclusion considers the prospect. 2. FRACTIONAL SYSTEMS BACKGROUND This part, recalls mathematical definition of non integer derivative, then usual methods to simulate fractional s. 2. - Non integer differentiation The non integer derivative concept of any order (non integer), was defined in the 9th century by Riemann Liouville. Fractional derivative of a function is defined by the ()
order derivative ( is the floor operator) of a fractional order integral Oustaloup(995) :,, the Euler's function is defined by :. (3) The Laplace Transform of the order derivative (with ) of a function relaxed at the initial instant is such as Oldham Spanier(974) : 2.2 Frequency-bounded integro differential operator synthesis Because of the natural limits of a physical of the /output signals (e.g. Shannon frequency signals spectrum), fractional operators are usually approximated with high rational order transfer function, over a frequency range [ ]. Thus, the integro-differential operator the rational transfer function have the same fractional behavior over this frequency range. The most commonly used approximation was suggested by Oustaloup(995). It consists in a recursive distribution of poles zeros over frequency range. Such a method naturally induces a degradation of the approximation around the frequencies. This side-effect is known as "boundary effect". The approximation error is decreased over by widening the frequency range over which the operator is synthesized, to [ ], where Boundary effect is considerably reduced by choosing Oustaloup(995). The approximation of the integrodifferential operator over the [ ] frequency range is given by the relation : where is an integer which is the number of pole-zero pair where, pulsations are given by : (7) is calculated in order to obtain a unitary gain at the frequency. Real parameters are defined by the integrodifferential order, such as (2) (4) (5) (6) (8) (9) () 2.2 Approximation time simulation of Havriliak Negami function In this paper, the Havriliak-Negami function Havriliak Negami(966), Havriliak Negami(967): () is used for identification synthesized for,. The method used for the synthesis time simulation of the Havriliak-Negami function was developed by Sommacal in Sommacal(27). Because the details of the approximation are, this part presents only the general idea. For futher explanations readers should refer to Sommacal(27). It is shown that the function () can be written as : with where is the number of s-roots of function. (2) (3) (4) Function is a product of classical Davidson-Cole functions which can be approximated using a recursive distribution of pole-zeros as presented above in the previous part of the paper. On the other h, is approximated using a non recursive distribution of pole-zeros. In this way, the Havriliak-Negami function is approximated on a frequency range as a product of pole-zeros. 3. EXPERIMENTAL PROTOCOL Dr Youssef Abdelmoumen his team are specialized in open-heart surgery. They provide the thermal acquired on sheep. The experiment was carried out at the PTIB platform of Xavier Arnozan Hospital. The general aim of this work is to study the variation of thermal characteristics of the lungs during extracorporeal circulation. Thus, the medical team recorded on a sheep the temperature of the lung during open-heart surgery before the ECC during the Lungs have a tree-like structure. From the trachea they are subdivided in series of bronchial tubes as shown in Fig.. The temperature inside the lungs is measured at the entry of the first three consecutive bronchial tubes (,, ). The temperature of the body is also measured. All these temperature are measured thanks to GI type thermocouple. These thermocouples are linked to a National Instruments PXI interface. A special software was designed with LABVIEW load into the PXI in order to acquire record the signals.
The measures are limited to these three locations because of the difficulty to accurately plant a sensor deeper into the lung. 4. SYSTEM IDENTIFICATION If ones want to study the variation of the thermal characteristics of the lungs during ECC, then one can identify two different models of the thermal transfer in the lungs, during normal condition of the organ during ECC, finally compare both models. However this paper presents only the first step which is the result of the identification of the models. The temperatures, are normalized by. Thus, two transfers are identified: where (5) (6) (7) Two different types of fractional models are used for the identification : with. trachea Fig. Rough diagram of lungs, the black dots ( approximate location of the thermal sensors ) show the (8) (9) model the dynamic behavior of the (s) transfer, while model. All models are identified with an output error structure. The vector of parameter is optimized using the non linear simplex optimization algorithm Subrahmanyam(989), Woods(985). As for the vector it is optimized using the oe function from the CRONE Toolbox of, the optimization algorithm used by this function is the Levenberg Marquadt one. The are sampled at, thus the Havriliak- Negami function is approximated over the frequency range: (2) The are divided in two parts, one used to identify the model parameters (identification ) the other to validate the value of the parameters (validation ). 4. Results The results of identification are shown on Fig. 2, 3, 4, 5, 6, 7, 8 9. On these figures the red curves are the model response using the Harviliak-Negami function, the green curves are the model responses using classical fractional models, the blue curves are the signals ( ), the black curves are the response ( ). These figures shows that the results of are the same. However the classical fractional models have one less parameter than the Havriliak-Negami models. Thus, for this application the models are sufficient more interesting..5 -.5 5 5 2 25 3 35 4.2 -.2 5 5 2 25 3 35 4 Fig. 2 System identification of before.5 -.5 Fig. 3 modeling error of modeling error, 5 5 2 25 3 35 4.5 -.5 -. 5 5 2 25 3 35 4, before
Temperature( C).5.5.5 -.5 -.5.5 5 5 2 25 3 35 4.5 -.5 5 5 2 25 3 35 4.5 5 5 2 25 3 35 4. -. 5 5 2 25 3 35 4 Fig. 4 System identification of during modeling error, Fig. 7 modeling error of, before.5.5.5 -.5.5 5 5 2 25 3 35 4.5 -.5 5 5 2 25 3 35 4 Fig. 5 modeling error of, during -.5.5 5 5 2 25 3 35 4.5 -.5 5 5 2 25 3 35 4 Fig. 8 System identification of during modeling error,.5 model model.5 -.5 -.5.5 5 5 2 25 3 35 4. -. 5 5 2 25 3 35 4.5 5 5 2 25 3 35 4.5 -.5 5 5 2 25 3 35 4 Fig. 6 System identification of before modeling error, Fig. 9 modeling error of, during
Because the models are identified as black box, it is difficult to give a physical or physiological meaning to their parameters. However, Tables show that the parameter is the one that varies the most. Table Parameters values of pre ECC ECC pre ECC ECC.5.65.48.92 -.32.2.3 -.6 3.4 3.4 8.79.78.24 e-3..9.87.67 2.27.25 Table 2 Parameters values of pre ECC ECC pre ECC ECC.2747.376.572.387.793.59.546.7256.8925.28 3.366.522.36.74.5379.538 4.2 Comparison between fractional models integer models. In order to emphasize the interest of the fractional identification, two non integer models ) ) are identified using the same. The transfer function of these models is: (2) The vector of parameters is optimized using the oe function of which means an output error model is applied the optimization algorithm is a gradient/hessian type. The values of the parameters identified are shown on table 2. Table 3 Parameters value of pre ECC ECC pre ECC ECC.4.65.22.28.28.2.64 -.5.8 3.4.6 -.4.46 e-3.2 -.6.97.67.95.34 To compare the accuracy of the fractional integer models the criterion is defined (22) where is the number of sample in which is the response of the, is the response of the model. The values obtained for are presented on tables 3, 4, 5 6. Table 4 Values of calculated for the model of the transfer between before Fractional model HN(s)/M(s) 7.6e-4/8.3e-4 7.3e-4/8.e-4 Integer model Table 5 Values of calculated for the model of the transfer between during Fractional model HN(s)/M(s).23/.8.35/.6 Integer model Table 6 Values of calculated for the model of the transfer between before Fractional model HN(s)/M(s) 9.5e-4/9.5e-4 8.8e-4/8.9e-4 Integer model Table 7 Values of calculated for the model of the transfer between during Fractional model HN(s)/M(s).72/.83.2/.4 Integer model Tables 5 to 8 show that for each case, the criterion is minimized by the fractional model. Thus the accuracy of the fractional models are better for a same number of parameters than the integer models. 5. CONCLUSION Eight models of bronchial tubes thermal transfer are identified considering two different conditions ie with or without A significant variation in the parameters of the models is observed. For now, these variations are not linked with a physical meaning. Comparison between the fractional models show that the Havriliak-Negami is not very suited for this application. Nevertheless the comparison between an integer model fractional models show the interest of the latter for black box identification. The main focus of the followings works will be to link the parameter of the model with a physical or physiological meaning. 6. REFERENCES Aoun, M., Malti, R., Levron, F., Oustaloup, A. (23). Orthonormal basis functions for modelling continuous-time fractional s. In Sysid-IFAC. Battaglia, J.L., Le Lay, L., Batsale, J.C., Oustaloup, A., Cois, O. (2). Utilisation de modèles d identification non entiers pour la résolution de problèmes inverses en conduction. International Journal of Thermal Sciences, 39(3), 374 389.
Battaglia, J.L.(22). A modal approach to solve inverse heat conduction problems, Inv. Prob. Eng. & Sci., 4 Gabano J.D. Poinot T. (2). Fractional modelling identification of s, Signal Processing, In Press. Havriliak, S. Negami, S. (966). A complex plane analysis of alpha dispersions in some polymers. Journal Polymer Science, 4(B), 99 7. Havriliak, S. Negami, S. (967). A complex plane representation of dielectric mechanical relaxation processes in some polymers. Polymer, 8, 6 2. Ionescu, C.-M., Hodrea, R., De Keyser, R. (2a). Variable time-delay estimation for anesthesia control during intensive care, IEEE Transactions on Biomedical Engineering, Vol. 58 (2), pp. 363-369. Ionescu, C.-M., Machado, J.A.T., De Keyser, R. (2b). Modeling of the lung impedance using a fractional-order ladder network with constant phase elements, IEEE Transactions on Biomedical Circuits Systems, Vol. 5(), pp. 83-89. Magin R.L. (2). Fractional calculus models of complex dynamics in biological tissues, Computers Mathematics with Applications; 59: 586593. Oldham, K. Spanier, J. (974). The fractional calculus. New-York London. Oustaloup, A. (995). La dérivation non entière: théorie, synthèse et applications. Hermès. Oustaloup, A., Cois, O., Le Lay, L. (25). Représentations et identification par modèle non entier. Pellet, M., Melchior, P., Petit, J., Cabelguen, J.M., Oustaloup, A. (2). Study of the peroneus digiti quarti muscle length effect on a motor unit fractional multi-model. In IEEE/ASME International Conference on Mechatronics Embedded Systems Applications (MESA). Sommacal, L. (27). Synthèse de la fonction d Havriliak- Negami pour l identification par modèle non entier et modélisation du système musculaire. Thèse de doctorat, Université Bordeaux, France. Sommacal, L., Melchior, J., Cabelguen, J.-M., Oustaloup, A., Ijspeert, A.J. (28). Fractional Multi-Models of the Gastrocnemius Frog Muscle, Journal of Vibration Control, Sage Publishing, Vol. 4(9), pp. 4543. Sommacal, L., Melchior, P., Cabelguen, J.M., Oustaloup, A., Ijspeert, A. (27a). Fractional Multimodels of the Gastrocnemius Muscle for Tetanus Pattern, chapter 4, 27 285. Advances in Fractional Calculus. Sommacal, L., Melchior, P., Dossat, A., Petit, J., Cabelguen, J.-M., Oustaloup, A., Ijspeert, A.J. (27b). Improvement of the muscle fractional multimodel for lowrate stimulation - Biomedical Signal Processing Control, Elsevier, Vol. 2(3), pp. 226-233. Subrahmanyam, M. (989). An extension to the simplex method to constrained nonlinear optimization. Journal of Optimization Theory Applications, 62(2), 3 39. Victor, S., Malti, R., Oustaloup, A. (29). Instrumental variable method with optimal fractionnal differentiation order for continuous-time identification. In 5th IFAC Symposium on System, SYSID. Victor, S., Melchior, P., Nelson-Gruel, D., Oustaloup, A. (28). Flatness control for linear fractional MIMO s: thermal application, 3rd IFAC FDA'8, Ankara, Turkey, 5-7 November, 28. Woods, D.J. (985). An interactive approach for solving multi-objective optimization problems (interactive computer, nelder-mead simplex algorithm, graphics). Houston, tx, usa, Rice University.