400
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-25, NO. 4, JULY 1978
[3] K. Marha, J. Musil, and H. Tuhi, Electromagnetic Fields and the
[41 [51 [6] [7]
[8]
[9] [10]
[11] [121
[13]
[14] [15] [16]
[17] [18] [19]
[201 [21] [22] [23]
[24l
Life Environment. San Francisco, CA: San Francisco Press, 1971. R. W. P. King and C. W. Harrison, Jr., "The transmission of electromagnetic waves and pulses into the earth," J. Appl. Phys., vol. 39, pp. 4444 -4451, Aug. 1968. H. S. Tuan and R. W. P. King, "Current in a scattering antenna embedded in a dissipative half-space," Radio Science, vol. 1, pp. 1309-1319, Nov. 1966. R. W. P. King, "Current distribution in an arbitrarily oriented receiving and scattering antenna," IEEE Trans. Antennas Propagat., vol. AP-20, pp. 152-159, March 1972. R. W. P. King, A. W. Glisson, S. Govind, R. D. Nevels, and J. 0. Prewitt, "Currents and charges induced in an arbitrarily oriented electrically thin conductor with length up to one and one-half wavelengths in a plane-wave field," IEEE Trans. Electromag. Compatib., vol. EMC-19, pp. 145-147, Aug. 1977. A. R. Shapiro, R. L. Lutomirski, and H. T. Yura, "Induced fields and heating within a cranial structure irradiated by an electromagnetic plane wave," IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 187-196, Feb. 1971. J. C. Lin, A. W. Guy, and C. C. Johnson, "Power deposition in a spherical model of man exposed to 1-20 MHz electromagnetic fields," IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 791-797, Dec. 1973. H. N. Kritikos and H. P. Schwan, "Hot spots generated in conducting spheres by electromagnetic waves and biological implications," IEEE Trans. Biomed. Eng., vol. BME-19, pp. 53-58, Jan. 1972. C. H. Durney, C. C. Johnson, and H. Massoudi, "Long-wavelength analysis of plane-wave irradiation of a prolate spheroid model of man," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 246-253, Feb. 1975. C. C. Johnson, C. H. Durney, and H. Massoudi, "Long-wavelength electromagnetic power absorption in prolate spheroidal models of man and animals," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 739-747, Sept. 1975. D. E. Livesay and K.-M. Chen. "Electromagnetic fields induced inside arbitrarily shaped biological bodies,"IEEE Trans. MicrowaveTheory Tech., vol. MTT-22, pp. 1273-1280, Dec. 1974. B. S. Guru and K.-M. Chen, "Experimental and theoretical studies on electromagnetic fields induced inside finite biological bodies," IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 433-440, July 1976. J. Van Bladel, "Some remarks on Green's dyadic for infinite space," IRE Trans. Antennas Propagat., vol. AP-9, pp. 563-566, Nov. 1961. L. M. Brekhovskikh, Waves in Layered Media. New York: Academic Press, 1960. R. W. P. King and C. W. Harrison, Jr., Antennas and Waves: A Modern Approach. Cambridge, MA: M.I.T. Press, 1969, Ch. 14. R. W. P. King and B. H. Sandler, "Subsurface communication between dipoles in general media," IEEE Trans. Antennas Propagat., vol. AP-26, pp. 770-775, July 1978. A. Bafios, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. R. W. P. King and L. D. Scott, "The cylindrical antenna as a probe for studying the electrical properties of media," IEEE Trans. Antennas Propagat., vol. AP-19, pp. 406-416, May 1971. R. W. P. King, P. F. Sforza, and T.I. S. Boak, III, "The current in a parasitic antenna in a dissipative medium," IEEE Trans. Antennas Propagat., vol. AP-22, pp. 809-814, Nov. 1974. R. W. P. King, K.-M. Lee, S. R. Mishra, and G. S. Smith, "Insulated linear antenna: Theory and experiment,"J. Appl. Phys., vol. 45, pp. 1688-1697, April 1974. R. W. P. King, K.-M. Lee, G. S. Smith, and S. R. Mishra, "Insulated linear antenna: Theory and experiment. II," J. Appl. Phys., vol. 46, pp. 1091-1098, March 1975.
R. W. P. King, "The many faces of the insulated antenna," Proc. IEEE, vol. 64, pp. 228-238, Feb. 1976. [25] R. W. P. King, "The insulated conductor as a scattering antenna in a relatively dense medium," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 327-330, May 1976. [26] R. W. P. King, "Embedded bare and insulated antennas," IEEE Trans. Biomed. Eng., vol. BME-24, pp. 253-260, May 197.7.
[27]
R. W. P. King, R. B. Mack, and S. S. Sandler,Arrays of Cylindrical Dipoles. New York: Cambridge University Press, 1968. [28] R. W. P. King, Fundamental Electromagnetic Theory. New York: Dover Publications, Inc., 1963, Ch. IV, Sec. 21.
Comments on "Controllability, Observability and Structural Identifiability of Multi Input and Multi Ouput Biological Compartmental Systems" JACQUES DELFORGE Abstract-In a recent paper,1 a criterion of structural identifiability for a system of biological compartments was given. Two comments are here made which show, in particular, that the proposed criterion is a necessary but not a sufficient condition for a finite number of solutions (not necessarily just one).
1. INTRODUCTION The problem of the structural identifiability of models, though very important as regards their validity, was raised only in 1970, by Bellman and Astrom [1]. A model is said to be identifiable if its unknown parameters can be uniquely determined from experimental results. In a recent paper [2], Cobelli and Romanin-Jacur deal with the problem of structural identifiability of models in the linear theory of biological compartments. After reviewing the method proposed in 1[ ], the authors seek to obtain very general criteria of structural identifiability. The discovery of such criteria would indeed be very useful, since it would open the way, for example, to a study of structural identifiability of models with many compartments, which at present is very difficult on account of the complexity of the analytical calculations needed. Bellman and Astrom emphasized from the start the difficulty of such an undertaking, and the progress made since 1970 has
in fact related only to details or to the demonstration of necessary but not sufficient conditions. The paper by Cobelli and Romanin-Jacur attacking this difficult problem is therefore important. Although a quick reading of their paper suggests that the problem is practically solved, a closer study shows that, unfortunately, this goal is not reached, for reasons now to be discussed. 2. THE CLASSICAL METHOD OF STRUCTURAL IDENTIFIABILITY STUDY In the linear theory of biological compartments, the problem is to find identifiability conditions for the unknown elements of the n X n square matrix A from observations y(t) of the state vector x(t) satisfying the differential equation
x(t)
=
Ax(t) + Bu(t)
x(to)= 0,
t [to, T1, E
(1)
(2)
Manuscript received October 12, 1976; revised January 24, 1977, and June 1, 1977. The author is with the Commissariata l'Energie Atomiquc, Centre d'Etudes de Saclay, Departement de Biologie, Gif-sur-Yvette, France. IC. Cobelli and G. Romanin-Jacur, IEEE Trans. Biomed. Eng., vol. BME-23, pp. 93-1 00, Mar. 1976.
Nucleaires
0018-9294/78/0700-0400$00.75 © 1978 IEEE
COMMUNICATIONS
401 1
2
Fig. 1.
Fig. 2.
y(t) = Cx(t), (3) where C is a m X n matrix, B a n X q matrix, and u(t) a postive vector function of dimension q, which may be chosen arbitrarily. Bellman and Astr6m [ 1 ] have proposed using transfer theory to investigate this problem. Briefly, if Y(s) is the Laplace transform of y(t), and U(s) that of u(t), we have Y(s) = G(s, A) U(s), (4) where G(s, A) is the transfer matrix defined by G(s, A ) = C[sIn - A I - I B, (5) In being the n X n identity matrix. For a given matrix A, the elements of the matrix G(s, A) are ratios of two polynomials in s, of degree not greater than n. The symbol {ai} will denote the set of coefficients in these polynomials. If the transfer matrix is known, the behaviour of the system can be completely described. The problem of identifying the matrix A from the experimental results y(t) can thus be reduced to that of identifying A from the transfer matrix. Each coefficient a i, which we now suppose known since it is found from the experimental results, is a non-linear combination (of degree not exceeding n) of unknown elements of the matrix A. Thus we obtain a set of non-linear equations. The matrix A is identifiable when this non-linear system has a unique solution. Here, I consider, one must be more careful than Cobelli and Romanin-Jacur, who conclude: "The system is structurally identifiable when the number of parameters that can be estimated from G(s)" (that is, the number of ai) "are not smaller than the numbers of the unknown system parameters" (that is, the number of unknown elements of the matrix A). I shall make two comments against this assertion. 3. COMMENT I
c. Then, according to Cobelli and Romanin-Jacur, the model is unique; but the explicit calculation of G(s) gives s +a + b +c
52 +s(a+b+c)+ab
(7)
The equations obtained by comparison of (6) and (7) are a + b + c = a1, a + b + c = a2, ab = aX3. (8) The first two equations are therefore in fact identical, since, if the system is soluble, the experimental identification of the coefficients ai must lead to equal values of a1 and a2. It is then clear that the equations (8) have an infinity of solutions, contrary to the expected result. Note also that the possible existence of more equations than unknowns is due to an interdependence of the equations, and not to excess information that would allow the experiments to be simplified or the model made more complex, as the authors state. 4. COMMENT 2 Let us suppose that the set of non-linear equations has as many independent equations as unknowns, i.e., that there is a finite number of solutions. There are cases where there is known a priori to be either one solution or an infinity of solutions (see, for example, [3] ). Here the problem is effectively solved and the model is structurally identifiable. In many other cases, however, there can exist several isolated solutions. For instance, consider the model shown in Fig. 2, with C = (1, l), BT= (1,O), and
-x 0 A =. x -y
The model is completely controllable and completely observThe fact of having a set of equations with the same number able. of unknowns as of equations is not a sufficient condition for The calculation of the transfer matrix gives the equations: having a finite number of solutions: all the equations must also x +y =a,. be independent, and a possible interdependence of the equations is not considered in the criteria proposed in [21 for calxy = a2. (9) culating the number of available equations. Here is an illustrative example: For the system shown schematically in Fig. 1, We see at once that x and y appear symmetrically in these equations, and deduce that the matrix with C= (1, 1), Br (1,O), and A=
-a
c
-Y 0 y -x
a -b-c
the model is completely controllable and completely observable. The criteria given in [2] lead to the analytical expression G(s) = 2
+
a,
S2 +Sa2+ a3
Thus there
are
three coefficients ati and three unknowns
(6) a,
b,
obtained by interchanging x and y in the matrix A leads to exactly the same non-linear equations (9), which therefore have at least two distinct solutions with a priori equal validity. 6. CONCLUSION
The problem of the structural identifiability of a linear model
402
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-25, NO. 4, JULY 1978
is by no means completely solved in the general case. At present there is no theorem giving the number of these isolated solutions, and, except for systematic numerical analysis, which is always very long and expensive, there is no general method of finding whether the known solutions are the only possible ones. The next step, defining experiments such that the results would indicate the only actual solution among the various solutions initially considered, cannot therefore be carried out with complete rigor.
REFERENCES [I] R. Bellman and K. J. Astr6m, On structural identifilability, Mathematical Biosciences 7, 3 29-339, 1970. [2] C. Cobelli and G. Romanin-Jacur, Controllability, observability and structural identifiability of multi input and multi output biological compartmental systems, IEEE Transactions on Biomedical Engineering BME-23, No. 2, 1976. 13] J. Delforge, The identifiability of a non-stationary linear system, International Journal of Systems Science. 8, 163-169, 1977.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-25, NO. 4, JULY 1978
[3] K. Marha, J. Musil, and H. Tuhi, Electromagnetic Fields and the
[41 [51 [6] [7]
[8]
[9] [10]
[11] [121
[13]
[14] [15] [16]
[17] [18] [19]
[201 [21] [22] [23]
[24l
Life Environment. San Francisco, CA: San Francisco Press, 1971. R. W. P. King and C. W. Harrison, Jr., "The transmission of electromagnetic waves and pulses into the earth," J. Appl. Phys., vol. 39, pp. 4444 -4451, Aug. 1968. H. S. Tuan and R. W. P. King, "Current in a scattering antenna embedded in a dissipative half-space," Radio Science, vol. 1, pp. 1309-1319, Nov. 1966. R. W. P. King, "Current distribution in an arbitrarily oriented receiving and scattering antenna," IEEE Trans. Antennas Propagat., vol. AP-20, pp. 152-159, March 1972. R. W. P. King, A. W. Glisson, S. Govind, R. D. Nevels, and J. 0. Prewitt, "Currents and charges induced in an arbitrarily oriented electrically thin conductor with length up to one and one-half wavelengths in a plane-wave field," IEEE Trans. Electromag. Compatib., vol. EMC-19, pp. 145-147, Aug. 1977. A. R. Shapiro, R. L. Lutomirski, and H. T. Yura, "Induced fields and heating within a cranial structure irradiated by an electromagnetic plane wave," IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 187-196, Feb. 1971. J. C. Lin, A. W. Guy, and C. C. Johnson, "Power deposition in a spherical model of man exposed to 1-20 MHz electromagnetic fields," IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 791-797, Dec. 1973. H. N. Kritikos and H. P. Schwan, "Hot spots generated in conducting spheres by electromagnetic waves and biological implications," IEEE Trans. Biomed. Eng., vol. BME-19, pp. 53-58, Jan. 1972. C. H. Durney, C. C. Johnson, and H. Massoudi, "Long-wavelength analysis of plane-wave irradiation of a prolate spheroid model of man," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 246-253, Feb. 1975. C. C. Johnson, C. H. Durney, and H. Massoudi, "Long-wavelength electromagnetic power absorption in prolate spheroidal models of man and animals," IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 739-747, Sept. 1975. D. E. Livesay and K.-M. Chen. "Electromagnetic fields induced inside arbitrarily shaped biological bodies,"IEEE Trans. MicrowaveTheory Tech., vol. MTT-22, pp. 1273-1280, Dec. 1974. B. S. Guru and K.-M. Chen, "Experimental and theoretical studies on electromagnetic fields induced inside finite biological bodies," IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 433-440, July 1976. J. Van Bladel, "Some remarks on Green's dyadic for infinite space," IRE Trans. Antennas Propagat., vol. AP-9, pp. 563-566, Nov. 1961. L. M. Brekhovskikh, Waves in Layered Media. New York: Academic Press, 1960. R. W. P. King and C. W. Harrison, Jr., Antennas and Waves: A Modern Approach. Cambridge, MA: M.I.T. Press, 1969, Ch. 14. R. W. P. King and B. H. Sandler, "Subsurface communication between dipoles in general media," IEEE Trans. Antennas Propagat., vol. AP-26, pp. 770-775, July 1978. A. Bafios, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. R. W. P. King and L. D. Scott, "The cylindrical antenna as a probe for studying the electrical properties of media," IEEE Trans. Antennas Propagat., vol. AP-19, pp. 406-416, May 1971. R. W. P. King, P. F. Sforza, and T.I. S. Boak, III, "The current in a parasitic antenna in a dissipative medium," IEEE Trans. Antennas Propagat., vol. AP-22, pp. 809-814, Nov. 1974. R. W. P. King, K.-M. Lee, S. R. Mishra, and G. S. Smith, "Insulated linear antenna: Theory and experiment,"J. Appl. Phys., vol. 45, pp. 1688-1697, April 1974. R. W. P. King, K.-M. Lee, G. S. Smith, and S. R. Mishra, "Insulated linear antenna: Theory and experiment. II," J. Appl. Phys., vol. 46, pp. 1091-1098, March 1975.
R. W. P. King, "The many faces of the insulated antenna," Proc. IEEE, vol. 64, pp. 228-238, Feb. 1976. [25] R. W. P. King, "The insulated conductor as a scattering antenna in a relatively dense medium," IEEE Trans. Antennas Propagat., vol. AP-24, pp. 327-330, May 1976. [26] R. W. P. King, "Embedded bare and insulated antennas," IEEE Trans. Biomed. Eng., vol. BME-24, pp. 253-260, May 197.7.
[27]
R. W. P. King, R. B. Mack, and S. S. Sandler,Arrays of Cylindrical Dipoles. New York: Cambridge University Press, 1968. [28] R. W. P. King, Fundamental Electromagnetic Theory. New York: Dover Publications, Inc., 1963, Ch. IV, Sec. 21.
Comments on "Controllability, Observability and Structural Identifiability of Multi Input and Multi Ouput Biological Compartmental Systems" JACQUES DELFORGE Abstract-In a recent paper,1 a criterion of structural identifiability for a system of biological compartments was given. Two comments are here made which show, in particular, that the proposed criterion is a necessary but not a sufficient condition for a finite number of solutions (not necessarily just one).
1. INTRODUCTION The problem of the structural identifiability of models, though very important as regards their validity, was raised only in 1970, by Bellman and Astrom [1]. A model is said to be identifiable if its unknown parameters can be uniquely determined from experimental results. In a recent paper [2], Cobelli and Romanin-Jacur deal with the problem of structural identifiability of models in the linear theory of biological compartments. After reviewing the method proposed in 1[ ], the authors seek to obtain very general criteria of structural identifiability. The discovery of such criteria would indeed be very useful, since it would open the way, for example, to a study of structural identifiability of models with many compartments, which at present is very difficult on account of the complexity of the analytical calculations needed. Bellman and Astrom emphasized from the start the difficulty of such an undertaking, and the progress made since 1970 has
in fact related only to details or to the demonstration of necessary but not sufficient conditions. The paper by Cobelli and Romanin-Jacur attacking this difficult problem is therefore important. Although a quick reading of their paper suggests that the problem is practically solved, a closer study shows that, unfortunately, this goal is not reached, for reasons now to be discussed. 2. THE CLASSICAL METHOD OF STRUCTURAL IDENTIFIABILITY STUDY In the linear theory of biological compartments, the problem is to find identifiability conditions for the unknown elements of the n X n square matrix A from observations y(t) of the state vector x(t) satisfying the differential equation
x(t)
=
Ax(t) + Bu(t)
x(to)= 0,
t [to, T1, E
(1)
(2)
Manuscript received October 12, 1976; revised January 24, 1977, and June 1, 1977. The author is with the Commissariata l'Energie Atomiquc, Centre d'Etudes de Saclay, Departement de Biologie, Gif-sur-Yvette, France. IC. Cobelli and G. Romanin-Jacur, IEEE Trans. Biomed. Eng., vol. BME-23, pp. 93-1 00, Mar. 1976.
Nucleaires
0018-9294/78/0700-0400$00.75 © 1978 IEEE
COMMUNICATIONS
401 1
2
Fig. 1.
Fig. 2.
y(t) = Cx(t), (3) where C is a m X n matrix, B a n X q matrix, and u(t) a postive vector function of dimension q, which may be chosen arbitrarily. Bellman and Astr6m [ 1 ] have proposed using transfer theory to investigate this problem. Briefly, if Y(s) is the Laplace transform of y(t), and U(s) that of u(t), we have Y(s) = G(s, A) U(s), (4) where G(s, A) is the transfer matrix defined by G(s, A ) = C[sIn - A I - I B, (5) In being the n X n identity matrix. For a given matrix A, the elements of the matrix G(s, A) are ratios of two polynomials in s, of degree not greater than n. The symbol {ai} will denote the set of coefficients in these polynomials. If the transfer matrix is known, the behaviour of the system can be completely described. The problem of identifying the matrix A from the experimental results y(t) can thus be reduced to that of identifying A from the transfer matrix. Each coefficient a i, which we now suppose known since it is found from the experimental results, is a non-linear combination (of degree not exceeding n) of unknown elements of the matrix A. Thus we obtain a set of non-linear equations. The matrix A is identifiable when this non-linear system has a unique solution. Here, I consider, one must be more careful than Cobelli and Romanin-Jacur, who conclude: "The system is structurally identifiable when the number of parameters that can be estimated from G(s)" (that is, the number of ai) "are not smaller than the numbers of the unknown system parameters" (that is, the number of unknown elements of the matrix A). I shall make two comments against this assertion. 3. COMMENT I
c. Then, according to Cobelli and Romanin-Jacur, the model is unique; but the explicit calculation of G(s) gives s +a + b +c
52 +s(a+b+c)+ab
(7)
The equations obtained by comparison of (6) and (7) are a + b + c = a1, a + b + c = a2, ab = aX3. (8) The first two equations are therefore in fact identical, since, if the system is soluble, the experimental identification of the coefficients ai must lead to equal values of a1 and a2. It is then clear that the equations (8) have an infinity of solutions, contrary to the expected result. Note also that the possible existence of more equations than unknowns is due to an interdependence of the equations, and not to excess information that would allow the experiments to be simplified or the model made more complex, as the authors state. 4. COMMENT 2 Let us suppose that the set of non-linear equations has as many independent equations as unknowns, i.e., that there is a finite number of solutions. There are cases where there is known a priori to be either one solution or an infinity of solutions (see, for example, [3] ). Here the problem is effectively solved and the model is structurally identifiable. In many other cases, however, there can exist several isolated solutions. For instance, consider the model shown in Fig. 2, with C = (1, l), BT= (1,O), and
-x 0 A =. x -y
The model is completely controllable and completely observThe fact of having a set of equations with the same number able. of unknowns as of equations is not a sufficient condition for The calculation of the transfer matrix gives the equations: having a finite number of solutions: all the equations must also x +y =a,. be independent, and a possible interdependence of the equations is not considered in the criteria proposed in [21 for calxy = a2. (9) culating the number of available equations. Here is an illustrative example: For the system shown schematically in Fig. 1, We see at once that x and y appear symmetrically in these equations, and deduce that the matrix with C= (1, 1), Br (1,O), and A=
-a
c
-Y 0 y -x
a -b-c
the model is completely controllable and completely observable. The criteria given in [2] lead to the analytical expression G(s) = 2
+
a,
S2 +Sa2+ a3
Thus there
are
three coefficients ati and three unknowns
(6) a,
b,
obtained by interchanging x and y in the matrix A leads to exactly the same non-linear equations (9), which therefore have at least two distinct solutions with a priori equal validity. 6. CONCLUSION
The problem of the structural identifiability of a linear model
402
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-25, NO. 4, JULY 1978
is by no means completely solved in the general case. At present there is no theorem giving the number of these isolated solutions, and, except for systematic numerical analysis, which is always very long and expensive, there is no general method of finding whether the known solutions are the only possible ones. The next step, defining experiments such that the results would indicate the only actual solution among the various solutions initially considered, cannot therefore be carried out with complete rigor.
REFERENCES [I] R. Bellman and K. J. Astr6m, On structural identifilability, Mathematical Biosciences 7, 3 29-339, 1970. [2] C. Cobelli and G. Romanin-Jacur, Controllability, observability and structural identifiability of multi input and multi output biological compartmental systems, IEEE Transactions on Biomedical Engineering BME-23, No. 2, 1976. 13] J. Delforge, The identifiability of a non-stationary linear system, International Journal of Systems Science. 8, 163-169, 1977.
