WO2008133538A1

WO2008133538A1 - A method for the non-invasive determination of density and poroelasticity parameters of the long bone

Info

Publication number: WO2008133538A1
Application number: PCT/PL2008/000032
Authority: WO
Inventors: Ryszard Uklejewski; Tomasz Czapski
Original assignee: Uniwersytet Im. Kazimierza Wielkiego W Bydgoszczy
Priority date: 2007-04-30
Filing date: 2008-04-28
Publication date: 2008-11-06
Also published as: EP2146641A1; PL382344A1

Abstract

The examined long bone is exposed to longitudinal elastic vibratiops, with a frequency range of a few Hz to a dozen kHz, advantageously up to 15 kHz, which in turn forces the flow of the ionic fluid filling the walls of its shaft, and in consequence generates an electric source voltage Estr which is a function of the relative velocity of liquid phase particles with respect to the sojio' phase of the wall of the shaft. On the basis of an electrical-computational model created earlier, a parametric database of various responses of the system is created, which provides a dense enough determination of the state of the examined long bone, and then, on the basis of the values of measured electrical signals and excitations the parametric identification of the system is performed, whereas the acousto-electric bone transmission line is represented as a system with mechanical and mechano-electrical couplings. The parametric identification of the tested long bone is performed wjth the 'aid artificial neural networks, in which the values of entered measured electric signals accompanying the transmission of longitudinal elastic vibrations in the tested long bone are processed.

Description

A method for the non-invasive determination of density and poroelasticity parameters of the long bone

A subject-matter of an invention is a method for the nqn-invasive determination of density and poroelasticity parameters of the long bone, i. e. parameters especially useful in the osteoporosis diagnostics and treatment monitoring.

In the book by R. Lorenc and J. Walecki (eds.), entitled "The diagnostics of osteoporosis" (Springer Verl. & Polish Sci. Publsh., Warsaw 1998), and also on the www.osteoforum.org.pl web pages the most important densitometric methods used currently to evaluate the density and the porosity of bones are presented. The listed methods include: radiological (DXA, pQCT), magnetic resonance, quantitative ultrasound (QUS) and biochemical markers used for estimation of osteogenesis and osteoresorption processes in the skeleton.

In the American patent description No. US 2003/0026385 a densitometric measurement method was presented which uses Roentgen rays and a two- dimensional detector of these rays, whereas in the patent description US 2006/0008050 a system of processors and a method for the creation of a densitometric model of a patient on the basis of roentgen tomography is given.

In the monograph by R. Uklejewski (Publish. House of the Inst. Biocybern. Biomed. Engng Polish Acad. Sci., Warsaw 1994.) the problem of transmission of longitudinal acoustoelastic vibrations along porous shafts of Iqng bones filled with physiological ionic fluid (electrolyte) was investigated and a mathematical description and electric model of the problem were given. A pulsating flow ionic pore fluid through a unit surface of cross-section area of long bone, caused by the cyclic elastic deformation of the material qf the solid phase of long bone, generates the electrical source voltage Estr, which is a function of the relative velocity of liquid phase particles with respect to the solid phase of the wall of the long bone shaft. The appropriate theoretical model for the generation of electrical currents and voltages due to elastic deformations of porous cortical bone filled with physiological electrolyte is obtained by combining the linear equations of electrokinetics, whicfi describe the coupled flow of ionic fluid in porous material caused by pressure gradients and by electric potential and pressure gradients, with the M. A. Biofs poroelasticity theory equations which describe the elastic deformation of porous material [Uklejewski R.: O efektach elektromechanicznych w pomwatej kosci zbitej wypefnionej pfynem fizjologicznym I efekcie akustoelektrycznym w trzonach kosci dtugich mokrych {On electromechanical effects in porous cortical bone filled with physiological fluid and on the acoustoelectric effect in wet long bone shafts] Publish. House of the Inst. Biocybern. Biomed. Engng Polish Acad. ScL, Warsaw 1994; Uklejewski R,, Kςdzia A., Rogala P.: Living porous bone as biomechatronic system In: POROMECHANICS - MABiot Centennial (eds: Y.N. Abousleiman, A.H-D. Cheng. F-J. UIm), p.21-25, Balkema Publ. Taylor & Francis Group, London - New York - Philadelphia -r Leiden - Singapore 2005; ISBN 04 1538 041 3)].

Extensive experimental investigations carried out by G.D. Scott and E. Korostoff published in: Oscillatory and step response: Electromechanical phenomena in human and bovine bone X Biomech. 1990, 23(2), 127-143, coustitute the experimental background and verification the above mentioned theoretical model of the elasto-electric phenomena in human cortical bone.

A purpose of the invention was to develop such a method of parametric identification which would enable especially the determination of the values of density and poroelastic parameters of human long bones on the basis of knowledge of values of electrical signals which accompany the propagation of longitudinal acoustoelastic waves in them-.

This purpose was achieved in accordance with the proposed invention, the nature of which is that the tested long bone is subjected to longitudinal elastic vibrations, advantageously acoustoelastic, with a frequency from a few Hz to a dozen of kHz, advantageously up to 15 kHz, which forces the flow of ionic fluid filling the wall of its shaft, and in consequence generates the electrical source voltage E^ as function of the relative velocity of fluid phase particles with respect to the solid phase of the shaft, and then, on the basis of an electrical- computational model created earlier, in which the long bone which transmits longitudinal elastic vibrations is treated as a multi-path, advantageously 3-path acoustoelectric bone transmission line, a parametric database of various responses of the system is created, which provides a dense enough determination of the state of the examined long bone, and then, oh the basis of the values of measured electrical signals and excitations the parametric identification of the system is performed, whereas the acousto-electric bone transmission line is represented as a system with mechanical and, mechanical and electrical couplings.

It is advantageous when the parametric base of particular system responses is created in a form of the matrix of electrical signals values which are assigned to the particular values of mechanical and electric immitances.

It is also advantageous when the parametric identification of the tested bone is conducted with the aid of at least one artificial neural network ANN, advantageously a double-layer Elman neural network, in which after previously entering a set of electrical signals values as an input and a set of immitances values assigned to them as an output, the entered values of measured electric signals accompanying the transmission of longitudinal elastic vibrations in the tested long bone are processed-. -

During the tests that were conducted it unexpectedly turned out that the electric signals which accompany the propagation of longitudinal acoustoelastic waves in a long bone shaft, which carry information about the field of velocity of intraosseous fluid flow in the pore space of bone and about the microgeometric structure of this pore space may be used to determine, with the use of artificial neural networks ANN, among other things, the density and the poroelastic parameters values of the bones Jn question. The presented method of osteoelectrodensitometry is completely new approach in the diagnostics of bone density and poroelastic parameters of normal and osteoporotic human long bones.

The elaborated numerical model of the problem of electrical potentials and currents generation in the long bone under the influence of longitudinal mechanical exication with a frequency from a few Hz to 1 kHz (the model with lumped parameters) and from above 1 kHz to a dozen kHz (the model with parameters distributed along the length of the long bone) enables the parametric identification of the model of the considered problem with the use of artificial neuronal networks.

The main application of the proposed method of electrical osteodensitometry deals with the determination of the values of bone density and poroelastic parameters of normal and osteoporotic human long bones. However, it is possible to extend the scope of potential applications of this method to other porous materials in which the solid phase material is an elastic dielectric and the fluid phase is a liquid electrolyte. Such materials include porous geomaterials filled with water, which is a natural electrolyte_/ or with petroleum. Geological boring provides cylindrically shaped samples of geomaterials; long bone shafts, which the proposed method applies to, have an approximately cylindrical shape.

Electrical signals induced by a cyclic elastic bone deformation may be non- invasively registered with the use of electrodes and devices used e.g. in electromyography, electrocardiography or electroencephalography. When compared with bone quality assessment by quantitative ultrasound (QUS ultrasonometers) the proposed method of electroosteodensitometry does not requires the use of a mechano-electrical receiving transducer for electrical monitoring the propagation of acoustoelectrical waves in long bones, such as piezoelectric ones, since in the electroosteodensitometry the natural mechano- electrical properties of cortical bone are used to transform mechanical quantities which describe the elastic wave propagation in bones into measurable electrical signals. The parametric identification of the system, i.e. of the tested long bone, carried in the proposed method with the use of artificial neural networks allows it to obtain, not only the density vajues of the long bones, but also information about values of many other parameters of the system.

E x a m p l e s

The invention will be described on the basis of example realization, whereas for a better presentation of computational issues a drawing was used, in which the individual figures represent:

- fig. 1 - type r four-terminal network, as an electrical equivalent for the one dimensional dynamic problem of the Biof s mechanics of elastic porous materials filled with a fluid;

- fig. 2 - electrical diagram of the dx element of the porous long bone shaft transmitting longitudinal harmonic elastic vibrations, where Estr means the electric source voltage, which is a function of the relative velocity of liquid phase particles with respect to the solid phase of the wall of the long bone shaft; fig.3 - series connection of multiport elements containing iterated LC element models and magnetic and capacitance couplings based on an implicit algorithm of the 2^nd degree, also called the trapeze method, with an excitation with a frequency of up to IkHz;

- fig.4 - series connection of a chain of four-terminal networks, containing an iterated model of C capacity with G conductance; and electric current sources distributed along the long bone shaft length;

- fig.5 - fragment of a two-layer Elman neural network with derailed

Activation functions used for the parametric identification of the system;

- fιg.6 - block diagram showing the consecutive stages of operation when using the method presented in the invention; - fιg.7 - the system responses in the form of electrical voltage and current waves which accompany the longitudinal harmonic elastic waves along the long bone shaft for the example 1;

- fig.8 - the system responses in the form of electrical voltage and current waves which accompany the longitudinal harmonic elastic waves along the long bone shaft for the example 2;

- fig.9, 10 - images of the learning error surface for the neural network;

- fig.11 - system responses in the form of electrical voltage and current waves which accompany the longitudinal harmonic elastic waves propagations along the long bone shaft for the example 4;

- fig.12 - system responses in the form of electrical voltage and current waves which accompany the longitudinal harmonic elastic waves propagating along the long bone shaft for the example 5.

Computational model with distributed parameters of the considered problem for the porous shaft of the long bone.

Electrical equivalent for the one dimensional dynamic problem of the Biot's mechanics of elastic porous materials filled with a fluid was represented on the fig. 1 as a r type four-terminal network with magnetic, conductance and capacitance couplings mapped as voltage sources in respectively transverse and longitudinal branches.

Electrical signals accompanying the propagation of harmonic acoμsto-elastic waves in the long bone shafts filled with physiological fluid.

On fig. 2 the electric diagram of the dx element of the porous long bone shaft filled with physiological fluid, transmitting longitudinal harmonic elastic vibrations was presented. The flow of ionic pore fluid

-Wi) through a unit surface generates in the bone the electric source voltage Estr, which is the function of the relative velocity of liquid phase particles with respect to the solid phase of the wall of the long bone shaft, given by the following relation: E_str = ZβJ_str = ZJA₂₁AJd₁(W₁ -W₁)

where:

Ze - electrical impedance of the long bone shaft per unit length;

S - cross-section area of the long bone shaft wall;

Jstr - density of electrokinetic streaming current;

An - coefficient of hydraulic permeability of the cortical bone of long bone shaft;

A21 - electrokinetic coefficient of streaming current.

The electrical source voltage Estr(x) of mechanical origin is a source of electrical signals in the system in question, whereas U(x) is the electrical voltage which accompanis the transmission of elastic vibrations along the long bone shaft, while Y^=G+jωC) is the admittance of the bone transmission line for the unit of length.

Computational model with lumped parameters for the porous shaft of the long bone transmitting longitudinal elastic vibrations in the frequency range up to 1 kHz.

Within the range of lower frequencies up to 1 kHz of vibrations transmitted in the human long bone shafts the wave phenomena do not have to be taken into account. With a known velocity of propagation of the elastic waves in the long bone shafts (V - approx. 3700 m/s) and a known length of long bone shafts (L α 0.5 m) the length of the elastic wave λ exceeds the lengthwise dimensions of the long bone. Thus, the long bone may be modeled as, a model with lumped parameters.

The model of the system presented on fig. 1 was transformed to a multiport element shown on fig. 3, in which the L and C parameters of the system were replaced with iterated computational models of, respectively, inductance and capacitance, in which the method of node potentials was used to solve this problem.

Next, on the fig. 4 the computational model for the 3^rd path (electric) of the bone acousto-electric transmission line, that is, the shaft of the porous long bone transmitting the longitudinal elastic vibrations, which forms the second part of the mechano-electrical transducer which is the bone. After the current equations are created and solved for the first four nodes, the electrical voltages and currents for the individual nodes of the system are determined.

The numerical values of the mechanical and electrical impedances and admittances of the system (Tab. 1, Tab.2) were determined on the basis of experimental results presented in the following studies:

[1] Hosokawa A. & Otani T.: Ultrasonic wave propagation in bovine cancellous bone. 3. Acoustical Society of America, 101(l),1997.

[2] Davis Ch. F.: On the mechanical properties of bones and a poroelastic theory of stresses in bone elements. University of Delaware, 1970.

[3] Fellah Z.E.A & Chapelon J.Y., Berger S., Lauriks W., Depollier C: Ultrasonic wave propagation in humen cancellous bone. Application of Biot theory. 3. Acoustical Society of America, 116(1), 2004.

[4] Sierpkowska 1, Toyras J., Hakulinen MA, Saarakkala S., Jurvelin 3., S., Lappalainen R.: Electrical and dielectrical properties of bovine trabecular bone - relationships with mechanical properties and mineraly density. Phys. Med.. Biol. 48(6), 775-786, 2003.

[5] Uklejewski R.: O efektach elektromechanicznych w porowatej kosci zbitej wypefnionej plynem fizjologicznym i efekcie akustoelektrycznym w trzonach kosci diugich mokrych [On electromechanical effects in dense porous bone filled with physiological fluid and on the acoustoelectric effect in the wet long bone shafts]. Publish. House of the Inst. Biocybern. Biomed. Engng Polish Acad. Sci., Warsaw 1994.

Tab. 1 Mechanical parameters per unit length of the long bone shaft

Tab. 2 Electrical resistivity (constant current) of the cortical bone with various concentrations of salt in the fluid filling the pores for three longitudinally different bone samples and the value of capacitance of the bone shaft for the unit of length.

Architecture and activation functions of the neural network used for the parametric identification of the system

In the neural network architecture: 10-80-10 was used (two neuron layers) and two activation functions, fig. 5. In the first layer the tansig activation function was used, and in the second the linear activation function purelin was used.

The method according to the invention presented on fig.6 is composed of the following steps:

(1) using a generator of excitation function in frequency range from a few Hz to 15 kHz;

(2) taking the experimentally established values of parameters of mechanical and electrical immittances;

(3) solving the numerical model for the problem of generation of electric voltages and currents in the long bone shaft transmitting longitudinal elastic vibrations;

(4) acquising the system's responses in the form of electric signals accompanying the longitudinal acoustic vibrations in the long bone;

(5) creating a parametric database of various system's responses in the form corresponding matrices of responses and of immittances:

Immittance_l - response_l Immittance_2 - response_2

Immittance_1000 - response_1000;

(6) teaching a neural network with the designed two layer structure through the use of the created parametric database;

(7) parametric identification of the tested long bone based on the taught artificial neural network. In the implementation of the method the following technical means were used:

- PC class computer;

- measuring probes;

- electrical signals preamplifier MICRO1401^mk n technical data with an analogue-digital card;

- MatLab software.

The electric signals generated by mechanical loading of the long bone may be measured non-invasively, with the use of insulated electrodes used in ECG or EMG measurements, and then sent to the electrical signals preamplifier with an analogue-digital card (eg. CED 1902 manufactured by CAMBRIDGE ELECTRONIC DESIGN LIMITED), which enables the real-time acquisition of data. The visualization of transformed signals was implemented based on CED Signal ver. 3 software, which was then used in conjunction with MatLab software to create a database of signal vectors Estr-

Example 1

Tab. 3 Assumed mechanical excitation parameters

Mechanical longitudinal excitation force applied on the input of the system

Amplitude of the mechanical i: =io [K?] longitudinal excitation force

Vibration frequency 500[Hz]

Initial condition U(χ,t = 0) = 0 for the 3^d path (electrical)

The assumed values of mechanical and electrical parameters of the system are given in Tab.l and Tab. 2.

On the plot shown in the fig.7 the system responses in the form of electric voltage and current waves which accompany the longitudinal harmonic elastic waves along the long bone shaft, were presented. Example 2

Tab. 4 Assumed mechanical excitation parameters

Mechanical longitudinal excitation force applied on the input of the system

Amplitude of the mechanical /₀ ^m = 10[Kr] longitudinal excitation force

Vibration frequency 5 [kHz]

Initial condition U(x,t = 0) = 0 for the 3^d path (electrical)

On the plot shown in the fig.8 the system responses in the form of electric voltage and current waves which accompany the longitudinal harmonic elastic waves along the long bone shaft were presented.

Results of teaching, testing and optimizing the Elman neural network.

Example 3

Tab.5 Response vector (vector of identified values of immitances of the system) and a fragment of the teaching samples set with indication to the proper response vector.

2. S200 4.5500 5.4500

Vector of identified values of 2.0600 5.6500 2.2200 immittances 4.5000 2.7400 2.7900

4.2800 3.1200 2.6100

4.6600 2.6700 1.8700

A fragment of the teaching samples of the parametric database The neural network has performed the test parametric identification with the error surface shown in fig.9.

In the network teaching process the neural network has performed the parametric identification of the system with_^ an error of approx. 0.01. In order to establish whether the neural network identifies the system's parameters with an optimum accuracy its structure was modified by entering new network functions, that is: moment coefficient me and error function sse.

After taking into account the changes in the neural network architecture the following parametric identification results were obtained (Tab. 6):

Tab.6 Response vector and a fragment of the teaching samples set with indication to the proper response vector obtained as result of modification of the neural network

-0.5101 -0.5100 1.1600 -0.0900

-0.4797 -0.4800 -1.1900 2.3000 0.3502 0.3500 -2.2600 -0.4300

-1.6698 -1.6700 0.6300 0.3600 0.2599

0.2600 0.3200 0.1100 2.4498 2.4500 1.5100 -1.2000

-0.0901 -0.0900 2.1100 1.4700 2.0301 2.0300 2.5800 2.2500 0.7900 0.7900 1.0300 3.5000 1.4199 _L 1.420O₁ 2.0100 2.1400

1.1300 4.0400 6.6800

2.5200 4.5500 5.4500

Vector of identified values 2.0600 5.6500 2.2200 of immittances

4.5000 2.7400 2.7900

4.2800 3.1200 2.6100

4.6600 2.6700 1.8700

3.3000 2.0100 0.4800

0.3800 4.1200 1.9800

0.7500 -0.8100 1.6400

Fragment of teaching samples set

Results of parametric identification of the system by using the designed and tested Elman's neural network

As a result of parametric identification of the system carried by a created, taught and tested Elman network the following values of system parameters were obtained in a form of a response vector - shown in the next examples 4 and 5, whereas mechanical and electric parameters were assumed in accordance with Tab. 1 and Tab. 2 and the mechanical excitation parameters in accordance with Tab. 3.

Example 4

The measured response of the system, in the form of electrical current and voltage (fig. 11) was introduced to the input of the neural network, and then, through the use of the network the parametric identification of the system was performed by determining: the vector of identified values of the system parameters shown in Tab.7 on the left side, and, on the right side, a fragment of immittance matrix of the parametric base of the system, on the basis of which the identification of the value of density and poroelastic parameters of the long bone was done.

Tab. 7 FRAGMENT OF N BWMTrTANCE MATRIX OF THE PARAMETRIC BASE OF THE SYSTEM

VECTOR OF PARMETEBS OBTAINED AS A RESULT OF PAKAMETWC IDENTIFICATION

1.0e3 * _ 1.0ε3

Lf= 0.00000000390032 [kG/tn²] 0.00000000000029 0.00000000000023 0.00000000000022 -0.00051000000000 O.00116000000000

1$ = 0.00000000193597 [kG/m²] 0.00000000000680 0.00000000000640 0.00000000000677 -0.00048000000000 -o.oousooooooooo

P_lI= 1.96500056628980 [kg/m³] 1.96500000000000 1.96500000000000 1.96500' 30000000 0.00035000000000 -0.00226000000000

Pn= 1.00000142662070 [kg/m³] i.oooooooooαoooo l.oooooooooooooo 1.0000C -10000000 -0.00167000000000 0.00063000000000 Pa =-0.96300063573199 [kg/m³] 0.96300000000000 0.96300000000000 0.9630C 10000000 0.00026000000000 0.00032000000000 k^H=-0.00000000266683 [m^J] 0.00000000000015 0.00000000000014 0.00000" 30000147 0.00245000000000 0.00151000000000 k^m= 0.00049999786608 0.00050000000000 0.00050000000000 0.00050000000000 -0.00009000000000 0.00211000000000

R = 3.B68S9265679202 [Ω/cm] 3.86900000000000 3.30000000000000 3.O0000O000O0000 0.00203000000000 0.00258000000000

0 0 0 0.00079000000000 0.00103000000000

0 0 0 0.00142000000000 0,00201000000000

where:

- poroelastic coefficient for the solid phase of the bone; - poroelastic coefficient for the fluid phase of the bone;

Ai - density of the solid phase of the bone; AΏ - density of the fluid phase of the bone;

Aa - coefficient of inertial coupling between the solid phase and the fluid phase, it has a negative value; k^H - hydraulic permeability of the bone; k^m - poroelastic coefficient of the coupling of bone phases; R - electric resistance of the porous long bone shaft per unit length.

Example 5

The measured response of the system, in the form of electric current and voltage (fig. 12) was introduced on the input of the neural network, and then, through the use of the network the parametric identification of the system was performed determining: the vector of identified values of the system parameters shown on the left side, and, on the right side, a fragment of immittance matrix of the parametric base of the system, on the basis of which the identification of the value of density and poroelastic parameters of the long bone was dane.

VECTOR OF PARMETERS OBTAINED AS A RESULT FRAGMENT OF THE MMnTANCE MATRIX OF OF PARAMETRIC IDENTIFICATION THE PARAMETRIC BASE OF THE SYSTEM 00029 0.00000000000023 0.00000000000022 -0.00051000000000 -0.00143000000000 0.00000000000680 0.00000000000640 0.00000000000677 -0.00043000000000 -0.00046000000000 1.90400000000000 1.96500000000000 1.96500000000000 0.00035000000000 -0.00052000000000 1.00000000000000 1.00000000000000 1.00000000000000 -0.00167000000000 0.00145000000000 0.9S300000000000 0.91300000000000 0.93300000000000 0.00026000000000 0 0.00000000000015 0.00000000000014 0.000000000001 7 0.00245000000000 0.00135000000000 00000 0.00050000000000 0.00050000000000 -0.00009000000000 0.00139000000000 00000 3.30000000000000 3.00000000000000 0.00203000000000 0.00081000000000 0 0 0 0.00079000000000 0.00003000000000

0 0 0 0.00142000000000 0.00139000000000

Where it is marked as in the example 4.

It was established that the density of the solid phase of the tested long bone

(#i), identified in the last example, significantly lower than in the previous example 4, corresponds to a long bone with a higher porosity.

Claims

1. A method for determining of density and poroelastic parameters of the long bones, basing on the use of the phenomenon of transmission of longitudinal elastic vibrations along the shafts of porous long bones filled with physiological electrolyte, characterized in that the tested long bone is exposed to longitudinal elastic vibrations, advantageously acousto- elastic, with a frequency from a few hertz (Hz) to a dozen kilohertz (kHz), advantageously up to 15 kHz, which in turn forces the flow of the ionic fluid filling the walls of its shaft, and in consequence generates an electric source voltage Estr which is a function of the relative velocity of liquid phase particles with respect to the solid phase of the wall of the shaft, and then, on the basis of an electrical-computational model created earlier, in which the long bone which transmits longitudinal elastic vibrations is treated as a multi-path, advantageously 3-path acoustoelectric bone transmission line, a parametric database of various responses of the system is created, which provides a dense enough determination of the state of the examined long bone, and then, on the basis of the values of measured electrical signals and excitation the parametric identification of the system is performed, whereas the acousto-electric bone transmission line is represented as a system with mechanical and mechanical and electric couplings.

2. A method according to the claim 1, characterized in that the parametric base of particular system responses is created in a form of the matrix of electrical signals values which are assigned to the particular values of the matrix of mechanical and electric immitances.

3. A method according to the claim 1 or 2, characterized in that the parametric identification of the tested long bone is performed with the aid of at least one artificial neural network, advantageously a double-layer Elman neural network, in which after previously entering a set of values for electrical signals on the input and a set of values for the immitance values assigned to them on the output, the values of in the next entered measured electric signals accompanying the transmission of longitudinal elastic vibrations in the tested long bone are processed.