Information theory framework to reconstruct Biot constants of trabecular bone from ultrasound Q. Grimal, P. Laugier M. Pakula Dpt. Structural Mechanics Institute of Environmental Mechanics Laboratoire d'Imagerie Biomedicale Universit Pierre et Marie Curie, Paris, France University of Granada, Spain and Applied Computer Science Email:
[email protected] Kazimierz Wielki University, Bydgoszcz, Poland Email: {quentin.grimal.pascal.laugier}@upmc.fr Email:
[email protected] G. Rus
simulate the ultrasonic transmission through a set of trabecular bone samples, as well as their plausible ranges,
Abstract-Reliable quantification of mechanical constants of bone tissue is an open issue with relevance for the diagnostic of bone quality disorders, such as osteoporosis. Two open questions are addressed here: the suitability of Biot's poroelasticity to explain the complex propagation patterns of ultrasound through
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to derive from the theoretical framework the parameter reconstructability and rank them in accordance,
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to expand the theoretical framework to merge informa tion provided by a two-parameter constitutive model to add knowledge to the reconstruction process, and
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to evaluate the degree of plausibility of Biot's the ory to explain the experimental measurements, alone compared to adding the two-parameter model.
trabecular bone, combined with a two-parameter homogeneiza tion model, and to determine the confidence intervals of the reconstruction. An information-theory based probabilistic inver sion framework is proposed for the first time to answer both questions. This work is aimed at (i) reconstructing the Biot-theory parameters that simulate the ultrasonic transmission through a set of trabecular bone samples, as well as their plausible ranges, (ii) evaluate which parameters are reconstructable and which not, as well as the degree of plausibility of Biot's theory to explain the experimental measurements, (iii) merge information provided
II.
by a two-parameter constitutive model to add knowledge to the reconstruction process, and evaluate the plausibility of this new hypothesis, again through the information-theory framework, and (iv) validate the reconstruction against independent tests.
I.
The information-theory reconstruction framework builds on the concept of combining information density functions from two independent sources: (1) experimental measurements and
INTRODUCTION
Experiment sample n
The unsolved challenge of modeling propagation of ultra sound through trabecular bone stems from the complexity of the medium, whereby the dispersion follows a non-monotonic scheme, and several wave components seem to travel at dif ferent phase velocities. These observations have led, together with histological observations, to the hypothesis that trabecular bone at the macroscopic level is a two-phase, anisotropic material composed of solid rod-like or plate-like skeleton filled in vivo with viscous fluid-like marrow [1]. Both the mechanical properties as well as the structure strongly influence wave properties such as attenuation and velocity. As theoretical basis for modeling elastic wave propagation in cancellous bone, mostly two-phase theory of dynamics of fluid-saturated porous materials proposed by Biot [2], [3] and extended by John son [4], [5] has been used. A reasonable agreement between Biot's theory and experiments has consistently been obtained when the phase velocity was considered [6], [7], [8], [9]. However, currently no consensus has been reached among the researchers working in the field about the relevance of Biot's theory for modeling of wave propagation.
Fig. 1.
to formulate and solve the information-theory inverse problem to reconstruct the Biot-theory parameters that
978-1-4799-8763-4/15/$31.00 ©2015 IEEE
Biot model
2 parameter model
P(hyp)
The specific objectives to answer the main goal of a reliable and rational reconstruction of model parameters given an experimenta and a set of existing idealization models are, •
METHODS
Flow chart of the information-theory probabilistic inverse problem
(2) mathematical models, over the same data (observations and
model parameters) under the idea of finding which ones are all true at the same time. The way these data and procedures are combined is outlined in the following flow-chart, which starts from the ingredients at the top and yields the answers at the bottom, from left to right. From top to bottom, we start with two sources of information: the experimental measurements and the model. The experimental measurements are gathered form an ultrasonic transmission setup (pulsed at a central frequency of 500kHz), was applied to 27 prismatic samples of human femoral condyle to measure the transmitted signal.
system parameters fulfill both propositions simultaneously, {r and fm}, as [l7],
f(O,M,H)
{r(O,M,H) and fm(O,M,H)} (2) r(O,M,H)fm(O,M,H)
The reconstructed probability for the model parameters providing the model hypothesis Hj is obtained from the joint probability f(O,M,H) in Equation 2 by extracting the marginal probability for all possible observations 0 E 0 and after some assumptions as,
M
l=1ij 10 f(O,M,H)dOdH 10 fO(O)fm(O,M,Hj)dO
The model to simulate the propagation is based on Fellah's [10] semianalytical ID layered model of Biot's [2] two-phase theory of dynamics of fluid saturating a porous solid,
where Pu Pf¢ - P12 · E and (1 - ¢)Ps - P12 , P22 represents dilatation of fluid and skeleton respectively. The other parameters in the equation (1), (2) and (3) are porosity ¢, tortuosity aT, permeability ko, fluid (Pf) and solid (Ps) densities and fluid viscosity 'fl. =
=
e
The elastic coefficients of saturated cancellous bone Q, R are related to the bulk moduli of each phases (fluid Kf and solid skeleton - Ks), bulk (Kb) and shear (N) moduli of porous frame as well as to the porosity by the well known Biot-Willis [11] formulas.
On the other hand, the model hypothesis probability f(H), understood as a measure of plausibility of hypothesis H [l7], is simply derived as the marginal probability of the posterior probability f(O,M,H) defined in Equation 2, after minor simplifications,
10 1M f(O,M,H)dMdO fO(H) j� 1 fO(O)fm(O,M,H)dMdO M
f(H)
P,
The concept of information density f is formulated and used as the tool to infer the sought nonlinear parameters, by a logical and operation (center of the flow chart) in the sense that both information are true at the same time [12], [l3], [14], [15]. From the resulting information, first, the hypothesis assumed about the model parameters will be ranked (bottom to the left) to optimize their choice, and, second, the model parameters will be reconstructed in a probabilistic way, i.e. providing not their values but their information plausibility functions (bottom to the right). Thus, we define the information contents provided by the observations as r( 0), and that provided by the model as fm ( 0, M,H), in the sense that the model couples values of model parameters M with observations 0, yielding true or false certainties f when the fed values in the model is fulfilled or not. Note that the latter may also depend on the hypothesis H we assume about the model, which in our case is a decision on which model parameters are treated as unknown for the search, or known and therefore fixed. In addition, a further hypothesis can be added by imposing the relationship between tissue density and tissue bulk and poisson constants given by the two-parameter homogenization model by Grimal [16]. Note that the hypothesis choice also conditions the number of unknowns. If we have two sources of information (probabilistic propositions) to infer information about the model parameters f(M), which are that originated by experimental observations of the system r, and that originated from a mathematical model of the system fm, the probabilistic logic conjunction operator allows to compute the information state that the
(3)
III.
(4)
RESULTS
Combining both Biot and two-parameter homogeneization models [16] yields an increased information gain, which is expressed in terms of an increased plausibility, either over a single sample, or combined over all samples, as summarized in Table I. This information is further increased by selecting Hypothesis
Sample '26'
All samples combined
Biot + two-parameter 80%
99.998%
Biot
20%
0.002%
PLAUSIBILITY GAIN WHEN INCORPORATING THE TWO-PARAMETER HOMOGENEIZATION MODEL TO BlOT'S MODEL. TABLE I.
the most plausible parameters to be kept as unknown, thus reducing the model complexity, and leaving the remaining parameters fixed at their most probable values. The resulting selection of reconstructable parameters is: porosity, tissue stiffness, trabecular characteristic length and density. Other constants cannot be reconstructed either because they are strongly coupled or have no independent effect on the signals, as illustrated in Figure 2. Since we deal with 8 parameters in Biot's model, the probability density occupies a 9-dimensional space, which is not easy to visualize. Fig. 3 captures four �2 --+ � slices of that space, keeping the remaining dimensions fixed at the reconstructed parameters, and for the probability den sity corresponding to one of the specimens and one of the 10 x 10 measuring points in it. The remaining probability maps behave similarly. A number of observations clarify the nature of the problems faced by the inverse problem: (i) the bottom left slice presents several local minima, which typically arises in dynamic problems, and justifies the choice of a full-range search technique. (ii) Both left plots present
Probability Density, ((m) [Ioglo-scale[
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Fig. 2. Example of probabilistic reconstruction of two parameter models of sample '26'. 10
oblonged valleys, meaning that every pair of parameters along the ridge have very similar plausibilities, indicating almost equally well matching experimental and numerical signals for different combinations of parameter values at the ridge (i.e. coupled parameters), which implies that the uncertainty in the recovered values will be low (good prediction) perpendicularly to the ridge direction, and high (poor prediction) along its direction. (iii) Some slices show a clear minimum at first sight, but with the problem of being too flat (the area covered by high plausibility values is much larger than in any other figure). (iv) No sharp peaks are found in the plausibility maps, which indicates that no parameter values provide a perfect matching between experimental and simulated features or signals. This is responsible for a high uncertainty in the reconstructed parameters and high sensitivity to noise and other errors. This illustrates the degree of ill-posedness that characterizes many inverse problems. Finally, the method is validated by comparing the recon structed parameters, together with their confidence intervals, against independent destructive tests on all samples, in Fig ure 4. IV.
CONCLUSIONS
An information theory based inversion scheme is proposed that allows not only to reconstruct the parameter values, but also to select the a priori information about them that maximizes the information gain. The inversion framework allows to incorporate more than one model to the experimental observations, in our case, Biot's theory and a two-parameter multiscale tissue constitutive model.
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Probability Density, ((m) [Iog -scale] lo
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Fig. 3. Selection of slices of the probability density maps, leaving the remaining parameters at the predicted value. Case of specimen 383. Cross: most probable value.
Model class optImIZation using the information contents criterion answers the question of which parameters are recon structable, and the fact that better correlations are obtained with independent tests across various samples validates that information is really gained and that the method allows to a priori select reconstructable models without the need to check with independent testing. Indeed, the technique shows that incorporating both models increases the information gain. In addition, it selects some parameters that should be removed from the search, as they not only are not reconstructable, but reduce the provided reconstruction information.
determination R2 between ultrasonically reconstructed and independent testing values, which supports the conclusion. However, some parameters can still be reconstructed with sufficient precision to discriminate between samples, with a clinical potential.
Porosity, 2 R =0.43 p SO
2 R =0.10 p
