Ultrasonic Measurement of the Reaction Kinetics of the...
Transcript of Ultrasonic Measurement of the Reaction Kinetics of the...
Ultrasonic Measurement of the ReactionKinetics of the Setting of Calcium Sulfate
Cements Using Implicit Calibration
Johan E. Carlson1 and Veli-Matti Taavitsainen2
1EISLAB, Luleå University of Technology, Luleå, Sweden
2Dept of Mathematics, EVTEK University of Applied Sciences, Espoo, Finland
P Background
P Ultrasonic measurement principle
P Reaction kinetics
P Implicit calibration
P Experimental results
P Conclusion
Outline
P The setting time of ceramic bone substitute materials is animportant property for both developers and end-users.
P There are models describing the reaction kinetics availablein the literature.
P The unknown parameters of these models can not bemeasured directly.
P We need some observations of an indirect quantity that isaffected by the mechanical changes in the material duringthe reaction.
Background
The measurement problem
P Ultrasound is a mechanical vibration, and as such, acousticproperties vary with changes in mechanical properties.
P We should be able to draw conclusions about the progressof the setting reaction by studying changes in acousticproperties.
P The observations are indirect, so we need to attack theproblem by some regression technique.
P But! We should make use of the physical and chemicalknowledge at hand.
P Solution: Implicit calibration!
Background (cont’d...)
Idea!
Ultrasonic measurement principle
The pulse-echo reflectometer
P The first echo, denoted p1(t) is recorded.
P The spectral amplitude of p1(t) is calculated, as
y = |(DFT(p1(t))|,
where DFT denotes the discrete Fourier transform
P During the setting reaction, pulses are measured every 20seconds. The corresponding spectral amplitude, yk, isstored as a row of a matrix Y.
Measurement principle (cont’d...)
CaSO 1
2 H O +
3
2 H O CaSO 2H O
4 2 2 4 2⋅ → ⋅
Reaction kinetics
The reaction
The reaction in this study is the setting of calcium sulfate hemi-hydrate (CSH) into calcium sulfate dihydrate (CSD)
Reaction kinetics (cont’d...)
The kinetics model
After a nucleation period t0, the reaction is described by the followingset of differential equations
where XI is the mass fraction of I’th species
P To explain the principle of implicit calibration, we shall usethe following notation
P Y is the matrix of measured amplitude spectra where the i'throw contains the spectrum measured at the i'thmeasurement time
P Xθ is the matrix of the modeled mass fractions using the
current estimates for the vector of kinetic parameters θ
P θ = [k t0 p]
Implicit calibration
Model
P In direct implicit calibration, the spectra are directly modeled(calibrated) by linear calibration using the modeled massfractions, i.e.
P Y = XθBθ + error
P Bθ is the matrix of multivariate calibration regression
coefficients, obtained by ridge regression using non-negativity constraints
Implicit calibration (cont’d...)
Model (direct implicit calibration)
P In indirect implicit calibration, the modeled mass fractionsare ‘remodeled’ (calibrated) by multivariate calibration usingthe spectra , i.e.
P Xθ = YB
θ + error
P Bθ is the matrix of multivariate calibration regression
coefficients, obtained by PLS regression
Implicit calibration (cont’d...)
Model (indirect implicit calibration)
1. Make an initial quess for θ
2. Using the current value of θ, solve (numerically) the kineticsystem of differential equations to obtain the matrix X
θ
3. Estimate the matrix Bθ using ridge regression with non-
negativity constraints and calculate the modeled spectraXθBθ and the residuals Y!X
θBθ
4. Calculate a new value for θ for minimizing the least squaresnorm of the residuals
5. Repeat the steps 2, 3 and 4 until the chosen convergencecriteria are met
Implicit calibration (cont’d...)
The algorithm: direct implicit calibration
1. Make an initial quess for θ
2. Using the current value of θ, solve (numerically) the kineticsystem of differential equations to obtain the matrix X
θ
3. Estimate the matrix Bθ using PLS regression and calculate
the calibrated mass fractions YBθ and the R2-value between
Xθ and YB
θ
4. Calculate a new value for θ for maximizing the the R2-valuebetween X
θ and YB
θ
5. Repeat the steps 2, 3 and 4 until the chosen convergencecriteria are met
Implicit calibration (cont’d...)
The algorithm: indirect implicit calibration
Experimental results
Indirect implicit calibration
0 50 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time [min]
indirect implicit calibration, dim = 1
CaSO4⋅1/2H
2O
H2O
CaSO4⋅2H
2O
0 50 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
time [min]
indirect implicit calibration, dim = 5
CaSO4⋅1/2H
2O
H2O
CaSO4⋅2H
2O
Experimental results (cont’d...)
Direct implicit calibration
0 5 100
0.050.1
t = 0
0 5 100
0.050.1
t = 11
0 5 100
0.050.1
t = 11.5
0 5 100
0.050.1
t = 12
0 5 100
0.050.1
t = 15
0 5 100
0.050.1
t = 20
0 5 100
0.050.1
t = 25
0 5 100
0.050.1
t = 30
0 5 100
0.050.1
t = 40
0 5 100
0.050.1
t = 50
0 5 100
0.050.1
t = 70
0 5 100
0.050.1
t = 90
Experimental results (cont’d...)
Comparison
Method k t0 SS R2
direct 0.777 10.9 0.0155 -indirect (dim = 1) 0.778 10.9 - 99.89 %indirect (dim = 2) 0.826 11.1 - 99.94 %indirect (dim = 3) 0.831 11.1 - 99.94 %indirect (dim = 4) 0.849 11.1 - 99.97 %indirect (dim = 5) 0.864 11.1 - 99.97 %
Experimental results (cont’d...)
Analysis and discussion
90
90
98
9899.3
99.799.85
k
t0
indirect implicit calibration, dim = 1
0.4 0.6 0.8 15
10
15
98
98
99.3
99.3
99.7
99.7
99.95
k
t0
indirect implicit calibration, dim = 5
0.4 0.6 0.8 15
10
15
P Ultrasound spectra change during the setting reaction ofcalcium sulfate cements.
P The parameters of a physical reaction kinetics model can beestimated using information in the spectral amplitudes of thepulses.
P This enables on-line monitoring of the reaction kinetics,using non-destructive evaluation.
Conclusions
P The authors wish to express their gratitude towards Dr. Malin Nilsson at BoneSupport AB for her valuablecomments.
Acknowledgements
Preparing to a hot discussion