Ultrasonic Measurement of the Reaction Kinetics 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 an important property for both developers and end-users.
P There are models describing the reaction kinetics available in the literature.
P The unknown parameters of these models can not be measured directly.
P We need some observations of an indirect quantity that is affected by the mechanical changes in the material during the reaction.
Background
The measurement problem
P Ultrasound is a mechanical vibration, and as such, acoustic properties vary with changes in mechanical properties.
P We should be able to draw conclusions about the progress of the setting reaction by studying changes in acoustic
properties.
P The observations are indirect, so we need to attack the problem by some regression technique.
P But! We should make use of the physical and chemical knowledge 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 20 seconds. The corresponding spectral amplitude, yk, is
stored 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 following set of differential equations
where XI is the mass fraction of I’th species
P To explain the principle of implicit calibration, we shall use the following notation
P Y is the matrix of measured amplitude spectra where the i'th row contains the spectrum measured at the i'th
measurement 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 mass
fractions, 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 fractions are ‘remodeled’ (calibrated) by multivariate calibration using the spectra , i.e.
P Xθ = YBθ + 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 kinetic system of differential equations to obtain the matrix Xθ
3. Estimate the matrix Bθ using ridge regression with non- negativity constraints and calculate the modeled spectra XθBθ and the residuals Y!XθBθ
4. Calculate a new value for θ for minimizing the least squares norm of the residuals
5. Repeat the steps 2, 3 and 4 until the chosen convergence criteria 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 kinetic system of differential equations to obtain the matrix Xθ
3. Estimate the matrix Bθ using PLS regression and calculate the calibrated mass fractions YBθ and the R2-value between Xθ and YBθ
4. Calculate a new value for θ for maximizing the the R2-value between Xθ and YBθ
5. Repeat the steps 2, 3 and 4 until the chosen convergence criteria are met
Implicit calibration (cont’d...)
The algorithm: indirect implicit calibration
Experimental results
Indirect implicit calibration
0 50 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
time [min]
indirect implicit calibration, dim = 1
CaSO4⋅1/2H2O H2O
CaSO4⋅2H2O
0 50 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
time [min]
indirect implicit calibration, dim = 5
CaSO4⋅1/2H2O H2O
CaSO4⋅2H2O
Experimental results (cont’d...)
Direct implicit calibration
0 5 10 0
0.05 0.1
t = 0
0 5 10 0
0.05 0.1
t = 11
0 5 10 0
0.05 0.1
t = 11.5
0 5 10 0
0.05 0.1
t = 12
0 5 10 0
0.05 0.1
t = 15
0 5 10 0
0.05 0.1
t = 20
0 5 10 0
0.05 0.1
t = 25
0 5 10 0
0.05 0.1
t = 30
0 5 10 0
0.05 0.1
t = 40
0 5 10 0
0.05 0.1
t = 50
0 5 10 0
0.05 0.1
t = 70
0 5 10 0
0.05 0.1
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
99.39899.799.85
k t0
indirect implicit calibration, dim = 1
0.4 0.6 0.8 1 5
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 1 5
10 15
P Ultrasound spectra change during the setting reaction of calcium sulfate cements.
P The parameters of a physical reaction kinetics model can be estimated using information in the spectral amplitudes of the pulses.
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 valuable comments.
Acknowledgements
Preparing to a hot discussion
