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