Q Leclère

N B Roozen

C Sandier

On the use of the H s estimator for the experimental assessment of transmissibility matrices

Keywords: Transmissibility matrix, MIMO systems, linear system estimation, H s estimator

The experimental estimation of frequency response functions characterizing SISO linear systems is a well established topic. Several estimators are defined in the literature, each estimator being optimal depending upon the assumptions with respect to the balance of noise between the input and output of the system. H 1 and H 2 have to be used in case of presence of noise on output and input, respectively. The H V or H s estimator is chosen if input and output are assumed to have equivalent SNR. These estimators are also established for MIMO linear systems, with additional difficulties due to the necessity of inversing cross spectral matrices. A transmissibility function is generally defined as a linear relationship between two outputs of a linear system. For SIMO systems, transmissibility functions are uniquely defined. The H s estimator is thus advised if both outputs are of equivalent SNR. In the case of MIMO systems, transmissibility functions are no more defined by the system only, it also depends on the input quantities. It is however possible to define a transmissibility matrix between two sets of outputs that is, under some assumptions, uniquely defined. This approach is especially the base of Operational Transfer Path analysis, an engineering method benefiting of a strong research effort in the last few years. This paper deals with the use of the application of MIMO system estimators to the experimental assessment of transmissibility matrices. Transmissibility matrices are generally estimated using a H 1 like approach in the literature. The possibility of using H 2 and H s is presented in this work, from the theoretical point of view and with numerical and practical illustrations.

Introduction

The transmissibility functions are generally defined as linear relationships between two reponses of a linear system. They have particular properties, in comparison with standard transfer functions representing classical excitationresponse relationships. The standard transfer function, for instance, is entirely defined by the studied system, while transmissibility between two responses depends also on the excitation configuration. Another important difference is that the standard transfer functions have peaks at the resonances of the system, while transmissibilities have peaks at frequencies corresponding to zeros of one of the considered response. Transmissibilities are thus more difficult to handle, but they are also easier to measure, because it is generally easier to mount a response sensor than an excitation sensor (which has to be inserted between the excitation device and the structure). That's why several transmissibility based methods have been developed in the literature, for instance in structural health monitoring [START_REF] Chesné | On the transmissibility functions and their use for damage localization[END_REF], output only modal analysis [START_REF] Devriendt | The use of transmissibility measurements in output-only modal analysis[END_REF][START_REF] Devriendt | An operational modal analysis approach based on parametrically identified multivariable transmissibilities[END_REF], or Operational Transfer Path Analysis [START_REF] Gajdatsy | Application of the transmissibility concept in transfer path analysis[END_REF][START_REF] Roozen | Operational transfer path analysis applied to a small gearbox test set-up[END_REF], in which the concept of transmissibility functions has to be extended to transmissibility matrices [START_REF] Ribeiro | On the generalisation of the transmissibility concept[END_REF] between two sets of responses. The present work focusses on the experimental estimation of such transmissibility matrices.

Several estimators are known for the experimental assessment of transfer functions between one input and one output : H 1 has to be used when the noise is on the output and H 2 when the noise is on the input [START_REF] Bendat | Engineering applications of correlation and spectral analysis[END_REF]. Another estimator, H s , has been proposed by Wicks and Vold [START_REF] Wicks | The hs frequency response function estimator[END_REF], based on a total least squares approach, that consider noise on both the input and the output. This estimator has been extended to MIMO systems in [START_REF] White | Analysis of the tls frequency response function estimator[END_REF] [START_REF] Tan | A non-parametric approach for linear system identification using principal component analysis[END_REF]. The H s approach seems particularly interesting for the estimation of transmissibility matrices, because the inputs and outputs of a transmissibility system are both responses, the SNR (Signal to Noise Ratio) has thus no reason to be higher on input or output responses.

The general principles of the transmissibility matrix approach is briefly treated in the first section of the paper. Then the concept of H 1 , H 2 , and H s estimates for transmissibility matrices is addressed from a theoretical point of view. The two last parts are dedicated to numerical and experimental illustrations, respectively.

N.B.: Throughout this paper, bold capitals are used for matrices (including vectors) and non-bold capitals for scalars. Note that all matrices are dependent upon frequency. For sake of brevity this dependency is not mentioned explicitly in the equations.

Transmissibility matrices : definition

Let us consider a linear dynamic system relating a set of n excitation dofs F (size n × 1) to two different sets of response dofs, named indicator dofs Y (n × 1) and output dofs X (m × 1) :

   X Y    =    H Φ    F (1)
where Φ and H are transfer matrices relating excitation dofs to response dofs. A linear relationship can be then defined between X and Y, under the condition of invertibility of Φ :

X = HΦ -1 Y = TY (2)
where T = HΦ -1 is the transmissibility matrix and where -1 denotes a matrix inverse. The existence of T thus depends on these two major conditions :

• the definition of a set of excitation dofs F

• the invertibility of matrix Φ relating Y to F 3 Transmissibility matrices : experimental estimation

Methodology

The relation (2) between inputs and outputs can be written using cross spectral matrices

G xy = TG yy (3) G xx = TG yx (4)
where element (p, q) of matrix G xy is the cross spectrum between the p th element of X and the q th element of Y. However, the cross-spectral matrices obtained for a single operational condition are in general non invertible. Matrices G yy and G yx are indeed rarely of full rank and even less often well conditioned. A potential solution to this problem is to assess transmissibility matrices from non-stationary operating conditions, like run-up or down.

One approach is to gather several steady-state operating condition in one system:

X K = [X 1 X 2 ...X k ] Y K = [Y 1 Y 2 ...Y k ]
where X i and Y i are response vectors obtained during one operating condition, using a phase reference sensor or more sophisticated techniques like Conditioned Spectral Analysis [START_REF] Bendat | Engineering applications of correlation and spectral analysis[END_REF] or Virtual Source Analysis [START_REF] Price | Virtual coherence : A digital signal processing technique for incoherent source identification[END_REF]. If more than one uncorrelated processes are identified (this is the case if the coherence function between channels is not close to unity), several response vectors can be extracted from each acquisition. Finally, cross spectral matrices can still be calculated from such results :

G xy = X K Y K ′ G yy = Y K Y K ′ G xx = X K X K ′
where ′ denotes the complex conjugate transpose. The experimental assessment of these matrices is not always easy, because the computation of T requires inversions of these matrices, that have to be consequently of full rank. This condition is rarely fulfilled using only one operating condition of the studied system ; as was said earlier, it is often necessary to gather information from several operating conditions, using several steady state operating points or run-up-down acquisitions.

H 1 and H 2 estimators

For a scalar transfer function, the H 1 and H 2 estimators are given by

H 1 [T ] = G xy G -1 yy (5) H 2 [T ] = G xx G -1 yx (6)
The H 1 and H 2 estimates of the transmissibility matrix are, by analogy to equations ( 5) and ( 6) and by inversion of systems (3) and (4), defined by

H 1 [T] = G xy G yy -1 (7)
H 2 [T] = G xx G yx + = G xx G yx (G yx G xy ) -1 (8)
where + denotes the pseudo-inverse. It is worth nothing that the estimator H 2 [T] requires m ≥ n, which is a necessary condition for (G yx G xy ) to be of full rank.

H s estimator

For a scalar transfer function, the H s estimator is based on the eigenvalue decomposition of

   s 2 G xx sG xy sG yx G yy    =    U x V x U y V y       λ 1 0 0 λ 2       U x V x U y V y    ′
where s is a positive scaling factor used to balance the magnitude of x and y.

Assuming that the smallest eigenvalue is representing noise, H s is defined as the ratio between contributions of the largest eigenvalue (λ 1 ) at x and y :

H s = U x sU y (9)
which is explicitely given by the both following formulas

H s = s 2 G xx -G yy + (s 2 G xx -G yy ) 2 + 4s 2 |G yx | 2 2s 2 G yx (10) = 2G xy G yy -s 2 G xx + (s 2 G xx -G yy ) 2 + 4s 2 |G yx | 2 (11)
It can be noted that eq. ( 10) is equivalent to H 2 when s → ∞, and that eq. ( 11) is equal to H 1 when s = 0.

The H s estimate of the transmissibility matrix is based on an analysis of the physical rank of the global cross spectral matrix. The system (1) can be formulated in terms of cross spectra :

G xyxy =    G xx G xy G yx G yy    =    H Φ    Sff    H Φ    ′ ( 12
)
The columns of G xyxy are thus linear combinations of the columns of [

H ′ Φ ′ ] ′ .
The rank of G xyxy can not be greater than the number of input loads. The H s estimate of the transmissibility matrix is obtained from the following scaled eigenvalue decomposition

   s x 0 0 s y       G xx G xy G yx G yy       s x 0 0 s y    =    U V    Λ    U V    ′ ( 13
)
where Λ is the diagonal matrix of eigenvalues, [U ′ V ′ ] ′ the matrix of eigenvectors, and s x and s y diagonal scaling matrices. Considering that the number of input loads is equal to n (as well as the number of indicators Y), the rank of the (scaled) G xyxy matrix is lower or equal to n. The m smallest singular values of Λ are thus considered as representing noise, and can be rejected:

   U n V n    Λ n    U n V n    ′ =    s x 0 0 s y       H Φ    S ff    H Φ    ′    s x 0 0 s y    , (14)
where Λ n the diagonal matrix of the n largest eigenvalues and [U n ′ V n ′ ] ′ are the n corresponding eigenvectors. Let us write for convenience the eigenvalue decomposition of the cross spectral matrix of unknown forces :

S ff = PΣP ′
System (14) can be written as follows :

     U n Λ n U n ′ = s x HPΣP ′ H ′ s x V n Λ n V n ′ = s y ΦPΣP ′ Φ ′ s y (15)
that can be written in a more simple way

     U n Λ n 1/2 = s x HPΣ 1/2 V n Λ n 1/2 = s y ΦPΣ 1/2 (16)
Then, Φ -1 and H are expressed as follows

     H = s x -1 U n Λ n 1/2 Σ -1/2 P ′ Φ -1 = PΣ 1/2 Λ n -1/2 V n -1 s y (17)
to finally obtain the expression of H s [T]

H s [T] = HΦ -1 = s x -1 U n V n -1 s y (18)
It can be noted that if response vectors are extracted from several operating conditions to build X K and Y K , the eigenvector decomposition of the whole matrix given by equation ( 13) can be replaced by the singular value decomposition :

   s x X K s y Y K    =    U V    A W ′ =    U V    Ψ, (19)
where

[U ′ V ′ ]    U V    = WW ′ = I,
and where A is the diagonal matrix of singular values. The product A W ′ = Ψ can be considered as a matrix of forces, and U and V as transfer matrices between Ψ and responses, respectively X K and Y K (cf. equation 1). Considering the number of forces exciting the structure equal to n as an a priori information, the m smallest singular values can be zeroed. Finally, noting [U n ′ V n ′ ] ′ the left singular vectors corresponding to the n largest singular values, the expression of H s [T] is the same as in equation ( 18). The computation of U n and V n leads indeed to equal results based on the eigen-decomposition of the whole cross spectral matrix (eq. 13) or from the singular value decomposition of the response vectors (eq. 20).

Effects of scaling matrices on H s

The truncation of singular values can be seen as a way to denoise measurements, because smallest zeroed ones are considered as representing noise. The denoised X K and Y K matrices, noted ỸK and ỸK , are then given by :

   XK ỸK    =    s x -1 U n s y -1 V n    A n W n ′ ( 20
)
with A n the diagonal matrix of the n largest singular values, and [U n ′ V n ′ ] ′ and W n the corresponding left and right singular vectors, respectively.

Let us consider scaling matrices s y = s y I and s x = s x I, with ǫ = s x /s y and m = n (as many response as indicator sensors) for the sake of simplicity. Then it can be shown that :

lim ǫ→0 A n = A Y lim ǫ→0 W n = W Y lim ǫ→0 V n = s y V Y lim ǫ→∞ A n = A X lim ǫ→∞ W n = W X lim ǫ→∞ U n = s x U X
where A Y and W Y are singular values and right singular vectors of Y K , and A X and W X are singular values and right singular vectors of X K , according to following SVDs

X K = U X A X W X ′ Y K = V Y A Y W Y ′
It means that the n largest singular values are governed by only Y K when ǫ → 0 and by only X K when ǫ → ∞. In the first case, ỸK is equal to Y K and rows of XK are projected on left singular vectors W Y . In the second case, XK is equal to X K and lines of ỸK are projected on left singular vectors W X :

lim ǫ→0    XK ỸK    =    X K W Y W Y ′ Y K    lim ǫ→∞    XK ỸK    =    X K Y K W X W X ′   
which means that in the former case the noise is considered as contaminating X K only, and in the latter case Y K only. Limits of U n and V n when ǫ → 0 or ∞ are identified from the previous equations :

lim ǫ→0 U n = s x X K W Y A Y -1 lim ǫ→∞ V n = s y Y K W X A X -1
Finally, the limits of

H s [T] = s x -1 U n V n -1 s y when ǫ → 0 or ∞ are obtained : lim ǫ→0 H s [T] = X K W Y A Y -1 V Y ′ = X K Y K + = H 1 [T] (21) lim ǫ→∞ H s [T] = U X A X (Y K W X ) -1 = X K X K ′ (Y K X K ′ ) -1 = H 2 [T] (22)
These results show that the H s estimator for transmissibility matrices has a similar behavior with respect to H 1 and H 2 than for scalar transmissibilities. When the weight s y of indicator sensors increases, H s gets similar to H 1 , and when the weight s x of response sensors increases, H s gets similar to H 2 . In the former case the SNR will be a priori considered to be higher on indicator sensors Y K (i.e. more noise on X K ) and in the latter case on X K (i.e. more noise on Y K ).

The correct scaling of the system has to be done with respect to noise, but also with respect to different units or overall levels of each sensor. This is crucial if different type of sensors are used, for instance accelerometers and microphones. In such a case, a global scaling has to be applied so that overall scaled levels are almost equal (see [START_REF] Leclere | Different least squares approaches to identify dynamic forces acting on an engine cylinder block[END_REF][START_REF] Leclere | Indirect measurement of main bearing loads in an operating diesel engine[END_REF] for details). If this step is not correctly carried out, then the H s estimator will be arbitrarily closer to H 1 or H 2 , not because of SNR balance assumptions but because of strong level differences due to the use of different units.

An indicator for the validity of H s

The validity of equation ( 14) depends on the hypothesis that the n largest eigenvalues are representing the signal, and the m smallest ones the noise. This hypothesis can be verified by inspecting the ratio between the n th eigenvalue, representing the energy of the smallest incoherent process being part of signal, and the (n+1) th eigenvalue representing the energy of largest noise component.

R = λ n λ n+1 = a 2 n a 2 n+1 ( 23
)
where λ k is the k th largest eigenvalue of Λ in equation ( 13) and a k is the k th largest singular value of A in equation ( 20). Note that this check could also be performed to verify the rank of the cross-spectral matrix in case an H 1 or an H 2 matrix estimate is used. Indeed, whilst for H 1 and H 2 only n singular values are extracted from G yy and G yx , respectively, it is also for these estimators of prime importance that the physical rank of the cross-spectral matrix is equal to n.

Numerical simulation

A numerical simulation has been conducted to validate the proposed approach.

An analytic model of a thin rectangular plate ((L 1 × L 2 ) = (0.6 × 0.5)m 2 ) has been used for simulations, with simply supported boundary conditions (physical parameters : aluminum, thickness 5mm, modal damping ratio 1%). Three excitation points ([x The estimation error is computed as a function of frequency for each estimator using Relative error E i (i=1,2,s) Fig. 2. 50Hz frequency band integrated relative errors of H 1 (dotted blue), H 2 (dash-dotted black) and H s (dashed red) estimates.

E i (f ) = ||T -H i (T)|| F ||T|| F (
This error is drawn in Figure 2 for i = 1, 2, s. It is clear that the error E s is significantly lower than E 1 and E 2 , except in low frequency (below 100Hz), where E 1 is the lowest one. This can be explained by a too low SNR in low frequency. If the SNR is too low, the eigenvalues representing noise in G xyxy can be higher than the eigenvalues representing the signal. In this case, keeping the n largest eigenvalues leads to an erroneous estimation of T. This can be illustrated by the value of the H s indicator R as introduced in section 3.5. This indicator R is equal to 0dB below 200Hz, which means that the n and n + 1 th eigenvalues are about the same level. It is thus hard to separate signal from noise. Above 200Hz, the ratio becomes significantly greater than 1, resulting in a better estimation. In the current case, an equal amount of noise is used on all outputs to simulate measurement noise. For this reason H s with equal scaling factors gives the best estimate, as can be seen from Figure 2. In case different signal-to-noise ratios are present for the indicator responses Y and the output responses X, a different scaling factor for the indicator responses Y and the output responses X, respectively, would in that situation give the best H s estimate.

Experimental illustration

An experimental validation has been carried out to validate the proposed approach. The experiment took place in two rooms which are acoustically connected with each other by means of an aluminum plate with a thickness of 1mm and a dimension of 60 x 40 cm. Twenty microphones -the set of output responses-were placed on the reception room side. The plate was excited from the emission room side by means of two shakers and one loudspeaker, constituting two structure borne paths and one airborne path. Two accelerometers were mounted on the plate, near shaker connection points, and one microphone was placed in the emission room : these three responses were chosen as indicators (the accelerometers for the two shakers and the microphone for the loudspeaker). In a first step, each source has been excited successively with white noise to measure directly transfer functions between inputs (signals sent to the shakers and loudspeaker) and responses, to build matrices H and Φ.

For each excitation configuration with only one active source, the transfer functions have been estimated using a H 1 approach, which is well suited for transfer function measurements with a low noise on inputs. This assumption seems reasonable because input signals are directly measured (without acoustic or vibration transmission) and because they are white (the energy is distributed continuously on the whole frequency range). The transmissibility matrix assessed with H and Φ (equation 2) is considered as the reference transmissibility matrix in the following. In a second step, the three physical sources were driven by three uncorrelated generators simultaneously to measure the whole response cross-spectral matrix G xyxy . H 1 and H 2 estimators of the transmissibility matrix have been assessed, as well ad H s . For the latter, a scaling has to be applied because different kinds of sensors are used (accelerometers and microphones). The scaling strategy was to normalize all measurement channel by its rms value, thus assuming that the SNR ratio is the same on all channels. One element of the transmissibility matrix is drawn in Figure 4. All estimators fit the reference transmissibility well on the whole frequency range, even if some errors are visible in low frequency. The normalized errors, which are computed by means of equation 24, are drawn in Figure 5. It is clear that the estimation error is significantly lower with the H s estimator on the whole frequency range. In low frequency, H s and H 2 are similar, and globally better than H 1 . In the high frequency range, H s is similar to H 1 , even slightly better, and both are significantly better than H 2 . It can be said globally that the H s estimator gives better results in this experiment than H 1 and H 2 . The applied scaling strategy (normalization of the global levels) was however very basic, and could be improved and optimized to obtain even more satisfying results. The relative error of H s is drawn in Figure 6 (top) in narrow frequency bands in the low frequency range, together with the R indicator defined in equation (23). It can be seen that the indicator is in good agreement with the estimation error : it can be said roughly that when the indicator is greater than 10dB, then the relative estimation error is below 10%. On the other hand, when the indicator drops to almost 0dB, then the relative error increases and can exceed 100%.

Conclusion

Transmissibility matrices are generally estimated using a H 1 like approach. The possibility to use H 2 and H s estimators for transmissibility matrices is presented. The H s approach is particularly interesting for the estimation of transmissibility matrices, because the inputs and outputs of a transmissibility system are both responses, the SNR (Signal to Noise Ratio) has thus no reason to be higher on input or output responses. The H s estimate of the transmissibility matrix is based on the eigenvalue decomposition of the global cross spectral matrix of input and output responses. After a rigorous derivation of the expressions to calculate H s , it is theoretically shown that H s tends to H 1 and H 2 , depending on the scaling ratios. It is worth nothing that the estimator H 2 only exists if the number of output dofs m is equal or larger than the number of indicator dofs n. Both a numerical simulation and a physical experiment have been conducted to validate the proposed transmissibility estimates. In the numerical simulation a simply supported rectangular plate was considered. In the experiment a plate was excited by means of two shakers and one loudspeaker, constituting two structure borne paths and one airborne path. Both accelerometers and microphones were used, which required some scaling of the data in order to optimize the H s estimate. For both the numerical simulation and the physical experiment it was found that H s behaves globally better than H 1 and H 2 . In addition, an indicator for the validity of H s is introduced, which checks that there is a clear separation between the n largest eigenvalues (representing the signal), and the m remaining eigenvalues (representing noise). By approximation, and at least for numerical and experimental illustrations presented in this work , it can be said that for values of the indicator larger than 10dB, the relative error in the transmissibility estimate is below 10%.

Appendix A : simulation of cross-spectral matrices of uncorrelated finite gaussian signals estimated by the periodogram method

The aim of this appendix is to explain how cross spectral matrices are randomly generated for measurement simulations. Studied signals are considered to be Gaussian :

x[n] = N (0 ; x) , where x stands for the rms value of the signal. The discrete Fourier Transform of x is given by

X k = 1 N N -1 n=0
x[n]e -j2πkn/N with the following expected value and variance

E(X k ) = 0 V(X k ) = x 2 /N
with E(X k ) = 0 and V(X k ) = x 2 /N . X k follows a complex gaussian law for k = {0, N/2}, the real and imaginary parts following real centered gaussian laws of variance x 2 /2N . For k = {0, N/2} , X k follows a real centered gaussian law of variance x 2 /N , but this case will not be treated here for the sake of brevity. The double sided instantaneous autospectrum for k = {0, N/2} is equal to

S i xx k = 2|X k | 2 = 2R (X k ) 2 + 2I (X k ) 2 = x 2 N χ 2 2
with the following expected value and variance

E(S i xx k ) = 2x 2 N V(S i xx k ) = 4x 4 N 2
The expected value and variance of the double sided instantaneous cross spectrum S i xy k = 2X k Y * k of two independent signals x and y are

E(S i xy k ) = 0 V(S i xy k ) = 4x 2 y 2 N 2
When applying the averaged periodogram method, auto and cross spectra are averaged over a number M of time windows :

S xx k = |X k | 2 M S xy k = X k Y k M
Assuming that M is sufficiently high to apply the central limit theorem, then averaged auto and cross spectra are following gaussian distributions :

S xx k = N 2x 2 N ; 2x 2 N √ M S xy k = N 0 ; 2x y N √ M , k ∈]0, N/2[

Fig. 1 .

 1 Fig. 1. #(3,1) element of the transmissibility matrix. reference T (Solid gray) H 1 (dotted blue), H 2 (dash-dotted black) and H s (dashed red) estimates

Fig. 3 .

 3 Fig. 3. H s indicator R given by equation (23)

Fig. 4 .

 4 Fig. 4. #(5,2) element of the transmissibility matrix, whole (top) and low (bottom) frequency ranges. Solid gray : T based on reference measurements, H 1 (dotted blue), H 2 (dash-dotted black) and H s (dashed red) estimates

Fig. 5 .

 5 Fig.5. 500Hz frequency band integrated relative errors of H 1 (dotted blue), H 2 (dash-dotted black) and H s (dashed red) estimates.

Fig. 6 .

 6 Fig. 6. Relative errors of H s [T ] (top) and corresponding SNR indicator (bottom) in the low frequency range.

  ) from noise-free H and Φ matrices, and estimated from noisy responses G xyxy . All response channels are accelerations, and the global SNR is the same on all responses. Scaling factors are thus chosen equal to unity for this numerical illustration. It can be seen that H s behaves globally better than H 1 and H 2 , particularly at low frequencies. In the low frequency range, the SNR is lower because the simulated response level is lower than in mid and high frequency. The global SNR is indeed fixed to 30dB, but the noise spectral density is constant (white noise) while the simulated accelerations are more energetic in high and mid frequency than in low frequency. The SNR spectrum is thus higher in mid and high frequency than in low frequency. Above 400Hz, the three estimations are in good agreement with the true transmissibility.

	1 , x 2 ] = [0.05, 0.01]; [0.05, 0.4]; [0.45, 0.25]) and six
	response points (same position as excitations for indicator responses Y and
	[x 1 , x 2 ] = [0.3, 0.2]; [0.5, 0.42]; [0.3, 0.1] for output responses X) have been con-
	sidered. The response cross spectral matrix G xyxy has been computed using
	equation (12), using transfer matrices H and Φ obtained with the analyti-
	cal model and a cross spectral matrix S ff of uncorrelated unitary excitations.
	Measurement noise is added on the cross spectral matrix, using the approach
	described in appendix A. Gaussian incoherent noise is added on each indicator
	and output responses, with rms values adjusted to get a 30dB SNR for all re-
	sponses and with periodogram parameters N = 3000, M = 100. One element
	of the transmissibility matrix is drawn in Figure 1, directly computed using
	equation (2

It is finally possible to simulate whole cross spectral matrices of uncorrelated signals using a gaussian random generator, from signals rms values, fixing values for N (number of samples of a time window) and M (number of time windows).


Nathalie Lemonne

Yann Lamarre

Marc Romana

Marie-Dominique Hardy- Dessources

François Lionnet

Xavier Waltz

Vanessa Tarer

Danielle Mougenel

Benoît Tressières

Marie-Laure Lalanne-Mistrih

Maryse Etienne-Julan

Philippe Connes
email: pconnes@yahoo.fr

Impaired blood rheology plays a role inthe chronic disorders associated with sickle cellhemoglobin C disease

Keywords: Sickle cell disease, sickle cell-hemoglobin C disease, blood viscosity, otologic disorders, retinopathy Blood viscosity in hemoglobin SC disease

Lionnet et al. [START_REF] Lionnet | Hemoglobin sickle cell disease complications: a clinical study of 179 cases[END_REF] recently reported a high prevalence of retinopathy (RET) and otologic disorders (OTD) in patients with sickle cell-hemoglobin C disease (SC), while a significant number of patients had renal diseases (mainly glomerulopathy; GLO) and osteonecrosis (OST). The pathophysiological processes of these complications in SC are not well defined, although blood hyperviscosity has been suspected, but never tested to the best of our knowledge, as responsible for several chronic complications in SC disease [START_REF] Lionnet | Hemoglobin sickle cell disease complications: a clinical study of 179 cases[END_REF][START_REF] Tripette | Red blood cell aggregation, aggregate strength and oxygen transport potential of blood are abnormal in both homozygous sickle cell anemia and sickle-hemoglobin C disease[END_REF] .The aim of this study was to analyze the associations between hematological and hemorheological parameters and chronic complications in adult SC patients.

Ninety consecutive adults with SC (M/F: 40/50; mean age: 38 ± 13 yr)were enrolled in the study. All patients were at steady state at study entry, i.e., no phlebotomy orblood transfusions in the previous three months, and absence of acute episodes (infection, vaso-occlusive crisis (VOC), acute chest syndrome (ACS), stroke, priapism) at least three months before enrollment. Pregnancy or breast feeding were also exclusion criteria. The study was conducted in accordance to the Declaration of Helsinki, and approved by the Regional Ethics Committee (registration number: 2010-A00244-35). Written informed consent was obtained from all participants.

History, presence of chronic disorders, and acute event occurrence during the previous year of study were obtained from retrospective chart review by two physicians.Patients under regular phlebotomy protocols, but without any phlebotomy in the 3 months preceding the study were identified. Phlebotomyis usually performed in symptomatic SC patientsto avoid recurrent acute events 3 ,and has been prescribedby some when hemoglobin and/or hematocrit rise above 11 g/dL or 32%, respectively,to prevent complications with suspected blood hyperviscosity [START_REF] Lionnet | Hemoglobin sickle cell disease complications: a clinical study of 179 cases[END_REF][START_REF] Lionnet | Venous and arterial viscosity-associated thrombosis in hemoglobin SC disease[END_REF] .

The optimal target hemoglobin level to reach under phlebotomy is unknown but our Sickle Cell Centre usually tries to reduce hemoglobin to 9.5-10.5 g/dL.In our study, SC patients with values greater than 11g/dL and 32% but no clinical assessment of blood viscosity were categorized as having "theoreticalhyperviscosity".

Measurements of hematological and hemolytic parameters (bilirubin, lactate dehydrogenase, aspartate aminotransferase) were performed using standard methods [START_REF] Lemonne | Does increased red blood cell deformability raises the risk for osteonecrosis in sickle cell anemia?[END_REF] .Blood viscosity, red blood cell (RBC) deformability, aggregation and disaggregation threshold (i.e., RBC aggregates strength) were measured as described [START_REF] Baskurt | New guidelines for hemorheological laboratory techniques[END_REF][START_REF] Lamarre | Hemorheological risk factors of acute chest syndrome and painful vasoocclusive crisis in children with sickle cell disease[END_REF] .

Unpaired Student's t-test and chi-square or Kappa coefficient test were used for continuous and categorical covariates, respectively. Association between several parameters was tested by Pearson correlation. The hemolytic component value was derived from hemolytic markers (bilirubin, lactate dehydrogenase, aspartate aminotransferase and reticulocytes) by principal component analysis [START_REF] Nouraie | The relationship between the severity of hemolysis, clinical manifestations and risk of death in 415 patients with sickle cell anemia in the US and Europe[END_REF] .

The most prevalent chronic complications in our SC cohort were RET (60%), GLO (micro-/macro-albuminuria, 40%), OST (31%), and OTD (20%). Leg ulcers, pulmonary hypertension and cerebral vasculopathy/stroke were extremely rare (2% each), as were ACS (2%) and VOC (5%) during the study period.Few males (8%) had a history of priapism.

Patients with OTD (OTD+) had higher RBC count (p<0.05), and a tendency to higher hemoglobin level than patients without OTD (OTD-; p<0.1; Table 1). Blood viscosity was increased by 9.1% in OTD+ compared to OTD-patients (p<0.05). SC RET+ patients had lower RBC deformability than RET-(p<0.05; Table 1). No association was observed between hematological or hemorheological parameters and OST or GLO. Nevertheless, OST+ and GLO+ patients were older than OST-and GLO-individuals (p<0.001 and p<0.01, respectively).An association was found between RET and OST, with higher frequency of OST+ in the RET+ (38%) than in the RET-(18%; p<0.05) groups. No association was found between the other complications.

In our cohort, 43% SC patients underwent phlebotomy, 74% of them on a regular basis (every 3 months). The frequency of RET+ and RET-patients treated by phlebotomy was not different, as 57.6% RET+ patients had phlebotomy vs. 42.4% RET-patients. Only a trend towards greater phlebotomy use was observed in OTD+ patients (64%; p<0.1). Most patients (88.6%) with "theoreticalhyperviscosity" (hemoglobin>11g/dL/hematocrit>32%)were phlebotomized (p<0.01). Comparing patients with or without theoreticalhyperviscosity demonstrated no significant difference in blood viscosity ( Elevated blood viscosity was hypothesized to cause ischemia at the labyrinthine artery level, leading to cochlear damage [START_REF] Lazarini | Idiopathic sudden sensorineural hearing loss: etiopathogenic aspects[END_REF] .This is in agreement with our cohort where OTD+ patients had higher blood viscosity than OTD-. However, OTD+ had only a trend towards higher hemoglobin and similar hematocrit as OTD-. Blood viscosity is influenced by several factors, including hematocrit and hemoglobin, hence their clinical use to prescribe phlebotomy.

However, this relationship was not significant in our cohort. Blood viscosity depends also on the rheological properties of RBCs: i.e., deformability and aggregation. For a given hemoglobin level, increased RBC deformability lowers blood viscosity while increased RBC aggregation causes a rise. The complex contribution of each hemorheological factor on blood viscosity makes that blood viscosity may be elevated in some patients despite "normal" hematocrit and hemoglobin levels.

More importantly, most patients with "theoretical hyperviscosity" did not have high blood viscosity, and only 44% of patients with measured "true hyperviscosity" were phlebotomized.

Thus, as periodic phlebotomy could be useful to decrease blood viscosity in hyperviscous SC patients as it is in patients with polycythemia vera [START_REF] Davis | Hearing improvement induced by phlebotomy in polycythemia[END_REF] or cyanotic congenital heart disease [START_REF] Perloff | Adults with cyanotic congenital heart disease: hematologic management[END_REF] , our findings strongly suggest that blood viscosity measurements would allow better identification of SC patients at risk for OTD.

In contrast to OTD, RET was not associated with blood hyperviscosity. Instead, RBC deformability was decreased by 10% in RET+ compared to RET-patients. RBC deformability is critical for optimal tissue perfusion and adequate blood flow in the micro-/ macrocirculation [START_REF] Baskurt | Blood rheology and hemodynamics[END_REF] , and reduced RBC deformability is associated with diabetic retinopathy [START_REF] Goldstein | Red blood cell membrane mechanical fluctuations in non-proliferative and proliferate diabetic retinopathy[END_REF][START_REF] Shin | Progressive impairment of erythrocyte deformability as indicator of microangiopathy in type 2 diabetes mellitus[END_REF] . The effects of phlebotomy on RBC deformability in SC patients with RET have never been investigated, calling for further studies to address this question.

In conclusion, our study provides new data on the pathophysiology of several frequent chronic complications in SC disease. They clearly show that the clinical use of hemoglobinand hematocrit as surrogates for high blood viscosityin SC patients is not satisfactory for establishing treatment or determining risk for OTD. A prospective study to evaluate the relationships between blood rheology and the occurrence of acute complications iswarranted.

funding for X.W. was supported by the association "Ensemble contre la drépanocytose" and by the regional council of Guadeloupe. Means ± SD. -= absence of complication and + = presence of complication.Sensorineural OTD (requiring hospitalization or not) were recorded using previously defined criteria [START_REF] Lionnet | Hemoglobin sickle cell disease complications: a clinical study of 179 cases[END_REF] . Criteria used to define VOC and ACS events (within the previous year of study) were similar to those used in previous studies performed in SCA adults [START_REF] Lamarre | Hemorheological risk factors of acute chest syndrome and painful vasoocclusive crisis in children with sickle cell disease[END_REF] .Criteria used to define osteonecrosis and glomerulopathy were similar to those used in previous studies [START_REF] Lemonne | Does increased red blood cell deformability raises the risk for osteonecrosis in sickle cell anemia?[END_REF][START_REF] Lamarre | Alpha thalassemia protects sickle cell anemia patients from macro-albuminuria through its effects on red blood cell rheological properties[END_REF] .Difference between groups (*p<0.05; **p<0.01; ***p<0.001). $ Statistical trend (p<0.1).

  7.50±1.03 vs 7.22±.35 cP,

	respectively),	whilehemoglobin	and	hematocritwere	higher	in	the
	theoreticalhyperviscositygroup (11.8±1.0 vs 11.2±1.3 g/dL and 32.3±2.5 vs 30.6±3.2 %,

respectively; p<0.01), as expected. Thecohort was divided according to the median measured viscosity, and patients with blood viscositygreater than median value were considered as having"true hyperviscosity".No significant association was found between theoreticaland truehyperviscosity (Kappa coefficient=0.09), or between blood viscosity and hemoglobin or hematocrit, (r=0.19 and r=0.18, respectively; p=0.16 in both cases).

Table 1 :

 1 Hematological and hemorheological determinants of otologic disorders, retinopathy, osteonecrosis and glomerulopathy in SC patients.

		Otologic disorders (OTD)		Retinopathy (RET)	Osteonecrosis (OST)	Glomerulopathy (GLO)
		OTD-	OTD+		RET-	RET+	OST-	OST+	GLO-	GLO+
		(n = 64)	(n = 16)		(n = 33)	(n = 49)	(n = 62)	(n = 28)	(n = 51)	(n = 34)
	Age (years)	40 ± 13	37 ± 14		39 ± 14	39 ± 13	35 ± 12	45 ± 13***	36 ± 13	43 ± 13**
	White blood cells (x10 9 L)	7.4 ± 2.6	7.0 ± 2.4		7.4 ± 2.4	7.4 ± 2.6	7.6 ± 3.0	7.0 ± 2.4	7.4 ± 2.9	7.6 ± 2.4
	Platelet count (x10 9 L)	306 ± 154	292 ± 151	298 ± 166	302 ± 145	308 ± 156	278 ± 139	301 ± 152	308 ± 152
	Red blood cells (x10 12 L)	4.3 ± 0.6	4.7 ± 0.7*	4.3 ± 0.7	4.4 ± 0.7	4.4 ± 0.7	4.4 ± 0.6	4.4 ± 0.6	4.3 ± 0.6
	Hemoglobin (g/dL)	11.2 ± 1.1	11.8 ± 1.3	$	11.3 ± 1.2	11.4 ± 1.2	11.4 ±1.3	11.3 ± 1.1	11.5 ± 1.1	11.2 ± 1.2
	Hematocrit (%)	31.0 ± 2.7	32.1 ± 3.8	31.0 ± 2.9	31.3 ± 3.1	31.1 ± 3.2	31.4 ± 2.9	31.4 ± 2.8	30.8 ± 3.1
	Hemolytic index (relative unit)	0.01 ± 0.98	-0.21 ± 0.96	-0.12 ± 1.07	0.02 ± 0.93	0.02 ± 0.99	-0.03 ± 1.03 -0.06 ± 0.94	0.08 ± 1.08
	Blood viscosity (cP)	7.38 ± 1.17	8.10 ± 0.85*	7.60 ± 1.10	7.28 ± 1.32	7.40 ± 1.14	7.50 ± 1.42	7.51 ±1.22	7.34 ± 1.16
	RBC deformability (a.u)	0.43 ± 0.07	0.41 ± 0.06	0.45 ± 0.06	0.41 ± 0.05*	0.43 ± 0.07	0.42 ±0.07	0.42 ± 0.07	0.43 ± 0.07
	RBC aggregation (%)	49 ± 8	47 ± 11		50 ± 8	48 ± 10	47 ± 8	50 ± 10	47 ± 10	50 ± 8
	RBC disaggregation threshold (s -1 )	320 ± 126	312 ± 132	317 ± 101	319 ± 140	306 ± 115	330 ± 141	312 ± 144	314 ± 91

Acknowledgments

We thank Dr. Martine Torres for critical review of the manuscript and editorial assistance.PhD funding for Y.L. was supported by the regional council of Guadeloupe. PhD Authorship and Disclosures: N.L., Y.L., M.R., M.D.H.D., B.T., M.L.L.M., M.E.J. and P.C. designed the protocol. N.L., Y.L., X.W., D.M. and P.C. performed the experiments. N.L., Y.L., M.R., M.D.H.D., F.L., X.W., V.T., B.T. and P.C. analyzed and interpreted the data. N.L., Y.L., M.R., M.D.H.D., F.L., X.W., V.T., D.M., B.T., M.L.L.M., M.E.J. and P.C. wrote the article.

We declare no conflict of interest.