Andrew T Young
email: atyoung@olemiss.edu

Matthew J Higgins
email: mhiggins@mgt.gatech.edu

Daniel Levy
email: levyda@mail.biu.ac.il

Sigma Convergence versus Beta Convergence: Evidence from U.S. County-Level Data

Keywords: sigma-convergence, beta-convergence, Solow growth model, speed of convergence, balanced growth, U.S. county-level data, income distribution, Gini coefficient, income equality

In this paper we outline (i) why -convergence may not accompany -convergence, (ii) discuss evidence of -convergence in the U.S., and (iii) use U.S. county-level data containing over 3,000 cross-sectional observations to demonstrate that -convergence cannot be detected at the county-level across the U.S., or within the large majority of the individual U.S. states considered separately. Indeed, in many cases statistically significant -divergence is found.

I. Introduction

One can distinguish between two types of convergence in growth empirics:convergence and -convergence. When the dispersion of real per capita income (henceforth, simply "income") across a group of economies falls over time, there isconvergence. When the partial correlation between growth in income over time and its initial level is negative, there is -convergence.

By the "convergence literature," economists typically refer to the large literature, typified by the seminal papers by [START_REF] Barro | Convergence[END_REF] and [START_REF] Mankiw | A Contribution to the Empirics of Economic Growth[END_REF], exploring -convergence. [START_REF] Sala-I-Martin | Regional Cohesion: Evidence and Theories of Regional Growth and Convergence[END_REF], p. 1326), surveying this literature, concludes that "the estimated speeds of [ -]convergence are so surprisingly similar across

[cross-sectional] data sets, that we can use a mnemonic rule: economies converge at a speed of two percent per year." In other words, economies close the gap between present levels of income and balanced growth levels by, on average, 2 percent annually. Panel data studies find even higher rates of -convergence -see [START_REF] Islam | Growth Empirics: A Panel Data Approach[END_REF] and Evans (1997a) -as do the county-level U.S. studies of [START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF] and [START_REF] Young | Heterogeneous Convergence[END_REF].

Despite the literature's stress on -convergence, economists have acknowledged that it is not a sufficient condition for -convergence. (For an early acknowledgement of this idea see Barro and Sala-i-Martin (1992, pp. 227-228)) Indeed, [START_REF] Quah | Galton's Fallacy and the Convergence Hypothesis[END_REF] and [START_REF] Friedman | Do Old Fallacies Ever Die[END_REF] both suggest that -convergence is of greater interest since it speaks directly as to whether the distribution of income across economies is becoming more equitable.

In this paper we demonstrate that -convergence is a necessary but not sufficient condition for -convergence. Then we discuss evidence of -convergence in the U.S.

using county-level data covering 1970 to 1998 and containing over 3,000 cross-sectional observations. We demonstrate, using the same data, that -convergence cannot be detected during that time period in the U.S. or within the large majority of the individual U.S. states considered separately. Indeed, in many cases statistically significantdivergence is found.

The paper is organized as follows. Section II demonstrates why -convergence need not accompany -convergence. Section III describes the U.S. county-level data that is the basis for our present ( -convergence) and previously reported ( -convergence) results. Section IV discusses the existing empirical evidence indicating thatconvergence exists in the U.S. Section V demonstrates that -convergence did not occur across the U.S., or within a majority of the individual U.S. states, from 1970 to 1998. Section VI reports Gini coefficients for the same county-level data that are consistent with a lack of -convergence. Section VII concludes.

II. -Convergence versus -Convergence

Following Sala-i-Martin's (1996) exposition, assume that -convergence holds for economies i = 1, ..., N.1 Natural log-income of the i-th economy can be approximated by

( ) ( ) ( ) , u y ln a y ln it t , i it + - + = -1 1 β (1)
where 0 < < 1 and u it has mean zero, finite variance, 2 u σ , and is independent over t and i. Because a is assumed constant across economies, balanced growth paths are identical. 2

Manipulating (1) yields,

( ) it t , i t , i it u y ln a y y ln + - = - - 1 1 β .
(1)' Thus, > 0 implies a negative correlation between growth and initial log income.

The sample variance of log income in t is given by

( ) [ ] 2 1 2 1 = - = N i t it t y ln N µ σ , ( 2
)
where t is the sample mean of (log) income. The sample variance is close to the population variance when N is large, and (1) can be used to derive the evolution of2 t σ :

( )

2 2 1 2 2 1 u t t σ σ β σ + - ≅ - .
(3)

Only if 0 < < 1 is the difference equation stable, so -convergence is necessary forconvergence.3 Given 0 < < 1, the steady-state variance is,

( ) ( ) [ ] 2 2 * 2 1 1 β σ σ - - = u .
(4) Thus, the cross-sectional dispersion falls with but rises with 2 u σ . Combining (3) and

(4) yields,

( ) ( ) [ ] ( ) * t t 2 2 2 1 2 2 1 1 1 σ β σ β σ - - + - = - , (5)
which is a first-order linear difference equation with constant coefficients. Its solution is

given by,

( ) ( ) ( ) ( ) t * t * t c 2 2 2 0 2 2 2 1 1 β σ σ β σ σ - + - - + = , ( 6
)
where c is an arbitrary constant. Thus, as long as 0 < < 1, we have |1 -| < 1, which implies that

( ) 0 1 2 = - ∞ → t t lim β . ( 7
)
This ensures the stability of 2 t σ because it implies that,

( ) * t t lim 2 2 σ σ = ∞ → . ( 8
)
Moreover, since (1 -) > 0, the approach to ( ) * 2 σ is monotonic.

It follows, therefore, that the variance will increase or decrease towards its steadystate value depending on the initial 2 0 σ . Intuitively, economies can be -converging towards one another while, at the same time, random shocks are pushing them apart.

Despite -convergence, if the initial dispersion of income levels is, by chance, small relative to the variance of random shocks then the dispersion of incomes will converge towards its steady-state value from below. 4Other scenarios where -convergence does not imply -convergence arise when the parameter a varies across economies. 5 Intuitively, consider two economies, A and B, where both economies begin at the same level of income. However, assume that B begins on its balanced growth path while A begins far below its balanced growth path, and assume that -convergence holds. The initial variance ( 2 0 σ ) will be zero, but 2 t σ will grow over time as A grows faster than B and approaches a higher balanced growth path. Indeed, -convergence is the reason for the increasing variance.6

III. U.S. County-Level Data

This paper utilizes the U.S. county-level data used by [START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF] and The definitions that are used for the components of personal income at the countylevel are essentially the same as those used for national measures. 8 For example, the BEA defines "personal income" as the sum of wage and salary disbursements, other labor income, proprietors' income (with inventory valuation and capital consumption adjustments), rental income (with capital consumption adjustment), personal dividend income and personal interest income.

IV. -Convergence: Existing Evidence

Many studies have documented -convergence in the U.S. [START_REF] Barro | Convergence[END_REF], Evans andKarras (1996a and1996b), [START_REF] Sala-I-Martin | Regional Cohesion: Evidence and Theories of Regional Growth and Convergence[END_REF]), and Evans (1997a[START_REF] Evans | Consistent Estimation of Growth Regressions[END_REF] find statistically significant -convergence effects using U.S. statelevel data. The present authors use U.S. county-level data to document statistically significant -convergence effects across the U.S. [START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF], and within many individual U.S. states in and of themselves [START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF]. See Table 1.

Using a consistent three stage least squares (3SLS) estimation method, we estimate the -convergence rate to be between 6.3 and 9.8 percent for the U.S. as a whole and, for individual U.S. states, -convergence rate point estimates range from just under 4 percent (California) to over 14 percent (Louisiana). (See Clearly, considerable evidence supports the existence of -convergence, which is a necessary condition for -convergence. Below we explore whether or notconvergence is occurring in the U.S. using the same county-level data that were used by [START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF] and [START_REF] Young | Heterogeneous Convergence[END_REF].

V.

-convergence

To our knowledge, the only study of U.S. regional -convergence is Tsionas (2000). He examines real Gross State Products (RGSPs) and finds that "…the cross sectional variance has fluctuated very little in the 20-year period from 1977 to 1996" (pp.

235-236). In contrast to Tsionas' data set, our data cover nearly a decade longer.

Moreover, we have over 3,000 county-level cross-sectional observations while Tsionas uses 50 state-level observations. 9 It turns out, nevertheless, that our findings are ultimately consistent with Tsionas'. Some have suggested that interpreting measures of dispersion may not be straightforward if the distributions are not unimodal, e.g., [START_REF] Quah | Empirics for Growth and Distribution: Stratification, Polarization, and Convergence Clubs[END_REF] and [START_REF] Desdoigts | Patterns of Economic Development[END_REF]. However, as Figure 2 demonstrates, for the U.S. county-level data the distribution of income is unimodal for both 1970 and 1998. Figure 2 also allows one to confirm, visually, that -convergence is not present.

VI. Has -divergence Implied Greater Income Inequality?

Another measure we report that is associated with -convergence (in the sense that it deals with the distribution of income) is the Gini coefficient associated with U.S. 9 As well, Tsionas apparently did not convert RGSPs into per capita measures. 10 There is one other state where the standard deviation is smaller for 1998: Kentucky. 11 Of note, of the 24 states with statistically significant increases in standard deviations, 16 of them appear in Table 1 for having statistically significant -convergence effects.

counties ' 1970 and 1998 (log) incomes: 0.0167 and 0.0165 respectively -a decrease of about 1.2 percent. See Table 3. Recall that Gini coefficient is a number between 0 (perfect equality) and 1 (perfect inequality).

Interestingly, at the county-level, although the distribution of U.S. per capita income became a bit more dispersed from 1970 to 1998, it became a bit more equal.12 However, the change in both the standard deviation and the Gini coefficient are small enough to suggest that both dispersion and equality remained essentially the same.

To try to understand further the evolution of the U.S. county-level income distribution, Table 3 summarizes two additional statistics computed from the 1970 and 1998 income distributions: skewness and kurtosis. From 1970 to 1998, the skewness of the distribution increased from -0.2244 (to the left) to 1.7240 (to the right). At the same time, kurtosis increased from 3.4334 to 10.3237, implying that the distribution became more peaked. This suggests that these two effects have been offsetting to a great extent.

VII. Conclusion

In this paper we show that -convergence is a necessary but not sufficient condition for -convergence. We discuss evidence of -convergence in the U.S. using county-level data for the 1970 to 1998 period. Using the same data, we show thatconvergence was not present during that time period in the U.S. or within a large majority of the individual U.S. states considered separately. In many cases, in fact, statistically significant -divergence is found.

What are we to make of the presence of -convergence and evidence ofdivergence? If the U.S. was approaching its steady-state income variance from below during the 1970 to 1998 period then, based on our discussion in Section II, two interpretations suggest themselves. 13 First, the initial distribution of income was narrow in 1970 relative to the distribution of balanced growth paths. Second, the 1970 (1998) draw of county-specific shocks had a small (large) sample variance relative to the population variance of shocks.

Going beyond (but still consistent with) the discussion of Section II, another interpretation is that the variance of the balanced growth paths themselves increased.

However, one may consider this second interpretation unlikely considering the relative institutional homogeneity of counties across the U.S. and especially within individual states where the same -convergence versus -convergence results hold in many cases.

A final -and perhaps least likely -interpretation is that rich counties tend to have balanced growth rates that are higher than those of poor counties. There is little reason to think, however, that the long-run growth rates of technological know-how are different across U.S. counties (and, again, especially within individual states).

In any case, the evolution of skewness and kurtosis suggests that there may be an underlying -convergence for a "majority club" of U.S. counties but that there is another "minority club" that is evolving into a long right-hand tail of the distribution, preventing -convergence in the aggregate. 13 A related issue, which we do not address in this paper directly, is whether or not the cross-sectional distribution of log per capita income is ergodic [START_REF] Evans | Using Cross-Country Variances to Evaluate Growth Theories[END_REF]. That would mean that the cross-sectional variance is stationary around a mean or is converging asymptotically toward a constant mean. Note: estimates are results originally reported in [START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF] and [START_REF] Young | Heterogeneous Convergence[END_REF]. Following Evans (1997b, footnote 17, p.16), we used . If the low value of the confidence interval was less than -T -1 , then the higher value was set equal to unity. It is clear from the above that confidence intervals computed using this procedure may be asymmetric around the point estimates.

  [START_REF] Young | Heterogeneous Convergence[END_REF] to study income growth from 1970 to 1998. The data set includes 3,058 county-level observations, and 50 individual state samples of various sizes, also at the county-level. See Figure1.The personal income measure we use is the definition used by the U.S. Bureau of Economic Analyses (BEA). It is adjusted to be net of government transfers and is expressed in per capita 1992 dollars using the U.S. GDP deflator. Population measures from the U.S. Census are used to construct per capita amounts. Real per capita income levels are expressed as natural logs and values are considered for both1970 and 1998. 7

Figure

  Figure 1: 3,058 U.S. Counties

Figure

  Figure 2: Distribution of U.S. Counties' Log Per Capita Incomes, 1970 vs 1998

  asymptotic rate of convergence. The confidence intervals (in parentheses) awere obtained in two steps. First, we obtained the end points of the confidence intervals by computing the interval the standard error associated with the estimate of . Next, these end-points were substituted into the expression,

Table 2

 2 reports 1970 and 1998 cross-sectional standard deviations of (log) income for the entire sample of U.S. counties, and for each of the 50 U.S. states; and the associated p-values for a variance ratio test of the null that the ratio of the two standard deviations is unity (against the two-tailed alternative). The 1998 standard deviation for the full U.S. sample (0.2887) is about 5.8 percent greater than that of 1970 (0.2728), a difference that is significant at the 1 percent level. In only 2 out of 50 states (Kansas and Oklahoma) is the 1998 standard deviation significantly less than that of 1970 (at the 10 percent level or better).

10

On the other hand, for 24 states the 1998 standard deviation is significantly larger (at the 10 percent level or better). Thus, for many individual states, as well as for the full U.S., -divergence occurred from

1970 to 1998. 11

Table 1 : Asymptotic Convergence Rates -Point Estimates & 95% Confidence Intervals

 1

	State	Number of Counties	OLS Estimates & 95% C.I. _	__ _3SLS Estimates & C.I. _
	United States	3,058	0.0239 (0.0224, 0.0255)	0.0658 (0.0632, 0.0981)
	Alabama	67	0.0424 (0.0036, 0.1080)	0.0931 (0.0492, 0.1466)
	Arkansas	74	0.0479 (0.0166, 0.1098)	0.0738 (0.0570, 0.1363)
	California	58	0.0457 (0.0046, 0.1249)	0.0375 (0.0178, 0.0868)
	Colorado	63	0.0166 (0.0031, 0.0384)	0.0759 (0.0426, 0.1009)
	Florida	67	0.0268 (0.0010, 0.1109)	0.0767 (0.0480, 0.1174)
	Georgia	159	0.0230 (0.0109, 0.0413)	0.1043 (0.0699, 0.1142)
	Idaho	44	0.0892 (0.0021, 0.1566)	0.0913 (0.0471, 0.1145)
	Illinois	102	0.0434 (0.0213, 0.1168)	0.0537 (0.0337, 0.1062)
	Indiana	92	0.0067 (-0.0054, 0.0245)	0.0622 (0.0354, 0.1221)
	Iowa	99	0.0570 (0.0224, 0.1176)	0.0574 (0.0175, 0.0954)
	Kansas	106	0.0560 (0.0360, 0.1086)	0.0639 (0.0434, 0.1228)
	Kentucky	120	0.0431 (0.0233, 0.0922)	0.1054 (0.0561, 0.1160)
	Louisiana	64	0.0341 (0.0128, 0.0955)	0.1555 (0.0989, 0.1940)
	Michigan	83	0.0121 (-0.0043, 0.0427)	0.1152 (0.0536, 0.1659)
	Minnesota	87	0.0202 (0.0053, 0.0459)	0.0454 (0.0305, 0.0719)
	Mississippi	82	0.0249 (0.0009, 0.1509)	0.1405 (0.0455, 0.1923)
	Missouri	115	0.0230 (0.0094, 0.0452)	0.0817 (0.0387, 0.1132)
	Montana	56	0.0359 (0.0099, 0.0996)	0.0865 (0.0367, 0.1566)
	New York	62	0.0111 (-0.0238, 0.0284)	0.0465 (0.0285, 0.0853)
	North Carolina	100	0.0228 (0.0078, 0.0491)	0.1302 (0.0966, 0.1574)
	North Dakota	53	0.0528 (0.0103, 0.1247)	0.0761 (0.0353, 0.1102)
	Ohio	88	0.0170 (-0.0005, 0.0520)	0.0503 (0.0299, 0.1059)
	Oklahoma	77	0.0415 (0.0139, 0.1136)	0.1152 (0.0574, 0.1437)
	Pennsylvania	67	0.0240 ( 0.0043, 0.0707)	0.0705 (0.0291, 0.1099)
	South Carolina	46	0.0142 (-0.0147, 0.1259)	0.0960 (0.0243, 0.1315)
	South Dakota	66	0.0274 (0.0250, 0.0300)	0.0406 (0.0184, 0.1144)
	Tennessee	97	0.0287 (0.0102, 0.0689)	0.0681 (0.0488, 0.1168)
	Texas	254	0.0312 (0.0208, 0.0458)	0.1170 (0.0675, 0.1564)
	Virginia	84	0.0047 (-0.0074, 0.0227)	0.0703 (0.0500, 0.1271)
	Washington	39	0.0518 (-.0119, 0.0971)	0.0845 (0.0448, 0.1449)
	West Virginia	55	0.0040 (-0.0184, 0.0199)	0.0634 (0.0466, 0.0972)
	Wisconsin	70	0.0270 (0.0077, 0.0716)	0.0390 (0.0231, 0.0688)

Table 2 : Standard Deviations U.S. Counties' Log Per Capita Incomes, 1970 vs 1998

 2

	1970 Per Capita Income 1998 Per Capita Income

Note: Per capita income figures are in natural log form. P-values are based on a variance ratio test where the null hypothesis is that the value of the ratio of the 1998 to 1970 standard deviations is unity (against the two-tailed alternative). Rows in boldface correspond to cases where the null hypothesis is rejected at the 10 percent level or better. Italics indicate, additionally, significant -convergence.

Table 3 : Summary Statistics for Distribution of U.S. Counties' Log Per Capita Incomes, 1970 vs 1998

 3 : per capita income figures are in natural log form.

	Statistic	1970 Per Capita Income 1998 Per Capita Income
	Standard Deviation	0.2728	0.2887
	Gini Coefficient	0.1666	0.1654
	Skewness	-0.2244	1.7240
	Kurtosis	3.4334	10.3237

Note

Barro and Sala-i-Martin (1991) present a similar exposition and intuitive discussion. More recently,[START_REF] Furceri | and -Convergence: A Mathematical Relation of Causality[END_REF] presents a related demonstration based on an OLS estimator of the coefficient on initial income.

This is the case of absolute -convergence; average growth rates of poor economies are unambiguously greater than those of rich economies. Allowing for different a i s, 0 < < 1 would imply the case of conditional -convergence; the average growth rate of an economy is an increasing function of its distance from its balanced growth level of income. This is a weaker case of -convergence and increases the set of possible scenarios where it does not imply -convergence. See the discussion at the end of this section.

If 0 the variance increases over time. If = 1 the variance is constant and if > 1 the partial correlation between (log) income and its previous-period value would be negative and the series would oscillate, potentially from positive to negative values and back (making little economic sense).

Note in (6) that the parameter governs the speed at which the variance approaches its steady-state value because, according to (1), it governs how long the effect of shocks persist.

This represents the case of conditional -convergence. In empirical applications the a i s are often modeled as linear functions of various economic and/or socio-demographic variables available to the researcher, e.g., see Table1in[START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF].

In real economies, -convergence would also depend on whether or not disturbances are correlated, and have constant variances, across time and economies.

For a more detailed discussion of the data, see[START_REF] Higgins | Growth and Convergence Across the U.S.: Evidence from County-Level Data[END_REF] or a data appendix available from the authors upon request. Also, see U.S.BEA (2001) for the personal income data concept and data gathering methods. The original data set contained 3,066 observations. Eight counties, however, were excluded from the data set for lack of data.

The data and their measurement methods are described in detail in "Local Area Personal Income, 1969-1992" published by the BEA under the Regional Accounts Data, February 2, 2001.

This statement is not to be confused with one concerning the distributions of U.S. individuals' incomes.


A Dorogoy

G Haiat

K Shemtov-Yona

D Rittel
email: merittel@technion.ac.il

MODELING ULTRASONIC WAVE PROPAGATION IN A DENTAL IMPLANT -BONE SYSTEM

Keywords: Dental implants, ultrasound, bone-implant interface, finite element, waves, stability

The evolution of the bone-implant interface reflects the implant osseointegration and bond strength, thereby determining the overall implant stability in the jawbone.

Quantitative ultrasound represents a promising alternative technique to characterize the interfacial integrity, precisely due to the fact that those waves propagate essentially along the bone-implant interface, and are therefore influenced by its state. This study reports a numerical investigation of ultrasonic wave propagation for a commercial implant-jawbone system in which the thickness and mechanical properties of the interfacial layer (corresponding to the interphase) are systematically varied through the application of a rule of mixtures, in order to mimic the evolution from a dominantly soft tissue -like medium up to a fully healed bone.

A simple figure of merit is devised in terms of an RMS-like (root mean square) factor based on the implant displacements, that evolves continuously and significantly with the bone "healing" process, thereby providing unequivocal information on the nature of the investigated bone-implant interface.

The results show that the wave propagation pattern is primarily dictated by the impedance mismatch rather than by the interface thickness. This study validates the concept of quantitative ultrasonic testing as a sensitive alternative to the widespread resonant frequency analysis, thereby opening the way for future sensitivity analyses that will address more refined bone-implant interface pathologies such as those observed in the clinical realm.

Introduction

The long-term success of dental implants depends on a long-lasting osseointegration and stable contact with the peri-implant bone. Osseointegration and implant stability depend on the quality of the surrounding bone and on the biomechanical properties of the bone-implant interface (BII) [START_REF] Albrektsson | Hard tissue implant interface[END_REF]. Today, long-term prognosis regarding implants is based on stability measurements. The evolution of the biomechanical properties of the bone-implant interface (BII), remains quite difficult to monitor in vivo by non-destructive methods. Yet, implant stability can be quantitatively assessed through non-invasive techniques, discussed in the sequel, that can be part of the clinical environment [START_REF] Gao | Biomechanical behaviours of the bone-implant interface: a review[END_REF]von Wilmowsky et al., 2014;[START_REF] Zanetti | Clinical Assessment of Dental Implant Stability During Follow-Up: What Is Actually Measured, and Perspectives[END_REF].

Despite their clear advantage as clinically easy to use devices, their reliability and correlation to the bone-implant properties remains to be established [START_REF] Aparicio | Validity and clinical significance of biomechanical testing of implant/bone interface[END_REF][START_REF] Atieh | Can Resonance Frequency Analysis Predict Failure Risk of Immediately Loaded Implants[END_REF][START_REF] Atieh | The prognostic accuracy of resonance frequency analysis in predicting failure risk of immediately restored implants[END_REF][START_REF] Manresa | The comparison between implant stability quotient and bone-implant contact revisited: an experiment in Beagle dog[END_REF][START_REF] Nkenke | Implant stability and histomorphometry: a correlation study in human cadavers using stepped cylinder implants[END_REF][START_REF] Zanetti | Clinical Assessment of Dental Implant Stability During Follow-Up: What Is Actually Measured, and Perspectives[END_REF].

Investigating of the biomechanical properties of the bone-implant interface can be done using quantitative ultrasound (QUS) [START_REF] De Almeida | Proposal for an Ultrasonic Tool to Monitor the Osseointegration of Dental Implants[END_REF]. The principle of the measurement relies on the dependence of ultrasonic wave propagation within the implant on the bone properties around the bone-implant interface (Mathieu et al., 2011b). The principle of the measurement was validated experimentally by showing the sensitivity of the echographic response of a planar BII to healing time using a coin-shaped implant model [START_REF] Mathieu | Influence of Healing Time on the Ultrasonic Response of the Bone-Implant Interface[END_REF]. Significant variations of the ultrasonic response of dental implants embedded in a bone substitute biomaterial were shown to occur when the implants are subjected to fatigue loading [START_REF] Vayron | Variation of the ultrasonic response of a dental implant embedded in tricalcium silicate-based cement under cyclic loading[END_REF]. Other in vitro studies proved the potentiality of QUS methods to assess the primary stability of dental implants inserted in bone phantom (Vayron et al., 2018a) and in bovine bone tissue (Vayron et al., 2014a).

A preclinical validation of the device in rabbits (Vayron et al., 2014b) and in sheep (Vayron et al., 2018b) was carried out and showed that i) the measurement was sensitive to healing time and ii) a significant correlation of the measurement with the bone-implant contact (BIC) ratio measured with histology. A finite element model was developed at the microscopic level to account for the effect of the surface roughness and of osseointegration phenomena of a planar bone-implant interface [START_REF] Hériveaux | Reflection of an ultrasonic wave on the boneimplant interface: A numerical study of the effect of the multiscale roughness[END_REF]. At the macroscopic level, the propagation of ultrasound in cylindrical implants was simulated using finite difference time domain simulation (Mathieu et al., 2011a) and finite element numerical simulations [START_REF] Vayron | Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment[END_REF], leading to a better performance of the device. Then, a 2-D axisymmetric finite element model was used to model the interaction between a dental implant and an ultrasonic wave considering a realistic implant geometry [START_REF] Vayron | Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A threedimensional finite element study[END_REF]. However, each bone properties were varied independently [START_REF] Vayron | Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A threedimensional finite element study[END_REF] while they all vary in parallel in clinical practice and it still remains difficult to determine the influence of the biomechanical properties corresponding to osseointegration phenomena and of the thickness of a bone layer surrounding an implant on its ultrasonic response.

In this work we consider a single generic implant geometry that is fully inserted in a jawbone section made of cortical and trabecular bone. A peri-implant bone layer is assigned with various stiffness values and widths to mimic the evolution of the peri-implant bone healing. The periimplant layer follows the geometry of the threaded implant. The implant is assumed to be fully bonded to the bone. The aim of this paper is to characterize the ultrasonic response of the dental implant using a finite element model for various kinds of peri-implant layer conditions.

The novelty of this investigation is in the consideration of a more realistic bone geometry with cortical bone tissue located all around the implant. It also considers the influence of the thickness of the peri-implant bone layer where osseointegration occur. This thickness is an important parameter for the implant success and in this work it follows the implant threads geometry which is more realistic. Moreover, the combined variation of the Young's modulus, Poisson's ratio and the mass density during the healing process is considered. A better indicator was defined and successfully used to characterize and quantify the vertical displacements of the upper face of the peg due to ultrasound excitation.

The usage of finite element modeling is mandatory in order to understand the influence of osseointegration on the ultrasonic response of an implant because the variation of the different bone properties is difficult to control experimentally.

Material and methods

Geometry and Assembly

An 2D axisymmetric model is used in the analyses, as assumed in [START_REF] Vayron | Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A threedimensional finite element study[END_REF]. The axisymmetric model was derived from a 3D model of the mandible bone and implant shown in Fig. 1a,b, similarly as in [START_REF] Dorogoy | Modeling dental implant insertion[END_REF][START_REF] Rittel | Modelling dental implant extraction by pullout and torque procedures[END_REF]. The 3D model is shown in Fig. The assembled axisymmetric model is comprised of 2 parts:

1) A mandible bone (Fig. 1d)

2) A dental implant bonded to an extension, subsequently referred to as "peg" (Fig. 1e).

The bone consists of an outer cortical bone (grey color in Fig 1c,d) having a thickness of ~2 mm, and of inner cancellous/trabecular bone (cream color in Fig. 1c,d). The implant and peg (Fig. 1e) are made of a commonly used titanium alloy ( Ti6Al4V).

An axisymmetric model was used because a very fine mesh is required to model ultrasonic wave propagation [START_REF] Drozdz | EFFICIENT FINITE ELEMENT MODELLING OF ULTRASOUND WAVES IN ELASTIC MEDIA[END_REF][START_REF] Egerton | A multiband approach for accurate numerical simulation of frequency dependent ultrasonic wave propagation in the time domain[END_REF] as discussed below, so that this model reduces significantly the computational size of the problem, as opposed to a prohibitively large full 3D model. Alternatively, a 2D model (e.g. plane strain) would grossly distort the geometry of the implant. The assumption of axial symmetry has a minor effect on the results as long as the reflecting pulses reaching the top of the peg are originating from the bone-implant interface (BII), the surrounding peri-implant layer and the trabecular bone. The model will lose its similarity to the real 3D bone at longer times for which reflections of shear wave from the cortical bone reach the top of the peg (See section 3.2 and Fig. 2 in the sequel).

Perfect bonding was prescribed between all contacting materials. It was assumed that a peri-implant bone layer is affected (damaged) by the implant insertion process. This weakened layer has an axisymmetric shape which follows the implant threads' shape as shown in Figs. 1f,g. The width w of this peri-implant layer was assumed to be w = 0.1 mm (Fig. 1f) and w = 0.2 mm (Fig. 1g), respectively.

Boundary conditions

A pressure P(t) is applied at the peg top face (Fig. 1c) and writes (Vayron et al., 2016a):

    2 41 0 ( ) sin 2 ft P t P ft e    (1)
where P0 = F/A , A is the area of the top peg face, and F=1N, which is chosen arbitrarily since the problem is fully linear. The frequency is 10 f MHz  . The amplitude is set to 0 for t > 2 μs.

The assembly is fixed at the bottom of the bone. Fully constrained ("encastre") conditions are applied at the center of the cortical bone (Fig. 1c).

Materials

For the sake of simplicity, all three parts of the model were assigned linear elastic and homogenous material properties. For the implant and peg, isotropic mechanical properties of Ti-6Al-4V ELI (American Society for Testing and and American Society for Testing and Materials, 2013) were used (see Table 1). Cortical and trabecular bone tissues were assumed to be isotropic with mechanical properties chosen according to (Vayron et al., 2015b)

(see Table 1, ξ=0).
It is assumed that immediately after the insertion, the peri-implant layer is made mostly of blood or soft tissues, for which water (liquid) properties are assumed for the sake of simplicity. During the healing process the layer becomes mineralized as a result of the osseointegration process. After a long time (typically several months), the properties of the peri-implant layer become identical to those of the host bone. A parameter ξ represents the amount of water in the mixture, 01  . For 1   (respectively 0   ), the mixture properties are those of water (respectively the host bone). The Young's modulus (E), the Poisson's ratio (ν) and the density ( ρ ), which are the input parameters for the finite element calculations, were assigned values according to Eqns. 2-4 (rule of mixtures).

 

 

1 water bone E E E       (2)     1 water bone          (3)     1 water bone          (4)
The Lamé coefficients (λ) and (µ), the shear modulus, are determined according to Eqns. 5-6:

              1 1 2 E           (5)         21 E      (6)
The longitudinal wave velocity ( CL ) and the shear wave velocity can be determined for each composition (mixture) according to Eqns. 7-8.

        2 L C         (7)       S C     (8)
Throughout this work, we considered 0.0 0.9   , rather than 1 to avoid modeling a pure water phase.

Mesh

The recommended mesh density for simulating the propagation of ultrasonic waves is [START_REF] Egerton | A multiband approach for accurate numerical simulation of frequency dependent ultrasonic wave propagation in the time domain[END_REF] :

0 / 20 20 S Cf dx   . ( 9
)
Where dx is the mesh size, λ0 is wave length, Cs is the shear wave velocity and f is the wave frequency. The mesh density is determined in terms of node spacing and not element side size [START_REF] Drozdz | EFFICIENT FINITE ELEMENT MODELLING OF ULTRASOUND WAVES IN ELASTIC MEDIA[END_REF]. We used quadratic triangular elements of type CAX6M. It was proved that for a given mesh density, quadratic quadrilateral elements strongly reduce the velocity error [START_REF] Drozdz | EFFICIENT FINITE ELEMENT MODELLING OF ULTRASOUND WAVES IN ELASTIC MEDIA[END_REF]. The quadratic elements which are used herein have mid-edge nodes and the distance between the nodes is estimated as half of the element side size. The smallest elements were applied on the trabecular peri-implant layer and have mesh density of 1.25 μm. Applying the recommended mesh density results in a mesh comprised of ~4 millions elements for w = 0.2 mm and ~3 millions elements for w = 0.1 mm, respectively.

Results

The total time simulated duration was 15 µs. A total of 13 cases were solved with a mesh density of dx=λ0/20 for values of ξ = 0.0, 0.3, 0.5, 0.6, 0.7, 0.8 and 0.9, for each peri-implant layer's thickness (0.1 and 0.2 mm).

The averaged vertical displacement at the top of the peg was monitored (numerically) during the first 15 µs from application of the ultrasonic pulse (Eqn. 1). Monitoring was done at 100 MHz, yielding 1500 equally spaced time points with Δt = 0.01 µs.

Wave propagation

The Tresca stress variation for case w = 0.1 mm and ξ = 0.5 is shown for illustration in Fig. 2

An indicator of the implant's ultrasonic response

In order to better characterize, and quantify to some extent the vertical displacements of the upper face of the peg, a new indicator that bears some similarity with the RMS, was defined, as shown in Eq. ( 10):

    * 2 * 0 t I t f t dt     , (10)
where f(t) are the above-mentioned vertical displacements.

This indicator is not a constant, as would be calculated from an RMS estimate, but rather a function of time (t * ). The units of this indicator are [mm 2. s] but they can be multiplied or normalized arbitrarily, hence we just mention values without units through the sequel.

Eqn. 10 is a modification of the indicator which was used by (for example) (Vayron et al., 2016b) in which the function is not squared and can be interpreted as

    * * 0 t I t f t dt   .
The squaring enhances the capability of characterization the effect of peri-implant properties.

Top peg average vertical displacements

Figure 3a shows the top peg average vertical displacement after application of the indicator (Eq. 9) for 0 < t * < 15 μs. Solid (respectively dashed) lines correspond to results obtained with w = 0.1 mm (respectively w = 0.2 mm). Two regions can be distinguished in Fig. 3a: 0 < t* < t0 and t* > t0. Until t* = t0 there is no difference between all cases because the top peg displacements are due to the applied load and reflected waves from the peg. At t* = t0, waves which are reflected by the BII reach the top of the peg, hence differences between the cases start to be increasingly visible.

As time increases, more waves are passing the BII and get to the top of the peg and the indicator values increase as well.

Figure 3b shows the results with translation to the axis (t',I') shown on Fig. 3a : The results shown in Fig. 3 show that the weaker the peri-implant layer, the higher the indicator value for all t*.At same time, it appears that the two peri-implant layer thicknesses of 0.1 and 0.2 mm do not influence significantly the indicator's values.

    * 2 * 00 0 ' t I t t f t dt I       (11)   0 2 0 0 t I f t dt     (12)
It can thus concluded that wave propagation is mostly dictated by the mechanical impedance mismatch at the BII and is just slightly affected by the peri-implant layer's thickness (impedance mismatch is the ratio between the multiplication of the density ρ and wave velocity C on both sides of an interface: 11

Figure 4:

The indicator values at 8 μs and 15 μs versus the healing process (ξ).

Summary and conclusions

A numerical simulation of the ultrasonic propagation in a dental implant is obtained from a model consisting of a single generic implant geometry, which is fully inserted in a jawbone section. The bone is made of cortical and trabecular bone tissues and the implant is fully bonded to the bone. A peri-implant bone layer was assigned various stiffness values and widths to mimic the various stages of osseointegration. The originality of the present paper, compared to (Mathieu et al., 2011a) [START_REF] Vayron | Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment[END_REF], is first to consider a realistic implant geometry. In addition, compared to [START_REF] Vayron | Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A threedimensional finite element study[END_REF], the present paper considers i) a combined variation of the Young's modulus, the Poisson's ratio and of the mass density through the parameter ξ, ii) a more realistic bone geometry with cortical bone tissue located all around the implant, which is more realistic than what was done in [START_REF] Vayron | Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A threedimensional finite element study[END_REF] and iii) the influence of the thickness w of the layer where osseointegration occurs, which is an important parameter for the implant success. The characterization is realized with the aid of a new Integral Indicator.

The results show that the weaker the peri-implant layer, the higher the indicator value for all t*. It also appears that the width of deteriorated peri-implant layer has a minor effect on the indicator's values. This fact indicates that the mechanical impedance mismatch between the implant and bone is the main cause for changes in the indicator values. This point is particularly interesting when comparing ultrasonic to resonant frequency measurements. The present results show the dominant contribution of the interfacial wave propagation with respect to the impedance mismatch, which confers a high sensitivity to the "measurements", in contrast with resonant frequency measurements for which the bone+interface+implant is a whole vibrating structure, of which the interface is only a minor component.

The gradients of the indicator values variation with ξ indicate that healing process might be more easily identified at the early stages than at the late stages of bone healing.

Since the differences in the indicator values between a fully healed and fully deteriorated bone are quite large (150-200%), it is concluded that monitoring US waves reflections can be successfully used to identify peri-implant bone healing.

  1a, while a side view from which the axisymmetric model was derived is shown in Fig 1b, and detailed in Fig 1c.

Figure 1 :

 1 Figure 1: a. Isometric view of exposed jaw bone with a dental implant. b. Side view of exposed jaw bone with a dental implant. c. The axisymmetric assembly. d. The mandible bone. e. The dental implant. f. Detail showing a peri-implant weakened layers of 0.1 mm wide. g. Detail showing a 0.2 mm wide peri-implant weakened layer.

  for the time interval of 1ms£ t £ 8ms . The axisymmetric model is rotated by 270 o to visualize the difference between the numerical model and a real bone which is shown in Figs. 1a,b. Note that the model proposed here is an approximation to the real model shown in Fig 1a,b. At time 0ms£ t £1ms , the shear wave propagates mostly along the peg+implant and reaches the upper BII. At 1ms£ t £ 2ms , some of the wave has started passing the BII. At 2ms£ t £ 8ms , the wave has propagated along the whole BII and passed the tip of the implant. The Tresca stress spreads significantly within the trabecular bone. The wave propagates always faster within the cortical bone (axial) than within the trabecular bone (radial).

Figure 2 :

 2 Figure 2: 3D visualization of the axisymmetric model at 18 s t s   showing the Tresca stress wave propagation.

Figure 3 :

 3 Figure 3: Indicator values applied to the top peg vertical displacements. a. The indicator Eqn. 9. b. The translated indicator values Eqns. 10-11.

  to sides of an interface. This ratio affects the relative reflection and transmission of a wave through a heterogeneous interface).It should be noted that for t* > Δt ( Δt = 7 μs), the recorded displacements at the top of the peg include reflections from the CTI (Cortical Trabecular interface) hence these displacements do not represent faithfully experimental displacements from a real bone (Fig.1a,b) which does not possess radial symmetry. Nevertheless, the results indicate that longer pulses will increase the Indicator values and therefore its usability.The indicator values at t * = 8 μs and t * = 15 μs are plotted against ξ [%] in Fig.5. The markers represent numerical results while the lines are fit to these results. Solid lines are for w = 0.1 mm and dash lines for w = 0.2 mm. Once again, it can be observed that the width of the damaged periimplant layer has a minor effect on the indicator values. The behaviour of the curves at t * = 8 μs and t * = 15 μs is similar. Two gradients β1 and β2 for ξ> 50% and ξ<50% can be identified. Since β1 > β2 it can be concluded that the indicator will be more sensitive in the early stages of healing while it might have a lower resolution as the healing process progresses. Nevertheless, the differences in the indicator values at each time between a fully healed and fully deteriorated bone is large (150-200%)this fact indicate that measurements of US waves can be used to identify healing.

Table 1 :

 1 Material properties. The peri-implant layers properties are determined according to the rule of mixtures (Eqns. 2-4).

	

[START_REF] Vayron | Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment[END_REF]

** American Society for Testing and and American Society for

Testing and Materials, 2013

Acknowledgement GH was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 682001, project ERC Consolidator Grant 2015 BoneImplant).

The authors declare they have no conflict of interests.