Ania ; Maciolek

Andrzej Drzewinski

Artur Barasinski

S Dietrich Drzewinski

A Drzewiński

A Macio

A Barasiński

Solvation forces in Ising films with long-range boundary fields: density-matrix renormalization-group study

Keywords: solvation force, Casimir effect, capillary condensation, finite-size scaling, Ising systems PACS numbers: 05.50.+q, 68.35.Rh, 68.08.Bc

Solvation forces in Ising films with long-range boundary

fields: density-matrix renormalization-group study.

I. INTRODUCTION

The solvation force f solv is the force per unit area between two surfaces, large colloidal particles or macromolecules due to the intervening fluid. It arises in the thermodynamic description of confined fluids as an excess pressure over the bulk value (fixed by the reservoir) and is conjugate to the distance L between the confining surfaces [1]. The study of this generalized force for simple and complex fluids is useful as it may play an important role in the field of colloid sciences, in self-assembling systems and in the biology of protein folding.

From a fundamental viewpoint, solvation forces present an abundance of interesting problems to investigate: the behaviour at the phase transitions of a confined fluid, such as capillary condensation [1][2][3][4], bridging [5][6][7], localization-delocalization transition [8,9], or layering [9], the structure at small separations L [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Christenson | [END_REF], or the dependence on the fluid-fluid interactions and the surface potentials [3,4,12]. An addition appeal lies in the fact that it may be measured experimentally by a number of different techniques [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Christenson | [END_REF]; e.g., the surface force apparatus, the atomic force microscope, or the total internal reflection microscopy (TIMR) [13]. By using the latter technique, the first direct determination of the universal contribution to f solv between a single colloidal particle and a planar surface in the presence of a near-critical binary liquid mixture was provided [14,15]. The universal contribution, called the critical Casimir force [16][START_REF] Krech | Casimir Effect in Critical Systems[END_REF][START_REF] Brankov | Theory of Critical Phenomena in Finite-Size Systems[END_REF][START_REF] Kardar | [END_REF], is induced by the confinement of critical fluctuations of the order parameter of the second order phase transition, e.g. the density of the fluid or the concentration of one species for a binary liquid mixture. At the bulk critical point this fluctuation-induced force has a leading power law decay f solv /(k B T c ) ∼ AL -d as L → ∞, where d is the spatial dimension of the system and A is the Casimir amplitude. The finitesize scaling theory predicts (see, e.g., Ref. [START_REF] Barber | Phase Transition and Critical Phenomena[END_REF]) that in the vicinity of the bulk critical point f solv is described by a scaling function that depends on the so-called universality class of the phase transition occurring in the bulk and on the geometry and surface universality classes of the confining surfaces [START_REF] Diehl | Phase Transition and Critical Phenomena[END_REF]. The critical Casimir force is a subject of recent considerable theoretical and experimental interest because at the submicron scale its strength and range is comparable with other interactions and because it can be easily controlled, including its sign, by changes of thermodynamic fields, such as the temperature, and by appropriate surface treatments.

As mentioned above, the solvation force is associated with fluid-fluid interactions and the surface potential. In the present paper we want to study the relevant effects of longranged, i.e., algebraically decaying forces on the near-critical behaviour of f solv between two parallel walls and test predictions of the critical finite-size scaling theory. Examples of such long-ranged forces are dispersion or van der Waals forces, dipolar forces, RKKY interactions, forces in charged systems, and elastic forces in solids. Here we utilize the equivalence between a lattice gas model of a fluid and the Ising model and consider Ising spin films subject to identical boundary fields decaying in the orthogonal direction at a distance j from the surface as -h 1 j -p , with p = 2, 3, 50 and h 1 ≥ 0. Using the density-matrix renormalization-group (DMRG) method, we calculate the solvation force in two-dimensional (2D) systems along various thermodynamic paths. The DMRG method is based on the transfer matrix approach and provides a numerically very efficient iterative truncation algorithm for constructing the effective transfer matrices for strips of fixed width and infinite length. The advantage of this method is that it can treat arbitrary fields that are coupled to the single spin variable.

Moreover, because the DMRG method gives quasi-exact results for the spectrum of the transfer matrix of the system, fluctuations are fully accounted for in this approach. Recall, that in 2D systems fluctuation effects are particularly strong. However, this method works only if the spin-spin interactions are nearest-neighbour and, at present, it is limited to the two-dimensional systems.

The studies carried out here complement and extend the earlier works (see Refs. [3,[START_REF] Drzewiński | [END_REF]23]). In Refs. [START_REF] Drzewiński | [END_REF]23] it was argued that the solvation (Casimir) force in the vicinity of the critical point is strongly influenced by capillary condensation, i.e., the shift of the bulk firstorder transition which occurs below the critical temperature T c . The residual condensation leads to the solvation force which is much more attractive at temperatures near T c and a reservoir densities slightly below the critical value (or compositions slightly away from the critical composition in a binary mixture) than the Casimir value (at the bulk critical point) for the same L. These predictions were supported by explicit calculations for 2D Ising films with short-ranged (contact) boundary fields. By using the DMRG method it was possible to study the region between the capillary and bulk critical points, i.e., for nonvanishing ordering field h < 0, which corresponds to the chemical potential difference µµ sat . In particular, the scaling functions of f solv along several isotherms were obtained showing that upon increasing the temperature towards T c a weakly rounded in 2D discontinuous jump of f solv on crossing the coexistence line [1,24] transforms gradually into a minimum. A location [START_REF] Drzewiński | [END_REF]. The similar behaviour was found in the field-theoretic model solved numerically in the mean-field approximation [25], again for the short-ranged boundary fields. In Ref. [3], the case of short-ranged fluid-fluid interaction and long-ranged wall-fluid potentials decaying as -Az -p for z → ∞ was considered for 2D Ising films and for the truncated Lennard-Jones fluid in a slit geometry. The discrete model was treated within the DMRG method whereas the continuum one by the nonlocal density functional theory. The study was focused on the asymptotic behaviour of the solvation force as L → ∞. Except for a high temperature Ising system, results for both models agree with the predictions from the analysis based on the wall-particle Ornstein-Zernike equations that f solv is repulsive and asymptotically decays with the same power law as the wall-fluid potential; this prediction holds away from the critical temperature and from any phase transition. For Ising films above T c the asymptotic behaviour was found to be of the higher order then that of the boundary field, i.e., L -(p+1) . This was explain by the specific symmetry of the order parameter with the spontaneous magnetization m * equal to zero above T c . Moreover, f solv was calculated along the bulk two-phase coexistence line slightly on the liquid side of this line and at the critical density for T > T c ; in the Ising system this path corresponds to the line H = 0. The obtained results imply that for ordering field H = 0 and L → ∞

f solv ∼ f reg solv + f sing solv = 2ρBL -p + (d -1)L -d ϑ(L/ξ τ ), ( 1
)
where ρ is the bulk density and B is related to the strength of the wall-fluid potential. For

Ising systems ρB is replaced by m * h 1 . ϑ(x) is a universal scaling function describing the contribution arising from critical fluctuations of a fluid. τ = (T -T c )/T c and ν is the critical exponent of the bulk correlation length ξ b . ϑ is vanishingly small away from the critical region and is negative for identical walls.

In the present study we consider the neighbourhood of the bulk critical point and investigate the scaling behaviour of the solvation force along the critical and two subcritical isotherms and along the pseudo-coexistence (capillary condensation) line slightly on the "liquid" and on the "gas" sides. A substantial progress in computer capacities allows to study sufficiently thick films to assure that the scaling limit is achieved. Here we consider strips of widths up to L = 700 lattice constants. According to general scaling arguments [4,[START_REF] Diehl | Phase Transition and Critical Phenomena[END_REF] the finite-size behaviour of the singular part of the solvation force is modified by the presence of the long-ranged substrate-wall potentials. However, if p > (d + 2η)/2 the universal behavior is expected to hold. In this case the long-ranged part of the boundary field is irrelevant in the RG sense with respect to a pure contact surface field [START_REF] Diehl | Phase Transition and Critical Phenomena[END_REF]. Here η is the critical exponent governing the algebraic decay of the two-point correlation function in the bulk and at T c . In the present case of the d = 2 Ising model η(d = 2) = 1/4 so that for p > 15/8 we expect to observe the power law L -2 . The scaling of the solvation force in films with long-ranged fluid-fluid and substrate-fluid potentials was analysed in Ref. [4] by using the general scaling arguments and mean-field theory. We will summarize the relevant conclusions of this analyses and relate our results to them in Sec. III.

Our paper is organized as follows. In Sec. II we introduce the model and describe the determination of the phase diagrams for various values of the parameter p. Sec. III contains our results. Finally, we give conclusions in Sec. IV.

II. MICROSCOPIC MODEL

We consider D = 2 Ising strips defined on a square lattice of size M × L, M → ∞ and subject to the same boundary fields on both sides. The lattice consists of L parallel rows at spacing a, so that the width of the strip is La; in the following we set a = 1. At each site there is an Ising spin variable taking the value σ k,j = ±1, where (k, j) labels the site.

The boundary surfaces are located in the rows j = 1 and j = L and periodic boundary conditions (PBCs) are assumed in the lateral x direction. The Hamiltonian of our model is

given by

H = -J   kj,k ′ j ′ σ k,j σ k ′ ,j ′ + L j=1 V ext j,L k σ k,j + H k,j σ k,j , (2)
where the first sum is over all nearest-neighbor pairs and the external potential is measured in units of J > 0. V ext j,L = V s j + V s L+1-j is the total boundary field experienced by a spin in row j; it is the sum of the two independent wall contributions. The single-boundary field with p > 0 and h 1 > 0. H is a bulk magnetic field. h 1 and H are dimensionless (see Eq. ( 1)).

V s j is taken to have the form V s j = h 1 j p (3)
As already mentioned this model is equivalent to the 2D lattice gas model of a twodimensional one-component fluid with a short-ranged interaction potential between the fluid particles and either short-ranged or long-ranged substrate potentials (see, e.g., Ref. [26]).

A. Phase diagram

In Fig. 1 we show the phase diagram for the present model calculated by using the DMRG method for a strip of width 600 and for three choices of the parameter p describing the decay of the boundary field: p = 2, 3, and 50. h 1 is chosen so that the scaling variable

x = h 1 * L 2 = 20000
, which for the short-ranged boundary fields is sufficient to ensure each system corresponds to the infinite surface field scaling limit [27]. The case p = 50 is expected to resemble the behavior corresponding to short-ranged surface forces. In this figure we display the various thermodynamic paths along which we have calculated the solvation force.

For surfaces which prefer the same bulk phase, the phenomenon equivalent to capillary condensation takes place. The pseudo-phase coexistence between phases of spin up and spin down occurs along the line H ca (T, L; p), which is given approximately by the analogue of the Kelvin equation [28]. For positive surface fields capillary condensation occurs at negative values H ca (T, L; p) of the bulk field H. The pseudo-coexistence lines have been identified

as those positions (H, T ) in the phase diagram where the total magnetization of the strip vanishes, i.e., L j=1 m j = 0 with m j = σ k,j . We have not attempted to localize the position of the pseudo-critical temperature T c,L which ends the pseudo-coexistence lines; the unique determination of the pseudo-critical point is not possible because non-analytic behaviour is rounded in 2D [29]. Notice, that for p = 3 and 50 and T > T c , the line defined by the zeros of the total magnetization moves to more negative values of H upon increasing temperatures.

For p = 2 we observe this trend already below T c . In Ref. [29] the shift of pseudo-phase coexistence lines H ca (T, L; p) for p = 2 and 3 relative to the short-ranged pseudo-phase coexistence line (p = 50) was analysed. For p = 50 and 3 also the scaling behaviour of the pseudo-coexistence line of capillary condensation was studied.

The thick solid line in Fig. 1 indicates the bulk phase coexistence line (H = 0, T < T c ) terminating at the bulk critical point (H = 0, T = T c ≃ 2.269J/k B ) (black circle). The 6 Before presenting our results let us summarize predictions for the scaling behaviour of the solvation force based on general scaling arguments [4]. One expects that the finitesize behaviour of the singular part of the solvation force is modified due to the presence of long-ranged substrate-fluid potentials:

f solv L d ≃ ϑ L/ξ τ , L/ξ H , (L/ξ 0 ) -ωs h 1 , (L/ξ 0 ) -ω g ω (4)
Here ξ τ (τ → ±0, H = 0) = ξ ± 0 |τ | -ν is the bulk correlation length at bulk coexistence H = 0 while ξ H (τ = 0, H) = ξ 0,H |H| -ν/∆ is the bulk correlation length at the critical temperature

T = T c . ω s = p -(d + 2 -η)/2
is the correction to scaling exponents due to the long-ranged tail of the substrate-fluid interactions. ω is the Wegner's [30] correction-to-scaling exponent for short-ranged systems, which is equal to 1/2 for 2D Ising model. g ω is a dimensionless nonuniversal scaling field. In the case of p = 50 that mimics the effect of the contact surface fields, the third argument of the scaling function in (4) should be replaced by Lh

ν/∆ 1 1
, where the surface gap exponent ∆ 1 = 1/2 for 2D Ising system. For large L and ω s > 0, one can expand the scaling function ϑ in (4)

f solv L d ≃ ϑ sr [L/ξ τ , L/ξ H ] + (L/ξ 0 ) -ωs h 1 ϑ lr s [L/ξ τ , L/ξ H ] (5)
+ (L/ξ 0 ) -ω g ω ϑ sr ω [L/ξ τ , L/ξ H ] .
Thus for L ≃ ξ τ , ξ H , ϑ lr s and ϑ sr ω represent corrections to the leading L-dependence provided by ϑ sr . For L/ξ ≫ 1 the scaling function ϑ sr decays exponentially, therefore in this regime ϑ lr s is the leading finite-size contribution in the singular behaviour of the force. As mentioned already in the Introduction, in the present case of the 2D Ising model η = 1/4 so that the expansion (5) should hold for p > 15/8, i.e., for all cases considered in our work.

However, one can argue that for sufficiently thin films the contribution of the longranged substrate-fluid potential is always important, also at the bulk critical point. Because the separation between two surfaces is finite and the substrate-fluid potentials are longranged and identical, there will be always some nonzero field acting at the center of the film. This effect can be interpreted as if the system experiences an external bulk field

H ef f = 2h 1 [L/(2ξ 0 )] -p
, although in fact it might be at the bulk coexistence curve H = 0.

The relevance of finite-size contributions due to this effective bulk field can be estimated by The total excess free energy per unit area for the case of identical surface fields h 1 = h 2 and non-vanishing bulk magnetic field H can be written as

f ex (L) ≡ L(f (L, T, H, h 1 ) -f b (T )) + 2f w (T, H, h 1 ) + f * (L, T, H, h 1 ) ( 6
)
where f is the free energy per site, f b is the bulk free energy, f w is the L-independent surface excess free energy contributed from each wall, and f * is the finite-size contribution to the free energy. All energies are measured in units of J and the temperature in units of J/k B .

f * , which vanishes for L → ∞, gives rise to the generalized force, which is analogous to the solvation force between the walls in the case of confined fluids,

f solv = -(∂f ex (L)/∂L) T,H,h 1 . (7)
In the transfer matrix approach the leading eigenvalue λ L of the transfer matrix T L

T L |v L = λ L |v L , (8)
gives the free energy per spin of an Ising strip as

βf (L) = - 1 L ln λ L . (9)
The components of the eigenvector |v L > related to the leading eigenvalue give the probabilities of various configurations. In order to calculate the size L dependence of the solvation force at fixed values of parameters (T, H, h 1 ) we calculate the excess free energy per unit area

f ex (L) ≡ (f -f b ) L at L 0 +1 and L 0 -1.
Having values f ex (L 0 +1) and f ex (L 0 -1) we approximate the derivative in eq.( 7) by a finite difference

f solv = -(1/2)(f ex (L 0 + 1)-f ex (L 0 -1)).
The case with vanishing external field H is relatively convenient to study because for the bulk free energy there exists an exact solution by Onsager [31]. For non-zero H, in order to find the bulk free energy for each thermodynamic state one has to perform the calculation 9 [32].

A. Thermodynamic path along the pseudo -phase coexistence line

A new computational problem arises when one wants to determine the solvation force on the thermodynamic path along the pseudo-coexistence line. In order to calculate the derivative of the excess free energy at certain L 0 the values of the free energy are necessary at L 0 + 1 and L 0 -1. According to the Kelvin equation, the pseudo-coexistence is shifted from the bulk position H = 0 proportionally to 1/L. Therefore the path ( 4) or ( 5) cannot be chosen to lie close to the pseudo-coexistence line corresponding to L 0 (see Fig. 2). Our way of defining the middle line between the L 0 + 1 and L 0 -1 pseudo-coexistence lines and the shift ∆ quarantees that for both L 0 + 1 and L 0 -1 one is on the same (liquid or gas) side of the above coexistence lines.

The solvation force scaled by L 2 as a function of the scaling variable L/ξ τ , evaluated along the pseudo-coexistence line slightly on its gas side (path ( 4)) and slightly on its liquid side (path ( 5)) is shown in Fig. 3 and Fig. 4, respectively. Data correspond to the fixed x = Lh 2 1 = 20000 and different values of L. Note that in the case of p = 50 which mimics a short-ranged contact surface field, x is the relevant scaling variable. Panels (a), (b) and (c) correspond to p = 2, 3, and 50, as depicted. On both sides of the pseudo-coexistence line and for all values of the parameter p, the force is attractive in the whole range of the studied scaling variable L/ξ τ .

On the gas side, data corresponding to different separations L do not collapse into a common curve, i.e., there is no scaling except for p = 3 and 50 near the minimum close to T c . Moreover, the solvation force f - solv is very weak -the weaker the larger value of the decay exponent p. For p = 3 and 50 the behaviour is qualitatively the same, we observe a slight linear variation of f - solv with a "dip" in the narrow interval below but close to T c . On the contrary, on the liquid side of the pseudo-coexistence line an excellent scaling has been found for p = 3 and 50. For p = 2 one can see deviations from scaling, especially below T c . The solvation force f + solv is much stronger than on the gas side -data shown in Fig. 4 to the scaling function divided by 100. The presence of the pseudo-capillary critical point manifests itself as a crossover between a rapid linear increase of the solvation force at low temperatures and a saturation at small values for high temperatures. In Ref. [23] it was argued that at the transition slightly on the liquid side f + solv ≈ 2H co (T )m * (T ) ≈ -2σ(T )/L. It follows that the amplitude of f + solv should increase in the same fashion as the interfacial tension as T increases at fixed L. For the 2D Ising model, the surface tension is given exactly by βJσ(T ) = 2(K -K * ), where K = J/k B T , K * = arcth(exp(-2K)), and [31]. Since σ(T ) ≃ -4K c τ , the scaling function should vary linearly with τ at fixed L, which agrees with our result for p = 50 and 3. The argument given in Ref. [23] is based on the macroscopic approximation for the total free energy which ignores interactions between surfaces, i.e., it is valid for L → ∞. For p = 2 such interactions cannot be ignored, even for large L. Therefore it is not surprising that we can see some deviations from the linear variation of f + solv in that case. For the gas side of the pseudo-coexistence curve, the analysis in Ref. [23] gives f - solv ≈ 0. What we have found, is a nonzero but very weak solvation force. It arises from the L-dependent part of the total free energy which is neglected in the previous argumentation. As mentioned in the Introduction, away from the bulk critical point the dominant contribution to the solvation force comes from the regular term which decays with L in the same way as the substrate fluid potential, i.e., ∼ L -p or exponentially in L for the case of short-range substrate potential -this explains the observed lack of the scaling behaviour. Finally, we note that on both sides of the pseudo-coexistence line f solv is attractive for all values of p. In contrast, in the same range of temperatures along the bulk coexistence line H = 0 the solvation force is repulsive for p = 2 and 3 and attractive for short-range boundary fields [3].

K c = J/(k B T c ) = ln(1 + √ 2)

B. Isotherms

As mentioned in the Introduction, the scaling behaviour of the solvation force along various isotherms was studied for 2D Ising model with short-range contact boundary fields in Ref. [23]. The conclusion was that in the range of the scaling variable L/ξ τ in which the solvation force displays the features of (weakly rounded) capillary condensation, i.e., the jump from values appropriate to a spin down (-) (gas) phase f - solv ≈ 0 to the negative values appropriate to a spin up (+) (liquid) phase f + solv ≈ 2Hm * (T ), scaling is not well obeyed. In the present study we have been able to perform calculations for much bigger systems than that considered in [23]. We find that along the isotherms corresponding to L/ξ τ ≈ 127 and ≈ 42, data for different values of the separation L collapse onto the common curve on the liquid side of the pseudo-coexistence line -see Fig. 5. This holds for all considered values of the parameter p. This is because on the liquid side of the pseudo-coexistence line the leading behaviour of the solvation force f + solv ≈ 2Hm * (T ), so that f solv L 2 is approximately a quadratic function of H 8/15 L. On the gas side, the leading behaviour is the regular part of the solvation force (see Eq. ( 1)). The maximum absolute value of the force is ∆f solv = f + solv -f - solv ≈ 2H co (T )m * (T ) ≈ -2σ(T )/L, which for a fixed temperature decreases as 1/L. The location of the abrupt change in the value of the solvation force depends slightly on L, as it should be for a weakly rounded transition. What is also characteristic is that on the liquid side of the pseudo-coexistence curve the scaling function varies only very little from one isotherm to another. As L/ξ τ decreases the jump of f solv gradually transforms into a minimum.

For p = 50, scaling along the critical isotherm is excellent in the whole range of the scaling variable |H| 8/15 L (see Fig. 6). For p = 2 and 3, data collapse only on the left shoulder of the minimum, which indicates some residual effect of the condensed phase. As one can expect the finite size effects increase with the range of the substrate potential. Notice also that for long-ranged boundary fields the maximal absolute value of the solvation force is strongly increased with respect to the one for the short-ranged one and that the force becomes repulsive for |H| 8/15 L → 0.

IV. CONCLUSIONS

We have presented DMRG results for the solvation force f solv for 2D Ising systems between identical walls with surface potentials decaying algebraically with the exponent p = 2, 3 and 50. The range of the substrate potential does not influence the qualitative behaviour of the solvation force. We found, in agreement with simple macroscopic arguments, that on the liquid side of the pseudo-coexistence line the behaviour of the solvation force is dominated by the bulk excess free energy associated with the (+) phase being metastable in the bulk. solvation force slightly on the liquid side of the pseudo-coexistence line. Contribution to the solvation force arising from the L-dependent regular terms in the free energy manifests itself strongly for p = 2 and on the gas side of the pseudo-coexistence line for all values of the decay exponent p. There, except in the vicinity of the pseudo-capillary critical point for p = 2, scaling does not hold and the solvation force is weak.

Close to the bulk critical temperature the solvation force changes its behaviour significantly, i.e., the jump of f solv transforms into a minimum, but the effects of (pseudo) capillary condensations remain strong. This manifests itself at small values of |H|

bulk phase.

For p = 2 and 3, along the pseudo-coexistence line on its gas side and along the isotherms for large values of |H| 8/15 L, the regular contribution (1) dominates the behaviour of the solvation force. Because m * (T )h 1 < 0, at these thermodynamic points f solv is attractive.

On the liquid side of the pseudo-coexistence line and far away from (pseudo) capillary condensation and bulk criticality, f solv becomes repulsive in agreement with (1) (see also Ref. [3]). Near the bulk critical point and close to the capillary condensation (pseudo) transition the solvation force is attractive. In the case of short-range-like boundary potential p = 50, f solv is always attractive.

2

 2

4 Page

 4

5 Page

 5

FIG. 1 :(

 1 FIG. 1: Phase diagram for a 2D Ising strip subject to identical boundary fields V ext j (see Eq. (3)) obtained by using DMRG for a strip width L = 600 and the amplitude h 1 for the boundary fields such that x = h 1 * L 2 = 20000. The thick solid line indicates the bulk coexistence line at H = 0 ending at the bulk critical point (black circle). The lines interpolating the symbols represent the pseudo-phase coexistence lines H ca (T, L; p) for three different values of the exponent p governing the algebraic decay of the boundary fields. Three isotherms along which the solvation force has been calculated are shown -paths (1)-(3). The other thermodynamic path (4) and (5) run parallel to the pseudo-coexistence line slightly on the "liquid" and on the "gas" side, respectively.

8 URL

 8 that H scales as HL ∆/ν . These finite-size contributions are negligible in the critical regime if 2 p+1 |h 1 | [L/ξ 0 ] ∆/ν-p ≪ 1. From that inequality one can identify the thickness of the film L crit such that for L ≤ L crit the effect due to the long-ranged substratefluid potentials is relevantL crit ≃ ξ 0 (2 p+1 |h 1 |) -1/(d-p-β/ν) .For 2D Ising model β/ν = 1/8 and ξ 0 ≈ 0.5673 therefore for p = 2 one obtains L crit ≃ 0.5673 (8|h 1 |) 8 , i.e., the long-ranged tails practically always matter. For p = 3, L crit ≃ 0.5673 (16|h 1 |) 8/9 ≃ 7 (|h 1 |) 8/9 which assuming |h 1 | = 10 gives L crit ≈ 50. The separation L is in units of a lattice constant.

4 FIG. 2 :

 42 FIG. 2: Phase diagram for an Ising strip subject to identical short-ranged (p = 50) boundary fields h 1 ∼ 6.32 for strip widths L = 500 and L = 502 corresponding to x = h 1 L 2 = 20000. The solid line denotes the middle line between both pseudo-coexistence lines defined as H mid (T ) = (H co (T ; L = 502) + H co (T ; L = 500)) /2. Dashed lines indicate the pseudo-coexistence lines, whereas the dashed-dot lines present the thermodynamic paths shifted with respect to the H mid line by ∆(T ) = H co (T ; L = 502) -H co (T ; L = 500).

11 Page 12 of 19 URL

 1119

12 Page 2 FIG. 3 :

 1223 FIG. 3: Scaled solvation force as a function of L/ξ τ = Lτ /ξ 0 calculated for different separations L along the line of pseudo-coexistence slightly on its gas side -the path (4). The amplitude of the substrate potential h 1 is chosen such that Lh 2 1 = 20000. Different panels show results for different values of the exponent p describing the algebraic decay of the substrate fluid potential. Scaling does not hold.

13 Page 2 FIG. 4 :

 1324 FIG.4: The same as in Fig.3but slightly on the liquid side of the pseudo-coexistence curve -the path(5). Scaling is very well obeyed.

14 Page 15 of 19 URL 2 FIG. 5 :

 141925 FIG. 5: Scaled solvation force as a function of |H| 8/15 L calculated for different separations L along 3 various isotherms. For L/ξ τ ≈ 127 (path (1)-bottom curve) and 42 (path (2)-middle curve), the sharp jumps to a strongly attractive solvation force as |H| 8/15 L is reduced indicate weakly rounded capillary condensation of the (+) (liquid) phase. Data corresponding to L = 601 (crosses), 501 (squares), and 401 (circles) follow common curves to the left from the jumps. Along the critical isotherm solvation force exhibits a minimum.

8 / 15 L 15 Page 2 FIG. 6 :

 8151526 FIG. 6: Scaled solvation force as a function of |H| 8/15 L calculated along the critical isotherm.Scaling is excellent for p = 50. For p = 3 and 2 data collapse on the left shoulder of the minimumto the right from the minimum we observe deviations from scaling. For p = 2 and 3, f solv becomes repulsive for small values of |H| 8/15 L. Symbols have the same meaning as in the Fig.5.

16 Page

 16

3 Page 4 of 19 URL: http://mc.manuscriptcentral.com/tandf/tmph

	Molecular Physics
	of this minimum follows roughly the continuation of the (pseudo) capillary condensation
	line towards T c
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

Page 7 of 19 URL: http://mc.manuscriptcentral.com/tandf/tmph

			Molecular Physics
	F o
	2	r P 2.05	2.1	2.15	2.2	2.25	2.3	2.35
		e e r
			R e v i e w
						O n
							l y

  for finite systems and extrapolate the data to L → ∞. We have determined the largest eigenvalue of T L for the strips (L × ∞) with free boundary conditions and widths L ranging from 380 to 680. Next the free energy was extrapolated to L → ∞ by means of the Bulirsch and Stoer method

	Molecular Physics
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y

Page 10 of 19 URL: http://mc.manuscriptcentral.com/tandf/tmph

correspond 10 Page 11 of 19 URL

		Molecular Physics
	F o r 2 2.05 -	2.1	k B T/J 2.15 2.2	2.25	2.3	2.35
	P
	e e r
		R e v i e w
				O n
					l y

: http://mc.manuscriptcentral.com/tandf/tmph

Acknowledgments

We dedicate this paper to Bob Evans on the occasion of his 65 birthday for being always a source of inspiration for us. We thank S. Dietrich for discussions. Numerical calculations were performed in WCSS Wroc law (Poland, grant 82).


S Passey

N Martin

D Player

Professor Mark P Lewis
email: mark.lewis@beds.ac.uk

Stretching skeletal muscle in vitro: does it replicate in vivo physiology?

Keywords: skeletal muscle model, stretch protocol, tissue engineering

Skeletal muscle is highly adaptable and responds to changes in loading through exercise or resistance training through a number of mechanisms resulting in increased muscle mass and changes in contractile phenotype. To further understand and study the molecular mechanisms underlying the adaptive response of muscle, a number of in vitro culture systems have been developed that utilise mechanical loading or stretching of the cultured muscle to recapitulate the adaptations observed in vivo. Here we review the use of such stretching regimes for engineered muscle constructs and assess how well these in vitro systems mimic in vivo muscle physiology and adaptation.

Introduction

Skeletal muscle tissue possesses a high degree of plasticity in that it has the ability to adapt specifically to the functional demands placed upon it or alterations in its environment by responding, for example, to changes in the levels of growth factors and cytokines or to changes in neuronal stimulation or mechanical loading such as is experienced during exercise or resistance training. At the physiological level, the muscle responds to such stimuli through increasing the synthesis of numerous proteins important for muscle contraction and metabolism to allow for the increased workload, leading to muscle hypertrophy. Various additional stimuli such as the availability of amino acids have been shown to potentiate this adaptive response [START_REF] Norton | Leucine regulates translation initiation of protein synthesis in skeletal muscle after exercise[END_REF][START_REF] Dreyer | Leucine-enriched essential amino acid and carbohydrate ingestion following resistance exercise enhances mTOR signaling and protein synthesis in human muscle[END_REF][START_REF] Tipton | Stimulation of muscle anabolism by resistance exercise and ingestion of leucine plus protein[END_REF]).

The study of the molecular mechanisms underlying muscle adaptation has traditionally relied on the use of human subjects and animal models. Studies using in vivo systems prove difficult for many reasons; and it has been suggested that the molecular responses observed in existing in vivo models may not accurately mimic the normal physiological muscle adaptation to exercise training [START_REF] Spangenburg | Changes in muscle mass with mechanical load: possible cellular mechanisms[END_REF]. For these reasons, suitable in vitro model systems that recapitulate the in vivo conditions and allow for accurate control of the experimental environment are highly desirable to allow detailed study of the molecular mechanisms of muscle adaptation and development.

Over the last 30 years, a number of in vitro model systems have been developed that show some similarities with muscle in vivo. The aim of this paper is to review the body of literature that have used these muscle stretching techniques, with reference to the ability of such regimes to mimic the effects seen in skeletal muscle in vivo.

In vitro skeletal muscle models

Skeletal muscle in vivo is formed of groups of myofibres attached to tendons longitudinally, and surrounded by a connective tissue matrix [START_REF] Mastaglia | Skeletal Muscle Pathology[END_REF]. Under normal physiological conditions the muscle fibres experience force in a longitudinal direction during embryogenesis and growth, and also during exercise or resistance training. It goes to follow that an ideal muscle stretching system would look to mimic the forces experienced by the muscle fibres in vitro and so would ideally exert uniaxial forces on the cultured cells.

However the different types of stretching systems available differ in producing uniaxial or multiaxial stretch, as illustrated in Figure 1A. Indeed the direction of the stretch forces is an important feature to consider when investigating muscle responses to strain, as it has been demonstrated that the signalling pathways activated differ in response to multiaxial vs. uniaxial stretch [START_REF] Hornberger | Intracellular signaling specificity in response to uniaxial vs. multiaxial stretch: implications for mechanotransduction[END_REF].

Figure 1 to be inserted here

The first-reported in vitro stretching model employed by Vandenburgh and Kaufman consisted of a stretching frame surrounding an elastic membrane on which chicken skeletal muscle myoblasts were plated and differentiated to form early myotubes in two dimensions, whilst being stretched under the multiaxial strain produced by the stretching frame [START_REF] Vandenburgh | In vitro model for stretch-induced hypertrophy of skeletal muscle[END_REF].

Work from the same laboratory further developed the Vandenburgh system through the addition of a computer controlled stepper motor [START_REF] Vandenburgh | A computerized mechanical cell stimulator for tissue culture: effects on skeletal muscle organogenesis[END_REF], which allowed the generation of various cyclical and ramp loading protocols (Figure 1B) that are used to model different strain types that skeletal muscles would be exposed to in vivo. The ability to vary the stretching protocols expanded the use of this system to mimic the effects of exercise, resistance training and growth and allow the study of the effects of different loading regimes on muscle physiology and the hypertrophy response. A similar system used by Iwata and colleagues uses a programmable motor to exert a uniaxial stretch on muscle cells cultured in 2D on an elastic silicone membrane [START_REF] Iwata | Uniaxial cyclic stretch-stimulated glucose transport is mediated by a ca-dependent mechanism in cultured skeletal muscle cells[END_REF]).

In the commercially available Flexercell system (Flexcell International Corporation, Hillsborough, NC, USA), cells are grown on flexible culture plates in two dimensions, which are deformed and stretched by a vacuum controlled by a computer, allowing varying regimens of strain to be employed [START_REF] Hubatsch | Mechanical stimulation increases expression of acetylcholinesterase in cultured myotubes[END_REF].

Muscle cells in vivo grow and differentiate in a 3-dimensional environment and their development is greatly influenced by the signals they receive from the surrounding cells and extracellular matrix (ECM) components. A number of 3D biomimetic model systems have been described with the intention of recreating aspects of the 3D environment and so promoting the formation of physiological muscle models. Such models include the use of collagen gel matrices [START_REF] Powell | Mechanical stimulation improves tissueengineered human skeletal muscle[END_REF][START_REF] Cheema | 3-D in vitro model of early skeletal muscle development[END_REF], or collagen gel foam sponges [START_REF] Auluck | A three-dimensional in vitro model system to study the adaptation of craniofacial skeletal muscle following mechanostimulation[END_REF] to provide a physiologically relevant matrix into which the muscle cells are seeded. Muscle constructs grown in such a 3D environment under uniaxial strain have been shown to exhibit a slow phenotype analogous to that of postural muscle in vivo (Mudera et al. 2010).

Alterations in muscle phenotype and metabolism

Hypertrophy

Skeletal muscle responds to increased load from resistance training or exercise through various physiological and molecular adaptations including increases in protein synthesis leading to muscle hypertrophy. Typically increases in muscle cross sectional area are observed in humans following resistance training programmes of six to sixteen weeks duration, with the extent of hypertrophy from 7.5% to 25% depending on the study participants, muscle type studied and exercise regime [START_REF] Mccall | Muscle fiber hypertrophy, hyperplasia, and capillary density in college men after resistance training[END_REF][START_REF] Kosek | Efficacy of 3 days/wk resistance training on myofiber hypertrophy and myogenic mechanisms in young vs. older adults[END_REF][START_REF] Holm | Changes in muscle size and MHC composition in response to resistance exercise with heavy and light loading intensity[END_REF][START_REF] Friedmann-Bette | Effects of strength training with eccentric overload on muscle adaptation in male athletes[END_REF]. In vivo animal studies have also demonstrated robust hypertrophic responses to muscle loading [START_REF] Goldberg | Work-induced growth of skeletal muscle in normal and hypophysectomized rats[END_REF][START_REF] Goldspink | Muscle growth in response to mechanical stimuli[END_REF].

Using the Vandenburgh model, the myotube diameter and creatine kinase activity both increase following in vitro stretching, indicating that similar hypertrophy responses are initiated by stretching using in vitro systems [START_REF] Vandenburgh | In vitro model for stretch-induced hypertrophy of skeletal muscle[END_REF][START_REF] Vandenburgh | Skeletal muscle growth is stimulated by intermittent stretch-relaxation in tissue culture[END_REF]. In vivo exercise also increases serum creatine kinase activity which tends to be most pronounced after eccentric muscle loading [START_REF] Nosaka | Variability in serum creatine kinase response after eccentric exercise of the elbow flexors[END_REF][START_REF] Nosaka | Delayed-onset muscle soreness does not reflect the magnitude of eccentric exercise-induced muscle damage[END_REF].

Using their 3D collagen gel system, [START_REF] Powell | Mechanical stimulation improves tissueengineered human skeletal muscle[END_REF] introduced cyclical stretching protocols to differentiated myotubes with 12% myotube hypertrophy being observed; again this data is comparable to muscle hypertrophy observed in vivo in response to exercise training.

Signalling through mTOR is required for the hypertrophy response [START_REF] Leger | Akt signalling through GSK-3beta, mTOR and Foxo1 is involved in human skeletal muscle hypertrophy and atrophy[END_REF][START_REF] Miyazaki | Cellular mechanisms regulating protein synthesis and skeletal muscle hypertrophy in animals[END_REF][START_REF] Dreyer | Resistance exercise increases leg muscle protein synthesis and mTOR signalling independent of sex[END_REF] and in vitro stretching systems have been used

to further investigate the molecular mechanisms of mTOR activation and signalling. Using the Iwata and Flexercell systems, similar involvement of the mTOR signalling pathway in the hypertrophy response to stretch has been observed following muscle stretch in C2C12 cells and in primary chick skeletal myotubes [START_REF] Hornberger | Mechanical stimuli regulate rapamycinsensitive signalling by a phosphoinositide 3-kinase-, protein kinase B-and growth factor-independent mechanism[END_REF][START_REF] Sasai | Involvement of PI3K/Akt/TOR pathway in stretchinduced hypertrophy of myotubes[END_REF].

The rate of protein synthesis in muscle in vivo actually decreases during exercise [START_REF] Bylund-Fellenius | Protein synthesis versus energy state in contracting muscles of perfused rat hindlimb[END_REF][START_REF] Dreyer | Resistance exercise increases AMPK activity and reduces 4E-BP1 phosphorylation and protein synthesis in human skeletal muscle[END_REF], and then increases again following exercise, an effect that is potentiated by increased availability of carbohydrate and amino acids [START_REF] Tipton | Stimulation of muscle anabolism by resistance exercise and ingestion of leucine plus protein[END_REF]. This effect has also been observed in vitro using the Flexercell system to apply stretch to L6 rat muscle cells, resulting in a suppression of myofibrillar protein synthesis by 40% over the initial 30 minute period which then returned to baseline at later timepoints [START_REF] Atherton | Cyclic stretch reduces myofibrillar protein synthesis despite increases in FAK and anabolic signalling in L6 cells[END_REF].

Neuromuscular adaptation

In addition to structural adaptations and hypertrophy of the muscle fibres themselves, some aspects of muscle adaptation in response to exercise are related to improved neuromuscular function and neural adaptation leading to improvements in muscle function [START_REF] Folland | The adaptations to strength training : morphological and neurological contributions to increased strength[END_REF]. For example, increased activity of acetylcholinesterase (AchE) in the neuromuscular junction is observed in vivo in rats following exercise training or muscle overload protocols [START_REF] Jasmin | Muscle acetylcholinesterase adapts to compensatory overload by a general increase in its molecular forms[END_REF][START_REF] Sveistrup | Chronic enhancement of neuromuscular activity increases acetylcholinesterase gene expression in skeletal muscle[END_REF]. Interestingly, in vitro studies using the Flexercell muscle stretching system have shown similar increases in AchE mRNA and protein following cyclical stretching protocols [START_REF] Hubatsch | Mechanical stimulation increases expression of acetylcholinesterase in cultured myotubes[END_REF]. Whilst the mechanism for this increase is not yet known, these alterations in the levels of neuromuscular junction proteins further suggest an adaptive response to exercise and stretch in vitro that mirrors that seen in vivo.

Myosin heavy chain composition

Skeletal muscle is heterogeneous in that it is composed of both fast and slow twitch muscle fibres, so-named on the basis of their contractile and metabolic phenotype. Fast twitch myofibres express type II Myosin Heavy Chain (MHC) isoforms and show a high contraction velocity, whereas slow-twitch fibres express Type I MHC isoforms and exhibit slower contraction velocities [START_REF] Bassel-Duby | Signaling pathways in skeletal muscle remodeling[END_REF]. The muscle fibre type found in an individual muscle relates to the physiological function of the muscle and the physical demands placed on it, and is also determined by the neuronal input the muscle receives [START_REF] Salmons | Significance of impulse activity in the transformation of skeletal muscle type[END_REF].

Human studies have demonstrated adaptive changes in MHC expression and phenotype in response to exercise, with a shift from fast MHC isoforms to slower isoforms being observed in response to endurance, resistance and even sprint training [START_REF] Jansson | Muscle adaptation to extreme endurance training in man[END_REF][START_REF] Hather | Influence of eccentric actions on skeletal muscle adaptations to resistance training[END_REF][START_REF] Coggan | Skeletal muscle adaptations to endurance training in 60-to 70-yr-old men and women[END_REF][START_REF] Adams | Skeletal muscle myosin heavy chain composition and resistance training[END_REF][START_REF] Allemeier | Effects of sprint cycle training on human skeletal muscle[END_REF][START_REF] Staron | Skeletal muscle adaptations during early phase of heavy-resistance training in men and women[END_REF]. Similar shifts in MHC expression have been observed in response to stretch in vitro using the Flexercell system, where a shift from the faster MHC IIb to the IIa and even MHC I isoforms was observed with increased stretch [START_REF] Sakiyama | Effects of stretching stress on the muscle contraction proteins of skeletal muscle myoblasts[END_REF][START_REF] Kurokawa | Effects of stretching stimulation with different rates on the expression of MyHC mRNA in mouse cultured myoblasts[END_REF]).

Glucose metabolism

Skeletal muscle is the major tissue involved in glucose uptake and homeostasis in vivo (Abdul-Ghani and DeFronzo 2010), and an increase in glucose uptake by the muscle is observed during exercise in vivo [START_REF] Rose | Skeletal muscle glucose uptake during exercise: how is it regulated?[END_REF] and in vitro. Hatfaludy et al, observed increases in both glucose uptake and lactate efflux by muscle cultures subjected to stretch-relaxation protocols using the Vandenburgh system [START_REF] Hatfaludy | Metabolic alterations induced in cultured skeletal muscle by stretch-relaxation activity[END_REF]) and Iwata et al. also described a calcium-dependent mechanism for glucose uptake by skeletal muscle cells following in vitro stretch [START_REF] Iwata | Uniaxial cyclic stretch-stimulated glucose transport is mediated by a ca-dependent mechanism in cultured skeletal muscle cells[END_REF]). In agreement with these results, evidence for calcium mediated glucose uptake following exercise in vivo is well documented [START_REF] Santos | Skeletal muscle pathways of contraction-enhanced glucose uptake[END_REF].

Muscle repair and regeneration

Growth factors and satellite cell activation

In vitro studies have also helped to increase our understanding of how muscle is repaired following acute muscle damage, such as may occur with active muscle stretch. Muscle repair and regeneration is thought to occur via the activation of muscle resident satellite cells; skeletal muscle stem cells which upon activation can differentiate to form myoblasts and can fuse with existing myofibres [START_REF] Seale | A new look at the origin, function, and "stem-cell" status of muscle satellite cells[END_REF].

Activation of satellite cells has been demonstrated in vivo following a bout of electrical stimulation [START_REF] Mackey | Assessment of satellite cell number and activity status in human skeletal muscle biopsies[END_REF]) and prevention of satellite cell proliferation through exposure to -irradiation has also been shown to inhibit muscle hypertrophy in rats, demonstrating a role for satellite cells in contributing nuclei to myofibres as part of the hypertrophy response [START_REF] Rosenblatt | Satellite cell activity is required for hypertrophy of overloaded adult rat muscle[END_REF]. Using the Flexercell system, satellite cell activation has been demonstrated in both single muscle fibres [START_REF] Wozniak | C-Met expression and mechanical activation of satellite cells on cultured muscle fibers[END_REF] and in cultured muscle cell lines as early as 2 hours poststretch [START_REF] Tatsumi | Mechanical stretch induces activation of skeletal muscle satellite cells in vitro[END_REF]. Hepatocyte growth factor (HGF) was identified as an important activator of satellite cells, as demonstrated by the observation of a dose-dependent decrease in the level of satellite cell activation in response to mechanical stretch with increasing concentrations of a blocking anti-HGF antibody [START_REF] Tatsumi | Mechanical stretch induces activation of skeletal muscle satellite cells in vitro[END_REF]. These results shed more light on the in vivo findings that direct injection of HGF into uninjured muscle of rats stimulates satellite cell activation [START_REF] Tatsumi | HGF/SF is present in normal adult skeletal muscle and is capable of activating satellite cells[END_REF]) and resistance training induces increases in serum HGF protein levels [START_REF] Shelmadine | Effects of 28 days of resistance exercise and consuming a commercially available pre-workout supplement, NO-Shotgun(R), on body composition, muscle strength and mass, markers of satellite cell activation, and clinical safety markers in males[END_REF].

Activation of satellite cells is regulated by various other growth factors, of which the splice variants of insulin-like growth factor 1 (IGF-1) appear to be of great importance. In vivo studies have shown mRNA levels of the IGF-1 splice variant mechano-growth factor (MGF, also known as IGF-1Ec in humans) to peak early after muscle damage/stretch, implicating MGF in the activation of muscle resident satellite cells [START_REF] Hill | Expression and splicing of the insulin-like growth factor gene in rodent muscle is associated with muscle satellite (stem) cell activation following local tissue damage[END_REF][START_REF] Mckay | Co-expression of IGF-1 family members with myogenic regulatory factors following acute damaging muscle-lengthening contractions in humans[END_REF]. These data are supported by similar observations in vitro, where increased levels of MGF mRNA have been observed in the 3D collagen model following stretch [START_REF] Cheema | Mechanical signals and IGF-I gene splicing in vitro in relation to development of skeletal muscle[END_REF].

Satellite cells and matrix remodelling

It is becoming increasingly obvious that muscle repair and satellite cell activation requires interaction with and remodelling of the surrounding matrix. These processes that are mediated by the matrix metalloproteinases (MMPs) and a number of in vivo studies have revealed that changes in MMP expression and activity accompany muscle repair and remodelling. A biphasic pattern of MMP expression in response to exercise or muscle damage has been described, beginning with an early increase in MMP-9 mRNA and protein levels immediately post exercise and followed by a later increase in MMP-2 levels [START_REF] Rullman | A single bout of exercise activates matrix metalloproteinase in human skeletal muscle[END_REF][START_REF] Rullman | Endurance exercise activates matrix metalloproteinases in human skeletal muscle[END_REF][START_REF] Zimowska | Distinct patterns of MMP-9 and MMP-2 activity in slow and fast twitch skeletal muscle regeneration in vivo[END_REF]).

In support of the in vivo findings, [START_REF] Lewis | Gelatinase-B (matrix metalloproteinase-9; MMP-9) secretion is involved in the migratory phase of human and murine muscle cell cultures[END_REF] showed that, in 2D, developing skeletal muscle exhibited MMP-2 activity throughout a 14 day culture, whereas MMP-9 activity was only present in an early culture (days 2-4). Similarly, using the 3D collagen system, MMP-2 activity increased following stretching in a load-dependent manner [START_REF] Auluck | A three-dimensional in vitro model system to study the adaptation of craniofacial skeletal muscle following mechanostimulation[END_REF].

Balancing the ECM-degrading activity of the MMPs is a family of inhibitor proteins, the tissue inhibitors of metalloproteinases (TIMPs) and, interestingly, an increase in TIMP-1 mRNA levels occurs at later time points following exercise, both in vivo [START_REF] Rullman | Endurance exercise activates matrix metalloproteinases in human skeletal muscle[END_REF] and in vitro as the skeletal muscle constructs matured [START_REF] Lewis | Gelatinase-B (matrix metalloproteinase-9; MMP-9) secretion is involved in the migratory phase of human and murine muscle cell cultures[END_REF]. This led to the suggestion that the early increase in MMP activity enabled ECM remodelling and satellite cell migration, and that at later times TIMP-1 expression increased to balance the actions of the MMPs [START_REF] Lewis | Gelatinase-B (matrix metalloproteinase-9; MMP-9) secretion is involved in the migratory phase of human and murine muscle cell cultures[END_REF]. This is further supported by additional in vitro data from [START_REF] Yamada | Matrix metalloproteinases are involved in mechanical stretch-induced activation of skeletal muscle satellite cells[END_REF] who demonstrated that the addition of TIMP-1 to the culture medium of cells during stretching with the Flexercell system attenuated HGF release and satellite cell activation.

Molecular regulation of satellite cells

The myogenic regulatory factors (MRFs) are a group of transcription factors that lie at the heart of skeletal muscle regeneration and myogenic differentiation, and consist of myf-5, MyoD, myogenin and MRF 4. At early stages of satellite cell activation myf-5 and MyoD are required, whereas myogenin and MRF4 appear to be involved at later points in the differentiation process [START_REF] Cornelison | Single-cell analysis of regulatory gene expression in quiescent and activated mouse skeletal muscle satellite cells[END_REF].

In vivo studies support the role of the MRFs in satellite cell activation, proliferation and differentiation; MyoD expression peaked at early points following exercise/muscle stimulation whereas myogenin expression peaked later [START_REF] Bickel | Time course of molecular responses of human skeletal muscle to acute bouts of resistance exercise[END_REF][START_REF] Mckay | Co-expression of IGF-1 family members with myogenic regulatory factors following acute damaging muscle-lengthening contractions in humans[END_REF]. In vitro experiments have given comparable results: the genes MyoD and myf-5 were up-regulated early in the stretching process (first 12 h), whereas myogenin and MRF-4 were up regulated after 24 h of stretching before returning to control levels thereafter (Abe et al.

2009).

Boonen et al. ( 2010), however, have presented seemingly contradictory results for the effects of stretching on muscle maturation and regeneration. Using both C2C12 cells and primary muscle precursor cells (MPCs) cultured either in 2D or in 3D fibrin-based constructs, the authors observed a decrease in MRF expression following intermittent stretching protocols performed using the FlexCell system. A decrease in muscle maturation was also described, determined by a reduction in the development of cross-striations and MHC mRNA transcripts in the stretched MPCs compared to static controls [START_REF] Boonen | Effects of a combined mechanical stimulation protocol: Value for skeletal muscle tissue engineering[END_REF]. The strain levels used in this study were lower than those commonly used for in vitro stretching studies [START_REF] Sakiyama | Effects of stretching stress on the muscle contraction proteins of skeletal muscle myoblasts[END_REF][START_REF] Kurokawa | Effects of stretching stimulation with different rates on the expression of MyHC mRNA in mouse cultured myoblasts[END_REF]) and this may explain some of the discrepancies between this data and other publications. The existence of such apparently contradictory data highlights the importance of further work to fully optimise in vitro muscle stretching systems, as many aspects including the cell culture matrix and environment (2D vs. 3D), the extent and speed of stretching and the duration of stretch are all likely to affect the adaptive responses of the cells and their maturation into functional muscle models.

Table 1 to be inserted here.

Conclusions.

Here we have briefly reviewed a range of in vitro engineered skeletal muscle models that utilise stretch regimes to mechanically exercise the muscle tissue in order to study the adaptive responses. Much progress has been made in developing such models, and already the application of such in vitro systems is helping to expand our understanding of muscle function and adaptation at the molecular level and compliment the findings of in vivo studies (summarised in Table 1).

It is an exciting time for muscle tissue engineering. In addition to further optimising the stretching regimes employed to mimic the physiological environment, future work will most likely aim to generate increasingly sophisticated human and mammalian muscle constructs and the integration of other aspects such as neuronal inputs. Future work in the field will be applying these in vitro muscle systems to delve ever deeper into the molecular world that governs muscle adaptation, repair and regeneration. showing example systems to generate multiaxial and uniaxial strain, (1) Multiaxial stretch using a stretching frame attached to a membrane on which cells are grown, expansion of the frame results in multiaxial strain within the membrane (redrawn from [START_REF] Vandenburgh | In vitro model for stretch-induced hypertrophy of skeletal muscle[END_REF]. (2) Multiaxial stretch using a vacuum to deform a membrane on which cells are grown.

(3) Uniaxial stretch using a stepper motor to increase uniaxial tension on cells seeded within a 3D collagen gel, as described by [START_REF] Cheema | Mechanical signals and IGF-I gene splicing in vitro in relation to development of skeletal muscle[END_REF]. The dotted lines in the stretch schematic of each system indicate the original pre-stretch position of the culture. (B)

Stretching regimes may involve cyclical loading or ramp loading protocols.

Figure 1

  Abdul-Ghani, M. A. and R. A. DeFronzo (2010). "Pathogenesis of insulin resistance in skeletal muscle." J Biomed Biotechnol 2010: 476279. Abe, S., S. Rhee, et al. (2009). "Effect of mechanical stretching on expressions of muscle specific transcription factors MyoD, Myf-5, myogenin and MRF4 in proliferated myoblasts." Anat Histol Embryol 38(4): 305-310.

Figure 1 .

 1 Figure 1. Variations in in vitro stretching models. (A) Different axes of strain generation,

Table 1 :

 1 Comparison of findings from in vivo studies and in vitro muscle stretching studies.

	Measure	In vivo	In vitro	In vivo	In vitro	In vitro
				References	References	stretch
						protocol
	Myofibre			McCall et al. 1996,	Vandenburgh et	Cyclical 20%
	diameter			Hom et al. (2008).	al. (1989a)	or 46% for up
						to 96h.
					Powell et al.	Ramp 10%
					(2002)	then cyclical
						5/10/15%
						over 8 days.
					Sasai et al. (2009) Cyclical 10% at
						0.16Hz for 72h
	Creatine			Nosaka et al.	Vandenburgh et	Cyclical 20%
	Kinase			1996, Nosaka et	al (1989a)	or 46% for up
	release			al. (2002)		to 96h.
	Protein	Initial	Initial	Bylund-Fellenius	Atherton et al.	Cyclical 15% at
	synthesis	followed by	followed by	et al. (1984),	(2009)	1Hz for 30 min
		above	to baseline	Dreyer et al.
		baseline	after stretch	(2006)
		after
		exercise
	MHC	Shift MHC	Shift MHC	Staron et al	Sakiyama et al.	Cyclical 15% at
		fast	fast	(1994), Adams et	(2005)	0.5Hz, for up
		slower	slower	al. (1993), Hather		to 5 days.
		isoforms	isoforms	et al. (1991), Jansson et al. (1977), Coggan et al. (1992), Allemeier (1994)	Kurokawa (2007)	Cyclical 15% at 0.1Hz, 0.5Hz or 0.9Hz over 5 days
	mTOR			Leger et al (2006),	Hornberger et al	Cyclical 15% at
	activation			Dreyer et al.	(2004)	1Hz for 10 min
				(2010).	Sasai et al. (2009) Cyclical 10% at
						0.16Hz for 72h
	Glucose			Rose and Richter	Hatfaludy et al.	Cyclical 20% at
	metabolism			(2005)	(1989)	0.25Hz for 60
						sec out of
						every 30 min.
					Iwata et al. (2007) Cyclical 10% at
						1Hz, for up to
						60min
	MMP	Acute	MMP2 at	Rullman et al	Auluck et al	Rapid ramp
	expression	MMP9,	6h of	(2009), Zimowska	(2005).	strain: 7.5% or
		chronic	stretching	et al. (2008).		15% for 6h.
		MMP2				Cyclical: 7.5%
						or 15%, 12
						cycles/h
	MRFs	MyoD	MyoD and	Bickel et al.	Abe et al (2009)	Cyclical 15% at
		early,	myf-5 early,	(2005), McKay et		0.5Hz, for up
		myogenin	myogenin	al. (2008)		to 48h
		later	and MRF-4
			later


Dr Mark S Pearce
email: m.s.pearce@ncl.ac.uk

Richard J Q Mcnally

Julie Day

S Murthy Korada

Steve Turner

Tim D Cheetham

Murthy Korada

David Allison

Mohammed Kibirige

Sir James Spence

Space-time clustering of elevated thyroid stimulating hormone levels

Keywords: Congenital hypothyroidism, TSH, incidence, seasonality

come   L'archive ouverte pluridisciplinaire

Introduction

Congenital hypothyroidism (CHT) is the most common congenital endocrine disorder with a world-wide incidence around 1 in 3,500 to 4,000 live births. Reduced thyroid hormone production in babies with CHT has a major detrimental effect on central nervous system development and growth. Prompt treatment with thyroxine will prevent these problems arising in the majority of babies.

Most babies with CHT have thyroid gland agenesis or dysgenesis with a poorly formed or absent gland. A minority (~20%) have a normally sited gland but an underlying single gene defect preventing the normal process of thyroid hormone production within the gland (dyshormonogenesis). Although germline mutations in thyroid transcription factors 1 and 2 (TTF-1 and TTF-2) and PAX-8 (paired box transcription-8) have been identified as aetiological risk factors for dysgenesis or agenesis, they only explain a small percentage of cases (around 2%) [START_REF] Vliet | Development of the thyroid gland: lessons from congenitally hypothyroid mice and men[END_REF]. It is important to highlight the fact that iodine deficiency is a well recognised and important cause of neonatal thyroid dysfunction in some parts of the world.

An increasing incidence of CHT has been suggested from an analyses of data including that from New York State [START_REF] Harris | Increase in congenital hypothyroidism in New York State and in the United States[END_REF] and Mexico [START_REF] Me | Birth prevalence of congenital hypothyroidism in Mexico[END_REF], although no similar increase was shown in a similar study of data from Quebec [START_REF] Deladoëy | Random variability in congenital hypothyroidism from thyroid dysgenesis over 16 years in Quebec[END_REF]. Previous studies in the United Kingdom have suggested that CHT is more prevalent in Asian sectors of the population and that prevalence has increased [START_REF] Hall | Congenital hypothyroidism, seasonality and consanguinity in the West Midlands, England[END_REF]. Studies also suggest that the prevalence of hypothyroidism in Scotland has increased [START_REF] Leese | Increasing prevalence and incidence of thyroid disease in Tayside, Scotland: the Thyroid Epidemiology Audit and Research Study (TEARS)[END_REF], with a study from the same area of Scotland demonstrating an increased population prevalence of hypothyroidism in young people compared to previously published rates [START_REF] Hunter | Prevalence and aetiology of hypothyroidism in the young[END_REF].

The incidence of CHT has also been suggested to vary seasonally in a number of studies in different geographical areas including the West Midlands of England [START_REF] Hall | Congenital hypothyroidism, seasonality and consanguinity in the West Midlands, England[END_REF],

Finland [START_REF] Virtanen | Aetiology of congenital hypothyroidism in Finland[END_REF], Japan [START_REF] Miyai | An analysis of the variation of incidence of congenital dysgenic hypothyroidism in various countries[END_REF][START_REF] Nakamizo | Seasonality in the incidence of congenital hypothyroidism in Japan[END_REF][START_REF] Miyai | Further studies on episodic occurrence of congenital dysgenetic hypothyroidism in Osaka, Japan[END_REF] and Australia [START_REF] Miyai | An analysis of the variation of incidence of congenital dysgenic hypothyroidism in various countries[END_REF]. However, seasonality also has been not observed in a number of other studies, including those in the North West of England [START_REF] Rosenthal | Congenital hypothyroidism: increased incidence in Asian families[END_REF], the Netherlands [START_REF] Reijneveld | No evidence for seasonality of congenital hypothyroidism in The Netherlands[END_REF], Saudi Arabia [START_REF] Henry | Screening for congenital hypothyroidism[END_REF], Canada [START_REF] Deladoëy | Random variability in congenital hypothyroidism from thyroid dysgenesis over 16 years in Quebec[END_REF], Norway, France and Switzerland [START_REF] Miyai | An analysis of the variation of incidence of congenital dysgenic hypothyroidism in various countries[END_REF]. Should seasonal variation in CHT risk exist, this would suggest that an unknown environmental factor may be involved in the disease's aetiology. Temporal trends in risk are usually unlikely to include genetic factors, unless either population shifts result in germline mutations being more prominent in a particular geographical area or environmental influences on germline mutations or epigenetic changes have increased in prevalence over the study period.

Circulating thyroid stimulating hormone (TSH) levels are measured as part of screening programmes for CHT across many parts of Europe, Japan and increasingly in North America. In the UK, neonatal screening for CHT began in 1979 in Scotland and in 1981 in the remainder of the UK after a recommendation from the UK Department of Health [START_REF]Screening for early detection of congenital hypothyroidism[END_REF]. This paper describes the incidence of elevated TSH levels in newborns in the North of England over an eleven-year period (1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) and examines whether seasonal variation in incidence exists in this geographical area.

Methods

Around 35,000 infants in the Northern Region of England, comprising North East England (the area from Teesside extending north into Northumberland) and North Cumbria, are screened every year in a single centre by measuring blood spot TSH levels. Data on all cases, including dates of birth, were available from 1994 to 2005 inclusive.

TSH as a surrogate for CHT

We opted to refer to 'TSH' rather than 'CHT' in this study for the following reasons.

1. The extent to which cases of suspected CHT are investigated will vary from one unit to another [START_REF] Jones | Improvement in screening performance and diagnosis of congenital hypothyroidism in Scotland 1979-2003[END_REF][START_REF] Ahmed | Initial evaluation of congenital hypothyroidism: a survey of general paediatricians in East Anglia[END_REF]. We felt that this was likely to be the case in our region of the UK as well.

2. There is no definitive test or tests that can identify the underlying thyroid gland abnormality in this condition. Even the combination of isotope scanning and ultrasonography does not reveal an underlying diagnosis in all infants [START_REF] Jones | Improvement in screening performance and diagnosis of congenital hypothyroidism in Scotland 1979-2003[END_REF][START_REF] Perry | Combined ultrasound and isotope scanning is more informative in the diagnosis of congenital hypothyroidism than single scanning[END_REF].

The sensitivity and specificity of tests such as isotope scanning is suboptimal

with potentially misleading information generated by factors such as early thyroxine therapy [START_REF] Perry | Combined ultrasound and isotope scanning is more informative in the diagnosis of congenital hypothyroidism than single scanning[END_REF].

Some studies have used thyroxine intervention as confirmation of underlying

CHT but the threshold for intervention will vary with time and from clinician to clinician and centre to centre [START_REF] Oppenheimer | A therapeutic controversy. Thyroid hormone treatment: when and what?[END_REF]. Hence some babies with 'raised' TSH values and thyroid hormone values within the laboratory normal range will be treated whilst others will not [START_REF] Hall | Congenital hypothyroidism, seasonality and consanguinity in the West Midlands, England[END_REF][START_REF] Jones | Improvement in screening performance and diagnosis of congenital hypothyroidism in Scotland 1979-2003[END_REF]. 5. Ultimately, biochemistry is the most important parameter; a baby with a raised TSH but normal imaging will require thyroxine treatment whilst a baby with normal biochemistry but abnormal imaging will not.

Sample processing and analysis

During the period of the study the screening centre moved between the Royal Victoria (Table 1). ACS-180 v DELFIA, n=682). Revised cut off values were established by comparing the results using scatter charts and using least squares linear regression. There was a highly significant correlation between the 2 methods when the assay was changed from RIA to ACS-180 in 1998 (p<0.001; r 2 = 0.94) and when it was changed from ACS-180 to DELFIA in 2003 (p<0.001; r 2 = 0.99). 100% agreement for screening passes and failures was obtained using the revised cut-off values.

Values identified as being greater than 20mU/l by the radioimmunassay method, 10mU/l by the ACS method and greater than 6 mU/l by the DELFIA method were followed by analysis of a repeat blood spot from the screening card. They were deemed to be screening failures if the final blood spot value was again greater than 10mU/l (ACS) or 6 mU/l (DELFIA). All infants where the paediatrician was subsequently notified of an increased value and hence were a neonatal screening test failure were included in the analysis.

Statistical analysis

The yearly incidence of elevated TSH values in newborns was calculated as the number of cases in each year per 100,000 live births born in the Northern Region.

Temporal changes in incidence were assessed using Poisson regression. Seasonal variation in incidence was assessed using the Edwards test for seasonality [START_REF] Edwards | The recognition and estimation of cyclic trends[END_REF] with an adjustment for variable month length. A p-value of less than 0.05 was considered statistically significant. All statistical analyses were performed using the statistical software package Stata, version 9.0 (StataCorp, Texas).

Approvals for this study were obtained from the Newcastle and North Tyneside Local

Research Ethics Committee and the Patient Information Advisory Group for England and Wales.

Results

Between The number of cases by month of birth is reported in table 3. Despite peak number of cases in May and in the August-October period, there was no significant evidence of seasonal variation in the number of cases (p=0.16). Nor was there evidence of seasonal variation with sex-specific seasonality analyses (p=0.17 for females and 0.59 for males).

Discussion

Despite advances made in identifying genetic risk markers for CHT, there remains a great deal to be explained in terms of the aetiology of the disease [START_REF] Vassart | Thyroid dysgenesis: multigenic or epigenetic… or both?[END_REF]. This study showed an increasing incidence of elevated TSH values in newborns in the Northern Region of England between 1994 and 2005, but did not find evidence of seasonal variation in the number of cases. Many other studies have depended primarily on biochemistry including TSH rather than other investigations when making a diagnosis of CHT [START_REF] Harris | Increase in congenital hypothyroidism in New York State and in the United States[END_REF]. Given the high risk of subclinical hypothyroidism and morphological abnormalities in 'false-positive' patients [START_REF] Calaciura | Subclinical hypothyroidism in early childhood: a frequent outcome of transient neonatal hyperthyrotropinemia[END_REF][START_REF] Daliva | Three-year follow-up of borderline congenital hypothyroidism[END_REF] we suspect that our figures for raised TSH will be closely linked to the number of actual cases of CHT. Unfortunately we do not have detailed information on outcome in these children because they were managed in more than 10 different hospitals by an even greater number of clinicians.

More detailed data were therefore not available to allow us to assess changes in permanent CHT or to analyse the data with respect to different aetiologies.

An increasing temporal trend in incidence of CHT has recently been reported in New

York State [START_REF] Harris | Increase in congenital hypothyroidism in New York State and in the United States[END_REF], with a 138% increase between 1978 and 2005. Excluding New York

State, nationwide United States data suggest a 73% increase between 1987 and 2002 [START_REF] Harris | Increase in congenital hypothyroidism in New York State and in the United States[END_REF]. We observed a 151% increase in raised TSH values between 1994 and 2003.

However, this is in contrast to research conducted in Quebec, Canada, where no changes in incidence were seen over a 16 year period [START_REF] Deladoëy | Random variability in congenital hypothyroidism from thyroid dysgenesis over 16 years in Quebec[END_REF]. A real temporal trend, aside from changes in diagnostic procedures which can lead to increases in incidence [START_REF] Corbetta | A 7-year experience with low blood TSH cutoff levels for neonatal screening reveals an unsuspected frequency of congenital hypothyroidism (CH)[END_REF],

suggests either an increasing exposure to an environmental risk factor, or a changing distribution of risk factors among the population. The incidence rates in this study dropped slightly after 2003 and it remains to be seen whether this is a true decline or simply random variation.

The division between a 'screen positive' and 'screen negative' in a screening programme such as this is not linked to robust outcome measures and the screening threshold and management of cases with mild thyroid dysfunction varies between regions. We were keen to establish that the change in incidence was not simply a reflection of change in assay methodology or laboratory practice (as opposed to seasonality where we would not expect assay change to have the same impact). All The different assay methods were rigorously compared to ensure that there would be no difference in the number of cases identified as a result of the change. To this end a large number of samples were analysed and there was no difference in screening passes or failures with 100% concordance. It is of note that the increasing incidence in raised TSH values was most striking during the period when TSH was measured only by the ACS method although we suspect that this represents a true increase because the rise had commenced prior to the change in methodology and continued after this changed to the DELFIA assay. Many other studies of CHT or elevated TSH levels have encountered similar issues regarding data interpretation as assay methodology has changed [START_REF] Jones | Improvement in screening performance and diagnosis of congenital hypothyroidism in Scotland 1979-2003[END_REF].

In terms of changes in the population structure, two previous studies from England have reported an increased incidence of CHT among Asians [START_REF] Hall | Congenital hypothyroidism, seasonality and consanguinity in the West Midlands, England[END_REF][START_REF] Rosenthal | Congenital hypothyroidism: increased incidence in Asian families[END_REF]. However, the Northern Region of England has a population of 3.1 million, of which less than 2% are from ethnic minorities, with low levels of migration [START_REF]Social focus in brief: Ethnicity[END_REF]. Therefore, while the data are not available to assess directly, it is unlikely that the increased incidence is related to changes in population structure and with it changes in genetic risk factor profiles. Exposure to environmental factors such as chemicals or increasing levels of other risk factors such as maternal iodine deficiency or high prenatal iodine exposures [START_REF] Nishiyama | Transient hypothyroidism or persistent hyperthyrotropinemia in neonates born to mothers with excessive iodine intake[END_REF] or low birth weight infants [START_REF] Korada | TSH levels in relation to gestation, birth weight and sex[END_REF], may be suggested by an increasing temporal trend, whereas infections or seasonally varying dietary factors or chemical exposures, such as dioxin and polychlorinated biphenyl [START_REF] Miller | Results from the Lake Michigan Mass Balance Study: Concentrations and fluxes of atmospheric polychlorinated biphenyls and trans-nonachlor[END_REF], may be suggested by evidence of seasonal variation in the number of cases. The potential role of a suboptimal maternal iodine status in some parts of the North of England should be highlighted and clearly warrants further study [START_REF] M S Kibirige | Prevalence of maternal dietary iodine insufficiency in the north east of England: implications for the fetus[END_REF]. We found little evidence of seasonal variation of elevated TSH levels in newborns, in line with a number of previous reports of no seasonality [START_REF] Deladoëy | Random variability in congenital hypothyroidism from thyroid dysgenesis over 16 years in Quebec[END_REF][START_REF] Miyai | An analysis of the variation of incidence of congenital dysgenic hypothyroidism in various countries[END_REF][START_REF] Rosenthal | Congenital hypothyroidism: increased incidence in Asian families[END_REF][START_REF] Reijneveld | No evidence for seasonality of congenital hypothyroidism in The Netherlands[END_REF][START_REF] Henry | Screening for congenital hypothyroidism[END_REF]. In contrast a number of previous studies have reported seasonal variation in a number of different geographical areas [START_REF] Hall | Congenital hypothyroidism, seasonality and consanguinity in the West Midlands, England[END_REF][START_REF] Virtanen | Aetiology of congenital hypothyroidism in Finland[END_REF][START_REF] Miyai | An analysis of the variation of incidence of congenital dysgenic hypothyroidism in various countries[END_REF][START_REF] Nakamizo | Seasonality in the incidence of congenital hypothyroidism in Japan[END_REF][START_REF] Miyai | Further studies on episodic occurrence of congenital dysgenetic hypothyroidism in Osaka, Japan[END_REF], including a different part of England [START_REF] Hall | Congenital hypothyroidism, seasonality and consanguinity in the West Midlands, England[END_REF]. Gu et al also found sex-specific seasonal patterns of incidence in Japan [START_REF] Gu | Seasonality of congenital hypothyroidism in Japan: gender-specific patterns and correlation with temperature[END_REF]. However, sex-specific analyses also showed little evidence of seasonal variation in this study. The issue of statistical power should be considered when interpreting our results and it is possible that with a larger sample a seasonal effect may have been found. It is also possible that differences in findings may reflect differences in the underlying populations. Our sex-ratio (F:M) of 1.3:1 was significantly less than the sex ratio of 2.8:1 previously shown in a study from Scotland that used thyroxine prescription data as a surrogate for hypothyroidism in children and young people [START_REF] Hunter | Prevalence and aetiology of hypothyroidism in the young[END_REF] and less than the ratio of 2.1:1 reported for cases of 'true' CHT from the same country [START_REF] Jones | Improvement in screening performance and diagnosis of congenital hypothyroidism in Scotland 1979-2003[END_REF]. This underlines the importance of taking factors such as iodine status into consideration in future work.

In conclusion, we have observed a significant increasing trend in the incidence of elevated TSH levels in newborns, a surrogate for increasing levels of CHT, since 1994. Whilst the reasons for the increase are unclear, it would appear from this analysis that seasonally varying factors are not involved. It is also unlikely to be due to a change in the population distribution of genetic risk factors, although environmental determinants of genetic mutations and epigenetic factors cannot be ruled out. Further research is required into the potential environmental determinants of increased CHT risk.

  Infirmary (RVI), Newcastle and the University Hospital of North Durham (UHND), Durham. The screening blood spot TSH method and cut off value for screening failure also changed during the study from a manual radioimmunassay method (1994-March 1998) to an ACS-(Beyer) chemiluminometric assay (April 1998-February 2003) and then a DELFIA (perkin Elmer) fluoroimmunometric assay (March 2003-present)

  births in the study region were screened in a single centre at any one time. The physical location of this centre changed originally in April 1998 from Newcastle to Durham and again in April 2005 from Durham to Newcastle. The move in 1998 also corresponded to a change in the laboratory assay with a further change in assay in 2003.

Table 1 . TSH newborn screening base, assay methodology and cut-off values during the study period 1994-March 1998 April 1998-

 1

		March 2003-	April 2005-
	February	March 2005
	2003

Inter-assay coefficients of variation (CV)'s for blood spot TSH assays were 16.5%, 9.4% and 8.8% for the manual radioimmunassay method at TSH values of 22.0, 38.6 and 74.1 mU/L respectively. Inter-assay CV's for blood spot TSH assays were 7% and 6% for the ACS chemiluminometric assay at TSH values of 14 mU/L and 68 mU/L respectively and 11% and 12% for the DELFIA fluoroimmunometric assay at TSH values of 16 mU/L and 60 mU/L respectively. Nine replicates over 9 analytical runs were used to calculate inter assay precision for the ACS assay and 42 replicates over 10 analytical runs for the DELFIA assay.

Table 2 . Annual number of cases and incidence of elevated TSH levels on newborn screening in the Northern Region of England, 1994-2005. Year Number of cases Incidence per 100,000 live births

 2 1994 and 2005 inclusive, there were 213 cases of elevated TSH values in newborns in the Northern Region of England. The ratio of female to male cases was

	1.3:1. Over the study period, the average annual incidence was 59.94 per 100,000 live
	births. The annual number of cases of TSH elevation in newborns in the Northern
	Region and annual incidence per 100,000 live births are shown in table 2. Incidence
	increased significantly over the study period (p<0.0001) from 37 per 100,000 in 1994
	to a peak of 92.8 in 2003.
	1994	15	37.12
	1995	11	32.26
	1996	13	38.61
	1997	5	15.21
	1998	16	49.99
	1999	20	64.65
	2000	15	50.63
	2001	22	76.11
	2002	26	89.02
	2003	28	92.84
	2004	22	70.93

Table 3 . Number of cases of elevated TSH levels on newborn screening by month of birth in the Northern Region of England, 1994-2005.

 3

	Month	Number of cases
	January	14
	February	14
	March	14
	April	17
	May	25
	June	19
	July	16
	August	20
	September	23
	October	20
	November	16
	December	15