B W J M C Neil

N R Thompson

D J Dunning

B Sheehy

High harmonic attosecond pulse train amplification in a Free Electron Laser

Keywords: numbers: 41.60.Cr, 42.65.Ky, 42.65.Re

It is shown using three dimensional simulations that the temporal structure of an attosecond pulse train, such as that generated via High Harmonic generation in noble gases, may be retained in a Free Electron Laser amplifier through to saturation using a Mode Locked Optical Klystron configuration. At wavelengths of ∼12 nm, a train of attosecond pulses of widths ∼300 attoseconds with peak powers in excess of 1 GW are predicted.

Introduction

Attosecond pulses from High Harmonic generation in noble gases can been used to great effect in capturing and controling ultra-fast events in atomic and molecular physics [1,2]. While many techniques require single attosecond pulses, methods that utilise trains of attosecond pulses, as generated in the normal mode of operation of an HH source, have also been employed to both capture and control electronic processes within atoms and molecules [3,4,5,6,7,8] High-gain free-electron lasers (FELs) operating at similar wavelengths typically have longer pulse durations but offer peak powers several orders of magnitude greater than the HH sources. If a FEL is seeded by and amplifies a HH source, the initial HH power P 0 , is exponentially amplified as a function of distance through the FEL interaction region as P (z) ≈ P 0 exp(z/ √ 3l g ) where l g = λ u /4πρ is the nominal gainlength, ρ is the FEL scaling parameter that determines the FEL coupling strength and efficiency and λ u is the period of the undulator [9]. However, due to the relative propagation of the radiation pulses through the amplifying electron bunch, such amplification of the HH source tends to 'wash out' the HH attosecond pulse train structure [10].

Several techniques have been proposed to achieve attosecond pulse durations in FELs (see e.g. [11,12,[START_REF] Penn | Proceedings of the 31st International Free Electron Laser Conference[END_REF][START_REF] Ding | [END_REF][START_REF] Mcneil | Proceedings of the 32nd International Free Electron Laser Conference[END_REF]). The widths of the light pulses generated are limited by the coherence length of the FEL interaction l c = λ r /4πρ, where λ r is the resonant FEL wavelength [9], and pulse widths tend to have durations l c /c > ∼ 2fs at the ∼12nm wavelengths considered here.

An alternative method, that applies the concepts from mode-locked cavity lasers in FEL amplifiers, has also been proposed [START_REF] Thompson | [END_REF] that may generate a train of pulses with durations not restricted by the FEL coherence length l c . In this technique, a series of spatiotemporal shifts are introduced between the radiation and the co-propagating electron bunch that define a set of axial radiation modes. The spatiotemporal shifts are achieved by delaying the electron bunch using magnetic chicanes inserted between undulator modules. Introducing a modulation of the electron beam properties at the mode spacing periodically suppresses or enhances the FEL interaction at different longitudinal positions along the beam, so that a train of equally spaced pulses develops aligned with the regions of strongest FEL interaction. In the frequency domain this is equivalent to the modes developing sidebands that overlap neighbouring modes, allowing mode-locking to occur and is analogous with mode-locking in conventional cavity lasers [START_REF] Siegman | Lasers[END_REF].

The spectrum generated by the XUV mode-coupled SASE system of [START_REF] Thompson | [END_REF] has a modal structure with mode-spacing ∆ω similar to the spacing of harmonics in a HH source. Equivalently, the temporal structure of the two systems form an attosecond pulse train. By matching the spectral/temporal structures of a mode-coupled FEL amplifier to an HH seed, it may therefore be possible to amplify the HH seed to saturation while retaining its attosecond train structure. Furthermore, it should be possible to achieve this without the need for introducing a modulation at the mode spacing as the HH attosecond seed already comprises a pulse train structure. In this paper it is demonstrated via 1D and 3D numerical simulations that this is the case.

Mode generation in an amplifier

The spatio-temporal shifts that generate the equally spaced modal structure in the amplifier spectrum are achieved by periodically delaying the electron bunch using magnetic chicanes between undulator modules [START_REF] Thompson | [END_REF], as shown schematically in fig. 1.

The notation used here follows that of [START_REF] Thompson | [END_REF]. The total propagation distance of a radiation wavefront ahead of the electron bunch per module in units of the cooperation length l c is s = s/l c = l + δ, where l is the similarly scaled slippage occurring in the undulator and δ is that within the chicane. The scaled FEL radiation frequency is defined as ω = (ω -ω r )/2ρω r where ω r is the resonant FEL frequency. The slippage enhancement factor is defined as S e = s/ l. In [START_REF] Thompson | [END_REF], a solution is obtained for the one dimensional wave equation describing the field evolution of a small, constant electron source term for a series of N undulator-chicane modules. For S e > 1 the spectrum takes the form of a sinc-function envelope (the single undulator module spectrum) modulated by a frequency comb centered at the scaled resonant frequency ω, with mode separation ∆ω = 2π/s corresponding to ∆ω = 2π/T s , where T s = s/c = sl c /c is the time taken for radiation to travel the slippage length. The number of modes under the central peak of the spectrum is shown to be N 0 = 2S e -1.

The delay provided by a four dipole chicane is of the order of its R 56 . The HH pulses are separated by one half the drive laser wavelength i.e. ∼400nm, so the R 56 of the slippage enhancing chicanes are of this order. By comparison, a bunch compression chicane used prior to electron injection into the FEL has an R 56 typically of several cm, and the slippage enhancing chicanes placed between the undulator modules are seen to be weaker by several orders of magnitude. Any CSR disruption caused by these chicanes can therefore assumed to be negligibly small.

Amplification of an HH seed

The temporal profile of the HH seed is a comb of attosecond pulses separated by half the wavelength of the drive laser λ d and modulated by the drive laser power envelope. The spectral and temporal structure of the radiation generated by the undulator-chicane system is matched to that of the HH seed by setting s = λd /2, where λd = λ d /l c is the scaled drive laser wavelength.

For a typical FEL parameter of ρ = 2 × 10 -3 , similar to that for an FEL operating in the XUV, then for undulator modules of 8 periods, l = 0.201. The value of the scaled slippage generated by the chicanes is chosen to be δ = 0.616 so that s = λd /2, matching the undulator-chicane system to the spectral and temporal structure of the HH seed. This gives a slippage enhancement factor of S e = 4.0625.

The HH seed was modelled in a similar way to that described in [10] with a FEL resonant wavelength chosen to be the 65th harmonic of the drive laser wavelength of λ d = 805 nm (e.g. Ti:Sapphire)i.e. λ r = λ d /65. The HH field was constructed by first summing over a weighted harmonic series modulated by an envelope determined by the Gaussian drive laser centered at t = t 0 and of width σ d ≈ 3fs:

E HH ∝ exp - (t -t 0 ) 2 2σ 2 d h E h e -i(ω h (t-t 0 )+φ h ) (1)
where h = 1, 3, 5, ..., ω h = hω d are the odd harmonics of the drive laser frequency and φ h and E h represent the phases and amplitudes of individual harmonic components. In a typical HH spectrum, the harmonic yield drops quickly over the first few harmonic orders, then much more slowly over a 'plateau region', up to a wavelength and intensity dependent cutoff order, beyond which the yield drops precipitously. Both the amplitude of particular harmonics and their relative phases are sensitive functions of the experimental conditions. We consider here a somewhat idealized HH source while retaining its essential features. The field amplitude dependence is assumed to be of the form:

E h = η hm/hp p η m hp hm -hp exp - ln(η p /η m ) h m -h p h (2)
where h m = 95 is the maximum harmonic, h p = 11 is the harmonic where the plateau region begins, η p = 5 × 10 -6 is the harmonic yield efficiency of the HH process at the beginning of the plateau and η m = 5 × 10 -7 that at the maximum harmonic. We choose this form to approximate the slow decline in harmonic yield of one to two orders of magnitude between the beginning of the plateau and the cutoff that has been observed in experiments [START_REF] Huillier | [END_REF]. A different smoothly falling function or small variations of individual harmonic amplitudes from this dependence does not significantly alter the attosecond character of the seed. Most numerical FEL simulation codes, like those used in this paper, average both the electron motion and the radiation field locally within the pulses over a radiation wavelength. The Nyquist theorem then limits the bandwidth of frequencies that can be properly modelled to ω r /2 < ω r < 3ω r /2 [10] which, for the resonant harmonic h = 65 considered here, corresponds to an harmonic range 33 < h < 95, all of which fall within the harmonic plateau region. We include only these harmonic orders in the simulation.

The attosecond character of the seed is sensitive to variations in the relative phases, φ h , of the harmonics [19]; when these phases vary, the pulse train can have more structure than that of the 'clean' pulse train of fig. 2, in which it was assumed φ h = 0 for all harmonic orders. While we use the phase-locked seed of fig. 2 in our simulations below for clarity, we have also varied these phases and run the simulations using a more structured seed and find, unsurprisingly, that the amplified output also has more structure, although a well defined pulse train is retained.

The system is modelled using a 1D code similar to that described in [START_REF] Mcneil | Proceedings of the 28th Int. FEL Conference[END_REF]. The HH seed is injected into the amplifying undulator-chicane modules with a cold electron beam. Note that no energy modulation of the electron beam was used. This is different from the mode-locked SASE case of [START_REF] Thompson | [END_REF] which starts up from noise and requires a beam energy modulation at the mode spacing ∆ω to generate an isolated train of equally spaced pulses. Here, the HH seed is modelled as a set of in-phase modes, hence it consists of a train of equally spaced pulses in the temporal domain. It will be seen that while this temporal structure is retained, there occurs a broadening of the individual pulses. The pulse durations attained are in agreement with the scaling observed in [START_REF] Thompson | [END_REF] for the pulse width, τ p ∝ 1/ √ N 0 , where N 0 is the number of amplified modes. After 22 undulator-chicane modules the amplified scaled radiation power and spectrum is as shown in Fig. 3. It is seen that the HH seed is amplified and the comb of temporal spikes is retained, but with increased pulse widths as discussed. This number of modules corresponds to a scaled FEL interaction length of 22 × l ≈ 4.4. This is just prior to saturation where scaled powers of |A| 2 ≈ 1 are achieved. However, the temporal structure of the pulses begins to break up at saturation. (Note that the dispersive strength of the chicanes D = R 56 /2l c ≈ 10 δ/6 reduces the gain length as discussed in [START_REF] Thompson | [END_REF]).

Optimising the system

In [START_REF] Thompson | [END_REF] it is demonstrated that the pulse widths in mode-locked SASE simulations are proportional to 1/ √ N 0 , this is in agreement with an analogous relation from modelocked conventional cavity lasers.

The maximum bandwidth of the amplified radiation spectrum is that of the single undulator module spectrum, with the minimum number of optical cycles per pulse approximately equal to the number of undulator periods per module. In order to attain narrower amplified HH pulses, the bandwidth of the system (and hence number of modes amplified) is increased by reducing the number of undulator periods per module. To reduce l while continuing to satisfy the condition s = λd /2, the value of δ is increased to keep s constant. The system is simulated for a case with the number of undulator periods per module set to four. The corresponding value of the slippage enhancement factor is S e = 8.125. The spectral power distributions for the two cases S e = 4.0625 and S e = 8.125 are shown in Figs. 3 and4 respectively. It is seen that the width of the spectral envelope of the amplified radiation is increased with reduced undulator module length l. The number of modes, N 0 , under the central peak of the spectrum increases, narrowing the temporal duration of the pulses. The variation of the peak intensity and pulse width with scaled distance through the undulator are plotted in Fig. 5 for the two cases S e = 4.0625 ( l = 0.201) and S e = 8.125 ( l = 0.101). For both cases, there is an initial rapid broadening since only the modal content that falls under the envelope of the single undulator spectrum is amplified. The pulse durations continue to increase more slowly during further amplification. This slow pulse broadening effect is attributed to slippage in the undulators. For S e = 8.125, 34 undulator-chicane modules are required to produce the same amplification of the 22 module S e = 4.0625 case. This corresponds to a reduction in the total undulator length from 4.4 to 3.4, which can be attributed to more effective seeding since a broader spectral range of the seed is amplified, and reduced gain length due to an increased number of chicanes.

For larger values of S e , the broadening of the modal sinc-function envelope becomes restricted by the Nyquist limit.

Energy Spread

The effects of energy spread on the interaction have been investigated using the 1D model by introducing a relative gaussian energy spread of width σ γ /γ = ρ/10 = 0.02% for the case S e = 8.125. The results close to saturation are plotted in Fig. 6, which is the equivalent of Fig. 4. While the cold beam case took 34 undulator-chicane modules, the case here took 42 modules. Note that the temporal and spectral structure of the radiation power remains largely unaffected by the energy spread. Larger energy spreads have also been simulated with the main effect being that a larger number of modules are required to achieve saturation.

Results of 3D Simulations

The system has also been modelled using the three-dimensional FEL simulation code Genesis 1.3 [START_REF] Reiche | [END_REF]. The main results of the 1D simulations are reproduced using the 3D code and show amplification of an HH seed pulse retaining its pulse train structure. The Genesis simulation code models both the radiation field and the electron beam in 3D and requires careful matching of the electron beam to the undulator/chicane lattice structure which additionally contains an electron beam quadrupole focussing lattice to maintain good electron-radiation field coupling through the long structure. The focussing lattice was optimised for normal FEL interaction in the absence of the chicane fields and it was found that this set-up also worked well when the chicanes were included to generate the radiation modes matched to the HH seed. Similar parameters as for the XUV mode-coupled FEL amplifier of [START_REF] Thompson | [END_REF] were used (i.e. without electron beam modulation) including an energy spread of width σ γ /γ = 0.01%. The seed power and spectrum show the attosecond pulse train structure and frequency comb, as seen in Fig. 7. The peak power of the seed is 3 MW in an envelope of σ d ≈ 25 fs. (This peak power is a reasonable estimate of what can be expected from HH sources in the present to near future [START_REF] Marangos | NLS Project: Conceptual Design Report Section 10[END_REF] and can be reduced by an order of magnitude while still retaining a good amplified pulse train structure that dominates competing SASE. Methods that enhance HH yields by orders of magnitude have been demonstrated [START_REF] Kim | [END_REF]24], although the impact of these methods on the attosecond character of the source has not been investigated) It was assumed that the wavelength of the Ti:Sapphire drive laser was λ d = 780 nm and the FEL was then tuned to be resonant with the 63rd harmonic of the drive laser with λ r = 12.4 nm. The repetition rate of the seed pulses is the same as that of the drive laser which could be expected to rise from the current 10's Hz to ∼1kHz in the next few years [START_REF] Marangos | NLS Project: Conceptual Design Report Section 10[END_REF].

Each undulator module had 4 periods of 3.1 cm and each chicane had a delay of 28 radiation wavelengths corresponding to a slippage enhancement S e = 8. These parameters for the undulator-chicane modules then match the mode spacing of the HH seed. The amplified seed approaching saturation, after 47 modules, is plotted in Fig. 8 and the pulse temporal and spectral properties agree well with the equivalent one-dimensional S e = 8.125 case of Fig. 6. The FWHM length of each pulse in the amplified train is 300 as. Further 3D simulations (not shown) with S e = 4 agreed well with the S e = 4.0625 case of Fig. 3. As may be expected from an amplifier seeded using a source of good temporal coherence with power well above noise level, the temporal coherence length of the amplified output is found, though not shown here, to be retained with little reduction.

P [W]

3.0 x10

Conclusions

It was shown, using 1D and 3D simulations, that the attosecond structure of the HH seed can be amplified to saturation using a Mode Locked Optical Klystron FEL amplifier configuration of [START_REF] Thompson | [END_REF] to increase the peak power by a factor ∼300. No pre-conditioning of the electron beam (e.g. energy or current modulation) was required. The only requirement is for an undulator-chicane modular structure, the radiation spectrum of which matches an HH seed source.

The maximum bandwidth of the amplified radiation is critical to achieving the minimum pulse durations (maximum mode number) from the system, and is set by the single undulator module spectrum. Reducing the number of undulator periods per module therefore appears to be the most effective route to achieving the shortest possible pulse durations. The introduction of a modulation to the electron bunch at the modespacing may well restrict the slow pulse broadening effect observed and so allow shorter pulse durations than those reported here. This will be investigated in future work. The behaviour of the MLOK system when modelled in a code allowing a greater spectral bandwidth, and so shorter pulse lengths, such as that of [25] will also be investigated.

Figure 1 .

 1 Figure 1. Schematic of the mode-coupled amplifier FEL interaction operating with an HH seed. The inset shows detail of the electron delay.

Figure 2 .

 2 Figure 2. Scaled longitudinal intensity profile top and middle (detail) as a function of the scaled distance within the pulse z1 measured in cooperation lengths, and scaled spectral power distribution for the HH seed (bottom).

Figure 3 .

 3 Figure 3. Scaled longitudinal intensity profile (top and middle) and scaled spectral power distribution of the amplified HH radiation for l = 0.201 (S e = 4.0625), after 22 undulator-chicane modules.

Figure 4 .

 4 Figure 4. Scaled longitudinal intensity profile (top and middle) and scaled spectral power distribution of the amplified HH radiation for l = 0.101 (S e = 8.125), after 34 undulator-chicane modules.

8 Figure 5 .

 85 Figure5. Scaled peak intensity (top) and pulse width in units of l c (bottom) plotted against the scaled interaction length, z[START_REF] Thompson | [END_REF], for the two cases S e = 4.0625 ( l = 0.201) and S e = 8.125 ( l = 0.101). It is seen that by reducing the number of undulator periods per module, both the pulse duration and the gain length are reduced.

Figure 6 .

 6 Figure 6. Scaled longitudinal intensity profile (top and middle) and scaled spectral power distribution of the amplified HH radiation for l = 0.101 (S e = 8.125) and an initial energy spread σ γ /γ = 0.02 %, after 42 undulator-chicane modules.

Figure 7 .

 7 Figure 7. Longitudinal intensity profile (top) and spectral power distribution (bottom) of the HH seed in 3D simulations.

Figure 8 .

 8 Figure 8. Longitudinal intensity profile (top and middle) and spectral power distribution (bottom) of the amplified HH radiation in 3D simulations, with S e = 8. The agreement with the equivalent 1D simulations shown in Fig. 6 is very good.

Acknowledgments

This work received support from Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy.


C Lazarou

B M Garraway

J Piilo

S Maniscalco

Generation of entanglement density within a reservoir

Keywords: numbers: 03.67.Mn, 03.65.Yz, 03.67.Bg

Introduction

In recent years, entanglement has attracted the attention of many physicists working in the area of quantum mechanics [1,2]. This is due to the ongoing research in the area of quantum information [START_REF] Nielsen | Quantum computation and quantum information[END_REF], and also because of the advances made in different experimental disciplines, such as in ion traps [START_REF] Leibfried | [END_REF] and Bose-Einstein condensation [5,6]. Developments in the field of cavity QED, where experiments in the strong coupling regime are carried out [7,8] provide plenty of motivation for studying quantum information and entanglement. Theoretical studies are also important in the context of atom-light interactions inside structured reservoirs [9] such as resonant cavities or photonic band gap materials. The theoretically predicted atom-photon bound state could also lead to entanglement and this can also be linked to another problem: that of atom-laser out-coupling from Bose-Einstein condensates [10][11][12], where analogous effects were predicted in the past.

Besides entangling atoms with each other, or atoms with a few modes, or modes with each other, it is clear that even a single atom can become entangled with a reservoir comprised of modes. The issue we address here is how to describe this kind of entanglement. In the analysis of Ref. [13] the reservoir is treated (in a limiting process) as a collective object, and the entanglement between the atom and each reservoir mode, or between the modes, is not considered. It is also possible to partition the reservoir [14] and then study entanglement between these partitions using a Weisskopf-Wigner theory for a decaying two-level atom.

In the present work we study entanglement between the atom and the reservoir by means of an entanglement density which can apply to non-Markovian systems, too. In our model, an atom interacts with a continuum of modes at zero temperature [START_REF] Haroche | Exploring the quantum[END_REF][START_REF] Breuer | The theory of open quantum systems[END_REF], and entanglement properties between the atom and the modes and also between the modes are considered. Using global entanglement as a measure of entanglement, we derive a pair of distributions that can be interpreted as densities of entanglement in terms of all the reservoir modes. Both distributions can be calculated in terms of the spectrum of reservoir excitation. With these two new measures of entanglement we can study in detail entanglement between the atom and the modes, and also between the modes.

In our analysis, we consider a Lorentzian reservoir and cover different dynamic regimes. For strong coupling we observe the periodic collapses and revivals of entanglement between the atom and the reservoir, which are associated with Rabi oscillations of the atom. Eventually, all the population leaves the atom and the reservoir becomes entangled. More precisely, two bundles of modes are excited, forming a "Belllike" state. The method developed here, in terms of entanglement distributions, can also be used when considering various types of structured reservoirs. For example, reservoirs with a density of modes characterized by a band gap [9,[START_REF] Garraway | [END_REF]18] can be treated with this method.

The paper is organized as follows. In section 2 we introduce a measure of entanglement, the global entanglement, and apply it to our system-reservoir states when the number of degrees of freedom is finite. In section 3 we introduce the density of entanglement for the limit of a bath with an infinite number of modes. This entanglement measure evolves according to our model system as introduced in section 4.

In section 5 we present our key results for the density of entanglement and we conclude in section 6.

Measures of multi-partite entanglement

Identifying and measuring entanglement in multi-partite systems presents various complications. Apart from the case of a two-qubit system, where entanglement can be identified both for a pure and a mixed state [19,20], multi-qubit entanglement is an open problem and to date several measures of entanglement have been proposed [1,2,[21][22][23][24].

For the analysis that follows we will be using the global entanglement [23] since this will enable us to deal with many modes (or qubits). The physical problem we consider is that of entanglement between an atom and a reservoir, and in the context of this problem we will also consider the entanglement between different reservoir modes. To quantify this we will take a discrete bath of reservoir modes. Since we will assume only a single excitation in each bath mode, we can treat the bath states as a set of qubits for the purpose of computing the entanglement, and then later take a continuum limit. With just one excitation in total in the system, this excitation may be in the atom, or in the bath, so that the state vector at all times will have the form

|ψ(t) = c 0 |0 + c a (t)|ψ a + λ c λ (t)|ψ λ , (1)
where |0 is the vacuum state

|0 = |0 a ⊗ |0000 • • • 000 . (2)
The other two states |ψ a and |ψ λ , correspond to the atom being excited

|ψ a = |1 a ⊗ |000 • • • 000 , (3)
or the mode λ of the reservoir being excited

|ψ λ = |0 a ⊗ |000 • • • 01 λ 0 • • • 000 . (4)
Recently this kind of approach has been utilized for the study of decoherence and entanglement decay in systems with one or two excitations (effectively at T = 0) . Some of this work examines the decay of a single excitation in a reservoir made from a continuum of modes [29-31, 42, 52] and other works examine entaglement with two or more qubits, but most still utilize the decay of a single excitation as considered in the present paper [25-28, 32-41, 43-46, 48, 50, 51].

If we started with a system of N qubits in a pure state |ψ , the global entanglement is defined as

Q(|ψ ) = 2 - 2 N N i=1 trρ 2 i , (5)
where ρ i is the reduced density matrix for the i-th qubit. When |ψ is a pure product state then Q(|ψ ) = 0. If we take an entangled state for |ψ , such as the GHZ state

|GHZ N = (|000 • • • + |111 • • • )/ √ 2, (6)
then Q(|ψ ) = 1. This example gives a maximum value for the global entanglement which is normalized such that 0 ≤ Q(|ψ ) ≤ 1. Another, more relevant example is the W-state

|W N = N j=1 |0 • • • 01 j 0 • • • 0 / √ N , (7)
for which the global entanglement goes to zero as 1/N for large N

Q(|ψ ) = 4(N -1)/N 2 . ( 8
)
An important property of the global entanglement is that it is equal to a sum over two-qubit concurrences [2]. More specifically, for pure states |ψ(t) [Equation ( 1)] we find from Equation ( 5), see the appendix for details, that

Q(|ψ ) = 2 N + 1 C 2 (t), (9)
where C 2 (t) reads

C 2 (t) = N λ=1 c 2 (ρ aλ ) + 1≤λ<µ≤N c 2 (ρ λµ ). (10)
The concurrence c 2 (ρ aλ ) is that for the two-qubit (reduced) density matrix ρ aλ [19,20]

ρ aλ = tr µ =λ {|ψ(t) ψ(t)|} (11)
between the atom and the λ-mode. The quantity ρ µλ is the corresponding density matrix for the modes µ and λ:

ρ µλ = tr a,κ =λ,µ {|ψ(t) ψ(t)|} . (12)
The two-qubit concurrence c(ρ) is equal to [19,20]

c(ρ) = max{0, λ 1 -λ 2 -λ 3 -λ 4 }, (13)
where λ j are the eigenvalues of the matrix

R = ρ(σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), (14)
in decreasing order, i.e. λ 1 > λ 2 > λ 3 > λ 4 , and σ y is the relevant Pauli matrix.

Is is important to note here that C 2 (t) is exactly equal to the square of the norm of the concurrence vector [24], which is one of the many proposed measures for multi-partite entanglement. Because of the connection between C 2 (t) and the two-qubit concurrence Equation (10), we shall refer to C 2 (t) simply as the concurrence.

For the remainder of this work we will focus only on C 2 (t) and its properties since, with the exception of the normalization factor 2/(N +1) in Equation ( 9), it is equivalent to the global entanglement. Furthermore, we will be considering a continuum of reservoir modes, i.e. N 1. In this limit, as we see in the following section, one can define a density of entanglement for continuous systems.

Density of entanglement

In the limit of continuous distribution for the reservoir modes, the summations in Equation ( 10) can be converted into integrals over the density of modes ρ λ , i.e. λ → dω λ ρ λ .

(15)

If we take this limit, the concurrence C 2 (t) can be expressed as the sum of two separate parts which ultimately involve either atomic population or reservoir mode populations:

C 2 (t) = ∞ -∞ dω λ E A (ω λ , t) + ∞ -∞ dω λ ∞ -∞ dω µ E R (ω λ , ω µ , t). ( 16
)
The two distributions E A and E R will form the entanglement densities and are defined in terms of the two qubit concurrences to be

E A (ω λ , t) = c 2 (ρ aλ )ρ(ω λ ), (17)
and

E R (ω λ , ω µ , t) = 1 2 c 2 (ρ λµ )ρ(ω λ )ρ(ω µ ), (18)
where ρ(ω) is the reservoir density of modes. The interpretation for these two functions is now very simple.

The distribution E A (ω λ , t) is the density of entanglement between the atom and all reservoir modes in the vicinity of mode ω λ i.e. E A (ω λ , t)dω λ is the total entanglement between the atom and all modes in the frequency interval ω λ and ω λ + dω λ . In the same way E R (ω λ , ω µ , t)dω µ gives the entanglement between the mode ω λ and modes in the frequency range ω µ to ω µ + dω µ .

For the remainder of this work, E A will be referred as the atom-mode density of entanglement and to E R as the mode-mode density of entanglement. The global entanglement Q(|ψ ), Equation (9), can be calculated from these two densities of entanglement, and thus a number of entanglement properties can be studied in terms of these two distributions. To this end it is important to note an interesting property for the two distributions E A and E R . One can show that both entanglement distributions can be written in terms of the spectrum of reservoir excitation [57]

S(ω λ , t) = ρ(ω λ )|c λ (t)| 2 , (19)
as

E A (ω λ , t) = 4|c a (t)| 2 S(ω λ , t), (20)
and

E R (ω λ , ω µ , t) = 2S(ω λ , t)S(ω µ , t) . (21)
Thus, both entanglement distributions can be derived from the reservoir excitation spectrum. Having the definition for the entanglement density we can now introduce a Hamiltonian and dynamics to consider the time-dependent properties of entanglement for an atom coupled to a Lorentzian reservoir.

Dynamical model

The model system consists of a two-level atom coupled to a reservoir of harmonic oscillators with annihilation and creation operators a λ and a † λ respectively. Within the rotating wave approximation the Hamiltonian reads ( = 1)

H = λ ω λ a † λ a λ + ω 0 |1 aa 1| + λ g λ a † λ |0 aa 1| + a λ |1 aa 0| , (22)
where g λ is the coupling between the mode λ and the atomic transition

|1 a → |0 a .
The atomic transition frequency is ω 0 whereas the λ-mode frequency is ω λ . This model will preserve the assumption of a single excitation which is built into the system states in Equation (1).

For the purposes of the analysis that follows, it is very useful to introduce the reservoir structure function D(ω λ ) which reflects the properties of the density of modes [START_REF] Garraway | [END_REF]. This is defined through

ρ λ (g λ ) 2 = Ω 2 0 2π D(ω λ ), (23)
and is normalized such that

∞ -∞ dωD(ω) = 2π. (24)
With this normalization a measure of the overall coupling strength is Ω 0 which is given by

Ω 2 0 = λ (g λ ) 2 . ( 25
)
Utilizing these assumptions, and a state vector of the form given by Equation (1), the Schrödinger equation in an interaction picture yields

i d dt ca = λ g λ e -iδ λ t cλ , (26)
i d dt cλ = g λ e iδ λ t ca , (27)
with the detuning between the atomic transition and the mode λ being given by

δ λ = ω λ -ω 0 . (28)
The interaction picture amplitudes ca and cλ are connected to c a and c λ via a timedependent transformation ca (t) = e iω 0 t c a (t), cλ (t) = e iω λ t c λ (t).

Apart from numerical integration of equations ( 26) and ( 27), one could use analytical methods to derive the dynamics. Examples of these are: the resolvent method [9], the Laplace transform [58,59] and that of the pseudomodes [START_REF] Garraway | [END_REF]18,42,60]. This latter method applies when the spectral function is analytic with poles in the lower complex plane. Then equations ( 26) and ( 27) can be replaced, in the continuum limit, by a set of equivalent equations where the atom now couples to a finite set of fictitious modes. Each of these modes has a one-to-one correspondence to the poles of D(ω).

For concreteness we consider Lorentzian reservoir structure functions of width Γ and peak frequency ω c , i.e.

D(ω) = Γ (ω -ω c ) 2 + (Γ/2) 2 . ( 30
)
This reservoir function has a simple pole at

z = ω c -iΓ/2, (31)
and for this D(ω) equations ( 26) and ( 27) are equivalent to two new equations

i d dt ca (t) = Ω 0 b(t), ( 32
) i d dt b(t) = (∆ -iΓ/2) b(t) + Ω 0 ca (t), (33)
where the atom-cavity detuning ∆ = ω c -ω 0 , and b(t) = e -iω 0 t b(t) is the pseudomode amplitude. This set of equations can be associated with a master equation, where the physical interpretation for the pseudomode is that of a leaking cavity mode coupled to the atomic transition [START_REF] Garraway | [END_REF]. These coupled ODEs are straightforward to solve, and in particular, for the case of resonance, ω c = ω 0 , one finds for the atomic amplitude c a (t):

ca (t) = ca (0)e -Γt 4 cosh (αt/4) + Γ α sinh (αt/4) . (34)
The modified decay rate α is given by

α = Γ 2 -(4Ω 0 ) 2 (35)
in terms of Γ and Ω 0 . For strong couplings, 4Ω 0 /Γ 1, the atomic population oscillates between the atom and the reservoir with a slowly decaying amplitude. In the long time limit all the energy is lost to the reservoir. If we decrease the coupling Ω 0 , the atom dissipates its energy faster, and for 4Ω 0 /Γ 1 the atom exponentially decays into the reservoir. These two types of behavior are illustrated in Figure 1 which is given as a reference point for the entanglement discussion in the next section. The figures show the atomic population for Ω 0 = 10Γ and Ω 0 = 0.1Γ plotted against time t.

Evolution of reservoir entanglement

Entanglement generation by decay

The interaction between the atom and the reservoir results in entanglement creation between the atom and the reservoir modes, and also between the reservoir modes. This latter entanglement is indirect and is due to the effective coupling between the modes as a result of their interaction with the atom. When considering our measure of the total entanglement, the concurrence C 2 (t) as a function of time (Figure 2), we see that for both strong and weak coupling the concurrence builds up to a maximum value equal to C 2 (t) = 2. In the case of strong coupling, Figure 2(a), the concurrence reaches the maximum value very quickly, and then oscillates in a way which is closely connected to the Rabi oscillations seen in Figure 1. The oscillations return to the maximum value of two, with a minimum value that also approaches two as time increases. If we compare the concurrence to the population in Figure 1 we see that every minimum in concurrence is matched by a peak in population of the atomic state. Likewise the maxima in concurrence are matched by minima in the atomic population: it seems that the decaying atom is very efficient at generating entanglement in the reservoir. At this point we note that at long times the energy has left the atom (see the population Figure 1) and the system is in an approximate product state of unexcited atom and bath states. The large value of the concurrence at these times indicates the presence of entanglement in the bath. In the long time limit the concurrence reaches a steady value C 2 (∞) = 2, a result that can be derived analytically if one uses Eqs. ( 16) (20) and ( 21) and the spectrum for t → ∞. This limit for the concurrence is the same in the weak coupling case, see Figure 2(b); the atom decays exponentially and the entanglement reaches a steady state monotonically. In the long time limit the atom again disentangles from the reservoir and entanglement is distributed only between the reservoir modes.

By examining the entanglement densities [Eqs. [START_REF] Garraway | [END_REF]18)] we can gain further insight into where the entanglement resides in this system and how it evolves over time. In Figure 3 we show the atom-bath density of entanglement E A (ω λ , t) for both strong and weak couplings. The strong coupling case shows a complex behavior. First a central peak of entanglement appears in the vicinity of ω λ ≈ ω 0 ; note the peak at δ λ ≈ 0 for short times in Figure 3(a). The entanglement is then transferred to Rabi sidebands at δ λ = δ ± ≈ ±Ω 0 . The sideband entanglement oscillations seen take place at half the 16), ( 20) and (21).

frequency of the central peak entanglement oscillations. Ultimately all the entanglement decays at long times, E A (ω λ , t) → 0, as there can be no entanglement between bath and atom when the atomic population approaches zero. At that point the entanglement indicated by the concurrence in Figure 2(a) must reside in reservoir mode entanglement which is examined in the next section, Sec. 5.2. Figure 3(a) shows that as Rabi sidebands develop in the reservoir excitation, the atom-bath entanglement moves from central frequencies to the sidebands at δ λ = δ ± . The period doubling of the central peak is due to the oscillations of the excitation there, combined with the oscillations of atomic population. The population of the sideband modes is more stable, but oscillations of the atomic population result in oscillations of entanglement there, too.

For weak coupling, Figure 3(b), the entanglement briefly resides across the whole reservoir structure. This is essentially because the coupling of the atom is over this same range. In weak coupling, however, Rabi sidebands do not develop. Instead the final population of bath modes will be over a relatively narrow frequency range well within the model Lorentzian profile, Equation (30). Since both atomic, and mode population is needed for atom-bath entanglement, the narrow central frequency region is entangled only for a short while and then decays.

Entanglement between reservoir modes

As already mentioned, coupling the atom and the reservoir modes will induce an indirect coupling between the reservoir modes. Because of this, the modes will entangle and it is important to consider the properties of the mode-mode density of entanglement E R . In the long time limit one can easily obtain analytic expressions for the spectrum of excitation in the reservoir and from that calculate the density of entanglement for t → ∞. Using the definition for the spectrum of reservoir excitation Equation ( 19)) with the solution 34 and Equation (27), we find that for t = ∞

S(ω λ , ∞) = Ω 2 0 D(ω λ ) 2π • δ 2 λ + (Γ/2) 2 (δ 2 λ -Ω 2 0 ) 2 + (Γ/2) 2 δ 2 λ , (36)
= Ω 2 0 Γ/2 π[(δ 2 λ -Ω 2 0 ) 2 + (Γ/2) 2 δ 2 λ ]
,

where, as before, δ λ = ω λ -ω 0 [Equation (28)]. Taking Equation ( 36) together with Equation ( 21) it is straightforward to calculate the mode-mode density of entanglement

for t → ∞ E R (ω λ , ω µ , ∞) = Ω 4 0 Γ 2 2π 2 [(δ 2 λ -Ω 2 0 ) 2 + (Γ/2) 2 δ 2 λ ] (δ 2 µ -Ω 2 0 ) 2 + (Γ/2) 2 δ 2 µ . (37)
Figure 4 shows the mode-mode density of entanglement for several couplings and in the limit t → ∞. In the strong coupling case, Figure 4(a), we see the formation of four sharp peaks which signify the existence of strong entanglement between the two symmetric modes δ ± as a result of the Rabi splitting. This picture has a simple and rather intuitive interpretation: the final state for the reservoir has the form of a Bell-like state between the two symmetric modes δ ± ≈ ±Ω 0 . First we note that the final state takes the approximate form

|ψ(∞) = |0 a ⊗ N λ=1 (P + (ω λ ) + P -(ω λ )) |ψ λ . (38)
where the two probability distributions |P ± (ω λ )| 2 are centered at δ λ = δ + and δ - respectively. For Ω 0 Γ, i.e. in the strong coupling regime, their width is very small which practically means that only the two modes δ ± are excited and, for this, the reservoir state can be approximately described by a Bell state of the form

|ψ(∞) ≈ |0 a ⊗ |0 ω =ω 0 +δ ± ⊗ |1 + 0 -+ e iφ |0 + 1 -/ √ 2, ( 39
)
where φ is an arbitrary phase factor. This picture applies only at long times. For short times the mode-mode density of entanglement will initially have a distribution that peaks in the vicinity of ω λ ≈ ω µ ≈ ω 0 . As time evolves this distribution breaks into four symmetrical peaks seen in Figure 4(a), as a result of the Rabi splitting.

For moderate and weaker couplings, Ω 0 = Γ and Ω 0 Γ, the simple patterns observed in Figure 4(a) for the mode-mode density of entanglement due to the Rabi splitting disappear. For example in Figure 4(b) the density of entanglement for t → ∞ and Ω 0 = Γ is plotted. From this we can see that, for moderate couplings, although the distribution peaks around δ µ = δ λ = δ ± , the entanglement spreads over a wider range of reservoir frequencies. For even weaker couplings, Figure 4(c), only modes in the vicinity of ω λ ≈ ω 0 are excited and thus the entanglement is created only between modes in this frequency range. This is the reason for having a peaked entanglement distribution with a centre at ω µ = ω λ = ω 0 .

Conclusion

In this work we have presented a method for studying entanglement between a twolevel atom and a large reservoir of modes such as may be found in, for example, a lossy cavity field. Our aim was to study entanglement between the atom and the reservoir, and within the reservoir itself. To do this we restricted our interest to a simple case where the reservoir modes can be treated as qubits. Using global entanglement, we derived the entanglement densities E A and E R for the reservoir modes. These distributions have been given in terms of the well known two-qubit concurrences, and can be calculated from the reservoir excitation spectrum. The entanglement densities are then used both for studying entanglement between the atom and the reservoir modes, and also between the modes.

In considering different dynamical regimes defined in terms of the coupling strength, we noticed that when strong interactions occur, the reservoir modes are entangled in a "Bell-like" state in the long time limit. This long time limit is a regime where no excitation remains in the atom and all the entanglement is amongst the reservoir modes. Since there is no direct interaction between the bath-modes, see the Hamiltonian 22, the final entanglement arises through indirect interaction. Another way of viewing this is that in the strong coupling regime we have a non-Markovian system. In such a case, as the atom decays, some information resides in part of the reservoir in a way that it can be returned to the atom later [34]. This allows the indirect coupling between the reservoir modes which creates the entanglement. In the weak coupling (Markovian) case, the information does not return to the atomic system and the reservoir entanglement cannot be created.

In the transition from strong to weak coupling, the "Bell-like" state of Figure 4(a) coalesces into a single peaked structure as was seen in Figure 4(c). Based on the interpretation given above we would expect that the transition to a single peak takes place as the system becomes Markovian rather than non-Markovian. In principle this could be tested with a measure of non-Markovianity [61].

In conclusion, when considering entanglement between an atom and a large reservoir, the analysis can be formulated in terms of entanglement density functions. These distributions can be associated with the spectrum of reservoir excitation, a quantity that can be measured, for example, in cavity QED experiments. Although the model considered here, an atom coupled to a Lorentzian reservoir, is rather simple, it can be extended to consider more complicated reservoir structures such as model photonic band gaps. Potentially, one could consider generalizations for the density of entanglement, and also extend the model beyond the assumption of a single excitation in the system.

Figure 1 .

 1 Figure 1. The atomic population |c a (t)| 2 as a function of time for strong coupling Ω 0 = 10Γ (solid line), and weak coupling Ω 0 = 0.1Γ (dashed line). Both results were obtained from Equation (34). The initial system state in this and the following figures is defined by c a (0) = 1, c 0 = 0 and all c λ (0) = 0. The detuning ∆ = 0.

Figure 2 .

 2 Figure 2. The concurrence C 2 (t) as a function of time for different couplings Ω 0 = 10Γ (a) and Ω 0 = 0.1Γ (b). The result was obtained from Eqs. (16), (20) and(21).

Figure 3 .

 3 Figure 3. The density of entanglement E A (ω λ , t) between atom and bath is plotted for different times t and reservoir mode frequencies ω λ . The coupling strength is: (a) Ω 0 = 10Γ; and (b) Ω 0 = 0.1Γ.

Figure 4 .

 4 Figure 4. The density of entanglement E R (ω λ , ω µ , t) between bath modes for t → ∞. The coupling strength is: (a) Ω 0 = 10Γ; (b) Ω 0 = Γ; and (c) Ω 0 = 0.1Γ.

Acknowledgments

This work has been supported in part by the European Commission's ITN project FASTQUAST. S.M. acknowledges financial support from the Turku Collegium of Science and Medicine, The Emil Aaltonen foundation and the Finnish Cultural Foundation. J.P. acknowledges financial support from the Magnus Ehrnrooth Foundation and the Academy of Finland (project 133682).

Appendix A. Global entanglement

For the completeness of our analysis, we show here how the global entanglement for the pure state |ψ(t) , Equation (1), can be derived. First, the state |ψ(t) can be written as a pure state of N + 1 qubits, i.e.

where j = a for the atom, or j = λ = 1,

where |χ 0 is the N -2-qubit vacuum state

and |χ k is the N -2-qubit state with a single excitation, i.e.

With these definitions for |χ 0 and |χ k , the reduced density matrix takes the form

where the basis states, starting from top and moving to bottom, are

The concurrence c(ρ ji ), Equation (13), for this density matrix reads

In order to calculate the global entanglement, we derive the single qubit density matrix from Equation (A.5) by tracing out the i-qubit

where the basis states are |0 j and |1 j . Using this we first calculate ρ 2 j and then its trace

which after using probability conservation becomes

Substituting in Equation ( 5) for the global entanglement we have

Separating the atom-mode terms and the mode-mode terms, and taking into account the symmetry of the two-qubit concurrence c 2 (ρ ji ) = c 2 (ρ ij ), (A.12) it takes the form which leads to the result of Eqs. ( 9) and ( 10).


Marleen Elles

Maruschka Kemper
email: e.m.kemper@amc.nl

Peter Merkus

Petra C Wierenga

Désiree B M Van Rijn-Bikker

Loraine Van Der Werff

Lie

Martin Huen

Offringa

Elles ; Kemper

Petra ; Van Rijn-Bikker

Van Der Werff

E M Kemper

M Merkus

P C Van Rijn

D Van Der Werff

L Lie-A-Huen

M Offringa

E M Kemper

TOWARDS EVIDENCE BASED PHARMACOTHERAPY IN CHILDREN

Keywords: Evidence based Pharmacotherapy, Off label, Regulation, Research infrastructure

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

SUMMARY

In daily practice is it difficult to find a registered drug for children, because about 70% of the drugs prescribed in children are not studied, off-label or unlicensed in this age group. Clinical trials have usually been performed in adults and then in daily practice dosages are adjusted for children without proper studies in that age group. In some countries national formularies are being established to overcome the existing variance in prescribing between physicians.

Complicating factors in finding the correct dosage for children include the heterogeneity between different age groups in the developmental stages of the organs influencing the absorption, distribution, metabolism and excretion as well as differences in body composition during growth. Growth may also influence the effects and adverse effects of a drug used in a child. For oral administration of drugs in children, the bioavailability, the taste, composition and the absence of toxic ingredients for that age group are additional important factors. The EU has recently introduced legislation to stimulate the pharmaceutical industry to investigate the pharmacological effect and safety of new medicines in children. In response to this legislation research networks are being established to provide the optimal infrastructure for paediatric drug investigation. The goal of this paper is to review the current problems in daily practice and to address the needs for evidence based pharmacotherapy in children.

INTRODUCTION

Before any medicine is approved for use in adults, the product must have undergone extensive testing including pre-clinical tests and clinical trials to ensure that it is safe and effective [START_REF] Caldwell | Clinical trials in children[END_REF].

The same is frequently not be true for medicines used to treat children. It is a major concern that about 70% of the medicines used in the care of children are not studied, off-label or unlicensed in one or more of their age groups. This often leaves no alternative to the prescriber than to use an unlicensed drug (for example a syrup made by the local pharmacy with a different formulation than originally licensed) or to use a drug off-label (a licensed drug used outside the license in terms of age, indication, route of administration, dose or contraindications) [START_REF] Conroy | Survey of unlicensed and off label drug use in paediatric wards in European countries. European Network for Drug Investigation in Children[END_REF][START_REF] Jong | Unapproved and off-label use of drugs in a children's hospital[END_REF][START_REF] Roberts | Pediatric drug labeling: improving the safety and efficacy of pediatric therapies[END_REF][START_REF] Cote | Drug development for children: the past, the present, hope for the future[END_REF]. In newborns this percentage of thus unlicensed and off-label use of medicines is even higher, up to 80-90% (2;3).

In the past children have suffered from serious toxic effects due to exposure to inadequately investigated medication. An example of such harm in children is the 'grey baby syndrome' after the administration of chloramphenicol in newborns, because a deficiency in glucuronide conjugation (the main pathway of chloramphenicol metabolism) in the first few days of life caused elevated and prolonged plasma concentrations of chloramphenicol [START_REF] Weiss | Chloramphenicol in the newborn infant. A physiologic explanation of its toxicity when given in excessive doses[END_REF]. Another example is the use of verapamil to treat infants with supraventricular tachycardia resulting in bradycardia, apnoea and cardiac death [START_REF] Garson | Medicolegal problems in the management of cardiac arrhythmias in children[END_REF]. In addition for some drugs higher drug doses are required in children in order to achieve the same target plasma concentration as in adults. E.g. children need a higher infusion rate of propofol compared to adult patients due to a higher clearance expressed as mg/kg/min in children compared with adults (8;9). Another issue is the absence of suitable, approved medicinal products to treat young children and infants, i.e., age appropriate formulations. Pharmaceutical companies have often been reluctant to invest in the reformulation of existing medicines for infants and young children. This is mainly because the market is small and therefore of lower commercial interest and studies can be difficult, long and expensive. Furthermore, a reason why pharmaceutical companies are reluctant to conduct research in children, and particular in neonates, may be the fear of litigation if there is an adverse outcome.

Thus there is still a gap between the performed research and the available evidence based medicines in paediatrics compared to those in adults [START_REF] Cohen | An absence of pediatric randomized controlled trials in general medical journals, 1985-2004[END_REF]. To address this lack of paediatric research, the United States and Europe have enacted legislation to encourage the investment of pharmaceutical companies in clinical trials involving children. In the field of paediatric anaesthesia there is an increasing number of publications, however, there are still issues which should be investigated. An example is the lacking information on the safety and pharmacokinetics of prolonged use of opioids in the mechanically ventilated newborn [START_REF] Bellu | Opioids for neonates receiving mechanical ventilation[END_REF], or the use of benzodiazepines. Another example is the use of anaesthetic agents and neuronal apoptosis, i.e. neonatal rats exposed to ketamine suffered widespread neuronal apoptosis and long-term memory deficits (12;13). The applicability of extrapolating rodent data to the care of human neonates continues to be debated (14;15), but should be taken into account in their usage. However, studies to evaluate those compounds would be difficult to perform due to the many confounding factors in these situations and the additional ethical and emotional aspects associated with studies in children.

The above often leaves no alternative to the prescriber than to use products in children without evidence-based information for a proper risk-benefit assessment. However, the Deleted: extremely expensive Deleted: , with difficulties to point a certain effect to the study drug and most likely unwillingness of ethical committees to endorse such studies. specifically commented on the fact that a practitioner should use a drug in a child if the indication is the same as for an adult as a child. In particular they suggested that it would be immoral to deprive a child of the benefits of the medication [START_REF]Uses of drugs not described in the package insert (off-label uses)[END_REF].

In this review we discuss the current challenges facing the pharmacotherapy in children and briefly review the regulatory and legal aspects.

LACK OF EVIDENCE BASED PHARMACOTHERAPY

A. Dosing

In daily practice, health care providers experience various challenges with regard to the prescription and the administration of medicines to children and the major area of difficulty is neonates and infants. In drug prescribing the physician often relies on different paediatric textbooks, the internet, personal hand-held devices, online formularies, and other sources, and differences exist between countries, hospitals and even between physicians in the same institution. Due to the lack of age and formulation specific research in children, the prescribing physician often relies upon the evidence from studies in adults, while effectiveness and adverse effects may be significantly different for children compared to adults [START_REF] Cramer | Children in reviews: methodological issues in child-relevant evidence syntheses[END_REF].

Children may differ from adults in physiology and disease pathophysiology which in turn may result in a difference in the pharmacodynamic and pharmacokinetic profile of a drug. These differences in profile may be a result of the various developmental stages of a child that influence the absorption, distribution, metabolism and clearance of medicines (1;18). The glomerular filtration capability of the kidney reaches its peak between 6 and 12 months after birth and this should be taken into account when prescribing medicines depending on renal clearance (19;20). An example of the effect of the immature kidney is the prolonged clearance of d-tubocurarine in neonates compared to infants and adults [START_REF] Fisher | Pharmacokinetics and pharmacodynamics of d-tubocurarine in infants, children, and adults[END_REF].

Due to maturation of phase I (e.g. cytochrome P450 families) and phase II (e.g. glucuronide conjugation) enzymes in the liver the metabolism of many drugs is reduced in newborns. This may result in reduced excretion and increased potential for toxicity and adverse effects (22;23). For example, a smaller dosage of fentanyl and morphine is needed in newborns, because of a prolonged clearance due to undeveloped metabolism of respectively, CYP3A4 and glucuronide conjugation (24;25). However, with morphine and fentanyl, other issues also play a role. Bolus dosing is determined by the volume of distribution and this is reduced for morphine in term postoperative neonates, so the bolus dose should be reduced in that situation [START_REF] Bouwmeester | Developmental pharmacokinetics of morphine and its metabolites in neonates, infants and young children[END_REF], but it is increased in premature ventilated neonates [START_REF] Anand | Morphine pharmacokinetics and pharmacodynamics in preterm and term neonates: secondary results from the NEOPAIN trial[END_REF] and the dose should then be increased. Maintenance dosing and infusion are determined by the clearance. A higher per kilogram bolus fentanyl dose is usually better tolerated in neonates than adults because the distribution volume is increased and redistribution rapid.

Another factor is the blood brain barrier function, which improves gradually with age, possibly only reaching maturity by full-term age and with small molecules thought to access foetal and neonatal brains more readily than adult brains. Kernicterus, for example, is more common in preterm neonates than in full-term neonates.

When prescribing drugs, especially to newborns and infants, also the pharmacodynamic response can be different from the response in adults. Even in older children the pharmacodynamic response may differ from adults and an example is the paradoxical reaction of midazolam (restlessness, excitement and agitation) when administered to children [START_REF] Massanari | Paradoxical reactions in children associated with midazolam use during endoscopy[END_REF]. The minimal alveolar concentration (MAC) for almost all vapours is also higher in infants than neonates or adolescents [START_REF] Lerman | Anesthetic requirements for halothane in young children 0-1 month and 1-6 months of age[END_REF][START_REF] Ledez | The minimum alveolar concentration (MAC) of isoflurane in preterm neonates[END_REF][START_REF] Taylor | Minimum alveolar concentration of desflurane and hemodynamic responses in neonates, infants, and children[END_REF]. The lack of evidence to guide prescribing and the unawareness of the role of developmental pharmacology in children may also lead to ineffective treatment. An example is the inadvertent but systematic underdosing of HIV infected children in the United Kingdom and Ireland with antiretrovirals. In their paper the authors identify three serious issues in prescribing antiretrovirals, which are relevant to paediatric prescribing in general: inadequate dosing before the incorrect recommendation at licensing was revised, incorrect guidance regarding dosage by weight or surface area, and lack of dose adjustment with growth [START_REF] Menson | Underdosing of antiretrovirals in UK and Irish children with HIV as an example of problems in prescribing medicines to children, 1997-2005: cohort study[END_REF].

To overcome the existing prescribing practice variance, and to provide adequate existing information, physicians and pharmacists have worked together in different countries to establish national formularies (32;33). These formularies are based on the available evidence in the literature combined with best-practice guidelines and opinions and experience of paediatric experts. In the process of preparing these national formularies, the gaps in knowledge have become apparent, and the positive result is that a problem driven research agenda has been developed.

B. Choosing route of administration

Besides prescribing the correct drug and dose, the formulation and route of administration are also important. Since oral solutions and suspensions are often preferred in newborns and young children, nurses have to measure small volumes accurately and mathematic errors can easily occur.

Especially with the administration of a concentrated solution, a small mistake in volume will result in a large dosage variation. This also is important when administrating drugs intravenously to children and Allegaert et al showed that adult vials of the antibiotic amikacin for paediatric use may result in dose inaccuracy and that a paediatric formulation improves dosing accuracy of amikacin [START_REF] Allegaert | Impact of a paediatric vial on the magnitude of systematic medication errors in neonates[END_REF]. However, it is important to emphasize that confusion and administration errors may occur when two formulations of a drug are available in a hospital as stressed for IV paracetamol formulations [START_REF] Allegaert | Not all intravenous paracetamol formulations are created equal[END_REF]. To prevent dosing errors in small children it is very important to have protocols for the preparation of drugs administered to small children and the need to at least double check the dosage with the patient.

Another concern is the need for a child friendly oral formulation appropriate for the child's age or the ability of the child to ingest solid dosage formulations. Moreover, children often refuse anything that does not taste or smell good to them. For adults the original taste and smell of the drug can be masked by film coating the tablet or by placing the 'offending' ingredients in a capsule. However, due to the lack of commercially available liquid formulations for children, hospital and community pharmacies often reformulate medicines designed for adults to a suitable liquid formulation or develop a new formulation from raw material [START_REF] Schirm | Lack of appropriate formulations of medicines for children in the community[END_REF].

When designing a (re)formulation it is important that all ingredients are compatible with each other and suitable for children. Besides the active ingredient, excipients (i.e. substances added to confer a suitable consistency or form to a drug such as bulk fillers, sweeteners, preservatives, coloring agents, surfactants, anti-oxidants, etc.) could lead to adverse drug events, altered absorption and/or bioavailability, or other symptoms [START_REF] Pawar | Issues in the formulation of drugs for oral use in children: role of excipients[END_REF]. This (re)formulation creates its own safety issues since absorption may be negatively affected by the excipients. A further concern is that the preservative is free of toxic adverse effects.

For example, benzyl alcohol as a preservative in intravascular flush solutions has been associated with a number of deaths and intraventricular hemorrhage in low-birth-weight infants [START_REF] Hiller | Benzyl alcohol toxicity: impact on mortality and intraventricular hemorrhage among very low birth weight infants[END_REF]. Hemolysis, central nervous system depression, hyperosmolality and lactic Unfortunately, despite the necessity of proper formulations for children in different age groups, the scientific literature has paid little attention to the formulation of drugs; only 37% of studies performed in children younger than 12 years old provided sufficient information about the formulation [START_REF] Standing | Poor formulation information in published pediatric drug trials[END_REF]. This is likely to be considered company-sensitive information, but information about the global contents of the excipients may still be useful.

C. Risks of off label use of medicines

In paediatric healthcare about 70% of the drugs used are either unlicensed or off-label and this percentage is even higher in newborns; up to 80-90% (2;3). These percentages are much higher than the off-label use of medication in adults, reportedly on average approximately 21% and with extremes for anticonvulsants (74%), antipsychotics (60%) and antibiotics (46%) (41;42). This is of course also a matter of concern as companies are evidently not performing trials in adults on new indications of their registered drug. Based upon the agedependent changes in pharmacokinetics and pharmacodynamics and, therefore, different mechanisms of drug metabolism, children, like the elderly, are considered a vulnerable group of patients in terms of patient safety. The use of unlicensed and off label drugs in these high risk groups of patients may result in unwanted adverse drug reactions (ADRs) [START_REF] Gill | Adverse drug reactions in a paediatric intensive care unit[END_REF][START_REF] Turner | Adverse drug reactions to unlicensed and offlabel drugs on paediatric wards: a prospective study[END_REF][START_REF] Horen | Adverse drug reactions and off-label drug use in paediatric outpatients[END_REF][START_REF] Neubert | The impact of unlicensed and off-label drug use on adverse drug reactions in paediatric patients[END_REF][START_REF] Schirm | Reported adverse drug reactions and their determinants in Dutch children outside the hospital[END_REF].

Although, the relation between ADRs and the prescription of unlicensed or off-label drugs is not yet well established, studies have suggested a 2 to 5 fold increase in risk for ADRs when those drugs are administered to children. Safety is also an issue in drug trials in children. A recent literature review showed that only 13 (2%) of 739 paediatric trials had a safety monitoring committee [START_REF] Sammons | Safety in paediatric clinical trials--a 7-year review[END_REF]. Five hundred twentythree (71%) trials reported adverse drug reactions (ADRs) and 151 (20%) of these trials reported a serious ADR. In about 11% of the trials a moderate or severe ADR occurred. The authors recommend that all future paediatric clinical trials should have a safety monitoring committee.

Full reporting of ADR's is of paramount to give insight in the actual adverse effects of medication, in adults and children. The national formularies make recommendations on how to monitor ADR's in children when prescribing unlicensed and off-label drugs. Paediatric professionals and also pharmacists need to stay alert for possible harm by an ADR and they need to be encouraged to report all ADRs to national pharmacovigilance centers. Through the collection and analyses of such data the extent of these ADRs can become apparent, and means to prevent them in the future can be developed. There is a growing societal awareness of medication safety for groups at risk, including children. Hospital management is responsible for the availability of a proper medication safety system where physicians, nurses and hospital pharmacists are encouraged to work together in a multidisciplinary team to prevent harm in their patients.

REGULATION

The lack of information on paediatric drugs has been caused, at least in part, that clinical trials in children are difficult, take long, are costly, and even are unethical. To improve and speed up the development of safe and effective medication for children, the legislators in the US and in the EU have enacted paediatric regulations (49;50). The implications are manifest. For each new drug that is developed in the European Union, the manufacturer is obliged to develop a Paediatric Investigation Plan (PIP); in return they will be rewarded with a six-month patent extension (51). The goal is to achieve this without children undergoing unnecessary studies, or delay in the registration of medicines in adults.

However, the FDA then requires the pharmaceutical industry to carry out that plan to the satisfaction of the FDA in order to obtain the six-month patent extension. This patent extension only applies to studies of drugs that have the same indication in children as in adults. It does not apply to paediatric-specific trials of drugs that are aimed at particular and unique paediatric populations. Most recently, the European ERA-NET PRIOMEDCHILD (work Package 5) project identified 216 research topics as key research priorities for medicines for children (52). The next step is to allocate public money as well as industry resources to this agenda.

In the USA there is already about ten years of experience with paediatric legislation. The 4). However, so far a substantial number of these studies have not yet been published [START_REF] Benjamin | Peer-reviewed publication of clinical trials completed for pediatric exclusivity[END_REF], but it is now required to publish the results on the FDA website.

An overall examination of the paediatric studies performed under this new legislation showed that the types of drugs studied tended to mirror those most commonly used by the adult market rather than drugs commonly used in children [START_REF] Boots | Stimulation programs for pediatric drug research--do children really benefit?[END_REF]. This reflects the lack of a genuine problem driven paediatric research agenda.

The National Institute of Child Health and Human Development (NICHD) and the National Institutes of Health (NIH) encourage studies in children. The Pediatric Research Equity Act and the Best Pharmaceuticals Act both earmarked money to the NIH to encourage study of drugs that were no longer patent protected, i.e., generic drugs. The European Medicines Agency, however, as stated on its website "installed paediatric-use marketing authorisation (PUMA) as a new type of marketing authorisation. It may be requested for a medicine which is already authorised, but no longer covered by intellectual property rights (patent, supplementary protection certificate), and which will be exclusively developed for use in children. This type of marketing authorisation will cover the indication and appropriate formulation for the paediatric population. A paediatric-use marketing authorisation will benefit from 10 years of market protection as a reward for the development in children."(51) Although the EU has followed the US to enact paediatric legislation there are still existing differences between the regulation of the EU and US. For pharmaceutical companies it is difficult to comply to both the requirements of the EU as the US and therefore harmonisation between the regulations would be an enormous step forward. Conducting clinical trials in children is more complex than in adults (Table 1). The target population of children suffering from a specific disease is generally smaller than in adults, different age categories need to be considered, specific drug formulations for different age categories are needed, accepted endpoints and validated outcome assessment tools are lacking and children should be studied in environments appropriate for children. Furthermore, the consent process is more complex and it is often difficult to convince parents of the importance of conducting research even in newborns and assuring them that the utmost will be done to protect the safety of their child (55;56). The consent process will be even more complex when emergency situations have to be studied, as for example the treatment of supraventricular tachycardia. It is therefore questionable if a prospective study would have prevented the bradycardia, apnoea and cardiac death after treatment of verapamil, the example already mentioned in the introduction and reported in case reports [START_REF] Garson | Medicolegal problems in the management of cardiac arrhythmias in children[END_REF]. European networks, investigators and centres with specific expertise in the performance of studies in the paediatric population. This aims to coordinate studies, to build up the necessary competencies at the European level, to increase cooperation and to avoid duplication of studies.

3.RESEARCH INFRASTRUCTURE/NETWORKS

CONCLUSION

There is an increasing awareness of the importance of evidence based medicine in paediatric anesthesia. However, good research into the safety and efficacy of medicines for children is still highly needed and is a shared responsibility of health care providers, pharmaceutical industry and parents. Standardization in the design of paediatric trials will contribute to the development of a methodologically valid and relevant evidence base for paediatric care. All future paediatric clinical trials should have a safety monitoring committee. Adequate reporting of the studies' findings and updating of existing national formularies will enable health care providers to translate the highly needed information into rational prescribing in their daily practice. Thank you for reviewing our manuscript entitled "Towards Evidence based pharmacotherapy in children." We are very grateful for the very quick review and the comments by this reviewer. It is obvious that this reviewer put an enormous effort in the details of our review to improve it.

The comments are incorporated throughout the text. In appendix 1 our responses to the comments of the reviewers are listed in detail. The changes in the manuscript are again done with track changes. The references are inserted with Reference Manager. We have chosen "Peadiatric Anaesthesia" as output style and hopefully the references are correctly this way.

We would like to resubmit our paper for reconsideration for publication in Paediatric Anaesthesia.

However, not without sincerely thanking the reviewers for the major improvements they made in this manuscript! With kindest regards, on behalf of all authors, Marleen Kemper [START_REF] Cramer | Children in reviews: methodological issues in child-relevant evidence syntheses[END_REF][START_REF] Kearns | Developmental pharmacology--drug disposition, action, and therapy in infants and children[END_REF][START_REF] Dewoskin | Renal clearance parameters for PBPK model analysis of early lifestage differences in the disposition of environmental toxicants[END_REF][START_REF] Rhodin | Human renal function maturation: a quantitative description using weight and postmenstrual age[END_REF][START_REF] Fisher | Pharmacokinetics and pharmacodynamics of d-tubocurarine in infants, children, and adults[END_REF][START_REF] Weiss Cf | Chloramphenicol in the newborn infant. A physiologic explanation of its toxicity when given in excessive doses[END_REF][START_REF] Johnson | The development of drug metabolising enzymes and their influence on the susceptibility to adverse drug reactions in children[END_REF][START_REF] Santeiro | Pharmacokinetics of continuous infusion fentanyl in newborns[END_REF][START_REF] Anand | Morphine pharmacokinetics and pharmacodynamics in preterm and term neonates: secondary results from the NEOPAIN trial[END_REF][START_REF] Bouwmeester | Developmental pharmacokinetics of morphine and its metabolites in neonates, infants and young children[END_REF][START_REF] Massanari | Paradoxical reactions in children associated with midazolam use during endoscopy[END_REF][START_REF] Lerman | Anesthetic requirements for halothane in young children 0-1 month and 1-6 months of age[END_REF][START_REF] Ledez | The minimum alveolar concentration (MAC) of isoflurane in preterm neonates[END_REF][START_REF] Taylor | Minimum alveolar concentration of desflurane and hemodynamic responses in neonates, infants, and children[END_REF][START_REF] Menson | Underdosing of antiretrovirals in UK and Irish children with HIV as an example of problems in prescribing medicines to children, 1997-2005: cohort study[END_REF][START_REF]Britisch National Formulary for Children)[END_REF], not glucuronide conjugation. We have to apologize for our statement that bupivacaine is metabolized by glucoronide conjugation. This information was generated from the Thompson Micromedex database. However, in the bupivacaine product information of Astra Zeneca, the metabolism of bupivacaine by CYP3A4 is mentioned as the main metabolic pathway of this drug as also stated by the reviewer. As this was not mentioned in the manuscript, we did not have to make changes.

I enjoyed reading this current revision.

There remain a few minor considerations.

1. I think you have to be careful equating unlicensed or off-label with not studied. We are in a bit of a predicament with this comment, because the other, previous reviewer insisted on changing unlicensed and off-label into not studied, with the argument that (completely) unlicensed medication is nt the case. In principle, we agree on the change requested by the present reviewer and we changed "not studied" in "not studied, unlicensed or off-label" in the summary and in the first paragraph of the introduction.

Please use drug concentrations rather than levels with refering to amount in blood or serum or plasma.

The document was searched for "level" and on page 3 "plasma levels of chloramphenicol" was changed in "plasma concentrations of chloramphenicol".

p15 L12 ore=or

This was changed on page 3 (P15 for the reviewer).

4. p16 L40. I agree that investigations of neuronal aptosis are difficult. However the quandry is being approached from 2 angles..a) a study looking at GA vs regional techniques in babies b) reviewing national data banks comparing neonates who had surgery with those who did not. I don't think you can say that ethical committeees will be unwilling to endorse such studies. To not do so could be argued unethical! Perhaps you should simply say that such studies are difficult etc We have deleted "with difficulties to point a certain effect to the study drug and most likely unwillingness of ethical committees to endorse such studies and this sentence is now: "However, studies to evaluate those compounds would be difficult to perform due to the many confounding factors in these situations and the additional ethical and emotional aspects associated with studies in children."

P17 L34 Efficacy is commonly confused with effectiveness. Efficacy in pharmacology is the max response on a dose or concentration response curve.

We agree on this and "efficacy was replaced by effectiveness" on p5 (for the reviewer page 17) line 34. The document was searched for "side effects" and this was changed in "adverse effect" on page 2, 5, 6, 8, 10 and in the table on page 15 (P14, P17, P18, P20, P22 and P27 for the reviewer).

Side effects =adverse effects

7. P18 L6 Please define CYP the first time it is used. Perhaps this line could read Due to maturing phase 1 (e.g. cytochrome P450 families) and phase II (e.g. glucuronide conjugation)... Line 6 on page 6 (P18 for the reviewer) was changed in: "Due to maturation of phase I (e.g. cytochrome P450 families) and phase II (e.g. glucuronide conjugation)"

glucuronide conjugation rather than glucuronidase

On page 6 glucuronidase was changed in glucoronide conjugation twice. 9. P18 L40 can differ=may differ. I would construct a second sentence rather than use and as a link...The minimal alvweolar concen...is also higher... "Can differ" was changed in "may differ" and we split the sentence as suggested by the reviewer. " The minimal alveolar concentration (MAC) for almost all vapours is also higher in infants than neonates or adolescents [START_REF] Lerman | Anesthetic requirements for halothane in young children 0-1 month and 1-6 months of age[END_REF][START_REF] Ledez | The minimum alveolar concentration (MAC) of isoflurane in preterm neonates[END_REF][START_REF] Taylor | Minimum alveolar concentration of desflurane and hemodynamic responses in neonates, infants, and children[END_REF].

10. P19 L 44. omit "causing a relative increase or decrease in assumed bioavailability"...It is dosing that is inaccurate; bioavailability is unchanged..."a paediatric formulation improved dosing accuracy of amikacin..." We agree, of course! On page 7 (P19 for the reviewer) the sentence "causing a relative increase or decrease in assumed bioavailability" was omitted and we added "accuracy" between "dosing" and "of".

P21 L14 the last phrase ...information about the global contents of the excipients may still be useful.

We have deleted "if the preparation procedures and the exact amounts of each constituent are not provided companies can still inform properly about the global contents of the excipients." and replaced it by "information about the global contents of the excipients may still be useful."

P21 L28 21% (xx)...no reference provided. Actually I thought the figure was around 50% for adults

We apologize for leaving out these references. We are referring to two publications: [START_REF] Radley | Off-label prescribing among office-based physicians[END_REF] verapamil, magnesium). Genetic polymorphisms that affect P-glycoprotein-related genes may explain differences in CNS-active drug sensitivity.

It may be simpler to say that the BBB changes with age and use kernicteris as an example.

The reviewer provides detailed knowledge and examples on the immature aspects of the BBB in neonatesand children, but subsequently suggests not to incorporate that information in the manuscript. We agree witht that suggestion and only replaced: "However, the underdeveloped blood brain barrier in neonates should also be taken into account with these compounds, because lipid insoluble compounds such as morphine as compared to fentanyl more easily pass the underdeveloped blood brain barrier and should therefore be reduced in dosage." into: "Another factor is the blood brain barrier function, which improves gradually with age, possibly only reaching maturity by full-term age and with small molecules thought to access foetal and neonatal brains more readily than adult brains. Kernicterus, for example, is more common in preterm neonates than in full-term neonates."

I think this a very useful paper that readers will find helpful and I appreciate all the efforts the authors have undertaken.

5 American

 5 Academy of Pediatrics in conjunction with the Food and Drug Administration have

  However, the underdeveloped blood brain barrier in neonates should also be taken into account with these compounds, because lipid insoluble compounds such as morphine as compared to fentanyl more easily pass the underdeveloped blood brain barrier and should therefore be reduced in dosage. ¶ Deleted: However, also Deleted: can Deleted: and also t

  reported after intravenous administration of propylene glycol, commonly used in parenteral medication[START_REF] Arulanantham | Central nervous system toxicity associated with ingestion of propylene glycol[END_REF].

  Deleted: if the preparation procedures and the exact amounts of each constituent are not provided companies can still inform properly about the global contents of the excipients.

  th of January in 2007, the European Commission enacted 'the EU Paediatric Regulation 1901/2006 on Medicinal Products for Paediatric Use'. The purpose of this new legislation is to improve the healthcare of children in Europe by providing data on the efficacy and safety of new agents and to support the development and availability of medicines. It will ensure that these products are of high quality and approved.

  introduction of the Food and Drug Administration Modernization Act (1997), the Best Pharmaceuticals for Children Act (2002), the Pediatric Research Equity Act (2003) and the Pediatric Medical Device Safety and Improvement Act (2007) resulted in a dramatic increase in the number of paediatric studies performed from a few dozen before 1997 up to more than

  To address these problems, properly resourced research infrastructures are needed. Recently, national research networks have been established or are evolving in several European countries, including the United Kingdom (Medicines for Children Research Network (MCRN)), The Netherlands (MCRN), Germany (the German Paediatric Research Network, PAED-net), Finland (Finnish Investigators Network for Pediatric Medicine,Finpedmed), France (Réseau d'Investigations Pédiatriques des Produits de Santé, RIPPS, translated into English The Investigation Network for Paediatric Health Products) and Belgium (Belgian Pediatric Drug Network, BPDN). These networks provide advice and support on all aspects of trial-design, patient recruitment, data collection and management involving trials of medicines for children.

  Regulation requires the European Medicines Agency (EMA) to coordinate the establishment of a European Paediatric Clinical Trials Network of existing national and

•

  Need to investigate different age groups • Specific formulation requirement • Late-onset adverse effects • Small numbers of patients • Lack of accepted endpoints & validated assessment tools • No research infrastructure • Convincing parents of the importance of paediatric research

  Author Bupivacaine clearance is through CYP3A4 (Mazoit JX Clin Pharmacokinet 2004; 43(1):

  Radley DC, Finkelstein SN, Stafford RS. Off-label prescribing among office-based physicians. Arch Intern Med 2006 May 8;166(9): 1021-1026. (42) Stafford RS. Regulating off-label drug use--rethinking the role of the FDA. Consequently, we might expect greater initial concentrations of morphine in neonates than in adults )or from meperidine), resulting in more pronounced respiratory depression in the former. Respiratory depression, measured by carbon dioxide response curves or by arterial oxygen tension are similar from 2 to 570 days of age at the same morphine blood concentration. The BBB theory in this particular circumstance lacks strong evidence. It is more likely that the increased neonatal respiratory depression after morphine is due to pharmacokinetic age-related changes.The BBB may have impact however, in other ways. Small molecules are thought to access foetal and neonatal brains more readily than in adults. BBB function improves gradually, possibly reaching maturity by full-term age. Kernicterus, for example, is more common in preterm neonates than in full-term neonates. In contrast to drugs bound to plasma proteins, unbound lipophilic drugs passively diffuse across the BBB equilibrating very quickly. This may contribute to bupivacaine's propensity for seizures in neonates. Decreased protein binding, as in the neonate, results in a greater proportion of unbound drug that is available for passive diffusion. In addition to passive diffusion, there are specific transport systems that mediate active transport. Pathological CNS conditions can cause BBB breakdown and alter these transport systems. Fentanyl is actively transported across the BBB by a saturable ATP-dependent process, while ATP-binding cassette proteins such as P-glycoprotein actively pump out opioids such as fentanyl and morphine. P-glycoprotein modulation significantly influences opioid brain distribution and onset time, magnitude and duration of analgesic response. Modulation may occur during disease processes, fever, or in the presence of other drugs (e.g.

	We inserted "the pharmaceutical industry".
	14. The BBB is an interesting example of differences between the adult and neonate.
	Unfortunately, the example you use of morphine to illustrate a leaky BBB is based on a poor
	paper by Way (Clin Pharmacol Ther 1965;6:454-61). Respiratory depression after morphine
	was greater than that after meperidine. This difference was attributed to greater brain
	concentrations of morphine because of the poorly developed BBB in the neonate. It was
	postulated that BBB permeability to water-soluble drugs such as morphine, change with
	maturation. However, the neonatal respiratory depression observed after morphine could
	have been explained by pharmacokinetic age-related changes. For example, the volume of
	distribution of morphine in term neonates 1-4 days (1.3 L/kg) is reduced compared with that
	in infants 8-60 days of age (1.8 L/kg) and in adults (2.8 L/kg). Meperidinne V is increased in
	neonates.
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	N Engl J Med
	2008 Apr 3;358(14): 1427-1429.
	We have clarified this part to: "reportedly on average approximately 21% and with extremes
	for anticonvulsants (74%), antipsychotics (60%) and antibiotics (46%) (41;42)"
	P22 L47 effectuated = enacted
	Enacted was inserted instead of effectuated.
	13. P23 L 16 requires the pharmaceutical industry to carry out...