MD Joost A M Labout

MD, PhD Liesbeth Duijts

Ankie Lebon

Ronald Groot

MD Albert Hofman

MD, PhD Vincent V W Jaddoe

MD, PhD Henri A Verbrugh

PhD Peter W M Hermans

MD, PhD Henriëtte A Moll
email: h.a.moll@erasmusmc.nl

MD, PhD Ronald De Groot

Risk factors for otitis media in children with special emphasis on the role of colonization with bacterial airway pathogens: the Generation R study

Keywords: odds ratio (OR), confidence interval (CI), adjusted odds ratio otitis media, risk factors, Streptococcus pneumoniae, Haemophilus influenzae and Moraxella catarrhalis

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

Abstract

Objective Acute otitis media is the most frequent diagnosis in children visiting physicians' offices. Risk factors for otitis media have been widely studied. Yet, the correlation between bacterial carriage and the development of otitis media is not entirely clear. Our aim was to study in a population-based prospective cohort the risk factors for otitis media in the second year of life with special emphasis on the role of colonization with Streptococcus pneumoniae, Haemophilus influenzae and Moraxella catarrhalis.

Patients and methods

The study was embedded in the Generation R Study. Data on risk factors and doctor-diagnosed otitis media were obtained by midwives, hospital registries and postal questionnaires in the whole cohort (n=7295). Nasopharyngeal swabs were obtained at the age of 1.5, 6 and 14 months in the focus cohort (n=1079).

Results

Of these children, 2515 (47.2%) suffered at least one period of otitis media in their second year of life. The occurrence of otitis media during the follow-up period in the first 6 months of life and between 6 and 12 months of age was associated with the risk of otitis media in the second year of life (aOR, 1.83 95% CI 1.24-2.71 and aOR 2.72, 95% CI 2.18-3.38, respectively). Having siblings was associated with an increased risk for otitis media in the second year of life (aOR 1.42, 95% CI 1.13-1.79). No associations were found between bacterial carriage in the first year of life and otitis media in the second year of life.

Introduction

Acute otitis media is one of the most common childhood infections and the leading cause for children to visit a doctor. ( 1

Patients and methods

Design

This study was embedded in the Generation R Study, a population-based prospective cohort study from fetal life until young adulthood. The Generation R Study has been described in detail previously. [START_REF] Jaddoe | The Generation R Study Biobank: a resource for epidemiological studies in children and their parents[END_REF][START_REF] Jaddoe | The Generation R Study: Design and cohort profile[END_REF] The cohort includes 9778 mothers and their children living in Rotterdam, the Netherlands. Detailed assessments of fetal and postnatal growth and development were conducted in 1,232 Dutch pregnant women and their children in the Generation R Focus Study. The Medical Ethics Committee of the Erasmus Medical Centre, Rotterdam, has approved the study. Written informed consent was obtained from all participants.

Measurements and outcome

We obtained data on birth weight, gestational age and gender from midwives and hospital registries. Information about maternal smoking, siblings, educational level of the mother, day care attendance, breast-feeding, pacifier use and otitis media in the first year of life was obtained by postal questionnaires at the infants' age of 6, 12 and 24 months. Mothers were asked whether their children suffered from fever in the prior period (no, yes), and subsequently whether this period of fever was accompanied by earache and whether they had visited a doctor. The diagnosis otitis media in the period from 0 to 6 months of age and the period from 6 to 12 months of age was defined as having at least one period of fever accompanied by earache for which a doctor was visited during any part of the 6 months before reaching the age of respectively 6 and 12 months. Likewise, the diagnosis of otitis media in the second year of life was defined as having at least one period of fever accompanied by earache for which a doctor was visited during any part of the year before reaching the age of 24 months. Furthermore, otitis-proneness was defined as having four or more episodes by 1 year of age, six or more by 2 years of age, or placement of myringotomy tubes, similar to the definition used by Faden et al. [START_REF] Faden | Nasopharyngeal flora in the first three years of life in normal and otitis-prone children[END_REF] Risk factors for otitis media in the second year of life were studied in the whole cohort.

In the Focus cohort, samples were taken at the infants' age of 1.5, 6, and 14 months to detect nasopharyngeal bacterial carriage. Nasopharyngeal samples were taken with rayon tipped dacron pernasal swabs (Copan Italia, Brescia, Italy) and transported in Amies transport medium and plated within 6 hours of sampling on a blood agar plate with 5% sheep blood, a chocolate agar plate and a Heamophilus selective agarplate. The plates are kept at 35 C in a CO 2 rich environment for two days. Bacterial growth was determined daily. All bacteria were determined by use of standard methods.

Analyses

In the whole cohort associations of birth weight, gestational age, gender, maternal smoking, siblings, educational level of mother, day care attendance, duration of breast-feeding, pacifier use and otitis media in the first year of life with otitis media in the second year of life were assessed by logistic regression models resulting in odds ratios (OR) with their 95% confidence interval (CI).

Additionally, a multivariate logistic regression model was performed to assess the association of each risk factor separately with otitis media in the second year of life was adjusted for all the other risk factors resulting in adjusted odds ratios (aOR). In the focus cohort association of bacterial carriage in the first year of life with otitis media in the second year of life was assessed for the airway pathogens S. pneumoniae, M. catarrhalis and H. influenzae. The analyses were performed using frequency of specific bacterial carriage in the first year of life, and at ages of first bacterial carriage. Furthermore, we did the same analysis with the three bacteria grouped as any airway pathogen. Additionally, regression models were adjusted including all determinants presented in Table 1. Furthermore, possible interaction between otitis media in the first year of life and bacterial carriage and the risk of otitis media in the second year of life was tested. According to the national Dutch vaccination policy all children were vaccinated against H.

Results

In the Generation R Cohort, 7295 children, with a delivery date from April 2002 until January 2006, participated in the postnatal phase of the study. Data on otitis media in the second year of life was available in 5323 (response rate 73%) children. Characteristics of the children of the whole cohort and their association with otitis media in the second year of life are presented in Table 1. Having siblings is associated with an increased risk of otitis media in the second year of life (aOR 1.42, CI 1.13-1.79). Furthermore, the occurrence of otitis media during the follow-up period between 0 to 6 months and the occurrence of otitis media during the follow-up period between 6 to 12 months of age was associated with the occurrence of otitis media in the second year of life (respectively, aOR 1.83, CI 1.24-2.71 and aOR 2.72, CI 2.18-3.38).

Bacterial carriage was studied in the focus cohort, in which 1079 children, with a delivery date from February 2003 until August 2005, participated. The children were planned to visit the Generation R Focus Study research centre at the age of 1.5 month (response rate 81.8%), 6 months (response rate 81.6%) and 14 months (response rate 80.0%). Of the infants 379 (43.3%) had at least one episode of otitis media in the second year of life.

In Table 2 the associations between the frequency of bacterial carriage in the first year of life and otitis media in the second year of life are presented. We analysed the association between bacterial carriage in the first year of life and otitis media in the second year of life in two different ways. First, we defined carriage by adding the three sampling time points and assessed whether the frequency of bacterial carriage was associated with otitis media. Second, we assessed whether early age of bacterial carriage was associated with otitis media (data not shown). Both analyses were done for the bacteria separately and for any airway pathogen. No associations were observed. In addition to this, we assessed whether bacterial carriage in the first year of life was associated with otitis-proneness. Again, no associations were observed (data not shown).

Possible interaction between otitis media in the first year of life and bacterial carriage was tested. The interaction term was not significant. In Table 3 we present the associations between otitis media in the first year of life (6-12 months) and otitis media in the second year of life stratified for different bacterial carriage states in the first year of life. Bacterial carriage did not significantly change the increased risk for otitis media in the second year of life in case previous otitis media (6-12 months) was registered.

Discussion

We observed a positive association between having siblings and otitis media. Also, we observed a tendency that low education of the mother might attribute to the risk for otitis media, while breast-feeding might be a protective factor. No associations were observed between birth weight, gestational age, gender, maternal smoking, day care attendance or pacifier use and otitis media in the second year of life. In a meta-analysis of risk factors for acute otitis media by Uhari et al., day care attendance was shown to be the most significant risk factor. ( 13) Furthermore, similar to the trend we observed, Uhari et al. showed a positive association between siblings and otitis media and a negative association between breast feeding and otitis media. This study, in contrast to our results, also showed that parental smoking and pacifier use was associated with otitis media.

We observed that otitis media in the first year of life is a risk factor for otitis media in the second year of life. Corbeel has stated that the first period of otitis media determines the risk for recurrence because inflammation causes dysfunction of the Eustachian tube in young children due to the small caliber and the horizontal direction and is, therefore, predisposing for a high risk for recurrent of otitis media. ( 14)

Next, we analyzed in the focus cohort the association between bacterial carriage and otitis media in the second year of life, and otitis-proneness. No associations were observed between frequency of bacterial carriage, age of first bacterial carriage or the different species of bacterial carriage and otitis media or otitis-proneness. In addition, we observed no role for bacterial carriage in recurrent otits media. Bacterial carriage in the first year of life is not associated with otitis media later on. In other studies the frequency of carriage with different bacteria was shown to be associated with otitis media. (4, 5, 7, 8) Faden et al.

have shown that carriage of S. pneumoniae, H. influenzae and M. catarrhalis was associated with otitis-proneness in 17 otitis prone children vs 17 non-otitis prone children, during healthy periods and during periods of otitis media, in children aged 0-36 months in. The association in healthy periods was only significant for H. influenzae carriage but not for the other two bacteria studied. [START_REF] Faden | Nasopharyngeal flora in the first three years of life in normal and otitis-prone children[END_REF] In another study in 157 infants, an association between carriage of H. influenzae, measured at 13 routine visits in the first year of life, and otitis-proneness was shown. [START_REF] Harabuchi | Nasopharyngeal colonization with nontypeable Haemophilus influenzae and recurrent otitis media. Tonawanda/Williamsville Pediatrics[END_REF] In another study in 306 children aged 0 to 24 months, an association between M. catarrhalis and otitis-proneness was observed. [START_REF] Faden | Epidemiology of Moraxella catarrhalis in children during the first 2 years of life: relationship to otitis media[END_REF] In this population they also showed an association between the frequency of carriage of the three airway pathogens and the frequency of otitis media episodes. ( 4 Taken together, it appears that the presence of newly acquired pathogenic bacteria in combination with (viral) upper respiratory tract infections increases the risk for otitis media, rather than the presence of pathogenic bacteria itself increases the risk for otitis media.

To appreciate the results some limitations of our study had to be considered. In contrast to other studies, our definition of otitis media was based on parental reported questionnaire data, we do not have a doctor verified diagnosis of otitis media. We observed that 49 (7.2%) of the children had at least one period of otitis media in their first six months of life, 247 (31.3%) had otitis media in their second six months of life and 379 (43.3%) of the children had at least one period of otitis media in their second year of life. 511 (71.8%) of the children had at least one period of otitis media by the age of 24 months. This might be an overestimation of the number of children experiencing otitis media, resulting in less contrast between children with and without otitis media. This might explain that we did not find significant risk factors for otitis media. On the other hand, children with fever and earache were only classified as otitis "yes" if they visited a doctor. To our opinion this makes our definition of otitis media reliable.

Conclusion: In our study, otitis media in the first year of life is an independent risk factor for otitis media in the second year of life. Surprisingly, bacterial carriage in the first year of life did not add to this risk. Moreover, no association was observed between bacterial carriage in the first year of life and otitis in the second year of life.

16.

Revai K, Mamidi D, Chonmaitree T.

( 4 - 8 )

 48 ) Most children experience at least one episode of otitis media; in 50-85% of the children acute otitis media is diagnosed at least once in the first 3 years of life. (2, 3) Risk factors for otitis media have widely been studied. In addition to host factors like birth weight, gestational age and craniofacial abnormalities, environmental factors like passive smoking, day care attendance, socioeconomic status, pacifier use and breastfeeding have been studied. Frequency of colonization with bacterial airway pathogens was shown to be associated with otitis media in the first years of life. Streptococcus pneumoniae, Haemophilus influenzae and Moraxella catarrhalis are the most frequently cultured pathogens in acute otitis media. Early age of colonization seems to increases the risk of otitis media. (4, 8, 9) Syrjänen et al. have suggested that carriage of newly acquired bacteria is associated with otitis media. (10) Our aim was to study in a population-based prospective cohort the risk factors for otitis media in children with special emphasis on the role of colonization with S. pneumoniae, H. influenzae and M. catarrhalis in the first year of life.

  influenzae B, but non of the children were vaccinated against pneumocci Differences of infant characteristics between infants with and without data on otitis media in the second year of life were assessed by the independent sample t-test for continuous normal distributed variables, non-parametric Mann-Whitney test for continuous non-normal distributed variables and the chi-square test for categorical variables. Differences of frequency of otitis media in the second year of life between groups with missing risk factors and complete data were assessed in the same way. All tests were carried out using a two-sided alpha level of 5%. The statistical analyses were performed using the Statistical Package of Social Sciences version 11.0 for Windows (SPSS Inc, Chicago, IL, USA).

  ) Prellner et al. did not show an increased risk for otitis media when S. pneumoniae, H. influenzae or M. catarrhalis were present. (15) Early age of first colonization was also shown to be associated with the risk for otitis media. (4, 8, 9) But, Syrjänen et al. have shown in their study on the temporal association between pneumococcal carriage and otitis media that not preceding pneumococcal carriage state, but newly acquired carriage is associated with otitis media. (10) Their data support the hypothesis that viral respiratory infection might enhance the acquisition of pneumococci. Recently, Revai et al. have shown an association between presence of pathogenic bacteria in the nasopharynx during upper respiratory tract infections and the risk for otitis media following the upper respiratory tract infection. (16)

Table 1 .

 1 Association of nasopharyngeal 324 bacterial colonization during upper respiratory tract infection and the 325 development of acute otitis media. Clin Infect Dis 2008;46(4):e34-7. 326 327 Population descriptive. , pacifier use (n=1717), otitis from 0 to 6 months (n=1722) and otitis from 6 to 12 months (n=815). No data was missing on gender. OR univariate model, aOR full model. Birthweight entered in analyses in Kg. *p<0,05, **p<0,01, ***p<0,001

	328

Table 2 .

 2 Association between bacterial carriage in the first year of life and otitis media in the second year of life. Otits media 12-24 months of age Or univariate model, aOR full model including all risk factors as shown in table 1.

		n (%)	no (n=384) yes (n=287)	OR	aOR
	Streptococcus pneumoniae
	never (reference)	316 (47,1)	47,1%	47,0%	reference	reference
	once	241 (35,9)	35,4%	36,6%	1,04 (0,74-1,45) 0,64 (0,35-1,17)
	2 or more	114 (17,0)	17,4%	16,4%	0,94 (0,61-1,45) 0,72 (0,33-1,57)
	Moraxella catarrhalis
	never (reference)	354 (52,8)	54,7%	50,2%	reference	reference
	once	240 (35,8)	33,1%	39,4%	1,30 (0,93-1,81) 1,57 (0,88-2,80)
	2 or more	77 (11,5)	12,2%	10,5%	0,93 (0,56-1,54) 0,91 (0,38-2,15)
	Haemophilus influenzae
	never (reference)	381 (56,8)	58,3%	54,7%	reference	reference
	1X	224 (33,4)	31,5%	35,9%	1,22 (0,87-1,69) 1,08 (0,60-1,96)
	once	66 (9,8)	10,2%	9,4%	0,99 (0,58-1,68) 1,16 (0,44-3,08)
	Airway pathogen
	never (reference)	136 (20,3)	20,6%	19,9%	reference	reference
	once	163 (24,3)	25,3%	23,0%	0,94 (0,59-1,50) 0,58 (0,25-1,35)
	2 or more	372 (55,4)	54,2%	57,1%	1,09 (0,73-1,63) 0,86 (0,40-1,84)

Table 3 .

 3 Association between otitis media in the first year of life (6-12 months) and otitis media in the second year of life stratified by carriage state of different bacteria.

Acknowledgements

The Generation R Study is conducted by the Erasmus Medical Center, Rotterdam in close collaboration with the School of Law and Faculty of Social Sciences of the Erasmus University Rotterdam, the Municipal Health Service Rotterdam area, the Rotterdam Homecare Foundation and the Stichting Trombosedienst & Artsenlaboratorium Rijnmond (STAR), Rotterdam. We gratefully acknowledge the contribution of general practitioners, hospitals, midwives and pharmacies in Rotterdam. The first phase of the Generation R Study was made possible by financial support from the Erasmus Medical Center, Rotterdam, the Erasmus University Rotterdam and the Netherlands Organization for Health Research and Development (Zon Mw). The present study was supported by an additional grant from Stichting W.H. Kröger, Rotterdam (Grant No. 00-048), and AGS Kinderstichting.


Dominick J O Mcintyre

Franklyn A Howe

Christophe Ladroue
email: c.ladroue@warwick.ac.uk

Fiona Lofts

Marion Stubbs

John R Griffiths

Can Localised 19 F magnetic resonance spectroscopy pharmacokinetics of 5FU in colorectal metastases predict clinical response?

Keywords:

or by combining it with different modulators such as leucovorin to enhance effectiveness. In addition, combination of 5FU with newer chemotherapies (for example irinotecan or oxaloplatin) has significantly improved the response rates (see for instance this recent review [START_REF] Segal | Evolving Treatment of Advanced Colon Cancer[END_REF]). However, still only 40-50% of patients respond to 5FU treatment, and the search for new therapeutic strategies is continuing.

After injection, 5FU is rapidly metabolised by two main pathways: the major endproduct is fluoro--alanine (FBAL), which is non-toxic; a smaller proportion is metabolised to toxic 5-fluoronucleotides (FNuct), primarily in tumour cells but also to a lesser extent in normal liver cells, which express low levels of the required enzymes.

5FU treatment is uniquely suited to monitoring by 19 F magnetic resonance spectroscopy (MRS) for several reasons: because the MRS sensitivity of fluorine is very high, because there is no natural background signal, and because MRS can distinguish the parent drug from its active and inactive metabolites in-vivo and with time resolution adequate for pharmacokinetic analysis of the drug metabolism.

In the search for predictive biomarkers of response to 5FU treatment, several clinical studies have been performed [START_REF] Presant | Association of intratumoral pharmacokinetics of fluorouracil with clinical response[END_REF][START_REF] Findlay | Measurement of plasma 5-fluorouracil by high-performance liquid chromatography with comparison of results to tissue drug levels observed using in vivo 19F magnetic resonance spectroscopy in patients on a protracted venous infusion with or without interferon[END_REF][START_REF] Wolf | Tumor Trapping of 5-Fluorouracil: In vivo 19F NMR Spectroscopic Pharmacokinetics in Tumor-Bearing Humans and Rabbits[END_REF][START_REF] Wolf | Non-invasive F-19-NMRS of 5fluorouracil in pharmacokinetics and pharmacodynamic studies[END_REF][START_REF] Wolf | 19F-MRS studies of fluorinated drugs in humans[END_REF]. It has been reported [START_REF] Presant | Association of intratumoral pharmacokinetics of fluorouracil with clinical response[END_REF] that in some patients with liver metastases (whom they termed "trappers") 5FU appears to be retained longer (t ½ >20 min) in tumours than in normal tissues, and that 5FU trapping is correlated with response; in other studies, however, no trapping or correlation between half-life and response was seen [START_REF] Kamm | 19Fmagnetic resonance spectroscopy in patients with liver metastases of colorectal cancer treated with 5-fluorouracil[END_REF][START_REF] Van Laarhoven | Prediction of chemotherapeutic response of colorectal liver metastases with dynamic gadolinium-DTPA-enhanced MRI and localized 19F MRS pharmacokinetic studies of 5-fluorouracil[END_REF].

In principle, the most attractive potential parameter for prediction of response to 5FU would appear to be direct detection of the toxic fluoronucleotide species (FNucts) in the tumour. FNuct signals were detected in the first MRS studies of 5FU in mouse tumours [START_REF] Stevens | 5fluorouracil metabolism monitored in vivo by 19F NMR[END_REF], and subsequent studies in animal and human tumour xenografts have shown that FNuct predicts response to 5FU treatment [START_REF] Mcsheehy | Carbogen breathing increases 5-fluorouracil uptake and cytotoxicity in hypoxic murine RIF-1 tumors: A magnetic resonance study in vivo[END_REF][START_REF] Sijens | 19F Magnetic Resonance Spectroscopy Studies of the Metabolism of 5-Fluorouracil in Murine RIF-1 Tumors and Liver[END_REF][START_REF] Mcsheehy | Prediction of 5-fluorouracil cytotoxicity towards the Walker carcinosarcoma using peak integrals of fluoronucleotides measured by MRS in vivo[END_REF]. Recent in-vivo and exvivo rodent studies have detected FNuct in both tumour tissue and normal liver [START_REF] Doi | Quantitative 19F imaging of nmol-level F-nucleotides/-sides from 5-FU with T2 mapping in mice at 9.4T[END_REF].

Although there have been a number of reports of FNuct detection in tumours in patients [START_REF] Kamm | 19Fmagnetic resonance spectroscopy in patients with liver metastases of colorectal cancer treated with 5-fluorouracil[END_REF][START_REF] Findlay | In vivo monitoring of fluoropyrimidine metabolites: magnetic resonance spectroscopy in the evaluation of 5-fluorouracil[END_REF] it has proved to be extremely difficult to measure them routinely. To our knowledge, no clinical studies have addressed whether the toxic metabolites formed from 5FU in vivo are correlated with response to treatment.

We have found that by summation of a time series of 19 F MRS spectra it is possible to detect fluoronucleotide signals in the livers of many patients treated with 5FU. In the present study we have used this method to monitor 5FU metabolism in 32 colorectal cancer patients, to see if the formation of fluoronucleotides correlated with clinical response. A novel spatial analysis was then developed to estimate the proportion of the detected signal due to metastases. The hepatic fluoronucleotide levels detected in each individual patient were then correlated with progression-free survival and with the volume of liver metastasis. A small cohort of patients underwent two scans, one breathing room air and one breathing carbogen (95%O 2 /5%CO 2 ) for five minutes to investigate any alteration of the 5FU kinetics resulting from carbogen breathing, a gas breathing challenge that could potentially improve drug delivery to tumours [START_REF] Mcsheehy | Carbogen breathing increases 5-fluorouracil uptake and cytotoxicity in hypoxic murine RIF-1 tumors: A magnetic resonance study in vivo[END_REF].

Patients and Methods

This research was carried out with informed consent from the patients and with full local ethics committee approval (Wandsworth Health Authority). Patients with histologically proven colorectal carcinoma, for whom chemotherapy was planned, were eligible for referral by the consulting clinician. Of the 32 patients enrolled, 18 were male and 14 were female, mean age 66 years, range 37-85 years. The primary focus was on patients with metastatic liver disease, but patients free from metastatic disease receiving 5FU in an adjuvant role were also studied to charac terise 5FU metabolism in normal liver. 15 of the subjects had liver metastases. The scans were performed between 1999 and 2004. Che motherapy Patients were prescribed 5FU containing regimens as a single agent with modulation by leucovorin. The 5FU was either administered as a single bolus each day for 5 days (MAYO regimen), as a weekly bolus, or on the de Gramont regimen which involves a loading bolus dose of 5FU followed by a 48 hour infusion.

Whichever the regimen, the patients received a bolus of 5FU (400-425mg/m 2 ) during the MR exam. Chemotherapy was prescribed either for the treatment of existing metastatic disease or as adjuvant therapy immediately after initial colorectal surgery to prevent the development of metastases.

Carbogen Breathing

A small cohort of patients (n=9) underwent two scans at 3-7 days apart. On the first scan, the bolus was administered while the patient was breathing room air freely. On the second scan, the patient was asked to breathe carbogen (95% O 2 /5%CO 2 ) delivered at 30 litres min -1 through a mask for five minutes, starting three minutes before the 5FU bolus delivery was initiated, and then subsequently air. This component of the study was designed to investigate any alteration of the 5FU kinetics resulting from carbogen breathing and thus the potential of carbogen for enhancing 5FU potency.

Response

Clinical response was assessed as Progression Free Survival (PFS) by the attending Medical Oncologist; this was defined as the time until the patient showed disease progression or died. At this point, the treatment of surviving patients was either stopped or switched to another regimen.

19 F MRS 19 F MRS and MR imaging were performed on a clinical 1.5T MR system (Signa: GE, Milwaukee, USA) with a dual-tuned 1 H/ 19 F surface coil (20 by 16 cm). The coil was semi-flexible and allowed conformation to the curvature of the chest for optimal coverage of the liver. Coronal and axial gradient echo images were first acquired to confirm correct positioning of the coil over the patient's liver. For patients found to have liver metastases, a respiratory-gated T 2 weighted axial fast spin-echo sequence was used to localise the metastases. A fluorinated silicone oil reference sample situated at the coil centre was used to calibrate the 19 F pulse power for each patient and to provide a reference signal for quantitative comparison of data between different patients. 19 F spectra were acquired using a 90° pulse at the coil centre with a 500ms repetition time, 16 kHz bandwidth and 1024 or 2048 data points. 32 averages with the reference signal on resonance were first acquired. The spectrometer frequency was then offset to observe the 5FU signal on resonance. As the bolus i.v administration of 5FU was started, spectral acquisitions were begun and acquired continuously for 32 minutes in 1-minute blocks.

Administration of 5FU lasted typically 1 minute, followed by a bolus of Leucovorin (20mg/m 2 ). 5FU was administered by a nurse who remained in the magnet room during the examination; for the cohort breathing carbogen during 5FU administration, the nurse also removed the patient's mask to allow free breathing of room air once the carbogen breathing was complete.

Data analysis

5FU data were analysed in the time-domain MRS fitting package jMRUI [START_REF] Naressi | Java-based graphical user interface for the MRUI quantitation package[END_REF][START_REF] Vanhamme | Improved method for accurate and efficient quantification of MRS data with use of prior knowledge[END_REF]. Data were averaged in two-minute blocks to give 16 time points. 5FU half-lives were evaluated by non-linear fitting of data points starting from the highest point reached, typically in the second block of data. Additionally, all spectra for a given patient were added together to give an average spectrum for the full time-course. This average spectrum was assessed for the presence of FNuct (which is rarely present in concentrations high enough to be detected in any single two-minute block, and often not detectable even in the average spectrum). Average spectra in which the SNR for FNuct from the jMRUI analysis was greater than 2.5 were defined to be FNuct-positive. An example is shown in Figure 1a

MRI tumor volume and MRS sensitivity analysis

Image data were analysed using a program custom-written in MATLAB 7.4 (The Mathworks, Inc, Natick, MA) that numerically simulated the sensitivity of the 19 F spectroscopy coil. The coil was located on the images by its internal position marker consisting of a tube of water. ROIs were drawn delineating any visible metastases, and all of the liver tissue visible within the field of view of the coil, from which ROI volumes were calculated. The coil sensitivity was computed at every voxel within each ROI. We then calculated the ratio of the metastasis volume (summed over all the metastasis ROIs) to the total liver volume, weighted voxelwise according to the 19 F MRS coil sensitivity. We refer to this ratio within this paper as the fractional metastasis signal (FMS); this measure represents the fraction of detected 19 F signal from metastases, assuming the metabolites to be uniformly distributed. The higher the FMS, the more likely that the 19 F measurements and kinetics are representative of the tumour (see Table 2).

Statistical analysis was carried out in GraphPad Prism 5.01 (GraphPad Software, San Diego, CA)

Results and Discussion

5FU Half-Life

The half-life of 5FU in the full patient cohort (n=32) was 8.0 +/-0.48 minutes (mean+/SEM) and the range was 4.0-15.5 minutes (see Table 1). None of the patients we studied had a 5FU half life of longer than 20 minutes, and therefore none would be defined as a 5FU "trapper" in the terminology of Presant et al [START_REF] Presant | Association of intratumoral pharmacokinetics of fluorouracil with clinical response[END_REF]. No statistical correlation was observed between PFS and 5FU half-life, either in the whole population or in the population with detectable hepatic metastases (p=0.31, R2=0.04 and p=0.18, R 2 =0.13 respectively).

Paired survival curves were analysed for the relationship between progression-free survival and the presence of liver metastases. As would be expected, the PFS was significantly longer in the group of patients without liver metastases (p=0.001, Figure 2a). Patients with and without liver metastases have been considered separately for the rest of the analysis to avoid any confounding effects from this variation in PFS, and to investigate the significance of spectroscopic data in understanding therapy of metastases. Retrospectively, we decided to investigate the proportion of tumour in the liver in a subset of patients where the volumes of the metastases were assessed from the images taken prior to the 5FU bolus. The purpose of this was two-fold: to investigate whether the signal from the metastasis was overwhelmed by the signal from the liver (thus masking the half-life of 5FU in the tumour tissue), and to investigate correlations between tumour volume and pharmacokinetic parameters.

MRI volume data (see Methods for details) were available for 16 subjects, 11 of whom had been clinically reported to have liver metastases and one to have liver cysts. Data were uninterpretable due to motion in 2 cases (1 reported to have metastases). Data were analysed for the remaining 14 subjects, and metastasis detection was consistent with the clinical report in all cases. For the 10 subjects with metastases, the FMS varied from 0.05 to 0.52 (mean 0.21, standard deviation 0.17). It should be noted that this group did not include the two patients with metastases who survived for longer than 5 months.

Both quantifiable 19 F MRS and quantifiable MRI were available for all these 14 subjects. Although none of them (like the other patients we studied) had a 5FU half life long enough (>20 mins) to be defined as a 5FU "trapper" [START_REF] Presant | Association of intratumoral pharmacokinetics of fluorouracil with clinical response[END_REF], we still investigated whether there was a correlation between the relatively short half lives that we measured and the hepatic metastatic load. If the metastases had trapped 5FU and hence had a longer 5FU half-life than normal liver, this should have given a correlation between FMS and the half-life measured by MRS. Notably, the two subjects with the highest FMS values of 0.49 and 0.52 had relatively long half-life values of 15.5 and 9.4 minutes (Table 2), with the other subjects showing half-life values ranging from 4.9 to 11.8 minutes. However, there was no significant correlation between FMS and the half-life of 5FU (R 2 =0.40, p=0.25). The other 8 subjects with metastases had values ranging from 0.05 to 0.27 (mean 0.14, standard deviation 0.09).

Interpretable MRI data were not available for all subjects with liver metastases, including the patients with the longest PFS; nevertheless, some useful inferences can be drawn from the available data. In the majority of the 14 subjects whose data could be analysed, the metastases represented a small proportion of the sensitive volume of the coil (FMS <0.3 in 8/10 patients with metastases, see Table 2), and therefore it is likely that the MRS data will be dominated by the kinetics of the metabolites in normal liver if their concentrations approach those in the tumours. There is no statistical relationship between metastatic load and 5FU half-life in this group, which is perhaps to be expected, since the patients where the metastases are the source of even half the signal are suffering a very high disease burden, and are not responding to treatment, thus would not be expected to fall into the group of patients showing long 5FU half-lives. Conversely, the group with smaller metastases may have tumour kinetics that were undetectable against the background normal liver metabolism.

Since there was evidence from animal studies (for example, McSheehy et al [START_REF] Mcsheehy | Carbogen breathing increases 5-fluorouracil uptake and cytotoxicity in hypoxic murine RIF-1 tumors: A magnetic resonance study in vivo[END_REF];

Kamm et al [START_REF] Kamm | Effect of carbogen breathing on the pharmacodynamics of 5-fluorouracil in a murine colon carcinoma[END_REF]) that carbogen (95%O 2 /5%CO 2 ) breathing could increase the uptake of 5FU into the liver metastases, we also invited several of the patients (n=9) to attend scanning twice, once breathing air during 5FU infusion and once breathing carbogen (see also Table 1). However, the mean 5FU half-life was found to be not significantly different (p=0.75, paired t-test, mean+/-SEM = 8.9+/-0.7 breathing carbogen, 8.7+/-0.9 breathing air) and this approach was not continued (see also [START_REF] Griffiths | Issues of normal tissue toxicity in patient and animal studies -Effect of carbogen breathing in rats after 5-fluorouracil treatment[END_REF] for two detailed case studies). There was also no significant difference between the levels of FNuct detected on the two scans (paired t-tests on semi-quantitative values, contingency table test on positive/negative classification). It has been proposed on the basis of preclinical studies that the enhanced effect of 5FU in combination with carbogen is a combination of two processes: acidification by the CO 2 component causing increased intracellular-extracellular ΔpH (which has been shown to increase 5FU uptake in isolated cells [START_REF] Ojugo | Influence of pH on the uptake of 5fluorouracil into isolated tumour cells[END_REF] and experimental tumours [START_REF] Guerquin-Kern | pH Dependence of 5-Fluorouracil Uptake Observed by in Vivo 31P and 19F Nuclear Magnetic Resonance Spectroscopy[END_REF]); and a trapping effect, where

tumour blood vessels open to allow increased blood flow then reclose, slowing the outflow of 5FU [START_REF] Mcsheehy | Carbogen breathing increases 5-fluorouracil uptake and cytotoxicity in hypoxic murine RIF-1 tumors: A magnetic resonance study in vivo[END_REF]. Such a mechanism appears not to apply to normal liver, and thus changes in tumour 5FU kinetics may be hidden by the signal from normal liver.

Howe et al [START_REF] Howe | Flow and oxygenation dependent (FLOOD) contrast MR imaging to monitor the response of rat tumors to carbogen breathing[END_REF] used 51-Cr radioactivity measurements to show that carbogen breathing increased tumour blood volume in the GH3 prolactinoma grown in rats, but liver blood volume did not increased. Similarly, Honess & Bleehen [START_REF] Honess | Perfusion changes in the RIF-1 tumour and normal tissues after carbogen and nicotinamide, individually and combined[END_REF] demonstrated that carbogen breathing resulted in 50-70% increases in perfusion in RIF-1 tumours, and showed a smaller but significant reduction of up to 20% in liver perfusion, depending on the carbogen delivery rate; thus an increased 5FU uptake in liver metastases could be masked by simultaneous decreases in liver uptake. A recent PET study with carbogen breathing of liver metastases in man [START_REF] Gupta | Carbogen and Nicotinamide Increase Blood Flow and 5-Fluorouracil Delivery but not 5-Fluorouracil Retention in Colorectal Cancer Metastases in Patients[END_REF] demonstrated increased perfusion and specific uptake value in metastases, accompanied by reduction in both parameters in normal liver, although the total area under the curve was not altered.

van Laarhoven et al [START_REF] Van Laarhoven | Carbogen breathing differentially enhances blood plasma volume and 5-fluorouracil uptake in two murine colon tumor models with a distinct vascular structure[END_REF] demonstrated that different colorectal tumour lines grown in mice respond differently to carbogen; compared to the well-differentiated C38 line, the poorly-differentiated C26a showed smaller changes in 5FU kinetics, but larger changes in response to drug treatment, which the authors ascribe primarily to vascular differences. Thus the absence in our study of a significant change of the 5FU half-life with carbogen breathing, as measured over the whole liver, may not necessarily mean there was no change in the 5FU kinetics in the metastases.

The previous observations of Wolf, Presant and co-workers have demonstrated improved survival in a substantial subgroup of patients [START_REF] Presant | Association of intratumoral pharmacokinetics of fluorouracil with clinical response[END_REF][START_REF] Wolf | Non-invasive F-19-NMRS of 5fluorouracil in pharmacokinetics and pharmacodynamic studies[END_REF] showing 5FU half-lives longer than 20 minutes (5FU "trappers"); 51 of their 99 evaluable subjects were classified as trappers, and 70% of these responded to chemotherapy, while none of the suitable to each patient's study was selected, the choice being dictated by the size of the tumor and its distance from the skin" [START_REF] Wolf | Non-invasive F-19-NMRS of 5fluorouracil in pharmacokinetics and pharmacodynamic studies[END_REF]. It is possible that a smaller probe more tightly coupled to a superficial liver metastasis would be more revealing of the tumour-specific pharmacokinetics than the larger coil used in the study presented here. Another explanation for the difference could be a different patient population; it may be that 5FU was administered to a group of patients with more advanced and /or less chemosensitive cancer in the present study, so that none of them achieved the responses found by Wolf et al. In addition, Wolf et al. studied some patients with primary breast cancer and other unspecified cancers as well as those with liver metastases; the responses are not broken down by tumour type. Fluoronucleotide Signals

There was no significant difference in the mean FNuct levels between the patients with liver metastases (mets+ (n=15) 0.49 0.12,) and those without (mets-(n=17), 0.59 0.14, p=0.61). Overall, these data suggest that patients vary greatly in FNuct level, but that substantial amounts of FNuct are generated in normal liver in some patients; detailed illustrative case studies of hepatic FNuct signals in two early patients from this study group in whom we verified the absence of liver metastases were previously presented in [START_REF] Vanhamme | Improved method for accurate and efficient quantification of MRS data with use of prior knowledge[END_REF].

Table 1 shows the PFS and FNuct data for each patient, and Figure 2a shows survival plots of the PFS for the groups with and without liver metastases. Figure 2b shows survival plots for PFS for patients with metastatic disease grouped by the presence or absence of FNuct in their averaged spectra. Remarkably, in this group of patients, the presence of FNuct in the spectra was associated with poorer outcome, measured as PFS (p=0.027). This is unexpected, as the presence of high levels of cytotoxic FNuct would intuitively be expected to result in higher cell kill and better outcome.

To analyse this further, we calculated correlations between PFS, FNuct level, FMS, and 5FU half-life in the metastasis-positive group (Table 3). FMS values are available for only half the cases. Intriguingly, the correlations between FNuct and the other three parameters are large, though only the positive correlation between FNuct and 5FU half-life is significant (p=0.016). The trend towards a negative correlation between FNuct and PFS is consistent with the poorer survival in the FNuct-positive group of patients. A plausible hypothesis is that in the metastasis-positive group, the level of FNuct is dominated by the volume of metastasis, and larger volumes of metastasis are correlated with poorer survival. In the met-free group, in contrast, the level of FNuct might relate to the levels of relevant enzymes in normal liver cells.

In conclusion, with the large coil used in this study, it is possible to detect FNuct in liver and tumour tissue. The similarity of 5FU half-life in patients with and without liver metastases suggests that the observed signal may be dominated by normal liver metabolism except when the metastatic load is very high; notably, the two subjects with the highest FMS values had long 5FU half-lives. Within the patient group with liver metastases, FNuct signal is positively correlated with 5FU half-life. There is no clear association between 5FU half-life and outcome, and the 5FU half-lives observed in this study are shorter than those associated with improved outcome in the literature.

Overall these results suggest that metabolism of 5FU in normal liver is confounding any association between FNuct and treatment response in the data. High FNuct levels appear to be associated with poorer outcome as measured by PFS; this is likely to reflect higher metastatic load since FNuct is more likely to be produced in tumour cells.

The use of 3T or higher power magnets should in future facilitate such studies b y reducing the minimum volume of interest that can be studied, and permitting the use of localised MRS methods. A localised 19 F MRS study at both 1.5T and 3T on the metabolism of capecitabine (a pro-drug of 5-FU) in liver metastases from colorectal cancer has been reported [START_REF] Van Laarhoven | In Vivo Monitoring of Capecitabine Metabolism in Human Liver by 19Fluorine Magnetic Resonance Spectroscopy at 1.5 and 3 Tesla Field Strength[END_REF]. Magnetic Resonance Spectroscopic Imaging (MRSI) was used at both field strengths to acquire a 3D grid of 8

Introduction 5 -

 5 Fluorouracil (5FU), after several decades of use, is still one of the most widely used cytotoxic agents for the treatment of several cancers including colorectal cancer. 5FU based chemotherapy improves overall and disease-free survival of patients with resected stage III colorectal cancer. Over the last 20 years the regimes of treatment have been modified at intervals either by changing the way the dose is given (bolus or infusion)

  Figure 1b. Additionally, FNuct was assessed semi-quantitatively by calculating the ratio

  non-trappers responded to chemotherapy. None of the 32 patients in the present study had a half-life longer than 15.5 minutes and thus none of them would have fallen into the "trapper" category. It is possible that this is in part due to differences in the coil designs used in the work of Wolf et al., and that used in the present study. The present study used a large coil (16cmx20cm) sensitive to the entire liver volume; the work of Wolf et al., used surface coils of 3, 7 or 15 cm in diameter, and the "surface coil most

Table 1 .

 1 x 8 x 8 voxels, with each voxel of 4 x 4x 4 cm. No FNuct signals were reported, but the 3T spectra had clearly improved signal/noise compared to the 1.5T spectra. The same group subsequently published a capecitabine study at 3T with absolute quantitation of19 F CSI data, using water 1 H CSI data acquired from the same coil tuned to 1 H[START_REF] Klomp | Quantitative 19 F MR spectroscopy at 3 T to detect heterogeneous capecitabine metabolism in human liver[END_REF], demonstrating considerable heterogeneity of capecitabine and its metabolites even within normal liver. An approach of this kind should make it possible to assess tumour 5FU kinetics with minimal contamination from normal liver; this would allow, for example, accurate assessment of the effect of carbogen breathing on 5FU kinetics in the tumour, and aggregating signals from multiple voxels may allow comparison of FNUct levels in tumour and normal liver. However, our current results suggest that there may still be a useful role for non-localised19 F spectroscopy as a prognostic marker, by enabling the straightforward detection of FNuct, which is present at very low concentrations distributed throughout the tissue. Clinical details, 5FU half-life values and progression-free survival (PFS) of patients.

	Patient	Age	Sex FU half-life (min)	Liver	FNuct	PFS
				(*with carbogen)	mets		Months
	A	73	M 8.9, 10.3*	+	+	2
	B	60	M 11.9, 8.9*	+	+	1
	C	62	M 15.5, 12.0*	+	+	1
	D	64	M 6.9	-(liver	+	30+
					cysts)
	E	79	M 9.4, 11.7*	+	-	2
	F	71	F	7.5, 8.08*	+	+	4
	G	51	F	7.4	+	-	4
	H	66	M 6.9	-	-	Adjuvant
							treatment.
							Discharged at
							5 years in full
							health
	I	59	F	5.9, 6.5*	+	-	3
	J	71	F	11.8	+	-	3

Table 2 .

 2 The fractional metastatic signal in the subset of patients with metastatic disease for whom interpretable T 2 -weighted MRI data were available(n=10). The FMS is the fraction of detected19 F signal which would result from metastases if 5FU and its metabolites were distributed uniformly throughout the liver and metastases.

	Patient	FU Half-life	FMS	Mets
		(min)
	A	8.9	0.05	+
	B	11.9	0.05	+
	C	15.5	0.49	+
	E	9.4	0.52	+
	F	7.5	0.19	+
	G	7.4	0.10	+
	I	5.9	0.21	+
	J	11.8	0.05	+
	M	9.3	0.27	+
	P	4.9	0.19	+

Table 3 (

 3 a) p-values (b) Pearson r-values for correlations between half-life, progression-free survival, FNuct levels (measured as FNuct/peak 5FU) and fractional metastasis signal in the patient group with liver metastatic disease

	Half-life	PFS	FNuct	FMS
	Half-life		0.17	0.016	0.25
	PFS	0.17		0.14	0.38
	FNuct	0.016	0.14		0.09
	FMS	0.25	0.38	0.09
	Table 3a
	Half-life	PFS	FNuct	FMS
	Half-life		-0.38	0.61	0.40
	PFS	-0.38		-0.40	-0.32
	FNuct	0.61	-0.40		0.56
	FMS	0.40	-0.32	0.56
	Table 3b

Acknowledge ments

We would like to acknowledge the support of The University of Cambridge, Cancer Research UK and Hutchison Whampoa Limited. The MRUI software package was kindly provided by the participants of the EU Network programmes: Human Capital and Mobility, CHRX-CT94-0432 and Training and Mobility of Researchers, ERB-FMRX-CT970160.u . The patient studies would not have been possible without the hard work of our research nurse, Natasha Price.


Jorge G Russo

Paul K Townsend

On the thermodynamics of moving bodies

We consider an Unruh-DeWitt particle detector, coupled to a massless scalar field, undergoing "acceleration with drift" in a (1 + 3)-dimensional Minkowski spacetime. We use this to model inertial motion in a (1 + 2)-dimensional Minkowski heat bath; in particular, motion within a 2-plane parallel (and near) to the horizon of a black 2-brane. We compute the angular response of the detector in its own rest frame. The response to particles arriving from within the 2-plane is isotropic and Planckian for zero drift velocity. For small drift velocities, and in the ultra-violet limit in which the excitations behave like a classical gas, the response is just Doppler shifted. However, we find discrepancies with the Doppler shifted formula in the infrared limit, and qualitatively different behaviour when the drift velocity is not small. We discuss possible explanations for this result and potential implications for observations of the cosmic microwave background radiation.

Introduction

Inertial motions are all equivalent in a vacuum but not in a radiation heat bath because the rest-frame of the radiation provides a preferred frame. Consider a Minkowski spacetime filled with a gas of massless spinless particles at temperature T . We shall assume that the particles of the gas are excitations of a scalar field φ which we may couple to a two-state Unruh-DeWitt particle detector with variable energy gap ω. If the detector is at rest with respect to an infinite heat bath, its response as a function of ω will be Planckian. This paper is concerned with the response of such a detector to inertial motion in the heat bath.

One motivation is potential relevance to the detection of the cosmic microwave background radiation (CMBR); although the massless particles in this case are spin 1 photons, this difference is not crucial for much of the physics. There is a standard theory in this case, which goes back to Pauli [START_REF] Pauli | Relativitats Theorie[END_REF] but was developed in the context of the CMBR by Peebles and Wilkinson [START_REF] Peebles | Comment on the Anisotropy of the Primeval Fireball[END_REF] and others [START_REF] Henry | Distribution of black body cavity radiation in a moving frame of reference[END_REF]: the background radiation is viewed as a classical gas of massless particles and the distribution in the moving frame is deduced from the Planck distribution in the rest frame by a Lorentz transformation of the individual particle trajectories; the result is the same (for massless particles) as one gets by an application of the relativistic Doppler shift. However, this theory is open to question because the description as a classical gas is valid only in the ultra-violet limit in which ω/T 1. We leave this point here for the moment, and turn to another motivation, of a more theoretical nature. This comes from consideration of the thermal physics of black branes. As for a static black hole, a static black p-brane with non-zero surface gravity κ radiates at the Hawking temperature T H = κ/(2π) and so can be in thermal equilibrium only if the temperature at transverse spatial infinity equals T H . The local temperature elsewhere is then given by the Tolman relation:

T = T H √ -k 2 , (1.1)
where k is the timelike Killing vector field such that k 2 = 0 on the horizon and k 2 = -1 at infinity. Assuming that the black p-brane is planar, this local temperature will be constant on the p-planes at a fixed distance r from the event horizon, and could be measured using a particle detector. As the temperature is a function of r, we should expect the detector's response to depend on the angle θ of incidence to the outward radial normal. However, if we fix θ = π/2, the radiation will be isotropic in all other directions; in effect we have a detector in a Minkowski heat bath of dimension (1 + p). We may again ask how the detector responds to inertial motion within this heat bath; in other words, how does it respond to motion parallel to the p-brane horizon?

If we assume that the detector is very close to the horizon then the effective spacetime geometry of a p-brane d-metric is the product of a (d -2)-sphere with a (2 + p) dimensional generalization of the flat Rindler metric, in which a static observer has proper acceleration a = κ/ √ -k 2 . In other words, we have a detector undergoing constant proper acceleration in a (2 + p)-dimensional Minkowski spacetime. For p = 0 (i.e. a black hole), Unruh showed [START_REF] Unruh | Notes on black hole evaporation[END_REF] that this acceleration causes the detector (the theory of which was subsequently clarified by DeWitt [START_REF] Dewitt | Quantum Gravity: the new synthesis[END_REF]) to respond as if it were in a heat bath at a temperature (the Unruh temperature) that coincides with the local temperature of the black hole:

T U ≡ a 2π = T H √ -k 2 (k 2 → 0) . (1.2)
In other words, the local temperature has a purely kinematical explanation near the horizon, as might be expected from the fact that there is actually nothing at the horizon. Unruh's result is expected to apply for all p > 0, but for odd p there is an apparent [START_REF] Takagi | On The Response Of A Rindler Particle Detector[END_REF] (but spurious [START_REF] Unruh | Accelerated monopole detector in odd spacetime dimensions[END_REF][START_REF] Takagi | Vacuum Noise And Stress Induced By Uniform Accelerator: Hawking-Unruh Effect In Rindler Manifold Of Arbitrary Dimensions[END_REF]) interchange of Bose and Fermi statistics, while for p > 2 the interaction of an Unruh-DeWitt detector with the scalar field is non-renormalizable. For these reasons, we shall restrict ourselves here to the case of p = 2, although some arguments will apply more generally. We thus have an accelerating particle detector in a 4-dimensional Minkowski spacetime. Because this acceleration picks out a particular direction, we should expect a direction dependent response1 but the response will be isotropic in a 2-plane orthogonal to the acceleration. In the context from which we extracted this picture, this 2-plane is one of the 2-planes parallel to a black 2-brane horizon, which we may conveniently think of as a (1+2)-dimensional braneworld at the uniform Unruh temperature implied by its acceleration in an orthogonal direction.

An obvious question that this raises is whether the model of a Minkowski heat bath as an accelerating brane can be 'abstracted' from the context described above and used more generally, perhaps in higher dimensions too, as a model for a Minkowski radiation heatbath [START_REF] Russo | Accelerating Branes and Brane Temperature[END_REF] (where by "radiation" we mean massless particles with zero chemical potential). If so, there could be implications for the detection of photons in the CMBR. The limits of applicability of this model are not clear to us, but it is interesting to note that it is an essential ingredient in the GEMS approach to black hole thermodynamics [START_REF] Deser | Mapping Hawking into Unruh thermal properties[END_REF]: if a static black hole spacetime is embedded globally, and isometrically, in a higher-dimensional flat spacetime, then a static observer anywhere, not just near the horizon, has an acceleration in the embedding spacetime that is related to the local temperature in thermal equilibrium by the Unruh formula. This is an observation, rather than a result, since it is far from clear why Unruh's formula should apply, but it is an observation that consistently gives correct results. We shall leave further discussion of these issues until the end of the paper.

Returning to our detector at fixed distance from a 2-brane horizon, we consider the effect on it of inertial motion within the plane in which it is constrained to move. This leads to consideration of the motion in a 4-dimensional Minkowski spacetime of a detector that undergoes constant proper acceleration in one direction along with a constant velocity 'drift' in an orthogonal direction. This is a stationary motion, in the sense that the extrinsic geometric invariants of the detector's worldline (curvature, torsion and hyper-torsion) are all constant; this implies that the detector's response is time-independent, at least in its own rest-frame. Stationary motions were classified by Letaw [START_REF] Letaw | Vacuum Excitation Of Noninertial Detectors On Stationary World Lines[END_REF] (more recent discussions can be found in [START_REF] Louko | How often does the Unruh-DeWitt detector click? Regularisation by a spatial profile[END_REF][START_REF] Russo | Relativistic Kinematics and Stationary Motions[END_REF]). Three of his six classes are inertial motion, accelerated motion and "acceleration with drift"; the other three all involve rotation. The response of a detector is direction dependent; in the case of acceleration with drift it will depend not only on the angle θ to the axis of the acceleration but also on the angle ϕ to the direction of the drift velocity in the θ = π/2 plane. The principal technical result of this paper is a computation of the angular response of an Unruh-DeWitt detector undergoing acceleration with drift in a four-dimensional Minkowski spacetime. We thus learn, from first principles, how an Unruh-DeWitt detector at fixed distance from a black 2-brane horizon responds to inertial motion parallel to the brane.

What result should we expect? If the detector is restricted to move within a plane at a uniform temperature, and to detect only radiation that is coming from other points on this plane, then we have a situation that is very similar to that encountered by a detector moving inertially in the CMBR, except for the one lower dimension of space. So should we expect the detector to see a distribution of massless particles that is just Doppler shifted relative to the rest-frame distribution detected by a static detector? One argument for this that applies in any dimension comes from consideration of the mode expansion for the scalar field φ. It is well-known that the thermal behaviour of a static detector in Rindler spacetime can be predicted from the non-trivial Bogoliubov transformation that relates the modes in the Rindler coordinates adapted to constant acceleration to those in the full Minkowski spacetime. We can similarly consider the mode expansion in coordinates adapted to acceleration with drift (adapted coordinate systems can be found explicitly for all stationary motions [START_REF] Letaw | The Stationary Coordinate Systems In Flat Space-Time[END_REF]). We shall present a version of this analysis, although the basic result is known [START_REF] Letaw | The Quantized Scalar Field In The Stationary Coordinate Systems Of Flat Space-Time[END_REF]: the modes are simply Doppler-shifted relative to those of pure acceleration since the additional Bogoliubov transformation associated with the drift motion is trivial (although this is not the full story [START_REF] Korsbakken | The Fulling-Unruh effect in general stationary accelerated frames[END_REF], as we shall explain). However, we find that the detector response reproduces the Doppler shifted Planck spectrum only to leading order in the drift velocity v in the ultra-violet limit ω/T 1. Following our presentation of this result, we make a detailed comparison with the Dopplershifted Planck spectrum. We then discuss the related issue of scalar mode expansions in coordinates adapted to the stationary motion of "acceleration with drift", and conclude with a discussion of the results, and their possible implications. We should mention here that similar issues have been addressed previously for rotational motion. Because this involves acceleration, it has been suggested that a version of the Unruh effect might be operative. However, the mode expansion analysis shows that the quantum vacuum in a rotating frame does not differ from the inertial frame, so one might expect the rotation to have no effect on a particle detector. It appears that this is indeed the case when modes of the scalar field are suitably restricted to take into account the fact that the coordinates in a rotating frame become singular at some critical radius [START_REF] Davies | The Rotating Quantum Vacuum[END_REF]. The physics of acceleration with drift is quite different, however. What appears to be most relevant in this case is that the drift motion effectively creates an ergo-region outside the acceleration horizon.

Detector response

In this section we compute the response of an Unruh DeWitt detector undergoing the stationary motion that we refer to as "acceleration with drift" in a four-dimensional Minkowski spacetime with metric

dX • dX = -dX 2 0 + dX 2 + dY 2 + dZ 2 . (2.1)
The worldline of the detector in this spacetime is given by

X 0 = a -1 sinh [aγτ ] , X = a -1 cosh [aγτ ] , Y = γvτ , Z = 0 , (2.2)
where γ = 1/ √ 1 -v 2 . The 4-acceleration A is non-zero for any v. For v = 0 we have the well-known trajectory of a detector undergoing constant proper acceleration |A| = a. When v = 0 we have "acceleration with drift" in which uniform motion with velocity v in the Y direction is superposed on constant proper acceleration |A| = γ 2 a in the X direction. For pure acceleration, the worldline has non-zero curvature but zero torsion, whereas the torsion is also non-zero for acceleration with drift [START_REF] Letaw | Vacuum Excitation Of Noninertial Detectors On Stationary World Lines[END_REF]. Equivalently, the relativistic jerk [START_REF] Russo | Accelerating Branes and Brane Temperature[END_REF][START_REF] Russo | Relativistic Kinematics and Stationary Motions[END_REF] is zero for pure acceleration but not for acceleration with drift. We therefore expect a v-dependence of the detector's response to the motion.

As we explained in the introduction, the motion of acceleration with drift is relevant to inertial motion in a heat bath if the latter is modeled as a brane at the Unruh temperature due to acceleration in an extra dimension, and we also explained how this model applies to motion near the horizon of a black 2-brane. For a 2-brane, the model is realized by the following family of embeddings of a 3-dimensional worldvolume with coordinates (τ, y, z) [START_REF] Russo | Relativistic Kinematics and Stationary Motions[END_REF]:

X 0 = a -1 sinh [aγ(τ + vy)] , X = a -1 cosh [aγ(τ + vy)] , Y = γ(y + vτ ) , Z = z , γ = 1/ 1 -v 2 (2.3)
A computation of the induced metric shows that the embedded spacetime is indeed Minkowski and also that the coordinates (τ, y, z) are cartesian. This Minkowski worldvolume can be viewed as the congruence of worldlines parametrized by position (y, z) on the brane. Each such worldline defines a stationary motion of acceleration with drift, with proper time τ . The embedding with non-zero v is obtained from that with v = 0 by a worldvolume Lorentz boost in the y direction.

Although the induced metric is not affected by this boost, the extrinsic geometry of the worldlines of the congruence is affected by it. In particular, the torsion (equivalently, relativistic jerk) of the worldlines of the congruence is zero only for v = 0. When account is taken of the Unruh temperature on the brane due to the acceleration, this fact can be seen as the kinematical equivalent of the fact that the rest frame of a heat bath provides a 'preferred' reference frame. We begin with some generalities. We assume a 'monopole' detector [START_REF] Dewitt | Quantum Gravity: the new synthesis[END_REF] with (variable) energy gap ω coupled to a free quantum scalar field Φ(X). The interaction is

S int = g dτ m(τ )Φ (X(τ )) , (2.4)
where m(τ ) is the detector's monopole moment operator, and X(τ ) is the detector's worldline, parametrized by proper time τ . The coupling constant g is dimensionless (this is a simplifying feature of four spacetime dimensions) and may be assumed small. The probability of excitation and accompanying emission of a massless particle of 4-momentum P = (k, k) into solid angle element dΩ may then be computed using first-order perturbation theory. Integrating over photon energy, and omitting a small dimensionless constant of proportionality, one has [START_REF] Birrell | Quantum Fields In Curved Space[END_REF][START_REF] Kolbenstvedt | Anisotropy of thermal radiation by accelerated detectors; an aberration effect[END_REF]

dP dΩ = ∞ 0 dk k ∞ -∞ dτ + ∞ -∞ dτ -exp 2i ωτ --i P • ∆X(τ, τ ) , (2.5)
where

∆X(τ, τ ) = X(τ ) -X(τ ) , τ ± = 1 2 τ ± τ . (2.6)
Henceforth, we choose units such that = 1, which means that all dimensionful quantities have dimensions of mass to some power.

Integrated response

From the braneworld perspective that we will eventually adopt, it would make no sense to integrate the detector's response over solid angle. However, for the general purpose of understanding the physics of detector response it is instructive to consider this integral. Setting = 1 in (2.5) we have

dΩ dP dΩ = dτ + Ṗ , Ṗ = 16π 3 ∞ -∞ dτ -e 2iωτ -G + , (2.7)
where Ṗ is the integrated excitation rate, determined by

G + = 4π 2 ∆X(τ, τ ) 2 -1 , (2.8)
which is the Wightman function evaluated for points on the detector's worldline specified by proper times τ and τ . The integral should vanish for inertial motion, for which

G + = -1/(16π 2 τ 2 -)
. The integrand has a pole at τ -= 0 but the usual 'iε' prescription will push this pole below the real axis so that it does not lie within the contour completed in the upper half of the complex τ --plane (we assume ω > 0). With this prescription, we recover the expected result that the detector remains in its ground state if it is moving inertially.

It has been proposed [START_REF] Costa | Temperature And Relativity[END_REF][START_REF] Landsberg | Laying the ghost of the relativistic temperature transformation[END_REF] that the response of an inertial detector in a (1+3)-dimensional Minkowski spacetime at temperature T is given by

ṖT ∝ ∞ -∞ dτ -e 2iωτ -G + T (τ -) , (2.9)
where G + T is the thermal Wightman function, again evaluated for points on an inertial worldline specified by proper times τ and τ :

G + T = - 1 16π 2 ∞ n=-∞ 1 (τ --inγ/2T ) 2 + (nγv/2T ) 2 .
(2.10)

Now there are poles inside the contour even for inertial motion and one gets a non-zero result that depends on the detector's velocity with respect to the heat bath. The 'scalar photon' number density may be read off from this result, and one finds that

n (3) (ω) dω = ωT 8π 2 γv log 1 -e -ωγ(1-v)/T 1 -e -ωγ(1+v)/T dω , (2.11)
where the (3) superscript indicates that this is a 3-space number density. Note the factor of v in the denominator, which ensures that the v → 0 limit is non-zero; naturally, this limit yields the usual Planckian distribution. For non-zero v, this result is exactly what one finds by integration over solid angle of the formula n

P P W (ω, n) dω dΩ = ω 2 16π 3 e ω/T eff -1 dω dΩ , T eff = T/γ [1 -v • n] , (3)
which is precisely the Doppler-shift result of Pauli, Peebles and Wilkinson (PPW) (after allowance is made for the fact that we here discuss "scalar photons" with only one polarization state). However, the starting formula (2.9) has not been derived from first principles. We now return to the conventional formula (2.7) for the integrated excitation rate, which is derived from first principles. This is time independent for any stationary motion since G + is then a function only of τ -. For "acceleration with drift" one finds that [START_REF] Letaw | Vacuum Excitation Of Noninertial Detectors On Stationary World Lines[END_REF]

G + = - a 2 16π 2 sinh 2 ξ -v 2 ξ 2 , ξ = aγτ -, (2.13)
as can be shown directly using (2.2). This yields a non-zero integrated excitation rate. One can arrive at the same result by first performing the k integral in (2.5). To this end, we choose spherical angles, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π, so that

P 0 = k , P X = k cos θ , P Y = k sin θ cos ϕ , P Z = k sin θ sin ϕ .
(2.14)

We then find that

dP dΩ = - a 2γ dτ + ∞ -∞ dξ e 2i(ω/aγ)ξ [Γ θ sinh ξ -vξ cos ϕ sin θ] 2 (2.15)
where Γ θ = cosh (aγτ + )cos θ sinh (aγτ + ) .

(2.16)

Integrating over solid angle, we recover the formula for the integrated excitation rate in terms of G + .

Angular response

We now focus on the formula (2.15), which gives the angular response of the detector integrated over time. Consider first the v = 0 case. We have, from (2.15) and (2.16),

dP dΩ v=0 = ∞ -∞ dτ + [Γ θ (τ + )] -2 d Ṗ dΩ v=0 (2.17) where d Ṗ dΩ v=0 = - a 2γ ∞ -∞ dξ e 2i(ω/aγ)ξ sinh 2 ξ . (2.18)
We would like to identify the integrand of the τ + integral with an excitation rate but this rate is τ + -dependent because of the [Γ θ ] -2 factor. As shown in [START_REF] Kolbenstvedt | Anisotropy of thermal radiation by accelerated detectors; an aberration effect[END_REF][START_REF] Brevik | Quantum Point Detector Moving Through A Dielectric Medium. 2. Constant Acceleration[END_REF], this factor arises because the detector has a time-dependent velocity dX/dT = tanh(aγτ + ) with respect to an inertial 'laboratory' frame. The two frames coincide at τ + = 0, so the time-independent excitation rate in the detector's frame is just (d Ṗ/dΩ) v=0 . The same logic applied for v = 0 yields the excitation rate

d Ṗ dΩ = - a 2γ ∞ -∞ dξ e 2i(ω/aγ)ξ [sinh ξ -vξ cos ϕ sin θ + iε] 2 , (2.19)
where we have now made explicit the iε prescription.

To compute the integral in (2.19), we complete the contour in the upper-half complex ξplane. For non-zero u ≡ v cos ϕ sin θ it encloses a finite number of poles at

ξ = ix , sin x = ux , x > 0 , (u ≡ v cos ϕ sin θ) . (2.20)
There is a unique pole at x = x 0 (u) when u > u 1 ≈ 0.13, and in this case the residue calculus yields

d Ṗ dΩ = πa γ c [(x 0 (u)] e -2ωx 0 (u) aγ , (2.21)
where

c [x] ≡ 2ωu -2ω cos x + aγ sin x aγ [u -cos x] 3 . (2.22)
At u = u 1 there is an extra double pole, which splits into two poles at x = x 1 > x 0 and x = x 1 > x 1 for u < u 1 . As long as u > u 2 , where u 2 < u 1 is a second critical value of u, there will be at most three poles. As u decreases further, new pairs of poles appear at a sequence of critical value values u of u, and for any non-zero v one finds that

d Ṗ dΩ = πa γ c(x 0 ) e - 2ωx 0 aγ + max =1 θ(u -u) c(x ) e - 2ωx aγ + c(x ) e - 2ωx aγ , (2.23)
where 2 max + 1 is the number of roots of sin x = ux. For u → 0 (e.g. as a result of v → 0) max → ∞ and the poles in the contour move to ξ = πki, thus reproducing the v = 0 result. Despite the step functions, d Ṗ/dΩ is a smooth function of u because the pairs of poles that that appear as u decreases though a critical value u give contributions of opposite signs that cancel at u = u . In the ultraviolet (UV) limit, the first term in (2.23) dominates, exponentially, so that (2.21) gives the asymptotic behaviour.

So far we have considered only positive u, but u becomes negative when 3π 2 > ϕ > π 2 . The equation f (u) ≡ sin x -ux = 0 then has no solution for u below a certain critical value u * , determined as follows. For u slightly above the critical value u * , there are two roots, which become a double root at u = u * . At this point u = u * also solves f (u) = cos x -u = 0. Combining both equations, we find the critical value at

-sin θ cos ϕ > v c v .
(2.25)

In other words, there is no response from the detector in a 'backward' cone, with forward axis defined by the detector velocity, of angle α such that

cos α = v c v .
(2.26)

The angle α goes to zero as v → v c , while α → arccos(v c ) ≈ 0.57π ≈ 77.5 • as v → 1. The cone closes at v = v c , and for v < v c the detector has a response in all directions. For u slightly above u * , the equation sin x -ux = 0 has two roots x 1 , x 2 and and the detector response is of the form

d Ṗ dΩ = πa γ c(x 1 ) e -2ωx 1 aγ + c(x 2 ) e -2ωx 2 aγ . (2.27)
Again it should be noted that there is no discontinuous behaviour: the two terms give contributions of opposite signs which cancel as u → u * .

Braneworld detector response and comparison with Doppler-shifted spectrum

We now want to apply our results to the braneworld approach to motion in a Minkowski heat bath. We therefore consider a detector restricted to detect photons arriving from directions within the brane. Recalling that θ is the angle that a photon's 3-momentum makes with the X-axis, we see that we must set θ = π/2. For convenience, we define

d Ḟ dϕ ≡ d Ṗ dΩ θ=π/2 = - a 2γ ∞ -∞ dξ e 2i(ω/aγ)ξ [sinh ξ -vξ cos ϕ + iε] 2 .
(2.28)

We have not been keeping track of an overall factor in the definition of the angular excitation rate of the detector, but this factor may now be fixed by considering the v → 0 limit, in which case we find that 1 8π

3 d Ḟ dϕ v=0 dω = ω dω 4π 2 e ω/T -1 , (2.29)
which is the Planck spectrum for the number density of 'scalar photons' in two space dimensions. More generally, therefore, we have

n (2) (ω, ϕ) = 1 8π 3 d Ḟ dϕ = - a 16π 3 γ ∞ -∞ dξ e 2i(ω/aγ)ξ [sinh ξ -vξ cos ϕ + iε] 2 .
(2.30)

The integral can be evaluated as before and the result is obtained from the previous formulas (2.21) and (2.23) by setting θ = π/2, so that u = v cos ϕ.

We shall be interested in comparing this result with what one finds from an application of the standard Pauli-Peebles-Wilkinson (PPW) theory for the distribution of massless particles in a radiation heat bath; for scalar particles in a 2-dimensional space, this gives

n (2) PPW (ω, ϕ) = ω 4π 2 e ω/T eff -1 , T eff = T/γ 1 -v cos ϕ , (2.31)
which is the 2-space dimensional analog of (2.12). We consider in turn the case of non-relativistic and relativistic velocities.

• Non-relativistic velocities. For v 1 we may expand the expression of (2.31) in powers of v to find that

n (2) PPW (ω, ϕ) = ω 4π 2 (e ω/T -1) + ω 2 v cos ϕ e -ω/T 4π 2 T 1 -e -ω/T 2 + O v 2 , (2.32)
How does this compare with (2.30) in the same approximation? Expanding the integrand in powers of v, we have

n (2) (ω, ϕ) = - a 16π 3 γ ∞ n=0 (n + 1) dξ e 2iωξ aγ (v ξ cos ϕ) n [sinh ξ + iε] n+2 .
(2.33)

We need only the n = 0, 1 terms in the sum. Using the residue calculus, and the Unruh formula 2πT = |A| = γ 2 a, we find that

n (2) (ω, ϕ) = ω 4π 2 (e ω/T -1) + ω 2 v cos ϕ e -ω/T 4π 2 T 1 -e -ω/T 2 Q(ω/T ) + O v 2 .
(2.34)

where

Q(ω/T ) = 1 -e -ω/T 1 + e -ω/T 2 1 - 2T ω + π 2 T 2 ω 2 - 2T ω e -ω/T . (2.35)
This differs from (2.32) by the additional factor Q multiplying the dipole term. This factor is a smooth monotonic function of ω/T with the UV and IR limits lim

ω/T →∞ Q = 1 , lim ω/T →0 Q = π 2 /4 . ( 2

.36)

We therefore find agreement with (2.32) in the UV limit, as might be expected on the grounds that the scalar field excitations behave like a classical gas of massless particles in this limit. Otherwise, there is a discrepancy in the factor multiplying the dipole term, which increases to about a factor of 2 in the IR limit.

• Relativistic velocities. In this case we need to return to the exact formula (2.30). The energy spectrum at ϕ = 0 as compared to ω n

PPW (ω, 0) is shown in fig. 1a for v = 0.04; in this case there are seven poles contributing to the sum (2.23). We see that the spectra are remarkably close at this velocity, differing slightly near the peak. The energy spectra continue to be close even for much higher velocities such as v = 0.15 where only a single pole contributes to the integral and the spectrum n (2) (ω, 0) is exactly given by (2.21) divided by 8π 3 . For this value of v the greatest deviation is at the peak of the spectrum and is of order 8 %. At higher velocities the difference becomes substantial; for example, for v = 0.8 the peak of the PPW energy spectrum is smaller by a factor of 5 and the PPW temperature (defined by the exponent of the tail) is higher by a factor of 1. Away from ϕ = 0 there are other significant effects. We return to formulas (2.21) and (2.23) evaluated at θ = π/2. For v > |u * | ≈ 0.2172 we find, for negative u, the same "cone of silence" within which the detector has no response because the integrand of (2.30) has no pole within the contour. This cone is now defined by ϕ > ϕ 0 , with cos ϕ 0 = u * /v. The "cone of silence" is obviously not a feature of the PPW formula (2.31).

Consider now u → 1, which implies both ϕ → 0 and v → 1. In this case there is a single pole which is located at x 0 = √ 6 √ 1 -v cos ϕ → 0; the leading behaviour is given by

n (2) (ω, ϕ) ∼ T 16πγ 3 √ 3 √ 2 exp - √ 6γω πT √ 1 -v cos ϕ (1 -v cos ϕ) 5 2
.

(2.37) This is significantly different from the equivalent limit of the formula (2.31).

Vacuum for acceleration with drift

It is well-known that the thermal spectrum registered by an accelerating detector can be explained, in a detector independent way, as a result of a difference of the Minkowski quantum vacuum to the (Fulling) vacuum of the Rindler spacetime. Specifically, the Bogoliubov transformation that relates the particle creation and annihilation operators in the two spacetimes is non-trivial, so that the Minkowski vacuum is a thermal state in the Rindler spacetime. It would be natural to suppose that our result for acceleration with drift has a similar explanation 2 , but the Bogoliubov transformation connecting the Fulling-Rindler vacuum with the analogous vacuum in a moving frame is trivial [START_REF] Letaw | The Quantized Scalar Field In The Stationary Coordinate Systems Of Flat Space-Time[END_REF]. It might seem from this result that the photons available for detection at non-zero drift velocity are just the same photons that were available for detection at zero velocity, but Doppler shifted. However, there is more to this problem than just the Bogoliubov transformation; in particular, it was shown in [START_REF] Letaw | The Stationary Coordinate Systems In Flat Space-Time[END_REF][START_REF] Korsbakken | The Fulling-Unruh effect in general stationary accelerated frames[END_REF] that the stationary motion we call "acceleration with drift" leads a geometry in which there is an ergo-region outside the acceleration horizon. We present here a summary of these results and then discuss the implications. This whole analysis can be carried out for arbitrary spacetime dimension D, in particular for the D = 4 case as well as for the D = 3 case that we have analyzed from the perspective of detector response.

We begin with the (D + 1)-dimensional Minkowski metric in the form

ds 2 = -dt 2 + dx 2 + dy 2 + |d z| 2 . (3.1)
where z = (z 1 , . . . z D-2 ). The change of coordinates

t = r sinh η , x = r cosh η , (3.2)
gives the Rindler metric

ds 2 = -r 2 dη 2 + dr 2 + dy 2 + |d z| 2 . (3.3)
These coordinates cover only two portions of the Minkowski spacetime, the positive ("right") and negative ("left") Rindler wedges,

R + = {x | x > |t|} , R -= {x | x < -|t|} ,
which correspond, respectively, to positive and negative r. In these regions the Killing vector field ∂ η = x∂ t + t∂ x is timelike, and a static observer at fixed r has proper constant proper acceleration a = 1/|r| (and proper time τ = a -1 η). Rindler coordinates are thus 'adapted' to such observers.

To obtain the metric in coordinates adapted to an observer in the Rindler spacetime who is drifting in the y direction with velocity v, we Lorentz boost the metric (3.3) in the y direction. Note that this boost is not an isometry; this is the kinematical counterpart of the statement that a thermal bath breaks Lorentz invariance. We perform the boost by setting

y = γ(ỹ -vr 0 η) , r 0 η = γ(r 0 η -v ỹ) , r 0 = a -1 , (3.4)
and rewriting the metric in terms of the new coordinates (η, ỹ, r, z):

ds 2 = -r 2 γ 2 (dη - v r 0 dỹ) 2 + dr 2 + γ 2 (dỹ -vr 0 dη) 2 + |d z| 2 (3.5) = -γ 2 (r 2 -v 2 r 2 0 )dη 2 + 2γ 2 v r 0 (r 2 -r 2 0 )dỹdη + γ 2 1 - r 2 v 2 r 2 0 dỹ 2 + dr 2 + |d z| 2 .
2 For example, there is an Unruh-type effect associated with constant velocity motion through a medium with refractive index n > 1 [START_REF] Brevik | Quantum Point Detector Moving Through A Dielectric Medium. 2. Constant Acceleration[END_REF].

This metric is stationary, with respect to the new time variable η, but it is not static. There is still a coordinate singularity at r = 0, which is again the acceleration horizon, but there is also a 'static limit' surface at r = vr 0 [START_REF] Korsbakken | The Fulling-Unruh effect in general stationary accelerated frames[END_REF]; the region between the horizon and this static limit surface is analogous to the ergo-region of a rotating black hole; see fig. 2. The hypersurface at r = r 0 is D-dimensional Minkowski spacetime in standard coordinates

ds 2 r=r 0 = -dτ 2 + dỹ 2 + |d z| 2 , (τ = r 0 η). (3.6)
A static detector in this subspace has proper acceleration γ 2 a in the larger space, where the γ 2 factor is a consequence of time dilation due to the uniform motion in the y direction. There is an ergo-region for the natural time-like Killing vector field between the acceleration horizon and the static-limit surface at r = vr0.

Consider now a complex massless scalar field φ in the (D +1)-dimensional Minkowski spacetime, obeying the wave equation φ = 0. The norm of any solution φ is defined as (we follow the review article [START_REF] Takagi | Vacuum Noise And Stress Induced By Uniform Accelerator: Hawking-Unruh Effect In Rindler Manifold Of Arbitrary Dimensions[END_REF])

||φ|| = -i dΣ µ (φ * ∂ µ φ -φ∂ µ φ * ) ( 3 . 7 )
where Σ is the volume element on an arbitrary spacelike hypersurface. A basis of solutions of positive norm is provided by the positive frequency plane parallel waves

U k = 1 [2ω k (2π) D ] 1/2 e i(-ω k t+kxx+kyy+ k• z) , ω k = k 2 x + k 2 y + |k| 2 . (3.8)
In Rindler coordinates, the solutions with positive norm are again those of positive frequency, but where this is now defined with respect to the Rindler time η; these modes are

u (σ) k = θ(σr) [2Ω(2π) D-1 ] 1/2 h (σ) k (r) exp i -σΩη + k y y + k • z , (3.9)
where σ > 0 for the positive Rindler wedge and σ < 0 for the negative Rindler wedge. Here we are using the fact that the timelike Killing vector ∂ η is future directed on R + and past directed on R -. In terms of the coordinate r * = log |r|, the equation of motion for the radial functions h (σ) k (r) takes the form of a one-dimensional Schrodinger equation:

- d 2 dr 2 * + µ 2 k e 2r * h (σ) k = Ω 2 h (σ) k , µ k = k 2 y + k 2 . (3.10)
At this point it is important to note that Ω is an arbitrary parameter that ranges from 0 to ∞, independently of the values of the momenta; the reason is that for any Ω > 0 the solution is always oscillatory for sufficiently small r * . The scalar field φ can be expanded either in terms of U k , U * k or in terms of u

(+) k , u (+) * k , u (-) k , u (-) * k
, as both sets constitute complete sets of modes3 :

φ = d D k (a k U k + ā † k U * k ) = ∞ 0 dΩ d D-1 k σ=± (b (σ) k u (σ) k + b(σ) † k u (σ) * k ) . (3.11)
The linear transformation from a k , a † k to b

(σ) k , b (σ)
k = d D k (α (σ) kk a k + β (σ) kk ā † k ) . (3.12)
The important implication is that the Minkowski vacuum state defined by

a k |0 M = āk |0 M = 0 ∀ k (3.13)
is not equivalent to the Fulling-Rindler vacuum state defined by

b (σ) k |0 R = b(σ) k |0 R = 0 ∀ σ, k . (3.14)
In particular, the expectation value of the Rindler particle number operator

N k = b (σ) † k b (σ) k in the Minkowski vacuum is d D k β kk β * k k , (3.15)
and as a result the Rindler observer is immersed in a thermal distribution of particles (see e.g. [START_REF] Takagi | Vacuum Noise And Stress Induced By Uniform Accelerator: Hawking-Unruh Effect In Rindler Manifold Of Arbitrary Dimensions[END_REF]).

We now aim to extend this analysis to the metric (3.5) adapted to the observer drifting with velocity v parallel to the acceleration horizon. As φ is a scalar field, we can obtain the solutions of the wave equation in the new coordinates from those in the original Rindler coordinates by simply making the substitution (3.4):

ũ(σ) k = θ(σr) [2Ω(2π) D-1 ] 1/2 h(σ) k (r) exp (i(-σ Ωη + ky ỹ + k • z)) (3.16)
where a Ω = γ(aΩ + σ vk y ) , ky = γ(k y + σ vaΩ) ,

The modes that were originally of positive (negative) norm still have positive (negative) norm, so the definition of the vacuum state is not affected by the boost [START_REF] Letaw | The Quantized Scalar Field In The Stationary Coordinate Systems Of Flat Space-Time[END_REF]. This result shows that there is no change in the definition of a "particle" as the drift velocity v is changed, so a distribution of particles at one velocity can be obtained from the distribution at zero velocity by the method of Pauli and Peebles and Wilkinson. From this perspective, it is a surprise that our explicit computation of the detector response yields a different distribution, at least for large drift velocity or, in the case of small small drift velocity, away from the UV limit. However, things are not quite so simple. As pointed out above, Ω and k y take independent values, 0 < Ω < ∞ and -∞ < k y < ∞. This has the implication that a mode of positive (negative) frequency before the Lorentz boost, becomes negative (positive) frequency after the boost if vk y < -Ω. This implies an instability of the vacuum [START_REF] Korsbakken | The Fulling-Unruh effect in general stationary accelerated frames[END_REF], which can also be understood from the fact that there is an ergo-region outside the acceleration horizon. Ergo-regions of stationary rotating black holes lead to a spontaneous emission of radiation that has the effect of reducing the angular momentum and hence shrinking the ergo-region. We expect the same here, with the difference that angular momentum becomes linear momentum. This could go some way to explaining why the spectrum of particles detected by a particle detector is not simply the Doppler-shifted version of the spectrum at zero velocity.

Discussion

Unruh has shown how the local temperature of a static black hole may be understood kinematically near its horizon from the response of a static particle detector to the acceleration needed to maintain it at a fixed distance from the horizon. The detector effectively accelerates in a 2-dimensional Minkowski space; equivalently, it is static in the 2-dimensional Rindler spacetime. Applying this insight to static p-branes, one has the additional possibility of considering a detector in motion parallel to the horizon -in particular, motion at constant velocity. The metric 'adapted' to such a detector is still stationary, so one expects a time-independent response. On the other hand, the motion near the horizon can be understood as one of "acceleration with drift" in a (2 + p)-dimensional Minkowski spacetime; as this is still a "stationary motion" (in the sense that all extrinsic geometric invariants of the detector's worldline are constant) one again expects a time-independent response. The latter interpretation allows a computation, from first principles, of the response of an Unruh-DeWitt monopole detector coupled to a massless scalar field. This is especially simple for p = 2, which is the case we consider. For zero drift velocity, the computation is a standard generalization of Unruh's original computation, and can be found in e.g. [START_REF] Birrell | Quantum Fields In Curved Space[END_REF]; the excitation probability rate of the detector, as a function of its energy gap, is Planckian. For non-zero drift velocity, the excitation probability rate is direction dependent; the result after integration over angles has been considered previously [START_REF] Kolbenstvedt | Anisotropy of thermal radiation by accelerated detectors; an aberration effect[END_REF] but our computation yields the full angular dependence.

Given this result, one can choose to consider only the response in directions orthogonal to the acceleration; equivalently in directions parallel to the 2-brane. From the perspective of a detector that is constrained to move in this plane, and to detect only particles that arrive from other points on it, it is effectively in a (1 + 2)-dimensional Minkowski radiation heat bath, at the Unruh temperature. We have called this the 'braneworld' model of a Minkowski heat bath.

For low drift velocities, and in the UV limit in which the modes of the scalar field can be viewed as massless particles of a classical gas, we find that the response spectrum of the detector is just a Doppler-shifted version of the Planck spectrum, characterized by an angular-dependent "effective temperature". This is just what would be predicted by the standard PPW theory of inertial motion in a radiation heat bath. However, our computation does not agree, numerically, with the Doppler-shift formula in the IR limit, and gives a qualitatively different answer at large drift velocities.

This result raises several issues. At the very least, it provides a further example of how a particle detector in stationary motion does not always respond in the way that one might expect. In the special case of pure acceleration, the thermal response of the detector is often attributed to the different global features of Minkowski and Rindler spacetimes, and of the resulting difference between the Minkowski and Fulling-Rindler vacua. However, this cannot be the whole story because there are more distinct stationary motions than there are distinct quantum vacua, at least if two vacua are considered equivalent if the Bogoliubov transformation connecting them is trivial. We have already mentioned that the vacuum for acceleration with drift is equivalent, in this sense, to the Fulling-Rindler vacuum. It was further shown in [START_REF] Letaw | The Quantized Scalar Field In The Stationary Coordinate Systems Of Flat Space-Time[END_REF] that for any stationary motion the quantum vacuum is equivalent, in this sense, to either the Minkowski vacuum or the Fulling-Rindler vacuum. In contrast, the local response of particle detectors on stationary worldlines is far more variable, being essentially different for each distinct stationary motion, and it is a challenge to understand why this is so (see e.g. [START_REF] Davies | The Rotating Quantum Vacuum[END_REF][START_REF] Sriramkumar | Probes of the vacuum structure of quantum fields in classical backgrounds[END_REF][START_REF] Korsbakken | The Fulling-Unruh effect in general stationary accelerated frames[END_REF]).

This issue was addressed for circular motion in [START_REF] Davies | The Rotating Quantum Vacuum[END_REF]. The quantum vacuum for a detector undergoing uniform motion in a circle is actually equivalent to the Minkowski vacuum, despite the centripetal acceleration, so why does the detector detect particles? One may guess that the answer must have to do with the coordinate singularity at a critical radius in static corotating cylindrical polar coordinates, and it was shown in [START_REF] Davies | The Rotating Quantum Vacuum[END_REF] that a co-rotating detector in the Minkowski vacuum does not detect particles if one assumes boundary conditions that remove the region of spacetime beyond the critical radius. For acceleration with drift, the resolution appears to be different. As we have mentioned, the appearance of an ergo-region outside the Rindler horizon is also expected to have an effect on a particle detector. However, it is difficult to understand how this could explain the "cone of silence" that we have found above the critical velocity v c ∼ 0.2172.

Although we arrived at the "accelerating braneworld" model of a radiation heat bath by consideration of motion near the horizon of a black brane, one may wonder whether it has a more general applicability. As we have pointed out in the Introduction, this model is an essential ingredient of the GEMS approach to the thermodynamics of asymptotically flat black holes because each static observer in the asymptotic Minkowski spacetime at the Hawking temperature T H has, from the GEMS perspective, constant proper acceleration 2πT H / in the embedding spacetime. Why the Unruh formula should 'work' in this context is something of a mystery; after all, the embedding spacetime is ostensibly no more than a mathematical construct, and there is no suggestion that the acceleration has any effect other than to produce a temperature. Nevertheless, the success of the GEMS program encourages the idea that the braneworld model of a radiation heat bath is of general applicability.

These considerations lead us to question the standard "Doppler-shift" theory of motion in a radiation heat bath. In the first place, the assumption that the energy eigenstates, of energy ω, of a quantum scalar field at temperature T behave like particles of a classical gas is valid only in the UV limit in which ω T . One might expect angular-dependent induced emission effects to become important in the opposite, IR, limit. This is indeed what we have found from the "accelerating braneworld" model, at least for small drift velocity. One may then ask what implication our results would have for the detection of photons in the CMBR if some analogous computation could be done in one higher dimension. To get some flavour of this, we conclude by exploring further the deviation of our formula (2.34) from the Doppler-shifted formula (2.31) for v = 0.002, which happens to be the approximate velocity of our local group of galaxies with respect to the CMBR rest frame. One finds that ∆n (2) /n (2) ∼ 10 -4 , with the largest deviation occurring at ϕ = π. However, if one focuses on the dipole term proportional to the detector velocity then one finds an appreciable discrepancy when ω ∼ T that climbs to about a factor of 2 as ω/T → 0, although there is still agreement in the UV limit ω/T → ∞. This is shown in fig. 3. A factor of 2 variation is fairly large, but it is possible that this factor would be found to be much smaller if we could compute the analogous result for motion in a (1 + 3)-dimensional Minkowski heat bath, in which case there would not be any obvious conflict with experimental data. One should also appreciate that since the CMBR experiment is calibrated on the dipole assuming a frequency independent dipole factor, a small frequency dependence might get interpreted as an anomaly elsewhere. Our results suggest that it might be worthwhile to look for such frequency dependence in the CMBR data, or to analyze the implications for calibrations of assuming frequency independence.

  tan x * = x * , x * ≈ 4.4934 , u * = cos x * ≈ -0.2172 . (2.24) Since u = v cos ϕ sin θ, this critical point exists only at velocities v > v c = |u * | ≈ 0.2172. At any velocity v > v c , there is no pole of the integrand of (2.19) within the contour whenever

Figure 1 :

 1 Figure 1: Spectra at (a) v = 0.04 and (b) 0.8, and ϕ = 0. The dashed line represents the Doppler-shifted spectrum.

Figure 2 :

 2 Figure 2: Spacetime diagram for a detector at r = r0 on a stationary trajectory of acceleration with drift velocity v.There is an ergo-region for the natural time-like Killing vector field between the acceleration horizon and the static-limit surface at r = vr0.

  † k is a Bogoliubov transformation, which mixes modes of positive and negative norm; it takes the form b (σ)

Figure 3 :

 3 Figure 3: Dipole strength registered by the detector in 1 + 2 dimensions relative to the Doppler shifted Planck spectrum. The plot is drawn at the CMBR temperature T = 2.725 K in the frequency range for which CMBR data (in 1+3 dimensions!) are available.

It actually turns out to be direction-independent in the detector's rest frame but this is a special feature of scalar radiation.

We remind the reader that the spacetime dimension is (D+1). The integration is over the D space dimensions.

Acknowledgements: We thank an anonymous referee of an earlier version of this paper for emphasizing the importance of the frequency dependence of the CMBR dipole moment. We also thank D. Kothawala for a useful discussion. JGR acknowledges support by research grants MCYT FPA 2007-66665, 2005SGR00564. PKT thanks the EPSRC for financial support, and the University of Barcelona for hospitality.