Miranda Jj

MD Geelhoed

MD PhD Vincent Wv Jaddoe

MD J J M Geelhoed
email: j.geelhoed@erasmusmc.nl

EARLY INFLUNCES ON CARDIOVASCULAR AND RENAL DEVELOPMENT

Keywords:

The hypothesis that a developmental component plays a role in subsequent disease initially arose from epidemiological studies relating birth size to both risk factors for cardiovascular disease and actual cardiovascular disease prevalence in later life. The findings that small size at birth is associated with an increased risk of cardiovascular disease have led to concerns about the effect size and the causality of the associations. However, recent studies have overcome most methodological flaws and suggested small effect sizes for these associations for the individual, but an potential important effect size on a population level. Various mechanisms underlying these associations have been hypothesized, including fetal undernutrition, genetic susceptibility and postnatal accelerated growth. The specific adverse exposures in fetal and early postnatal life leading to cardiovascular disease in adult life are not yet fully understood. Current studies suggest that both environmental and genetic factors in various periods of life may underlie the complex associations of fetal growth retardation and low birth weight with cardiovascular disease in later life. To estimate the population effect size and to identify the underlying mechanisms, well-designed epidemiological studies are needed. This review is focused on specific adverse fetal exposures, cardiovascular adaptations and perspectives for new studies.

INTRODUCTION

Many epidemiological studies from different geographical regions demonstrated consistent association between low birth weight and the risk of cardiovascular disease. [START_REF] Huxley | Is birth weight a risk factor for ischemic heart disease in later life?[END_REF][START_REF] Osmond | Early growth and death from cardiovascular disease in women[END_REF][START_REF] Lawlor | Birth weight is inversely associated with incident coronary heart disease and stroke among individuals born in the 1950s: findings from the Aberdeen Children of the 1950s prospective cohort study[END_REF] These associations can not be explained by preterm birth. [START_REF] Barker | Fetal origins of coronary heart disease[END_REF][START_REF] Barker | The relation of small head circumference and thinness at birth to death from cardiovascular disease in adult life[END_REF] Also, they seem to be independent of influences in adult life including social class, obesity and smoking habits. [START_REF] Frankel | Birthweight, body-mass index in middle age, and incident coronary heart disease[END_REF][START_REF] Stampfer | Birth weight and risk of cardiovascular disease in a cohort of women followed up since 1976[END_REF] The mechanisms underlying these associations are not well known. It has been suggested that developmental adaptations due to suboptimal fetal nutrition permanently program the fetus and lead to an increased risk of coronary heart disease many decades later. [START_REF] Barker | Infant mortality, childhood nutrition, and ischaemic heart disease in England and Wales[END_REF] It has also been argued that the associations between size at birth and later disease could primarily be the result from common genetic influences. Based on more recent studies, a more general developmental plasticity hypothesis has been proposed. [START_REF] Gluckman | Effect of in utero and early-life conditions on adult health and disease[END_REF] Developmental plasticity is defined as the ability of an organism to develop in various ways, depending on the particular environment or setting. [START_REF] Bateson | Developmental plasticity and human health[END_REF] In this process, early environmental influences induces anatomical, physiological and biochemical adaptations in later life. These adaptations may be beneficial for short term survival but could have adverse long term consequences. This latter conceptual basis is extended with the Predictive Adaptive Response (PAR) theory, stating that long-term consequences of these early environmental influences may be especially harmful if the actual postnatal, mature, environment differs from the environment predicted during the plastic, developmental, phase. [START_REF] Gluckman | Life-long echoes--a critical analysis of the developmental origins of adult disease model[END_REF] The greater the degree of mismatch, the greater the disturbance in physiology and the greater the risk of disease.

Studies in humans generally use birth weight as measure of adverse fetal exposure.

Birth weight might indeed be an indicator of the quality of the intra-uterine environment. [START_REF] Brooks | Birth weight: nature or nurture?[END_REF] However, the same birth weight might be the result of different growth patterns and exposures in fetal life. Furthermore, a period of compensatory growth will follow after nutritional deficit in utero. The most rapid growth acceleration in low birth weight children occurs in the first weeks after birth. [START_REF] Hokken-Koelega | Children born small for gestational age: do they catch up?[END_REF] This accelerated growth rate might also have important consequences later in life. [START_REF] Eriksson | Catch-up growth in childhood and death from coronary heart disease: longitudinal study[END_REF] These findings suggest that children with both restricted fetal and infant growth and accelerated childhood weight gain are at particular risk of cardiovascular disease in adult life and its risk factors. [START_REF] Eriksson | Early growth and coronary heart disease in later life: longitudinal study[END_REF][START_REF] Roseboom | Coronary heart disease after prenatal exposure to the Dutch famine, 1944-45[END_REF] In this paper, we discuss epidemiological studies designed to identify mechanisms underlying the associations of fetal growth retardation and low birth weight with the development of cardiovascular disease in adulthood. We will focus on specific adverse fetal exposures, cardiovascular adaptations and perspectives for new studies.

SPECIFIC FETAL EXPOSURES

Fetal undernutrition

Maternal anthropometrics, maternal diet and placenta function

The fetal nutrition supply line includes maternal anthropometrics, diet and placenta function.

Maternal anthropometrics during pregnancy are related to their nutritional and health status. [START_REF] Bergmann | Energy intake and net weight gain in pregnant women according to body mass index (BMI) status[END_REF][START_REF] Siega-Riz | Biological determinants of pregnancy weight gain in a Filipino population[END_REF] Several studies have shown that maternal anthropometric factors such as prepregnancy body mass index and weight gain during pregnancy are associated with fetal growth characteristics and adverse pregnancy outcomes. [START_REF] Kabiru | Obstetric outcomes associated with increase in BMI category during pregnancy[END_REF][START_REF] Jensen | Pregnancy outcome and prepregnancy body mass index in 2459 glucose-tolerant Danish women[END_REF][START_REF] Ay | Maternal anthropometrics are associated with fetal size in different periods of pregnancy and at birth. The Generation R Study[END_REF] Thus, the maternal nutritional and health status during pregnancy may lead to an adverse fetal environment and might affect fetal growth and development (Table 1).

Studies examining the direct effect of maternal anthropometrics on cardiovascular outcomes in the offspring demonstrated conflicting results (Table 1). Most of these studies were based on retrospective cohorts. We found in a prospective cohort study that maternal weight gain during pregnancy is associated with larger left ventricular mass at the age of 6 months, independent of maternal weight just before pregnancy. [START_REF] Geelhoed | Maternal anthropometrics in pregnancy are associated with left ventricular mass in infancy. The generation R study[END_REF] Additionally, a recent study showed that greater maternal gestational weight gain is associated with greater offspring body mass index into early adulthood and that this may translate into higher systolic blood pressure in the offspring. [START_REF] Mamun | Associations of gestational weight gain with offspring body mass index and blood pressure at 21 years of age: evidence from a birth cohort study[END_REF] Several underlying biological pathways may explain the associations between maternal anthropometrics during pregnancy and vascular development and cardiac growth and development postnatally. One explanation could be that the usual increase in insulin resistance seen in late pregnancy is higher in mothers who show a marked increase in weight during pregnancy, resulting in increased fetal body and cardiac growth. [START_REF] King | Maternal obesity, metabolism, and pregnancy outcomes[END_REF][START_REF] Fowden | The role of insulin in prenatal growth[END_REF] Further studies, in which measures of insulin resistance such as insulin, glucose or HbA1c levels in pregnancy have been measured, are needed to identify the potential role of insulin resistance for fetal cardiovascular development. The major limitation of weight gain is that the total amount depends on both an increase in fluid and fat mass. Also, it is not clear whether maternal pre-pregnancy weight or weight gain during pregnancy affects cardiovascular development in the offspring.

The fetal nutrition supply line may also be directly affected by dietary intake of the mother during pregnancy. Follow-up of persons who were conceived during the Dutch famine during the second World War demonstrated a doubled risk and an earlier onset of coronary artery disease among subjects who were exposed to famine during fetal life. [START_REF] Roseboom | Coronary heart disease after prenatal exposure to the Dutch famine, 1944-45[END_REF] These results were independent of size at birth and current smoking and social economic status and suggest that severe maternal malnutrition during early gestation contributes to the occurrence of coronary artery disease in the offspring. However, these results could not be replicated in a cohort of adults who were exposed as fetus to severe famine during the siege of Leningrad. [START_REF] Sparen | Long term mortality after severe starvation during the siege of Leningrad: prospective cohort study[END_REF] The associations of less extreme variations in maternal intake of macronutrients and micronutrients with the development of risk factors for cardiovascular disease in the offspring have also been studied. Prospective cohort studies showed that variation in maternal total protein intake during pregnancy does not program offspring blood pressure already in infancy or in adolescence. [START_REF] Hawkesworth | Maternal protein-energy supplementation does not affect adolescent blood pressure in The Gambia[END_REF][START_REF] Huh | Maternal protein intake is not associated with infant blood pressure[END_REF][START_REF] Kinra | Effect of integration of supplemental nutrition with public health programmes in pregnancy and early childhood on cardiovascular risk in rural Indian adolescents: long term follow-up of Hyderabad nutrition trial[END_REF] Stronger relations have been reported between micronutrients and risk factors for cardiovascular disease in the offspring. Maternal calcium supplementation during pregnancy has been described to be associated with lower systolic blood pressure in the offspring. [START_REF] Belizan | Long-term effect of calcium supplementation during pregnancy on the blood pressure of offspring: follow up of a randomised controlled trial[END_REF][START_REF] Gillman | Maternal calcium intake and offspring blood pressure[END_REF][START_REF] Van Mierlo | Blood pressure response to calcium supplementation: a meta-analysis of randomized controlled trials[END_REF] Although, in some follow-up studies the effect disappeared later in childhood and no association were reported in twins. [START_REF] Bakker | Maternal calcium intake during pregnancy and blood pressure in the offspring at age 3 years: a followup analysis of the Project Viva cohort[END_REF][START_REF] Morley | Maternal calcium supplementation and cardiovascular risk factors in twin offspring[END_REF] Therefore, it is still unclear whether ensuring adequate calcium intake among pregnant women could be a way to prevent hypertension in the next generation. Low folate and high homocystein levels, in combination or independently, have been shown to be risk factors for endothelial dysfunction and cardiovascular disease. [START_REF] Moat | Folate, homocysteine, endothelial function and cardiovascular disease[END_REF][START_REF] De Bree | Folic acid improves vascular reactivity in humans: a meta-analysis of randomized controlled trials[END_REF] In healthy infants, a relationship between maternal folate levels during pregnancy and vascular endothelial function was demonstrated. [START_REF] Martin | Endothelial function in newborn infants is related to folate levels and birth weight[END_REF] More recently developed statistical approaches, such as dietary patterns of maternal nutrition intake during pregnancy, may give more detailed inside in the development of risk factors for cardiovascular disease later in life.

Placental function, including placental weight and haemodynamic function is one of the most important determinants of the fetal supply line. Size of the placenta reflects only an indirect measure of its capacity to transfer nutrients to the fetus, but it is strongly associated with fetal size at birth. [START_REF] Walker | Pre-eclampsia[END_REF] A review focused on the associations of placental weight with the risk of cardiovascular disease in the offspring and reported no consistent associations. [START_REF] Godfrey | Fetal nutrition and adult disease[END_REF] Preeclampsia is considered as the most extreme form of haemodynamic placental dysfunction.

According to recent studies, women with prior preeclamptic pregnancies are at increased risk of cardiovascular disease. [START_REF] Smith | Pregnancy complications and maternal risk of ischaemic heart disease: a retrospective cohort study of 129,290 births[END_REF][START_REF] Callaway | The association of hypertensive disorders of pregnancy with weight gain over the subsequent 21 years: findings from a prospective cohort study[END_REF] Furthermore, preeclampsia has been associated with elevated blood pressure in offspring during childhood and adolescence. [START_REF] Tenhola | Blood pressure, serum lipids, fasting insulin, and adrenal hormones in 12-year-old children born with maternal preeclampsia[END_REF] Fetal growth restriction due to placental dysfunction or common genetic variants may at least partly explain these associations. A milder form of placental haemodynamic dysfunction is reflected by the resistance in the umbilical artery. An increased umbilical artery resistance is associated with fetal growth restriction and low birth weight. [START_REF] Baschat | Fetal growth restriction due to placental disease[END_REF]

Maternal smoking during pregnancy

Maternal smoking during pregnancy is strongly associated with fetal growth retardation. [START_REF] Higgins | Smoking in pregnancy[END_REF][START_REF] Ong | Size at birth and early childhood growth in relation to maternal smoking, parity and infant breast-feeding: longitudinal birth cohort study and analysis[END_REF][START_REF] Jaddoe | Maternal smoking and fetal growth characteristics in different periods of pregnancy: the generation R study[END_REF] The effects of smoking on placental vessels could be due to nicotine or hypoxia. This association is partly mediated by restricted blood flow in the vascular beds of the placenta, due to increased resistance of the umbilical-placental circulation. [START_REF] Larsen | Stereologic examination of placentas from mothers who smoke during pregnancy[END_REF][START_REF] Morrow | Maternal cigarette smoking: the effects on umbilical and uterine blood flow velocity[END_REF] Fetal exposure to maternal smoking might also have adverse and persistent consequences for cardiovascular growth and development. Recent studies showed associations between intra-uterine exposure to maternal smoking and high blood pressure in childhood and adulthood. [START_REF] Blake | Maternal cigarette smoking during pregnancy, low birth weight and subsequent blood pressure in early childhood[END_REF][START_REF] Brion | Modifiable maternal exposures and offspring blood pressure: a review of epidemiological studies of maternal age, diet, and smoking[END_REF] However, the association of maternal smoking and offspring blood pressure might be confounded by a comprehensive range of indicators of social economic position. Confounding by social and familial factors is further supported by the similarity of maternal and paternal smoking effects, suggesting that modest differences in childhood blood pressure associated with maternal smoking could be the result of not only biological effects on the intrauterine environment, but also common adverse familial factors. [START_REF] Brion | Similar associations of parental prenatal smoking suggest child blood pressure is not influenced by intrauterine effects[END_REF] Studies focused on fetal cardiovascular development may unravel these associations. We found in a prospective cohort study that maternal smoking during pregnancy is associated with placental and fetal haemodynamic adaptations indicating increased arterial resistance. [START_REF] Geelhoed | Early influences on cardiovascular and renal development[END_REF] These fetal haemodynamic adaptations were subsequently associated with fetal growth retardation and changes in postnatal cardiac structures. Similarly, maternal smoking in pregnancy has also been suggested to be directly associated with changes in fetal cardiac dimensions and volumes. [START_REF] Lampl | Growth patterns of the heart and kidney suggest inter-organ collaboration in facultative fetal growth[END_REF][START_REF] Severi | Intrauterine growth retardation and fetal cardiac function[END_REF]

Genetic susceptibility

The association of fetal growth restriction and low birth weight with the increased risk of cardiovascular disease may also be explained by common genetic variants related to insulin sensititivity or angiogenesis (Table 2). [START_REF] Hattersley | The fetal insulin hypothesis: an alternative explanation of the association of low birthweight with diabetes and vascular disease[END_REF] Insulin and insulin-like growth factors are important fetal growth factors. [START_REF] Fowden | The insulin-like growth factors and feto-placental growth[END_REF] Genetic factors related to insulin or insulin-like growth factors production and sensitivity may lead to both impaired fetal growth and to type 2 diabetes and cardiovascular disease in later life. [START_REF] Hattersley | The fetal insulin hypothesis: an alternative explanation of the association of low birthweight with diabetes and vascular disease[END_REF] Previously, Single Nucleotide Polymorphisms (SNPs) in the IGF1 gene and the INS VNTR gene have been suggested to be associated with fetal and postnatal growth. However, these effects seem not to be consistent. [START_REF] Geelhoed | Variation in the IGF1 gene and growth in foetal life and infancy. The Generation R Study[END_REF][START_REF] Do | Insulin gene variable number of tandem repeats is not associated with weight from fetal life until infancy: the Generation R Study[END_REF] Common polymorphisms of type 2 diabetes gene PPARγ2 may also explain previously suggested associations of growth in early life with the risk of cardiovascular disease in later life. Two large birth cohort studies found no association between the PPARγ2 polymorphism and birth weight. [START_REF] Bennett | No evidence that established type 2 diabetes susceptibility variants in the PPARG and KCNJ11 genes have pleiotropic effects on early growth[END_REF][START_REF] Pfab | Fetal and maternal peroxisome proliferatoractivated receptor gamma2 Pro12Ala does not influence birth weight[END_REF] We recently showed that the PPARγ2 Ala12 allele is associated with an increased growth rate in early life. This effect was modified by the duration of breastfeeding. [START_REF] Do | Breast-feeding modifies the association of PPARgamma2 polymorphism Pro12Ala with growth in early life: the Generation R Study[END_REF] In addition to the PPARγ2 polymorphism, other common type 2 diabetes genetic susceptibility variants seem to affect size at birth directly through the fetal genotype.

Risk alleles at CDKAL1 and HHEX-IDE were both associated with reduced birth weight.

These findings suggest the associations between low birth weight and type 2 diabetes might at least partly explained by common genetic variants. [START_REF] Freathy | Type 2 Diabetes Risk Alleles are Associated with Reduced Size at Birth[END_REF] Furthermore, it has been demonstrated that the PPARγ2 polymorphism modifies the associations of low birth weight with lipid levels and hypertension. [START_REF] Eriksson | The effects of the Pro12Ala polymorphism of the PPARgamma-2 gene on lipid metabolism interact with body size at birth[END_REF][START_REF] Yliharsila | Interactions between peroxisome proliferator-activated receptor-gamma 2 gene polymorphisms and size at birth on blood pressure and the use of antihypertensive medication[END_REF] Glucocorticoids are important regulators of cardiovascular function and metabolism.

Studies in rats showed that activity of placental 11β-hydroxysteroid dehydrogenase type 2, which converts physiological glucocorticoids to inactive products, correlates positively with birth weight and negatively with placental weight. [START_REF] Benediktsson | Glucocorticoid exposure in utero: new model for adult hypertension[END_REF] In addition, administration of low-dose dexamethasone to pregnant rats not only reduces birth weight but also leads to high blood pressure in young adult offspring. [START_REF] Benediktsson | Glucocorticoid exposure in utero: new model for adult hypertension[END_REF] In human studies, it is demonstrated that fetuses with the greatest exposure to maternal glucocorticoids have low birth weight and high placental weight and might be at a higher risk of subsequent hypertension. [START_REF] Barker | Fetal and placental size and risk of hypertension in adult life[END_REF] Increased exposure to cortisol in adults leads to increased risks of cardiovascular disease, type 2 diabetes and obesity. [START_REF] Gluckman | Effect of in utero and early-life conditions on adult health and disease[END_REF][START_REF] Whincup | Birth weight and risk of type 2 diabetes: a systematic review[END_REF] Thus higher exposure to glucocorticoids in fetal and early postnatal life might affect cardiovascular development in fetal life and early childhood. The effects of these hormones, including cortisol, are mediated by glucocorticoid receptors. Glucocorticoid receptor gene (NR3C1) SNPs may explain part of the associations between growth characteristics in early life and disease in adulthood by increasing glucocorticoid sensitivity in the fetus for maternal glucocorticoids. A previous study in adults found an association between the GR-9β polymorphism and an increased risk of cardiovascular disease. [START_REF] Van Den Akker | Glucocorticoid receptor gene and risk of cardiovascular disease[END_REF] We found an association of this GR-9β polymorphism with increased left ventricular mass and systolic blood pressure in children aged two years. [START_REF] Geelhoed | Early influences on cardiovascular and renal development[END_REF] No associations were observed between this GR-9β polymorphism and fetal and early postnatal growth. [START_REF] Geelhoed | Glucocorticoid receptor gene polymorphisms do not affect growth in fetal and early postnatal life. The Generation R Study[END_REF] Other recent identified glucocorticoid related polymorphisms, such as the brain-derived neurotrophic factor (BDNF) and the mineral corticoid gene, may affect fetal and postnatal growth by influencing the glucocorticoid metabolism.

Thus fare, results of studies focused on the associations of common genetic variants related to both early growth and the risk of cardiovascular disease later in life are inconsistent.

Further systematic searches for common genetic variants by means of genome-wide association studies will enable us to obtain a more complete understanding of what genes and polymorphisms are involved in both growth in fetal life and infancy and development of cardiovascular disease in adulthood.

CARDIOVASCULAR ADAPTATIONS

Cardiac development

Stimuli for cardiac development

It has been demonstrated that fetuses have stiffer fetal ventricles than neonates and the diastolic filling patterns in normally grown fetuses mimic those of the diseased adult heart. [START_REF] Veille | Ventricular filling patterns of the right and left ventricles in normally grown fetuses: a longitudinal follow-up study from early intrauterine life to age 1 year[END_REF] Left ventricular elastic compliance increases with gestational age and left ventricular stiffness significantly decreases. In growth restricted fetuses, this process may be affected. [START_REF] Veille | Ventricular filling patterns of the right and left ventricles in normally grown fetuses: a longitudinal follow-up study from early intrauterine life to age 1 year[END_REF] Furthermore, reduced end-diastolic flow velocities in fetal umbilical and maternal uteroplacental arteries have been associated with increased peripheral vascular resistance. [START_REF] Schulman | The clinical implications of Doppler ultrasound analysis of the uterine and umbilical arteries[END_REF] This increase in peripheral arterial resistance and subsequently in fetal cardiac afterload may lead to an increase in fetal cardiac performance and to persistent structural left ventricular changes. [START_REF] Rizzo | Doppler echocardiographic assessment of atrioventricular velocity waveforms in normal and small-for-gestational-age fetuses[END_REF] Decreased fetal growth was also associated with fetal adaptations in cardiac function in the whole range of fetal growth. [START_REF] Verburg | Fetal hemodynamic adaptive changes related to intrauterine growth: the Generation R Study[END_REF] Furthermore, birth weight and measures of placental vascular resistance and fetal cardiac output were associated with left cardiac structures until the age of two years. [START_REF] Geelhoed | Cardiac structures track during the first 2 years of life and are associated with fetal growth and hemodynamics: the Generation R Study[END_REF] This is in line with a previous prospective cohort study in children which demonstrated positive associations of birth weight with total coronary artery diameter, aortic root diameter and left ventricular outflow tract diameter in children aged 9 years. [START_REF] Jiang | Birth weight and cardiac structure in children[END_REF] However, inverse associations between birth weight and left ventricular mass in adolescents and adults have been demonstrated. [START_REF] Vijayakumar | Birth weight, weight at one year, and left ventricular mass in adult life[END_REF][START_REF] Zureik | Weights at birth and in early infancy, systolic pressure, and left ventricular structure in subjects aged 8 to 24 years[END_REF] It should be investigated whether these relations persist during later life and are related to the development of cardiovascular disease in adulthood.

Adverse cardiac adaptations

Studies in children showed that left ventricular mass tracks from childhood to adulthood. [START_REF] Janz | Predicting heart growth during puberty: The Muscatine Study[END_REF][START_REF] Urbina | Effect of body size, ponderosity, and blood pressure on left ventricular growth in children and young adults in the Bogalusa Heart Study[END_REF] This implies that children with smaller left cardiac structures in early postnatal life tend to keep their relatively smaller left cardiac structures in childhood. [START_REF] Geelhoed | Cardiac structures track during the first 2 years of life and are associated with fetal growth and hemodynamics: the Generation R Study[END_REF][START_REF] Janz | Predicting heart growth during puberty: The Muscatine Study[END_REF][START_REF] Urbina | Effect of body size, ponderosity, and blood pressure on left ventricular growth in children and young adults in the Bogalusa Heart Study[END_REF] However, a relatively smaller left ventricle and aortic root may lead to insufficient cardiac functioning for increasing metabolic demands in postnatal life. Subsequently, the heart may respond by growth and adverse remodeling. Since the number of heart cells is established largely in fetal life, adaptation and growth of existing cells may eventually lead to left ventricular dysfunction and hypertrophy. Left ventricular hypertrophy is a strong and independent risk factor of cardiovascular morbidity and mortality in adulthood. [START_REF] Levy | Prognostic implications of echocardiographically determined left ventricular mass in the Framingham Heart Study[END_REF][START_REF] Toprak | Relation of childhood risk factors to left ventricular hypertrophy (eccentric or concentric) in relatively young adulthood (from the Bogalusa Heart Study)[END_REF] Thus fetal exposures affecting cardiac development may have life long consequences.

Renal development

Stimuli for renal development

Nephrons start to form from day 30 of gestation. [START_REF] Nigram | Development and maturation of the kidney. Brenner and Rector's the Kidney[END_REF] Numerous factors, including the reninangiotensin system (RAS), various growth factors, apoptosis and an adequate supply of nutrients are required for nephrogenesis. [START_REF] Guron | An intact renin-angiotensin system is a prerequisite for normal renal development[END_REF][START_REF] Hammerman | Growth factors in renal development[END_REF][START_REF] Merlet-Benichou | Influence of fetal environment on kidney development[END_REF] Nephrogenesis continues until 36 weeks of gestation and the induction of nephron number ceases thereafter. [START_REF] Hinchliffe | Human intrauterine renal growth expressed in absolute number of glomeruli assessed by the disector method and Cavalieri principle[END_REF][START_REF] Nyengaard | Glomerular number and size in relation to age, kidney weight, and body surface in normal man[END_REF] On average ±750,000 nephrons per kidney are present, with a wide interindividual range (250,000 -2,000,000). [START_REF] Hughson | Glomerular number and size in autopsy kidneys: the relationship to birth weight[END_REF][START_REF] Hughson | Hypertension, glomerular number, and birth weight in African Americans and white subjects in the southeastern United States[END_REF][START_REF] Keller | Nephron number in patients with primary hypertension[END_REF][START_REF] Nyengaard | Glomerular number and size in relation to age, kidney weight, and body surface in normal man[END_REF] Animal studies have shown that various determinants of fetal nutrition including low protein intake, relative vitamin A deficiency, reduced placenta perfusion and administration of steroids in late pregnancy cause fetal growth restriction, smaller kidneys and a permanent reduced nephron number. [START_REF] Langley-Evans | Fetal exposure to a maternal low protein diet impairs nephrogenesis and promotes hypertension in the rat[END_REF][START_REF] Lelievre-Pegorier | Mild vitamin A deficiency leads to inborn nephron deficit in the rat[END_REF][START_REF] Bassan | Experimental intrauterine growth retardation alters renal development[END_REF][START_REF] Celsi | Prenatal dexamethasone causes oligonephronia, sodium retention, and higher blood pressure in the offspring[END_REF] Human studies demonstrated that low birth weight infants have lower kidney weight with a reduced number of nephrons. [START_REF] Hinchliffe | The effect of intrauterine growth retardation on the development of renal nephrons[END_REF][START_REF] Manalich | Relationship between weight at birth and the number and size of renal glomeruli in humans: a histomorphometric study[END_REF] It was also demonstrated that hypertensive subjects have lower nephron numbers. 100 The best surrogate measure for assessing nephron number in epidemiological studies appears to be kidney weight or size measured by ultrasound. [START_REF] Nyengaard | Glomerular number and size in relation to age, kidney weight, and body surface in normal man[END_REF] A few studies investigated renal length or volume in early life.

Being small for gestational age (SGA) is associated with small kidneys at birth and impaired kidney growth in early childhood. 101 Kidney length in preterm SGA infants follows closely the other anthropometric parameters during the first year of life. 102 SGA term infants had shorter kidney length at birth compared to appropriate for gestational age infants, but a similar length from 3 rd to 24 th month of life. 103 This may represent either an accelerated renal maturation or early compensatory kidney hypertrophy in the SGA infants. Maternal anthropometrics, fetal abdominal circumference, fetal blood flow redistribution, and raised placental resistance are associated with both third trimester fetal kidney volume and kidney volume at the age of two years. 104,105 Since, kidney size tracks from third trimester of pregnancy to early childhood, these adaptations may have persistent consequences. 104

Adverse renal adaptations

The kidneys respond to this reduced number of nephrons by hyperperfusion and remodelling. 106 According to the hyperfiltration theory, this will lead to more sodium reabsorption, increased systolic blood pressure and albuminuria. 107 This process may be in favor of short-term renal function but may eventually lead to glomerular hypertrophy and damage. 106 Finally, this may predispose the individual to renal failure and hypertension. It has been shown that low birth weight is associated with early onset chronic renal failure. In subjects aged less than 50 years, those who weighted less than 2.5 kg at birth had a higher risk for end-stage renal disease than people who weighted 3-3.5 kg at birth. 108 This association was shown in all groups of primary causes of end stage renal failure in adults (hypertension, diabetes and other causes). Studies in younger subjects have focused on urine albumin excretion, a predictor of cardiovascular and renal disease in diabetic and non-diabetic subjects. 109 Low birth weight is associated with microalbuminuria in children and adults independent of blood pressure and measures of insulin resistance. 110,111 The pathway leading from small kidneys to hypertension may include the renin-angiotensin system, which has been demonstrated to be altered in the early phase of primary hypertension. 112 An increased activity of the renin-angiotensin system could be a compensatory mechanism in a decreased number of nephrons in order to maintain normal renal filtration. It has been demonstrated that renin activity in umbilical cord blood is inversely related with the size of the kidney at birth. 113

Vascular development

Haemodynamic stimuli for fetal vascular development

Fetal growth retardation leads to a preferential blood flow to the brain and heart, which deprives other organs of adequate oxygen and nutrients supply. This brain-sparing suggests organ specific vasodilatation and vasoconstriction and has been demonstrated in growth restricted fetuses. 114 In addition, we recently demonstrated that decreased fetal growth is associated with adaptive fetal cardiovascular changes. [START_REF] Verburg | Fetal hemodynamic adaptive changes related to intrauterine growth: the Generation R Study[END_REF] These changes already occured before the onset of clinically apparent fetal growth restriction. Future studies are needed to identify whether these adaptations in fetal haemodynamics have consequences for the development of cardiovascular disease in later life.

Endothelial function

The endothelium controls vascular tone, coagulation and inflammatory responses. 115 Endothelial dysfunction is an early event of atherosclerosis, preceding structural changes in the vascular wall. 116 Atherosclerosis is thought to begin in childhood and to develop silently before clinical events such as myocardial infarction or stroke occur. Many studies demonstrated atherosclerotic wall thickening in the arteries of children with cardiovascular risk factors using ultrasound imaging. 117 Furthermore, studies have shown that risk factors for cardiovascular disease measured in childhood are track into adulthood. 118,119 There is limited direct evidence that risk factors measured in childhood are predictive of atherosclerosis in adulthood. 120,121 Only a few studies examined the associations of fetal and maternal factors during pregnancy and vascular changes in childhood. Low maternal folate levels during pregnancy and low birth weight are demonstrated to be associated with vascular endothelial dysfunction in newborn infants. [START_REF] Martin | Endothelial function in newborn infants is related to folate levels and birth weight[END_REF]122,123 Further follow up studies are needed.

Arterial stiffness

Flow may determine vascular growth in the fetal cardiovascular system and thereby a reduction in flow may alter later vascular behaviour. 124 One mechanism that could underlie the association of low birth weight with raised blood pressure may be a suboptimal development of the fetal cardiovascular system due to these circulatory changes, thus increasing the stiffness of the vessel wall and comprising the cardiovascular function. 125 Several studies have attempted to test this hypothesis by investigating the relationship between birth weight and indicators of arterial stiffness such as pulse wave velocity and pulse pressure, but with conflicting results. [126][127][128][129][130] Another mechanism could be that these haemodynamic adaptive changes in the fetal circulation, which occur in a critical period of blood vessel development, may influence rates of elastin synthesis. 129 This may result in a reduced compliance of the large arteries, leading to higher pulse and mean blood pressures.

Furthermore, the process of aging causes a gradual loss of elastin and replacement by collagen, which amplifies the increase in blood pressure. 130 Persistent alteration in conduit artery function and arterial stiffness may predispose a person to hypertension and cardiovascular disease in adulthood.

PERSPECTIVES Study designs

The associations of fetal growth retardation and low birth weight with cardiovascular disease

are not yet fully understood. Methodological issues, such as the role of potential confounders in the associations and the population effect size, should be further examined. Well-designed epidemiological studies are necessary to overcome current methodological issues. Populationbased prospective cohort studies starting in the preconceptional period or in early fetal life, in which the offspring is followed from early fetal life until young adulthood, seem to be the most suitable epidemiological design (Figure 1). This epidemiological design is the best design for assuring quality of the data, taking account for potential confounders and identifying growth patterns at risk. Recently, various population-based prospective cohort studies have been started worldwide. [131][132][133][134] However, a limitation of this design is the long period needed for the studied adult disease to develop. The design of a retrospective cohort study may lead to an earlier availability of data according to cardiovascular disease. However, this design will not take account of all potential confounders and will probably not be able to study the effects of fetal or early postnatal influences.

The effects of many adverse fetal exposures due to maternal life style habits can only be studied in observational designs. However, for assessing the effects of nutritional exposures on fetal growth restriction and the risk of cardiovascular disease, randomized controlled trials would be the design of choice. For example the impact of calcium, folate and other micronutrients on offspring birth weight and blood pressure is examined in randomized controlled trials. [START_REF] Belizan | Long-term effect of calcium supplementation during pregnancy on the blood pressure of offspring: follow up of a randomised controlled trial[END_REF]135,136 Randomized controlled trials should be able to overcome current methodological issues and identify mechanisms in the causal pathway that underlie the associations.

Detailed exposure studies

Nutrition

In Western countries, maternal variation in dietary patterns and intake of micronutrients seems more relevant than a restricted total energy and macronutrient intake for having an effect on fetal growth retardation. 137 Future studies should focus on the intake of particular macronutrients and micronutrients, including calcium, folic acid and homocystein, in specific periods of pregnancy. Identifying critical periods in pregnancy will result in detailed dietary advices aimed at these specific time points in pregnancy. Additionally, dietary patterns during the preconception period have been associated with biomarkers concentrations and related to complex diseases, such as cardiovascular disease. 138,139 Overall dietary patterns may be easier for the public to interpret or translate into diets.

Genome-wide association studies

Common genetic variants may explain the associations between growth in early life and diseases in adulthood. Large scale genome-wide association studies (GWAS) have revealed links between DNA sequence variation and a growing range of diseases and continuous traits, such as cardiovascular disease and type 2 diabetes. [140][141][142] The effect of these common genetic variants on growth characteristics and growth patterns may already be present in fetal and early postnatal life. Relating these genetic loci to early growth patterns might identify genes related to both fetal growth retardation and cardiovascular disease.

Epigenetics

Epigenetics is understood as the heritable changes in gene expression potential that are not caused by changes in DNA sequence. 143,144 DNA methylation is one of the best understood epigenetic modification and is a key epigenetic contributor to maintenance of gene silencing.

CpG islands are mostly located in the control regions of genes, for example the promoter region of actively transcribed gene, and are generally unmethylated. Methylation of this promoter region leads to decreased binding of transcription factors and thereby to a reduced gene expression. 145 Maternal diet has been shown to dynamically affect DNA methylation status. 146 Methyl-supplemented diets include the methyl donors such as folic acid, vitamin B12, choline, L-methionine and zinc. Studies in sheep showed associations between reductions in the availability of Vitamin B12 and folate during the periconceptual period and alterations in methylation status of CpG islands in the offspring. 147 This may lead to widespread epigenetic modifications to the genome of the offspring. In humans, lower DNA methylation of the IGF2 gene was reported in adults who have been exposed to the Dutch Famine in the periconceptional period. 148 These results suggest that early-life environmental conditions can cause epigenetic changes that persist throughout life and could lead to the development of risk factors for hypertension and cardiovascular disease in adult offspring.

However, future research is needed to better understand the role of epigenetic dysregulation in the associations between adverse fetal nutritional exposures and diseases in adult life. 149 Since the periconceptional period is a crucial period for establishing and maintaining epigenetic marks, cohort studies starting in the preconceptional period would give important knowledge. 150

Paternal influences

We discussed substantial affects on fetal programming by maternal prenatal behaviors, specifically by maternal smoking, maternal diet and maternal genetic susceptibility. However, paternal influences, for example paternal genetic susceptibility, could also underlie the pathways of fetal growth retardation and low birth weight to the development of cardiovascular disease in adulthood. Also, comparing maternal and paternal effects provides a method of separating intra-uterine effects from associations related to familial or environmental factors. Similar associations of maternal and paternal exposures with offspring health outcomes suggest that common factors related to the family or environment may drive the associations, rather than intrauterine effects only. Future research is required to compare associations of both maternal and paternal exposures during pregnancy with components of offspring health.

Detailed cardiovascular development studies

To disentangle the mechanisms underlying the associations of fetal growth restriction and low birth weight with the development of cardiovascular disease, future research should focus on early markers of cardiovascular adaptation. Good repeatability and reproducibility of most left cardiac structures in children measured by 2D ultrasound is previously demonstrated in largescale multicenter studies in young children. 151,152 However, newer imaging techniques, such as three-dimensional echocardiography and magnetic resonance imaging, can more precisely calculate physiologic variables of interest. 153 In the future, these new imaging techniques offer great opportunities for detailed cardiovascular measurements in epidemiological research.

Thereby, subtle developmental cardiovascular changes can be identified already in fetal or early postnatal life. Furthermore, focusing on other biomarkers of cardiac development, including detailed repeated measurements of right cardiac structures, may give newer inside in the pathway leading to the development of cardiovascular disease.

Although there is limited direct evidence that risk factors measured in childhood are predictive of atherosclerosis in adulthood 154,155 , less is known about the associations of fetal and maternal factors during pregnancy and vascular changes in childhood. Endothelial function and structural arterial changes, including arterial resistance and arterial stiffness, can be measured noninvasively en early childhood with high resolution ultrasound to measure brachial artery flow-mediated dilatation and carotid artery intima-media thickness, respectively. 156,157 Furthermore, future studies should focus on additional markers of vascular structures and function, including retinal arteriolar narrowing. This may lead to peripheral vascular resistance and thereby predispose a person to the development of hypertension and cardiovascular disease. 158

Clinical implications and future research

The associations between low birth weight and cardiovascular disease in later life seem to be one of the most intriguing and controversial epidemiological findings from the last decade and might have clinical and public health implications. However, the effect size of low birth weight on cardiovascular disease and blood pressure in adulthood presented in current studies seems to be small. [START_REF] Huxley | Is birth weight a risk factor for ischemic heart disease in later life?[END_REF]159 Furthermore, the specific adverse exposures and underlying mechanisms are not known. Recently, new areas of research in cardiovascular epidemiology have already become prominent. The focus has increasingly been on genes implicated in cardiovascular diseases. [160][161][162][163][164] However, the focus of most of the studies in cardiovascular epidemiology is still on putative risk factors [165][166][167][168][169][170][171][172][173] , with particular focus on nutrition 174-177 , on socio-economic factors and health inequalities [178][179][180][181][182][183][184][185][186][187][188][189][190][191][192][193] , and, more recently, on physical activity [194][195][196][197] . In addition, studies focus on methods of cardiovascular studies [198][199][200][201][202][203] and on frequency measures and trends in risks of cardiovascular morbidity and mortality [204][205][206][207][208][209][210][211][212][213] . For further exploring the underlying mechanisms and to assess potentials for clinical implications, research has to move onwards from birth weight association studies to adverse fetal exposures and detailed early cardiovascular development studies. Since plasticity operates across the entire range of environment and leads to multigenerational cycles of disease, any rational approach to health care should start early in life and take a cross-generational perspective. 214 Data of prospective cohort studies confirm the existence of a window of opportunity for intervention in early childhood. 215,216 The growing awareness of the importance of the preconceptual period, when nutrition can have long-lasting effects without causing any change in birth weight, underscores the importance of healthy nutrition during the prepregnancy period. Research should also focus on identifying individuals at greater risk of later poor health and develop strategies aimed at these specific individuals.

Conclusion

The associations between low birth weigh and cardiovascular disease are present within the normal and physiologic ranges, and suggest that specific exposures in different periods of fetal and early postnatal life have permanent consequences for cardiovascular growth and function. The mechanisms underlying these associations are not known, but may include environmental and genetic determinants. Future studies should be focused on these mechanisms and epigenetic modifications of specific genes related to cardiovascular development. Furthermore, studies should focus on detailed cardiovascular adaptive responses in fetal life and early childhood by use of new imaging and functional techniques. Eventually results from these studies may lead to improved health in childhood and adulthood by promoting a better fetal environment.

Figure 1 .

 1 Figure 1. A developmental origins study model, studying the fetal origins of cardiovascular disease in

First

Table 1 .

 1 Maternal anthropometrics and diet in association with cardiovascular disease later in life

	Maternal anthropometric characteristic
		Cardiovascular outcome
	during pregnancy
	Maternal anthropometric
		Increased death rates from coronary heart
	Short stature
		disease
	Low triceps skin fold thickness	Increased blood pressure
	High weight gain during pregnancy	Increased left ventricular mass
	Maternal diet
		Increased risk and earlier onset coronary artery
	Total energy intake
		disease
	Low protein intake	Increased blood pressure
	Low calcium intake	Increased blood pressure
	Low folate intake	Endothelial dysfunction

Table 2 .

 2 Common genetic variants (PPARγ2 and GR gene) studied to explain the associations between low birth weight with type 2 diabetes and cardiovascular disease later in life

ACKNOWLEDGEMENTS

The Generation R Study is conducted by the Erasmus Medical Center in close collaboration with the School of Law and Faculty of Social Sciences of the Erasmus University Rotterdam, the Municipal Health Service Rotterdam area, Rotterdam, the Rotterdam Homecare Foundation, Rotterdam and the Stichting Trombosedienst & Artsenlaboratorium Rijnmond (STAR), Rotterdam. We gratefully acknowledge the contribution of general practitioners, hospitals, midwives and pharmacies in Rotterdam.

DISCLOSURE OF INTERESTS

FUNDING SOURCES

This work was supported by the Erasmus Medical Center, Rotterdam; and the Erasmus University Rotterdam; and the Netherlands Organization for Health Research and Development (ZonMw).

Conflict of Interest: none declared. Van Rossum et al (2003) 220 G-allele of the Bcl-I polymorphism is associated with increased glucocorticoid sensitivity and lower BMI. Rosmond et al (2000) 221 G-allele of the Bcl-I polymorphism is associated with increased abdominal obesity and higher cortisol levels in GG-carriers compared to CC-carriers.

Bueman


Pierre Gaillard
email: pierre.gaillard@u-bourgogne.fr

From finite-gap solutions of KdV in terms of theta functions to solitons and positons

From finite-gap solutions of KdV in terms of theta functions to solitons and positons

The KdV equation and solutions in terms of theta functions

We consider the Riemann surface Γ of the algebraic curve defined by ω 2 = 2g+1 j=1 (z -E j ), with E j = E k , j = k. Let D be some divisor D = g j=1 P j , P j ∈ Γ. The so-called finite gap solution of the KdV equation

u t = 6uu x -u xxx (1)
can be expressed in the form [START_REF] Its | Hill's operator with finitely many gaps[END_REF] u(x, t) = -2 d 2 dx 2 ln θ(xg

+ tv + l) + C. (2)
We recall briefly, the notations. In [START_REF] Beutler | What do solitons, breathers and positons have in common?[END_REF], θ is the Riemann function defined by

θ(z) = k∈Z g exp{πi(Bk|k) + 2πi(k|z)}, (3) 1
constructed from the matrix of the B-periods of the surface Γ, and the vectors g, v, l are defined by g j = 2ic j1 , (4)

v j = 8i( c j1 2 2g+1 k=1 E k + c j2 ), (5)
l j = - g k=1 P k ∞ dU j + j 2 - 1 2 g k=1 B kj , (6)
C = 2g+1 k=1 E k -2 g k=1 a k zdU k , (7)
the coefficients c jk being relating with abelian differential dU j by

dU j = g k=1 c jk z g-k 2g+1 k=1 (z -E k ) dz, (8)
and coefficients c jk can be obtained by solving the system of linear equations

a k dU j = δ jk , 1 ≤ j ≤ g, 1 ≤ k ≤ g.

Degeneracy of solutions

We suppose that E j are real, E m < E j if m < j and try to evaluate the limits of all objects in formula ( 2) when E 2m , E 2m+1 tends to -α m , -α m = κ 2 m , κ m > 0, for 1 ≤ m ≤ g, and E 1 tends to 0 (these ideas were first presented by A. Its and V.B. Matveev, exposed for example in [START_REF] Belokolos | Algebro-geometric approach to nonlinear integrable equations[END_REF]).

Limit of P

(z) = 2g+1 j=1 (z -E j )
The limit of

P (z) = 2g+1 j=1 (z-E j ) is evidently equal to P (z) = z g j=1 (z+α j ) 2 2.0.2 Limit of dU m = g k=1 c mk z g-k 2g+1 k=1 (z-E k ) dz The limit of dU m is equal to dU m = ϕm(z) √ z g j=1 (z+α j ) dz, where ϕ m (z) = g k=1 cmk z g-k .
The normalization condition takes the form in the limit

a k dU j → 2πiϕ j (-α k ) κ k m =k (α m -α k ) = δ kj , (9)
which proves that the numbers -α m , m = k are the zeros of the polynomials ϕ k (z), and so ϕ k (z) can be written as ϕ k (z) = ck1 m =k (z + α m ). By (9), we get in the limit ck1 = κ k 2πi .

So d Ũk = κ k 2πi √ z(z + α k ) dz 2.0.3 Limit of v k and g k
By identification of the powers of z g-2 in ( 10)

φk = c k1 l =k (z + α l ) = g j=1 c kj z g-j , (10)
we get in the limit

ck1 g l=1 -α l + ck2 = κ 3 k 2πi .
So we have the limit values of v k and g k :

ṽk = 4 π κ 3 k and gk = 1 π κ k .
2.0.4 Limit of U j (P ) and B mk

For

λ 0 = -α m = κ 2 m , I = 0 λ 0 dU k → 1 2
Bmk . The integral I can be easily evaluate along the real axis on the upper sheet of surface Γ and we get

I → i 2π ln κ m + κ k κ m -κ k .
So we have the limit values of matrix B :

Bmk = i π ln κ m + κ k κ m -κ k .
So iB kk tends to -∞. As previously, we have

P ∞ dU j → - i 2π ln κ j - √ z P κ j + √ z P . (11)

Limit of argument of exponential in θ(p)

Let us denote A the argument of θ(p) = k∈Z g exp{πi(Bk|k) + 2πi(k|p)}.

A can be rewritten in the form

A = πi g j=1 B jj k j (k j -1) + 2πi j>m B mj k m k j + g j=1 πi(2p j + B jj )k j . ( 12
)
Using the inequality k j (k j -1) ≥ 0 for all k ∈ Z g and the fact that iB kk tends to -∞, we can reduce the limit θ of θ(p) to a finite sum taken over vectors k ∈ Z g such that each k j must be equal to 0 or 1. So, A can be rewritten in the form

A = πi g j=1 B jj k j (k j -1) + 2πi j>m B mj k m k j + g j=1 k j [2πi(g j x + v j t) -πi(-j + 2 g k=1 P k ∞ dU j + m =j B mj )].
In other words

A = πi g j=1 B jj k j (k j -1) + 2πi j>m B mj k m k j + g j=1 k j Q j ,
with Q j = 2πi(g j x + v j t) + β j and

β j = -πi(-j + 2 g k=1 P k ∞ dU j + m =j B mj ).
The quantity β j has a finite limit value βj independent from x and t.

Limit of θ(p)

By means of the inequality K j (K j -1) ≥ 0 for all K ∈ Z g and the previous relation iB kk tends to -∞, it turns out that the limit θ of θ(xg + tv + l) reduce to a finite sum taken over vectors k ∈ Z g with the property that each k j must be equal to 0 or 1.

θ = k∈Z g , k j =0 or 1 exp{ m>j 2 ln κ m -κ j κ m + κ j k m k j + i( g j=1 2κ j x + 8κ 3 j t + 2κ j x j + πj - m =j i ln κ m + κ j κ m -κ j )k j }, with x j = x j (κ j ) = 1 2iκ j g k=1 ln κ j - √ z k κ j + √ z k .
It can be rewritten as θ = J⊂{1,...,g} j∈J j,k∈J j<k

κ j -κ k κ j + κ k 2 exp 2i j∈J (κ j x + 4κ 3 j t + κ j x j ). j∈J k =j κ j + κ k κ j -κ k .( 13
)
Using the equality j,k∈J j<k

κ j -κ k κ j + κ k 2 j∈J k =j κ j + κ k κ j -κ k = j∈J k / ∈J κ j + κ k κ j -κ k , it can be reduced to θ = J⊂{1,...,g} j∈J j∈J k / ∈J κ j + κ k κ j -κ k exp 2i j∈J (κ j x + 4κ 3 j t + κ j x j ). ( 14
)
3 1-positon of order 1

A 1-positon of order 1 is given by (see [START_REF] Matveev | Asymptotics of the multipositon-soliton τ function of the Korteweg-de Vries equation and the supertransparency[END_REF]),

u = -2∂ 2 x log W (φ, ∂ K φ),
where W is the classical wronskian, and φ the function

φ = φ(x, K) = sin(K(x + x 1 (K) + 4K 2 t))
In this case

W = 1 2 (sin 2Θ -2Kγ), where Θ = K(x + x 1 (K) + 4K 2 t) and γ = ∂ K Θ.
We consider here a Riemann surface of genus 2.

Using the previous section, when we take the limit as E j tend to K 2 j the function θ tends to θ which takes the form

θ = 1 -K 2 +K 1 K 2 -K 1 exp 2i(K 1 + 4K 3 1 t + K 1 x 1 (K 1 )) + K 2 +K 1 K 2 -K 1 exp 2i(K 2 + 4K 3 2 t + K 2 x 2 (K 2 )) -exp 2i((K 1 + K 2 ) + 4(K 3 1 + K 3 2 )t + K 1 x 1 (K 1 ) + K 2 x 2 (K 2 )).
It can be rewritten as θ

= 1 + (K 2 + K 1 ) exp 2i(K 2 +4K 3 2 t+K 2 x 2 (K 2 ))-exp 2i(K 1 +4K 3 1 t+K 1 x 1 (K 1 )) K 2 -K 1 -exp 2i((K 1 + K 2 ) + 4(K 3 1 + K 3 2 )t + K 1 x 1 (K 1 ) + K 2 x 2 (K 2 )). (15)
Now, it is clear that when K 2 tends to

K 1 = K, we get θ = 1 + 2Ki exp(2iΘ)2i∂ K Θ -exp(4iΘ). ( 16
)
It can be reduced to

θ = -2i exp(2iΘ)(sin 2Θ -2Kγ) = -4i exp(2iΘ) × W. ( 17
)
Since Θ is a linear function of x, we recover exactly the 1-positon

u(x, t) = -2∂ 2 x ln θ = -2∂ 2 x W (φ, , ∂ K φ) = 16K 2 sin Θ(sin Θ -Kγ cos Θ) (sin 2Θ -2Kγ) 2 .
4 1-positon of order 2

We define a 1-positon of order 2 by (see [START_REF] Matveev | Asymptotics of the multipositon-soliton τ function of the Korteweg-de Vries equation and the supertransparency[END_REF]),

u = -2∂ 2 x log W (φ, ∂ K φ, ∂ 2 K φ)
, where W is the classical wronskian, and φ the function

φ = φ(x, K) = sin(K(x + x 1 (K) + 4K 2 t))
In this case

W = 4γ 2 K 2 cos Θ -2Kγ sin Θ -48K 3 t sin Θ -sin 2Θ sin Θ, where Θ = K(x + x 1 (K) + 4K 2 t) and γ = ∂ K Θ.
Here we consider a Riemann surface of genus 3. We can write the function θ using the previous section. When we take the limit as E j tend to K 2 j the function θ tends to θ which takes the form θ

= 1 -K 1 +K 2 K 1 -K 2 K 1 +K 3 K 1 -K 3 exp 2i(K 1 + 4K 3 1 t + K 1 x 1 (K)) + K 2 +K 1 K 2 -K 1 K 2 +K 3 K 2 -K 3 exp 2i(K 2 + 4K 3 2 t + K 2 x 2 (K)) -K 3 +K 1 K 3 -K 1 K 3 +K 2 K 3 -K 2 exp 2i(K 3 + 4K 3 3 t + K 3 x 3 (K)) -K 1 +K 3 K 1 -K 3 K 2 +K 3 K 2 -K 3 exp 2i(K 1 + 4K 3 1 t + K 1 x 1 (K) + K 2 + 4K 3 2 t + K 2 x 2 (K)) + K 1 +K 2 K 1 -K 2 K 3 +K 2 K 3 -K 2 exp 2i(K 1 + 4K 3 1 t + K 1 x 1 (K) + K 3 + 4K 3 3 t + K 3 x 3 (K)) -K 2 +K 1 K 2 -K 1 K 3 +K 1 K 3 -K 1 exp 2i(K 2 + 4K 3 2 t + K 2 x 2 (K) + K 3 + 4K 3 3 t + K 3 x 3 (K)) + exp 2i(K 1 + 4K 3 1 t + K 1 x 1 (K) + K 2 + 4K 3 2 t + K 2 x 21 (K) + K 3 + 4K 3 3 t + K 3 x 3 (K)) 6
Now, it is clear that when K 2 and K 3 tends to

K 1 = K, we get θ = 1 -exp(2iΘ) -2K 2 (-4γ 2 + 48iKγt) exp(2iΘ) -4iKγ exp(2iΘ) + 16K 2 γ 2 exp(4iΘ) -exp(4iΘ) + exp(6iΘ) + 4iKγ exp(4iΘ) + 2K 2 (-4γ 2 + 48iγKt) exp(4iΘ).
It can be reduced to

θ = -4 exp(3iΘ)(4γ 2 K 2 cos Θ -2Kγ sin Θ -48K 3 t sin Θ -sin 2Θ sin Θ).( 18
)
As Θ is linear in x, we recover exactly the 1-positon of order 2

u(x, t) = -2∂ 2 x ln θ = -2∂ 2 x W (φ, ∂ K φ, ∂ 2 K φ).
5 From Theta to Wronskian

From Theta to Fredholm

We consider the following matrix A = (a jk ) 1≤j,k≤N defined by

a jk = l =k K l + K j K l -K k exp(i(K j x + 8K 3 j t + 2K j x j ), (19)
where x j is an arbitrary parameter. Then det(I + A) has the following form

det(I + A) = J⊂{1,...,N } j∈J j∈J k / ∈J K j + K k K j -K k exp(2i j∈J (K j x + 4K 3 j t + K j x j ).( 20
)
By the previous section, θ = J⊂{1,...,g} j∈J j∈J k / ∈J

κ j + κ k κ j -κ k exp 2i j∈J (κ j x + 4κ 3 j t + κ j x j ). ( 21
)
If we compare the expression [START_REF] Van Diejen | Formulas for q-Spherical functions using inverse scattering theory of reflectionless Jacobi operators[END_REF] to [START_REF] Van Diejen | Integrabilty of difference Calogero-Moser systems[END_REF], we have clearly the equality θ = det(I + A).

It remains to find the link between this Fredholm determinant and a certain wronskian.

From Fredholm to Wronskians

In this section, we consider the following functions

φ j (x) = sin(K j x + 4K 3 j t + K j x j ), (23)
where K j are real numbers such that K 1 ≤ . . . ≤ K N , and x j an arbitrary constant independent of x.

We use the following notations :

θ j = K j x + 4K 3 j t + K j x j . W = W (φ j , . . . , φ N ) is the classical Wronskian W = det[(∂ j-1 x φ i ) i, j∈[1,...,N ]
]. We consider the matrix A = (a jk ) j, k∈[1,...,N ] defined by

a jk = l =k K l + K j K l -K k exp(i(K j x + 8K 3 j t + 2K j x j ). ( 24
)
Then we have the following statement Theorem 5.1

det(I + A) = 2 N i N (N +5) 2 exp(i N j=1 θ j ) N j=2 j-1 i=1 (K j -K i ) W (φ 1 , . . . , φ N ) (25)
Proof : We start to remove the factor (2i) -1 e iθ j in each row j in the Wronskian W for 1 ≤  ≤ N . Then

W = N j=1 e iθ j (2i) -N × W 1 , (26)
with

W 1 = (1 -e -2iθ 1 ) iK 1 (1 + e -2iθ 1 ) . . . (iK 1 ) N -1 (1 + (-1) N e -2iθ 1 ) (1 -e -2iθ 2 ) iK 2 (1 + e -2iθ 2 ) . . . (iK 2 ) N -1 (1 + (-1) N e -2iθ 2 ) . . . . . . . . . . . . (1 -e -2iθ N ) iK N (1 + e -2iθ N ) . . . (iK N ) N -1 (1 + (-1) N e -2iθ 2N )
The determinant W 1 can be written as

W 1 = det(α jk e j + β jk ),
where α jk = (-1) k (iK j ) k-1 , e j = e -2iθ j , and

β jk = (iK j ) k-1 . Denoting U = (α ij ) i, j∈[1,...,N ] , V = (β ij ) i, j∈[1,...,N ] , the determinant of U is clearly equal to det(U ) = (-1) N (N +1) 2 (i) N (N -1) 2 N ≥l>m≥1 (K l -K m ). ( 27
)
Then we use the following Lemma

Lemma 5.1 Let A = (a ij ) i, j∈[1,...,N ] , B = (b ij ) i, j∈[1,...,N ] , (H ij ) i, j∈[1,...,N ]
, the matrix formed by replacing the jth row of A by the ith row of B Then

det(a ij x i + b ij ) = det(a ij ) × det(δ ij x i + det(H ij ) det(a ij ) ) ( 28
)
Proof : For à = (ã ij ) i, j∈[1,...,N ] the matrix of cofactors of A, we have the well known formula

A × t à = det A × I. So it is clear that det( Ã) = (det(A)) N -1 . The general term of the product (c ij ) i,j∈[1,..,N ] = (a ij x i + b ij ) i,j∈[1,..,N ] ×(ã ij ) i,j∈[1,..,N ]
can be written as

c ij = N s=1 (a is x i + b is ) × ãjs = x i N s=1 a is ãjs + N s=1 b is ãjs = δ ij det(A)x i + det(H ij ). We get det(c ij ) = det(a ij x i + b ij ) × (det(A)) N -1 = (det(A)) N × det(δ ij x i + det(H ij ) det(A) ). Thus det(a ij x i + b ij ) = det(A) × det(δ ij x i + det(H ij ) det(A) ). 2
Using the previous lemma (28), we get :

det(α ij e i + β ij ) = det(α ij ) × det(δ ij e i + det(H ij ) det(α ij ) ),
where (H ij ) i, j∈[1,...,N ] is the matrix formed by replacing the jth row of U by the ith row of V defined previously. We compute det(H ij ) and we get det(H ij ) = (-1)

N (N +1) 2 +1 (i) N (N -1) 2 N ≥l>m≥1, l =j, m =j (K l -K m ) l<j (K k -K l ) l>j (K k -K l ).( 29
)
We can simplify the quotient q = det(H ij ) det(α ij ) :

q = l =k (K l + K k ) l =k (K l -K k ) . ( 30
)
So det(δ jk e j + det(H jk ) det(α jk ) ) can be expressed as

det(δ jk e j + det(H jk ) det(α jk ) ) = N j=1 e -2iθ j det(δ jk + l =k K l + K k K l -K k e 2iθ j ),
and therefore

det(δ jk e j + det(H jk ) det(α jk ) ) = N j=1
e -2iθ j det(I + A).

The Wronskian can be written as

W (φ 1 , . . . , φ N ) = N j=1 e iθ j (2i) -N (-1) N (N +1) 2 (i) N (N -1) 2 N j=2 j-1 i=1 (K j -K i ) N j=1 e -2iθ j det(I+A) It follows that det(I + A) = e i N j=1 θ j (2) N (i) N (N +5) 2 N j=2 j-1 i=1 (K j -K i ) W (φ 1 , . . . , φ N ) (31) 2 6

Positons of arbitrary order

Now it is clear how to get multi-positons. It is the same strategy used for the preceding examples.

If we want to get the following general positon

u = -2∂ 2 x ln W (φ 1 , . . . , φ (k 1 ) 1 , φ 2 , . . . , φ (k 2 )
2 , . . . , φ l , , . . . , φ

(k l ) l ), (32)
we consider a Riemann surface of genus g = l j=1 k j + l. As defined in the first section, we suppose that E j are real, E m ≤ E j if m < j and we evaluate the limits of all objects in formula (2). We consider

K i > 0, for 1 ≤ m ≤ g such that K m ≤ K j if m < j.
First, we choose the following limits :

E 1 tends to 0. for 1 ≤ i ≤ k 1 + 1, E 2i , E 2i+1 tends to K 2 i ; for 1 ≤ i ≤ k 2 + 1, E 2(k 1 +1)+2i , E 2(k 1 +1)+2i+1 tends to K 2 k 1 +1+i ; we continue until; for 1 ≤ i ≤ k l +1, E 2(k 1 +...k l-1 +l-1)+2i , E 2(k 1 +...k l-1 +l-1)+2i+1 tends to K 2 k 1 +.
..+k l-1 +l-1+i . Then from the results (20), ( 22) and (25), we get θ = det(I+A) = J⊂{1,...,g} j∈J

(-1) j j∈J k / ∈J K j + K k K j -K k exp j∈J (i(2K j x+8K 3 j t+2K j x j ). = 2 g i g(g+5) 2 exp(i g j=1 θ j ) g j=2 j-1 i=1 (K j -K i )
W (ϕ 1 , . . . , ϕ l ), with θ j = K j x + 4K 3 j + K j x j , and ϕ j = sin(θ j ).

We use the following notations :

for 1 ≤ i ≤ k 1 + 1, ϕ i = ϕ(K i ); for 1 ≤ i ≤ k 2 + 1, ϕ k 1 +1+i = ϕ(K k 1 +1+i
); and so on until; for 1 ≤ i ≤ k l + 1, ϕ k 1 +...+k l-1 +l-1+i = ϕ(K k 1 +...+k l-1 +l-1+i ).

We make here the following choice for K j , 1 ≤ j ≤ g : for 1

≤ i ≤ k 1 + 1, K i = κ 1 + (i -1)h; for 1 ≤ i ≤ k 2 + 1, K k 1 +1+i = κ 2 + (i -1)h);
and so on until; for 1

≤ i ≤ k l + 1, K k 1 +...+k l-1 +l-1+i = κ l + (i -1)h.
Then we consider the the classical difference derivative operator ∆ h defined by the formula :

∆ h f (x) = f (x + h) -f (x) h .
It is easy to prove that

∆ j h f (x) = 1 h j j k=0 (-1) k C k j f (x + (j -k)h), (33)
and it is obvious that for any function f (x) ∈ C j , (i.e. having j continuous derivatives), lim h→0

∆ j h f (x) = f (j) (x). ( 34
)
We consider W := W (ϕ 1 , . . . , ϕ 2 , . . . , ϕ l ).

Combining the columns, W can be written as

W = l i=1 h k i (k i +1) 2
×W (ϕ(κ 1 ), ∆ϕ(κ 1 ), . . . , ∆ k1 ϕ(κ 1 ), ϕ(κ 2 ), ∆ϕ(κ 2 ), . . . , ∆ k2 ϕ(κ 2 ), . . . , ϕ(κ l ), ∆ϕ(κ l ), . . . , ∆ k l ϕ(κ l )).

Then θ can be expressed as θ = c×W (ϕ(κ 1 ), ∆ϕ(κ 1 ), . . . , ∆ k1 ϕ(κ 1 ), ϕ(κ 2 ), ∆ϕ(κ 2 ), . . . , ∆ ϕ(κ 2 ), . . . , ϕ(κ l ), ∆ϕ(κ l ), . . . , ∆ k1 ϕ(κ l )),

with c = c 1 × c 2 .
The coefficient c 1 defined by

c 1 = 2 g i g(g+5) 2 exp(i g j=1 θ j )
is such that θ j is linear in x, and so verify 2∂ 2

x ln c 1 = 0. The coefficient c 2 is defined by

c 2 = 1 l j=1 2! . . . k i ! k 2 +1 m=1 k 1 +1 i=1 (K k 1 +1+m -K i ) . . . k l +1 m=1 k 1 +...+k l-1 +l-1 i=1 (K k 1 +k l-1 +l-1+m -K i )
.

By definition of the terms K j , the coefficient c 2 tends to a finite value independent of x when h tends to 0, and so verify 2∂ 2 x ln c 2 = 0. If we denote φ j = ϕ(κ j ), then when h tends to 0 W (ϕ(κ 1 ), ∆ϕ(κ 1 ), . . . , ∆ k 1 ϕ(κ 1 ), ϕ(κ 2 ), ∆ϕ(κ 2 ), . . . , ∆ k 2 ϕ(κ 2 ), . . . , ϕ(κ l ), ∆ϕ(κ l ), . . . , ∆ k 1 ϕ(κ l )) tends to W (φ 1 , . . . , φ

(k 1 ) 1 , φ 2 , . . . , φ (k 2 ) 2 . . . , φ l , , . . . , φ (k l ) l ). So we get when h tends to 0 u(x, t) = -2∂ 2 x ln( θ) = -2∂ 2 x ln(W ) = -2∂ 2 x ln W (φ 1 , . . . , φ (k 1 ) 1 , φ 2 , . . . , φ (k 2 ) 2 . . . . , φ l , , . . . , φ (k l ) l ).
We get clearly in the limit the positon defined by (32).

Conclusion

• This work takes its origin in the seminal paper of A. Its and V.B. Matveev in 1975 [START_REF] Its | Hill's operator with finitely many gaps[END_REF], in the study of V.B. Matveev in 1976 of abelian functions and solitons [START_REF] Matveev | Abelian functions and solitons[END_REF] and other papers like [START_REF] Matveev | Asymptotics of mulpositon-solution τ function of the Korteweg-de Vries equation and the supertransparency[END_REF], [START_REF] Matveev | Theory of positons[END_REF], [START_REF] Matveev | Multi positons solutions of the KdV equation. Positonpositon and soliton-positon interactions[END_REF], [START_REF] Matveev | Positon-positon and soliton-positon collisions : KdV case[END_REF], [START_REF] Matveev | Positon : slowly decreasing analogues of solitons[END_REF]. This result is based on two remarks. First, it was essential to express the degenerate θ function into an explicit Fredholm determinant; this remark was initiated by the works of Kirillov and Van Diejen [START_REF] Van Diejen | Kirillov Determinantal formulas for zonal spherical functions on hyperboloids[END_REF]. The second step was to get the transformation of the Fredholm determinant into a Wronskian.

• As a byproduct, we get the result for the case of a soliton; it is more simple.

If we want to get the following soliton u = 2∂ 2 x ln W (φ 1 , . . . , φ l ),

we consider a Riemann surface of genus g = l.

We choose E j real such that E m < E j if m < j.

We consider K i > 0, for 1 ≤ m ≤ g. We choose the following limits : E 1 tends to 0. for 1 ≤ i ≤ l, E 2i , E 2i+1 tends to -α i = K 2 i ; Then we get clearly in the limit the soliton defined by (35).

• We can also mentioned that we can get the famous DPT potential.

Let n and d be some non negative integers, m = n + d, If we choose the coefficients K j defined by

K p = p, if d = 0 , 1 ≤ p ≤ m -n, K j = n -m + 2j, if n = 0 , m -n + 1 ≤ j ≤ m,
Taking t = 0 and x j = 0, then u = 2∂ 2 x ln W (φ 1 , . . . , φ m ), is exactly the DPT potential m(m + 1) sin 2 (x) + n(n + 1) cos 2 (x) .

Its a consequence of a previous work. We refer the reader to the paper [START_REF] Gaillard | Wronskian addition formula and its applications[END_REF] for the details or to an another forthcoming publication containing a different approach.