A Fasano
email: fasano@math.unifi.it

M A Herrero
email: miguelherrero@mat.ucm.es

J M López
email: jmlopez@uniovi.es

E Medina
email: elena.medina@uca.es

On the dynamics of the growth plate in primary ossification

In this work a mathematical model for the interaction of two key signalling molecules in rat tibia ossification is presented and discussed. The molecules under consideration are Indian hedgehog (Ihh) and parathyroid hormone-related peptide (PTHrP). These are known to be major agents in the dynamics of the so-called Growth Plate, where transition from pristine cartilage to advancing bone takes place. Our model consists in a steady-state linear approximation to a Reaction-Diffusion system where only diffusion and absorption mechanisms are retained. Estimates on some system parameters are given, on the basis of the knowledge of a few measurable quantities. This allows for explicitly solving our model, whereupon a discussion on robustness and regulatory properties thereof is provided. In particular, we show that the size of the Proliferative Zone in the Growth Plate is rather insensitive to variations in the flux coefficients for Ihh and PTHrP at their boundaries. Besides, we also show that the model is also insensitive to large changes in the (comparatively small) critical value of the PTHrP concentration which marks the transition form Proliferative to Hyperthropic Regions within the Growth Plate. These results hold irrespective of the particular diffusivities selected for Ihh and PTHrP.

Introduction

Ossification, the formation of bone tissue from a preeexisting cartilage template, is a key stage in vertebrate development. The skeleton, which in humans contains over two hundred pieces of different sizes and shapes, performs a number of important functions. For instance, it provides a protecting frame for internal organs, and it transmits the force produced by muscle contraction in the course of movement. In this paper we investigate a key ossification process, whereby bone tissue is produced behind an advancing front and moves into a pre-existing cartilage template. More precisely, we aim at gaining insight into the manner in which cartilage cells (chondrocytes) determine the size of the Proliferative Zone (PZ), a cartilage region where cell proliferation takes place before chondrocytes undergo hypertrophy as a previous step to their replacement by invading bone tissue. The Proliferative Zone is located between two interfaces, separating it from the Resting Zone (RZ), where chondrocytes are generated, and from the Hyperthropic Zone (HZ), where proliferation is arrested, and the former cells enlarge in size before being overrun by advancing bone.

To describe this process, we shall make use of a comparatively simple mathematical model, to be explained and discussed below. The core of the model is a reaction-diffusion system describing how chemical signaling regulates the cell activity via an internal loop, which eventually determines the growth dynamics. Among the specific goals of the present study, we list: (i) to estimate some of the basic parameters of the process from the knowledge of measurable quantities, and (ii) to describe how the size of the Proliferative Zone depends upon variations on such parameters. As a preliminary step, we next provide a short summary of the basic features of ossification processes. be strictly coordinated with primary ossification, to eventually result in almost total replacement of cartilage by bone, except for a thin region near the distal boundaries where periarticular cartilage remains. This last is crucial to allow for movable joints between adjacent bones. Both processes are represented in Figure 1 below. The reader is referred to [START_REF] Hunziker | Mechanism of longitudinal bone growth and its regulation by growth plate chondrocyte[END_REF], [START_REF] Rivas | Structural stages in the development of the long bones and epiphysis[END_REF], [START_REF] Kronenberg | Developmental regulation of the growth plate[END_REF], [START_REF] Herrero | Bone formation: biological aspects and modelling problems[END_REF] and [START_REF] Mackie | Endochondral ossification: How cartilage is converted into bone in the developing skeleton[END_REF] for additional details.

Figure 1: Diagram of stages of intramembranous (A) and endochondral (B) ossification. Mesenchymal cells (brown) are the precursor cells in both cases. In intramembranous ossification (A), mesenchymal cells directly differentiate into osteoblasts (purple) and these cells begin the secretion of mineralized matrix (orange). When a osteoblast is completely surrounded by matrix it differentiates to mature osteocytes (red). In a subsequent stage (inset A) most cells have differentiated into osteocytes and undifferentiated cells are solely located at the border of the bone. In endochondral ossification (B), mesenchymal cells differentiate into an intermediate cell type, the chondrocyte (blue) which form a temporary structure that is subsequently invaded by osteoblasts at both ends of the bone according to a well-defined spatial pattern (inset B).

Longitudinal bone growth thus depends on the activity of the growth plate that in particular leads to continuous production of new cell and cartilage matrix. While this phenomenon simultaneously proceeds in both opposite directions from the equatorial region, from now on we shall refer only to one of these two directions for definiteness. Chondrocytes of the growth plate are arranged into columnar stacks whose axes cluster about a vertical axis of anisotrophy. In a vertical column the chondrocyte life cycle is recapitulated, since chondrocytes display a continuous structural variation that reflects successive maturational stages of a cell-differentiation cycle (cf. Figure 2). Maturation occurs synchronously for cells located at the same horizontal level in different columns and this gives rise to distinct horizontal strata. The layer nearest the epiphysis (i.e., the more distal part, farther form the centre of the cartilage template) is narrow and irregularly contoured and is referred to as the Resting Zone. Chondrocytes in this zone are small in size, irregularly arranged and rarely divide. Further toward the metaphysis (i.e, the equatorial zone), chondrocytes undergo rapid division which increases the cell pool, thus contributing to the widening of the Growth Plate. Daughter chondrocytes from a cell division process lie one above the other, and this is of major importance for the formation of vertical columns and unidirectional growth of the bone. After having run through a certain number of cell divisions, chondrocytes lose their capacity to divide and begin to enlarge to become hypertrophic cells. Cell volume increase during hypertrophy results in particular in Growth Plate widening. Matrix synthesis during both proliferative and hypertrophic stages is a third component of cartilage expansion. In isolation, chondrocyte proliferation, cell-volume increase and matrix synthesis would lead to progressive enlarging of the Growth Plate. However, cartilage increase is balanced by a coordinated process of cartilage resorption and replacement by osseous tissue at a moving front, the epiphysealmetaphyseal interface. A strict coordination between the proceses of cartilage enlargement, cartilage resorption and osseous tissue formation is required for normal bone growth.

The dynamics of the Growth Plate: The Ihh/PTHrP regulatory loop

Bone growth rate is dependent on the chondrogenic activity of the Growth Plate (GP). Different growth rates in different bones and/or in distinct developmental stages of the same bone are determined by differences in the rate of progression of chondrocytes through their differentiation cycle. Differentiation of Growth Plate chondrocytes involves a basic sequence of steps each of them having a wide range of quantitative variation. Parameters like the cell cycle time, the number of cellular divisions before the beginning of hypertrophy, the degree of organization into columns of cells, the time duration of hypertrophy, the rate of increase in cell volume during hypertrophy, the net matrix production and the rate of matrix mineralization could vary at different stages, and this determines different bone growth rates.

The extent of the different phases of the chondrocyte differentiation process as well as the replacement of cartilage by bone must be tightly regulated. Regulation must ensure not only the generation of the correct pattern of chondrocyte maturation but also the accurate quantitative values for each stage. In this way, it is well known that although regulation of Growth Plate is under a set of systemic, long-range control mechanisms, locally mediated control systems play a key role in GP evolution [START_REF] Nilsson | Endocrine regulation of the growth plate[END_REF]. There are many different factors affecting the activity of chondrocytes, and it is the complex interplay of all these factors that determines their behaviour. The Indian hedgehog (Ihh)/ parathyroid hormone-related protein (PTHrP) signaling pathway is a key regulator of these processes by forming a feedback loop to control the rate at which chondrocytes undergo maturation [START_REF] Kronenberg | Developmental regulation of the growth plate[END_REF], [START_REF] Shimo | Indian hedgehog and syndecans-3 coregulate chondrocyte proliferation and function during chick limb skeletogenesis[END_REF], [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF]. Indeed, it is well known that PTHrP is produced by periarticular chondrocytes and perichondral cells of the articular surface and acts directly on columnar proliferating chondrocytes to maintain the proliferative state and to block hypertrophic differentiation. Chondrocytes located close to the periarticular perichondrium (i.e, farther from the equatorial region) are exposed to high levels of PTHrP because of their relative proximity to the source, and such high concentrations inhibit chondrocyte hypertrophy. By contrast, chondrocytes farther from the articular surface are exposed to a lower concentration of PTHrP and thus undergo hypertrophic differentiation. PTHrP synthesis depends on, and is positively regulated by, Ihh which is expressed when proliferating chondrocytes differentiate into prehypertrophic chondrocytes. Thus, Ihh signals the relative position of prehypertrophic and hypertrophic chondrocytes to the periarticular Growth Plate. See references [START_REF] Shimo | Indian hedgehog and syndecans-3 coregulate chondrocyte proliferation and function during chick limb skeletogenesis[END_REF], [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF] for further details on this regulatory loop, which is summarised in Figure 3 below.

As to the precedings of this work, a number of mathematical models have been proposed to describe chemical signalling among the various regions of the Growth Plate. Among them, we should mention [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF] and [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF] where Reaction-Diffusion systems for the interaction among Ihh and PTHrP are postulated. While these articles have provided a basic inspiration for this work, our approach and that followed in [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF] and [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF] differ in that we have selected a simpler model, for which all parameters can be measured from experimental data with the help of the solutions of the mathematical model itself. Thus no further indirect estimates are required in order to fully discuss the behaviour of solutions, which yield the predicted values for the Ihh and PTHrP concentrations along the Proliferative Zone.

At this stage it should be noted that however important chemical signalling can be, it only constitutes one of the several mechanisms involved in the process of bone formation through endochondral ossification. For instance, a step further has been given in [START_REF] Garzón-Alvarado | A reaction-diffusion model for long bones growth[END_REF], where the coupling between cell density and chemical cues production has been considered. These authors formulated a model in which cell dynamics proceeds along a preferential, anisotropic direction (as it actually happens in the actual ossification process) which is in turn coupled to the Ihh and PTHrP concentrations, the latter being regulated by a suitable Reaction-Diffusion (RD) system. This work opens the way for further exploring the interaction among RD signalling models and the mechanics of bone growth (see also [START_REF] Moreo | Bone ingrowth on the surface of endosseous implants. Part 1: Mathematical model[END_REF] for recent related work in a different context). However, the number of parameters appearing in the model is considerably larger than in our case, which leaves open the question of how these could be possibly determined from available experimental data. We wish also mention that RD models (without mechanical coupling) have also been used in [START_REF] Garzón-Alvarado | Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism[END_REF] to describe secondary ossification, a process which, as we have remarked before, occurs at a different place than (but is finely tuned to) primary ossification, which is our main concern here.

Aim of the paper

The aim of the present work is to develop a mathematical model to describe the activity of the Ihh / PTHrP regulatory loop in the Proliferative Zone of the Growth Plate, and the comparative importance of the agents involved, as a function of a small number of key parameters.

More precisely, the plan of this paper is as follows. In Section 2 below, we describe and solve our mathematical model for the Ihh/PTHrP interaction in the Proliferative Zone (PZ). To this end some experimental results needed to identify parameters in the model are either recalled or provided.

We then discuss in Section 3 how the length of the Proliferative Zone is affected by small variations in the model coefficients, or by the Ihh/PTHrP concentrations at the RZ/PZ and PZ/HZ interfaces. In particular, we show that variations in the flow parameters in the model are largely damped out by the Ihh/PTHrP feedback loop, whereas fluctuations of the concentrations of such molecules at the RZ/PZ interface have opposite effects on the length of the PZ region. Moreover, the impact of variations in a particularly small parameter (c P H , the concentration of PTHrP at the PZ/HZ interface) is assessed. Our results are presented for three different age groups [START_REF] Minina | Bmp AND Ihh/PTHrP signalling interact to coordinate chondrocyte proliferation and differentiation[END_REF]35 and 80 days old rats). Such choice is motivated by the well-known fact that the Growth Plate dynamics is strongly age-dependent, which is in turn related to the influence of additional signalling molecules (e.g. sexual hormones) which are not dealt with This mechanism self-regulates the distance between the boundaries of the Resting Zone RZ (PTHrP-expressing area) and the Hyperthrophic Zone HZ (Ihh-expressing area). In our approach, the RZ/PZ and the PZ/HZ transition regions, which are assumed to be thin enough, are considered as signalling interfaces, a terminology to be kept in the sequel.

here. A number of conclusions are then listed in our final Section 4. On the other hand, as we shall presently see, our model relies only on absorption and diffusion as mass transport mechanisms, thus discarding other such processes as convection. A discussion on the negligible character of the latter in our current setting is postponed to Appendix A at the end of the paper.

The dynamics of the Growth Plate

We next state, and solve, the mathematical model that we propose to describe the Ihh/PTHrP signalling loop at the Growth Plate.

A remarkable fact concerning primary ossification is that many key aspects of such process can be considered in a one-dimensional setting, since chondrocytes are arranged along vertical columns, and lines at same distance from the advancing bone front correspond to a similar cell maturation state (cf. for instance [START_REF] Kronenberg | Developmental regulation of the growth plate[END_REF]).

In what follows we shall keep track of the Growth Plate dynamics along a single column of chondrocytes, but all quantities entering the model should be considered as averages over the whole set of columns. We focus our attention on the GP region, and more precisely in the Proliferative Zone (PZ) therein. For modelling purposes, it is convenient to select a reference frame in which the boundary between the PZ/HZ boundary remains at rest. More precisely, we take a longitudinal coordinate ẑ such that ẑ = 0 is located at the position of such an interface. On the other hand, the RZ-PZ transition is assumed to occur at ẑ = Γ P , with HZ ending at ẑ = -Γ H . In this reference frame, chondrocytes move from right to left. This situation is depicted in Figure 4 below:

* + ] * S ] ] ,KK 37+U3 3= += 5= %RQH IURQW

A mathematical model for Ihh/PTHrP interaction at the Proliferative Region

To describe in a precise manner the nature of the interaction between Ihh and PTHrP signalling in the Growth Plate, some quantitative models have been considered (cf. for instance [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF], [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF], [START_REF] Garzón-Alvarado | A reaction-diffusion model for long bones growth[END_REF], [START_REF] Moreo | Bone ingrowth on the surface of endosseous implants. Part 1: Mathematical model[END_REF], [START_REF] Garzón-Alvarado | Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism[END_REF]). As in those articles, transport of PTHrP and Ihh will be assumed to occur basically by diffusion through the liquid permeating the ECM, and by absorption. The latter process has two basic components. The first one takes place over chondrocytes, and is regulated by the kinetics of binding to specific receptors at the cell membrane. On the other hand, absorption takes also place at the ECM, the corresponding rate being in principle different from that for receptor binding. These two types of mechanisms will be represented by means of a single, effective absorption term in the model that follows. On its turn, convection will be assumed to be negligible. Justification for this last hypothesis is postponed to Appendix A.

We next state our mass balance equations in the proliferative zone. To this end, we respectively denote by ĉp , ĉi the concentrations of PTHrP and Ihh and write:

-D d 2 ĉi dẑ 2 + δ i ĉi = 0, 0 < ẑ < Γ P , (2.1)
-D d 2 ĉp dẑ 2 + δ p ĉp = 0, 0 < ẑ < Γ P , (2.2)
together with the boundary conditions:

-D d ĉi dẑ ẑ = 0 = γi , (2.3)
d ĉi dẑ ẑ = ΓP = 0, (2.4)
D d ĉp dẑ ẑ = ΓP = αp ĉi (Γ P ). (2.5)
At the transition between the PZ and HZ regions we assume that

ĉp (0) = c P H , (2.6)
where c P H , the concentration of PTHrP at the PZ/HZ interface is considered as a model parameter, and:

d ĉp dẑ ẑ = 0 = 0. (2.7)
Before solving the problem consisting in (2.1)-(2.7) some remarks on the motivation thereof are in order. To begin with, (2.1), (2.2) represent a steady-state approximation to a diffusion-absorption process. As in previous related work (cf. for instance [START_REF] Williams | Solute Transport in Growth Plate Cartilage: In Vitro and In Vivo[END_REF]) the diffusion coefficients for Ihh and PTHrP have been assumed equal for simplicity. This hypothesis is motivated by the fact that both substances possess similar molecular weights (cf. [START_REF] Kalderon | Similarities between the Hedgehog and Wnt signalling pathways[END_REF], [START_REF] Saha | Signal dynamics in Sonic Hedgehog tissue patterning[END_REF], [START_REF] Edwards | Parathyroid hormone-related protein (PTHrP) in breast cancer and in benign breast tissue[END_REF]), although no direct estimates on their diffusivities along the Growth Plate seem to be available; for instance, the value for the common diffusion coefficient selected in [START_REF] Williams | Solute Transport in Growth Plate Cartilage: In Vitro and In Vivo[END_REF] (and recalled in (2.26) below) has been determined for bovine albumin, a substance of similar molecular weight to that of Ihh and PTHrP. There are, however, authors who have selected different diffusivities for both substances in their studies (see for instance [START_REF] Garzón-Alvarado | A reaction-diffusion model for long bones growth[END_REF], [START_REF] Garzón-Alvarado | Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism[END_REF]). In those works, the authors deal with models including a large number of parameters, whose values cannot be determined in a unique way, and propose diffusivity choices which, being in some cases arbitrary, are compatible with standard activation-inhibition assumptions as explained for instance in [START_REF] Meinhardt | Models of biological pattern formation[END_REF]. An important fact concerning our approach is that, as we shall discuss in detail later on, our basic results concerning regulatory properties of the Proliferating Zone are independent of the particular values chosen for these coefficients, and in particular selecting both diffusivities equal or different does not change such results. Condition (2.3) corresponds to the assumption that a flux of Ihh is produced at the moment chondrocytes leave PZ to enter HZ, that is, at the PZ/HZ interface (see [START_REF] Kobayashi | PTHrP and Indian hedgehog control differentiation of growth plate chondrocytes at multiple steps[END_REF], [START_REF] Vortkamp | Regulation of rate of cartilage differentiation by Indian Hedgehog and PTH-related protein[END_REF], [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF]). Since flux is thought to occur at such precise threshold, it is natural to assume it to be constant, as we shall do in the sequel. On its turn, (2.4) states that Ihh diffusion beyond ẑ = Γ P is assumed to be negligible (the absence of flux can be justified by the occurrence of secondary ossification, a process somehow symmetric taking place within the epiphysis). Condition (2.5) states that, at the RZ/PZ boundary, a flux of PTHrP is produced, which is proportional to the concentration of Ihh there. This hypothesis corresponds to current assumptions on the way in which PTHrP production at the interface of the Resting Zone is stimulated by Ihh concentration there (see for instance [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF], [START_REF] Kobayashi | PTHrP and Indian hedgehog control differentiation of growth plate chondrocytes at multiple steps[END_REF], [START_REF] Vortkamp | Regulation of rate of cartilage differentiation by Indian Hedgehog and PTH-related protein[END_REF], [START_REF] Minina | Bmp AND Ihh/PTHrP signalling interact to coordinate chondrocyte proliferation and differentiation[END_REF]) .

Finally, by (2.7) we mean that diffusion of PTHrP is basically symmetric in PZ and in HZ.

Solving the PZ model

To proceed further, we introduce non-dimensional variables as follows:

z = ẑ Γ P , c i = ĉi c P H , c p = ĉp c P H , γ i = γi Γ P D c P H , α p = αp Γ P D . (2.8)
A quick check reveals that, in the new variables (2.8), (2.1)-(2.7) transforms into

d 2 c i dz 2 = Ω 2 i c i for 0 < z < 1 where Ω 2 i = Γ 2 P δ i D = τ D δ i , (2.9)
with τ D being a characteristic diffusive time given by τ

D = Γ 2 P D , d 2 c p dz 2 = Ω 2 p c p for 0 < z < 1 where Ω 2 p = Γ 2 P δ p D = τ D δ p , (2.10)
dc i dz z = 0 = -γ i , (2.11
)

dc i dz z = 1 = 0, (2.12
)

dc p dz (1) = α p c i (1), (2.13)
c p (0) = 1, (2.14)
dc p dz (0) = 0. (2.15)
Equations (2.9)-(2.15) can be explicitly solved. Indeed, on trying

c i (z) = a 1 sinh(Ω i z) + a 2 cosh(Ω i z) and c p (z) = b 1 sinh(Ω p z) + b 2 cosh(Ω p z) where sinh(s) = e s -e -s 2 , cosh(s) = e s + e -s 2
, one obtains after some elementary calculations that:

c i (z) = γ i Ω i (coth Ω i cosh(Ω i z) -sinh(Ω i z)) , (2.16
)

c p (z) = cosh(Ω p z), (2.17)
where coth Ω i = cosh Ω i sinh Ω i . A link between c i (z) and c p (z) is given by (2.13)

which yields Ω i Ω p sinh Ω i sinh Ω p = α p γ i . ( 2
, ω i = δ i D 1 2
.

(2.19)

Estimating parameters from experimental data

We next set out to estimate the coefficients in (2.16)-(2.18), as well as in the corresponding dimensional formulae. To this end we shall begin by making use of some available experimental results. More precisely the following values have been reported in [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF]; see also [START_REF] Zhao | Expression of parathyroid hormone -related peptide (PthrP) and its receptor (PTH1R) during the histogenesis of cartilage and bone in the chicken mandibular process[END_REF], [START_REF] Pathi | Comparative biological responses to human Sonic, Indian and desert hedgehog[END_REF] and [START_REF] Shimo | Indian hedgehog and syndecans-3 coregulate chondrocyte proliferation and function during chick limb skeletogenesis[END_REF]. (2.21) Plugging (2.21) into (2.16) yields: Here SD stands for the standard deviation. To derive Table 1, three age groups of male Sprague-Dawley rats were used 21, 35 and 80 days after birth. Tissue collection and processing were performed as described in [START_REF] Alvarez | Different bone growth rates are associated with changes in the expression patterns of types II and X collagens and collagenase 3 in proximal growth plates of the rat tibia[END_REF]. Analysis of longitudinal bone growth rates and proliferative activity were achieved. Proliferating chondrocytes were immunocytochemically identified by the 5-bromodeoxyuridine (BrdU) method. Proliferative activity was quantified by the percentage of BrdU-labeled cells within the proliferative stratum. Histomorphological methods were carried out as previously reported. Basically, sections were stained with toluidine blue and photographed in the light microscope first at a low magnification (X230). On these prints, growth plates were divided into stem cell, proliferating and hypertrophic zones according to morphological criteria. The hypertrophic zone was subdivided into an upper (H1) and lower (H2) half of equal size. The heights the growth plate and its different zones were estimated by point counting.

γ i Ω i coth Ω i = 5 • 10 3 , γ i Ω i sinh Ω i = 2 • 10 3 , ( 2
As to the diffusion coefficient D, we select for definiteness

D = 50 μ m 2 /s. (2.26)
The choice of this figure is motivated by the value reported in the experimental study conducted in [START_REF] Williams | Solute Transport in Growth Plate Cartilage: In Vitro and In Vivo[END_REF]. It should be noted, however, that in such work the diffusivity of bovine albumin in the Growth Plate was considered. Based on the similarity of molecular weights, it was assumed that Ihh and PTHrP diffusivities should be similar to that of such substance. We are not aware, though, of any direct study on Ihh/PTHrP diffusion properties along the Growth Plate. As we have already remarked at the end of paragraph 2.1, we shall presently show that our key results are not affected by the particular choice made in (2.26). Summing all the results up we obtain the estimates:

γi = 0.199 μ M • μm • s -1 , 0.224 μ M • μm • s -1 ,
δ p = D Ω 2 p Γ 2 P , δ i = D Ω 2 i Γ 2 P , whence
δ i = 3.79 • 10 -3 s -1 , 4
.76 • 10 -3 s -1 and 6.49 • 10 -3 s -1 , δ p = 11.7 • 10 -2 s -1 , 14.7 • 10 -2 s -1 and 2.00 • 10 -1 s -1 .

(2.30)

for the ages of 21, 35 and 80 days respectively.

The size of the PZ region: regulatory properties

In this Section we begin by discussing how variations on parameters appearing on our Ihh/PTHrP model reflect on the length of PZ. To this end, we shall use a perturbative approach, and consider in each case small changes in the coefficients involved.

Dependence of Γ P on small perturbations of the flux coefficients

We first show that the size of PZ is robust (that is, comparatively insensitive) with respect to small variations in the flux coefficients in (2.1)-(2.7). To this end, let us denote by Γ 0 P the values of Γ P corresponding to the values of αp , γi specified in our former Section. In particular for the ages of 21, 35 and 80 days we have seen that Γ 0 P = 180, 160.6 and 137.5 μ m respectively. Suppose now that αp , γi undergo small perturbations, so that their values are changed to αp (1 + p ), γi (1 + η i ) with 0 < p , η i 1. Let us denote by Γ 0 P the original length of PZ and by Γ 0 P (1 + ξ) that of the resulting PZ corresponding to the new flux coefficient values. In the perturbed case, (2.19) reads:

ω p ω i sinh ω p Γ 0 P (1 + ξ) sinh ω i Γ 0 P (1 + ξ) = αp (1 + p ) γi (1 + η i ) D 2 c P H .
Expanding both sides by Taylor's formula, we obtain ω i ω p sinh(ω p Γ 0 P ) sinh(ω i Γ 0 P ) + ξω p Γ 0 P cosh(ω p Γ 0 P ) sinh(ω i Γ 0 P )+

ξω i Γ 0 P cosh(ω i Γ 0 P ) sinh(ω p Γ 0 P ) + O(ξ 2 ) = αp γi D 2 c P H (1 + p + η i + p η i ) .
(3.1) Thus, retaining only linear terms in (3.1), and substracting (2.19) from the resulting expression (with Γ P replaced by Γ (0) P ), we eventually obtain that, to the lowest order:

ξ Γ 0 P ω p coth(ω p Γ 0 P ) + ω i coth(ω i Γ 0 P ) = p + η i . (3.2)
Recalling that

Γ 0 P ω p coth(ω p Γ 0 P ) + Γ 0 P ω i coth(ω i Γ 0 P ) = Ω p coth Ω p + Ω i coth Ω i , (3.3)
and therefore the relative perturbation in the length Γ 0 P is one order of magnitude inferior to the relative perturbations in the flux coefficients αp , γi .

We next remark that the robustness result described in (3.4) is independent of the particular choice of equal diffusivities made in (2.26). Suppose for instance that we now consider that effective diffusivities of Ihh and PTHrP are respectively given by D i and D p . Then, we have to replace D by D i in (2.1) and (2.3), and by D p in (2.2) and (2.5). If we next use the non-dimensional variables z, c i and c p given by the three first equations in (2.8), we obtain the system (2.9)- (2.15) where

Ω 2 i = Γ 2 P δ i D i , Ω 2 p = Γ 2 P δ p D p , γ i = γi Γ P D i c P H , α p = αp Γ P D p . ( 3
ξ = p + η i Ω p coth Ω p + Ω i coth Ω i ,
which coincides with (3.4) independently of the values of the diffusivities.

Dependence of Γ P on small perturbations of the boundary concentrations of PTHrP and Ihh

Suppose now that the concentrations of PTHrP and Ihh undergo small variations at the boundary of the proliferative zone. As before, let us denote by Γ 0 P the original length of PZ and by Γ 0 P (1+ξ) that of the resulting PZ corresponding to the new concentration values. Let the new non-dimensional concentrations be given by

c i (0) = c i (0) (1 + δ 0 ), c i (1) = c i (1) (1 + δ 1 ), c p (1) = c p (1) (1 + η), (3.6)
where we are denoting with the superscript ¯those variables which correspond to the perturbed situation. Note also that the one-dimensional length z in (3.6) has been obtained by rescaling with respect to the new PZ size. Here we leave c P H unperturbed, postponing the analysis of the corresponding perturbation to the following subsection. On writing equations (2.16) and (2.17)

cosh Ω i = c i (0) c i (1)
, cosh Ω p = c p (1), both in the perturbed and unperturbed case, one readily obtains

cosh Ω i = cosh Ω i 1 + δ 0 1 + δ 1 = cosh Ω i (1 + δ 0 -δ 1 ) + lower order terms, cosh Ω p = cosh Ω p (1 + η).
(3.7) Besides, from (2.8) and (2.16) in ẑ = 0 and ẑ = Γ P , we obtain that γi = D

Γ (0) P Ω i sinh Ω i ĉi (Γ (0)
P ), (3.8) and taking into account this equation in the perturbed and unperturbed cases we also see that γi

γi = Ω i sinh Ω i Ω i sinh Ω i 1 + δ 1 1 + ξ . (3.9)
On the other hand, it follows from (2.19) that

Ω p Ω i (Γ 0 P ) 2 (1 + ξ) 2 sinh Ω p sinh Ω i = αp γi D 2 c P H , (3.10)
that using (3.9) can be rewritten as

Ω p Ω i sinh Ω p sinh Ω i = αp γi D 2 c P H (Γ 0 P ) 2 (1 + ξ) (1 + δ 1 ), ( 3
Ω p = Ω p + η (coth Ω p ) = Ω p 1 + η Ω p , and sinh Ω p = sinh(Ω p + η) = sinh Ω p [1 + (coth Ω p )η] = sinh Ω p (1 + η),
(where we used that for values of Ω p close to Ω p = 8.70 we approximately have that coth Ω p = 1), equation (3.12) reduces to

1 + η Ω p (1 + η) = (1 + δ 1 ) (1 + ξ).
Retaining only leading terms we arrive at

ξ = 1 Ω p + 1 η -δ 1 . (3.13)
Then, in this range of approximations, the perturbed Γ P length is sensitive to variations on c i (1) and c p (1), but not to variations of the Ihh concentration at the PZ/HZ interface. Again, this result is independent of the fact that the diffusivities are taken equal or different, and is also independent of their particular values. Indeed, by recalling (3.5), it is clear that if we take diffusivities D i and D p for Ihh and PTHrP respectively, we need to replace D by D i into (3.8) and D 2 by D i D p into (3.10) and (3.11). However, equations (3.9), (3.12) and (3.13) remain invariant.

Negative regulation: sensitivity with respect to c P H

We have already remarked that parameter c P H stands out among other parameters in the model since it is comparatively quite small. Hence, any estimate on its precise value has to be taken with caution. Then, the question naturally arises of ascertaining whether changes in its value have a significant impact on the behavior of the solutions of the model. We shall consider below a particular case of the corresponding sensitivity analysis. More precisely,we next study the way in which changes in c P H translate into changes for δ p and αp to keep the length of PZ invariant. To do that we consider values for c P H within the range c P H ∈ (10 -5 μM, 10 -3 μM ) and obtain from our solution the required values for δ p and αp . The results are given in Figures 5 and6 and in tables 2, 3 and 4 where δ p is given in s -1 and αp in μ m/s. It is worth pointing out that, in order to keep the length Γ P constant, considerable variations in the value of c P H can be compensated by means of comparatively small changes in the model parameters being considered.

We remark that Figures 5,6 are important in the framework of our work, whose aim is not just to propose a mathematical model for the dynamics of the growth plate, but also to understand how critical the role of the threshold c P H is, since reliable quantitative information on it is actually scarce. Such figures show that the regulatory loop is not very sensitive to c P H in a considerably large range, in the sense that if for instance c P H varies by a factor of 2 in the range 0.0004 ÷ 0.008 the adjustment required on δ p , αp to achieve the same value of Γ p is less than 10 %.

Finally, note that both parameters δ p and αp , depends only of the diffusivity D p . From (3.5) and (2.18) we can write

δ p = D p Γ 2 p Ω 2 p , αp = D p Γ p γ i (Ω i Ω p sinh Ω i sinh Ω p ) = D p c P H Γ p ĉi (Γ p ) Ω p sinh Ω p ,
where Ω p is determined by the equation

cosh Ω p = ĉp (Γ p ) c P H .
Thus, a different choice for D p implies different values for the parameters δ p and αp (both parameters are multiplied by Dp D ). Nevertheless the form of the graphics in figures 5 and 6, and our main conclusions here, namely, the little sensitivity of the model to the precise value of c P H are independent of the value of D p .

Discussion and concluding remarks

This work is concerned with a relevant aspect of primary ossification in rat tibia, a characteristic example of long bone formation in vertebrates. More precisely, a mathematical model for the interaction of two key signalling molecules, Ihh and PTHrP, has been presented and discussed. While other substances as Bone Morphogenetic Proteins, BMPs ( [START_REF] Minina | Bmp AND Ihh/PTHrP signalling interact to coordinate chondrocyte proliferation and differentiation[END_REF]) or Vascular Endothelial Growth Factor, VGEF ( [START_REF] Colnot | Distinguishing the contributions of the perichondrium, cartilage and vascular endothelium to skeletal development[END_REF]) have been reported as playing various roles in ossification, a consensus remains concerning the fundamental importance of the Ihh/PTHrP feedback loop in the process under consideration ( [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF], [START_REF] Kobayashi | PTHrP and Indian hedgehog control differentiation of growth plate chondrocytes at multiple steps[END_REF], [START_REF] Vortkamp | Regulation of rate of cartilage differentiation by Indian Hedgehog and PTH-related protein[END_REF], [START_REF] Minina | Bmp AND Ihh/PTHrP signalling interact to coordinate chondrocyte proliferation and differentiation[END_REF]). We have examined in detail the way in which Ihh and PTHrP act on chondrocytes in the Proliferative Zone (PZ). This is a part of the Growth Plate, the moving region where the transition from an advancing bone front to an invaded cartilage template takes place.

Our study is based on current knowledge about the manner in which the Ihh/PTHrP feedback loop takes place ( [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF], [START_REF] Kobayashi | PTHrP and Indian hedgehog control differentiation of growth plate chondrocytes at multiple steps[END_REF], [START_REF] Vortkamp | Regulation of rate of cartilage differentiation by Indian Hedgehog and PTH-related protein[END_REF], [START_REF] Minina | Bmp AND Ihh/PTHrP signalling interact to coordinate chondrocyte proliferation and differentiation[END_REF]). More precisely, we assume that Ihh stimulates the production of PTHrP at the Resting Zone, RZ. From there, a decreasing PTHrP gradient is established, that keeps chondrocytes at a proliferating stage (thus determining the length Γ P of PZ), until PTHrP concentration falls below a threshold value denoted by c P H . The position (z = 0 in our current notation) where this value is attained marks the transition to the Hyperthrophic Zone, HZ. To keep the feedback loop operating, cells at a thin region near z = 0 produce a flux of Ihh. This last gives raise to a gradient of Ihh, which reaches up to the RZ, and regulates the production of PTHrP there, thus closing the loop.

In this article, an attempt has been made to present a mathematical model for the Ihh/PTHrP loop as simple as possible, so as to allow for explicit resolution in terms of the parameters involved. It should be considered that the Growth Plate of postnatal animals is a highly complex three-dimensional structure, whose experimental study presents many technical difficulties. As a result, experimental parameter determination is quite difficult, and in many cases a considerable degree of uncertainty remains about the accuracy of the corresponding measurements. For instance, in [START_REF] Williams | Solute Transport in Growth Plate Cartilage: In Vitro and In Vivo[END_REF] estimates on the Ihh and PTHrP diffusivities were indirectly obtained by using bovine albumin fragments having similar molecular weight to that of Ihh or PTHrP. Although this is a widely used methodology, the question remains to ascertain whether the interaction of Ihh/PTHrP with the extracellular matrix (ECM), or with other ligands, might result in different effective diffusivities in the actual in vivo process. This makes all the more remarkable the fact, that we have already stressed, that the regulatory properties described by formulae (3.4) and (3.13) are independent of the particular choice of Ihh and PTHrP diffusivities made. This remark on the limits of experimental measurement also applies to the values reported in (2.20). In particular, a sharp difference appears there among the scale in which most constants are estimated (μ M) and that in which c P H is presented (n M). Concerning the latter, measuring in vivo so small a value for a concentration at a particular location can only be accepted, we believe, after allowing for a wide fluctuation range. As a consequence, it seems that experimental estimates of that particular parameter have to be understood up to considerable fluctuations, even by one (or more) orders of magnitude.

A main goal of our work has been to obtain a mathematical model involving a reduced number of parameters, whose values are either experimentally derived in this work or alternatively could be found in the literature, and to analyze the dependence of solutions on changes in parameter values. In our view, uncertainty about parameter validation is a bonus to look for simple models. For instance, consideration of nonlinear kinetics (as those of Michaelis-Menten or Hill types) will lead to inclusion of further parameters that should then be estimated somehow. Needless to say, simplicity might result acceptable as far as conclusions derived from the model are consistent with, or represent a (motivated) challenge to, our current knowledge about the functioning of the Ihh/PTHrP loop. Bearing this statement in mind, we briefly recall below some of the results derived from the analysis of our model.

To begin with, we have considered the question of the efficiency of the Ihh/PTHrP loop to regulate the size of the Growth Plate. A way to address such issue consists in describing how perturbations in that loop affect to the length of the Proliferative Zone. We have addressed this issue in various manners. First, we have discussed in Section 3 how the length of PZ changes under small variations in the model data, which are likely to occur, even in physiological circumstances. In particular, one of the main conclusions of this work is that small changes in the flux parameters related to Ihh and PThrP production at the signalling interfaces RZ/PZ and PZ/HZ are efficiently damped out, since they induce modifications on the PZ length that are at least one order of magnitude lower than the perturbations (see (3.4)).

On the other hand, we have shown that, if Ihh concentration (respectively, PTHrP concentration) undergoes small variations at both signalling interfaces (respectively, at the RZ/PZ the length of PZ itself experiences small changes, which are roughly proportional to those experienced by the concentrations. Another important conclusion of this work is that, as shown in (3.13) the length of PZ is rather insensitive to small fluctuations in the Ihh concentration at the PZ/HZ interface. Moreover, as the same formula shows, a (small) increase in Ihh concentration at the RZ/PZ interface tends to reduce the length of PZ, whereas an increase of PTHrP there has the opposite effect. Note that such results hint at the robustness of the model under consideration. As has been already stressed, such conclusions are independent of the particular diffusivity coefficients taken for Ihh and PTHrP. It is worth remarking that such predictions seem to be in agreement with regulation properties of the Growth Plate obtained form experimental studies (cf. for instance [START_REF] Vortkamp | Regulation of rate of cartilage differentiation by Indian Hedgehog and PTH-related protein[END_REF], [START_REF] Weir | Targeted overexpression of parathyroid hormone-related peptide in chondrocytes causes chondrodysplasia and delayed endochondral bone formation[END_REF]). In particular, it has been observed in [START_REF] Vortkamp | Regulation of rate of cartilage differentiation by Indian Hedgehog and PTH-related protein[END_REF] that misexpression of Ihh prevents proliferating chondrocytes to enter into the hypertrophic phase, and that markers of hypertrophic cartilage are suppressed by Ihh. Furthermore, targeted disruption of the PTHrP gene in mice has an effect opposite to that of overexpressing Ihh. More precisely, in PTHrP-deficient mice, the transition from the proliferative to the hypertrophic zone is accelerated, which leads to a premature ossification. All these findings are compatible with our results above. It should be noted in this respect that when Ihh concentration is increased at the RZ/PZ interface, but the concentration of PTHrP is not modified there, the flux condition (2.5) implies a steeper slope for the PTHrP profile, thereby shortening the length of PZ. In fact (3.13) provides an estimate on the relative small perturbations of both signalling cues to allow for enlarging of the PZ.

A second way to obtain insight into the regulatory properties implicit in our model consists in estimating how modifications, not necessarily small, in the model parameters should be counterbalanced in order to keep the length of PZ constant. We have addressed such question in a particular case, and we have shown that even variations of up to two orders of magnitude in the value of c P H can be accommodated by means of rather small changes in some of the model parameters in order to keep Γ P constant. This strongly suggests that our model is rather insensitive to changes in the value of c P H , which is by far the parameter less accessible to experimental measurement.

At this juncture, it is worth remarking on how our results are related to previous modelling work in the field. The approach in this article is similar to that in references [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF] and [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF] (see also [START_REF] Garzón-Alvarado | A reaction-diffusion model for long bones growth[END_REF] and [START_REF] Garzón-Alvarado | Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism[END_REF]). In particular, in [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF] three signalling molecules (Ihh, PTHrP and VGEF) were considered, although only the first two actually were assumed to operate at the PZ. A time-dependent, nonlinear reaction-diffusion model was considered in [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF] and numerically simulated there. In that model, molecule synthesis (described by means of a Hill-type dynamics) plays an important role. Since some of the parameters therein involved remain unknown, they had to be externally fitted from what was referred to as a stable Growth Plate model. In particular, it was observed that increased (respectively, decreased) PTHrP synthesis rates result in longer (respectively, shorter) PZ. No significant change was observed in the HZ length. A similar result was obtained concerning Ihh synthesis rates. A related approach was then pursued in [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF], where this time only equilibrium conditions were considered, as in our current work here. Moreover, the Ihh/PTHrP concentration profiles in the PZ were assumed to be strictly linear. Again, some of the parameters in the model remained unknown (although useful references for some of them, including our c P H , were provided). In this way, a number of relevant hints about the way in which PZ is regulated were discussed. Concerning HZ, uncertainty remains about the validity of the proposed behaviour, since some of the estimates provided in [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF] seem to yield unrealistic values. We point out however that our work, which has considerably benefited from [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF] and [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF], can be considered as a further step in that modelling direction. In particular the mathematical formalism has been further simplified, as to allow for an explicit solution instead of a numerical one (which, as seen in [START_REF] Brouwers | Can the growth factors PTHrP, Ihh and VGEF, together regulate the development of a long bone?[END_REF], [START_REF] Van Donkelaar | The PTHrP-Ihh fedback loop in the embryonic growth plate allows PTHrP to control hypertrophy and Ihh to regulate proliferation[END_REF] often shows what seem to be spurious oscillations).

We conclude this Section by observing that, however important they are for endochondral ossification, Ihh and PTHrP are merely two, out of a larger set, of signalling molecules, and in order to gain further insight into the dynamics of the Growth Plate, more complex models have surely to be considered. These should account for more sophisticated biochemical circuitry (that could possibly incorporate the Ihh/PTHrP loop in a modular way), but also for a detailed analysis of the mechanical aspects of Growth Plate regulation. In particular, it would be quite interesting to check the validity of our results for the cases considered , for instance, in [START_REF] Garzón-Alvarado | A reaction-diffusion model for long bones growth[END_REF] and [START_REF] Garzón-Alvarado | Appearance and location of secondary ossification centres may be explained by a reaction-diffusion mechanism[END_REF]. We intend to address these issues in future work. for 21, 35 and 80 days old animals, respectively. We now take advantage of the estimates for the proliferation rates provided in Table 5 The labelling index (LI) was calculated as the percentage of nuclear profiles which are labelled by means of BrdU marking techniques. Differences between LI at 1 and 3 hours were used to estimate the time needed to duplicate the total number of proliferating chondrocytes (cell cycle time).

By Table 5 we have that

Figure 2 :

 2 Figure 2: (Left) Section of the proximal tibial Growth Plate showing the different sizes of hypertrophic (bottom) and proliferative (top) chondrocytes. (Right) A magnification of one of the vertical columns which make up the Growth Plate, distinctly showing different types of chondrocytes along a vertical direction

Figure 3 :

 3 Figure 3: Ihh is expressed in early hypertrophic chondrocytes and induces expression of PTHrP by resting chondrocytes. PTHrP keeps chondrocytes in a proliferating state, inhibiting expression of Ihh. Chondrocytes undergo hypertrophic differentiation and begin to express Ihh only when PTHrP drops below a threshold level. Thus Ihh and PTHrP form a negative feedback loop since increased production of Ihh results in its subsequent reduction by the system.This mechanism self-regulates the distance between the boundaries of the Resting Zone RZ (PTHrP-expressing area) and the Hyperthrophic Zone HZ (Ihh-expressing area). In our approach, the RZ/PZ and the PZ/HZ transition regions, which are assumed to be thin enough, are considered as signalling interfaces, a terminology to be kept in the sequel.

Figure 4 :

 4 Figure 4: From right to left, Resting Zone (RZ) and Proliferative Zone (PZ) are shown. Note that decreasing values of ẑ lead to transition from the Proliferative to the Hypertrophic Zone. Ihh/PTHrP gradients are represented in the PZ Region

  ĉi (0) = 0.5 μ M; ĉi (Γ P ) = 0.2 μ M; ĉp (Γ P ) = 0.3 μ M; ĉp (0) = c P H = 0.1 n M.

( 2 . 20 )

 220 Note that the value assigned to c P H is significantly smaller than the remaining three figures. Recalling (2.8), one readily deduces from (2.20) that c i (0) = 5 • 10 3 ; c i (1) = 2 • 10 3 ; c p (1) = 3 • 10 3 ; c p (0) = 1.

Figure 5 :Figure 6 :

 56 Figure 5: δp as a function of c pH for 21 (bottom), 35 (middle) and 80 (top) days of age respectively. Dots correspond to the value selected in (2.20)

5 •

 5 3600 s -1 = 1.34 • 10 -5 s -1 , for 21 days old rats,χ P = 1 25.9 • 3600 s -1 = 1.07 • 10 -5 s -1 , for 35 days old rats, χ P = 1 10.6 • 3600 s -1 = 2.62 • 10 -5 s -1 , for 80 days old rats, 10 4 s, 9.3 • 10 4 s and 3.8 • 10 4 s. term in the left of (A.7) results to be much smaller than the remaining ones, which are of size O(1) by the rescaling made in (A.4). The other age groups are dealt with similarly. If we take different diffusivities D i and D p in equations (A.1) and (A.2) it is easily checked that the above quotients are multiplied by D Di and D Dp for the equations for c i and c p respectively. It the follows that, if D i and D p are smaller than D, convective effects remain negligible whenever such diffusivities are up to one order of magnitude smaller than D. Convective effects can always be discarded for values of D i and D p larger than D.

  and we have already recalled an estimate for c P H (cf. (2.20)), we shift our attention to Γ P , we shall make use of the following values

	Γ P = 180 μ m	for the age of 21 days,
	Γ P = 160.6 μ m	for the age of 35 days,	(2.25)
	Γ P = 137.5 μ m	for the age of 80 days.
	This figures are obtained from experimental results for rat tibia in three age
	groups, which are reported below in Table 1
				.22)
	whence
	cosh Ω i = 2.5,
	which in turns yields the estimate
	Ω i = 1.57.	(2.23)
	Moreover from (2.22) and (2.23) we obtain:
	γ i = 7.18 • 10 3 .	(2.24)
	We are now in a position to estimate our first dimensional parameter γi in (2.3).
	Since by (2.8)
	γi =	D c P H Γ P	γ i ,

TABLE 1 21 days of age 35 days of age 80 days of age

 1

	Parameter	MEAN	SD	MEAN	SD	MEAN	SD
	Growth plate height (μ m)	736,3	61,4	589,5	45,2	297,7	25,1
	Height (μ m) of the Resting	47,5	7,8	38,4	5,1	25,6	7,2
	Height (μ m) of Proliferating	179,8	12,1	160,6	9,2	137,5	14,9
	Height (μ m) of Upper Hypertrophic	254,5	27,9	195,2	19,8	67,3	15,9
	Height (μ m) of Lower Hypertrophic	254,5	27,9	195,2	19,8	67,3	15,9

  and 0.261 μ M • μm • s -1 , (2.27) for the ages of 21, 35 and 80 days respectively. In a similar manner, we deduce from (2.17) that cosh Ω p = 3 • 10 3 , whence we take

			Ω p = 8.70.	(2.28)
	We now estimate α p from (2.18) to obtain
			α p = 13.0,
	whence αp = α p	D Γ P	can be taken to be
	αp = 3.62 μ m • s -1 , 4.06 μ m • s -1 , and 4.75 μ m • s -1 ,	(2.29)
	for the ages of 21, 35 and 80 days respectively. Finally δ i and δ p are determined
	through the relations (2.9)-(2.10) as

Table 2 : 21 days of age

 2

	cPH 10 -5	4 • 10 -5	7 • 10 -5	10 -4	2 • 10 -4	3 • 10 -4	5 • 10 -4	7.5 • 10 -4	10 -3
	δp	0.187	0.143	0.127	0.117	0.0989	0.0892	0.0776	0.0690	0.0631
	αp	4.58	4.01	3.77	3.62	3.34	3.17	2.95	2.79	2.67

Table 3 : 35 days of age cPH

 3

		10 -5	4 • 10 -5	7 • 10 -5	10 -4	2 • 10 -4	3 • 10 -4	5 • 10 -4	7.5 • 10 -4	10 -3
	δp	0.235	0.179	0.159	0.147	0.124	0.112	0.0974	0.0866	0.0793
	αp	5.14	4.49	4.23	4.06	3.74	3.55	3.31	3.12	2.99

Table 4 : 80 days of age

 4

	cPH	10 -5	4 10 -5	7 10 -5	10 -4	2 10 -4	3 10 -4	5 10 -4	7.5 10 -4	10 -3
	δp	0.320	0.245	0.217	0.200	0.170	0.153	0.133	0.118	0.108
	αp	6.00	5.24	4.94	4.75	4.37	4.15	3.87	3.65	3.49

TABLE 5 LI

 5 below

					Duplication
	. 1hr.	LI. 3hr.	difference	Duplication	time (hr.)	Cell cycle
	(%)	(%)	(%)	(percentage)	for marked	(in hours)
					chondrocytes
	21 days
	13.4	20.8	7.4	55.2	3.6	27.0
	±2.9	±3.9
	35 days
	16.3	25.9	9.6	58.9	3.4	20.8
	±3.6	±5.1
	80 days
	7.9	10.6	2.7	34.2	5.9	74.1
	±1.7	±2.2

1.1. Endochondral ossification: a process leading to long bone formationAccording to the manner in which it proceeds, mammalian ossification can be divided in two main types. A first one is intramembranous ossification, where bones grow by peripheral addition of new bone tissue at some osteogenic fronts. This is the way in which skull bones are made. A second type, endochondral ossification, is responsible for the formation of most vertebrate bones starting from a cartilage template. It involves two tightly coupled phenomena: continuous and vectorial production of cartilage tissue, and replacement of part of the cartilage by bone tissue through an invasive process. Endochondral ossification allows for a high rate of longitudinal growth, as required for the formation of long bones. In this last case, two coordinated stages can be distinguished, and are briefly recalled below.Shortly after cartilage differentiation, chondrocytes in the central region of the template start growing larger in size, thus undergoing hypertrophy. Hypertrophic chondrocytes secrete, and organize, a particular type of extracellular matrix, ECM. Then periosteal buds containing blood vessels and mesenchymal cells invade the middle of the template. Such angiogenic process is instrumental in replacing cartilage by bone, and creates a primary ossification centre, that subsequently moves simultaneously into the distal areas of the cartilage template. This process is called primary ossification, and the whole zone including RZ, PZ, HZ is commonly termed as the Growth Plate (GP).Once primary ossification sets in, cartilaginous areas at both distal sides develop an angiogenic process of their own. In this way, blood vessels penetrate distal peripheries to move inwards, thus producing secondary ossification centres at both ends of the template, which subsequently evolve much as primary centres do. This is known as secondary ossification, which happens to

Acknowledgements: This work has been partially supported by MICINN Research Grant MTM2008-01867, FIS2008-00200. The authors are grateful to two anonymous referees for a number of suggestions which have led to an improvement in the presentation of our results.

Appendix A: Discarding convective effects

We present below an argument that shows that convective effects can be safely neglected when compared with reaction-absorption ones in the model described in Section 2. To this end, we keep to the notation in subsection 2.2 and consider the more general form of equations (2.1)-(2.2) given by

which clearly reduce to (2.1)-(2.2) when the convective velocity v vanishes. Note that v = v(ẑ) denotes the cell velocity in the chosen frame of reference so that

where χ P is the cell proliferation rate within PZ. To proceed further, we make use of the following non-dimensional variables which can be considered as a generalization of (2.8).

, where v 0 = v(0) = χ P Γ P .

(A.4) Using (A.4), (A.1) transforms into

We next make use of the diffusive time τ D (defined in (2.10)) and the convective time τ C defined below

On setting Ω 2 i = δ i τ D (cf. (2.9)) it follows from (A.6) that (A.5) can be rewritten as

If we now take D = 50 μ m 2 /s (see Section 2) and Γ P as given in (2.25) we have τ D = 648 s, 516 s, and 378 s, (A.8)


MD Panagiotis ; Anagnostis
email: anagnwstis.pan@yahoo.gr

Fotini Adamidou

Stergios A Polyzos

Zoe Efstathiadou

Athanasios Panagiotou

Marina Kita

Pituitary incidentalomas: a single center experience

Keywords: pituitary incidentaloma, pituitary adenoma, prolactinoma, hypopituitarism

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

• The prevalence of pituitary incidentalomas in autopsy series ranges from 3-27%

• Non-functioning macroadenomas constitute the "lion's share", manifesting usually with extrasellar extension.

• The percentage of macroadenomas presenting with hypopituitarism ranges from 0-41%

What's new?

• Prolactinomas and acromegaly should always be excluded as they may be underdiagnosed • Hypopituitarism (partial or panhypopituitarism) is common in patients harbouring macroadenomas • Neurosurgery may improve visual field defects but hypopituitarism may be still present

Introduction

Pituitary incidentalomas (PIs) constitute a subgroup of pituitary adenomas detected incidentally during radiological evaluation for signs and symptoms unrelated to pituitary disease [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF][START_REF] Soule | The evaluation and management of subclinical pituitary disease[END_REF]. The estimated prevalence of PIs in autopsy series ranges from 3-27% [START_REF] Burrow | Microadenomas of the pituitary and abnormal sellar tomograms in an unselected autopsy series[END_REF][START_REF] Teramoto | Incidental pituitary lesions in 1,000 unselected autopsy specimens[END_REF][START_REF] Molitch | The pituitary "incidentaloma[END_REF][START_REF] Tomita | Pituitary adenomas and granular cell tumors. Incidence, cell type, and location of tumor in 100 pituitary glands at autopsy[END_REF]. Interestingly, a study of 100 normal volunteers who were examined by high-resolution magnetic resonance imaging (MRI) of the pituitary gland, revealed that 10% had lesions of decreased signal intensity after contrast administration, suggestive of microadenomas (3-6 mm) [START_REF] Hall | Pituitary magnetic resonance imaging in normal human volunteers: occult adenomas in the general population[END_REF]. Incidentally found macroadenomas are generally rare [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF][START_REF] Soule | The evaluation and management of subclinical pituitary disease[END_REF].

The majority of PIs are hormonally inactive adenomas in both sexes [START_REF] Soule | The evaluation and management of subclinical pituitary disease[END_REF][START_REF] Burrow | Microadenomas of the pituitary and abnormal sellar tomograms in an unselected autopsy series[END_REF].

Notably, non-functioning pituitary adenomas stain often positive for gonadotropins or their subunits (α-and β-subunit) or less frequently for adrenocorticotropic hormone (ACTH), growth hormone (GH), PRL or thyroid stimulating hormone (TSH), thus regarded respectively as silent corticotroph, somatotroph, lactotroph, thyrotroph or even mixed adenomas [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF]. The higher prevalence rate of PIs with increasing age might be associated with age-related decline in target gland hormones (especially sex hormones) levels and subsequent lack of feedback suppression leading to permanent stimulation which is the crucial step in the pituitary tumour development [START_REF] Kastelan | High prevalence rate of pituitary incidentaloma: is it associated with the age-related decline of the sex hormones levels?[END_REF].

When a PI is discovered, the clinician should decide whether he/she should proceed to surgical resection or not. This decision is based on 2 crucial issues. The first is whether compression of neighbouring structures exists leading to visual field defects and/or pituitary hormone deficiencies. The second is the potential of hormonal hypersecretion by the tumor [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF][START_REF] Soule | The evaluation and management of subclinical pituitary disease[END_REF]. In either case, except for prolactinomas which generally correspond well to dopamine agonists, neurosurgery is recommended. For the remainder, careful surveillance including regular endocrinological and The aim of the present study was to describe the clinical and functional characteristics as well as the natural course of PIs in a cohort of patients, who were followed-up in a single centre.

Patients and methods

This was a retrospective observational study conducted at the department of Endocrinology in Hippokration General Hospital (Thessaloniki, Greece), a secondary referral center for endocrinology and diabetes. Data from patients enrolled between 1982 and 2009 were extracted from the department's electronic database, reviewed and analyzed. Inclusion criteria were: 1) pituitary adenomas detected on occasion of MRI or CT of the brain performed for unrelated reasons and 2) at least one complete clinical and biochemical assessment. Patients complaining for symptoms associated with a sellar mass (such as visual field defects, symptoms of hypopituitarism or anterior pituitary hormone excess) at presentation were excluded.

All patients underwent clinical examination and visual field perimetry, baseline endocrinological assessment for PRL, GH, luteinizing hormone/follicle stimulating hormone (LH/FSH), ACTH and TSH, as well as target hormone assessment (estradiol, testosterone, serum cortisol, and free T4). GH deficiency was diagnosed when one of the following was present: levels of insulin growth factor-1 (IGF-1) were below the reference range for age and sex and/or an insufficient rise in GH levels after stimulation tests. detected in case of a morning serum cortisol level <3µg/dl or an insufficient increase (<18 µg/dl) in cortisol levels during a cosyntropin-stimulation test or a glucagonstimulation test, while TSH deficinecy was determined by low or 'inappropriately normal' TSH with thyroid hormone levels (free T4 and/or free T3) below the normal reference range. A threshold for stalk-effect hyperprolactinemia was set at >100 ng/ml. The patients were reevaluated at least yearly by clinical examination, routine biochemical and hormonal testing, both basic and dynamic, and by magnetic resonance imaging (MRI) or computerized tomography (CT) scan. CT was performed in cases when MRI was contraindicated or unavailable. Macroadenomas were defined when the maximum tumor diameter was >1 cm, while microadenomas when maximum diameter was ≤1 cm. Mass enlargement was defined as an increase of ≥0.5 cm in at least one dimension during follow-up. Earlier evaluation was performed in cases of significant alterations in clinical status. The duration of follow-up was estimated from the dates of first and last imaging.

Statistical analysis

Data for continuous variables are presented as mean ± standard error of the mean (SE). Data for categorical variables are presented as numbers and/or percentages. Kolmogorov-Smirnov test was used to check the normality of distribution of continuous variables. Chi-square test or Fischer's exact test were used to compare categorical variables. Independent-sample T-test or Mann-Whitney test was used to identify differences between two independent groups of continuous variables. A p value less than 0.05 was considered statistically significant in all the above tests. Statistical analysis was performed with SPSS 13.0 for Windows (SPSS Inc., Chicago, Illinois, USA).

Results

Three hundred and ninety-three patients with a pituitary adenoma were retrieved from our electronic database. Sixty-one [38 males (62%), 23 females (38%)] met the study criteria. Men were harbouring a macroadenoma more frequently than women (90% vs. 65%, p=0.043). The mean age at diagnosis was 53±2 years (range 16-90).

The mean size of adenomas was 20±2 mm (range 4-44). Twelve PIs (20%) were microadenomas, 48 (78%) macroadenomas and 1 lesion (2%) was described as a Rathke's cyst. The patients' characteristics at initial assessment are presented in table 1. The indications for imaging were: chronic and persistent headache, dizziness, stroke, syncope and other neurological disorders, loss of visual acuity, head trauma and encephalitis (table 2).

Based on endocrinological and histological evaluation (where available), 47 PIs (77%) were non-functioning adenomas, 11 (18%) prolactinomas, 2 (3%) GHsecreting adenomas, and 1 (2%) Rathke's cyst. Extrasellar extension was observed in 37 patients (77% of the macroadenomas, 61% of the whole). In particular, 20 (33%) had suprasellar, 4 (7%) parasellar and 13 (21%) supra-and parasellar extension (table 1). Follow-up data were available for 40 patients (66% of the whole) with a mean duration time of 48±8 months (range 0-222).

At presentation, visual field defects were detected in 22/49 (45%) of the cases with available data (all with macroadenomas), while chronic headache was present in 31/54 (57%), 4 of which were harbouring microadenomas. PRL levels at presentation were 76±16 ng/ml. Deficiency of ≥1 pituitary axes was identified in 61%. More specifically, LH/FSH deficiency was detected in 30%, ACTH in 18% and TSH in 16%, whereas panhypopituitarism in 12%. GH Surgical approach was selected for 24 patients (48% of those with available data).

Of these, 6 (25%) were operated twice and 3 (13%) thrice, due to tumor recurrence and deterioration of visual acuity and visual fields. At presentation, the patients not subjected to surgery had significantly lower rates of extrasellar extension (38% vs. 82% respectively, p=0.041) and number of pituitary axes affected (0.5±0.3 vs. 1.8±0.3, p=0.004). Postoperative tumor remnants were evident in 20 (87%), while in 3 (13%) the adenomas were totally resected. Mass recurrence developed in 11 cases during follow-up (48% of the surgically treated), 9 of which showed gradual enlargement. In 1 case the increase in size was due to intra-adenoma haemorrhage.

Histological examination was available for 8 PIs, 7 of which were immunohistochemically negative non-functioning ("null" cell) adenomas and 1 hypophysiocytoma.

Twelve patients with visual field disturbances underwent neurosurgery, 6 of which (50%) presented significant improvement [2 (17%) restored to normal during follow-up]. In 5/12 (42%) visual fields remained unchanged and worsened in 1 (8%).

Surgery was not performed in 10 patients with visual field defects due to the following reasons: 4 were prolactinomas and received therapy with dopamine agonists, 2 patients were of advanced age and 4 patients were lost to follow-up, after surgical management recommendation. Ten of 12 patients with chronic headache at However, diabetes insipidus developed in 2/23 (9%) of the patients postoperatively.

Hormonal deficiencies were significantly higher postoperatively compared to those at initial presentation: GH deficiency in 67% (p<0.001), LH/FSH in 70% (p<0.001), ACTH in 70% (p=0.001) and TSH in 70% (p=0.004), while panhypopituitarism in 57% (p=0.004) (figure 1). Of note, 2 patients developed improvement in pituitary function after surgery.

Five patients (8% of the whole, 21% of the surgically treated) underwent conventional radiotherapy, and 1 stereotactic (γ-knife) radiosurgery. Radiotherapy lead to a significant decrease in tumor size in 1 patient whereas the adenoma remained stable in 2 (duration of follow-up 3 and 7 years) and enlarged in 1 (10 years of follow-up). All these PIs were non-functioning adenomas. Dopamine agonists (DA) were administered in 3 patients with non-functioning PIs postoperatively without improvement in tumor volume. Of 8 patients with prolactinomas and followup data, DA resulted in mass decrease in 2 PIs (both macroadenomas), while 6 remained stable (1 microadenoma).

Of 9 patients with hormonally inactive PIs who were managed conservatively, tumor growth was observed in 1 (11%) macroadenoma. Of these patients, 6 were microadenomas (5 remained stable and 1 decreased in size), and 3 were macroadenomas, 2 of which did not show any alterations in size during surveillance.

Discussion

By definition of PIs, all patients included in our study presented with signs or symptoms which eventually resulted in a head MRI scan rather than a pituitary scan.

We concluded that non-functioning macroadenomas constitute the "lion's share", other studies [START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF][START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Sanno | A survey of pituitary incidentaloma in Japan[END_REF][START_REF] Day | Retrospective multicentric study of pituitary incidentalomas[END_REF], although in 1 study the majority were microadenomas [START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF].

Before the year 1995 most imaging examinations of the brain were performed by CT.

Interestingly, some studies [START_REF] Donovan | The natural history of the pituitary incidentaloma[END_REF][START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF] have reported data about PIs with CT scan alone.

MRI was available after this year. We acknowledge that the number of microadenomas might be higher in the present as well as in other series [START_REF] Donovan | The natural history of the pituitary incidentaloma[END_REF][START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF].

However, the number of these patients in our study was small (10 cases, 9 macroadenomas and 1 microadenoma, most of them with follow-up times of 8-20 years) and, thus, we do not consider this factor to have changed dramatically our results. In our series, PIs appeared more frequently in men, although this was not confirmed by others [START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Sanno | A survey of pituitary incidentaloma in Japan[END_REF][START_REF] Day | Retrospective multicentric study of pituitary incidentalomas[END_REF]. In addition, men were more likely to harbour a macrothan a microadenoma in comparison with women, in accordance with previous reports [START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Day | Retrospective multicentric study of pituitary incidentalomas[END_REF].

It is of note that it may be difficult to distinguish between overt PRL production and hyperprolactinemia due to stalk effect from a macroadenoma. Although it is quite arbitrary, macroprolactinomas usually present with PRL levels >200 ng/ml (4,000 mU/l) [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF]. A study correlated the serum PRL levels with the histological diagnosis in a series of 128 pituitary adenomas. The authors concluded that PRL levels >8,000 mU/l were compatible with a true prolactinoma, while hypeprolactinemia <3,000 mU/l should be regarded as a result of a non-fucntional tumor. For intermediate values both conditions can be considered [START_REF] Bevan | Misinterpretation of prolactin levels leading to management errors in patients with sellar enlargement[END_REF]. In a large series of 226 histologically confirmed cases with non-functioning pituitary macroadenomas, 98.7% had serum PRL <2000 mU/l, providing a quite safe threshold above which a prolactinoma should be suspected [START_REF] Karavitaki | Do the limits of serum prolactin in disconnection hyperprolactinaemia need re-definition? A study of 226 patients with histologically verified non-functioning pituitary macroadenoma[END_REF]. Furthermore, the 2 cases of acromegaly detected in our study indicate the importance of screening for pituitary hormone overproduction in patients with PIs. In both cases acromegaly was not suspected initially and these patients underwent radiological evaluation of the brain on occasion of stroke and head injury. The possibility of acromegaly in patients with PIs was also indicated by others [START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF].

Hormonal deficiencies at initial presentation were not negligible, with GH and gonadotropin deficiencies being the most frequent. Partial or panhypopituitarism has also been reported in various series [START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF][START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Day | Retrospective multicentric study of pituitary incidentalomas[END_REF]. It is clear that routine hormonal screening is necessary to reveal hypopituitarism. Assessment of both baseline anterior pituitary hormones and their target hormones is strongly recommended, along with stimulation tests when necessary [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF][START_REF] Soule | The evaluation and management of subclinical pituitary disease[END_REF][START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF][START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Sanno | A survey of pituitary incidentaloma in Japan[END_REF][START_REF] Day | Retrospective multicentric study of pituitary incidentalomas[END_REF]. All patients in our study presenting with hypopituitarism were carrying macroadenomas. The percentage of macroadenomas presenting with hypopituitarism ranges from 0-41% [START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF]. However, microadenomas may not be completely harmless. In a recent study of patients with non-secreting pituitary microadenomas, half of them demonstrated abnormal response to GH-releasing hormone (GHRH)-arginine stimulation test despite normal IGF-1 levels, as well as partial deficiencies of ≥1 of the other anterior pituitary hormones [START_REF] Yuen | Prevalence of GH and other anterior pituitary hormone deficiencies in adults with nonsecreting pituitary microadenomas and normal serum IGF-1 levels[END_REF]. About half of our patients underwent neurosurgery, based on the criteria presented above. Surgical approach appeared to be beneficial in terms of visual field impairment and resulted in amelioration of headaches in the majority of them, but further anterior pituitary hormone deterioration was observed, as well as diabetes insipidus. Surgery has been reported to improve visual function in about 80% of the cases with macroadenomas, but, as in our study, hypopituitarism will still be present in a considerable proportion [START_REF] Dekkers | Treatment and follow-up of clinically nonfunctioning pituitary macroadenomas[END_REF]. The low success rates by surgical approach in the present series are mainly attributable to the surgeon's experience and the fact that our patients were operated by different surgeons in various centers. It must be stated that without a public policy of regionalization of care for pituitary adenomas, the choice of neurosurgeon and hospital center is mainly driven by patients' socioeconomic status.

In addition, total resection of the adenomas was not achieved in most cases of our series and about half of them relapsed in size during follow-up. In studies of patients without postoperative radiotherapy, regrowth rates ranged between 6 and 46%. Even after prophylactic postoperative radiotherapy, regrowth was observed in 0-36% [START_REF] Dekkers | Treatment and follow-up of clinically nonfunctioning pituitary macroadenomas[END_REF].

In our study, radiotherapy as well as dopamine agonist administration, did not exert favourable outcomes with respect to tumor volume, although the duration of followup for the former was short and the number of patients inadequate for safe conclusions.

Regarding the natural course of non-operated patients, no syndrome of hormone excess was observed during follow-up. With respect to size, 11% of the patients conservatively managed demonstrated mass enlargement during follow-up. A similar proportion of PIs decreased in size and the vast majority remained stable during follow-up, although the number of patients was relatively small. Nonetheless, these findings are in accordance with various series, although some did not involve PIs in general, but non-functioning pituitary adenomas. In these studies, about 83% of microadenomas and 63% of macroadenomas remained stable during surveillance, 11% and 24% respectively increased in size, whereas 6% and 13% decreased respectively [START_REF] Donovan | The natural history of the pituitary incidentaloma[END_REF][START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF][START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Sanno | A survey of pituitary incidentaloma in Japan[END_REF][START_REF] Day | Retrospective multicentric study of pituitary incidentalomas[END_REF][START_REF] Nishizawa | Therapeutic strategy for incidentally found pituitary tumors ("pituitary incidentalomas[END_REF][START_REF] Igarashi | Long-term magnetic resonance imaging followup of asymptomatic sellar tumors. --their natural history and surgical indications[END_REF][START_REF] Arita | Natural course of incidentally found nonfunctioning pituitary adenoma, with special reference to pituitary apoplexy during follow-up examination[END_REF][START_REF] Dekkers | The natural course of nonfunctioning pituitary macroadenomas[END_REF][START_REF] Karavitaki | What is the natural history of nonoperated nonfunctioning pituitary adenomas?[END_REF]. Growth rate of these lesions has been estimated at 0.6 mm per year (210 and longer follow-up times may reveal greater proportions of tumor enlargement. It has been suggested that with longer follow-up about 50% of macroadenomas may increase in size [START_REF] Dekkers | Treatment and follow-up of clinically nonfunctioning pituitary macroadenomas[END_REF]. Despite meticulous data recording in the database of our department throughout the years, we acknowledge the following limitations to our study: 1) it constitutes a retrospective study 2) complete biochemical and imaging assessment was not available for each year in all patients; 3) the GH axis was not adequately explored in some patients of the series, due to non performance of insulin hypoglycaemia test.

However, long-term follow-up enabled us to make some relative secure assumptions.

In conclusion, patients with PIs usually follow a benign course, since the majority constitutes non-functioning adenomas which remain stable in size. Surgery is recommended in cases of masses attached to the optic chiasm especially in young patients, for tumors that increase in size and when hypopituitarism is present. This approach may be beneficial in terms of visual dysfunction, but hypopituitarism may worsen postoperatively. For non-operated patients, conservative management is a

  to be quite safe, since the potential of tumor enlargement is low[START_REF] Molitch | Pituitary tumours: pituitary incidentalomas[END_REF][START_REF] Soule | The evaluation and management of subclinical pituitary disease[END_REF][START_REF] Donovan | The natural history of the pituitary incidentaloma[END_REF][START_REF] Reincke | The 'incidentaloma' of the pituitary gland. Is neurosurgery required?[END_REF][START_REF] Feldkamp | Incidentally discovered pituitary lesions: high frequency of macroadenomas and hormone-secreting adenomas -results of a prospective study[END_REF][START_REF] Sanno | A survey of pituitary incidentaloma in Japan[END_REF].

  after surgery (p=0.041, compared to initial assessment).

  extrasellar extension. These findings are in accordance with

  reasonable and safe. PIs, although benign in nature, need individualized evaluation for pituitary hormonal over-or undersecretion and a treatment decision based on the clinical, morphological and functional characteristics of the tumor as well as patient preference and local neurosurgical expertise. In any case, long term follow-up is recommended. , PA and FA; Data analysis/interpretation, SAP and ZE; Critical revision of article, PA; Approval of article, MK.

Figure 1 .

 1 Figure 1. Hormonal deficiencies during follow-up.

table 1 )

 1 deficiency was present in 47% at initial diagnosis. No patient had evidence of diabetes insipidus at initial assessment . Patients with microadenoma had significantly lower PRL levels at presentation compared with those with macroadenoma (29.6±16.4 vs. 88.9±19.7 ng/ml, p=0.026). Similarly, the age of diagnosis (42±4 vs. 56±2 years, p=0.008) and

	8
	(the number of pituitary axes affected at presentation (0.6±0.2 vs. 1.1±0.2, p=0.04)
	were lower in patients with microadenomas compared with those with
	macroadenomas.
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y
	8

Table 1 . Patients' characteristics at initial presentation

 1

	Male/Female

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	P
	e
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	R
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Table 2 . Reasons leading to pituitary imaging

 2

	Headaches	26 (43%)
	Dizziness	11 (18%)
	Stroke	10 (16%)
	Syncope	4 (7%)
	Epilepsy-seizures	3 (5%)
	Other neurological disorders (Parkinson's	7 (11%)
	disease, multiple sclerosis, astasia)
	Visual acuity (without visual field) disorders	2 (3%)
	Head trauma	3 (5%)
	Encephalitis	1 (2%)

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	F o
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	R e e w i v 254x190mm (96 x 96 DPI)
	O n
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Conflict of interest

This paper was written independently. The authors did not receive financial or professional help with the preparation of the manuscript. The authors have given talks, attended conferences and participated in advisory boards and trials sponsored by various pharmaceutical companies.

Authors' contributions