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The authors have declared that no competing interests exist.

Conceived and designed the experiments: PM. Performed the experiments: CM. Analyzed the data: CM PM. Contributed reagents/materials/analysis tools: BDM GBM. Wrote the paper: CM BDM GBM PM.

The ways in which embryo development can vary across individuals of a population determine how genetic variation translates into adult phenotypic variation. The study of developmental variation has been hampered by the lack of quantitative methods for the joint analysis of embryo shape and the spatial distribution of cellular activity within the developing embryo geometry. By drawing from the strength of geometric morphometrics and pixel/voxel-based image analysis, we present a new approach for the biometric analysis of two-dimensional and three-dimensional embryonic images. Well-differentiated structures are described in terms of their shape, whereas structures with diffuse boundaries, such as emerging cell condensations or molecular gradients, are described as spatial patterns of intensities. We applied this approach to microscopic images of the tail fins of larval and juvenile rainbow trout. Inter-individual variation of shape and cell density was found highly spatially structured across the tail fin and temporally dynamic throughout the investigated period.

Despite the rapidly growing knowledge of the mechanisms underlying embryological development, little is known about how development varies across the individuals of a population. The variational properties of development determine how genetic and environmental variation translate into phenotypic variation in postnatal and adult individuals

The study of developmental variation has been hampered by the difficulties of measuring the geometry of developing embryos jointly with the spatial patterns of tissue formation and cellular activity. Yet an integrated understanding of organ formation and evolutionary change requires the coordinated study of gene expression, cellular activity, and organismal geometry

Geometric morphometrics is the state-of-the-art method for biological shape analysis

Statistical image analysis based on the gray values or color values of image elements (two-dimensional “pixels” or three-dimensional “voxels”) is frequently used in medical imaging

In the new method we term

GMIA thus starts with a careful, manual or semi-automatic representation of homologous, well-defined anatomical point locations, curves, and surfaces by the assignment of a dense set of landmarks and semilandmarks. The positions of the semilandmarks are estimated by the sliding landmark algorithm, which minimizes the “bending energy” of the thin-plate spline interpolation, a measure of local form difference, between the specimens and their sample average

This approach yields two complementary sets of data: (1) the Procrustes shape variables, describing variation of well-differentiated anatomical structures, and (2) the texture of the registered images (i.e., the pixel or voxel values), representing variation in the spatial distribution of imaged tissue properties. Statistics and resulting visualizations can be computed separately for shape and texture, and also jointly for both.

This two-step approach resembles the separate parameterization of shape and “shape-free” texture in active appearance models and related techniques, which have found wide application in face recognition and some areas of medical imaging

To illustrate this approach, we use a sample of 20 larval and juvenile specimens of rainbow trout (

Tail fin of a 21 dpf

Our sample consists of 20 rainbow trout specimens (

We recorded 4 anatomical landmarks and 95 semilandmarks on each of the 20 images (

The positions of the semilandmarks along their corresponding curves were computed by the sliding landmark algorithm

In most applications of the sliding landmark algorithm, the curves start and end with anatomical landmarks that constrain the sliding of the semilandmarks. As the curves in our application were all open, we computed the average landmark configuration only once and iteratively slid the semilandmarks against this average. This guaranteed convergence to a non-degenerate mean shape.

The 20 configurations of landmarks and semilandmarks were superimposed by Generalized Procrustes Analysis

Group mean shapes are estimated by averages of the shape coordinates. A low-dimensional ordination of shape space was computed by a between-group principal component analysis

All images were registered to the sample mean shape based on the measured landmarks and semilandmarks by using the thin-plate spline interpolation

Mean cell density was computed as the average of the RGB values of the registered images, computed separately for every color channel of every pixel. For calculating group mean differences and variances of image texture, we transformed the RGB values of each pixel into a scalar value by using the average of the RGB values of each pixel. This average corresponds to the brightness of the pixel and was interpreted as cell density in our application. Group mean differences and variances of image texture (cell density) were visualized by color maps. As for shape, the pattern of individual and group mean differences in texture were ordinated by between-group principal component analysis. Principal components of images are also referred to as eigenimages in the image analysis literature

Group mean shape and mean texture were jointly visualized by unwarping the average texture to the corresponding average shape. Group mean differences for both shape and texture were displayed by superimposing a TPS deformation grid and a color map of texture differences.

A joint ordination analysis of shape and texture is not possible via PCA because the scaling of the pixel values relative to the shape coordinates is ambiguous. While both shape and texture can be separately equipped with a Euclidean metric (but see also

All analyses and visualizations of shape and texture were computed in Mathematica 9.0 (Wolfram Research Inc., Champaign, IL, USA)

After estimating the semilandmark positions and superimposing the configurations by Generalized Procrustes Analysis, the resulting shape coordinates were averaged for each of the three age groups in order to estimate the

Average fin shape (first row), average cell density (second row), and average fin shape together with average cell density (third row) for each of the three age groups (the three columns). The structures measured by (semi)landmarks – the outlines of the fin fold, the musculature, and the notochord – are perfectly registered, but note also how precisely the fin rays, which emerge at 40 dpf, are registered by the TPS interpolation, even though they are not measured by landmarks.

After registering all images to the same shape, texture can likewise be averaged for the three age groups in order to estimate age-specific image texture, which, in our case, represents the

In

The differences between the age groups, i.e., the

(A) Between 21 dpf and 40 dpf and (B) between 40 dpf and 56 dpf. The deformation grid shows how the shape of the fin changed from a rounded to a more triangular shape during both periods. Changes of cell density are represented by the color map. From 21 to 40 dpf, cell density increased (blue) in the fin, whereas it decreased (red) in the musculature and the notochord. From 40 to 56 dpf, cell density increased in the fin rays and decreased in between.

The spatial pattern of

The spatial pattern of variation in cell density is shown by a color map for each age group. (A) 21 dpf, (B) 40 dpf, (C) 56 dpf. Variation in cell density was not uniform across the tail fin and differed between the age groups. At 21 dpf, the variance was concentrated in the fin fold, whereas cell density was very similar across all individuals in the notochord and in the musculature. At 40 dpf and 56 dpf, variance between the fin rays was higher than variance of cell density in the fin rays.

The multivariate pattern of individual differences can be assessed by an ordination analysis that yields a low-dimensional diagram in which the distances between the individuals approximate a certain measure of multivariate (dis)similarity. The shape metric typically used in geometric morphometrics is Procrustes distance (approximated by the Euclidean distance between the sets of shape coordinates

We used a between-group principal component analysis

Principal component analyses of (A) fin shape and (B) cell density, as well as (C) a joint ordination of both fin shape and cell density based on a scaled partial least squares analysis

Quantitative developmental studies, such as our example, typically are exploratory; statistical tests of the usual null-hypotheses do not bear much biological relevance here. However, as already obvious from the ordination analyses, permutation tests indicated significant group mean differences in shape as well as in image texture between the age groups (

Population models in evolutionary theory, genetics, and epidemiology are based on quantitative representations of phenotypic and genetic variation. Yet variation in embryological development and growth – the processes translating genetic variation into phenotypic variation – is still poorly understood. In fact, it has rarely been addressed empirically. Nevertheless, it has been argued on theoretical grounds that the properties of development assume a major role in shaping phenotypic variation within and across populations

The study of developmental variation has been hampered by difficulties with the quantification of embryological traits. Our morphometric approach combines the strengths of geometric morphometrics and pixel- or voxel-based image analysis. Well-defined tissue structures are described by the shape of their boundaries, whereas diffuse spatial patterns, such as cell condensations or molecular gradients, are described as scalar fields that are extracted from the texture of the registered images. We demonstrated how to separately analyze embryological shape and image texture (cell density), and we also outlined a strategy for their joint analysis. The anatomical structures are perfectly registered by our method and, hence, group averages of anatomical structures have well-defined boundaries. Variance in image texture is concentrated within the structures, not at their misaligned boundaries

In our application of GMIA we analyzed two-dimensional microscopic images. The method can equally be used with other imaging methods, including 3D imaging such as confocal microscopy and micro-CT

Several components of the GMIA approach, most notably the registration of the images in between the landmarks, were based on the TPS interpolation. This algorithm has proved powerful in multiple morphometric contexts, and it worked excellently for registering the fin rays in our application. But the algorithm originated in material physics

By applying GMIA to a sample of rainbow trout tail fins, we were able to demonstrate how average fin shape and average cell density changed within a period of 35 days. These changes include the emergence of the fin rays as novel tissue structures between 21 and 40 dpf, a process associated with a general shift of cell density from the musculature to the fin fold (

In order to verify these results, we produced histological sections along the frontal plane for one fish of each age class. Between 21 and 40 dpf, the tail fin became thicker and the density of mesenchymal cells increased. Also, the number of mucous glands increased. The myotomes decreased in width relative to the notochord. This corresponds well to our results of increasing (projected) cell density in the fin fold and decreasing density in the musculature. Between 40 and 56 dpf, the relative thickness of the myotomes increased again, associated with an increase of visible muscle fibers by approximately 50%. The fin rays and the in-between connective tissue were well differentiated at 40 and 56 dpf. Again, this is represented by our results (

Highly variable embryonic structures, resulting from variation in the onset, tempo, or mode of developmental processes, are particularly responsive to environmental or genetic disturbances and, hence, are promising candidates for experimental studies. These loci of developmental variation constitute the pattern of phenotypic variation that is subject to natural selection during development and adulthood. But because tissues differ in their degrees of canalization, variation generated at early developmental stages is not equally maintained during later stages and may induce different evolutionary dynamics (see also

GMIA presents itself as a powerful biometric tool for studying variation in organismal shape in concert with variation in the spatial patterns of various tissue properties. This approach may foster research in an emerging field of biomedical science, the study of developmental variation, which, at the same time, is central to any formal connection between evolutionary and developmental biology. More generally, the inherent strategy underlying GMIA can be used to study the statistical properties of various scalar fields, vector fields, or even tensor fields embedded within in the geometries of different organisms. This includes color patterns, such as in butterfly wings or in human facial skin, or even biomechanical properties, such as the distribution of stress and strain in samples of adult or subadult individuals (e.g.,

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We thank Fred Bookstein, Hans Nemeschkal, and Harald Ahnelt for discussion, and Elisabeth Rauscher for preparing the histological sections.