Which direction to take!?

One of the most important things palaeontologists and taxonomists do is the description of new species or fossils. Focussing on dinosaurs, because they’re the ones I know the best, there is a whole host of descriptive anatomy to get your head around. It’s not just the names of the bones; it’s also the names of the parts and structures within bones, including muscle scars and hypothesised muscles that attached to them. As well as this, you have to describe the relative position and spatial relationships between these elements to build a 3-dimensional image of a fossil based on descriptive terminology. This final part comes with a host of orientation related terminology, and can be incredibly confusing to decipher. At request from Sam Barnett (@Palaeosam), here’s an attempt to break it down, so that whenever you’re reading a description of a new species, you’ll hopefully be able to figure out some of the position-related jargon scientists have used!

Note, that while these can refer also to specific parts of the body, they can be used as relative terms too (e.g., the scutes are positioned dorsally, and the scutes are dorsal to the vertebral column).

One small step for digital Palaeontology

The time of digital technology is upon us. No scientific domain is embracing it’s fast-paced and dynamic progression more so than Palaeontology. One such realm that is exploding with new studies and enrapturing the minds of people and the global media is the increasing possibility to digitise and manipulate three-dimensional fossils. Surface laser-scanning, C-T scanning and mechanical digitizers are all commonplace now in palaeontological studies. The implications of such techniques are far-reaching, from reconstructing robotic dinosaurs (see video), to understanding vertebrate biomechanics at an intricate level. Other palaeontologists digitally reconstruct the internal anatomy of various organisms; for example, in the Herefordshire deposits in the UK, digital models are recreated from exquisitely preserved fossils within nodules to look at the evolution of the internal structures  that were pivotal in the evolution of extant hyperdiverse invertebrate groups, such as arthropods.

It is pretty well established that the fossil record is fraught with completeness issues. I covered the problem of this in a previous post in terms of understanding biodiversity patterns in deep geological time, in the context of lineage completeness. Another problem however is individual specimen completeness. Several authors have attempted to compensate for this secondary level of ‘bias’, using various quantitative metrics, and use these to guide assessments of biodiversity through time in specific lineages (e.g., sauropod dinosaurs). Another problem is that often, fossils have been ‘squished’ and distorted by the weight of successive layers of rock over the thousands or millions of years they have been buried for. This is a problem which is typically found in dinosaur skulls, making them somewhat resemble Imhotep in The Mummy (this may be fictional).

Ugly beyond all reason, possibly as a result of post-mortem decay. "You talking to me?!"

Geometric morphometrics is something that I’ve mentioned in previous posts. It sounds awful, the  very mention of it usually enough to put people off or smash a keyboard upside your head. But thanks to several review papers, the basic concepts are now much easier to grasp and apply to a variety of scientific hypotheses. Statistics are quantitative, easy to record, less subjective than qualitative statements, and available for repeated manipulation through a wide variety of methods. The integration of geometry-based analysis is now commonplace in almost every aspect of Palaeontology, intimately coupled with an increase in the availability of digital techniques. The fact that you don’t have to damage unique specimens during the processes (usually) is a bonus too!

The latest analysis, and a critical study for palaeontologists and museum curators around the world, uses geometry-based reconstruction of a poorly-preserved fossil to digitally reconstruct missing or distorted parts. And the best part about it, is that it’s fully open access (including all supplementary videos); the comment that “this method does not require specialised software or artistic expertise” is perhaps a bit misleading, as you firstly need a fossil and a CT scanner (or a previous scan), a pretty beasty PC, and the software mentioned is hardly cheap (Rhinoceros is €195 for a student license, and for Geomagic the cheapest price I could find was $8000). The actual software used (Mimics) appears to be free, but I’m still awaiting confirmation for downloading. Additional software, such as MeshLab and Autodesk Maya are freeware, at least for trial versions.

Clack et al. set out to build a method of digital reconstruction that builds upon previous methods, giving greater geometric accuracy. The methods revolve around using a digital mesh obtained through laser or C-T scanning as a model for a landmark-based geometric reconstruction. The sample specimen is a vertebra from the infamous tetrapod fossil Acanthostega. Only one half of the vertebra is actually preserved, therefore this was digitally reconstructed and attached to its mirror image, creating a bilaterally symmetrical three-dimensional element.

Landmark selection involved a mixture of Type 1 and Type 2 landmarks; that is topographically homologous points, mixed with sites of geometric significance, such as local maxima or minima of curvature. These were used as the basis for constructing a surficial grid of contour lines describing the medial and lateral geometry of the neural spine. Videos of the processes involved are actually available online, embedded within the article, a really awesome and useful addition, making the whole methodology more transparent and easier to replicate, should you wish. There’s not really much else to say about the methodology; the processes, such as modelling and surface extrapolation are laid out systematically and reasonably easy to understand for anyone with an understanding of the concepts of geometry and fossils.

The resultant reconstructions are high quality, smooth and geometrically faithful in representing the original vertebra in three dimensions, free of any taphonomic deformation or distortion, and with missing parts accurately reproduced. The groups of models created are validated using Procrustes superimposition and principal components analysis, two standard statistical techniques. The first two principal components do appear to have a low explanatory power however (PC-1 = 24.3%), which may be an issue relating to the complexity in the form of the vertebra. The authors are right to discount the use of the thin-plate spline technique, as this is known to be misleading in that the deformation patterns it produces are homogeneous with respect to the landmark configuration, leading to potentially false morphological variation in areas of no data, something which is largely overlooked.

Acanthostega model reconstruction, half-fish half-muppet; Copyright - Eliot Goldfinger

The advantages of the techniques explored here are in the handling style of the models, and their statistical power and accuracy. Furthermore, anyone can conduct or replicate these methods, providing they have access to an initial CT scan. The potential applications are numerous too: digital models of reconstructed elements can give more accurate parameters for biomechanics where data may have been previously extrapolated in a subjective or qualitative manner; it may yield hitherto unknown data for character construction, which may in turn increase the validity of phylogenetic analysis. The landmark mapping procedure may need refinement in terms of increasing the number of points, such as by using semi-landmarks, which will more accurately reconstruct the surface geometry and open the way for other statistical procedures.

The study represents a great step forward though in accurate specimen reconstruction, and reveals another field in which the power of geometric morphometric techniques is unparalleled. A limitation could be that to reconstruct missing parts, you have to have an idea of what the gross geometry is, meaning at least one half of a bilaterally symmetrical element must be present. This means that if you wanted to reconstruct the neural spine for example, it would be impossible if the whole part was absent, even if the entire centrum was preserved. This is something that could be integrated in future using close relatives of the species that are being reconstructed.

Quantitative Shape Analysis 2: Data Collection – easier than you think!

“If you want to inspire confidence, give plenty of statistics. It does not matter that they should be accurate, or even intelligible, as long as there is enough of them.” Lewis Carroll (1832-1898)

 

The last post on this series gave an introduction to the background and significance of quantitative shape analysis. I conveyed the use of landmarks, or geometric co-ordinates, as the basis for statistical analysis of shape. The last article finished by stating this article would discuss different methods of geometric morphometric analysis, but I forgot one crucial step: Data Collection! Here, I present a simple and efficient way of collecting data for use as the basis for a range of geometric morphometric analyses.

Following is an example of data collection from a simple coursework study I did last year, looking at cranial allometry in carnivores. Firstly, you need a target or hypothesis for your analysis. The target here was to use exemplar carnivorous mammal species to look at shape variation in the skull, and to interpret in terms of form and dietary function. The first decision to make is what points to use as your landmark data. I’ll use a hypothetical skull as an example.

Chosen selection of landmarks - you can chose any, as long as they are topographically correspondent between all specimens as described in the last post

 

Each one of these landmarks represents a specific topographically correspondent point amongst all specimens in the sample. For the sake of simplicity in this example, assume that the lower jaw and the cranium are a single module. The landmarks can be defined as such:

Cranial landmarks (right-lateral aspect; red)

1. Posterior extremity of occipital margin (type 3)

2. Tympanic aperture (centre) (type 1)

3. Posteroventral extremity of occipital condyle (type 3)

4. Ventral extremity of dorsal postorbital process (type 3)

5. Rostral extremity of orbital periphery (type 3)

6. Mid-point on ventral maxillary margin between premolars and canines (type 3)

7. Ventral deflection in dorsal margin (maximum curvature) [rostral to postorbital] (type 2)

8. Dorsal expansion in dorsal margin (maximum curvature) [posterior to external nares] (type 2)

9. Anterior extremity of premaxilla (type 3)

10. Dorsal extremity of dorsal margin (type 3)

11. Ventral extremity of zygamatic arch-jugal suture (type 1)

12. Position of distal border on posterior-most tooth (maxillary) (type 1)

 

Lower jaw landmarks (right-lateral aspect; blue)

13. Posterior extremity of angular process (type 3)

14. Posterior-most (distal) extent of dentary molars (type 3)

15. Mid-point on ventral dentary margin between premolars and canines (type 3)

16. Anterior extremity of dentary (type 3)

17. Point of posterodorsal deflection of ventral margin, culminating in angular process (type 2)

18. Ventral pinnacle of coronoid process (dentary) (type 2)

 

This is just a hypothetical example to show landmark positions and how to define them. Real data is freely available for almost anything on the internet. A series of sample images can be easily obtained through MorphBank, for example. If anyone reading this would like, I can send them a copy of the images I used for this coursework as a trial data set – just drop me a quick message with your email address.

Converting these landmarks into usable geometric data is possible through a number of image modification programs. A good one to use is ImageJ, freely available on the web. An important thing to note at this point is that within your image collection, every one you import into this program or any other must be angularly identical, or as close as possible (e.g., all of a precise lateral view of a skull).

Using ImageJ you can simply import an image with pre-defined landmarks as above, use the ‘Point’ tool to click on the landmark, hit ctrl-m (or use the Analyse-Measure tab), and hey-presto, you have the two-dimensional geometric co-ordinate of that point in a table! Consecutive points can then be added to this table for each specimen. Do this for all points per sample in a pre-defined numerical sequence (as indicated above), then simply export to an Excel spread-sheet. A rather nifty thing you can then do is plot them as a graph, and you’ll see a landmark representation of your image (awesomeness of this depends on how many landmarks you use). Repeat for all samples, and you have a comparable data set. Simple eh! Note that this can be done free of scale, so you don’t need to measure any lengths or inter-landmark distances. A future post will cover how to compensate for this in quantitative shape analysis. What you want to end up with at this stage is a single spread-sheet, with a labelled tab for each specimen, and containing a series of geometric co-ordinates that are topographically related between specimens.

There are of course more techniques using more complex software and data imaging methods (using surface or outline data, laser scans etc.), but typically these will not be accessible to the general public. The above procedure is a convenient and free method of obtaining a decent and workable initial data set, without having to spend endless time in a museum collection or laboratory.

So, now you know the procedure, nothing should stop anyone from going out there and collecting a data set, constructing a series of landmarks and digitally obtaining their geometric co-ordinates. Right? Next time, I’ll actually discuss how to assemble this data into a format that you can use to input to some free software, and several analyses you can then conduct with this software (e.g., Principal Components Analysis).

Why Geometric Morphometrics Kicks Ass 1

Those of you who have read my recent articles will probably have noticed the phrase ‘geometric morphometrics’ a few times. When mentioned to people, the usual reaction is to melt or run away screaming satanic verse and tearing chunks of hair out (pers. obs). This is largely due to the pretty intense mathematical basis behind the huge variety of implementable statistical procedures, which range from simple linear regressions to more complex 3D extended-eigensurface analyses. Each of these essentially provides a quantitative method of analysis of biological structures that can be interpreted in terms of biological function, a pretty crucial aspect of both zoology and palaeontology. To get to grips with the necessary analytical tools, it’s not really important to dig into the fundamentals such as how to construct a covariance matrix – these are well-defined mathematical concepts. The aim of the following few posts is to break down what is one of the most powerful yet under-used tools available for bio-structural analysis. What I’d also like to achieve is some kind of informal discussion about ideas in which geometric morphometrics can be applied to simple ‘pilot’ analyses based on freely available information, such as photographs from Morphbank. This first post will deal with the initial concepts, and future articles will provide examples of the different methods and tools available. Hopefully you will find this useful, and begin to openly develop and understand the processes involved.

Geometric morphometrics, as you might infer from the name, is the statistical analysis of form using geometric co-ordinates, or Cartesian landmarks. Form is defined here as the total dimensionality resulting from both shape and size. Size is the totality of spatial dimensions within a form, and shape is defined as the aspect of a form’s geometry that remains after scale (i.e., size), position (translation) and rotation have been normalised. Shape is essentially a localised metric for describing variation of spatial dimensions. Distinguishing between these is actually pretty crucial, as typically in an analysis you will want to differentiate between size and shape. Recently, the field has accelerated in strength due to the ability obtain three-dimensional structures such as skulls using techniques such as computer-tomography (CT) scanning. This considerably increases the information available for geometric morphometricians, and has led to numerous concurrent methodological adaptations in order to rigorously process available data. Using 3D techniques is kind of like a ‘total evidence’ approach to form analysis.

My personal opinion is that geometric morphometrics completely out-strips traditional morphometrics in terms of theoretical strength, methodology, and explanatory power. Consider simple linear measurements for starters, in for example, describing a lateral view of any mammalian hind-limb. You can imagine all sorts of bisecting, parallel, oblique and orthogonal measurements that would aid reconstruction of the form. Collecting these measurements and the relative angles would be time-consuming however, especially if you were looking for example, at sexual dimorphism of the femoral head in an antelope population. The world’s supply of coffee would be extinct before completion. However, with a simple photograph and the right software, breaking down a femur into a geometric outline or surface that you can use for all sorts of morphometric wizardry takes seconds (not including the months it takes until you are granted access to specimens).

Ratios are also a statistical over-simplification. The combination of measurements that can produce the same ratio is constrained only by the size of an object, and furthermore, ratios are a gross under-estimate of the potential geometric complexity of an object; try and imagine modelling a sine-wave with a linear ratio (or as complex a morphological structure as you want). Not going to happen is it.

The core of geometric morphometrics revolves around the assessment of allometry. Allometry is an ubiquitous aspect of nature, describing how organisms change their form. Discovering and interpreting allometry is the proximate target of most investigations, with the null hypothesis being isometry: no shape variation with respect to size. Typical investigable targets include detection of heterochronic trends, called paedomorphosis or peramorphosis, relating to the timing of acquisition of certain structures in an organism’s or species’ history (essential for evo-devo analyses).

Landmarks form the principal units of analysis for geometric morphometrics. Landmarks are formally known as Bookstein shape co-ordinates, with a defined Cartesian geometric position (i.e., x, y, z variables). Landmarks represent a subset of possible locations or distances, based on the nature of sampling. The only problems with this approach include difficulties in recognising landmarks, missing data, and possible redundance of data due to over-lapping inter-landmark spacings.

The data coverage available through morphometrics is exquisite! Each point is an individual landmark

REALLY IMPORTANT: Landmarks are NOT homologous sensu stricto. They represent topographically correspondent ‘characters’ – you should be able to write down the exact location in an unambiguous manner. This is really important when it comes to the biological interpretation of data.

There are three types of Cartesian landmark. Although not necessarily that important, it can provide an idea of how geometrically faithful a representation of an object you have.

Type 1: These represent the juxtaposition of biological components, such as sutures.

Type 2: These represent geometric aspects of form, such as local maxima or minima of curvature. These and type 1 landmarks are typically used in structure-based analyses.

Type 3: These are co-ordinate dependant equidistant interpolations, such as mid-points between two type 1 landmarks. They are also known as semi-landmarks, and are typically used in refining shapes such as profile outlines.

Landmarks form the cornerstone of all geometric morphometric analyses. What they provide you with is an unambiguous and quantitative dataset, that most importantly is highly informative in terms of biological structure. With landmark data, the wealth of potential modes of analysis at your disposal is phenomenal, as are the available software packages. One I would highly recommend is the tps series that can be found here, as well as a rather comprehensive overview of all things geometric.

In the mean-time, I wish everyone here an awesome 2012, and try not to get apocalypsed/raptured. I’ve popped a few references at the bottom here regarding the recent application of morphometrics in the field of vertebrate zoology, definitely worth reading a few just to get to grips with how scientists are currently using the techniques. I also strongly recommend the PalaeoMath series by Norm MacLeod, freely available here through the Palaeontological Association. It’s good stuff, and includes data so you can try your own analyses!

Next time: Principal Components Analysis, Principal Co-ordinates Analysis, and Procrustes superimposition.

Barden, H. E. and Maidment, S. C. R. (2011) Evidence for sexual dimorphism in the stegosaurian dinosaur Kentrosaurus aethiopicus from the Upper Triassic of Tanzania, Journal of Vertebrate Palaeontology, 31(3), 641-651

Brusatte et al. (2011) The evolution of cranial form and function in theropod dinosaurs: insights from geometric morphometrics, Journal of Evolutionary Biology, DOI: 10.1111/j.1420-9101.2011.02427.x

Goswami, A., Milne, N. and Wroe, S. (2010) Biting through constraints: cranial morphology, disparity and convergence across living and fossil carnivorous mammals, Proceedings of the Royal Society B, doi:10.1098/rspb.2010.2031

Hadley, C., Milne, N. and Schmitt, L. H. (2009) A three-dimensional geometric morphometric analysis of variation in cranial size and shape in tammar wallaby (Macropus eugenii) populations, Australian Journal of Zoology, 57, 337-345