Chaos, Solitons & Fractals 57 (2013) 19–23

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

Letter

Multi-fractal nature of human left ventricular trabeculae:Possible biomechanical role?

0960-0779/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.chaos.2013.08.005

⇑ Corresponding author. Tel.: +1 (970) 491 3706; fax: +1 (970) 4913827.

E-mail address: lakshmi.dasi@colostate.edu (L.P. Dasi).

Brandon Moore, Lakshmi Prasad Dasi ⇑Department of Mechanical Engineering, Colorado State University, Room A103D Engineering, 1374 Campus Delivery, Fort Collins, CO 80523-1374, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 April 2013Accepted 6 August 2013

Ventricular systolic and diastolic dysfunctions represent a large portion of healthcare prob-lems in the United States. Many of these problems are caused and/or characterized by theiraltered fluid–structure mechanics. The structure of the left ventricle in particular is com-plex with time dependent multi-scale geometric complexity. At relatively small scales,one facet that is still not well understood is the role of trabeculae in the pumping functionof the left ventricle. We utilize fractal geometry tools to help characterize the complexity ofthe inner surface of the left ventricle at different times during the cardiac cycle. A high-res-olution three dimensional model of the time dependent ventricular geometry was con-structed from computed tomography (CT) images in a human. The scale dependentfractal dimension of the ventricle was determined using the box-counting algorithm overthe cardiac cycle. It is shown that the trabeculae may indeed play an integral role in thebiomechanics of pumping by regulating the mechanical leverage available to the cardiacmuscle fibers.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

There are over 80 million people in the United Stateswith some form of cardiovascular disease [1]. Symptomsof these diseases are often caused by an alteration in bloodflow, and these flow patterns are determined by anatomi-cal structure.

In this study, we are interested in examining the geo-metrical properties of the left ventricle of the human heartas they pertain to pumping function. Such properties in-clude size and overall shape as well as the more detailedstructure of the inner surface of the ventricle, which con-tains fingerlike projections known as ventricular trabecu-lae carneae. These trabeculations are formed during earlyembryonic development when woven cardiac fibers are‘‘compacted’’ into a solid, continuous structure [2,3]. How-ever in some cases, such as the noncompacted ventricle,trabeculae are so numerous and prominent that the entire

interior of the ventricle is filled with myocardial tissue andappears somewhat like a sponge. Therefore, since the nor-mal ventricle displays a balance between a smooth and a‘‘spongy’’ interior, there could be a mechanical purpose oftrabeculae.

The shape of the ventricle plays a large role in its func-tion. Often modeled as an oblate spheroid [4] or prolateellipsoid [5], the ventricle’s shape is optimized to achievethe pressures that are needed to pump blood throughoutthe body. In addition, the thickness of the myocardium var-ies at different points along the ventricle wall. This is be-cause different geometries require more tension, andtherefore more muscular tissue, to achieve the sameamount of pressure. For example, the ventricle wall is verythick near its base and rather thin at its apex, which is duein large part to the radius of curvature at these two loca-tions. The law of Laplace states that more tension is re-quired in a flatter region, or a region with a larger radiusof curvature, which is why the base wall is thicker thanthe more sharply curved apex [6].

In addition to the mechanical role played by the overallshape of the left ventricle, we argue that there could also

20 B. Moore, L.P. Dasi / Chaos, Solitons & Fractals 57 (2013) 19–23

be a mechanical reason for the presence of the trabeculaecarneae. These features project into the ventricle chamberand therefore could alter the complex fluid dynamics ofblood that passes through it [7]. Another factor presentedhere is the ability of such a rough surface to expel bloodmerely by changing its geometric conformation, whichwould otherwise not be possible with a smooth surface.

Many studies have examined the effects of cardiac fibergeometry [8,9] and mechanics [10–13] on energy expendi-ture of the left ventricle. Such studies have included thehelical structure of heart fibers as well as their impact onthe relationship between contraction and relaxation[14,15]. Other studies have examined the fractal natureof intricate geometries. Catrakis et al. used fractal dimen-sion as a measure to quantify the complexity of turbulentscalar interfaces [16,17], while Meier used this measureto monitor changes in arterial branching due to coronarystenosis and hypotension [18]. However, we are unawareof any studies that have used fractal geometry as a mea-sure to help describe the role of trabeculae in cardiacpumping mechanics.

This paper shows that the trabeculae carneae may havea significant mechanical role in pumping mechanics andthat this can be elucidated through the use of fractal geom-etry. This finding is significant in biomechanics as it ex-plores the notion of pressure–volume work done by ageometrically complex interface through a change in itsfractal-dimension.

2. Fractal characteristics of ventricular trabeculaecarneae

We consider the ventricular-blood interface as a scaledependent multi-fractal surface whose fractal dimension,DðkÞ is a function of scale size, k. A 3D model of this surface(see Fig. 1) was acquired at ten equally spaced phasesthroughout the cardiac cycle using high-resolution cardiacCT imaging at a spatial resolution of 0.77 mm/pixel. Thesubject used in this study was a 51 year old male with noabnormality in ventricular pumping characteristics. Dis-cretized 3D models were created for each cardiac phaseby segmenting out the blood-endocardium interface usingmimics software (Materialize Inc., Plymouth, MI). Fig. 1shows both the raw CT images and the 3D surfaces visual-izing the complexity of the ventricular-blood interface. Themeshed models were then imported into MATLAB and ana-lyzed using an in-house code to implement the box-count-ing algorithm, which estimates the multi-fractaldimension DðkÞ ¼ � d log NðkÞ

d log k . Where NðkÞ is the number ofboxes of size ðkÞ, needed to fully cover the interface. DðkÞwas determined for a range of scales, from approximately0–50% of the total/longest length scale of the ventricle.The DðkÞ profiles obtained as a function of cardiac phaserepresent the temporal variations in the multi-fractalproperties of the ventricle-blood interface.

Fig. 2 shows the DðkÞ profiles of the ventricle during 5key phases of the cardiac cycle. As seen in the figure, thegeometric dimension of the ventricle is clearly scaledependent. The concave down profiles indicate that thegeometric complexity is highest at intermediate scales

corresponding to where the profile achieves a maximum.From the figure DðkÞ is maximum at relatively low scalesaround k between 3 and 6 mm corresponding to the ana-tomical trabecular structures that exist at these lengthscales. Scale lengths below this range show the fractaldimension approaches the value of 2.0 consistent with asmooth endocardial surface. At the large scales aboveabout k of 20 mm, the calculation of fractal dimensionusing the box-counting algorithm is unreliable due to therelatively large box sizes compared to the size of the ven-tricle. Between k of 15–20 mm the fractal dimension is be-low the value of 2.0 indicating that the surfacerepresentation with such large boxes begin to have signif-icant holes. Thus at such large scales the large scale varia-tions in the surface force a fractal dimension lower than2.0. A general trend appears to emerge in fractal dimensionduring the cardiac cycle. During ventricular contraction(systole), the fractal dimension appears to decrease. Thistrend reverses during diastole.

To better understand the changing complexity in thesmall-scale geometry of the ventricle throughout systoleand diastole, it is necessary to examine its fractal dimen-sion at different times during these processes. This canbe done by binning the DðkÞ profile into the small-scaleand the large-scale by spatially averaging the DðkÞ curvesshown in Fig. 2. Small scales are defined as ðkÞ < 10 mmand the large scales are defined as ðkÞ > 10 mm. The aver-age and standard deviation within these defined scale sizesare shown in Fig. 3. As can be seen in this figure, the fractaldimension of the ventricle’s surface at the small scale sig-nificantly relates to the cardiac pumping cycle. Note thesignificant decrease in fractal dimension fairly steadily(p < 0.003) during systole and then increase at a slightlyslower rate during diastole. However at the large-scalesthere is no significant variation in fractal dimensionthroughout the cardiac cycle.

In order to more clearly understand the changing geom-etry of the ventricle, 2D slices were extracted about boththe long and short axes (Fig. 4) for each of the ten cardiacphases. Note that the figure only shows 5 phases. In theseseries of images the surface can be seen changing from avery rough texture to a relatively smoother one during sys-tole and then transitioning back during diastole. Theseimages also provide visual evidence of how small scale fea-tures and overall wall motion work in conjunction whenthe ventricle contracts. It can be seen that small cavitiesalong the wall of the ventricle formed by trabeculae flattenout, thus expelling blood, during systole. This trend is mostevident from early to mid systole, at which point the wallsof the ventricle begin to translate inward. Both of thesetrends (flattening and translation) are approximately mir-rored during diastole. However the re-appearance of thetrabecular structures appears to occur much later towardsend diastole.

The cyclic changes in the fractal dimension duringsystole and diastole can be quantitatively related to theventricular contraction by correlating it to the ventricularvolume. Using the box-counting algorithm, the volume ofthe ventricle was computed at cardiac phase. As the boxsize decreased, the volume increased to reach a constantat sufficiently small box size. In order to create a

Septal Wall

Free WallAor�c Valve

Mitral Valve

Fig. 1. Imaging slices and extracted 3D model.

λ (mm)

Dim

ensi

on,D(λ)

0 5 10 15 20

1.9

2

2.1

2.2 End DiastoleEarly SystoleMid SystoleEnd SystoleEarly Diastole

Fig. 2. Plot of fractal dimension vs scale at different times during thecardiac cycle.

t / TO

Dim

ensi

on,D(λ)

0 0.2 0.4 0.6 0.8 11.9

2

2.1

2.2 Small ScaleLarge Scale

P < 0.003*

Fig. 3. Plot of fractal dimension vs time at two different scales.

B. Moore, L.P. Dasi / Chaos, Solitons & Fractals 57 (2013) 19–23 21

fully-closed object, the superior portion of the ventricle,where the mitral and aortic valves are, was artificially‘‘walled-off’’ with a plane perpendicular to the long axisof the ventricle. This was done at the same location forall time steps so as to cancel out any possible error thatit could have induced. Fig. 5 shows the variation in thefractal dimension of the small-scale features as a functionof normalized ventricular stroke volume. It can be clearlyseen that the decrease in fractal dimension during systoleis not exactly mirrored during diastole. As seen in the fig-ure, the systolic decrease in fractal dimension appearsmore gradual with the most rapid reduction occurringmid-way during the stroke. In contrast, most of the com-plexity is recovered during diastole only in the last 30%of the stroke.

3. Discussion

From the above presented results, it is clear that there isa significant change in small-scale geometrical featurescorresponding to the trabeculae as the heart pumps. Giventhe overall change in volume over time during the cardiaccycle, the data presented above demonstrates that thereexists a mechanical role for the trabecular in cardiacpumping mechanics.

The first aspect that will be explored is the effect ofscale on fractal dimension. Since the ventricle is a multi-fractal, it has different dimensions at different scales. Useof the box-counting algorithm at a certain scale only givesresults that describe features around the size of that scale.For example, the ventricle is approximately 120 mL in vol-ume at the end of diastole. In this context, a small scale fea-ture might be represented by a pocket or cavity in theventricle wall that encompasses about 5 mL of space,whereas a large scale feature would be something moredescriptive of the overall geometry such as the curvatureat the base or apex. The same would hold true for projec-tions, rather than cavities, of similar size.

Since there appears to be a dynamic relationship be-tween fractal dimension and volume, we offer a physicalreason to explain this finding. The result in Fig. 5 is dy-namic as opposed to kinematic due to the clearly evidenthysteresis in the fractal dimension cycle. A pure kinematicrelation would have had no hysteresis which would havesupported no active role of the ventricular trabeculae.The physical argument is that when a solid–liquid multi-fractal interface changes its dimension value, it must causea displacement, and therefore perform pressure–volumework. This can be understood by examining the definitionof fractal dimension as described by the box-countingalgorithm [19]. As its name suggests, this algorithminvolves dividing a space into a grid of boxes and thencounting the number of boxes that contain at least a partof the fractal surface. Repeating this process for a numberof different box sizes and utilizing the relationship

Early Systole Mid Systole End Systole Early Diastole Late Diastole

Fig. 4. 2D slices along the long axis (top) and short axis (bottom) of the ventricle at different times throughout the cardiac cycle.

(a)

(b)

22 B. Moore, L.P. Dasi / Chaos, Solitons & Fractals 57 (2013) 19–23

DðkÞ ¼ � d log NðkÞd log k yields the fractal dimension of the surface.

Since k is an independent variable, a change in DðkÞ meansthere must be a change in the number of boxes (i.e.N � kDðkÞ. For this to occur, a fractal surface must alter itsgeometry to take up either more or less space. Since afractal surface is made up of a number of features thatdetermine its space-filling capacity, it must alter these fea-tures to change its box-count, and therefore dimension. Asurface can achieve this in 3 ways: by changing (a) featurefrequency, (b) feature size, or (c) feature complexity. Illus-trations of these three ways are depicted in Fig. 6. A changein feature frequency as shown in Fig. 6a will not yield achange in volume of a region bounded by a fractal surface.To provide a heuristic argument in support of this, letsvisualize a fractal interface embedded in 2D appearing asa triangle wave with amplitude h, frequency s, and span-ning an overall length of L as shown in Fig. 6a. Then thevolume occupied under each wave is V ¼ 1

2 bh where b ¼ Ls

Therefore total volume, Vtot ¼ Vs ¼ 12 Lh, which is frequency

independent. However, Fig. 6b and c illustrate ways a frac-tal interface can change its dimension and cause a volumedisplacement. Since Vtot is dependent on the amplitude, h,of a feature, it shows that variations in feature size indeedcause a volume displacement. Fig. 6c is self-explanatory asit shows that a change in feature shape or complexity leads

% Stroke Volume

Dim

ensi

on,D(λ)

0 20 40 60 80 1002

2.1

2.2

Fig. 5. Plot of fractal dimension vs volume for small scales.

to volume change as this can be represented either byaddition or deletion of features.

The above fundamental physical arguments support thenotion that trabeculae carneae play a role in pumping.These features are essentially small blood-containing cavi-ties along the inner surface of the left ventricle. As the ven-tricle contracts, these pockets flatten out thus expelling theblood that they once contained, as shown by the reductionof fractal dimension in Fig. 5. If the trabeculae were notpresent, then the only method of blood expulsion would

(c)

Fig. 6. Illustration of fractal-dimension changes resulting in volumedisplacement.

B. Moore, L.P. Dasi / Chaos, Solitons & Fractals 57 (2013) 19–23 23

be purely by translation of the ventricle walls. Note thatwall translation occurs simultaneously with the flatteningof trabeculae. Fig. 5 also shows that the trabeculae remainflattened through late diastole.

These combined methods of blood displacement, via‘‘squeezing pockets’’ and wall translation could help theheart pump in a manner similar to the way a mechanicaltransmission system operates. In the case of the transmis-sion, large forces are initially needed to initiate movement,and this is accomplished at the expense of displacement.However, with the gain of momentum, the gear ratio canbe shifted in order to yield larger displacement withoutthe need for such large forces. The observations presentedhere are very similar. Systolic ejection of blood appears tobe achieved by changing the small-scale geometric fea-tures to minimize mechanical effort. Since the inner curva-ture of these features are large, the relative mechanicalforces in the tangential direction need not be great to de-velop a significant fluid pressure within the cavities thusreducing the effort to initial ejection. Once this has oc-curred, the ventricular walls can then move inward in or-der to expel the rest of the blood contained in theventricle. During diastole however, there is no contributionto mechanical effort through this physical mechanism.Therefore we do not see the trabeculae projections re-ap-pear until late diastole.

In summary, we have presented new insight into thepotential physical role of the complex anatomical featurescalled trabecular that are present on the ventricle-bloodinterface in the human heart. Given the complexity of theirgeometry, the tools of fractal geometry have been utilizedto quantitatively describe the anatomical changes in tra-beculae geometry over the cardiac cycle. We show thatthe significant changes in multi-fractal dimension of theventricle corresponding to the trabeculae length scales de-scribe an efficient manner to initiate ejection during earlysystole. These results offer a unique perspective into thecomplex biomechanics of cardiac pumping. Future workis needed in understanding these characteristics with re-spect to factors such as age, gender, status of heart diseaseetc. in large patient databases in-order to effectively assessany clinical relevance of the underlying biomechanicsuncovered. The present work only offers a novel hypothe-sis regarding the potential mechanical relevance of trabec-ulae in cardiac pumping.

4. Limitations

Future work is obviously needed to better understandthe clinical relevance of the above findings with respectto factors such as age, gender, status of heart disease etc.in large patient databases. One limitation of our study isthat it only offers a novel hypothesis regarding the poten-tially new mechanical role of VTCs in cardiac pumping. An-other major limitation of this study is the unitary samplesize that was used. While this yields a lack of information

regarding patient-to-patient variability, fundamental geo-metrical characteristics of the left ventricle have still beencharacterized and discussed in relation to pumpingmechanics.

Acknowledgements

We acknowledge Dr. John Oshinski at the Departmentof Radiology, Emory University for providing the high-res-olution CT-scan of the human ventricle. The authors haveno conflict of interest.

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