The Quest for the Effective Myocardial...

Luciano TERESIDip. Matematica e Fisica, Università Roma Tre

Mathematical Physiology of Cardiac, Skeletal and Smooth Muscles

Centro Ricerca Matematica Ennio De Giorgi, Pisa

5-8 October 2015

The Quest for the Effective Myocardial Architecture

Research Group

A. Di Carlo1, Roma Tre (Physics)A. Evangelista2, Sapienza Roma (Cardioloy, Echography)S. Gabriele, Roma Tre (Solid Mechanics)P. Nardinocchi3, Sapienza Roma (Soft Solid Mechanics)P. Piras3, Sapienza Roma (Biology)P.E. Puddu2, Sapienza Roma (Cardiology)C. Torromeo2, Sapienza Roma (Cardiology)L. Teresi1, Roma Tre (Continuum Physics)V. Varano1, Roma Tre (Solid Mechanics)

1Dip. Matematica e Fisica2Dip. Scienze Cardiovascolari, Respiratorie Nefrologiche e Geriatriche, 3Dip. Ingegneria Strutturale e Geotecnica4Dip. Meccanica e Aeronautica & Dip. Ingegneria Strutturale e Geotecnica

Michael H. Dickinson et al. How Animals Move: An Integrative View, Science 288, 100 (2000).

be estimated from their muscle length andcross-sectional area (112–114). Even for swim-ming scallops, a simple locomotor system in-volving one joint and a single power muscle(Fig. 3A), comparison of in vitro and in vivomuscle performance is complicated by hydro-dynamic effects and remains controversial(108, 115, 116). Given advances in transducertechnology, the number of preparations inwhich it is possible to measure in vivo workloops should increase in coming years, expand-ing even further our understanding of muscle’sdiverse roles in locomotion.

ProspectiveIntegrative and comparative approaches haveidentified several general principles of animallocomotion, which surprisingly, apply toswimming, flying, and running. The way inwhich animals exert forces on the externalworld often allows mechanical energy fromone locomotor cycle to be stored and recov-ered for use in another. The generality of

energy-storage mechanisms in differentmodes of locomotion is just beginning to beexplored. Forces lateral to the direction ofmovement are often larger than one mightexpect for efficient locomotion, but they mayenhance stability, and their modulation is es-sential for active maneuvers. Mechanisms ofnonsteady locomotion, including starting,stopping, and turning, are emerging areas ofinterest. Technological advances have en-abled the nascent studies of locomotion innatural environments and the mechanical in-teractions of organisms with their environ-ment. Animals use their musculoskeletal sys-tems for a variety of behaviors and, as aconsequence, are not necessarily optimizedfor locomotion. In nature, unlike in the labo-ratory, straight-line, steady-speed locomotionis the exception rather than the rule. Further,environmental forces make extreme demandson the musculoskeletal system of some loco-moting animals. The control system that en-ables animals to actively steer in the face of

these changing conditions combines bothneural and mechanical feedback with feedfor-ward control and pattern-generating circuits.The interface between these modes of controloffers a rich area for exploration. Finally,methods adapted from muscle physiology,combined with measurements of locomotormechanics, have revealed many mechanicalfunctions of muscle during locomotion.

The many recent advances in the studyof molecular motors are just beginning tobe integrated into an understanding of lo-comotion at the cellular scale. Molecularbiology and genetic engineering tech-niques, such as site-directed mutagenesis,are already being used to link the structureof individual molecules to locomotor per-formance at the organismal level (117,118). With a more thorough understandingof muscle function, systems-level control,interactions with the environment, and en-ergy transfer acting at the organismal level,locomotor biomechanics is now poised to

Fig. 3. Muscles canact as motors, brakes,springs, and struts.Muscles that generatepositive power (mo-tors) during locomo-tion and the areawithin associated workloops are indicated inred. Muscles that ab-sorb power during lo-comotion (brakes) andthe area within associ-ated work loops are in-dicated in blue. Mus-cles that act as springsof variable stiffnessare indicated in green.Muscles that act totransmit the forces(struts) are shown inblack. (A) Scallopswimming provides asimple example of amuscle generating posi-tive work to act as amotor. The cycle be-gins in the lower rightcorner of the loop, when the gape of the shell is maximal. Activation of themuscle (indicated in the scallop by the red rectangle) causes a rise in forceand subsequent shortening producing the pressure to drive a jet of waterthat propels the animal. At the upper left, the muscle begins to deactivate,force declines, and shortening continues. In the lower left, the muscle is fullydeactivated and force is minimal. Along the lower border of the loop, theshells are opened by passive recoil of elastic hinge ligaments. The areaenclosed within the loop is equal to the work done (product of force andlength change) by the muscle during each cycle. The counterclockwise workloop and red color indicate that the muscle generates positive power duringlocomotion. Adapted from (108) with permission from Company of Biolo-gists Ltd. (B) The pectoralis muscle of birds generates the positive powerrequired to fly. In pigeons, it has been possible to measure in vivo work loopswith strain gages bonded to bones near the muscle attachment point (force)and sonomicrometric crystals implanted at the ends of muscle fibers(length). Adapted from (100) with permission from Company of BiologistsLtd. (C) In running cockroaches, some muscles that anatomically appear tobe suited for shortening and producing power instead act as brakes andabsorb energy because of their large strains. Adapted from (96) with permis-

sion from Company of Biologists Ltd. (D) In flies, an intrinsic wing muscleacts to steer and direct the power produced by the primary flight muscles.Changes in activation phase alter the dynamic stiffness of the muscle andproduce alterations in wing motion. Adapted from figure 11 of “The controlof wing kinematics by two steering muscles of the blowfly (Calliphoravicina)” (98), copyright Springer-Verlag. (E) In swimming fish, the function ofmuscles varies within a tail-beat cycle and has been investigated with avariety of techniques in a diversity of species. In some fish designs, early ina beat, the cranial muscle fibers shorten and produce power, which istransmitted by more caudal muscle fibers acting as struts. As the beatcontinues, the fibers that were previously acting as struts change their roleto power-producing motors. The cartoon at the top shows a fish from theside. Beneath it are views from above the fish at two points in thetail-beat cycle. Adapted from (123) with permission from Company ofBiologists Ltd. (F) In vivo muscle force and length measurements inrunning turkeys indicate a dual role for the gastrocnemius muscle. Itgenerates positive power during uphill running but acts as a strut duringlevel running, which allows the springlike tendons to store and recoverenergy. Adapted from (106).

www.sciencemag.org SCIENCE VOL 288 7 APRIL 2000 105

M O V E M E N T : M O L E C U L A R T O R O B O T I C

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How Animals Move

LENGTH-DEPENDENT ACTIVATION OF CARDIAC MUSCLE/ter Keurs et al. 711

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1.6Sarcomere length

FIGURE 7 The relations between tension and sarco-mere length at [Ca2+]o = 2.5 m.M (circles) and 0.5 mM(triangles) obtained from three muscles. The relationsbetween active tension and length from contractions inwhich sarcomere length was held constant (filled sym-bols) did not differ from the sarcomere length-total ten-sion relations derived from isometric contractions of themuscle (unfilled symbols) during which 8-14% shorten-ing occurred.

and Strobeck, 1978; Pollack et al., 1977) and ino-tropic interventions influence both the central sar-comeres and the damaged regions.

The experiments on trabeculae in this study showshortening of sarcomeres in a uniform central regionof the specimen (Figs. 2 and 4) at the expense ofdamaged regions near the clamps. Elimination ofsarcomere shortening revealed two features of themechanisms involved in contraction: (1) tensiondevelopment consists of a rapid rise of tension to amaximum attained within 60 msec followed by atension plateau which precedes a gradual decline oftension; (2) the relation between peak tension andsarcomere length depends on the instantaneous sar-comere length (Fig. 7), and on the inotropic state ofthe muscle.

Three earlier studies of tension development dur-ing contractions in which the length of the contrac-tile element was held constant are well known.Brady (1971) used an ingenious technique in whichthe relation between length and tension of the serieselastic element was measured first. Contractile ele-ment length was kept constant during subsequent

100-

50-

mN.mm'

TENSION

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16 2b 2.4 pm— * • SARCOMERE LENGTH

FIGURE 8 Relation between passive and active tensionand sarcomere length from isometric contractions of themuscle at [Ca2+]o = 2.5 and 0.5 mM at low rate ofstimulation and after frequency potentiation in two mus-cles. The relations were derived from measurements inthe following order: high [Ca2*]o and a low rate ofstimulation (unfilled circles), high [Ca2+]o after fre-quency potentiation (filled squares), low [Ca2+]o and alow rate of stimulation (unfilled triangles), low [Ca2+]ofollowing frequency potentiation (asterisks), high [Ca2+Jo and a low rate of stimulation following the series atlow [Ca2+]o (unfilled squares). The response to frequencypotentiation at low [Ca2*]o is illustrated in the inset.Upper trace: force (F) 0.1 mN/div, negative deflectionsare event markers; lower trace: sarcomere length (SL)0.02 [im/div, the bar indicates 2.00 nm, upward deflec-tions indicate shortening; time base: 20 sec/div. A seriesof 20 stimuli is delivered in 4 seconds followed by a 30-second period of rest. The next contraction starts at SL= 2.15 iim and develops 220% of the force at low rate of

stimulation; shortening is 0.08 \an less than control. Thesecond stimulus is again delivered after an interval of30 seconds and is followed by stimuli at intervals of 5seconds. Tension decays to control value in about 10beats. The relation between tension and sarcomerelength following potentiation at low [Ca*]o (asterisks) isidentical to the relations at high [Ca2+]o.

contractions by conversion of the measured tensionto series elastic element length changes imposed onthe muscle by a computer-controlled muscle puller.Julian et al. (1976) used the spot follower technique(Gordon et al., 1966) to control central segmentlength of rabbit papillary muscle dynamically, and

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active force

activated muscle

H.E.D.J. ter Keurs et al., Circ. Res. 1980

Tension development and sarcomere length in rat cardiac trabeculae. Evidence of length-dependent activation

passive response

4

Tension-Length Relations at Sarcomere Scale

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thereafter decreases and becomes zero during diastolicfilling. On the other hand LV Ep starts increasing after theinitiation of LV filling as the LV volume increases. Itreaches its maximum value at the end-of-filling phase,remains constant during isovolumic contraction, andthereafter decreases during ejection (as the LV volumedecreases). While the generation of Ea helps us to explainthe development of the LV pressure increase during isovo-lumic contraction, the decrease of Ea during diastole helpsus to explain the decrease in LV pressure during early fill-ing. The incorporation of both Ep and Ea helps us toexplain the LV pressure changes during the filling andejection phases.

MethodsData acquisitionThe subjects in this study were studied in a resting recum-bent (baseline) state, after premedication with 100–500mg of sodium pentobarbital by retrograde aortic catheter-ization. Left ventricular chamber pressure was measuredby a pigtail catheter and Statham P23Eb pressure trans-ducer; the pressure was recorded during ventriculography.Angiography was performed by injecting 30–36 ml of75% sodium diatrizoate into the LV at 10 to 12 ml/s. It hasbeen found by using biplane angiocardiograms that calcu-

lated orthogonal chamber diameters are nearly identical[16]. These findings are used to justify the use of single-plane cine techniques, which allow for beat-to-beat anal-ysis of the chamber dimensions.

For our study, monoplane cineangiocardiograms wererecorded in a RAO 30° projection from a 9 in image inten-sifier using 35 mm film at 50 frames/s using INTEGRISAllura 9 system at the Nation Heart Centre (NHC), Singa-pore. Automated LV analysis was carried out to calculateLV volume and myocardial wall thickness. The LV data,derived from the cineangiographic films and depicted inFigure 2 consists of measured volume and myocardialthickness of the chamber as well as the correspondingpressure. All measurements are corrected for geometricdistortion due to the respective recordings systems.

In Figure 2, it is noted that during the early filling phase,LV pressure decreases even though LV volume increases.This phenomenon is defined as the 'LV suction effect',which will be explained later by using our concepts ofactive and passive elastances. This phenomenon is alsodepicted in Figure 3 and Table 1.

A case study of measured LV pressure, volume and wall thickness during a cardiac cycleFigure 2A case study of measured LV pressure, volume and wall thickness during a cardiac cycle. An example of a patient's measured LV pressure, volume and wall thickness during a cardiac cycle; t = 0-0.08s is the isovolumic contrac-tion phase, t = 0.08s-0.32s is the ejection phase, t = 0.32s-0.40s is the isovolumic relaxation phase, and t = 0.40s-0.72s is the filling phase. Note that even after 0.4 s, the LV pressure still continues to decrease from 17 mmHg (at 0.4s, at start of filling) to 8 mmHg at 0.44s.

Relationship between LV volume and pressure for the data of figure 2Figure 3Relationship between LV volume and pressure for the data of figure 2. Relationship between LV volume and pressure for the data of Figure 1. Points (21–36) constitute the filling phase, (1–5) constitute the isovolumic contraction phase, (5–17) constitute the ejection phase, and (17–21) con-stitute the isovolumic relaxation phase. Note that after point 21, the LV pressure decreases; this characterizes LV suction effect.

isovolumic

contraction

isovolumic

relaxation

ejection

(systole)filling

(diastole)

AV close

MV open

AV open

MV close

ejection

filling

contraction

relaxation

VES

(volume end systole)

VED

(volume end diastole)

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thereafter decreases and becomes zero during diastolicfilling. On the other hand LV Ep starts increasing after theinitiation of LV filling as the LV volume increases. Itreaches its maximum value at the end-of-filling phase,remains constant during isovolumic contraction, andthereafter decreases during ejection (as the LV volumedecreases). While the generation of Ea helps us to explainthe development of the LV pressure increase during isovo-lumic contraction, the decrease of Ea during diastole helpsus to explain the decrease in LV pressure during early fill-ing. The incorporation of both Ep and Ea helps us toexplain the LV pressure changes during the filling andejection phases.

MethodsData acquisitionThe subjects in this study were studied in a resting recum-bent (baseline) state, after premedication with 100–500mg of sodium pentobarbital by retrograde aortic catheter-ization. Left ventricular chamber pressure was measuredby a pigtail catheter and Statham P23Eb pressure trans-ducer; the pressure was recorded during ventriculography.Angiography was performed by injecting 30–36 ml of75% sodium diatrizoate into the LV at 10 to 12 ml/s. It hasbeen found by using biplane angiocardiograms that calcu-

lated orthogonal chamber diameters are nearly identical[16]. These findings are used to justify the use of single-plane cine techniques, which allow for beat-to-beat anal-ysis of the chamber dimensions.

For our study, monoplane cineangiocardiograms wererecorded in a RAO 30° projection from a 9 in image inten-sifier using 35 mm film at 50 frames/s using INTEGRISAllura 9 system at the Nation Heart Centre (NHC), Singa-pore. Automated LV analysis was carried out to calculateLV volume and myocardial wall thickness. The LV data,derived from the cineangiographic films and depicted inFigure 2 consists of measured volume and myocardialthickness of the chamber as well as the correspondingpressure. All measurements are corrected for geometricdistortion due to the respective recordings systems.

In Figure 2, it is noted that during the early filling phase,LV pressure decreases even though LV volume increases.This phenomenon is defined as the 'LV suction effect',which will be explained later by using our concepts ofactive and passive elastances. This phenomenon is alsodepicted in Figure 3 and Table 1.

A case study of measured LV pressure, volume and wall thickness during a cardiac cycleFigure 2A case study of measured LV pressure, volume and wall thickness during a cardiac cycle. An example of a patient's measured LV pressure, volume and wall thickness during a cardiac cycle; t = 0-0.08s is the isovolumic contrac-tion phase, t = 0.08s-0.32s is the ejection phase, t = 0.32s-0.40s is the isovolumic relaxation phase, and t = 0.40s-0.72s is the filling phase. Note that even after 0.4 s, the LV pressure still continues to decrease from 17 mmHg (at 0.4s, at start of filling) to 8 mmHg at 0.44s.

Relationship between LV volume and pressure for the data of figure 2Figure 3Relationship between LV volume and pressure for the data of figure 2. Relationship between LV volume and pressure for the data of Figure 1. Points (21–36) constitute the filling phase, (1–5) constitute the isovolumic contraction phase, (5–17) constitute the ejection phase, and (17–21) con-stitute the isovolumic relaxation phase. Note that after point 21, the LV pressure decreases; this characterizes LV suction effect.

isovolumic

contraction

isovolumic

relaxation

ejection

(systole)filling

(diastole)

AV close

MV open

AV open

MV close

ejection

filling

contraction

relaxation

VES

(volume end systole)

VED

(volume end diastole)

Pressure-Volume loop in a human left ventricle

L. Zhong, D.N. Ghista, E.Y.K. Ng and S.T. Lim, Passive and active ventricularelastances of the left ventricle, BioMedical Engineering On Line. 2005.

4 / 40

L. Zhong, D.N. Ghista, E.Y.K. Ng and S.T. Lim, Passive and active ventricular elastances of the left ventricle, BioMedical Engineering On Line. 2005.

Pressure-Volume at Organ Scale

Figure 2. Confocal microscopy of cardiac muscle.(A) Single ventricular myocyte viewed by confocal microscopy. Image prepared by combining a stack of serial optical sections

through the cell. Myofibrils (seen as striations) visualised by immunostaining with an antibody against !-actinin (a component of themyofibril Z-bands).

(B) Distribution of vinculin in a section of cardiac muscle revealed by immunofluorescence labelling. The image shows numerouscells like that in panel (A), joined together at intercalated disks (id). Vinculin is seen as a series of bright dots along the lateral surfacesof the plasma membrane (series of small arrows), marking the sites of costameres. Vinculin extends from the surface plasmamembrane along the transverse tubule membranes penetrating into the cell (seen as striations) and is also abundant in the transversesegments of the disk, where the myofibrils join the plasma membrane via fasciae adherentes junctions.

(C) Immunoconfocal localization of dystrophin (a peripheral membrane protein) and "-dystroglycan (an integral membrane protein)by double labelling. Both these proteins show a continuous distribution at the lateral plasma membrane in the rodent myocardiumviewed here, and form part of the structure of the transverse tubules penetrating into the cell. Unlike vinculin, dystrophin, and"-dystroglycan are not present at the intercalated disks. (Micrographs by Shirley Stevenson and Stephen Rothery).

My favorite cell

Single ventricular myocyte viewed by confocal microscopy. Myofibrils (striations) visualised by immunostaining with an antibody against alpha-actinin, a component of the myofibril Z-bands.

N. J. Severs The cardiac muscle cell, BioEssays 22:188–199, (2000) John Wiley & Sons, Inc.

Cardio myocite

high-K! solution containing collagenase (Sigma Blend Type L: 1 mg/ml,10 min) for digestion (note: a small but controlled content of calciumis beneficial for ensuring steady and reproducible enzyme activity; seecomposition in Table 1).

Next, ventricles were harvested by cutting along the atrioventri-cular border, chopped into small cubes ("1–2 mm3), and gentlyagitated in oxygenated high-K! solution (free of enzyme). Thesupernatant containing ventricular cells was collected after tissuefragments were allowed to settle, and the remaining tissue wasresuspended in fresh high-K! solution. This procedure was repeatedsix times until little or no visible tissue remained.

The supernatant, containing cells, was passed through a 100-#mnylon mesh (Cadish Precision Meshes, London, UK) and centrifugedfor 1 min at 16 g (PK121R; ALC, Cologno Monzese, Italy) using asoft-start/soft-stop protocol. After centrifugation, the supernatant wasdiscarded and the cell pellet was carefully resuspended in storagesolution (50 ml of Tyrode solution, with 50 mg of bovine serumalbumin and 0.83 mg of trypsin inhibitor to terminate further enzymeactivity).

Experimental Setup

Carbon fibers. CF (12–14 #m in diameter), kindly provided byProf. J.-Y. Le Guennec (University of Tours) were mounted in glass

capillaries pulled from 1.16-mm inner diameter/2.0-mm outer diam-eter glass tubes (GC200F-10; Harvard Apparatus, Holliston, MA).The final section (1.2–1.4 mm) of the glass capillary, containing theCF, was bent by 30° to allow near-parallel alignment of the CF withthe bottom of the perfusion chamber (Fig. 1, right). The CF protrudingfrom the glass capillary was trimmed to 1.20 mm and fixed in positionusing cyanoacrylate adhesive.

CF position control. Each CF is mounted to a position controlsystem with six regularly exploited degrees of freedom (explainedbelow). Before use, CF are aligned near-horizontal (a minor slant isrequired to ensure that the CF tip is slightly lower than the rim of theglass capillary that holds the CF) and near-parallel to each other(again, a slight angle is needed to allow CF tip proximity withoutinterference of the two glass capillaries holding them; Fig. 1). This islargely achieved by fitting the glass tube (with the CF at its pulledend) in a modified patch-clamp pipette holder (rubber seal compres-sion fixing), which provides rotational control along the glass tube (1stdegree of freedom). This, in conjunction with connecting the twopipette holders at a slight angle to each other, ensures that the CF tipscan be aligned near-parallel to the bath and to each other.

For use, CF are lowered into the bath with a rotational coupling(roughly aligned with the x-axis of the system’s coordinate space; 2nddegree of freedom) between the pipette holder and the hydraulicmanipulators that are used for CF fine positioning. The correctcombination of height of the attachment point, length of the glasstube, angle of rotation of pipette holder, and angle of capillary tipensures seamless alignment and positioning of CF in the bath near theoptical path.

For cell attachment, CF are manipulated using one Narishigeminiature hydraulic manipulator on each side (SM-28; Narishige,Tokyo, Japan) with 5-mm travel in x/y/z directions (3rd to 5th degreesof freedom). Once CF positioning and cell attachment have beencompleted, cells are lifted off the glass coverslip (see below for furtherdetail).

From this point in time, bidirectional position control along thex-axis is implemented via a pair of high-accuracy PZT (P-621.1CL;Physik Instrumente, Karlsruhe/Palmbach, Germany; 6th degree offreedom). Each PZT, carrying one CF assembly, is fixed (withopposite intrinsic directionality) on sleighs, mounted on an optical-

Table 1. Composition of solutions used inthe isolation procedure

CompositionTyrode

SolutionCa2!-FreeSolution

High-K!

Solution

NaCl, mM 140.0 140.0 4.0KCl, mM 5.4 5.4 10.0K-glutamate, mM 130.0MgCl2, mM 1.0 1.0 1.0CaCl2, mM 1.8 0.025EGTA, mM 1.0HEPES, mM 5.0 5.0 5.0Glucose, mM 11.0 11.0 11.0pH 7.4 (with NaOH) 7.4 (with NaOH) 7.2 (with KOH)

Fig. 1. Overview of stretching device.Sledges, mounted to an optical railing sys-tem, allow for coarse manual positioning andsupport precise alignment of piezo transla-tors (PZT) that are used for fine controlduring force-length (FL) manipulations inparallel to the x-axis of the coaligned opticaland perfusion system components (double-headed arrows). Attached to the PZT are3-axis hydraulic micromanipulators for car-bon fiber (CF) handling during attachment tothe cell (triple arrows). Two further degreesof freedom are provided 1) by rotationalcouplings along the axes of the glass pipettes(from which the CF protrude, right inset)and 2) by a spring-loaded joint that couplesthe CF holder to the x-axis head stage of thehydraulic manipulators (see semicircular ar-rows; numbers relate to the degrees of free-dom described in text).

H1488 DYNAMIC FORCE-LENGTH CLAMP OF CARDIOMYOCYTES

AJP-Heart Circ Physiol • VOL 292 • MARCH 2007 • www.ajpheart.org

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G. Iribe, M. Helmes, P. Kohl. Force-length relations in isolated intact cardiomyocytes subjected to dynamic changes in mechanical load. Am J Physiol Heart Circ Physiol (2007).

Force-Length Relations for Cardio-myocytes


that it lumps the symmetrical movement of both PZT into a singleparameter).

For phase 1 (isometric contraction), both PZT are moved outwardto keep cell length and position constant. The PZT command isidentical to roughly the first quarter of the curve labeledA ! [!SourceSignal] (t) in Fig. 4, right (i.e., an inverted and tunedversion of the source signal) until the time T1–2, at which active forcereaches a user-defined afterload. Up to this point, the PZT commandsignal (heavy solid line in Fig. 4, right) [Command](t) can bedescribed as follows:

if 0 ! t ! T1–2

"Command#$t% " A ! "!SourceSignal#$t%

where [!SourceSignal](t) is the inverted source signal at each point intime t (obtained during auxotonic contraction, as described above) andA is the command signal gain that yields isometric contraction.

For phase 2 (isotonic shortening), both PZT are moved inwardto keep CF bending constant, following the shape of the sourcesignal with a gain B that yields isotonic behavior, until the timepoint at which the source signal reaches peak contraction (Tmax), sothat:

if T1–2 # t ! Tmax

"Command#$t% " "Command]$T1–2% $ |B ! "SourceSignal#$t%

$ B ! "SourceSignal#$T1–2%|

For phase 3 (isometric relaxation), both PZT are moved furtherinward to avoid distension of the relaxing cell by the force stored inthe CF, bent during the previous contraction. This is done by follow-ing the shape of the inverted source signal with a gain C that supportsisometric relaxation, until the time T3–4, at which the command signalmeets a value corresponding to an isotonic contraction initiated fromthe original end-diastolic point:

if Tmax # t ! T3–4

"Command#$t% " "Command#$Tmax% $ |C ! "!SourceSignal#$t%

$ C ! "!SourceSignal#$Tmax%|

where C is the gain that supports an isometric state.For phase 4 (isotonic lengthening), both PZT are moved outward

again to lengthen the cell while keeping CF bending (i.e., force)constant. The PZT command in this phase is identical to roughly thelast quarter of the active control sequence labeled D![SourceSignal] (t)in Fig. 3C.

if T3–4&t

[Command]$t% " D ! "SourceSignal#$t%

where D is the gain that yields relengthening of the cell at constantforce.

Statistics

All values are presented as means ' SE. Two-way analysis ofvariance was used for statistical analyses, and P & 0.05 was consid-ered to indicate a significant difference between means.

RESULTS

Cell-CF Attachment

CF attach glue-free to cardiac cells (19). The actual mech-anism of CF-cell adhesion is not currently understood in detail,but cell adhesion is usually strong enough to support passivediastolic distension and auxotonic contraction studies (3). Priorresearch involving isometric contractions (where more signif-icant forces occur between cell and CF) was limited to workingat minute preloads (end-diastolic SL near slack values; Ref.

Fig. 3. PZT command (top; (LP) and related force (middle) and length (bottom; (LF) curves for auxotonic (A), isometric (B), isotonic (C), and physiologicalwork-loop style contractions (D). The length change during auxotonic contraction (bottom left) is referred to throughout as the “source signal.”



on May 10, 2007


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Sarcomere exercise


internal shortening was variable; contractions that were alsoisometric at the SL level could sometimes be measured, butusually internal shortening took place. Sarcomere shorteningwas reproducible when switching back and forth betweenvarying preloads and afterloads, so although there was internalsarcomere shortening, there appeared to be no slippage ofCF-myocyte attachment points. In addition, internal sarcomereshortening is fairly homogeneous in the area between CF.Regional internal sarcomere shortening during isometric con-traction was 3.29 ! 1.10% adjacent to CF and 5.05 ! 1.02%in the center, between CF (n " 16). Using “upside down”-mounted CF (Fig. 1, “front view”), we further found thatinternal shortening is also reasonably homogeneous in thez-direction, with values of 4.31 ! 0.68% near the cell surfaceto which CF are attached and 5.21 ! 1.00% near the opposite“free” surface of the cell (n " 9).

FL Relationship

We traced FL relation curves to identify ESFLR, indepen-dently varying preload and/or afterload. Between interventions,cells were routinely released to record auxotonic contractionsfrom a range of preloads (see Fig. 5A). These were comparedwith preintervention reference ESFLR to assess maintainedcell mechanical performance (instability of which was anexclusion criterion for data analysis). The observed consis-tency of ESFLR throughout experiments with dynamic FLcontrol suggests that effects of slow force responses to stretch(1) are negligible in this setting. For all results presented in thiscommunication, ESFLR was steady over experimental periodsof up to 20 min.

Figure 5A shows superimposed FL relation curves of auxo-tonic contractions (reference ESFLR) and of isometric contrac-tions from a range of preloads (end-diastolic SL: 1.85 to 1.97#m). Figure 5B shows FL relation curves with afterloads

Fig. 5. FL relation curves of auxotonic, isometric, and isotonic contractions withvaried preloads and afterloads and their end-systolic FL relation (ESFLR). A:superimposed FL curves of auxotonic contractions (no PZT command; shadedlines) and of isometric contraction (solid lines), obtained with varied preloadsranging from 1.85- to 1.97-#m end-diastolic sarcomere length (EDSL). TheESFLR was near linear for all contractions, regardless of mode of contraction(dashed line). B: superimposed FL contraction curves with varying afterload(from isometric to isotonic), obtained at low (SL 1.90 #m; shaded lines) andhigh preloads (SL 1.97 #m; solid lines). ESFLR was near linear and indepen-dent of preload and afterload (dashed line).

Fig. 6. FL relation curves of auxotonic and work-loop style contractions withvaried preload and afterload. A: superimposed FL curves of auxotonic con-tractions (shaded lines) and work-loops (solid lines), initiated from varyingpreloads (SL 1.85 to 2.04 #m). ESFLR was load and contraction modeindependent and near linear (dashed line). B: superimposed FL curves ofauxotonic contractions (shaded lines) with varied preload (SL 1.82 to 1.97 #m)and physiological work-loop style contractions (solid lines) initiated fromconstant preload (SL 1.90 #m) but working against varying afterloads. ESFLRremains independent of contraction mode and near linear line (dashed line).



on May 10, 2007


ownloaded from

G. Iribe, M. Helmes, P. Kohl. Force-length relations in isolated intact cardiomyocytes subjected to dynamic changes in mechanical load. Am J Physiol Heart Circ Physiol (2007).

internal shortening was variable; contractions that were alsoisometric at the SL level could sometimes be measured, butusually internal shortening took place. Sarcomere shorteningwas reproducible when switching back and forth betweenvarying preloads and afterloads, so although there was internalsarcomere shortening, there appeared to be no slippage ofCF-myocyte attachment points. In addition, internal sarcomereshortening is fairly homogeneous in the area between CF.Regional internal sarcomere shortening during isometric con-traction was 3.29 ! 1.10% adjacent to CF and 5.05 ! 1.02%in the center, between CF (n " 16). Using “upside down”-mounted CF (Fig. 1, “front view”), we further found thatinternal shortening is also reasonably homogeneous in thez-direction, with values of 4.31 ! 0.68% near the cell surfaceto which CF are attached and 5.21 ! 1.00% near the opposite“free” surface of the cell (n " 9).

FL Relationship

We traced FL relation curves to identify ESFLR, indepen-dently varying preload and/or afterload. Between interventions,cells were routinely released to record auxotonic contractionsfrom a range of preloads (see Fig. 5A). These were comparedwith preintervention reference ESFLR to assess maintainedcell mechanical performance (instability of which was anexclusion criterion for data analysis). The observed consis-tency of ESFLR throughout experiments with dynamic FLcontrol suggests that effects of slow force responses to stretch(1) are negligible in this setting. For all results presented in thiscommunication, ESFLR was steady over experimental periodsof up to 20 min.

Figure 5A shows superimposed FL relation curves of auxo-tonic contractions (reference ESFLR) and of isometric contrac-tions from a range of preloads (end-diastolic SL: 1.85 to 1.97#m). Figure 5B shows FL relation curves with afterloads

Fig. 5. FL relation curves of auxotonic, isometric, and isotonic contractions withvaried preloads and afterloads and their end-systolic FL relation (ESFLR). A:superimposed FL curves of auxotonic contractions (no PZT command; shadedlines) and of isometric contraction (solid lines), obtained with varied preloadsranging from 1.85- to 1.97-#m end-diastolic sarcomere length (EDSL). TheESFLR was near linear for all contractions, regardless of mode of contraction(dashed line). B: superimposed FL contraction curves with varying afterload(from isometric to isotonic), obtained at low (SL 1.90 #m; shaded lines) andhigh preloads (SL 1.97 #m; solid lines). ESFLR was near linear and indepen-dent of preload and afterload (dashed line).

Fig. 6. FL relation curves of auxotonic and work-loop style contractions withvaried preload and afterload. A: superimposed FL curves of auxotonic con-tractions (shaded lines) and work-loops (solid lines), initiated from varyingpreloads (SL 1.85 to 2.04 #m). ESFLR was load and contraction modeindependent and near linear (dashed line). B: superimposed FL curves ofauxotonic contractions (shaded lines) with varied preload (SL 1.82 to 1.97 #m)and physiological work-loop style contractions (solid lines) initiated fromconstant preload (SL 1.90 #m) but working against varying afterloads. ESFLRremains independent of contraction mode and near linear line (dashed line).



on May 10, 2007


ownloaded from

Work-loop

varyingpre-load

varyingafter-load

Isometric / Isotonic

no PZTcontrol

PV Loop for a Spherical Heart

50 100 150Vol HmlL0

50

100

150

Prex HmmHgL

1

2

3

4

Figure 6: A typical PV loop of a normal human pa-tient, as measured in (15), with our ESPVR and ED-PVR curves superimposed, as equation (4.9) dictates;the large blue dots correspond to the same four keypoints shown in figure 1.

In figure 7, bottom, the time course of the contractioncycle is represented: the end–diastolic state corre-sponds to "

c

= 1 and the end–systolic state to "c,es

=0.766. Corresponding to these values, equations (4.9)gives the EDPVR and ESPVR curves which are rep-resented in figure 6 (blu and green solid line, respec-tively) as superimposed on the PV loop extractedby (15). Moreover, the transition from EDPVR toESPVR may be derived through a generalization toany point during the cardiac cycle of the procedureused to extract the EDPVR and the ESPVR rela-tionships from equation (4.9). The pressure–volumecurves corresponding to a few intermediate points inthe cycle are shown in figure 6 (red lines). Let usnote the following key points.

• The EDPVR curve is intrinsically nonlinear, asit is expected and as equation (4.9) rules. More-over, it shows that, at subphysiological volumeranges, increasingly negative pressures are re-quired to reduce volume. Nevertheless the sim-plicity of the theoretical model, a likely (see (1))

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL0

20

40

60

80

100

120

140Pressure HmmHgL

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05Contraction

Figure 7: A typical pressure cycle VS time (top, mea-sured from (15)) and the corresponding contraction(bottom) as given by equation (4.9).

e↵ect is caught by the model.

• The volume axis intercept vs

of the EDPVRcurve corresponds to the slack chamber at zerostress which is determined at a ventricular pres-sure of 0 mmHg. As noted in (1), this interceptdi↵ers from the volume axis intercept of the ES-PVR which represents the volume of the cham-ber at zero stress and at maximum contraction.

• According to the experiments, (see figure 5 in(1)), figure 6 shows a smooth transition fromthe EDPVR to ESPVR curves corresponding toa sti↵ening of the chamber due to the muscleactivation.

• Actually, the volume axis intercepts of thepressure–volume curves at di↵erent values ofcontraction di↵er one from the other. In otherwords, the volume axis intercept is not totallyindependent of the contraction state (2).

9

• Figure 6 shows a non linear ESPVR. Indeed, theESPVR is in general non linear even if reason-ably linear relationships are commonly used tocharacterize properties of the chamber with mus-cles in a state of activation (see equation (2.1)).

As well as a new point of view in looking at pressure–volume loops, our model allows the association of anyPV loop with a contraction–volume loop. It capturesat a glance a hidden property of the left chamber andgives an idea of the contractility of the left ventriclearound all the key phases of the cycle. Precisely, fig-ure 8 shows the contraction–volume loop correspond-ing to the numeric PV loop described in figure 6. Itshows some key interesting features such as a vari-ability of the contraction measure in a range whichis in agreement with scientific literature (2) and astrong similarity to the experimental loop in figure 3.

80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4

Figure 8: Contraction along the cardiac cycle as func-tion of volume.

The following cartoon sketches e�ciently our point ofview. Blue balls aim to represent the visible states ofthe left chamber in the four key points of the cycle.Correspondingly, red balls represent the contractedand stress–free states. As an example, let us followwhat happens from point 1 to point 2. During the iso-volumic contraction, the visible volume of the ballsis roughly the same (136ml versus 132ml), that is,red balls 1 and 2 have almost the same radius. Atthe same time, at a hidden layer something is occur-ring and the red ball 2 is smaller than the red ball 1

(identifying the slack state of the chamber) becausea contraction is acting and making the di↵erence be-tween states 1 and 2.

r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27

Figure 9: Schematic of LV contraction. The top rowshows the visible state of the LV in the four key pointsof the cycle; the bottom row shows the correspondingcontracted, but unloaded, LV (actually, the first pointrepresents the slack state). All the measures are incm.

Finally, let us introduce the deformation " withrespect to the end–diastolic state; the correspondingstrains ("2 � 1)/2 along the cardiac cycle are shownin figure 10 and, interestingly, are in agreement with(22) where circumferential strains at di↵erent pointsof the left ventricle are measured.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain

Figure 10: Visible strains vs time along the cardiaccycle.

10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10



80 90 100 110 120 130 140 150Volume HmlL0.7

0.8

0.9

1.0

1.1Contraction

1

2

3

4




r1 = 3.19 r2 = 3.16 r3 = 2.75 r4 = 2.78

rc,1 = 2.57 r

c,2 = 2.31 rc,3 = 1.97 r

c,4 = 2.27



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time HsL-0.20

-0.15

-0.10

-0.05

0.00Visible Strain


10

actual shape

range of “ground states”

ground state

P. Nardinocchi, L. Teresi, V. Varano. A simplified mechanical modeling for myocardial contractions and the ventricular pressure–volume relationships. Mechanics Research Communications (2011).

0.8

1

0.9

filling

ejection

Ciclo della contrazione

Working Hypothesis

With motion we mean shape change.

Growth can be considered as a motion, too.

Living matter realizes motions at almost no stress.

How to model a change of the zero stress state?

Free Swelling of Hydrogels

A Lesson From Wood

micrometer scale

making it unsuitable for uses in which it is likely to experience high stresses. Thelarger microfibril angle also means that the wood has a higher longitudinal shrink-age on drying, but a lower transverse shrinkage. This explains the distortion ondrying of pieces of wood containing both normal and compression wood. In severecases the S3 wall layer is absent and the S2 layer contains splits which lie parallel tothe microfibril angle.

The structure of tension wood fibres is less clear-cut than that of compressionwood tracheids. Tension wood fibres are longer than those in normal wood and havebeen found to contain a lower proportion of lignin than normal wood, giving it awhiter appearance. They are most commonly described as lacking an S3 layer andhaving variable amounts of the S2 remaining, inside which is a gelatinous layercomposed mainly of hydrated cellulose microfibrils (Norberg and Meier 1966)oriented almost parallel to the long axis of the fibre. This gives the wood aglistening gelatinous appearance when wet. Variations on this theme have beenreported, however. For example, Faruya et al. (1970) reported the presence of agelatinous layer in fibres of Populus euroamericana which had retained their S3layer, while Cote et al. (1969) reported an S3 layer inside a gelatinous layer. Moreand more variations on the structure of tension wood fibres are being reported, withnumerous species apparently lacking a gelatinous layer in their fibre walls.

These structural variations, which are adapted to the mechanical role the fibresand tracheids must fulfil, are of course, a normal response by the tree to enhance its

Fig. 1.3 Cell structure ofnormal (mature) wood,juvenile wood andcompression wood (fromJozsa and Middleton 1994with thanks toFPInnovations, Canada)

1 Introduction 7

1 - Middle lamella (mostly lignin) 2 - Primary wall (randomly orientated microfibrils) 3 - Outer wall (microfibrils oriented in 2 directions not parallel to cell axis) 4 - Middle wall (microfibrils orientated almost parallel to cell axis, also thickest layer) 5 - Inner wall (same as 3) 6 - Pore (or lumen)

Image from: Gardiner B., Barnett J., Saranpa P., Gril J. (Editors), The Biology of Reaction Wood, Springer (2014)

Micro(n) Architecture + Anisotropic Swelling

Microfibrils orientation controls the anisotropy of swelling

normal wood tension wood

or gelatinous layer because of its cellulose content, and translucent and jelly-likeappearance. Although gelatinous fibres can often be detected on unstained sectionsit is preferable to use a staining method to highlight the occurrence of G-layer suchas fast-green/safranin (Chow 1946), safranin/astra blue (Jourez et al. 2001) or AzurII© (Clair et al. 2003).

The cell wall organization of gelatinous fibres can show some variation, both inthe same species and in different species (Fig. 2.9). Actually the literature aboundsin sometimes conflicting observations on gelatinous fibre morphology, linked, forexample, to the species in question, the area sampled in the tree or in the ring, or thepresence of axis eccentricity (Jourez et al. 2001). In the same way that ordinaryfibres show a three-layered structure in their secondary wall with the S1, S2 and S3layers, gelatinous fibres can show various patterns, i.e. S1+G, S1+S2+G,S1+S2+S3+G. Onaka (1949) referred to three types of gelatinous layer which maycorrespond to the ones cited above. However, he has indicated that each type is tobe found in certain genera or families, whereas the present observations havedemonstrated the occurrence of more than one type in the same tree or particularspecimen (Wardrop and Dadswell 1955; Araki et al. 1983).

Besides the above variations in structure that can appear in any specimencontaining gelatinous fibres, a variation in the intensity of the development of thegelatinous layer exists inside the same tree and expressed through the thickness ofthe gelatinous layer. However, the border effect observed by Clair et al. (2005b)brings doubt to this point (Fig. 2.10). Their study shows that during cross section-ing, some major changes occur in the G-layer thickness and the transverse shapenear the surface. Further results by Clair et al. (2005a) clearly demonstrate that theuse of transverse cross sections for anatomical observations of tension woodcontaining a G-layer can be misleading. Most standard methods for sectioningwood samples do not include embedding, but perform sectioning on softened

Fig. 2.9 Schematic models for the cell wall structures of fibres in normal wood (a) and tensionwood (b–d), redrawn fromWardrop and Dadswell (1955). Solid lines indicate cellulose microfibrilorientation. (a) Normal fibres do not develop a G-layer. (b) G-layer where S2 and S3 layers developnormally. (c) S3 layer replaced with G-layer. (d) G-layer forms as the innermost layer next to theS3 layer (Kwon 2008)

2 Morphology, Anatomy and Ultrastructure of Reaction Wood 21

Image from: Gardiner B., Barnett J., Saranpa P., Gril J. (Editors), The Biology of Reaction Wood, Springer (2014)

P. Nardinocchi, M. Pezzulla, L. Teresi. Steady and transient analysis of anisotropic swelling in fibered gels. JAP (2015)

normal woodreaction wood

leaning trunk

compression wood

long & thin

tension wood

short & wide

compressed fibers

stretched fibers

Reaction Wood

Le curve ESPVR ed EDPVRPressure-Volume Loop: Atrium VS Ventricle

4

3

1

21

5

45

let us focus on this part of the PV loop

Ventricle PV loopAtrium PV loop

Is the atrium an effective pump?

1. Distortions

1.1 Within the theory of elasticity, distortions are introduced to model a changeof the ground state, or zero–stress state, of a body element;they are described by a tensor-valued map B 3 x 7! F

o

(x) 2 Lin

x

Fo

(x)

1.2 Distortions have a two-fold nature:Kinematics: they add 9 degrees of freedom.Dynamics: they describe a ground state.“F

o

cannot even be conceived without the standard notion of stress andsome constitutive information on it.”

A. DiCarlo, S. Quiligotti. Growth & Balance. Mech. Res. Communications, 2002.

6 / 40

Distortions - 1

1.3 “Cut & Distort”Distortions act point-wise; moreover, we consider smooth tensor fields.

x1

x2

x3

x1

x2

x3

Fo

(x1)

Fo

(x2)

Fo

(x3)

A key question is: can we glue each other the distorted volume elements?

7 / 40

Distortions - 2

1.4 Can we glue each other the distorted volume elements? NO!Distortions describe how volume elements “would like” to be placed; usually,one cannot do what he likes.

x1

x2

x3

x1

x2

x3

Fo

(x1)

Fo

(x2)

Fo

(x3)

To glue all the volume elements we must add further strain.

8 / 40

Distortions - 3

2. Elastic Strain

2.1 Given a material fiber e, the elastic deformation Fe

measures the differencebetween its distorted image F

o

e and its actual state Fe.

f = Fo

eFo

e

F

Fe = Fe

f

Fe

= FF�1o

2.1 The elastic energy ' has to be a function of the elastic strain Ce

= F>e

Fe

:

' = '(Ce

) .

9 / 40

Elastic Strains

3. Metric

3.1 Metric: from ancient Greek µ"⌧⇢o⌫ = to measure.We consider a placement f : B ! E of a body B into the Euclidean space E;f maps body points into positions; its gradient rf maps (tangent) vectors.

x

dx

f

y = f(x)

dy = rf dx

rf

The (square of the) length of the line element dy is given by:

dy · dy = rf dx ·rf dx = (rf)

>rf dx · dx = C dx · dx .

The symmetric tensor C := (rf)

>rf measures the change in length andangle of line elements: it is the natural metric on the configuration f(B) of B.

3.2 The elastic metric is given by the elastic strain Ce

:

Ce

= F>e

Fe

= F�>o

(rf)

>rf F�1o

= F�>o

CF�1o

11 / 40

Metric - 2

P. Nardinocchi, L. Teresi, V. Varano. The elastic metric: A review of elasticity with large distortions, Int. J. of Non-Linear Mechanics (2015)

3.3 Back to the key question: can we glue together the distorted volumeelements, without adding any further strain?This question can be rephrased in the realm of differential geometry:given the Riemannian metric C

o

induced by a distortions field Fo

:

Co

= F>o

Fo

,

we may ask: is Co

an Euclidean metric? We may put it even better.3.4 The key question from a differential geometry point of view:

is there a placement f : B ! E such that:

(rf)

>rf = Co

?

A positive answer to this question is of paramount importance because, ifsuch is the case, the volume elements distorted by F

o

can be glued togetherwithout any further strain:

(rf)

>rf = Co

) Ce

= I ) '(I) = 0

You can realize extremely large deformation at zero elastic energy.

12 / 40

Metric - 2

4. Where we are now

4.1 We have:• a body B;

• a placement of the body f : B ! E into the Euclidean space;

• a natural metric C = (rf)

>rf induced by f ;

• a distortions field Fo

;

• the metric induced by distortions Co

= F>o

Fo

;

• the elastic strain Ce

= F�>o

CF�1o

;

• an elastic energy ' = '(Ce

).

Moreover, we have a key question:can we realize the distortions without varying the elastic energy?

4.2 What we need? A way to characterize such a kind of distortions.

13 / 40

Intermediate Summary

6. A neat example

6.1 Let us consider a flat 2D manifold (a sheet of paper): it may happen that agiven distortion cannot be realized, without further strain, whilemaintaining the sheet on a flat plane

F

Fo

Fe

6.2 Two strategies:1) add further strain to each surface element, and stay in the plane;2) give up staying plane, and enjoy the full 3D Euclidean space.

18 / 40

A Neat Example

Hot to See a Metric?

(a ) el l ip t ic geometry

--

(b ) Eucl idean geometry

(c) hyperbo l i c geometry

Figure 10.1 The three homogeneous two-dimensional geometries.

MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 1001 6

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.


--







--






J.R. Weeks (Freelance mathematician), The Shape of Space, Marcel Dekker 2002

creases slit insert a t r iangle

into the slit

Figure 10.3 (See Exercise 10.1) Tape equilateral triangles together so that seven meet at each vertex. The inset shows a

good way to get started.



creases slit insert a t r iangle

into the slit

Figure 10.3 (See Exercise 10.1) Tape equilateral triangles together so that seven meet at each vertex. The inset shows a

good way to get started.



Hot to Play with Metric?

J.R. Weeks (Freelance mathematician), The Shape of Space, Marcel Dekker 2002

5.3 The Curvature of SpaceA key intrinsic property of a manifold is its Riemann Curvature R.The Riemann curvature “explores” the manifold, “sailing” through it; giventhree vector fields u, v (the routes) and w (the vessel), you have

R(u,v)w = ru rv w �rv ru w �r[u,v] w .

When the vector w is parallel transported around a loop in a Euclideanspace, it will always return to its original position. However, this does nothold in the general case.

5.4 A test on the metric.The Riemann Curvature R can be represented in terms of a metric C:

R = R(C) .

R(C) measures the extent to which a metric C is not locally isometric to aEuclidean space.

16 / 40

The Curvature of Space - 1

5.5 A Remarkable Theorem.A placement f : B ! E of a body B into the Euclidean space E is dubbedimmersion in the jargon of differential geometry.The remarkable theorem says: if B is simply connected, then

R(C) = 0 , there exists an immersion f such that: (rf)

>rf = C .

Moreover, if B is connected, then f is unique up to isometries.

5.6 A test on distortionsThe previous theorem is a powerful tool to test the metric C

o

= F>o

Fo

induced by distortions:

R(Co

) = 0 , there exists f ) distortions are realizable for free .

In short:R(C

o

) = 0 ) '(Ce

) = 0 .

P.G. Ciarlet, An Introduction to Differential Geometry with Applications toElasticity, Lecture Notes, 2005.

17 / 40

The Curvature of Space - 2

7.2 Representation of the nematic distortions:The mesogen direction is represented by the nematic tensor field

N :=n⌦ n , with n the director field (|n| = 1 .

given that, we may have:

7.2.1 Transversely-isotropic distortions:

Uo

= �k N+ �? (I�N) .

N

7.2.2 Orthotropic distortions:

Uo

= �k N+ � e⌦ e+ µ e⇤ ⌦ e⇤ . with n · e = 0 , e⇤ = n⇥ e .

The intensity of the distortions is measured by the scalar fields �k, �?, �, µ;these parameters = 1 in the isotropic phase, and are 6= 1 in the nematic one.

21 / 40

Nematic Distortions

10. Hybrid

10.1 Hybrid geometry: N is on vertical planes.

e1

e3

e2

#(z)

# = #(z)

n(#) = cos(#) e1 + sin(#) e3

N(#) = n(#)⌦ n(#)

Uo

(#) = �k N(#) + �? (I�N(#))

Co

(#) = �

2k N(#) + �

2? (I�N(#))

10.2 Our key question:can we realize the distortions without varying the elastic energy?

29 / 40

9.1 Hybrid Nematic Elastomers: N stays vertical

Y. Sawaa, K. Urayama, T. Takigawa, A. DeSimone, L. Teresi, Thermally DrivenGiant Bending of Liquid Crystal Elastomer Films with Hybrid Alignment,Macromolecules, 2010.

9.2 Chiral Nematic Elastomers: N stays horizontal

Y. Sawaa, F. Ye, K. Urayama, T. Takigawa, V. Gimenez-Pinto, R.L.B. Selinger, J.V.Selinger, Shape selection of twist-nematic-elastomer ribbons, PNAS, 2011.

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Planar Nematic Distortions

10.5 Second answer: with a nearly linear variation it is possible.Being C

o

a plane metric (N stays on the vertical plane), the associateRiemann curvature R has only one non trivial component:

R(Co

) = 2 (�

2k��

2?) ( cos

2#(z)�sin

2#(z) )

⇣ ✓d#(z)

dz

◆2

+cos#(z) sin#(z)

d

2#(z)

dz

2

⌘.

The differential equation R(Co

) = 0 with boundary conditions#(�h/2) = ⇡/2, #(h/2) = 0 has a simple solution:

#(z) =

1

2

arccos

✓2

h

z

◆) '(C

e

) = 0 for any bending.

Compatible (red) and linear (black) nematic direction along the thickness.

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8. Cylinder with orthotropic distortions

8.1 Can we realize orthotropic distortion on a cylinder?The answer is no, unless we admit the strains �k, �, µ to vary radially:

Uo

= �k(r)N(r,#) + �(r) e(r,#)⌦ e(r,#) + µ(r) e⇤(r,#)⌦ e⇤(r,#) .

22 / 40

Orthotropic Distortions - 1

8.2 We consider a candidate immersion:to satisfy the test with the Riemann curvature

R(Co

) = 0 , with Co

= U2o

,

the immersion should respect the symmetries of Fo

; thus, we try:

f(#, r, z) = ⇢(r) [ cos(⌧ z)n(#) + sin(⌧ z) t(#) ] + f3(r, z) e3 .

It is worth noting that the body is not simply connected: the condition ofvanishing Riemann curvature is necessary but not sufficient for ensuring thecompatibility; this is the reason for starting from a candidate immersion.

23 / 40


8.3 Problem: the intensities of the distortion �(r), �(r), µ(r) must satisfy:

(rf)

>rf = �

2(r) e(✓)⌦ e(✓) + µ

2(r) e⇤(✓)⌦ e⇤(✓) + �

2k(r)n(✓)⌦ n(✓) .

This is a system of six differential equations in six unknown functions,three for the candidate immersion, three for the distortion intensities.

8.4 Solution: distortions must behave like that:

24 / 40


8.5 The global (stress-free) configurations:

P. Nardinocchi, L. Teresi, V. Varano, Strain induced shape formation in fibredcylindrical tubes, J. Mechanics and Physics of Solids, 2012.

25 / 40

Compatible Stress-free Configurations

P. Nardinocchi, L. Teresi, V. Varano, Strain induced shape formation in fibred cylindrical tubes,

J. Mechanics and Physics of Solids, 2012.

Fetal Heart

D. Srivastava, E.N. Olson. A genetic blueprint for cardiac developmentNature 407, 221-226 (2000)

Helical Ventricular Myocardial Band

F. Torrent-Guasp. Ventricular Myocardia Band: Structure and Function


between anterior and posterior papillary muscles, the formerbeing completely encircled by the hand.

Finally, we came to the aspect of the dissection when theHVMB is unraveled and finally ready to be stretched out tobecome a single straight myocardial band (Fig. 7(24)). Asimple 908 rotation around the apex unravels the apical loopsegments (Fig. 7(25)). Additional 1808 rotation around thecentral fold unravels the basal and the apical loops of theHVMB (Fig. 7(26)). The HVMB of Torrent-Guasp now appears inits full extent and beauty, with pulmonary artery at one andthe aorta at the opposite side (Fig. 7(27)).

Fig. 8 depicts the most important phases of the dissectionprocedure (see also Video 2—available only in online versionof this article).

The elegance and astounding simplicity of this dissectionis reflected in the capacity to easily reverse these unravelingsteps, with ready re-establishment of the well-known three-dimensional ventricular architecture that existed prior to thebeginning of dissection (Fig. 8 and Video 2).

The HVMB is divided in two loops, each of themcomprising two segments (Fig. 9D). The central 1808-foldof the HVMB defines two loops: the basal loop (from the rootof the pulmonary artery to the beginning of the centralfold—i.e. to the anterior papillary muscle) and the apicalloop (from the beginning of the central fold to the root ofthe aorta) (Fig. 9B and D). Each of these two loops is furtherdivided in two segments. The posterior interventricularsulcus topographically coincides with the posterior linearborder of the RV cavity and divides the basal loop into twosegments: the right segment — coinciding with the RV freewall and the left segment — coinciding with the LV free wall(Fig. 9B and D). The right segment also defines the outer(nonseptal) border of the tricuspid orifice and the leftsegment defines the outer (nonseptal) border of the mitralorifice. These borders are common targets in AV surgicalannuloplastic procedures.

The apical loop is also divided in two segments. After the1808 twist (at the central fold of the HVMB), the descendant

M.J. Kocica et al. / European Journal of Cardio-thoracic Surgery 29S (2006) S21—S40S28

Fig. 8. The most important phases of the dissection procedure (bovine heart). Bottom: Torrent-Guasp’s home-laboratory, where the HVMB concept was borne andwhere lots of people were able to learn Torrent-Guasp’s dissections from the very source (photo by M.J. Kocica, 2002).

by on July 28, 2009 ejcts.ctsnetjournals.orgDownloaded from

M.J. Kocica et al. / European Journal of Cardio-thoracic Surgery 29S (2006) S21—S40


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Saturday,29 September 2012

M.J. Kocica et al. The helical ventricular myocardial band: global, three-dimensional, functional architectureof the ventricular myocardium. European Journal of Cardio-thoracic Surgery 29S (2006)

Fibres Architecture

The Three-Dimensional Arrangementof the Myocytes Aggregated

Together Within the MammalianVentricular Myocardium

MORTEN SMERUP,1* EVA NIELSEN,1 PETER AGGER,1 JESPER FRANDSEN,2

PETER VESTERGAARD-POULSEN,2 JOHNNIE ANDERSEN,3

JENS NYENGAARD,3 MICHAEL PEDERSEN,4 STEFFEN RINGGAARD,4

VIBEKE HJORTDAL,1 PAUL P. LUNKENHEIMER,5 AND ROBERT H. ANDERSON6

1Department of Cardiothoracic & Vascular Surgery, Aarhus University Hospital,Skejby, Denmark

2Center for Functionally Integrative Neuroscience, Aarhus University Hospital,Aarhus Sygehus, Denmark

3Stereology and EM Laboratory and MIND Center, Aarhus University, Denmark4MR Research Center, Aarhus University Hospital, Aarhus, Denmark

5Klinik und Poliklinik fur Thorax-, Herz- und Gefasschirurgie, University Munster,Munster, Germany

6Cardiac Unit, Institute of Child Health, University College, London, UK

ABSTRACTAlthough myocardial architecture has been investigated extensively,

as yet no evidence exists for the anatomic segregation of discrete myocar-dial pathways. We performed post-mortem diffusion tensor imaging on 14pig hearts. Pathway tracking was done from 22 standardized voxelgroups from within the left ventricle, the left ventricular papillarymuscles, and the right ventricular outflow tract. We generated pathwayswith comparable patterns in the different hearts when tracking from allchosen voxels. We were unable to demonstrate discrete circular or longi-tudinal pathways, nor to trace any solitary tract of myocardial cellsextending throughout the ventricular mass. Instead, each pathway pos-sessed endocardial, midwall, and epicardial components, merging one intoanother in consistent fashion. Endocardial tracks, when followed towardsthe basal or apical parts of the left ventricle, changed smoothly their heli-cal and transmural angulations, becoming continuous with circular path-ways in the midwall, these circular tracks further transforming into epi-cardial tracks, again by smooth change of the helical and transmuralangles. Tracks originating from voxels in the papillary muscles behavedsimilarly to endocardial tracks. This is the first study to show myocardialpathways that run through the mammalian left and right ventricles in ahighly reproducible manner according to varying local helical and trans-mural intrusion angles. The patterns generated are an inherent featureof the three-dimensional arrangement of the individual myocytes aggre-gated within the walls, differing according to the regional orientation andbranching of individual myocytes. We found no evidence to support theexistence of individual muscles or bands. Anat Rec, 292:1–11,2009. ! 2008 Wiley-Liss, Inc.

*Correspondence to: Morten Smerup, Department of Cardio-thoracic & Vascular Surgery, and Clinical Institute, AarhusUniversity Hospital, Skejby, Brendstrupgaardsvej, 8200 AarhusN, Denmark. Fax: 14589496016.E-mail: [email protected]

Received 25 March 2008; Accepted 27 August 2008

DOI 10.1002/ar.20798Published online 2 December 2008 in Wiley InterScience (www.interscience.wiley.com).

! 2008 WILEY-LISS, INC.

THE ANATOMICAL RECORD 292:1–11 (2009)

Fibres Architecture

Diffusion Tensor Magnetic Resonance Imaging

Sampling: 128 x 128 x 128 = 2M voxels, resolution 1.3 mm

Key Findings:

First, that the averaged long axis of myocytes confined within small volumes of the ventricular wall is not exclusively tangential, but rather intrudes or extrudes to a varying extent from the epicardium to the endocardium.

Second, that the myocytes are aggregated within the fibrous matrix so as to produce a potentially infinite number of pathways throughout the myocardium, albeit that the number and vectors of the intercellular connec- tions permit identification of a distribution of diversion for any chosen small myocardial volume.

Smerup et al.,The Anatomical Record, 292:1–11 (2009)

dependent upon two factors. The first is the orientationin space of the long axis of the myocyte, which is its lon-gitudinal vector. The second is the net vector of theaggregated myocytes, the branches with adjacent myo-

cytes determining the strength of communication in anythree-dimensional direction. In addition, when examin-ing the myocardium with a technique such as diffusiontensor imaging, spatial resolution plays an important

Fig. 7. Track progression from a transmural segment of the left ventricular basal free wall. The wallspans four voxels, the red being endocardial, the green and blue belonging to the midwall and the yellowbeing epicardial. Also here tracks smoothly change helical and transmural intrusion angles to travelbetween endocardial, midwall, and epicardial positions.

9MYOCARDIAL THREE-DIMENSIONAL ARRANGEMENTFibres Architecture (pig heart)

Heuristic Problems in Defining theThree-Dimensional Arrangement of

the Ventricular MyocytesROBERT H. ANDERSON,1* SIEW YEN HO,2

DAMIAN SANCHEZ-QUINTANA,3 KLAUS REDMANN,4 AND

PAUL P. LUNKENHEIMER4

1Cardiac Unit, Institute of Child Health, University College, London, United Kingdom2Department of Paediatrics, National Heart and Lung Institute, Royal Brompton

Campus, Imperial College, London, United Kingdom3Department of Anatomy, Universidad de Extramadura, Badajoz, Spain

4Experimental Thoraco-, Vascular and Heart Surgery,University Hospital, Munster, Germany

ABSTRACTThere is lack of consensus concerning the three-dimensional arrangement

of the myocytes within the ventricular muscle masses. Bioengineers are seek-ing to model the structure of the heart. Although the success of such modelsdepends on the accuracy of the anatomic evidence, most of them have beenbased on concepts that are far from anatomical reality, which ignore manysignificant previous accounts of anatomy presented over the past 400 years.During the 19th century, Pettigrew emphasized that the heart was built on thebasis of a modified blood vessel rather than in the form of skeletal muscles.This fact was reemphasized by Lev and Simkins as well as Grant in the 20thcentury, but the caveats listed by these authors have been ignored by propo-nents of two current concepts, which state either that the myocardium isarranged in the form of a “unique myocardial band,” or that the walls of theventricles are sequestrated in uniform fashion by laminar sheets of fibroustissue extending from epicardium to endocardium. These two concepts arethemselves incompatible and are further at variance with the majority ofanatomic studies, which have emphasized the regional heterogeneity to befound in the three-dimensional packing of the myocytes within a supportingmatrix of fibrous tissue. We reemphasize the significance of this three-dimen-sional muscular mesh, showing how the presence of intruding aggregates ofmyocytes extending in oblique transmural fashion also contends against thenotion that all myocytes are orientated with their long axes parallel to theepicardial and enodcardial surfaces. Anat Rec Part A, 288A:579–586, 2006.© 2006 Wiley-Liss, Inc.

Key words: myocardium; cardiodynamics; fibrous matrix;perimysium

It is well recognized that, in terms of their histology,myocytes can be voluntary, involuntary, or cardiac. Thedifferent types of muscle differ not only in their micro-scopic appearances, but also in their function (Bozler,1948; Burnstock, 1970; Williams et al., 1989). Cardiacmuscle is intermediate in both structure and functionbetween the striated and smooth variants. Initiallythought also to be syncytial in nature (Williams et al.,1989), it is now accepted that the atrial and ventricularmuscular masses are made up of millions of individualmyocytes set in axially coupled endless chains embedded

in a supporting matrix of fibrous tissue (Criscione et al.,2005). Each myocyte nonetheless is an entity in itself.

*Correspondence to: Robert H. Anderson, Cardiac Unit, Institute ofChild Health, 30 Guilford Street, London WC1N 1EH, United King-dom. Fax: 44-20-7905-2324. E-mail: [email protected]

Received 24 November 2005; Accepted 13 January 2006DOI 10.1002/ar.a.20330Published online 3 May 2006 in Wiley InterScience(www.interscience.wiley.com).

THE ANATOMICAL RECORD PART A 288A:579–586 (2006)

© 2006 WILEY-LISS, INC.

Fibres Architecture

Secondary StructureSecondary Structures

First eigenspaceof TD-MRI

Second eigenspaceof TD-MRI

“Y” branching of cardio-myocytes

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Saturday,29 September 2012

Muscle cells:

cardiac

skeletal

smooth

Basic Classification of Muscles

Could it be that Myocardial Fibers do not Exist?

Could it be that cardiomyocites are arranged in order to keep under control the amount of elastic energy stored at

the end of systole and to use it for an efficient filling?

Provocative Questions?

The Quest for the Effective Myocardial...

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