A simple model for myocardial changes in a failing heart

International Journal of Non-Linear Mechanics 68 (2015) 132–145

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International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm

A simple model for myocardial changes in a failing heart John A. Shaw a,n, Kannan Dasharathi b, Alan S. Wineman c, Ming-Sing Si d a

Department of Aerospace Engineering, University of Michigan, 1320 Beal Ave, Ann Arbor, MI 48109, USA Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA d Department of Cardiac Surgery, University of Michigan, 1540 E. Hospital Dr. SPC 4204, Ann Arbor, MI 48109, USA b c

art ic l e i nf o

a b s t r a c t

Article history: Received 19 April 2014 Received in revised form 25 June 2014 Accepted 25 June 2014 Available online 1 August 2014

In a simplified setting, a multi-network model for remodeling in the left ventricle (LV) is developed that can mimic various pathologies of the heart. The model is an extension of the simple model introduced by Nardinocchi and Teresi [9], Nardinocchi et al. [10,11] that results in an algebraic relation for LV pressure– volume–contraction. We considered two networks, the original tissue and a new tissue, each of which has its own volume fraction, stress-free reference configuration, elastic properties, and contractility. This is used to explore the consequences of microstructural changes in the muscle tissue on LV function in terms of the pressure–volume loop during a single cardiac cycle. Special attention is paid to the stroke volume, which is directly related to cardiac output, and changes in LV wall stress caused by various disease states, including wall thinning (dilated cardiomyopathy), wall thickening (hypertrophic cardiomyopathy), contractility degradation, and stiffness changes (scarring). Various scenarios are considered that are of clinical relevance, and the extent and nature of remodeling that could lead to heart failure are identified. & 2014 Published by Elsevier Ltd.

Keywords: Biomechanics Heart failure Multi-network hyperelasticity

1. Introduction Heart failure is a medical condition with significant individual and societal impact [1,2]. Approximately 1 in 9 deaths in the USA are associated with congestive heart failure (CHF) and 50% of adults diagnosed with CHF will die in 5 years [1–3]. Further, up to 5% of neonates born with congenital heart disease have single ventricle heart disease with a deficient ventricular mass that cannot support a biventricular repair and undergo surgical palliation within the first 2 years of life [4–6]. This palliation results in a decreased life expectancy, a severely limited exercise capacity and significant predisposition to end-stage CHF later in life [7]. Therefore gaining further understanding of the mechanics of heart failure may have significant implications for the guidance and design of treatment strategies. Although there is better understanding of cardiac disease at the molecular and cellular levels, there remains a definite need for simple mathematical models at the organ level to aid physiologists and clinicians. Many different approaches and models used to analyze heart failure and ventricular mechanics have been proposed since the 1970s, yet even a n

Corresponding author. Tel.: þ 1 734 764 3395; fax: þ1 734 764 0578. E-mail addresses: [email protected] (J.A. Shaw), [email protected] (K. Dasharathi), [email protected] (A.S. Wineman), [email protected] (M.-S. Si). http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.015 0020-7462/& 2014 Published by Elsevier Ltd.

basic pressure–volume loop analysis of the heart has not yet been standardized [8]. Our aim is to describe and model the mechanical behavior of the left ventricle (LV) in the setting of certain cardiomyopathies and to explore the conditions under which ventricular function degrades to critical levels. The recent mathematical model of the heart by Nardinocchi and Teresi [9], Nardinocchi et al. [10,11] (henceforth termed the NTV model) is appealing in its simplicity to capture the essential features of the pressure–volume cycle of the LV. The goal of the NTV is mathematical tractability, while ignoring many well known complicating and individualized aspects of actual physiology. Our extension to the NTV model investigates the global implications of remodeling in the myocardium that may result in changes in LV function and cardiac output. More precise and complex models of LV function exist [12–14]; however, they all require detailed numerical methods and large scale computations. The NTV model decomposes muscle deformation into two parts [9]: (1) a change of reference configuration due to electrochemically induced contraction (active deformation), and (2) the purely elastic deformation due to applied (external) loads from this new reference configuration (mechanical deformation). This interpretation is entirely consistent with the multi-network framework for materials undergoing microstructural changes introduced by Wineman and Rajagopal [15,16], in which a constitutive

J.A. Shaw et al. / International Journal of Non-Linear Mechanics 68 (2015) 132–145

theory was developed for elastomers undergoing scission and crosslinking of their macromolecular structure due to large deformation and elevated temperatures. Further extension to account for dissolution and re-assembly of macromolecular networks in natural rubber was carried out by Shaw et al. [17]. The consequences of macromolecular structure changes on the mechanical response were explored in a number of studies on elastomers [18–23], fiber reinforced composites with elastic matrix and fibers [24], and a study of the mechanical interaction between a contracting muscle and a deformable body [25]. This notion of active deformation formed the basis of the NTV model for the study of myocardial contractions [10,11]. In particular, the LV was modeled as a thin-walled spherical container composed of homogeneous muscle tissue undergoing uniform circumferential stretch. The actual stretch ratio in the muscle tissue was the product of a mechanical stretch ratio and one representing muscle contraction. When the muscle contracted, the radius of the spherical container attempted to decrease, but encountered resistance by the contained blood. This led to mechanical stretch and stress in the container wall and pressure on the blood. The muscle tissue forming the container was described as a non-linear elastic material for which the stress was given in terms of a mechanical stretch ratio. With this approach, it was possible to discuss ventricular pressure– volume–contractility response under various conditions without the complications arising from the specific architecture of the extracellular matrix and muscle fibers as well as their individual constitutive properties. This special issue is dedicated to the memory of our colleague Alan Gent, who was a master at distilling complex problems to their essential features and providing an insightful, simplified analysis. We offer our simple model of LV function in the same spirit. The NTV model provides a convenient and intuitive means to introduce and explore additional modeling ideas without the complications arising from heterogeneous physiological, spatial, and biomechanics details. The particular idea of interest here is that of microstructural change of muscle tissue which is analogous to scission and crosslinking of a deforming elastomeric network structure. Inspired by the NTV model, the purpose of the present work is to introduce microstructural changes and to study their consequences. The microstructural change could be due to excessively large deformations or non-mechanical effects arising from various disease states. The model can be used to simulate LV wall thickening (hypertrophy) or LV circumferential wall lengthening (dilated cardiomyopathy), as well as changes in effective contractility that create modified pressure–volume relationships. The derivation of the baseline model is provided in Section 2, and the baseline structural response is reviewed in Section 3. Our modified constitutive equation for microstructural change and the altered structural response are derived in Section 4. The results of a numerical parametric study and discussion of its implications are then provided in Section 5.

2. The baseline model This section presents the essential ingredients of the simplified model of the LV developed in Nardinocchi et al. [10,11], with a slight modification. The LV is modeled as a thin-walled hollow sphere that undergoes a radially symmetric motion due to muscle tissue contraction and a uniform time-dependent pressure p applied to its internal surface from the contained blood. The hollow sphere is idealized as a membrane with slack, or unpressurized, radius Rs . In the current state, its radius is r, the pressure over its internal surface is p, and the biaxial Cauchy tensile stress is

133

σ distributed uniformly over the wall thickness h. Except where needed for clarity, explicit indication of time as an independent variable is suppressed for notational simplicity. Force balance of a hemispherical portion of the hollow sphere in the current configuration is expressed by the relation p¼2

σh r

:

ð1Þ

Although the LV wall is a composite material formed of muscle, extracellular matrix (collagen), and vasculature, it is treated as an equivalent homogeneous, isotropic material that supports a mean equi-biaxial stress σ . Thus, this heart model is composed of material elements, each of which is envisioned as a representative volume element (RVE) whose response can, in concept, be derived from that of its components by a micromechanical analysis. Thus, from here on we use the terminology ‘muscle tissue’ or ‘material element’ interchangeably to represent the homogenized material element of muscle tissue. Each material element undergoes equal biaxial circumferential stretch. Its reference configuration is considered to be stress free and is described as being slack. Let Ls and ℓ denote the slack and current lengths, respectively.1 The circumferential stretch ratio

λ¼

ℓ Ls

ð2Þ

is associated with the deformation from the slack state to the current state, called the visible (or total) deformation. The muscle fiber will contract when subjected to an electrochemical stimulus if it is not constrained by an externally applied force. Let Lc denote the (time-dependent) contracted length when the muscle tissue is stimulated but unstressed. The circumferential contraction stretch ratio

λc ¼

Lc Ls

ð3Þ

is associated with the deformation induced by the stimulus, called the active deformation ð0 o λc r1Þ. The circumferential stretch ratio

λm ¼

ℓ Lc

ð4Þ

is associated with the deformation from the contracted state to the current state. It is called the mechanical stretch ratio, which is the passive, purely elastic stretching associated with the applied stress. Eqs. (2)–(4) imply the decomposition

λ ¼ λm λc ;

ð5Þ

which interprets the total deformation as an active deformation from the slack to a stress free contracted state followed by a mechanical deformation from the contracted state to the current state. The slack, contracted and visible lengths of the muscle tissue correspond to distinct radii and contained volumes of the spherical surface, which enter into the pressure–volume relationship. This correspondence leads, on using v ¼ 4π r 3 =3, to the following relations:

λc ¼


1=3 Rc Vc ¼ ; Rs Vs

λm ¼


1=3
 r v 1 v 1=3 ¼ ¼ ; Rc Vc λc V s

ð6aÞ

ð6bÞ

1 Upper case variables are used to denote reference, or stress free, configurations, while corresponding lower case variables denote current (loaded) configurations.

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λ¼

J.A. Shaw et al. / International Journal of Non-Linear Mechanics 68 (2015) 132–145


1=3 r v ¼ : Rs Vs

ð6cÞ

The muscle tissue is assumed to respond as a non-linear elastic material, and the biaxial response function assumed in Nardinocchi et al. [10,11] is

σ h ¼ Y½εðλm Þ3 ;

ð7Þ

where Y is a measure of the elastic stiffness and

εðλm Þ ¼ 12 ½λ2m  1

ð8Þ

is the Green–St. Venant strain based on the mechanical stretch ratio. Combining Eqs. (1), (6b), (7) and (8) gives the pressure– volume–contraction relation #3
 "
 Y 4π 1=3 1 v 2=3 p¼  1 : ð9Þ 4 3v λ2 V s c

We note that Y is a structural parameter, rather than a material constant since it does not have the usual interpretation of an elastic modulus for the following reasons: (a) at λm ¼ 1 ðσ ¼ 0Þ the tangent modulus is zero ðdσ =dλm ¼ 0Þ, independent of Y, so Y does not represent a linearized proportionality constant between stress and strain about the reference (slack) configuration, (b) Y as used here relates a referential strain measure ε to the current (Cauchy) stress, which is unconventional, and (c) its units are ‘force/unit length’ rather than ‘force/unit area’. To avoid these issues, we prefer to use a conventional hyperelastic constitutive model. Among the many suitable choices, we chose the Ogden hyperelastic model [26] where in the principal frame the strain energy density (energy/unit reference volume) for isotropic material is n

W 0 ðλm;1 ; λm;2 ; λm;3 Þ ¼ ∑

k¼1

μk αk α α ½λ þ λ k þ λ k  3; αk m;1 m;2 m;3

ð10Þ

in terms of the mechanical principal stretch ratios fλm;1 ; λm;2 ; λm;3 g. For isochoric (volume preserving) deformation, the principal Cauchy stresses are

σ i ¼ q þ λm;i

∂W 0 ; ∂λm;i

i ¼ 1; 2; 3;

ð11Þ

and q is a constant arising from the incompressibility constraint, here solved by setting the out-of-plane stress to zero, σ 3 ¼ 0. For simplicity, we chose a ‘one-term’ Ogden model (setting n ¼1, μ1 ¼ μ, α1 ¼ α). Under isochoric deformation (λm;1 ¼ λm;2 ¼ λm and λm;3 ¼ λm 2 ) the equi-biaxial stress ðσ ¼ σ 1 ¼ σ 2 Þ is

σ ¼ σ^ ðλm ; μ; αÞ ¼ μ½λαm  λm 2α :

ð12Þ

The response function σ^ takes the mechanical stretch ratio λm as its argument and has two parameters, the coefficient μ in stress units and a dimensionless exponent α. For simple tension, the Young's modulus (linearized about λm ¼ 1) is E ¼ 32 μα;

ð13Þ

which is required to have a positive value. Under isochoric 2 deformation, the current wall thickness is h ¼ λ H s where H s is the slack wall thickness. Using Eq. (6c), the pressure–volume relation is then "
α=3
  2α=3 # Hs V s v v α 2α p¼2 μ λc  λc : ð14Þ Vs Vs Rs v

3. Baseline structural response Fig. 1a shows the LV p–v cycle data (discrete points) of a normotensive individual, taken from Zhong et al. [27]. The portions

of the pressure–volume loop with solid and open circles shown in Fig. 1 represent the respective systolic and diastolic segments of the cardiac cycle. Table 1 provides the parameter values that were chosen to give reasonable fits of this healthy LV (baseline behavior) in terms of the end-diastolic pressure volume relation (EDPVR, λc ¼ 1) and the end-systolic pressure volume relation (ESPVR, λc ¼ 0:795). The fitted parameters correspond to a slack LV volume V s ¼ 85:4 ml and Young's modulus of E¼3.08 kPa. For reference, the gray lines show EDPVR and ESPVR results from the NTV model using the parameters in Nardinocchi et al. [10], which are not very different from those obtained by our model. Referring to the magnified view in Fig. 1b, systole begins at point A where the end-diastolic volume ðvA Þ and pressure ðpA Þ intersect the EDPVR curve. LV contraction begins with the mitral valve closed, and the pressure rises nearly vertically (isovolumetric contraction) until the aortic valve opens near point B when the LV pressure exceeds the proximal aortic pressure. At this point blood is ejected from the LV, the enclosed volume of the LV decreases, and the pressure continues to increase due to inertia of blood and the systemic arterial (peripheral) vascular resistance, yet at a progressively decreasing rate. At point C0 , the pressure has reached a maximum, contraction is essentially complete, and the volume– pressure lies near the ESPVR curve ðvC ; pC Þ. Between points C0 and C, a small volume of blood continues to be ejected due to inertial effects and the pressure drops slightly until C when the aortic valve closes (marking the end of systole). Diastole begins near point C. Between points C and D the LV begins to relax, so the pressure decreases in a nearly isovolumetric way since both the aortic and mitral valves are closed. Once the LV pressure drops below that of the left atrium (LA) the mitral valve opens near point D. Filling then starts, the volume rapidly increases, and the pressure drops further to its minimum, called LV suction. At D0 the LA contracts to complete the filling (so-called ‘atrial kick’) and the pressure rises slightly along the EDPVR until A, at which point the entire cycle repeats. Fig. 2a shows again the LV pressure ðpÞ and blood volume ðvÞ data taken from Zhong et al. [27], now plotted versus time for a single cardiac cycle. The respective end-systole and end-diastole LV pressures are about pC ¼ 120 mmHg and pA ¼ 20 mmHg. We note that normal EDP is o 12 mmHg in children and adults, so the high EDP according to the data does reflect some diastolic dysfunction. The respective minimum and maximum LV volumes are about vC ¼ 85:5 ml and vA ¼ 136:7 ml, giving a stroke volume of Δv ¼ vA  vC ¼ 51:2 ml. As was done in Nardinocchi et al. [10], the ðv; pÞ data at each time instant can be substituted as known quantities into Eq. (14) to solve for the corresponding contraction stretch ratio ðλc Þ, and its calculated time history is plotted in Fig. 2b, according to the right-hand vertical scale. Additionally, we calculated the LV wall stress [both the Cauchy stress (σ , solid line and open circles) and the referential stress (Π ¼ σ =λ, dashed line)] at each time instant, and the time history is shown in the same plot according to the left-hand vertical scale. The former stress measure is the circumferential force per current wall area, while the latter is the circumferential force per unit slack (reference) wall area. The reason why the referential stress is shown in Fig. 2b is that we consider it potentially more relevant than the Cauchy stress. The reference wall area is a measure of the number of ‘links’ (muscle fibers and other connective tissue) that support the current force, so the referential stress is a measure of the force/number of links that (on average at least) would be relevant for a failure criterion due to over stressing. The changing area due to stretching and thinning of the LV wall inherent in the Cauchy stress would seem to be unimportant. The maximum calculated stress is σ ¼ 18:8 kPa ðΠ ¼ 16:8 kPaÞ between points B and C0 . Interestingly, maximum stress does not occur at the end of systole when the pressure is maximum, but instead earlier

J.A. Shaw et al. / International Journal of Non-Linear Mechanics 68 (2015) 132–145

135

Fig. 1. (a) Baseline model of a healthy LV, showing end-diastolic and end-systolic pressure–volume relations (EDPVR and ESPVR, respectively) using parameters of Table 1. Solid black lines correspond to Eq. (14) using the Ogden constitutive model, and the original NTV results [10] are shown by gray lines for comparison. Overlaid p–v cycle data was taken from Zhong et al. [27]. (b) Magnified view of data and significant events and stages during the cardiac cycle.

Table 1 Baseline parameter values. Ogden coefficient Ogden exponent Contraction stretch ratio Slack sphere mean radius Slack wall thickness End diastolic pressure (EDP) End systolic pressure (ESP)

μ α λc Rs Hs pA pC

 1:710 mm Hg 9 0:795 to 1 2:732 cm 1:489 cm 20 mm Hg 120 mm Hg

during systole near the point of maximum ejection rate (where v_ ¼ dv=dt is minimum, see Fig. 2c). Fig. 2d shows the corresponding time history of the mechanical _ ¼  pv, _ calculated from the discrete ðv; pÞ data pumping power, W using a trapezoidal rule to calculate each increment in mechanical work ΔW divided by the time increment Δt ¼ 0:02 s between data _ ¼ 5 J=s (Watts) also points. The maximum power of about W occurs near the time of maximum stress. For reference, the total (positive) work output during systole (path ABC) is about W AC ¼ 0:74 J, and the recovered work during diastole (path CDA) is about W CA ¼  0:10 J. Thus, the unrecovered work (energy hysteresis) is about W AC þ W CA ¼ 0:64 J, which corresponds to the shaded area within the p–v loop of Fig. 1b. Fig. 3a shows a series of pressure–volume relations from the baseline model for several different contraction stretch ratios in the range 0:795 r λc r 1. The shaded box shows an idealized pressure–volume loop (ABCD) that provides a rough approximation, avoiding the need to calculate detailed hemodynamics (although this could be done if desired, using for example the

lumped model of Virag and Lulić [28]). According to the model fit, the stroke volume is Δv ¼ 51:6 ml (only slightly different than the measured value, 51.2 ml), the systolic work done is then simply W AC   pC Δv ¼ 0:83 J (expectedly higher than but comparable to the measured value of 0.74 J), and the hysteresis work is  ðpC  pA ÞΔv ¼ 0:69 J (compared to the actual 0.64 J). We will use this idealized construction later as a simplified, yet consistent, way to assess stress changes as the LV degrades and remodels. Thus, we fix the end-diastolic and end-systolic pressures at ðpA ; pC Þ ¼ ð20; 120Þ mmHg and treat the LV volume as a function ^ λc Þ by numerically of pressure and contraction stretch ratio v ¼ vðp; inverting Eq. (14), or later Eq. (24). Fig. 3b provides a plot of the ^ A ; 1Þ, the end-systolic end-diastolic volume (constant) at vA ¼ vðp ^ C ; λc;min Þ, and the stroke volume Δv ¼ vA  vC as a volume vC ¼ vðp function of λc . During systole, the contraction starts at λc ¼ 1 and decreases monotonically ðdλc =dt o 0Þ, as shown in Fig. 3b. A threshold contraction at λc o 0:906 is necessary before any positive stroke volume is achieved. Using Eqs. (1) and (12), Fig. 3c shows the constitutive trajectories (biaxial Cauchy stress-stretch) corresponding to the idealized box ABCD of Fig. 3a.

4. Multinetwork model Our extension to the NTV model is to consider modifications of the constitutive response that might result from microstructural changes and remodeling of the muscle tissue. Our prior work, in the context of non-biological materials such as elastomers, considered the effects of microstructural changes due to scission,

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J.A. Shaw et al. / International Journal of Non-Linear Mechanics 68 (2015) 132–145

Fig. 2. Baseline behavior. (a) Time history of LV pressure and volume data during one cardiac cycle (p–v data from [27]). (b) Calculated histories of LV wall stress (Cauchy stress σ and referential stress Π) and contraction stretch ratio ðλc Þ from the baseline model. (c) LV volume rate (calculated from data). (d) Mechanical pumping power (calculated from data).

Fig. 3. Baseline model. (a) Pressure–volume (p–v) relations at several contraction stretch ratios 0:795 r λc r 1. The shaded box is the idealized p–v loop. (b) Characteristic volumes, end-diastolic volume (EDV, vA ), end-systolic volume (ESV, vC ), and stroke volume ðΔv ¼ vA  vC Þ versus contraction stretch ratio during systole (A to C). (c) Biaxial Cauchy stress-stretch ððσλÞ relations at four contraction stretch ratios corresponding to the corner points ABCD in Fig. 3a.

J.A. Shaw et al. / International Journal of Non-Linear Mechanics 68 (2015) 132–145

recoiling and re-crosslinking of the underlying macromolecular networks. These changes were due to large deformation [15], elevated temperature [16], or their combined effects with other chemical processes such as oxidative scission [17]. The macroscopic manifestation was a modified (perhaps continually evolving) structural response in terms of changes in elastic stiffness and the accumulation of permanent strain upon removal of the mechanical load. While the underlying microstructural mechanisms and their kinetics that we have considered in elastomers are different from those relevant to cardiac muscle contraction, the ideas and mathematical treatment at the continuum level can be adapted in the present context for remodeling of myocardial tissue due to disease. Conceptually, the multi-network theory developed in Wineman and Rajagopal [15], Rajagopal and Wineman [16], Shaw et al. [17], Demirkoparan et al. [24] considered microstructural changes while the material was in a deformed configuration due to scission of underlying macro molecular network junctions within some volume fraction of the initial macromolecular network. These macromolecules could then recoil and re-crosslink to form a new network, now having a stressfree reference configuration corresponding to the current configuration at which it was created. Upon further loading, the new network contributed to the total stress based on its elastic response from the reference configuration at which it was formed (relative stretching) and its volume fraction in the RVE. The process occurred continuously, i.e. an increment of volume fraction could undergo the scission/recrosslinking process in each increment of deformation or time. At each instant the material could be thought of as a mixture of macromolecular networks each with its own reference configuration, and perhaps, distinct properties. Constitutive equations, developed to describe this process of combined chemical change and mechanical deformation, were used in analyses to show that the microstructural change resulted in permanent set upon removal of external loads, modified material properties, and altered structural response. In this work, the modeling of scission/re-crosslinking occurred continuously and resulted in constitutive equations with history integrals. As the purpose of the simplified theory presented here is the same as in Nardinocchi et al. [10,11], e.g. to explore the physical consequences of microstructural change and active deformation without undue complexity, the continuous process will be approximated by a discrete one in terms of two networks, the original network and a remodeled network. The additional feature we introduce here is that disease processes may change the passive elastic properties, contractility, and reference configuration of muscle tissue networks. Network 1 refers to material that is derived from the original tissue, having the same reference configuration, passive elastic properties, and contractility. Network 2 refers to new growth or remodeled material, having a potentially different reference configuration, passive elastic properties, and contractility. We take the Voigt homogenization viewpoint, where the two networks support the load in parallel and are locally subject to simultaneous inplane biaxial deformations (see Fig. 4). Considering a diametrical section across the resulting double-layer sphere, the forces carried in Network 1 and Network 2 are f f

ð1Þ

¼ 2π rhσ

ð2Þ

¼ 2π rh σ n

ð1Þ

;

ð2Þ

ð15aÞ ;

ð15bÞ

where σ ð1Þ and σ ð2Þ are the equi-biaxial Cauchy stresses in the n respective networks, and h and h are the corresponding (current) 2 network thicknesses. Force balance with the internal pressure

requires f

ð1Þ

þf

ð2Þ

137

¼ π r 2 p, giving the pressure–stress relation

2 n p ¼ ½hσ ð1Þ þ h σ ð2Þ : r

ð16Þ

The initial volume of solid tissue in its reference (slack) configuration is V 0 ¼ 4π R2s H 0 . At some later time after microstructural changes have occurred, we suppose the solid volume occupied by Network 1 in its slack configuration may be different V ð1Þ ¼ 4π R2s H s , and the volume occupied by Network 2 in its slack configuration is V ð2Þ ¼ 4π ðRns Þ2 H ns . Assuming that contraction and mechanical deformation in both networks are isochoric, these material volumes are the same as those in the current configuran tion V ð1Þ ¼ 4π r 2 h and V ð2Þ ¼ 4π r 2 h . The volume fractions of the respective material networks (relative to the initial reference volume) are then b¼

V ð1Þ H s 2 h ¼ ¼λ ; H0 V0 H0

ð17aÞ

n V ð2Þ n H 2h ¼ ðλs Þ2 s ¼ λ : V0 H0 H0 n

n

b ¼

ð17bÞ

n

Solving for h and h and substituting into Eq. (16) give p¼

2 H0

λ3 Rs

½bσ ð1Þ þ b σ ð2Þ : n

ð18Þ

Considering microstructural changes in Network 1, degradation (scission) of the original tissue might arise from necrosis of myocytes and/or severing of connective junctions between myocytes or with the extracellular matrix, implying that 0 o b o 1. Alternatively, considering hypertrophy or growth of wall thickness, the value could increase to values greater than unity (b 4 1). Effectively, Eq. (17a) suggests the interpretation that b is a dimensionless parameter that captures the relative change in effective load carrying thickness of Network 1 from its original value H 0 to a new slack thickness H s (see again Fig. 4). The process of scission implies that bðtÞ is time dependent with the initial condition bð0Þ ¼ 1 and that it evolves with time. This makes H s ðtÞ time dependent as well, but we will suppress the explicit time arguments in the interest of notation simplicity. In fact, the numerical results that will be shown in Section 5 consider selected values of b, avoiding for now the question of how long it took to reach that value. Let t n denote a time when a second material (Network 2) forms as a layer of tissue acting in parallel with Network 1 to carry load. One can envision post-injury scarring or adaptive growth (remodeled tissue) as possible scenarios. Network 2 is assumed to form in a stress-free state when the length of Network 1 tissue has been deformed to length ℓðt n Þ ¼ Lns , which we define to be the slack length of Network 2. The stretch ratio at that instant is λðt n Þ ¼ Lns =Ls ¼ λns . Since at some later time t 4 t n the current lengths of the newly formed tissue and the original tissue coincide, the stretch ratio of the new tissue (Network 2) relative to its slack length is

λn ¼

ℓ λ ¼ : Lns λns

ð19Þ

In response to electrochemical stimulus, the new tissue may also contract, but perhaps by a different amount. The contracted unstressed length in Network 2 is Lnc , so the contraction stretch ratio is Ln Ls

λnc ¼ cn ;

ð20Þ

and the mechanical stretch ratio is Unless explicit superscripts ðÞð1Þ and ðÞð2Þ are used, Network 2 variables and parameters will have an asterisk superscript ðÞn to distinguish them from those associated with Network 1. 2

ℓ Lc

λnm ¼ n ;

ð21Þ

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J.A. Shaw et al. / International Journal of Non-Linear Mechanics 68 (2015) 132–145

Fig. 4. Intermediate configurations and the current representative volume element (RVE) of multi-network tissue under equi-biaxial stretching.

giving the decomposition

λn ¼ λnc λnm :

Π ð2Þ ¼ ð22Þ

Thus, the following expressions for Network 2 are analogous to Eqs. (6a) and (6b)
n 1=3 Rn V λnc ¼ cn ¼ cn ; ð23aÞ Rs Vs

λnm ¼


 r 1 v 1=3 ; n ¼ n Rc λc V ns

ð23bÞ

where V ns is the enclosed slack volume of the new tissue when it forms at time t n . Using the response function Eq. (12) with σ ð1Þ ¼ σ^ ðλm ; μ; αÞ and σ ð2Þ ¼ σ^ ðλnm ; μn ; αn Þ and substituting Eqs. (6b) and (23b) into Eq. (18), results in the final pressure–volume– contraction relation " ! !#

 H0 V s 1 v 1=3 1 v 1=3 n n n ^ ^ bσ p¼2 ; μ; α þb σ n ;μ ;α : Rs v λc V s λc V ns

ð2Þ

f r h ð2Þ σ ð2Þ λs ð2Þ ¼ σ ¼ n ¼ σ : λ 2π Rns H ns Rns H ns λ n

n

ð25bÞ

In terms of these referential stresses, the pressure–stress relation becomes " # n 2 H0 b ð2Þ ð1Þ p¼ 2 b Π þ nΠ : ð26Þ λs λ Rs

5. Altered structural responses In this section, we explore the consequences of Eq. (24) on gross mechanical features of the cardiac cycle. Cardiac output is normally defined as the stroke volume multiplied by heart rate, so the focus here is the stroke volume ðΔvÞ for a single contraction. The following is a parameter study to investigate the influence of various parameters on the p–v structural response and to illustrate how the modified model might be used to mimic certain pathologies of the LV myocardium.

ð24Þ This relation can now be used to discuss how microstructural change of muscle tissue may affect the pressure–volume–contracn tion relationship for the LV. When b ¼ 1 and b ¼ 0 it reduces to Eq. (14). We note that Nardinocchi et al. [10,11] did consider the effects of a reduction in effective stiffness by treating Y as an evolving parameter, thereby capturing the effects of scission without remodeling. Here, we take b as the evolving parameter (holding constitutive properties constant), which is entirely equivalent in the absence of remodeling or new growth. We have n interpreted b and b as volume ‘fractions’, but this need not imply n that networks are conserved (as in b þ b ¼ 1) in general, since degradation of the original tissue in terms of volume (or mass) reduction may or may not be offset by a congruent change in volume of new tissue. As mentioned previously, the referential stress (now in two networks) is better suited than the Cauchy stress to assess the likelihood of mechanical damage. Accordingly, using Eqs. (6c), (15) and (17) the referential stresses in Network 1 and Network 2 are, respectively,

Π ð1Þ ¼

ð1Þ

f r h ð1Þ σ ð1Þ ¼ σ ¼ ; 2π Rs H s Rs H s λ

ð25aÞ

5.1. Modified Network 1 response Here, we consider the implications of microstructural changes in Network 1 without (for now) any existence of a second network, n i.e., no remodeling ðb ¼ 0Þ. The slack volume V s and passive elastic properties μ and α of Network 1 are held constant at their previous values. This leaves two possible changes in Network 1 behavior, either changes in contractility (modeled by the minimum λc ) or changes in network stiffness by degradation (modeled by b). The effects of contractility were already seen in Fig. 3, where any change in the minimum value of λc would have a profound effect on the stroke volume ðΔvÞ. Decreased contractility (minimum λc 4 0:795), for example, would severely degrade cardiac output, unless the heart rate increased to compensate for the smaller Δv per cycle. 5.1.1. Network 1 scission Fig. 5 shows the consequences of scission in terms of a reduction in LV wall thickness ðbÞ. The ESPVR and EDPVR curves become more compliant as b decreases, and end-points A and C shift to larger volumes if the end-diastolic and end-systolic pressures remain fixed at their previous values (see shaded solid line box in Fig. 5a). Constant ESP is a reasonable assumption as

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Fig. 5. Network 1 scission. (a) End-systolic and end-diastolic pressure–volume relations (ESPVR and EDPVR, respectively) for selected values of Network 1 volume fractions 0 o b r 1. The shaded boxes are idealized p–v loops for b ¼ 0:4 at fixed end-systolic pressure (ESP, pC ) and either (i) end-diastolic pressure (EDP, pA , solid line) or (ii) enddiastolic volume (EDV, vA , dotted line). (b) Characteristic LV volumes as a function of b, showing vA , vC , and stroke volume Δv ¼ vA  vC .

there is a minimum physiologic systolic pressure needed for endorgan perfusion. Continuous curves for the changes in vA (EDV, solid line), vC (ESV, black line) and Δv ¼ vA  vC (bold solid line) as a function of b are shown in Fig. 5b. As b decreases from unity, vA increases at a greater ‘rate’ than vC resulting in an increase in stroke volume Δv. At moderate levels of scission, say 0:6 ob o 1, this increase in Δv is relatively shallow and nearly linear, but at extreme scission levels ðb o 0:3Þ the increase accelerates dramatically. While the above is one possible scenario, another is shown by the shaded dotted line box in Fig. 5a where the end-diastolic volume is held fixed (along with the end-systolic pressure). In this case the end-diastolic pressure pA decreases in proportion (exactly linear) with LV wall thickness b. The dotted lines in Fig. 5b show the fixed vA and the consequent reduction in Δv as b decreases. To maintain cardiac output this decrease in stroke volume would be compensated by an elevated heart rate (up to a ceiling rate). At moderate levels of scission, the reduction in Δv is gentle, but at extreme scission levels the reduction monotonically accelerates and even reaches zero as vC overtakes vA at about b ¼ 0:1. Returning to the first scenario, which presumed that end-pressures remain the same, Fig. 5 shows that as the LV becomes more compliant diastolic function improves and stroke volume Δv increases. One should not conclude, however, that a loss in LV stiffness is altogether a beneficial development. To maintain constant end-pressures, the stress in the ‘degraded’ LV wall must increase, as shown in Fig. 6. The calculated referential wall stress ðΠ Þ is plotted (solid lines) against contraction stretch ratio ðλc Þ during the idealized cardiac cycle for selected values of b. The black points show the calculated data corresponding to the actual measurements (Fig. 2a). The black line ðb ¼ 1Þ from the idealized p–v box fits the data reasonably well, except at the upper corner during systole (at B, near λc ¼ 0:906) where it overshoots the discrete data. Nevertheless, the trend with decreasing b is clear, showing a significant rise in maximum stress (point B) by almost three-fold from 20 kPa at b ¼ 1 to 57 kPa at b ¼ 0:4. A continuous plot of characteristic reference stresses at A, B, C, D against b is provided in Fig. 6b. The two curves at B and C bracket the likely maximum stress, and they show how the stress grows to unsustainable levels as degradation occurs (decreasing b). These findings of increased wall stress during network degradation are of clinical relevance. Situations leading to elevated myocardial wall stress increase resistance through the coronary microcirculation, leading to direct consequences on myocardial contractility and relaxation [29]. For these reasons, this scenario, i.e., loss in LV stiffness from normal values, may be unrealistic since we cannot envision any clinical situations where significant network degradation occurs yet contractility is preserved. On the other hand, this situation

might mimic favorable remodeling from abnormal LV stiffness after relief of a ventricular outflow obstruction (e.g., aortic or subaortic stenosis), where regression of LV mass results in wall thinning approaching normal values. This remodeling is accompanied by an increase in ventricular compliance, end-systolic volume and stroke volume. 5.1.2. Network 1 hypertrophy Fig. 7 considers the alternative case where b increases, rather than decreases. In this case, the EDPVR and ESPVR curves become stiffer, perhaps mimicking the gross effects of LV wall thickening, or hypertrophy (ignoring, of course, valve function, LV outlet obstruction, myocardial disarray, and/or reduction in interior volume that can often accompany hypertrophic cardiomyopathies). We show an extreme case in Fig. 7a where b ¼ 2 is compared to the baseline behavior ðb ¼ 1Þ. With no change in contractility or end-diastolic pressure (Fig. 7a, solid box), the endpoint volumes shift to lesser values. The stroke volume is now somewhat less than that at baseline (45.6 ml versus 51.6 ml). The stroke volume Δv (solid line) is plotted over the full range 0 o b r2 in Fig. 7b, which shows a gentle decline as b increases within 1 r b r 2. Alternative, perhaps more relevant, scenarios assuming either fixed end-diastolic volume (dotted box) or fixed stroke volume (dashed box) are also shown in Fig. 7a. The corresponding stroke volume for fixed end-diastolic volume is the dotted line in Fig. 7b, showing a gentle increase for b 4 1. In all scenarios, the changes in stroke volume are relatively small for b 4 1, suggesting that this level of stiffening (diastolic dysfunction) could probably be tolerated, in the absence of other complicating factors. Given one of these volumetric assumptions, the end-diastolic pressure must change as demonstrated in Fig. 7c (dashed or dotted lines). The trends agree well with the hemodynamic sequelae of hypertrophic cardiomyopathy, where to achieve adequate stroke volume in the setting of increased LV wall thickness, an increased filling pressure (end-diastolic pressure) is required [30]. 5.2. Remodeling of Network 1 to Network 2 Now, we consider LV remodeling where the original tissue (Network 1) is gradually converted to modified tissue (Network 2) of different properties. Although not an essential limitation, the following examples will assume that the rate of scission (degradation) of Network 1 is equal to the rate of formation (healing) of n Network 2, such that b ¼ 1  b. We treat the fraction of second n n network, b , as the disease parameter, where b ¼ 0 represents no

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Fig. 6. Network 1 scission. (a) Referential stress Π versus contraction stretch ratio λc corresponding to idealized p–v boxes at fixed ESP and EDP (see solid line box in Fig. 5a) at selected volume fractions b r 1. Discrete points are calculated from the data of Zhong et al. [27]. (b) Referential stresses at characteristic end-points, showing the dramatic increases in LV wall stress levels as scission proceeds (decreasing b).

Fig. 7. Network 1 hypertrophy. (a) End-systolic and end-diastolic pressure–volume relations (ESPVR and EDPVR, respectively) for selected volume fractions 1 r b r 2. The boxes show idealized p–v loops at b ¼ 2 for fixed end-systolic pressure (pC ) and either (i) end-diastolic pressure (pA , solid line), (ii) stroke volume (Δv, dashed line) or (iii) end-diastolic volume (vA , dotted line). (b) Stroke volumes Δv ¼ vA  vC as a function of b over the range 0 o b r 2. (c) Corresponding end-diastolic pressures.

n

disease and b ¼ 1 represents complete remodeling of the LV. Similar to Network 1, the passive properties of Network 2 are defined by a slack volume (V ns ) and elastic parameters (μn , αn ), and the active property of Network 2 is described by its minimum n contraction stretch ratio (λc ). In all cases below, the properties of Network 1 ðV s ; μ; α; λc Þ are unchanged from their baseline values (Table 1).

5.2.1. Altered slack volume and contractility We start by assuming that Network 2 forms with the same stiffness (μn ¼ μ and αn ¼ α) as Network 1, but with a larger slack volume ðV ns 4 V s Þ. Network 2 probably forms with a slack volume somewhat between EDV and ESV in the normal LV, so we choose the average value ðV ns ¼ 111:8 mlÞ which corresponds to a slack radius of Network 2 about 9.4% larger than that of Network 1. This V ns value is also close to the LV volume at maximum wall stress in the original tissue (see again Fig. 2). Fig. 8a provides a series of ESPVR (thin lines) and EDPVR (bold lines) curves for several values n of b for the case where contractility of Network 2 is the same as n n Network 1 (minimum λc ¼ λc ¼ 0:795). The baseline case ðb ¼ 0Þ is n shown by black lines. As b increases, all curves shift to the right (and become more compliant) due to the larger slack volume of Network 2. The shaded box in Fig. 8a shows the case of 80%

n

remodeling (b ¼ 0:8) from Network 1 to Network 2. With no change in contractility or effective stiffness, one can see that the stroke volume increases relative to the baseline case, since remodeling just creates a larger LV. Fig. 8b provides a similar series of ESPVR and EDPVR curves, but where the contractility, minimum (end-systolic) contraction stretch ratio of Network 2 n (minimum λc ¼ 0:90) is poorer than in Network 1. Compared to Fig. 8a, the ESPVR curves in Fig. 8b are shifted to the right, while the EDPVR curves are unchanged. This causes a decrease in stroke volume for the same extent of remodeling. As shown in the above two examples, a poorer contractility in Network 2 tissue has serious consequences on the stroke volume. The end-diastolic volume ðvA Þ and end-systolic volume ðvC Þ are n plotted in Fig. 8c as a function of fraction of Network 2 ðb ¼ 1  bÞ. n The dotted curve is the EDV (vA , at λc ¼ λc ¼ 1) at fixed EDP, which n has a gentle increase as remodeling progresses (increasing b ). The solid lines show a series of ESV ðvC Þ curves, depending on the contractility of Network 2 tissue. The lowest curve is the case where there is no change in contractility from the original tissue n (minimum λc ¼ 0:795), and it has a similar upward trend to the diastolic curve. As the contractility of Network 2 worsens, minin mum λc ¼ f0:8; 0:85; 0:9; 0:95; 1g, this upward trend is more pron nounced, although little difference is seen between them until b exceeds about 0.4, or so. The stroke volume ðΔvÞ is plotted in

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ml

ml

141

ml

ml

Fig. 8. Remodeling of Network 1 to Network 2. (a), (b) Progression of ESPVR (thin lines) and EDPVR (bold lines) curves during LV remodeling, where original tissue (Network n 1, volume fraction b, slack volume V s ¼ 85:4 ml) is converted to new tissue (Network 2, volume fraction b ¼ 1  b, larger slack volume V ns ¼ 111:8 ml, same elastic stiffness). Idealized p–v loops are shown at 80% remodeling for fixed ESP & EDP for (a) no change in contractility, (b) poorer contractility in Network 2. (c) End-diastolic volume n (vA , dashed line) and end-systolic volume (vC , solid lines) versus Network 2 volume fraction (b ), for several values of minimum contraction stretch ratio (λnc in Network 2). (d) Stroke volume Δv ¼ vA  vC at fixed end-systolic and end-diastolic pressures. Open circles indicate local maxima.

n

Fig. 8d against b for the same set of λc values. Interestingly, the n case with no contractility change (λc ¼ 0:795) exhibits an increased stroke volume with remodeling. Again, this just corresponds to a larger, but functionally normal LV, that results in an increased cardiac output. This situation is seen in early adaptive remodeling when a normal ventricle is subjected to additional volume loading and has to consequently increase ventricular cardiac output; e.g., with a ventricular septal defect or valvular n regurgitation. The case λc ¼ 0:83 (dashed curve) results in the n n same stroke volume at b ¼ 1 as at b ¼ 0. This shows that a modest amount of contractility change (systolic degradation) can be offset by increased LV filling, which is a recognized mode of compensation in the early stages of heart failure. Larger degradan tion of Network 2 contractility (minimum λc Z 0:83), however, n results in a dramatic reduction in stroke volumes as b approaches unity. The stroke volume is relatively constant for remodeling fractions less than 40%, or so, but a rapid drop off begins at about 60% to 80% remodeling, depending on the severity of Network 2 contractility degradation. The maximum referential stresses (at point B in the idealized loop) in the two networks, during the type of remodeling just shown, are provided in Fig. 9. It shows how the maximum stress in ð1Þ the original tissue network Π B increases monotonically and n accelerates as remodeling progresses (increasing b ). In the best case if the contractility in Network 2 is as good as in Network 1 n (minimum λc ¼ λc ¼ 0:795), the upturn is only delayed somewhat. Thus, this elevated stress represents a significant mechanical driving force for the conversion of tissue to Network 2. The stress ð2Þ in the remodeled network Π B starts small and gradually increases during remodeling. As expected, more stress is carried in Network 2 the better its contractility. Interestingly, in the best case λnc ¼ 0:795 at 100% conversion, the LV wall stress is somewhat ð2Þ higher than it was originally (final Π B ¼ 27:4 kPa compared to n

Fig. 9. Referential stresses in Network 1 and Network 2 at point B in the idealized p–v loop during remodeling (corresponding to Fig. 8). ð1Þ

initial Π B ¼ 20:2 kPa), since internal pressure has a mechanical advantage in a larger LV of the same (reference) wall thickness. For the two worst contractility cases, the Network 2 curves are terminated just before 100% remodeling where the stroke volume in Fig. 8d reaches zero. Of course, these are optimistic end points since LV stroke volume would have decreased to a degree (that could not be compensated by an elevated heart rate) much earlier. In this model, we have not identified what specific disease conditions may have caused remodeling. At this point, it is also important to emphasize that we have assumed no change in mechanical stiffness between the original tissue and the remodeled tissue, and this will be considered in Section 5.2.2. Nevertheless, Fig. 8d clearly shows that a reduction in stroke volume occurs only if there exists a sufficient degradation in contractility in the remodeled tissue (as expected), and it seems to suggest that a significant amount of remodeling (i.e., greater than 50%) can be

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Fig. 10. Stroke volumes during remodeling of Network 1 to Network 2 at fixed ESP and EDP for bounding values of Network 2 slack volume: (a) smallest V ns ¼ 86:1 ml (normal LV ESV), (b) largest V ns ¼ 138 ml (normal LV EDV).

tolerated before the stroke volume degrades appreciably. LV function is remarkably robust in the face of small to moderate amounts of remodeling and contractility degradation. In other words, from a gross mechanical performance perspective, a noticeable reduction in stroke volume/cycle (in the absence of valve issues) requires both (1) a large amount (extent) of remodeling, and (2) significantly degraded contractility in the new tissue. These represent targets for cardiomyopathy treatment: (1) minimizing, halting or even reversing adverse remodeling (e.g., correction of volume loading lesions), and (2) preservation or improvement in contractility (e.g., infusion of inotropic agents). Clearly, the values cited above and the particular curves shown could be sensitive to our choice of modeling parameters. In particular, the choice of slack volume for Network 2 ðV ns Þ is somewhat speculative, selected only as an intermediate value between ESV and EDV in the normal LV. To bracket the range of possibilities, Fig. 10a and b n shows stroke volume curves versus remodeling (b ) assuming two extreme values, V ns ¼ 86:1 ml (normal heart ESV) and V ns ¼ 138 ml (normal heart EDV), respectively. Fig. 10a shows little change in Δv if λnc ¼ λc ¼ 0:795 across the entire range 0 r bn r 1, since V ns ¼ 86:1 ml is quite close to V s ¼ 85:4 ml. Any significant degradation in contractility in Network 2, however, causes an early and significant drop in Δv with bn , and this slack volume ðV ns ¼ 86:1 mlÞ seems to represent the most severe case. The opposing case of V ns ¼ 138 ml (Fig. 10b) is qualitatively similar to that of V ns ¼ 111:8 ml shown previously in Fig. 8d, relatively benign except in the case of significant loss of contractility beyond 90% remodeling. Again, however, one must keep in mind that the wall stress may become unsustainable (not shown here, but see again Fig. 9), thereby leading to significant contractility degradation.

5.2.2. Scarring The final aspect of our parameter study is to consider the implications of stiffness changes that might accompany remodeling. The end result of various disease processes such as inflammation, infection and ischemia lead to tissue damage, loss and replacement with scar, thereby resulting in tissue with radically different elastic properties. For simplicity, we take the classical view that scar tissue (Network 2) is effectively inert with no contractility. The normal expectation is that scar tissue would be stiffer than the native tissue, so we consider cases where μn Z μ. The average ESV-EDV value ðV ns ¼ 111:8 mlÞ is again selected for the slack volume of the scar tissue. Fig. 11a shows sets of modified EDPVR and ESPVR curves (colored n lines) at 50% remodeling ðb ¼ b ¼ 0:5Þ for several different relative values of Network 2 stiffness in the range, 1 r μn =μ r 10 (keeping αn ¼ α fixed), along with the nominal curves (gray lines) for

reference. These stiffness values bracket a range of ‘scar’ stiffness from equally compliant to stiffer than that of the native tissue. The contractile stretch ratio for Network 2 is set to unity in all curves n ðλc ¼ 1Þ, and again for simplicity, conservation of networks is n assumed according to b þb ¼ 1. As one expects, the resulting EDPVR curves (bold lines) diverge from one another, with smaller enddiastolic volumes for larger scar (Network 2) stiffnesses. The ESPVR curves (thin lines), however, essentially lie on top of one another, except at very low pressures that are not physiologically relevant. The end-systolic volumes at 120 mmHg are all quite close, which n holds true generally across the range 0 r b o 0:9 (not shown). n The corresponding stroke volumes over the entire range of b are shown in Fig. 11b. The topmost curve (black), where the elastic stiffness of the scar is the same as the original tissue ðμn ¼ μÞ, shows n a relatively constant stroke volume over the range 0 r b o 0:8, but n drops steeply at larger values of b . This case might mimic the mechanical behavior of necrotic tissue soon after injury before fibrosis and scarring have formed, or in the situation of the formation of a LV aneurysm post infarction (although this occurs in a localized area). It is again interesting that significant remodeling can occur before the stroke volume is adversely affected, yet a ‘cliff’ does exists beyond which stroke volume degrades very rapidly. However, as Network 2 becomes stiffer than Network 1, more consistent with the notion of a fully developed scar, the plateau in the stroke volume disappears and stroke volumes immediately trend downward even n for low values of b . The stroke volume monotonically decreases at an accelerating ‘rate’ as remodeling occurs. In the most extreme case considered (μn ¼ 10μ) a 30% decrease in stroke volume is reached before about 40% remodeling, so relatively little remodeling creates problems rather quickly. The overall results, therefore, suggest that a relatively compliant scar in the myocardium is much more benign (up to a point) than that of a stiff scar. As shown in Fig. 11c, however, one must recognize that the maximum stress in Network 1 at point B increases substantially during this process. Thus, excessive mechanical stress will likely accelerate the remodeling process considerably. The stresses in Network 2 material are much smaller than those in Network 1, and only the stiffest scar is able to contribute much to n delay (in b ) the upturn in stress within the original tissue. The effects of scarring on the end-diastolic pressure are shown in Fig. 12 for the alternative scenario where the stroke volume Δv ¼ 51:6 ml is maintained (and constant end-systolic pressure). The trends in the end-diastolic pressure ðpA Þ in Fig. 12 are nonlinear, roughly the inverse of those seen in the stroke volume curves of Fig. 11b. A relatively compliant scar ðμn ¼ μÞ exhibits a slight decrease, a local minimum, then a steep rise in end-diastolic n pressure as remodeling nears completion ðb 4 0:9Þ. A stiffer scar ðμn 4 μÞ, by contrast, has no local minimum, and instead increases and accelerates monotonically. The stiffest scar considered ðμn ¼ 10μÞ

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n

Fig. 11. The influence of scarring. Remodeling from Network 1 to Network 2 (b ¼ 1  b, V ns ¼ 111:8 ml, λnc ¼ 1), considering scars of several relative stiffnesses. (a) ESPVR and n n EDPVR curves at 50% remodeling (colored lines). Gray dashed lines show the initial curves at b ¼ 0. (b) Stroke volumes during remodeling (increasing b ). (c) Network 1 and ð2Þ Network 2 referential stresses (Π ð1Þ and Π , respectively) at point B in the idealized p–v loop during remodeling. (For interpretation of the references to color in this figure B B caption, the reader is referred to the web version of this paper.)

Fig. 12. The influence of scarring ðλnc ¼ 1Þ on the end-diastolic pressure ðpA Þ if the stroke volume is held constant at its baseline value ðΔv ¼ 51:6 mlÞ. The end-systolic pressure is held fixed at pC ¼ 120 mmHg. The curves show the dependence of pA on n volume fraction of scarring ðb Þ and the stiffness of scar tissue ðμn Þ relative to original tissue stiffness ðμÞ. The slack volume of scar tissue (Network 2) is the same intermediate value as assumed previously ðV ns ¼ 111:8 mlÞ.

n

exhibits an increase in pA immediately even at low values of b , and no plateau-like behavior is observed. Of course, the full range of pressures shown is unrealistic, since excessively large end-diastolic pressures ð 430 mmHgÞ are associated with pulmonary hypertension, pulmonary hemorrhage and right ventricular failure.

6. Summary and conclusions This work introduces a modification of the NTV model for LV contractions to account for microstructural changes in the muscle

tissue. As in the original NTV model, the LV is idealized as a thinwalled, isotropic sphere of muscle tissue, consisting of homogenized mixture of muscle fibers, extracellular matrix, and vasculature. The mechanism of microstructural change is viewed as scission, growth, and/or conversion of the underlying tissue network to one of different properties. For simplicity, we considered two networks, the original tissue (Network 1) and a new tissue (Network 2), to simulate various disease states. At some time t n , Network 1 may have undergone scission (wall thinning) or hypertrophy (wall thickening) in response to injury or growth, respectively. Concurrently, Network 2 may form (healing, or scarring), which has a reference (slack) configuration that coincides with the configuration of Network 1 at t n . At some later time t 4 t n , both Network 1 and Network 2 occupy the same current configuration, and the resultant muscle tissue is the superposition of the two networks. Each network has its own material parameters, e.g. unstressed configuration, volume fraction, stiffness, and contractility. Different choices for the parameters in the two networks correspond to various pathological changes in the LV: hypertrophy (wall thickening), dilated cardiomyopathy (wall lengthening), stiffening due to scarring, and modified contractility. Numerous results are presented that show the influence of these parameters on the EDPVR and ESPVR curves, the pressure–volume relation and stroke volume during a contraction, and stresses in the LV wall. The set of results in Section 3 reviews the baseline behavior of our LV model, which is a slight modification of the original NTV model. Interestingly, the maximum LV wall stress occurs, not at the point of maximum contraction (end of systole), but earlier when the volumetric ejection rate is maximum (Fig. 2). For simplicity of analysis, an idealized rectangular pressure–volume

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loop is introduced (Fig. 3) that gives a reasonable approximation to the hysteresis associated with an actual pressure–volume loop. This is then used to evaluate the influence of parameters on LV function, especially the stroke volume (related to cardiac output). The set of results in Section 5.1 examines the consequences of scission (network loss) and growth (network gain) of the original network without the formation of a second network. It is shown that if the end-systolic pressure (ESP) and end-diastolic pressure (EDP) and contractility (minimum contraction stretch ratio) remain constant, wall thinning (scission) causes LV enlargement yet an increased stroke volume (Fig. 5). The LV is adversely affected by this degradation, however, because the maximum wall stress increases rapidly with the amount of degradation (Fig. 6). This would accelerate scission kinetics and affect coronary microcirculation and degrade contractility. Conversely, wall thickening causes a relatively gentle reduction in stroke volume at constant end-pressures (Fig. 7). Alternative scenarios were also considered, where end-systolic pressure was held fixed and either the enddiastolic volume (EDV) or stroke volume ðΔvÞ were held fixed, and the trends were reversed – wall thinning causes a stroke volume decrease (and EDP decrease), while wall thickening causes a stroke volume increase (and EDP increase). Either scenario could be relevant, depending on the type and stage of the disease. The results presented in Section 5.2 explore the influence of volume fraction, slack configuration, contractility, and stiffness of the second network. When the volume fraction of Network 1 is converted to that of Network 2 and there is no change in stiffness or contractility, the LV enlarges due to the larger slack volume of Network 2 and the stroke volume increases at constant EDP (Fig. 8). Fig. 8d shows an interesting result that occurs with changes in both degradation and contractility at constant EDP, where the stroke volume is relatively unaffected up to about 40% network conversion regardless of the contractility of Network 2. Significant reductions in stroke volume are observed only beyond 80% conversion and for significantly worse contractility in Network 2, at which point the stroke volume undergoes a precipitous decrease. A similar result is shown in Fig. 11b in the extreme case when Network 2 has no contractility (scar) if it is as compliant as the original tissue. For relatively stiff scars, however, the stroke volume decreases immediately and then accelerates downward with increasing conversion. If, on the other hand, the stroke volume is held constant the end-diastolic pressure must increase (Fig. 12), which is often observed clinically. A very stiff scar appears to be the most severe case with regard to LV function, although all cases are accompanied by significant increases in maximum wall stress (Fig. 11c). These results demonstrate the information that can be obtained from the current model for myocardial changes. However, two points should be made. First, our multi-network model, like the original NTV model, is a toy model that represents only the simplest features of a more complex model that would be required for proper physiological fidelity of the myocardium. The results presented here can provide useful insight into interpreting the connection between particular microstructural changes and the ventricular pressure– volume relation. However, because of the mechanical and material modeling assumptions involved, this insight should be regarded as qualitative rather than quantitative. Second, the results presented here should be interpreted as snapshots in time in which each value of a parameter represents its current value, without consideration of the kinetic process that led to the microstructural change. In a sense, we have presented a ‘state-space’ model to assess how much and what type of remodeling could be tolerated before heart failure. We have not answered the important question of how rapidly these changes would take place, but the current model should provide guidance for the development of suitable kinetic laws for remodeling of the LV, which we leave for future work.

Acknowledgments We (J.S. and A.W.) significantly benefitted from discussions with Alan Gent in our prior research on degradation of elastomeric materials in extreme environments. We are indebted to his encouragement and insight and consider it a privilege to contribute this paper in the special issue of IJNM in Alan Gent's honor. Our motivation for this particular foray into biomechanics was largely driven by personal experience with a remarkable girl, Madeline Clair Shaw (daughter of J.S. and former patient of M.S.), who suffered unexpected and sudden heart failure at age 2 and overcame extreme obstacles to survive until age 4. Thus, it is with love and pride that this paper is dedicated to the memory of Maddie.

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