Residual Stress Estimates from Multi-cut1

Opening Angles of the Left Ventricle2

Xin Zhuan and Xiaoyu Luo∗

School of Mathematics and Statistics, University of Glasgow,

Glasgow G12 8SQ, UK.

3

Abstract—Residual stress tensor has an essential influence on the mechanical behaviour4

of soft tissues and can be particularly useful in evaluating growth and remodelling of the5

heart and arteries. In many previous models, it has been assumed that a single radial cut6

can release the residual stress in a ring of the artery or left ventricle. However, experiments7

by Omens and his colleagues [14], on mouse hearts, have shown that this is not the case.8

In this work, we have developed models of multiple cuts to estimate the residual stress in9

the left ventricle and compared with the one-cut model. Both two- and four-cut models are10

considered. Given that the collagen fibres are normally coiled in the absence of loading, we11

use the isotropic part of the Holzapfel-Ogden strain energy function to model the unloaded12

myocardium. Importantly, we demonstrate that both radial and circumferential cuts are13

required to release the residual stress in the left ventricle; using multiple radial cuts alone is14

not sufficient. The estimated residual hoop stress from the new model is around 8 to 9 times15

greater than that of a single-cut model. We also show that the results are not significantly16

different using homogeneous or heterogeneous material models. Although in principle infinite17

cuts are required to release the residual stress, we find four cuts seem to be sufficient as the18

∗Correspondence e-mail: xiaoyu.luo@glasgow.ac.uk

1

model agrees well with experimental measurements of the myocardial thickness. Indeed,19

even the two-cut model already gives a reasonable estimate of the maximum residual hoop20

stress. Finally, we explain that the multiple cuts approach also applies to arteries.21

Keywords—Residual stress, opening angles, multiple cuts, soft tissue, heterogeneity, left22

ventricle.23

2

1 INTRODUCTION24

Living tissues in the heart continuously interact with their bio-environment, reshape and25

rearrange their constituents under chemical, mechanical or genetic stimuli during their life26

cycles. In the mature period, the tissues of a healthy heart remain in a homeostatic state.27

However, heart diseases disrupt this balance, and cause the tissues to grow and remodel.28

Physiologically, exercise may also induce healthy and reversible growth and remodelling.29

An important ingredient in evaluating the mechanics involved in the cardiovascular system30

is knowledge of the solid mechanical properties of the soft tissues involved, including the31

components of the heart, such as the left ventricle, henceforth abbreviated as LV. A particular32

aspect is that the tissues of the heart are residually stressed, so that when the external33

loading is removed, residual stresses remain in the material. However, residual stresses, which34

are generally assumed to result from growth and remodeling, are imprecisely characterized35

(experimentally) at present, and how best to include the important effects of residual stresses36

in cardiovascular applications therefore presents a modelling challenge.37

Over the last century [5], various hypotheses on the growth and remodelling response to38

mechanical loading have been put forward, with particular success in arteries. In conventional39

elasticity theory the existence of a stress-free reference configuration that coincides with40

the unloaded configuration is normally assumed [13]. In the present context, however, the41

unloaded configuration is not stress free, but is residually stressed. The residual stress can42

be estimated using the so-called opening angle method [2, 9], in which an opening angle43

indicative of the extent of the residual stress can be measured after a single radial cut of an44

unloaded arterial ring. Using the opened configuration as the reference configuration, the45

residual stress in a cylindrical artery model can then be estimated [2,21]. This methodology46

has been extended to multiple cuts by Taber and Humphrey [20], and used in a two-layered47

arterial model by Holzapfel et al. [6].48

Inclusion of residual stress is important in modelling the mechanics of soft tissues for a num-49

ber of reasons: (a) in nonlinear elasticity theory the stress state in the reference configuration50

3

can have a substantial effect on the subsequent response to loads, and omission of the resid-51

ual stress can lead to significantly different stress predictions under load [4, 11, 17]; (b) in52

biological tissues, the growth and remodelling process will significantly affect the (residual)53

stress statement in living tissue [1,10]; (c) while the detailed process of local growth is diffi-54

cult to measure, residual stress, on the other hand, may be estimated from experiments, as55

demonstrated by the opening angle measurement. Thus, estimates of the residual stress at56

particular time instants could provide useful information about the growth history of living57

tissues.58

However, most work that includes residual stress in complex organs, such as the arteries59

and heart, assumed that a simple radial cut can release all the residual stresses [18, 22–26].60

However, this assumption is not supported by all experiments. For example, Omens et61

al. [14] showed that residual stress in a primary mouse heart could be further released by a62

circumferential cut following the initial radial cut, as illustrated in Fig. 1, They also showed63

that the opening angles are location-dependent, with greater values at heart apex. This64

implies that the single-cut opening angle configurations does not correspond to the stress-65

free configuration, and, as is well known, can only be considered as approximately stress-free.66

Holzapfel and Ogden [8] used a three-layer (adventitia, media and intimal) model to study67

residual stress in the artery, where they found each layer has a different opening angle when68

cutting open separately. In other words, it is impossible to release all the residual stress from69

a single radial cut across the three layers. By developing a model to count for the three-layer70

structure of the artery, the estimated residual stress is much greater than treating the artery71

as a single-layer model.72

Myocardium does not have such a distinct layer structure; however, it is clear from the73

experiment by Omens et al. [14] that multiple cuts need to be considered when studying74

residual stress in the heart. Inspired by the finding in [14] and the three-layer modelling75

work by Holzapfel and Ogden [8] on arteries, in this paper, we estimate the residual stress76

distribution across the wall of an intact mature heart using multiple cuts in a simplified77

4

heart model. Unlike the approach in [8] where the three distinct artery layers are modelled78

with different material properties, we know that any material property change across the79

myocardium must be smooth and that the transmural stress distributions are continuous.80

These considerations are taken into account in the current work.81

Figure 1: A typical short-axis apical segment of a mouse heart before and after cuts [14].

The initial intact segment, shown in A, was about 2 mm thick. The same segment after

a single radial cut and a further circumferential cut is shown in B and C, respectively. In

particular, the endocardial segment has reversed its curvature, in C. Note that the definition

of the opening angle in [14] follows that in Chuong and Fung [2], which is different from that

used in the present paper. Reproduced from [14] with permission.

2 METHODOLOGY82

There is experimental evidence that the collagen fibres are coiled and wavy in their unloaded

state in arteries [28, 29] and heart [16]. Similarly to previous studies [8, 22], we assume that

collagen fibres are not in tension in the unloaded configuration of the myocardium. Therefore,

5

for modelling the residual stress, we use the isotropic part of the invariant-based constitutive

law for the myocardium developed by Holzapfel and Ogden [7]:

Ψ =a

2b{exp[b(I1 − 3)]− 1}, (1)

where I1 = trC = tr(FTF), and a, b are material constants. The Cauchy stress tensor is

then [7]

σ = −pI + 2∂Ψ

∂I1

B, (2)

where B = FFT is the left Cauchy–Green deformation tensor.83

For simplicity, we model the LV as an incompressible single-layered cylindrical tube. We84

consider different scenarios based on different numbers of cuts and assume that the stress85

in the tube in the absence of loads on its curved surfaces can be released by either a single86

(radial) cut or multiple cuts (a radial cut followed by one or three circumferential cuts). We87

also assume that all the cut segments retain their cylindrical configurations, each with its88

own opening angle, i.e. each is a circular cylindrical sector.89

2.1 One cut: radial90

We take basis vectors {er, eθ, ez} to correspond to the local radial, circumferential and

longitudinal directions, respectively, in the intact circular cylindrical ring. For a single

(radial) cut, the opening angle approach has been well described [2, 6] for arteries, but is

summarised briefly here for completeness. Let the geometry of the right-hand panel in Fig. 2

represent a stress-free configuration B2, which is assumed to be a circular cylindrical sector

described by cylindrical polar coordinates {R,Θ, Z} as

R(i) 6 R 6 R(o),α2

26 Θ 6 2π − α2

2, 0 6 Z 6 L, (3)

where R(i), R(o), and L denote the inner and outer radii, and the tube length, respectively,91

and α2 is the opening angle. Let {ER,EΘ,EZ} be the associated cylindrical polar basis92

vectors in B2.93

6

angle in B2. The outer radius in B3 is

r(o) =

√R(o)2 −R(i)2

kλ(3)z

+ r(i)2. (8)

(a) (b)

B3 B2

F(3)α2

Figure 2: 1-cut model: cylindrical model of the LV after a single radial cut of the intact

unloaded configuration B3 into a stress-free sector B2.

The corresponding deformation gradient (from B2 to the intact ring configuration B3),

denoted F(3), is given by

F(3) = λ(3)1 er ⊗ ER + λ

(3)2 eθ ⊗ EΘ + λ(3)

z ez ⊗ EZ , (9)

where

λ(3)1 =

R

rkλ(3)z

, λ(3)2 =

kr

R(10)

and {ER,EΘ,EZ} are cylindrical polar axes in B2. It follows that the invariants I1 and I4

(with the superscript (3) omitted temporarily for simplicity) are given by

I1 = λ21 + λ2

2 + λ2z, I4 = λ2

2 cos2 γ + λ2z sin2 γ, (11)

where the components of f0 and f are (0, cos γ, sin γ)T and (0, λ2 cos γ, λz sin γ)T, respectively,82

and γ is the angle defining the fibre orientation with respect to the circumferential direction83

in B2.84

6

Figure 2: (a) Cross-section of a circular cylindrical LV model with no loading on its circular

boundaries in configuration B3; (b) Stress-free circular cylindrical sector B2 after a single

radial cut from B3. Note that maintenance of B3 as a circular cylindrical configuration

requires axial and torsional loads. The deformation gradient from B2 to B3 is denoted F(3).

The (isochoric) deformation from B2 to the intact configuration B3 is then expressed as

x = rer + zez, (4)

where, by incompressibility,

r =

√R2 −R(i)2

kλ(3)z

+ r(i)2, θ = k(Θ− α2/2), z = λ(3)

z Z, (5)

r(i) being the inner radius in the configuration B3, λ(3)z = l/L is the constant axial stretch

from B2 to B3, l is the cylinder length in B3, and k = 2π/(2π − α2) is a measure of the

opening angle in B2. The outer radius in B3 is

r(o) =

√R(o)2 −R(i)2

kλ(3)z

+ r(i)2. (6)

The corresponding deformation gradient (from B2 to B3), denoted F(3), is given by

F(3) = λ(3)1 er ⊗ ER + λ

(3)2 eθ ⊗ EΘ + λ(3)

z ez ⊗ EZ , (7)

where

λ(3)1 =

R

rkλ(3)z

, λ(3)2 =

kr

R. (8)

7

It follows that the invariants I1 (with the superscript (3) omitted temporarily for simplicity)

is given by

I1 = λ21 + λ2

2 + λ2z. (9)

The components of the Cauchy stress tensor in B3 are then94

σrr = −p+ 2∂Ψ

∂I1

λ21, (10)

σθθ = −p+ 2∂Ψ

∂I1

λ22, (11)

σzz = −p+ 2∂Ψ

∂I1

λ2z, (12)

where p is the Lagrangian multiplier. In the absence of body forces the stress components

σrr and σθθ in B3 satisfy the equilibrium equation divσ = 0, which, for the considered

deformation, reduces to

dσrrdr

+σrr − σθθ

r= 0, (13)

and the associated zero-traction boundary conditions are σrr = 0 for r = r(i), r(o).95

On integration and use of the latter boundary conditions equation (13) gives∫ r(o)

r(i)

σθθ − σrrr

dr = 0, (14)

which, on substitution from (10) and (11), can be used to obtain r(i) in B3 (and r(o) from96

(5)) when the initial radii R(i) and R(o) and k and γ are known. Hence, given Ψ, all the97

Cauchy stress components can be obtained explicitly in B3.98

It should be emphasized that σ is not strictly a residual stress since the presence of the

components σzz requires appropriate non-zero boundary conditions, whereas true residual

stress is associated with zero-traction boundary conditions. However, the axial force N

required to maintain the circular cylindrical configuration B3 [6]

N = 2π

∫ r(o)

r(i)σzzrdr = π

∫ r(o)

r(i)(2σzz − σrr − σθθ)rdr, (15)

is non-zero but small. Following [8], we adjust the axial stress so that σzz − Nπ[(r(o))2−(r(i))2]

is99

the approximate measure of the residual axial stress. The adjusted residual axial stress has100

its mean removed so that the resulting axial load vanishes.101

8

(a) (b) (c)

B3 B2 B1

F(3) F(2)

α2 α(2)1 α

(1)1

outerouter

innerinner

Figure 3: The 2-cut model: (a) cylindrical model of the LV as the intact ring in B3, (b) after

a radial cut to B2, and (c) followed by a circumferential cut to B1. Notice that the inner

segment in B1 has a negative curvature, as in [14]. The red curve, at the mid-wall radius

R = (R(i) +R(o))/2 in (b), separates the inner and outer sectors which become the separate

inner and outer sectors in B1 after the circumferential cut.

2-Cut Model95

In the 2-cut model, following a radial cut, a circumferential cut is performed around the

mid-wall in B2 at radius R = (R(i) +R(o))/2. It is assumed now that residual stress remains

after the radial cut but is removed after the circumferential cut and that no axial deformation

is associated with the second cut. The resulting stress-free configuration B1 is depicted in

Fig. 3. We note, in particular, that following the circumferential cut the inner segment has a

negative curvature. The geometry in B1 is described in terms of cylindrical polar coordinates

{R,Θ, Z}, with subscripts 1 and 2 corresponding to the inner and outer sectors, respectively.

Thus,

R(i)1 6 R1 6 R

(o)1 , −(π − α

(1)1

2) 6 Θ1 6 π − α

(1)1

2, 0 6 Z1 6 L, (19)

R(i)2 6 R2 6 R

(o)2 ,

α(2)1

26 Θ2 6 2π − α

(2)1

2, 0 6 Z2 6 L, (20)

where R(i)j , R

(o)j , α

(j)1 , j = 1, 2, and L denote the inner and outer radii, the opening angles,96

and the tube length in B1. In B2, R(i)1 and R

(i)2 both become R, while R

(o)1 and R

(o)2 translate97

8

Figure 3: The two-cut model (a) cylindrical model of the LV as the intact ring in B3, (b)

after a radial cut to B2, and (c) followed by a circumferential cut to B1. Notice that the

inner sector in B1 has a negative curvature, as in [14]. The red curve, at the mid-wall radius

R = (R(i) +R(o))/2 in (b), separates the inner and outer sectors which become the separate

inner and outer sectrors in B1 after the circumferential cut. Appropriate axial and torsional

loads are required to maintain the shapes in B2 and B3.

2.2 Two cuts: radial and circumferential102

For the two-cut model we consider the separation of the sector in B2 into two separate103

circular cylindrical sectors (inner and outer) by means of a circumferential cut around the104

mid-wall in B2 at radius R = (R(i) + R(o))/2. The two new sectors form the configuration105

B1, which is now taken as the stress-free reference configuration. Thus, B2 is no longer106

stress free but requires torsional and axial loads to maintain its circular cylindrical shape, It107

is, however, residually stressed in the sense that there is no traction on its curved surfaces.108

The transition from B2 to B3 is now different different from that in the one-cut model. The109

stress in B3 is calculated from the constitutive laws based on the reference configuration B1110

with the appropriate deformation gradients from the two sectors in B1 to B3. The transition111

from B1 to B2 to B3 is depicted in Fig. 3.112

We note, in particular, that the inner sector in B1 has a negative curvature. The geometry

in B1 is described in terms of cylindrical polar coordinates {R,Θ, Z}, with subscripts 1 and

9

2 corresponding to the inner and outer sectors, respectively. Thus,

R(i)1 6 R1 6 R

(o)1 , −π +

α(1)1

26 Θ1 6 π − α

(1)1

2, 0 6 Z1 6 L, (16)

R(i)2 6 R2 6 R

(o)2 ,

α(2)1

26 Θ2 6 2π − α

(2)1

2, 0 6 Z2 6 L, (17)

where R(i)j , R

(o)j , α

(j)1 , j = 1, 2, and L denote the inner and outer radii, the opening angles,113

and the tube length in B1. In B2, R(i)1 and R

(i)2 both become R, while R

(o)1 and R

(o)2 translate114

to R(i) and R(o), respectively, the opening angle is α2 and the axial length l.115

For each sector, the isochoric deformation from B1 to B2 can be written as

X = RER + ZEZ , (18)

with

R =

√√√√R(i)1

2 −R21

k1λ(21)z

+ R2, Θ = π − k1Θ1, Z = λ(21)z Z1 (19)

R =

√√√√R22 −R(i)

2

2

k2λ(22)z

+ R2, Θ = k2(Θ2 − π) + π, Z = λ(22)z Z2 (20)

for the inner and outer sectors, respectively, where kj = (2π−α2)/(2π−α(j)1 ), j = 1, 2. Note116

that the negative curvature of the inner sector in B1 depicted in Fig. 3 is captured by the117

expression for R in (19), and that structural compatibility in B2 is ensured since the two118

expressions for R match at R.119

As indicated in Fig. 3 the deformation gradient from B1 to B2 is denoted F(2), which is

shorthand notation for the two separate deformation gradients from the two sectors in B1

to B2. These are denoted F(2j), j = 1, 2, and given by

F(2j) = λ(2j)1 er ⊗ ER + λ

(2j)2 eθ ⊗ EΘ + λ(2j)

z ez ⊗ EZ , (21)

where

λ(2j)1 =

Rj

kjRλ(2j)z

, λ(2j)2 =

kjR

Rj

, λ(2j)z = λ

(2j)1

−1λ

(2j)2

−1, j = 1, 2. (22)

10

Similarly to the one-cut model, the equilibrium equation in B2 yields

dσRRdR

+σRR − σΘΘ

R= 0,

∫ R(o)

R(i)

σΘΘ − σRRR

dR = 0. (23)

Equation (23)1 can be rearranged as

σΘΘ =d

dR(RσRR), (24)

from which it follows, on use of the zero-traction boundary conditions σRR = 0 on R(i) and

R(o), that ∫ R(o)

R(i)

σΘΘdR = 0, (25)

i.e. the mean value of σΘΘ through the thickness is zero.120

It is assumed that is there is no bending moment on the faces Θ = α2/2 and Θ = 2π−α2/2

in B2, which yields ∫ R(o)

R(i)

σΘΘRdR = 0. (26)

Substitution of σΘΘ from (24) into this equation followed by integration by parts and a

further application of the zero-traction boundary conditions leads to

∫ R(o)

R(i)

σRRRdR = 0. (27)

Equations (23)2, (26) and (27) are solved with equations (19) and (20) to obtain the radii121

R(i) and R(o), and the angle α2 of the sector in B2.122

The deformation gradient associated with the transition F1→3 from B1 to B3 has the form123

F(3)F(2), where F(3) is given by equation (7) from the one-cut approach, and F(2) is either F(21)124

or F(22), as given in equation (21). Note that F(3)F(21) and F(3)F(22) are the values obtained125

for the inner and outer sectors, respectively, and these must match at the interface R, i.e. the126

deformation gradient must be continuous in B2, which implies that k1/R(i)1 = k2/R

(i)2 at the127

interface. Enforcing of this requirement will ensure that when traction continuity is applied128

p is continuous and hence that all the stress components are continuous, in particular that129

σΘΘ is continuous.130

11

Hence, for the two-cut model, we require geometric information, e.g. R(i)1 , R

(i)2 , α1, α2 in131

B1, from experiments (Fig.1C). Then the geometric information of B2 is obtained from the132

two-cut model. The residual stress of the intact-ring configuration, B3, can now be solved133

using the same equilibrium equations as (23), (26) and (27), except the deformation gradient134

is now F(3)F(2).135

An expression for the required axial load N for the intact ring is obtained from a formula136

similar to that in (15), and the corresponding residual axial stress is adjusted to make the137

resulting axial load vanish.138

2.3 Four cuts: one radial and three circumferential139

The effect of two further circumferential cuts, one in each of the two separated sectors, is140

now considered in order to assess if there is any significant change in the resulting calculated141

residual stress compared with that obtained with a single circumferential cut, although this142

is not a test that has been carried out experimentally. For definiteness we consider taking143

a circumferential cut along the mid-wall of each of the two sectors of the two-cut model,144

leading to the four separate sectors depicted in Fig. 4. In this model we assume that the145

resulting configuration B0 is stress-free with no further reversal of the curvature, so that the146

two sectors in B1 are no longer stress free.147

B0

α(4)0

α(3)0

α(2)0 α

(1)0

outer II

outer I inner II

inner I

Figure 4: The stress-free configuration B0 consisting of the four sectors obtained by circum-

ferential cuts of the two sectors in B1.

mation gradients from B1 to B2 which also have to be continuous, and the deformation137

gradient in B3 likewise has to be continuous. The consequences of these constraints now138

need to be evaluated.139

The deformations from the four sectors in B0 to the two sectors in B1 are described by

R1 =

√√√√ρ2 − ρ(i)1

2

k11λ(11)z

+ R21, Θ1 = k11φ, Z1 = λ(11)

z ζ, (inner I), (35)

R1 =

√√√√ρ2 − ρ(o)2

2

k12λ(12)z

+ R21, Θ1 = k12φ, Z1 = λ(12)

z ζ, (inner II), (36)

R2 =

√√√√ρ2 − ρ(o)3

2

k23λ(23)z

+ R22, Θ2 = k23(φ− π) + π, Z2 = λ(23)

z ζ, (outer I), (37)

R2 =

√√√√ρ2 − ρ(i)4

2

k24λ(24)z

+ R22, Θ2 = k24(φ− π) + π, Z2 = λ(24)

z ζ, (outer II), (38)

where

k1n = (2π − α(1)1 )/(2π − α(n)

0 ), n = 1, 2, k2n = (2π − α(2)1 )/(2π − α(n)

0 ), n = 3, 4,

λ(1n)1 =

R1

ρk1nλ(1n)z

, λ(1n)2 =

k1nρ

R1

, n = 1, 2,

12

Figure 4: The stress-free configuration B0 consisting of the four sectors obtained by circum-

ferential cuts of the two sectors in B1.

12

Each of the four sectors in B0 is described in terms of cylindrical polar coordinates {ρ, φ, ζ}148

according to149

ρ(i)1 6 ρ 6 ρ

(o)1 , −π +

α(1)0

26 φ 6 π − α

(1)0

2, 0 6 ζ 6 L (inner I), (28)

ρ(i)2 6 ρ 6 ρ

(o)2 , −π +

α(2)0

26 φ 6 π − α

(2)0

2, 0 6 ζ 6 L (inner II), (29)

ρ(i)3 6 ρ 6 ρ

(o)3 ,

α(3)0

26 φ 6 2π − α

(3)0

2, 0 6 ζ 6 L (outer I), (30)

ρ(i)4 6 ρ 6 ρ

(o)4 ,

α(4)0

26 φ 6 2π − α

(4)0

2, 0 6 ζ 6 L (outer II), (31)

where ρ(i)n , ρ

(o)n , α

(n)0 , n = 1, 2, 3, 4, and L are the internal radii, the external radii, opening150

angles, and the lengths of the four sectors in B0, and the notations I and II are identified151

in Fig. 4.152

In B1, the geometries of the two sectors are described in terms of polar coordinates {R,Θ, Z},153

with indices 1 and 2, as in equations (16) and (17). Next, in B2, R(i), R(o), α2, l denote the154

internal and external radii, the opening angle and the length of the single sector according155

to (19)–(20).156

The deformations from the four sectors in B0 to the two sectors in B1 are described by

R1 =

√√√√ρ2 − ρ(i)1

2

k11λ(11)z

+ R21, Θ1 = k11φ, Z1 = λ(11)

z ζ, (inner I), (32)

R1 =

√√√√ρ2 − ρ(o)2

2

k12λ(12)z

+ R21, Θ1 = k12φ, Z1 = λ(12)

z ζ, (inner II), (33)

R2 =

√√√√ρ2 − ρ(o)3

2

k23λ(23)z

+ R22, Θ2 = k23(φ− π) + π, Z2 = λ(23)

z ζ, (outer I), (34)

R2 =

√√√√ρ2 − ρ(i)4

2

k24λ(24)z

+ R22, Θ2 = k24(φ− π) + π, Z2 = λ(24)

z ζ, (outer II), (35)

where

k1n = (2π − α(1)1 )/(2π − α(n)

0 ), n = 1, 2, k2n = (2π − α(2)1 )/(2π − α(n)

0 ), n = 3, 4,

13

λ(1n)1 =

R1

ρk1nλ(1n)z

, λ(1n)2 =

k1nρ

R1

, λ(1n)z = λ

(1n)1

−1λ

(1n)2

−1, n = 1, 2,

λ(2n)1 =

R2

ρk2nλ(2n)z

, λ(2n)2 =

k2nρ

R2

, λ(2n)z = λ

(2n)1

−1λ

(2n)2

−1, n = 3, 4,

and

Rj =1

2(R

(i)j +R

(o)j ), j = 1, 2.

The deformation gradients from B0 in Fig. 4 to B1 in Fig. 3 are

F(1n) = λ(1n)1 ER ⊗ eρ + λ

(1n)2 EΘ ⊗ eφ + λ(1n)

z EZ ⊗ eζ , n = 1, 2, (36)

F(2n) = λ(2n)1 ER ⊗ eρ + λ

(2n)2 EΘ ⊗ eφ + λ(2n)

z EZ ⊗ eζ , n = 3, 4. (37)

In this model there are four separate deformation gradients from B0 to B1, pairs of which157

have to be continuous in B1. Then, the two separate deformation gradients from B1 to B2158

also have to be continuous, and the deformation gradient from B2 to B3 likewise has to be159

continuous. The transformations between the various internal and external radii are listed160

in Table 1.161

Table 1: Transformations between the various internal and external radii in the different

configurations.

B0 B1 B2 B3

ρ(i)1 → R1

ρ(o)1 → R

(o)1 → R(i) → r(i)

ρ(i)2 → R

(i)1 → 1

2(R(i) +R(o))

ρ(o)2 → R1

ρ(i)3 → R

(i)2 → 1

2(R(i) +R(o))

ρ(o)3 → R2

ρ(i)4 → R2

ρ(o)4 → R

(o)2 → R(o) → r(o)

14

In B1, the radial equilibrium equation for each of the two sectors yields

∫ R(o)j

R(i)j

σΘΘ − σRRR

dR = 0, j = 1, 2. (38)

The radial traction σRR should be continuous across each interface Rj, j = 1, 2, and hence,

since the deformation gradient is required to be continuous, p is continuous and the other

stress components are also continuous. For the considered deformation, with λz given, con-

tinuity of both the radial and circumferential stresses guarantees that both p and the de-

formation (as measured by λ2) are continuous. Note that continuity of the circumferential

stress at the interfaces in B1 requires that

σΘΘ is continuous for R1 = R1 (R2 = R2) in the inner (outer) sector. (39)

Also, similarly to the two-cut model,

∫ R(o)j

R(i)j

σΘΘRdR =

∫ R(o)j

R(i)j

σRRRdR = 0, j = 1, 2. (40)

The corresponding deformation gradient from B1 to B2 is F0→2 = F(2)F(1), where again

F(2) is either F(21) or F(22), F(1) is the appropriate F(1n), n = 1, 2, or F(2n), n = 3, 4, and the

governing equations for the single sector are

∫ R(o)

R(i)

σΘΘ − σRRR

dR = 0,

∫ R(o)

R(i)

σΘΘRdR =

∫ R(o)

R(i)

σRRRdR = 0, (41)

as in (23)2, (26) and (27).162

This model involves eight independent equations (38)–(41), and eight unknown geometrical163

parameters in B0: ρ(i)n , α

(n)0 , n = 1, 2, 3, 4. The required external axial load is then calculated164

similar to (15), and made to vannish by adjuing the residual axial stress.165

In summary, for the four-cut model, we require geometric information of both B1 and B2166

from experiments. Once we obtain all the details, e.g, opening angles and radii of all sectors in167

B0, we estimate the (residual) stress components σrr and σθθ in the intact-ring configuration168

B3, with the total deformation gradient F0→3 = F(3)F(2)F(1).169

15

3 RESULTS170

3.1 Modelling Parameters171

The reference configuration is different for the different models, so we need to define the172

parameters according to each specific model considered. For B2 in the one-cut model, we173

assume that the axial stretch has the constant value λ(3)z = 1.14, and R(i) = 2.06, R(o) = 3.20174

α2 = 65◦ are estimated from the experiments [14].175

For the two-cut model, in addition to the parameters used in the one-cut model, we use176

additional measurements in B1 of the two-cut model in [14]: R(i)1 = 5.28, R

(i)2 = 1.97, α

(1)1 =177

268◦ and α(2)1 = 180◦. We also assume that there is no axial stretch in the transformation178

from B1 to B2, i.e. λ(2j)z = 1, j=1,2.179

For the four-cut model, in addition to the parameters used in the two-cut model, we need180

more geometrical information in B1, which is again estimated from the measurements [14]181

as listed in Table 2. We also assume that λ(1n)z = λ

(2n)z = 1 for the deformation from B0 to182

B1, i.e. no axial deformation occurs as a result of the circumferential cuts.183

Table 2: Measured geometrical input for the four-cut model, estimated from [14].

Configuration B1 Configuration B2

R(i)1 = 5.28 1

2(R(i) +R(o))

R(o)1 = 6.29 R(i) = 2.06

α(1)1 = 268◦ α2 = 65◦

R(i)2 = 1.97 1

2(R(i) +R(o))

R(o)2 = 3.04 R(o) = 3.20

α(2)1 = 180◦ α2 = 65◦

Initially, we consider a homogeneous myocardium model, for which the material parameters184

of the constitutive law (1) are fitted to the data of mice [15]. This gives a = 2.21kPa, and185

b = 1.8.186

16

However, the dramatic difference in the maximum hoop stress between the single-cut and187

multiple-cut models raises a question about the rationale of considering homogeneous ma-188

terial properties. Novak et al. [12] showed that canine myocardium is heterogeneous with189

location dependent material properties. In particular, they showed that the mid myocardium190

is softer than epicardium or endocardium, although they didn’t find differences in regional191

(anterior wall vs septum) stiffness. We therefore also consider an inhomogeneous myocardium192

model, and fit the data from [12] with the strain-energy function (1).193

The parameters are spatially dependent, as shown in Fig. 5. Since our model is for mice,194

and there are no experimental data on the heterogeneous properties of mice myocardium,195

we take the spatial variation of the canine data, but keep the mean values of the mice data196

from our fitted parameters, and include these in our calculations.197

Figure 5: The heterogeneous material parameters a (left), and b (right), fitted to experiments.

The red dashed lines indicate the constants used for the homogeneous models, and the red

solid lines are for the heterogeneous mice models, with the spatial distributions taken from

the canine data in [12] (black solid lines).

3.2 Results for the Homogeneous Myocardium Model198

The geometry of the intact ring predicted in the three different models is summarized in199

Table 3, and compared with the measured data in [14].200

17

It is clear that the agreement of the estimated radius and thickness of the unloaded configu-201

ration gets better as the number of cuts increases. Although in principle the true zero-stress202

configuration requires infinite cuts, Table 3 suggests that two or four cuts provides a good203

approximation of the zero-stressed configuration, given that the measured geometry is not204

exactly circular, although it is assumed to be circular in each model.205

Table 3: Computed intact ring for the homogeneous models, compared to measurements [14].

Measurements (mm) One-cut model Two-cut model Four-cut model

r(i) = 0.565 0.555 0.347 0.364

r(o) = 2.185 2.147 1.9561 1.9950

r(o) − r(i) = 1.620 1.5920 1.6438 1.6310

Difference in thickness 1.73% 1.47 % 0.68%

The components of the residual stress distribution in the intact-ring configuration B3 from206

the three different models are shown in Fig. 6. The residual axial stress for each model is207

adjusted to remove the impact of the non-zero values of N (=5.67× 10−3, 1.71× 10−3, and208

1.42×10−3, respectively, in the one-cut, two-cut, and four-cut models.) Although the overall209

distributions are similar in all these models, there is marked difference in the magnitudes210

of the hoop stresses. In particular, the maximum σθθ is 1.75 kPa, 17.13 kPa, and 17.15211

kPa, respectively, for the one-cut, two-cut, and four-cut models. Comparing to the one-cut212

model, the ratio of the maximum hoop stresses over the single cut is about 9.78 times for213

the two-cut model, and 9.80 times for the four-cut model. We also notice that although214

the four-cut model gives much smoother stress distribution, the two-cut model leads to a215

similar magnitude of the maximum hoop stress. This suggests that the significant rise in the216

residual stress is due to the negative curvature at the first circumferential cut.217

18

(a) (b)

(c)

Figure 6: Distribution of the residual stress components in the intact ring from (a) single cut, (b)

two-cut, and (c) four-cut models based on the homogenous material assumption.

3.3 Results for the Heterogeneous Myocardium Model218

The results of the intact ring from the heterogeneous myocardium model are computed.219

Again, the residual axial stress for each model is adjusted to remove the impact of the non-220

zero values of N (=4.59 × 10−3, 1.35 × 10−3, and 1.09 × 10−3, respectively, in the one-cut,221

two-cut, and four-cut models.) . All the stess compoenets are plotted in Fig. 7, which shows222

that the maximum σθθ is 1.85 kPa, 16.12 kPa, and 16.63 kPa, respectively, for the one-cut,223

two-cut, and four-cut models. The ratio of the maximum hoop stresses over the single cut224

is slightly lower than that of the homogeneous material, at about 8.71 times for the two-cut225

19

model and 8.73 times for the four-cut model. Again, apart from smoother and somewhat226

different stress distributions, the four-cut model predicts very similar maximum hoop stress227

as the two-cut model.228

(a) (b)

(c)

Figure 7: Distribution of the residual stress components from (a) one-cut, (b) two-cut, and (c)

four-cut models based on the heterogeneous myocardium assumption.

20

4 DISCUSSION229

The issue of multiple cuts has been studied before. In particular, Fung suggested that230

one radial cut might be sufficient to release all the residual stress. This was supported by231

his experiments which showed that after two or more radial cuts, no obvious deformation232

occurs from the one-radial-cut configuration of the arteries [3]. Using our models we can233

show, however, that multiple radial cuts do not indeed release more residual stress, since due234

to the symmetry of the considered geometry, once a radial cut is made, no further elastic235

deformation can occur after more radial cuts.236

In other words, the deformation gradient in each of our different models is independent of the237

azimuthal angle and the solutions are also independent of this angle. However, this does not238

indicate that the one-cut configuration is a stress-free one. To further release residual stress239

circumferential cuts following a radial cut are necessary. We note that even with multiple240

cuts, we may not release all the residual stresses. In principle, only infinite cuts can release all241

the residual stresses. This explains why the stress distributions are not smooth and appear242

to have deflections around where the cuts are, since we have to assume the configuration243

after two or four cuts is stress-free. This process could be improved with more cuts, though244

more than four cuts cannot be modelled in this paper without further experimental data.245

However, the curves in the four-cut model are almost smooth, the required axial load to246

maintain the cylindrical shape of the intact ring reduces is smaller, and the predicted intact247

ring thickness agrees very well with the measurements. All these tentatively suggest that the248

four-cut model is already a good approximation for the residual stresses. Indeed, in terms of249

estimating the residual stress magnitude, even the two-cut model seems to be good enough.250

The fact that our two-cut and four-cut models predict a much higher (about 8-9 times!)251

hoop stress is perhaps not unexpected given the large negative curvature revealed by the252

experiments [14]. The increased value of the residual hoop stress agrees with the rough253

estimation by Omens et al. [14] based on a simplified concentric cylindrical shell model,254

where they showed that the maximum loop stress was about 20kPa, which is close to our255

21

estimation of about 17kPa.256

We argue that the significant stress underestimate of the one-cut model does not just occur257

in the heart models. In the residual stress modelling of arteries, by treating the artery wall as258

three separate layers (intima, media and adventitia), and measuring the opening angles for259

each of the three layers, Holzapfel and Ogden [8] have essentially developed a three-cut model260

(one radial cut followed by two circumferential cuts, in this case separating the layers with261

different properties). Their model is also heterogeneous, as different material parameters are262

used for different layers. We now compare the residual stress distribution across the artery263

wall in Fig. 8 using their three-cut approach and the one-cut model. The results from the264

one-cut (or one-layer) model are reproduced here using the same parameters as in [8]. The265

maximum values of the hoop stress and their ratio to the one-cut model results in different266

layers of the three-cut model are listed in Table 4, which shows that the ratio of the hoop267

stress in the different layers ranges from 24 to 50 times. This difference is even greater than268

that of the mouse heart.269

In this study, we assume that the deformation gradient is diagonal. Note this is in general not270

true for the heart under loading [27]. However, since residual stresses are estimated from the271

unloaded configuration, the fibres are in general coiled and do not bear the load. Therefore272

the myocardium behaves like an isotropic material, and the radial, circumferential and axial273

directions are the principal directions of the deformation, i.e. the deformation gradient is274

diagonal in these directions. This is different from artery modelling, when the two families275

of fibres are assumed symmetrical about the axial direction. Hence, even when loaded with276

pressure, tensioned fibres do not alter the principal directions of the deformation. Therefore,277

the diagonal deformation gradient (e.g. [8]) is assumed in arteries for a different reason.278

Finally, we would like to state the limitations of the study. We have assumed that cross-279

sections of the heart are cylindrical, and all the cut segments retain their cylindrical config-280

urations, each with its own opening angle, i.e. each is a circular cylindrical sector. Needless281

to say, this is not always true, particularly in opening angle studies of arteries, where a282

22

single cut of an arterial ring can have a non-uniform curvature [19]. In addition, with the283

cylindrical assumption, we cannot model the experimental observation that opening angles284

are location-dependent, with higher values at heart apex [14]. Indeed, the “true” residual285

stress can only be achieved through infinite cuts using the opening angle method, which is286

not practical. We believe the residual stress estimation, although an approximation, is a287

step forward to the physiological range compared with using the single cut opening angle288

method.289

(a) (b)

Figure 8: Residual stress distributions through the intima, media and adventitia of the artery

wall as functions of the radial coordinate r: (a) residual stress is recomputed here using a one-cut

model, all other parameters being the same, so that the comparison can be made with the multi-

layer approach, and (b) the original result with layer separations from [8], reused with permission.

Table 4: Residual stress computed using a single-cut and the original HO model [8]

Max. hoop stress (kPa) one-cut model three-cut model [8] Ratio of stresses

Adventitia 0.57 15.08 26.45

Media 0.47 23.74 50.51

Intima -0.97 -23.20 23.91

23

5 CONCLUSION290

Based on experimental observations that a single radial cut does not release all residual291

stress, we have used multiple cuts to estimate the residual stress distributions in a mouse292

left ventricle model. Our results show that both radial and circumferential cuts are required293

to release the residual stresses in the middle wall of the left ventricle. Remarkably, using294

radial cuts alone leads to a significant underestimate of the residual stress, which will be295

around 8 to 9 times greater if estimated on the basis of combined radial and circumferential296

cuts. Similar findings apply to arteries based on the model in [8]. We remark that the results297

are not significantly different using the homogeneous or heterogeneous material models. In298

addition, although the stress distributions are different and much smoother in the four-cut299

model, the two-cut model can already estimate the maximum hoop residual stress quite300

satisfactorily.301

Conflict of Interest302

There is no conflict of interest to declare.303

ACKNOWLEDGMENTS304

We would like to thank Ray Ogden and Jay Humphrey for a helpful discussions. XZ is305

sponsored by the China Scholarship Council Studentship and the Fee Waiver Programme306

at the University of Glasgow. XYL is grateful for the support by the UK Engineering and307

Physical Sciences Research Council grants (EP/S020950, EP/S014284, and EP/N014642).308

24

References309

1. Ambrosi, D., G. A. Ateshian, E. M. Arruda, S. C. Cowin, J. Dumais, A. Goriely, G. A.310

Holzapfel, J. D. Humphrey, R. Kemkemer, E. Kuhl, et al.. Perspectives on biological311

growth and remodeling. J. Mech. Phys. Solids. 59:863–883, 2011.312

2. Chuong, C. J., and Y. C. Fung. Residual stress in arteries In: G.W. Schmid-Schonbein,313

S.L-Y. Woo and B.W. Zweifach (eds), Frontiers In Biomechanics 117–129, 1986.314

3. Fung, Y. C., and S. Q. Liu. Change of residual strains in arteries due to hypertrophy315

caused by aortic constriction. Circ. Res. 65:1340–1349, 1989.316

4. Hoger, A. On the residual stress possible in an elastic body with material symmetry.317

Arch. Ration. Mech. Anal. 88:271–289, 1985.318

5. Hsu, F. H. The influences of mechanical loads on the form of a growing elastic body. J.319

Biomech. 1:303–311, 1968.320

6. Holzapfel, G. A., T. C. Gasser, and R. W. Ogden. A new constitutive framework for321

arterial wall mechanics and a comparative study of material models. J. Elasticity. 61:1–322

48, 2000.323

7. Holzapfel, G. A., and R. W. Ogden. Constitutive modelling of passive myocardium: a324

structurally based framework for material characterization. Philos. Trans. R. Soc. A325

367:3445–3475, 2009.326

8. Holzapfel, G. A., and R. W. Ogden. Modelling the layer-specific three-dimensional327

residual stresses in arteries, with an application to the human aorta. J. R. Soc. Interface.328

7:787–799, 2010.329

9. Liu, S. Q., and Y. C. Fung. Zero-stress states of arteries. J. Biomech. Eng. 110:82–84,330

1988.331

25

10. Menzel, A., and E. Kuhl. Frontiers in growth and remodeling. Mech. Res. Commun.332

42:1–14, 2012.333

11. Merodio, J., R. W. Ogden, and J. Rodrıguez. The influence of residual stress on finite334

deformation elastic response. Int. J. Non-Lin. Mech. 56:43–49, 2013.335

12. Novak, V. P., F. C. P. Yin, and J. D. Humphrey. Regional mechanical properties of336

passive myocardium. J. Biomech. 27:403–412, 1994.337

13. Ogden, R. W. Non-Linear Elastic Deformation. New York, NY: Dover Publications.338

1997.339

14. Omens, J. H., A. D. McCulloch, and J. C. Criscione. Complex distributions of resid-340

ual stress and strain in the mouse left ventricle: experimental and theoretical models.341

Biomech. Model. Mechanobiol. 1:267–277, 2003.342

15. Omens, J. H., D. A. MacKenna, and A. D. McCulloch. Measurement of strain and343

analysis of stress in resting rat left ventricular myocardium. J. Biomech. 26:665–676,344

1993.345

16. Omens, J. H., T. R, Miller, and J. W. Covell, Relationship between passive tissue346

strain and collagen uncoiling during healing of infarcted myocardium, Cardiovascular347

research,33:351–358, 1997348

17. Shams, M., M. Destrade, and R. W. Ogden. Initial stresses in elastic solids: constitutive349

laws and acoustoelasticity. Wave Motion 48:552–567, 2011.350

18. Rodriguez, EDWARD K and Omens, JEFFREY H and Waldman, LK and McCul-351

loch, AD Effect of residual stress on transmural sarcomere length distributions in rat352

left ventricle American Journal of Physiology-Heart and Circulatory Physiology 264353

(4):H1048–H1056, 1993.354

19. T. Sigaeva, M. Destrade, E. Di Martino. Multisector method for arteries: The residual355

stresses of circumferential rings with non-trivial openings, Journal of the Royal Society356

26

Interface, 20190023. 2019.357

20. Taber, L. A., and J. D. Humphrey. Stress-modulated growth, residual stress, and vascular358

heterogeneity. J. Biomech. Eng. 123:528–535, 2001.359

21. Taber, L. A., N. Hu, P. Tomas, E. B. Clark, and B. Keller. Residual strain in the360

ventricle of the stage 16-24 chick embryo. Circ. Res. 72:455–462, 1993.361

22. Wang, H. M., X. Y. Luo, H. Gao, R. W. Ogden, B. E. Griffith, C. Berry, and T. J.362

Wang. A modified Holzapfel–Ogden law for a residually stressed finite strain model of363

the human left ventricle in diastole. Biomech. Model. Mechanobiol. 13:99–113, 2014.364

23. Weis, Sara M and Emery, Jeffrey L and Becker, K David and McBride Jr, Daniel J365

and Omens, Jeffrey H and McCulloch, Andrew D Myocardial mechanics and collagen366

structure in the osteogenesis imperfecta murine (oim) Circulation research. 87:663–669,367

2000.368

24. Costa, Kevin D and May-Newman, Karen and Farr, Dyan and ODell, Walter G and369

McCulloch, Andrew D and Omens, Jeffrey H Three-dimensional residual strain in mi-370

danterior canine left ventricle American Journal of Physiology-Heart and Circulatory371

Physiology. 273:H1968–H1976, 1997.372

25. Nevo, Erez and Lanir, Yoram The effect of residual strain on the diastolic function of the373

left ventricle as predicted by a structural model Journal of biomechanics. 27: 1433–1446,374

1994.375

26. Grobbel, Marissa R and Shavik, Sheikh Mohammad and Darios, Emma and Watts,376

Stephanie W and Lee, Lik Chuan and Roccabianca, Sara Contribution of left ventric-377

ular residual stress by myocytes and collagen: existence of inter-constituent mechanical378

interaction Biomechanics and modeling in mechanobiology. 17: 985–999, 2018.379

27. , Guccione, Julius M and McCulloch, Andrew D and Waldman, LK Passive mate-380

rial properties of intact ventricular myocardium determined from a cylindrical model381

=American Society of Mechanical Engineers, 113: 42–55, 1991.382

27

28. Clark, John M and Glagov, Seymour Transmural organization of the arterial media.383

The lamellar unit revisited. Arteriosclerosis: An Official Journal of the American Heart384

Association, Inc. 5:19–34, 1985.385

29. Dingemans, Koert P and Teeling, Peter and Lagendijk, Jaap H and Becker, Anton386

E Extracellular matrix of the human aortic media: an ultrastructural histochemical387

and immunohistochemical study of the adult aortic media The Anatomical Record: An388

Official Publication of the American Association of Anatomists. 258:1–14, 2000.389

28