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Modelling of viscoelasticity in pressure-volume curve of an intact gallbladder

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Abstract

Like other organs such as artery, bladder and left ventricle, human intact gallbladders (GBs) possess viscoelasticity/hysteresis in pressure-volume curves during in vitro or in vivo dynamic experiments made by using saline infusion and withdrawal cycle to simulate GB physiological emptying-filling cycle in normal and diseased conditions. However, such a viscoelastic property of GBs has not been modelled and analysed so far. A non-linear discrete viscous model and a passive elastic model were proposed to identify the elastic, active and viscous pressure responses in the experimental pressure-volume data of an intact GB under passive and active conditions found in the literature in the paper. It turns out that the elastic, viscous and active pressure responses can be separated in less than 2% error from the pressure-volume curves. The peak active state in the GB occurs at 30% dimensionless volume. The GB stimulated with cholecystokinin (CCK) or treated with indomethacin is subject to almost constant stiffness at low dimensionless volume (≤ 70%) but quick increasing stiffness at high dimensionless volume (>70%) and a larger work-to-energy ratio (0.57–0.61) compared with the normal GB in the passive state. The models are sensitive to the change in the biomechanical property of the GBs stimulated or treated with hormonal or pharmacological agents, showing a potential in clinical application. These results may contribute fresh content to the biomechanics of GBs and be helpful to GB disease diagnosis.

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Abbreviations

a 1, a 2, a 3 :

Model constants of viscosity in the passive state

b 1, b 2, b 3 :

Model constants of viscosity in the active state

c 0, c 1, c 2, c 3 :

Model constants of GB pressure-volume curve

E :

Elastic energy stored in GB wall during its elastic deformation, N.m

F :

Objective function

i :

An experimental point

k :

Stiffness of GB pressure-volume curve

m 1 :

Model constant of viscosity in the passive state

m 2 :

Model constant of viscosity in the active state

n 1, n 2 :

Model constants of GB pressure-volume curve

N :

Number of experimental points

p :

Pressure, mmHg or cm H2O

t :

Time, min

t 1, t 2, t 3 :

Time moments at the start of infusion, end of infusion and end of withdrawal, min

v :

Dimensionless GB volume

V :

GB bile volume, ml

W :

Work done on the viscous response of GB wall in an infusion-withdrawal cycle, N.m

γ :

Work-to-energy ratio, γ = W/E

ε :

Error in pressure in passive and active states, %

η :

Viscosity of dashpot

1:

Passive state

2:

Active state

e:

Elastic

exp:

Experimental

f:

Infusion

max:

Maximum value

p:

Pressure

v:

Viscous

w:

Withdrawal

3D:

Three-dimensional

ABP:

Acalculous biliary pain

CCK:

Cholecystokinin

EF:

Ejection fraction

GB:

Gallbladder

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Appendix A: Validation of proposed GB pressure-volume model

Appendix A: Validation of proposed GB pressure-volume model

Traditionally, experimental GB pressure-volume curves are considered as a parabola of volume either in the passive state or in the active state [72]. However, this mathematical model does not seem to be general. A more general mathematical model is put forward by Eq. (13), which is written as the following in terms of GB volume:

$$ {p}_e={p}_{e,\mathit{\exp}}(0)+{c}_0V+{c}_1{V}^{n_1}+{c}_2{e}^{c_3{V}^{n_2}} $$
(19)

where pe, exp(0) is the GB pressure at V=0 in an experiment and the model constants c0, c1, c2, c3, n1 and n2 will be optimized based on the experimental data in [72,73,74]. The first three terms in Eq. (19) represent the GB response to a small variation in volume, while the last term is the response to a large change in volume.

The experimental data and Eq.(19) are programmed in MATLAB® R2018b by employing its lsqnonlin function in terms of the trust-region-reflective optimization algorithm. The accuracy of the optimization is assessed by the error εp defined as

$$ {\varepsilon}_p=\frac{\sqrt{\sum_{i=1}^N{\left[{p}_e\left({V}_i\right)-{p}_{e,\mathit{\exp}}\left({V}_i\right)\right]}^2}}{\sum_{i=1}^N{p}_{e,\mathit{\exp}}\left({V}_i\right)}\times 100\% $$
(20)

where pe, exp(Vi) and pe(Vi) are the experimental GB pressure and the pressure predicted with Eq.(20) at an experimental point i, N is the number of total experimental points in a GB pressure-volume test.

The five model constants optimized, and the corresponding errors based on the experimental data in [72,73,74] are listed in Table 2. The comparison between measurements and predictions is demonstrated in Fig. 9. These errors are less than 4%, and the excellent agreement has been achieved between them. Thus, the mathematical model works well in fitting a variety of GB pressure-volume curves in passive and active states.

Table 2 Model constants in curve fitting of experimental GB pressure volumes for rabbit, opossum, guinea pig and human
Full size table
Fig. 9
figure 9

The comparison of GB pressure-volume curves predicted by using the mathematical model with the experimental data. a Pressure-volume curves of rabbit and guinea pig in the passive state [73]. b Pressure-volume curves of opossum in passive and active states [72]. c Pressure-volume curves of patients with acute cholecystitis in the passive state [74]

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Li, W. Modelling of viscoelasticity in pressure-volume curve of an intact gallbladder. Mech Soft Mater 2, 8 (2020). https://doi.org/10.1007/s42558-020-00023-6

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