Stress-shielding, growth and remodeling of pulmonary artery reinforced with copolymer scaffold and transposed into aortic position

Abstract

Ross operation, i.e., the use of autologous pulmonary artery to replace diseased aortic valve, has been recently at the center of a vivid debate regarding its unjust underuse in the surgical practice. Keystone of the procedure regards the use of an autologous biologically available graft which would preserve the anticoagulative and tissue homeostatic functions normally exerted by the native leaflets and would harmoniously integrate in the vascular system, allowing for progressive somatic growth of aortic structures. With this respect, recently, some of the authors have successfully pioneered a large animal model of transposition of pulmonary artery in systemic pressure load in order to reproduce the clinical scenario in which this procedure might be applied and allow for the development and testing of different devices or techniques to improve the pulmonary autograft (PA) performance, by testing a bioresorbable mesh for PA reinforcement. In the present work, to support and supplement the in vivo animal experimentation, a mathematical model is developed in order to simulate the biomechanical changes in pulmonary artery subjected to systemic pressure load and reinforced with a combination of resorbable and auxetic synthetic materials. The positive biological effects on vessel wall remodeling, the regional somatic growth phenomena and prevention of dilatative degeneration have been analyzed. The theoretical outcomes show that a virtuous biomechanical cooperation between biological and synthetic materials takes place, stress-shielding guiding the physiological arterialization of vessel walls, consequently determining the overall success of the autograft system.

Appendices

Appendix A: Details on the in vivo animal model

The experimental model of transposition of the pulmonary trunk as autograft in aortic position has been developed and performed under cardiopulmonary bypass in 20 growing lambs (Nappi et al. 2014, 2015b, c). Technical and anatomical issues imposed reimplantation of the PA in the descending aorta, with the pulmonary trunk being replaced by a homograft from another lamb of the same age and weight (Fig. 3a). The age of the animals at the moment of the implant was 2 months (8–10 weeks), and baseline mean weight was about 21$\pm$3 kg, allowing to observe the progression of the autograft diameter during the period of fastest growth. Morphometric and cardiovascular parameters were comparable preoperatively among animals. There was no difference in hemoglobin levels and ventricular function. The group of 20 lambs was divided into two subgroups: a control group (n$=$10), subjected to ordinary PA transposition, and a group of 10 animals in which the PA was reinforced with an external synthetic semiresorbable armored scaffold (prosthetic). All animal experiments have been performed in respect of the guidelines for animal care and handling, and the protocol was approved by the institutional animal care committee.

Semiresorbable copolymer scaffold The experimental design of the device consisted of an internal bioresorbable scaffold made with Polydioxanone (PDS), arranged in a frame of hexagonal cells, externally coupled with a nonresorbable layer of e-PTFE, having an auxetic behavior. The mesh structure and arrangement were specifically designed in order to constrain the excessive enlargement of the vascular graft diameter by also carrying wall mechanical stress while accommodating its natural longitudinal growth by embracing the root of the aorta. For this purpose, the unit cells of the PDS and e-PTFE plies have been, respectively, positioned as sketched in Fig. 2.

Surgical Model Lambs were premedicated with ketamine (25mg / kg via intramuscular injection), and anesthesia was guaranteed by the injection of sodium thiopentothal (6–8mg / kg) via the internal jugular. Animal received 100mg of lidocaine intravenously as prophylaxis against rhythm disturbance. After endotracheal intubation, ventilation was provided up to animal awakening and the anesthesia was maintained with inhalation isoflurane (1–2.5$\%$). The electrocardiogram was monitored, and chest was prepped and shaved. The heart was approached via left thoracotomy. After opening the pericardium, the right atrium was exposed for cannulation and the trunk of the pulmonary artery was dissected free from its right ventricular origin up to its bifurcation in the pulmonary arteries. The same was done for the descending thoracic aorta, and a region distal to the portion of choice for the PA transposition was cannulated. Approximately 8 cm of the descending thoracic aorta was left for clamp positioning and to perform the anastomosis with the pulmonary artery trunk under optimal conditions. Heparin (3mg/kg) was administered intravenously, and cardiopulmonary bypass was started between right atrium and descending aorta. The cerebral circulation of the animal was guaranteed on a beating heart. A 3-cm tract of pulmonary artery trunk was transposed into the descending aorta with an end-to-end anastomosis in 5-0 prolene. A fresh pulmonary artery homograft, explanted from animals killed on the same day for another experimental study, was inserted to reconstruct the right outflow tract, with a proximal and distal end-to-end anastomosis in 5-0 prolene, as in the Ross operation. Left thoracotomy was closed and aspiration drainage left in place. Before implantation, in the experimental group, the PA has been reinforced with PDS and e-PTFE meshes according to the study design. The resorbable mesh was prepared at the operative table (time 10$\pm$2 min). Meshes used in this study were cut into a rectangle measuring 20mm in height matching with the height of autograft and rolled out on a metallic candle and then reassured by a suture to create a cylinder with an internal diameter of 10mm (20mm in height in 10mm diameter directly adherent to the PA). The autograft was then inserted into the fibrillar cylinder and was anastomosed suturing both its margins and those of the prosthetic structure to the pulmonary autograft trunk. The mesh was oriented to allow maximal extensibility in the longitudinal direction and minimal transverse extensibility. All animals survived to the procedure and did not experience surgical complications. A case of PA initial rupture and thrombosis occurred at 6-month follow-up in the control group, without causing animal decease. Procedure did not pose particular technical challenges. At 6 months, the lambs weight was doubled ( 21$\pm$3kg at day 0 and 55$\pm$10kg at 6 months), suggesting a normal growth process. The animal model was mainly focused on the development of an effective and reproducible model of pulmonary autograft transposition into arterial system with the aim to study the behavior of the autograft and develop suitable strategies to prevent its future dilation, which represents one of the major drawback of this operation.

Appendix B:Thermodynamic forces associated with the growth model

The theoretical model presents a kinematic description of the growth process based on the deformation gradient decomposition into an elastic and a growth tensor, whose components are determined through both the evolution Eq. (2.2) and the introduction of an anisotropy exponent see (2.3). The evolution equation represents a kinematic relation for the body elementary growing volume; in particular, by introducing a growth volumetric source (and sink) term $r_g$, under the assumptions of no mass fluxes and constant density $\rho$ (which implies a pure volumetric growth), the mass conservation for the elastically incompressible body (i.e., $J_e=1$) can be expressed by:

$$\begin{aligned} \frac{d}{dt}\left( dm\right) =r_g dv\quad \text {or}\quad \frac{d\,J_g}{dt}=J_g\,\rho ^{-1}\,r_g \end{aligned}$$

(4.1)

Table 2 Synoptic table of data and employed parameters

In order to describe the growth behavior of the experimental animal models and reproduce the effects of the physiological growth on the stresses in vessel walls, a logistic rate has been assumed in the form of (2.35). This hypothesis implies that growth and the other model variables are uncoupled and thermodynamic conjugate forces have to be derived for the dissipative problem at hand. To make this, the constitutive laws involving the Piola-Kirchhoff stress in (2.10) and the remodeling Eq. (2.11) have been here derived from a dissipation principle, by following the approach by Lubarda and Hoger (2002) and Olsson and Klarbring (2008), in this way in turn determining the explicit face of the growth-associated forces. In particular, under the hypothesis of isothermal process, the balance of energy can be written by taking into account a contribution to the growth which represents a metabolic energy supply per unit mass, say $\varepsilon _g$, and a vector of driving forces $\mathbf {k}$ responsible of the remodeling-associated microstructural changes. In this way, one finally obtains:

$$\begin{aligned} \frac{d}{dt}&\int _\mathcal {V}\,\rho \left( \frac{1}{2}\mathbf {v}\cdot \mathbf {v} u\right) dv\nonumber \\ =&\int _\mathcal {V}\,\left( \rho \frac{d}{dt}\left( \frac{1}{2}\mathbf {v}\cdot \mathbf {v}\right) +\varvec{\sigma }:\mathbf {d}+\mathbf {k}\cdot \dot{\varvec{\gamma }}\right) dv\nonumber \\&+ \int _\mathcal {V}\,r_g\left( \frac{1}{2}\mathbf {v}\cdot \mathbf {v}+ u\right) dv+\int _\mathcal {V}\,\varepsilon _g\,r_g\,dv \end{aligned}$$

(4.2)

u and $\mathcal {V}$ being the internal energy per unit current mass and the current volume measure, respectively. Also, $\mathbf {v}$ is the velocity vector and $\mathbf {d}=sym(\dot{\mathbf {F}}\mathbf {F}^{-1})$ is the symmetrical velocity gradient, the other quantities $\varvec{\sigma }$ and $\dot{\varvec{\gamma }}$ defining the Cauchy stress tensor and the rate of the remodeling parameter vector, as also specified in the main text. By using (4.1), the balance of energy (4.2) reduces to:

$$\begin{aligned} \int _\mathcal {V}\,\rho \, \frac{du}{dt}\, dv=\int _\mathcal {V}\,\left( \varvec{\sigma }:\mathbf {d}+\mathbf {k}\cdot \dot{\varvec{\gamma }}\right) dv+\int _\mathcal {V}\,\varepsilon _g\,r_g\,dv \end{aligned}$$

(4.3)

The total internal dissipation per unit initial mass can be instead accounted by introducing two thermodynamic forces $f_g$ and $\varvec{f}_{\gamma }$, respectively, conjugated to the rates $r_g$ and $\dot{\varvec{\gamma }}$: In such a way, the rate of dissipation is written down:

$$\begin{aligned} \int _\mathcal {V}\, \theta \frac{ds}{dt}\rho \, dv=\int _\mathcal {V}\,\left( f_g\,\rho ^{-1}r_g+\varvec{f}_{\gamma }\cdot \dot{\varvec{\gamma }}\right) \,dv \end{aligned}$$

(4.4)

where s is the entropy per unit current mass and $\theta$ is the absolute temperature. The second law of thermodynamics requires the right side of (4.4) to be nonnegative. By combining the energy Eq. (4.3) and the entropy Eq. (4.4), the free energy per unit volume $\psi =\rho \left( u-\theta \,s\right)$ can be thus obtained as a function upon the elastic deformation $\mathbf {F}_e$ and the remodeling parameters $\varvec{\gamma }$, as also established in (2.9). At the end, it results:

$$\begin{aligned} \frac{d}{dt}&\int _\mathcal {V}\, \psi \, dv\nonumber \\&=\int _\mathcal {V}\,\left( \varvec{\sigma }:\mathbf {d}+\left( \mathbf {k}-\varvec{f}_{\gamma }\right) \cdot \dot{\varvec{\gamma }}+\left( \rho \varepsilon _g-f_g\right) \rho ^{-1}r_g \right) dv \end{aligned}$$

(4.5)

$$\begin{aligned} \text {or}\nonumber \\ \frac{d}{dt}&\int _{\mathcal {V}^0}\, J_g \psi \, dV^0\nonumber \\&=\int _{\mathcal {V}^0}\,\left( \mathbf {P}:\dot{\mathbf {F}}+J_g\left( \mathbf {k}-\varvec{f}_{\gamma }\right) \cdot \dot{\varvec{\gamma }}+\left( \rho \varepsilon _g-f_g\right) \dot{J}_g \right) dV^0 \end{aligned}$$

(4.6)

${\mathcal {V}^0}$ denoting the referential volume. By means of the localization theorem and exploiting the deformation multiplicative decomposition $\mathbf {F}=\mathbf {F}_e\mathbf {F}_g$, one has:

$$\begin{aligned}&J_g\frac{\partial \psi }{\partial \mathbf {F}_e}:\dot{\mathbf {F}}_e+J_g\frac{\partial \psi }{\partial \varvec{\gamma }}\cdot \dot{\varvec{\gamma }}+\psi \dot{J}_g\nonumber \\&\quad =\mathbf {P}\mathbf {F}_g^T:\dot{\mathbf {F}}_e+\mathbf {F}_e^T\mathbf {P}:\dot{\mathbf {F}}_g \nonumber \\&\qquad +J_g\left( \mathbf {k}-\varvec{f}_{\gamma }\right) \cdot \dot{\varvec{\gamma }}+\left( \rho \varepsilon _g-f_g\right) \dot{J}_g \end{aligned}$$

(4.7)

A direct comparison of the terms at both sides of (4.7) leads to:

$$\begin{aligned}&\mathbf {P}=J_g\frac{\partial \psi }{\partial \mathbf {F}_e}\mathbf {F}_g^{-T}\nonumber \\&\varvec{f}_{\gamma }=\mathbf {k}-\frac{\partial \psi }{\partial \varvec{\gamma }}\nonumber \\&f_g=\rho \varepsilon _g+\Sigma \end{aligned}$$

(4.8)

from which it follows that the growth-conjugate force is the result of the interplay of metabolic (e.g., biochemical) and mechanical factors, with $\Sigma =\varvec{\Sigma }:\mathbf {I}$ being the trace of the Eshelby-like stress tensor related to the change in domain variations induced by the volumetric growth $\varvec{\Sigma }=\mathbf {F}_e^T\partial \psi /\partial \mathbf {F}_e-\psi \mathbf {I}$, as obtained for example by Ambrosi and Guillou 2007 and Olsson and Klarbring 2008. As a consequence, the dissipation inequality, derivable by imposing the second member of (4.4) to be nonnegative, can be split into two independent contributions:

$$\begin{aligned}&\int _{\mathcal {V}^0}\left( \mathbf {K}-J_g\frac{\partial \psi }{\partial \varvec{\gamma }}\right) \cdot \dot{\varvec{\gamma }}\,dV^0 \,\ge 0\end{aligned}$$

(4.9)

$$\begin{aligned}&\int _{\mathcal {V}^0}\left( \rho \varepsilon _g+\Sigma \right) \dot{J}_g\,dV^0\,\ge 0 \end{aligned}$$

(4.10)

The first one leads to the form of the (2.11) through the assumption $\dot{\varvec{\gamma }}=c_{\gamma }\left( \mathbf {K}-J_g{\partial \psi }/{\partial \varvec{\gamma }}\right) , c_{\gamma }>0$ and $\mathbf {K}=J_g\mathbf {k}$ being the referential force which drives the remodeling process. The second inequality is a pressure–volume relationship (vanishing in the pure remodeling case, $\dot{J}_g = 0$). Its validity implies that the growth case $\dot{J}_g>0$ is characterized by the presence of a pressure responsible of the domain expansion and by an adequate amount of metabolic energy, convertible into mass growth, that is assumed to be indefinitely available; vice versa, the resorption case $\dot{J}_g < 0$ can be associated with the lack of energy supply and to the presence of stresses contracting the volume domain.