A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis

Journal of Biomechanics xxx (2017) xxx–xxx

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A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis Tianqi Wang a, Fuyou Liang a,b,⇑, Zunqiang Zhou c, Lu Shi d a

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai Jiao Tong University, Shanghai 200240, China c Department of Surgery, Shanghai Jiao Tong University Affiliated Sixth People’s Hospital, Shanghai 200233, China d Institute of Underwater Technology, Shanghai Jiao Tong University, Shanghai 200231, China b

a r t i c l e

i n f o

Article history: Accepted 25 September 2017 Available online xxxx Keywords: Computational model Portal hypertension Cirrhosis Hepatic venous pressure gradient (HVPG) Portal pressure gradient (PPG)

a b s t r a c t Measurement of hepatic venous pressure gradient (HVPG) is currently widely adopted to provide an estimate of portal pressure gradient (PPG) in the diagnosis and treatment of portal hypertension associated with liver cirrhosis. Despite the well-documented clinical utility of HVPG, it remains poorly understood how the relationship between HVPG and PPG is affected by factors involved in the pathogenesis and progression of cirrhosis. In the study, a computational model of the hepatic circulation calibrated to in vivo data was developed to simulate the procedure of HVPG measurement and quantitatively investigate the error of HVPG relative to PPG under various pathophysiological conditions. Obtained results confirmed the clinical consensus that HVPG is applicable to the assessment of portal hypertension caused by increased vascular resistance located primarily at the sinusoidal and postsinusoidal sites rather than at the presinusoidal site. On the other hand, our study demonstrated that the accuracy of HVPG measurement was influenced by many factors related to hepatic hemodynamics even in the case of sinusoidal portal hypertension. For instance, varying presinusoidal portal vascular resistance significantly altered the difference between HVPG and PPG, while an enhancement in portosystemic collateral flow tended to improve the accuracy of HVPG measurement. Moreover, it was found that presinusoidal and postsinusoidal vascular resistances interfered with each other with respect to their influence on HVPG measurement. These findings suggest that one should take into account patient-specific pathological conditions in order to achieve a better understanding and utilization of HVPG in the clinical practice. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Portal hypertension (PH) is a common hemodynamic manifestation of chronic liver diseases, especially those associated with liver cirrhosis (Ge and Runyon, 2016). Accordingly, portal pressure has been widely demonstrated as an important parameter for evaluating the severity of cirrhosis and predicting the risk of complications (Berzigotti et al., 2013). A direct measurement of portal pressure is, however, clinically impractical due to its invasive nature and complex operation. As a less invasive method, the measurement of hepatic venous pressure gradient (HVPG) is nowadays widely employed to provide an estimate of portal pressure gradient (PPG, which is the difference between the portal

⇑ Corresponding author at: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, China. E-mail address: [email protected] (F. Liang).

pressure and the pressure in the hepatic vein or inferior vena cava (Huang et al., 2017)) in the management of cirrhosis. HVPG is, by definition, the difference between the wedged hepatic venous pressure (WHVP) (measured with one of the hepatic veins being occluded by a balloon catheter) and the free hepatic venous pressure (FHVP) (Kumar et al., 2008). A meta-analysis demonstrated that HVPG compared reasonably with PPG in patients with different degrees of pH (Thalheimer et al., 2005). In the past decades, HVPG has been extensively demonstrated to be a robust surrogate marker for clinical end points, and HVPG measurement has been increasingly applied to various aspects of clinical hepatology, including assessment of pH, risk stratification, monitoring of the efficacy of medical treatment and prediction of complications (Berzigotti et al., 2013; Bosch et al., 2009; Merkel and Montagnese, 2011). Despite the well-documented clinical utility of HVPG, the application of HVPG measurement is confined to specific classes of liver disease where PH is caused by increased vascular resistances

https://doi.org/10.1016/j.jbiomech.2017.09.023 0021-9290/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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T. Wang et al. / Journal of Biomechanics xxx (2017) xxx–xxx

located predominantly in the sinusoidal and postsinusoidal portions, such as in alcoholic liver disease and in hepatitis C- or hepatitis B-related cirrhosis (Wongcharatrawee and Groszmann, 2000; Berzigotti et al., 2013). For presinusoidal PH caused by portal vein thrombosis, nodular regenerative hyperplasia or early stage of primary biliary cirrhosis, HVPG was found to correlate poorly with PPG and hence cannot be used to assess portal pressure (Berzigotti et al., 2013). Despite the conceptual clarification on the pathological conditions under which HVPG is applicable, the difference between HVPG and PPG has been found to differ considerably even among patients with the same pathogenesis (Deplano et al., 1999; Perello et al., 1999). The phenomenon may be ascribed to the complex vascular changes in the context of chronic liver disease. It has been pointed out that the increase in portal vascular resistance often involves both the presinusoidal and sinusoidal areas in chronic hepatitis or alcoholic liver injury and the location and severity of pathological vascular alterations vary dynamically with the progression of liver disease (Orrego et al., 1981; PomierLayrargues et al., 1985). Due to the lack of practical means for patient-specifically measuring the intrahepatic distribution of vascular resistance and associated hemodynamic variables, it is challenging for clinical studies to elucidate the respective influence of different pathological factors on the accuracy of HVPG as an estimate of PPG. In comparison with in vivo measurements which are usually restricted by accessible regions, detectable bio-information or accuracy of measurement, computational modeling permits a quantitative deterministic study on hepatic hemodynamics on the basis of a systemic integration of all major vascular components of the hepatic circulation, thereby providing a more practical means for investigating the reliability of HVPG under various pathological conditions. Computational models have been employed to study various hepatic hemodynamic phenomena in the literature (Chu and Reddy, 1992; Mynard, 2011; Rypins et al., 1987). In recent years, models were increasingly applied to address issues raised in clinical practice. For instance, a lumped-parameter model that includes various generations of hepatic vessels was developed to study liver hemodynamics during hypothermic machine perfusion (Debbaut et al., 2011; Debbaut, 2013). A simplified lumped-parameter model of the hepatic circulation was employed to investigate hemodynamic changes induced by the transjugular intrahepatic portosystemic shunt (Ho et al., 2013). More recently, lumped-parameter models of the hepatic circulation coupled with the cardiovascular system were constructed to simulate intrahepatic hemodynamic changes associated with partial liver ablation surgery (Audebert et al., 2017a, 2017b). Despite the wide application of computational models to various problems related to hepatic hemodynamics, no models have been utilized to investigate HVPG. The purpose of the present study was therefore to construct a computational model of the hepatic circulation and apply it to quantitatively investigate the sensitivity of HVPG to various factors involved in chronic liver disease. The model was firstly calibrated to in vivo data acquired from patients with liver cirrhosis to simulate the baseline hemodynamic features in cirrhosis. Subsequently, a series of numerical experiments were carried out to quantify the changes of HVPG relative to PPG in response to variations in major factors that affect hepatic hemodynamics.

2. Methods Hemodynamic changes associated with chronic liver disease are characterized by not only the elevation of portal pressure but also the redistribution of blood flow over the body (Mcavoy et al., 2016). Such hemodynamic characteristics cannot be fully

accounted for by a standalone local model of the hepatic circulation whose boundary conditions are usually fixed rather than interactive with systemic hemodynamics. In particular, the occlusion of a hepatic vein in HVPG measurement will induce hemodynamic changes in both the hepatic circulation and the adjacent vascular systems. Therefore, modeling the hepatic circulation together with other parts of the cardiovascular system would be expected to yield a more reasonable tool for simulating HVPG measurement and addressing its sensitivity to various cardiovascular properties. Following the concept, the hepatic circulation was represented by a lumped-parameter (0D) model coupled with a 0D-1D multi-scale model of the cardiovascular system in the present study (see Fig. 1). In the contents that follow, emphasis is placed on describing the modeling of the hepatic circulation. For details on the modeling of the systemic circulation, readers are referred to Liang et al. (2009a, 2009b). 2.1. Lumped-parameter modeling of the hepatic circulation Three major vascular systems were included in the model of the hepatic circulation, namely, the hepatic arterial system, portal venous system and hepatic venous system, which were respectively ramified into eight subsystems according to the classical Couinaud nomenclature where the liver is divided into eight segments (Bismuth, 1982; Lafortune et al., 1991; Mahadevan, 2014) (see Fig. 1). Vessels in each vascular subsystem were further compartmentalized into several vascular compartments to account for hemodynamic variations across the liver. Each vascular compartment was represented by a set of lumped parameters (i.e., resistance, compliance and inertance) that represent viscous resistance, elasticity of vascular wall and inertia of blood, respectively (Alastruey et al., 2008; Liang and Liu, 2006; Olufsen and Nadim, 2004). It is noted that although each liver segment owns its individual blood supply, the interfaces between different segments are not completely insulative because anastomoses (named as intersinusoidal communications) always exist at the microvascular level (Wongcharatrawee and Groszmann, 2000). To incorporate the feature, resistances were created between the sinusoidal compartments of adjacent liver segments (see Fig. 1 for details on intersinusoidal communications). Moreover, portosystemic collaterals are often found in patients with portal hypertension (Kim et al., 2000), which were herein represented by a resistance linking the portal vein and the inferior vena cava. The governing equations of each vascular compartment were derived from the principle of mass and momentum conservation (Alastruey et al., 2008). All governing equations of the hepatic circulation formed an ordinary differential equation system, which was solved together with the governing equations of other cardiovascular portions with the numerical methods proposed in Liang et al. (2009a). 2.2. Parameter assignment 2.2.1. Baseline model parameters for a normal liver The resistances, compliances and inertances of the first two generations of the hepatic artery (HA), portal vein (PV) and hepatic vein (HV) were estimated based on the measured geometrical data (Debbaut et al., 2011). Resistances in the microvascular compartments were estimated to enable the model to simulate the general hemodynamic characteristics in a normal liver, such as pressure variations from HA/PV to HV and flow distributions among the eight liver segments. Herein, the blood flow rate to each liver segment was assumed to be proportional to the anatomical volume of the segment reported in Mise et al. (2014). The total vascular compliance in a normal liver of an adult was estimated to be about 55

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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T. Wang et al. / Journal of Biomechanics xxx (2017) xxx–xxx

0-D model

1-D model Arterioles/ capillaries/ venules/ veins

Superior vena cava

Other arteries Pulmonary circulation Hepatic artery

Inferior vena cava

Right atrium

Right ventricle

Left atrium

Left ventricle

Aorta Splanchnic arteries

Portosystemic collaterals

Left PV

Right PV

Splanchnic vessels (0-D model)

Portal vein (PV)

Intersinusoidal communications Couinaud nomenclature Sinusoids

Portal capillaries

Portal venules

Portal veins

Left HV Left HA

Middle HV

Inferior vena cava

Right HA

Hepatic artery (HA)

Right HV

Hepatic veins

Hepatic venules

Hepatic capillaries

Hepatic arterioles

Hepatic arteries

Fig. 1. Schematic description of the computational model of the hepatic circulation coupled with the cardiovascular system. The hepatic circulation is divided into eight components (according to the classical Couinaud nomenclature), with each consisting of a portal venous subsystem and a hepatic arterial subsystem which merge at hepatic sinusoids. Hepatic blood flows distal to sinusoids are directed to the inferior vena cava through hepatic venules and three hepatic veins. Intersinusoidal communications are represented by resistances introduced between sinusoidal compartments. Moreover, the portal vein and the inferior vena cava are connected with a resistance to represent portosystemic venous collaterals usually found in patients with portal hypertension. The 1-D (one-dimensional) model represents the arterial tree constituted by fifty-five large arteries responsible for distributing cardiac output to major organs/tissues in the human body. For more details on the modeling of the systemic circulation, please refer to Liang et al. (2009b).

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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ml mmHg
1 according to the data reported in Kjekshus et al. (1997), which is close to the value used in a previous study (Mynard and Smolich, 2015). The allocation of compliance among the HA, PV and HV vascular systems was assumed to be 1%:27%:72%, with the microvascular compliance in each vascular system being distributed to the eight liver segments in proportion to the segment volumes. The total inertances of the HA, PV and HV vascular systems were estimated respectively so that their values are comparable in order of magnitude and proportional relation to those reported in a previous model study (Ho et al., 2013). The longitudinal distribution of the microvascular inertance along each vascular system was empirically determined based on the data reported in Liang et al. (2009b). It is noted that identical microvascular inertances were assigned to the eight liver segments for simple purpose. 2.2.2. Baseline model parameters for a cirrhotic liver Increased intrahepatic vascular resistance represents a major causative factor for portal hypertension in cirrhosis, which may occur at the presinusoidal site, sinusoids, postsinusoidal site, or a mix of them depending on the underlying pathogenesis (Groszmann and Atterbury, 1982; Sanyal et al., 2008). In this study, two types of portal hypertension (PH) were modeled, namely, presinusoidal PH (e.g., in the early stage of primary biliary cirrhosis) and sinusoidal PH (e.g., in alcoholic cirrhosis) (Wongcharatrawee and Groszmann, 2000). The former was modeled by increasing the resistances of the portal venules and capillaries (presinusoidal portal vessels) by 400% relative to the

reference values assigned to a normal liver, while the latter by increasing the resistances of intersinusoidal communications (sinusoidal vessels) and hepatic venules (postsinusoidal vessels) by 300% (Roskams et al., 2003). In each case, vascular resistances in all the liver segments were elevated to the same extent. For the hepatic microvessels whose resistances were elevated to simulate PH, their compliances were empirically reduced by 25%, while with the inertances being maintained at the default values. Moreover, model parameters used in the model of the systemic circulation were adjusted as well to simulate the hyperdynamic blood flow conditions associated with PH, which will be detailed in the section that follows. 2.3. Model calibration The values of model parameters must be tuned so as to yield models capable of simulating the in vivo data measured in healthy subjects with normal portal pressure and cirrhotic patients with PH. In vivo data available in Mcavoy et al. (2016) mainly included blood flow rates in the portal vein and large arteries, other hemodynamic data were either estimated or derived from clinical data reported elsewhere (Hadaegh et al., 2012; Lebrec et al., 1997; Moreno et al., 1967; Tandon et al., 2010). Heart rate was fixed at 72 beats/min and 88 beats/min for healthy subjects and cirrhotic patients, respectively (Mcavoy et al., 2016). Clinical data-based parameter tuning was performed with a parameter optimization algorithm proposed in Liang et al. (2015) (see Appendix A for more details). The resulting model parameter values are summarized in

Table 1 Comparisons of model-simulated hemodynamic variables with in vivo measurements ([1] Mcavoy et al., 2016; [2] Hadaegh et al., 2012; [3] Tandon et al., 2010; [4] Lebrec et al., 1997; [5] Moreno et al., 1967) under the normal condition and the cirrhotic condition (with sinusoidal portal hypertension). For hemodynamic variables whose ranges of distribution rather than the population-averaged values are given in clinical studies, mid values were estimated and used as the references in the parameter tuning procedure. For instance, the reference values of portal pressure and vena cava pressure were set to be 10mmHg vs 20mmHg, 6mmHg vs 7.5mmHg for the normal and cirrhotic conditions, respectively. The reference value of systemic arterial pressure was fixed at 89mmHg for both conditions. Note that all the blood pressures reported herein refer to the mean pressure over a cardiac cycle. The superscript ‘*’ denotes the estimated reference values for flow rates not derivable from relevant clinical studies. Hemodynamic variables

Vessel

In vivo measurement (Healthy)

Simulation (Healthy)

In vivo measurement (Cirrhotic)

Simulation (Cirrhotic)

Flow rate (ml/s)

L. & R. carotid arteries L. & R. renal arteries Super mesenteric artery Hepatic artery Portal vein Proximal abdominal aorta L. & R. subclavian arteries Intercoastal arteries Splenic artery Gastric artery Inferior mesenteric artery

10.80 ± 2.13[1] 10.40 ± 3.65[1] 3.65 ± 2.15[1] 4.18 ± 2.47[1] 12.93 ± 5.82[1] 55.12 ± 13.28[1]

10.78 10.38 3.65 4.17 12.93 55.02

9.22 ± 4.10[1] 5.03 ± 1.87[1] 7.48 ± 4.20[1] 6.72 ± 4.87[1] 15.78 ± 7.20[1] 55.38 ± 12.98[1]

9.20 5.03 7.48 6.68 15.77 58.30

12.61* 8.00* 2.32* 4.07* 2.90*

12.59 7.99 2.31 4.06 2.90

14.35* 9.00* 3.00* 4.70* 3.60*

14.34 8.99 3.00 4.69 3.60

Systemic artery Portal vein Inferior vena cava

91 ± 11[2] 7–12[4] 2–7[4]

88.24 9.86 6.06

84–95[3] 8–30[4] 6.6–11.7[5]

89.02 20.30 7.44

Pressure (mmHg)

Table 2 Model-simulated flow distributions under the normal condition and the cirrhotic condition (with sinusoidal portal hypertension) among the eight liver segments compared with measured volume proportions (Mise et al., 2014). The absolute flow rate through each liver segment is the sum of the flow rates through the portal venous system and hepatic arterial system. The superscript ‘*’ denotes the adjusted flow rate proportions since the sum of the measured volume proportions is not strictly equal to 100%. Segment number

Volume proportion (%)

Flow rate (Normal) (ml/s)

Flow rate (Cirrhotic) (ml/s)

Flow rate proportion (%)

I II III IV V VI VII VIII

4.0 7.9 9.5 13.6 12.6 7.9 16.8 26.1

0.6882 1.4919 1.7664 2.3304 2.1517 1.3489 2.8683 4.4563

0.9026 1.9567 2.3165 3.0576 2.8255 1.7718 3.7681 5.8527

4.0 8.7* 10.3* 13.6 12.6 7.9 16.8 26.1

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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2.4.1. Simulation of HVPG measurement The measurement of HVPG involves the use of a balloon catheter to occlude the right hepatic vein according to the clinical guidelines (Groszmann and Wongcharatrawee, 2004), which was herein simulated by increasing the hepatic venous resistance to an extremely large value. Simulation was firstly run for several cardiac cycles with the intact model to obtain a converged solution of the free hepatic venous pressure (FHVP) and portal pressure (PP). Subsequently, the right hepatic vein was occluded and the simulation was continued for about 60 s to obtain a stable wedged hepatic venous pressure (WHVP). HVPG was calculated as the difference between WHVP and FHVP, while PPG was taken to be PP subtracted by FHVP. 2.4.2. Parameter sensitivity analysis The model calibrated to the cirrhotic patients with sinusoidal PH was used as the reference model for parameter sensitivity analysis. The model parameters selected for sensitivity analysis were those representing the clinically-demonstrated factors involved in the pathogenesis and progression of cirrhosis. For instance, in the case of sinusoidal PH, apart from the increase in sinusoidal and postsinusoidal vascular resistances, the presinusoidal portal vascular resistance may deviate from the normal state and vary among patients (Wongcharatrawee and Groszmann, 2000). The resistance of hepatic arterial system is often changed as a consequence of the hepatic arterial buffer response (HABR) to altered flow rate through the portal system in cirrhosis (Abshagen et al., 2015). In addition, the amount of portosystemic collateral flow (resulting from portosystemic collateral vessels or shunt intervention) and blood perfusion to the portal system from the splanchnic organs may change with the progression of cirrhosis (Debbaut, 2013). As a consequence, six model parameters corresponding to the aforementioned factors were selected, namely, postsinusoidal vascular resistance (Rpts), intersinusoidal communication resistance (Risc), presinusoidal portal vascular resistance (Rpxs), hepatic arteriolar resistance (Rhar), portosystemic collateral resistance (Rpsc) and total vascular resistance of splanchnic organs (Rspl). Each of these parameters was varied respectively from
50% to +50% (in an interval of 10%) relative to its reference value and correspondingly a set of simulations were carried out to quantify its impact on the error of HVPG relative to PPG. For model parameters which may vary in a larger range according to the pathophysiology of cirrhosis, the ranges of parameter variation were widened. Herein, the upper bound of Rpxs variation was increased up to 400% to represent the enhanced pathological change of presinusoidal portal vessels, and the lower bound of Rpsc variation was reduced to
85% to account for the effect of portosystemic shunt intervention. Moreover, parameter sensitivity analyses that involve paired parameter variations were performed as well. Here, Rpxs was varied in combination with other four parameters (i.e., Rspl, Rhar, Rpsc and Rpts).

EHVPG ¼

HVPG
PPG  100%: PPG

ð1Þ

Here, the smaller the magnitude of EHVPG is, the more accurate is HVPG as an estimate of PPG.

3. Results 3.1. Simulated intrahepatic pressure variation under normal and cirrhotic conditions Simulations were performed using the models calibrated to the normal condition and the cirrhotic condition. For the cirrhotic condition, both presinusoidal PH and sinusoidal PH were simulated with the methods described in ‘Section 2.2.2’. Fig. 2 shows that the simulated portal pressure and sinusoidal pressure under the normal condition are 9.9 mmHg and 8.1 mmHg respectively, both falling in the normal ranges reported in Eipel et al. (2010) and Lebrec et al. (1997). Under the two types of cirrhotic condition, despite the comparable increase in portal pressure, the simulated sinusoidal pressure differed remarkably, which was close to the normal value (10.2 mmHg vs 8.1 mmHg) in the case of presinusoidal PH, while significantly higher (18.2 mmHg vs 8.1 mmHg) in the case of sinusoidal PH. Accordingly, the simulated hepatic arteriolar pressure with sinusoidal PH was considerably higher than the normal value.

a

100

Normal condition Presinusoidal PH Sinusoidal PH

80

Pressure (mmHg)

2.4. Numerical experiments

The error (EHVPG) of HVPG relative to PPG was evaluated by taking the percentage difference of HVPG and PPG normalized by PPG.

60 40 20 0 Hepatic arteries Arterioles

Sinusoids

Hepatic veins

IVC

Position

b

24

Normal condition Presinusoidal PH Sinusoidal PH

21

Pressure (mmHg)

Appendix B. The values of other parameters were maintained intact as reported in a previous study (Liang et al., 2009b). Table 1 shows that the model simulations agree reasonably with the in vivo measurements. From Table 1, the characteristics of hyperdynamic circulation in cirrhosis are evident, such as increased blood flows through the splanchnic organs and the hepatic arterial system, reduced blood flow to the kidneys and preserved brain perfusion. Table 2 shows that the simulated flow rates through the eight liver segments are proportional to the segment volumes under both the normal condition and the cirrhotic condition.

18 15 12 9 6 Portal veins

Venules

Sinusoids

Hepatic veins

IVC

Position Fig. 2. Simulated intrahepatic pressure variations along (a) the hepatic arterial system and (b) the portal venous system (in liver segment IV and middle hepatic vein) under various conditions. The simulated portal pressures for the presinusoidal portal hypertension (PH) condition and the sinusoidal portal hypertension (PH) condition are both much higher than that simulated for the normal condition, however, significantly increased pressure at the sinusoidal level is predicted only in the case of sinusoidal PH.

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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3.2. Results of the simulation of HVPG measurement The procedure of HVPG measurement was simulated for the normal, presinusoidal PH and sinusoidal PH conditions, respectively. Fig. 3(a)–(c) shows the time histories of the changes in hepatic venous pressure (HVP) and PP during the procedure. HVP was rapidly elevated following the occlusion of the hepatic vein under all conditions, resulting in a WHVP higher than the FHVP. However, the difference between WHVP and FHVP differed significantly among the three cases. An elevation of PP was also detected, but much small in magnitude compared with the elevation of HVP. HVPG and PPG calculated based on the simulated results are compared in Fig. 3(d). HVPG was closest to PPG with sinusoidal PH, followed by the case of normal condition. However, HVPG significantly underestimated PPG under the presinusoidal PH condition. 3.3. Results of parameter sensitivity analysis Fig. 4 shows the simulated changes of HVPG, PPG and their relative error (EHVPG) with the variations in six model parameters. EHVPG was sensitive most evidently to Rpxs, secondarily to Rspl, Rhar and Risc. Relatively, varying Rpts and Rpsc had mild influence on EHVPG. When Rpsc was reduced lower than 50% of the reference value, EHVPG continued to decrease (see Fig. 4(e)). As Rpxs was increased by over 50%, EHVPG kept decreasing, converting from positive to negative when Rpxs was increased by over 100% (see Fig. 4 (f)). Fig. 5 illustrates the contour maps of EHVPG when Rpxs was varied in combination with other model parameters. The nonlinear behavior of EHVPG was most remarkable for the pair of Rpxs and Rpts, where increasing Rpts attenuated the sensitivity of EHVPG to Rpxs, whereas increasing Rpxs augmented the sensitivity of EHVPG to Rpts. The sensitivity of EHVPG to Rpxs was influenced by varying

a

14

Rspl and Rhar as well, although the degree was low compared with the case of varying Rpts. Relatively, varying Rpsc had little influence on the sensitivity of EHVPG to Rpxs. 4. Discussion A computational model of the hepatic circulation coupled with the cardiovascular system was constructed to provide quantitative insights into the reliability of HVPG as an estimate of PPG. Modelbased simulations of the procedure of HVPG measurement showed that occluding the right hepatic vein led to different degrees of increase in HVP and different magnitudes of EHVPG depending on the pathological conditions of the hepatic circulation (see Fig. 3). For instance, the magnitude of EHVPG was smallest with sinusoidal PH, whereas extremely large with presinusoidal PH (see Fig. 3(d)). From the clinical point of view, a HVPG lower than 5 mmHg indicates normal portal pressure (Kumar et al., 2008), and the threshold value of HVPG indicative of clinically significant PH is 10–12 mmHg (Merkel and Montagnese, 2011). The simulated HVPG (= 14.4 mmHg) with sinusoidal PH properly reflected the state of portal hypertension, whereas, the simulated HVPG (= 4.1 mmHg) with presinusoidal PH was lower than 5 mmHg although the real PPG (= 13.0 mmHg) was higher than the threshold value and hence will result in a false ‘normal portal pressure’ diagnosis. These results indicate that HVPG can be used to diagnose PH in the case of sinusoidal rather than presinusoidal PH, which confirms the wellestablished clinical consensus regarding the conditions under which HVPG is applicable (Wongcharatrawee and Groszmann, 2000). Further numerical experiments performed based on the sinusoidal PH model revealed that EHVPG was subject to the influence of various factors, which implies the variability of the accuracy of HVPG measurement among patients with different pathological changes in the hepatic circulation.

b

Normal condition

27 Presinusoidal PH

24

12

Pressure (mmHg)

Pressure (mmHg)

21 PP

10

WHVP

8 FHVP 6 4

Free

Occluded

PP

18 15

WHVP

12 FHVP

9 6 3

2

Occluded

Free

0 0

30

60

90

120

0

30

Time (s)

c

28

d

Sinusoidal PH

WHVP

PP

16 12 8

FHVP

4 0

Pressure (mmHg)

Pressure (mmHg)

21

PPG HVPG

18

24 20

60

90

Time (s)

15

EHVPG= 12.5%

EHVPG= -68.4%

12 9 6

EHVPG= -18.3%

3

-4

Occluded

Free

0

-8 0

30

60 Time (s)

90

Norm

al co

nditio

n Pres

inuso

idal P

H Sinu

soida

l PH

Fig. 3. Simulated time histories of the changes in hepatic venous pressure and portal pressure during HVPG measurement: (a) normal condition, (b) presinusoidal portal hypertension (PH) and (c) sinusoidal portal hypertension (PH). The corresponding comparisons of HVPG and PPG are illustrated in panel (d).

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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T. Wang et al. / Journal of Biomechanics xxx (2017) xxx–xxx

20

b

16

Splanchnic vascular resistance (Rspl)

18

HVPG PPG EHVPG

14

12

10 12 8 10 -60

-40

-20

0

20

40

16

14

12

13

-40

Variation in Rspl (%)

c

16

14

15

12 -60

60

HVPG PPG EHVPG

EHVPG (%)

16

Pressure (mmHg)

14

16

Hepatic arteriolar resistance (Rhar)

17

18

EHVPG (%)

Pressure (mmHg)

a

-20

0

20

10 60

40

Variation in Rhar (%)

d

16

Intersinusoidal communication resistance (Risc)

20

Postsinusoidal vascular resistance (Rpts)

16

13

12 -60

-40

-20

0

20

10

40

Pressure (mmHg)

HVPG PPG EHVPG

14 16 12

14 12

10 HVPG PPG EHVPG

10

-60

-40

Portosystemic collateral resistance (Rpsc)

f

14

20

12

12

11

10

10 HVPG PPG EHVPG

8 -60 -50 -40

-20

0

20

40

60

40

28 21

9

6 -80

20

60

Variation in Rpsc (%)

HVPG PPG EHVPG

18

14 7

16

0 -7

14

EHVPG (%)

14

8

0

Presinusoidal portal vascular resistance (Rpxs)

13

EHVPG (%)

Pressure (mmHg)

16

-100

-20

Variation in Rpts (%)

Pressure (mmHg)

18

8

8

8 60

Variation in Risc (%)

e

EHVPG (%)

12

14

EHVPG (%)

Pressure (mmHg)

18

14

15

-14 -21

12

-28 -100

0

50 100

200

300

400

Variation in Rpxs (%)

Fig. 4. Results of single parameter sensitivity analysis (changes in HVPG, PPG and EHVPG plotted against variations in model parameters): (a) variation in splanchnic vascular resistance (Rspl); (b) variation in hepatic arteriolar resistance (Rhar); (c) variation in intersinusoidal communication resistance (Risc); (d) variation in postsinusoidal vascular resistance (Rpts); (e) variation in portosystemic collateral resistance (Rpsc); and (f) variation in presinusoidal portal vascular resistance (Rpxs). The first four model parameters are varied between
50% and 50% relative to the reference values, whereas the ranges of variations in the last two parameters are extended to
85% and 400% respectively to account for wider pathophysiological conditions, as highlighted by the gray shadows.

While PH can be phenomenologically categorized into two major families (i.e., sinusoidal PH and presinusoidal PH), pathological factors underlying PH are complex (Berzigotti et al., 2013). The increase in intrahepatic vascular resistance usually involves various sites depending on the underlying pathogenesis, and the pattern and severity of vascular changes may change following the progression of liver disease (Wongcharatrawee and Groszmann, 2000). Our study demonstrated that EHVPG could be altered by varying many factors even in the case of pH dominated by increased vascular resistances in the sinusoidal and postsinusoidal portions. For instance, it was found that increasing Rpxs reduced the error of HVPG, but the trend was reversed when Rpxs was increased by over 100% of the normal value. In another word, HVPG overestimates PPG with low Rpxs, whereas underestimates PPG with high Rpxs. The results may partly explain the clinical observation that

EHVPG appeared both positive and negative when investigated in patients with sinusoidal PH whose presinusoidal vessels might be at different pathological states (Thalheimer et al., 2005). In light of the general vascular changes associated with cirrhosis, the decreases in Rspl and Rhar (related to hyperdynamic circulation) (Sanyal et al., 2008) seem to play counteracting roles in influencing EHVPG, whereas the impairment in intersinusoidal communication flow (represented by increasing Risc) (van Leeuwen et al., 1990) tends to enlarge EHVPG. Moreover, decreasing Rpsc was found to reduce EHVPG, which implies that the validity of HVPG measurement holds despite enhanced portosystemic collateral flow resulting from severe cirrhosis or portosystemic shunt intervention. The results presented in Fig. 5 further demonstrate the mutual effects on EHVPG of different pairs of model parameters. For instance, varying Rpts only had moderate influence on EHVPG

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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T. Wang et al. / Journal of Biomechanics xxx (2017) xxx–xxx

a

40% 7.5

-4.5

2.0

20% -0.50

-8.5

-13

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0 -17

12

-25

-20% 16

-40%

Variation in Rhar

Variation in Rspl

b

3.5

40%

-26

20% 10 6.0

0 14

-22

-20% 18

-10

-40%

-29

-2.0 -20%

60%

140%

220%

300%

380%

-20%

60%

Variation in Rpxs

-6.0

220%

300%

380%

Variation in Rpxs

c

d

5.5

40%

40%

1.5

-16

20% 8.5

-12

-20

0 13

-24

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Variation in Rpts

4.5 0.50 -3.5 -7.5

Variation in Rpsc

140%

-18

-14

-2.5

-6.5

20% -11

9.5

0 14

-15

-19

-23

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-31

-40%

-40%

-35

-20%

60%

140%

220%

300%

380%

-20%

60%

140%

220%

300%

-39

380%

Variation in Rpxs

Variation in Rpxs

Fig. 5. Contour map plots of EHVPG corresponding to intercrossing variations in two model parameters: (a) Rspl and Rpxs; (b) Rhar and Rpxs; (c) Rpsc and Rpxs; and (d) Rpts and Rpxs. The values of EHVPG are illustrated by gray levels combined with contour lines (labeled with the values of EHVPG).

under the reference cirrhotic condition (see Fig. 4(d)), however, the sensitivity of EHVPG to Rpts rose significantly with the increase in Rpxs (see Fig. 5(d)). In summary, our study suggests that the application of HVPG is basically valid with sinusoidal PH, although the effects of multiple pathological factors should be comprehensively considered in order to achieve a better understanding and utilization of HVPG, especially in patients suspected of considerable pathological changes in presinusoidal portal vessels. The study is subject to certain limitations. The model has been developed based on population-averaged data, which renders the model unable to simulate the patient-specific procedure of HVPG measurement. Accordingly, the presented findings will rather serve as theoretical references for guiding the utilization of HVPG than predict the accuracy of HVPG measurement in a specific patient. Sensitivity analyses have been performed with respect to each individual model parameter or the combination of two parameters. Under in vivo conditions, however, various pathological factors might change simultaneously (Sanyal et al., 2008; Wongcharatrawee and Groszmann, 2000), which may lead to more complex HVPG-PPG relationships than those revealed in our study. Moreover, given the fact that a clinically relevant threshold value of EHVPG has not been determined by large-scale clinical studies, it is difficult to identify the pathological conditions under which EHVPG becomes clinically unacceptable through numerical experiments. Further studies, such as well-devised clinical studies, computational studies with global parameter sensitivity analysis, are awaited to further address these issues.

sinusoidal portal hypertension and, in the meantime, revealed the considerable influence of various pathological factors on the accuracy of HVPG measurement. In particular, the study demonstrated that a large variation in presinusoidal portal vascular resistance could significantly alter the error of HVPG relative to PPG, which highlights the importance of considering the pathological state of presinusoidal portal vessels in clinical application of HVPG measurement.

5. Conclusions

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A computational model of the hepatic circulation was developed to quantitatively study the relationship between HVPG and PPG under various pathological conditions. Obtained results confirmed the validity of HVPG as an estimate of PPG in the case of

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Conflict of interest statement There are no conflicts of interest in this study from any of the authors. Acknowledgement The study was supported by the National Natural Science Foundation of China (Grant no. 81370438) and the SJTU MedicalEngineering Cross-cutting Research Project (Grant no. YG2016MS09). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.jbiomech.2017.09. 023.

Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023

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Please cite this article in press as: Wang, T., et al. A computational model of the hepatic circulation applied to analyze the sensitivity of hepatic venous pressure gradient (HVPG) in liver cirrhosis. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.09.023