Self-Similarities of Pulmonary Arterial Tree and a New Integrated Model of Pulmonary Circulation with the Name of Fractal Phasic Perfusion (FPP) Model

Pulmonary arterial hypertension (PAH) has become an important topic of basic and clinical research in recent years. Morphologic researches have shown that specific PAH-lesions are located in the lobular small muscular arteries and correlate with hemodynamic measurements. However, it still remains to be shown how do pathological changes of the small arteries in the lobule develop to PAH. Based on both the fractal properties of pulmonary arterial tree and asynchronous phasic contractions of lobular arterial muscles under the evenness of the pulmonary capillary pressure (PCP) in the lung, the author has constructed an integrated model of pulmonary circulation which has produced a mathematical relationship between the mean pulmonary arterial pressure (MPAP) and the cardiac output (CO). By use of the expression between MPAP and CO, it has been able to explain the pathogenesis of PAH in terms of statistical changes among regional and temporal perfusions in the lung. In order to detect clinically the early stage of PAH, the author has suggested that it is important to establish the pulmonary functional imaging of regional and temporal perfusions.


Introduction
Pulmonary arterial hypertension (PAH) is a rare condition but has now become an important topic of basic and clinical research because that both have led to an ongoing development of specific medical therapies improving symptoms and stabilizing the patient. This rise in interest is partly due to the fact that pulmonary vascular remodeling with pulmonary hypertension is often encountered in various chronic lung diseases including collagen vascular lung diseases [1]. The morphologic study of diseased lungs from patients with PAH represents an important www.videleaf.com landmark on the chart of pathophysiologic concepts. Characteristic lesions in lungs of patients with PAH do not concern the larger pulmonary arteries of elastic type, but that typical obliterating PAH-lesions are found in pre-acinar and intra-acinar small arteries. Recently, Stacher and coworkers presented a large analysis of 62 PAH cases and 28 control subjects with systemically sampled lung tissue obtained at the time of lung transplantation, concluding also that the intima and intima plus media fractional thicknesses of small arteries were increased in the PAH group versus the control lungs and correlated with pulmonary hemodynamic measurements [2].
Hemodynamic measurements of pulmonary circulation are usually measured with a triple lumen balloon-tipped thermodilution fluid-filled catheter (Figure 2) [3]. The catheter is inserted into a central vein and floated through the right heart chambers into the pulmonary artery under constant pressure wave monitoring. Pulmonary vascular pressures including the mean pulmonary arterial pressure (MPAP) are measured in the pulmonary arterial trunk when the lungs are at functional residual capacity. Therefore there seems a great difference in the levels between the sites of pathological arterial lesions and of MPAP measurements. To translate these morphologic observations into hemodynamic measurements in clinical practices concerning both early detection of PAH and accurate monitoring of effectiveness of therapy, it seems necessary to answer the basic problem how do pathological changes of the small arteries make increase in MPAP.
In this study, I have proposed a new integrated model of pulmonary circulation named "Fractal Phasic Perfusion (FPP) model" based on the fractal properties of elastic pulmonary arterial tree and asynchronous phasic contractions of lobular arterial muscles which regulate corresponding lobular perfusion under the evenness of the pulmonary capillary pressure (PCP) in the lung. By use of FPP model, I have proposed a statistical mechanism of increase in MPAP in subjects with pre-acinar and intra-acinar arterial lesions. In order to understand the development of pulmonary hypertension in pulmonary arterial diseases, I have also suggested the necessity of pulmonary 4 www.videleaf.com functional images representing regional and temporal distribution of perfusions in the lung.

Self-similarities of Pulmonary Vascular Structure and Fractal Power Laws
The pulmonary lobes are composed of many lobules (Miller's secondary lobules, Figure 1), which are integrated into the whole lung by the pulmonary arterial tree (PAT) accompanying with the bronchial tree (BT). By applying the rules of integration to PAT with BT, we assumed a power relationship between the diameters at a bifurcation ( 1 r , 2 r , and 3 r ) in Figure 1A, as follows: 1 2 3 n n n r r r . Biologists with an interest in the selfsimilarities of biological branching structures including PAT have found Horton's branching law to be applicable to PAT [4,5]. They implemented Horsfield's version of Horton's branching law as another assumption as follows:  when the edges of order j and order k come together at a vertex, the third edge is assigned to one order greater than the greater of j and k, or to j + 1 if j = k. (C) There is the self-similarity of branching known as a statistical law of Horton among branches with Horsfield ordering. Horsfield's version of Horton's law implies a geometrical self-similarity (modified from ref. [4]). www.videleaf.com Fluid dynamic parameters of the arterial tree as the conduit system are derived in power functions of the radius, including the fractal dimension in the exponents. Suwa and Takahashi [7] showed fluid dynamic relationships based on the fractal property in the length-radius relationship of various arterial branching including PAT as follows:

Asynchronous Phasic Perfusion of Lobules
Muscular artery or arterioles appear along the arterial branches of less than 2,000 m in diameter [6]. Thick bands of smooth muscle have been found in the wall of lobular artery, which regulates perfusion of corresponding lobule in time. Krahl directly observed asynchronous lobular perfusion through a window on the ribcage of rabbits in vivo [7].

Evenness of Pulmonary Capillary Pressure (PCP)
Right heart catheterization with flow-directed balloon-tipped catheter measures successively the right atrial pressure (RAP), the pulmonary arterial pressure (PAP) and the occluded PAP (OPAP). Arterial occlusion creates a stop-flow condition. A measurement of pulmonary capillary pressure (PCP) can be obtained by the analysis of a PAP decay curve after balloon occlusion (Figure 2) [8]. A pressure decay curve is made of a first fast component which corresponds to the stop of flow through an arterial resistance, and a shorter component, which corresponds to the emptying of the compliant capillaries through a venous resistance. The intersection between the two components of the PAP decay curve offers an estimate of PCP (Figure 2) [9]. Measurements of PCP from the analysis of the PAP decay curve after balloon occlusion have showed the evenness of PCP in the whole lung.  1) and (2)). Thus, the pressure drop P along a branch is constant or even because of i+n=4 as follows,

Grouping of Lobules by Weibel's Number
It is generally accepted that the pattern of branching of the pulmonary artery closely follows that of bronchial tree down to the lobular branch. [6] Thus, each lobular branch of artery can be numbered according to accompanying bronchial generation of Weibel (the branching number from the trachea of zero). [10] When a lobule is supplied by a lobular artery of Weibel's number w, the pressure in the lobular artery (P l ) is expressed by the equation as follows, l P PAP w P    because of the evenness of pressure drop (Eq.(4)). If a PAP decay curve can be obtained by arterial occlusion at the lobular artery, an additional Weibel's number w' in the lobule would be obtained as follows,   ' / i w P PCP P    . Therefore, the simple relationship has been obtained among PAP, PCP, w and w' as follows, , where x=w+w' is named the modified Weibel's number. It is important to note that the parameter w is determined by the pattern of branching of elastic arterial tree, and that the parameter w' correlates with pathological conditions of intralobular muscular arterioles as well. Thus, grouping the lobules by x can characterize the anatomico-pathological conditions of arteries including intra-lobular arterioles. 9 www.videleaf.com

Pulmonary Arterial Flow and Phasic Lobular Perfusions
The pulmonary arterial flow is measured as Q(t) at the time of t, when we can see lobular perfusions in the number of s(t) and the relation is ( ) . When x is assigned for each lobular perfusion, the flow Q(t) is rewritten by the following, , where s x (t) is the number of lobules with x where we can see a perfusion at the time of t.

Relationship between MPAP and CO
Pulmonary vascular pressures and flows are usually measured with triple lumen balloon-tipped thermodilution fluid-filled catheter. The catheter is inserted into a central vein and floated through the right heart cambers into the pulmonary artery under constant pressure wave monitoring (Figure 2). The pulmonary arterial catheter provides successive measurements of a right arterial pressure (RAP), a right ventricular pressure (PVP), and a pulmonary arterial pressure (PAP), and an occluded PAP. Catheter measurements also provide pulmonary arterial pressure (PAP) and flow waves (Q). Based on the set of measurements of PAP and Q during a single stroked time (), the mean pulmonary arterial pressure (MPAP) has been obtained as following calculations according to Eq.(5), , where t x is the perfusion time seen in the lobules with x during the period of  , and r xt is the correlation coefficient of variables x and t x . Then, MPAP is expressed by the following equation, This expression of Eq. (10) is the "Fractal Phasic Perfusion (FPP)" model of pulmonary circulation.

Discussion
This study has been carried out the assumption that pulmonary arterial tree is entirely prescribed by a single fractal dimension. A study has shown that it is not true in some cases, where the system appears to be ruled by two different dimensions for large and small diameters [11]. However, single fractal dimensions of pulmonary vascular structure have been reported by many studies for mammals including human [12]. In a recent through analysis of the pulmonary vascular trees conceived as fractal structures it has been shown that the fractal dimension of both arteries and veins is about 2.7 [5,13].
The pulmonary circulation is a high flow and low pressure circuit. This property of pulmonary circulation has been www.videleaf.com characterized by recruitment and distension of vascular tree. After observations of Krahl [7], the lobular arterial muscle bundle has been recognized as a sphincter to regulate corresponding lobular perfusion. In fractal tree of the elastic pulmonary artery the diameter of a branch is related with corresponding flow through it, therefore the number of lobules supplied by the branch would determine the corresponding flow and its diameter as well. If a number of asynchronous phasic contractions of lobular arterial smooth muscles should appear simultaneously, the phenomena of recruitment and distention would be observed according to the pattern of simultaneous lobular perfusions. Thus, the recruitment and distension are recognizable as different faces of harmonized asynchronous contractions of lobular arterial muscles, which would show various regional patterns of pulmonary perfusion.
Many cardiac and pulmonary diseases are associated with an abnormal increase in pulmonary arterial pressures (PAH) [1] and obliterating lesions of pre-and intra-acinar arteriolar lesions have been found in patients with PAH. However, it is still unknown how these lesions in the lobule develop to PAH. According to FPP model of this study, the obliterating lesions of the intimal layer of arterioles in a lobule make the number of x higher, and the distribution of x would change higher in its mean or variation (Figure 3), then MPAP would increase according to Eq.(10). The variation coefficient of ( x / x ) in Eq. (10)   Heart and pulmonary arterial (Ventriculo-arterial) coupling is recognizable through a pressure-flow (PAP-Q) relationship. [3] In the case of FPP model, PAP-Q loop can be translated into the relationship between variables x and t x in the lung (Figure 4). The correlation coefficient r xt is recognizable by the ratio of the area in the PAP-Q loop (A Loop ) to the area of PAP-Q square (A Square ) by using the following equation, Since the ratio of A Loop /A Square indicates the efficiency of energy transmission through elastic pulmonary arterial tree from the right cardiac ventricle to pulmonary parenchyma, the condition of ventriculo-arterial coupling would be recognizable through the correlation coefficient r xt . www.videleaf.com The variation coefficient of distribution in regional perfusion   t t  is not able to estimate from the physiological measurements but only from the functional images of regional and temporal perfusion in the lung. New active researches have begun to develop and utilize advanced imaging technology in order to measure regional and temporal changes of perfusion in the lung [14]. Further advances in technology to measure regional and temporal perfusions of the lung concerning   t t  would be able to estimate the distribution of perfusion in the whole lung, and we hope to get it in coming years. When we can estimate the distribution of regional perfusions through new functional images, the effectiveness of FPP model would be