Virtual Construction of Human Lung

Transcription

1 Review Forma, 14, , 1999 Virtual Costructio of Huma Lug Ryuji TAKAKI 1 ad Hiroko KITAOKA 2 1 Tokyo Uiversity of Agriculture ad Techology, Kogaei, Tokyo , Japa takaki@cc.tuat.ac.jp 2 Faculty of Medicie, Osaka Uiversity, Suita, Osaka , Japa kitaokah@image.med.osaka-u.ac.jp (Received November 19, 1999; Accepted December 27, 1999) Keywords: Airway, Brachig System, Lug, Space Fillig, Acius Abstract. A algorithm to produce a brachig system is proposed i order to simulate the huma lug. This system has costraits that the ed poits of the airway should fill the 3D space uiformly, that the eergy to supply air to the ed poits should be small ad that the quatity of material to costruct this system should be small. Several rules are set up to satisfy these costraits ad the system is costructed virtually by the use of microcomputer. The optimum oe amog resultig airways proves to be quite similar to the real lug. 1. Itroductio Problems how to fill 3D space are ot cofied to those i crystallography, particle dyamics, or cell dyamics. Fie structures i orgas i livig systems also provide iterestig problems as space fillig oes. It would be quite atural to assume that these structures are costructed almost determiistically so that fuctios eeded i orgas are created effectively ad that biological systems possess a abilities to produce such structures via morphogeesis. Sice geeral approach to this problem is ot expected at the preset stage, it would be a meaigful way to make case studies for various orgas or parts of biological systems, i which their forms ad fuctios are described mathemathically ad certai relatios are discussed betwee them. The lug treated here is a orga to supply air to all elemets fillig the space i itself through a brachig ductal system, to make gas exchage there ad to trasport blood both through arteries ad veis. It is a orga with the largest volume i huma body, but 90% of its volume is occupied by the air. WEIBEL (1963) ad HORSFIELD et al. (1971) proposed structural airway models icludig airway dimesios ad coectivity, which do ot iclude iformatio of the spatial structure ad are looked upo as 1D models. 2D geometric airway models were proposed by NELSON ad MANCHESTER (1988) ad by MARTINEN et al. (1994). Recetly, GLENNY ad ROBERTSON (1995) ad PARKER et al. (1997) proposed 3D models of the pulmoary arterial system for simulatig blood flow distributios i the lug. They proved 309

2 310 R. TAKAKI ad H. KITAOKA to be useful for predictio of lug fuctios, while brachig structures i these models are symmetric ulike i the real lug. KITAOKA et al. (1999) proposed a ew algorithm to create the brachig system of the huma lug airway. The purpose of this paper is to explai the basic ideas ad result of this work. 2. Algorithm for Geeratig Airway 2.1. Geometrical priciples At ay brachig geeratio the airway has two kids of geometrical characteristics. Oe is cocered to geometrical properties of itself, i.e. its locatio, legth, thickess ad directio. Aother is cocered to those of a regio i the lug govered by a brach, i.e. its volume ad cotour shape. Now, it is required that the air should be supplied equally to ay regio, hece the brachig should me made so that termial braches fill the space uiformly ad the amout of fluid delivery through a brach should be proportioal to the volume of the regio govered by the brach. These requiremets lead to rules to determie brachig agles, diameters ad legths of braches, directios of plaes o which lie the mother ad daughters Dyamical priciples The dyamical characteristics of the airway are the flow rate ad the mometum coservatios durig brachig. I additio the viscous dissipatio of eergy at the brachig is a importat factor. Sice the air compressibility is eglected i real situatios, the volume flow rate should be coserved at the brach poit. The mometum of air is also coserved so that the duct does ot receive excess stress. This requiremet meas that the mother ad daughter braches lie o the same plae. Some past studies treated the eergy dissipatio i ductal systems (MURRAY, 1926; GROAT, 1948; SUWA et al., 1963; KAMIYA et al., 1974). They derived the followig formula for diameters of the paret d 0 ad daughters d 1, d 2 ad also that for brachig agles θ 1, θ 2 (see Fig. 1): d = d + d, ( ) ( ) = + ( ) ( ) 4/ 4/ 4/ 4/ cos θ1 = + r r r r, cos θ, ( ) 2 / / 2r 21 r where is a costat called a diameter expoet, ad r is a flow dividig ratio (0 r 0.5). The miimum eergy loss requires = 3 (MURRAY, 1926; KAMIYA et al., 1974), while morphometric data vary from 2.6 to Some aatomical coditios The mode of brachig is assumed to be dichotomous. The outer shape of the lug should be give to defie the 3D space i which a airway is costructed. The ratio of legth to diameter of a brach is assumed to be 3 accordig to morphological data. The iitial

3 Virtual Costructio of Huma Lug 311 Fig. 1. Geometrical defiitio sketch. coditio for brachig, i.e. the locatios ad the diameters of the mai truk ad a few geeratios from it are give based o aatomical data. The braches with flow rates less tha a threshold is assumed to be a termial. Beyod it the gas diffusio domiates the process of air trasport The algorithm with morphogeetic rules Based o the cosideratios metioed above we propose the followig 9 rules for costructig brachig system: Rule 1: Brachig is dichotomous. Rule 2: The paret ad its two daughters lie i the same plae (brachig plae). Rule 3: The flow rate is coserved after brachig. Rule 4: The regio supplied by a paret is divided ito two daughter regios by a plae (space dividig plae), which is perpedicular to the brachig plae. Rule 5: The flow dividig ratio r is equal to the volume dividig ratio of daughter regios. Rule 6: Diameters ad brachig agles of the two daughters are determied by Eqs. (1) ad (2). Rule 7: The legth of each daughter brach is three times its diameter. Rule 8: The brachig plae is perpedicular to the precedig brachig plae. Rule 9: The brachig process termiates wheever the flow rate becomes less tha a specified threshold or the brach exteds out of its ow regio. Some of these rules are modified if resultig airway is too ufavorable (for precise refer KITAOKA et al. (1999)).

4 312 R. T AKAKI ad H. KITAOKA Resultig airway tree is evaluated by the use of the followig five quatities. They are (i) failure flow rate, i.e. sum of flow rates i braches which termiated because of extedig out of their ow regio, (ii) variatio of the desity of termials defied by the ratio of the mea to the stadard deviatio, (iii) variatio of aciar volumes, (iv) volume ratio of the total airway volume to the total lug volume, ad (v) eergy loss per termial. Evaluatio of airway tree is made by a simple sum of these five quatities. 3. Results of Simulatio Figure 2 shows a aterior ad a right lateral view of this 3D model with the mostly optimum coditios. The rules have several parameters for modificatio, as oted above, ad there is a room to look for the best result. It should be oted that this model resembles the real airway remarkably. It has 27,306 termial braches, ad the mea geeratio umbers dow to termials is 17.6 ± 3.4 varyig from the miimum 8 to the maximum 32. We also tried cases with other geometric shapes for outer boudary, such as a sphere, a cylider ad a cube. The results were similar to those obtaied with the lug shape. This suggests that the algorithm is quite robust ad capable of geeratig efficiet tree structures i may differet orga shapes. Fig. 2. Resultig 3D airway tree with the best choices of parameters. Left: aterior view, Right: right lateral view. The parameters to produce this model are slightly differet from those used i the previous study (KITAOKA et al., 1999).

5 4. Coclusios ad Discussio Virtual Costructio of Huma Lug 313 The success of the preset lug model suggests that the algorithm proposed here is quite reasoable. It should be oted also that resultig trees exhibit self-similar properties, which is caused by the idepedece of the algorithm o the size of regio except for braches close to outer boudaries. Some discussios are made o the preset algorithm. First, the determiistic algorithm ca ot produce a iter-idividual variatios or a iter-species differeces. However, itroductio of fluctuatios ito parameters such as legth-diameter ratio will accout for these variatios. Whe a outer boudary of the lug was give very faithfully to the real oe, the model provided a large failure flow rate (Qf). It is caused by the existece of cocave surfaces due to the existeces of the heart ad the aorta, where brachig stopped too early (Rule 9). I the preset algorithm costractio of brachig system is associated with a space divisio. I other words, a hierarchical space divisio, associated with material trasport, ca be produced by costructig a brachig system. This ature is cosidered to be commo i all biological brachig systems. The preset authors would like to express their cordial thaks to Prof. B. Suki of Bosto Uiversity ad to Prof. H. Itoh for their helps durig this work. Mr. M. Kashiwabara helped us i preparig computed images, for which the preset authors are thakful. REFERENCES GLENNY, R. B. ad ROBERTSON, H. T. (1995) J. Appl. Physiol., 79, GROAT, R. A. (1948) Fed. Proc., 7, 45. HORSFIELD, K., DART, G., OLSON, D. E., FILLEY, G. F., ad CUMMING, G. (1971) J. Appl. Physiol., 31, KAMIYA, A., TOGAWA, T. ad YAMAMPTO, A. (1974) Bull. Math. Biol., 36, KITAOKA, H., TAKAKI, R. ad SUKI, B. (1999) J. Appl. Physiol., 87, MARTINEN, T. B., YANG, Y. ad DOLOVICH, M. (1994) J. Thoracic Imagig, 9, MURRAY, C. D. (1926) I. Proc. Natl. Acad. Sci. USA, 12, NELSON, T. R. ad MANCHESTER, D. K. (1988) IEEE Tras. Med. Imagig, 7, PARKER, J. C., CAVE, C. B., ARDELL, J. L., HAMM, C. R. ad WILLIAMS, S. G. (1997) J. Appl. Physiol., 83, SUWA, N., NIWA, T., FUKASAWA, H. ad SASAKI, Y. (1963) Tohoku J. Exp. Med., 79, TAKAHASHI, T. (1986) Sciece o Form (Proc. 1st It. Symp. Sciece o Form, Tsukuba, 1985), ed. S. Ishizaka et al., KTK Sci. Publ. ad Reidel Sci. Co., WEIBEL, E. R. (1963) Morphometry of the Huma Lug, Academic, New York.