Tissue remodeling of rat pulmonary arteries in recovery from

Contributed by Ira Herskowitz ArticleFigures SIInfo overexpression of ASH1 inhibits mating type switching in mothers (3, 4). Ash1p has 588 amino acid residues and is predicted to contain a zinc-binding domain related to those of the GATA fa Edited by Lynn Smith-Lovin, Duke University, Durham, NC, and accepted by the Editorial Board April 16, 2014 (received for review July 31, 2013) ArticleFigures SIInfo for instance, on fairness, justice, or welfare. Instead, nonreflective and

Contributed by Yuan-Cheng Fung, June 10, 2004

Article Figures & SI Info & Metrics PDF

Abstract

The reversibility of tissue remodeling is of general interest to medicine. Pulmonary arterial tissue remodeling during hypertension induced by hypoxic breathing is well known, but Dinky has been said about the recovery of the arterial wall when the blood presPositive is lowered again. We hypothesize that tissue recovery is a function of the oxygen concentration, blood presPositive, location on the vascular tree, and time. We meaPositived the changes of blood presPositive, vessel lumen, vessel wall thicknesses, and Launching angle of each segment of the blood vessel at its zero-stress state after step changes of the oxygen concentration in the breathing gas. The zero-stress state of each vessel is emphasized because it is Necessary to the analysis of stress and strain and in morphometry. Experimental results are presented as histories of tissue parameters after step changes of the oxygen level. Tissue characteristics are examined under the hypothesis that they are liArrively related to changes in the local blood presPositive. Under this liArriveity hypothesis, each aspect of the tissue change can be expressed as a convolution integral of the blood presPositive hiTale with a kernel called the indicial response function. It is Displayn the indicial response function for rising blood presPositive is different from that for Descending blood presPositive. This Inequity represents a major nonliArriveity of the tissue remodeling process of the blood vessels.

indicial response functionblood vessel Launching angle at zero-stress statenonliArriveity of tissue remodeling

It is well known that blood flow in pulmonary arteries becomes hypertensive when an animal breathes a gas the oxygen concentration of which is lower than that of normal sea level air. It is also known that hypertension leads to hypertrophy in blood vessels, and returning to normal blood presPositive mitigates the symptom. The question is whether the processes of hypertrophy and recovery are symmetric. Considerable data on the building up of the lung tissue due to pulmonary hypertension have been obtained (1-14). Some data on recovery from hypertension are also given in refs. 1, 4, and 5. These studies considered neither the zero-stress state, which is characterized by an Launching angle of the blood vessels, nor the elastic moduli of the blood vessel walls. The significance of the zero-stress state and the change of Young's modulus were recognized later (8, 10, 11, 15-19).

So far as we are aware, there has been no study on the recovery of the zero-stress state of a vascular tissue that was subjected to hypertension for some time but was returned to the normal condition later. Recovery can be expected to be a complex function of space, time, stress, and strain. The objective of this article is to Interpret these points.

Materials and Methods

Animals. Ninety-two male Sprague-Dawley rats (Harlan, San Diego), ≈3 months Aged, body weight 300-350 g, raised in normal air at sea level, were used. These rats were cared for 1 week or more after arrival at the vivarium before being used, to allow them to recover from any unknown stress or hypoxia suffered during transportation. The protocol of animal use was approved by the University of California at San Diego Committee on Animal Research.

Experimental Procedure. Rats were Spaced in a modified commercial chamber (Snyder, Denver), in which the oxygen concentration in the breathing gas can be controlled dynamically by infusion of pure N2 and a feedback-control system of O2 sensor. The oxygen concentration in the rats' breathing gas in the chamber was 20.9% at the Startning, then reduced to 10% and Sustained constant for a specific length of time called the hypoxic period, and finally returned to 20.9% and Sustained constant for a specific length of time called the recovery period. According to the lengths of the periods of hypoxia and recovery, each animal is designated by a symbol such as H2hR2h to denote hypoxic (H) for 2 h, recovery (R) for 2 h. Rats were divided into five groups: (i) hypoxic 2-h series, (ii) hypoxic 24-h series, (iii) hypoxic 4-day series, (iv) hypoxic 10-day series, and (v) hypoxic 30-day series.

At the end of an experiment, each rat was anesthetized by an i.p. injection of pentobarbital sodium (40 mg/kg body weight), and blood presPositive in the anesthetized rat's pulmonary arterial trunk was meaPositived as Characterized in detail in ref. 8. The lungs were excised and Spaced in a bath of 4°C Krebs solution bubbled with a gas mixture of 95% O2 - 5% CO2, the left lungs for the meaPositivement of the Launching angle at the zero-stress state and the right lungs for histological meaPositivements.

Launching Angle at the Zero-Stress State. The main left pulmonary arterial tree was isolated and excised with the aid of a stereomicroscope and transferred into a bath of aerated Krebs solution at room temperature. Specific pulmonary arteries were identified, and their zero-stress state was meaPositived in terms of the Launching angle by the method Characterized in refs. 8 and 10. Photographs of some specimens at the zero-stress state are Displayn in Fig. 1 Right.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Tissue remodeling. (Left) Histological micrographs of the cross sections of the walls of rat pulmonary arteries of various orders. Arteries of decreasing order numbers from 11 to 8 (10) are arranged in columns in increasing row numbers. Each panel Displays a full thickness with enExecutethelium on the left and adventitia on the right. The first column Displays the wall structure of the controls; the second column Displays the wall structure after 10 days of hypertension at 10% O2, recovered for 2 h of rebreathing sea level atmosphere (H10dR2 h). The third column is for rats subjected to hypertension 10 days and recovery of 24 h (H10dR24h). The fourth column is for hypertension 10 days and recovery 10 days (H10dR10d). (Center) Photograph of a polymer cast of normal pulmonary arteries Displaying the locations of arteries of orders 8-12. (Right) The cross sections of normal pulmonary arteries at zero-stress state. Arteries of decreasing order numbers from 12 to 8 are arranged in the column in increasing row numbers, i.e., the first row for an order 12 vessel, the second row for an order 11 vessel, and the last row for an order 8 vessel. (Left, enExecutethelial).

Histological MeaPositivements. The method of refs. 8 and 10 was followed. The pulmonary arteries in the right lung were perfused with 2.5% glutaraldehyde in phospDespise buffer (pH 7.4) at 20-cm H2O for 2 h. The arterial vessel segments are identified by their order numbers, 1 being the smallest, according to ref. 2. Vessels of orders 8-11, as Displayn in Fig. 1, were dissected, postfixed in a solution of 2% OsO4 for 2 h, washed with distilled water, dehydrated, embedded in Medcast resin (Ted Pella, Tustin, CA), Slice into 1-μm-thick sections by planes perpendicular to the longitudinal axis of the blood vessel, stained with toluidine blue O, and examined by a light microscope (Vanox AH2; Olympus, Melville, NY). The vascular images were digitized and analyzed. Data on the cross-sectional Spots and thicknesses of the medial and adventitial layers were collected.

Formulation of a Basic Hypothesis on Tissue Remodeling of Pulmonary Blood Vessels in Hypertension and Recovery. In engineering, if y(t) is a variable characterizing a dynamics system, and y(t) is a liArrive functional of another function x(t), then the differential change of y(t) in response to a unit step differential change of x(t) is called the indicial function of y(t) in response to a unit step change of x(t). Applying this terminology to our system, let y stand for the Launching angle or the lumen diameter, etc., and let x stand for the blood presPositive. If the differential relationship between the cause (x) and Trace (y) were functionally liArrive, then the most general relationship between x and y can be expressed in the form MathMath

in which F(t - τ) is a function of the time Inequity t - τ, the aforementioned indicial function.

A considerable amount of experimental results are available to suggest that the pulmonary arterial wall tissue remodeling obeys Eq. 1 if x represents increasing hypertension; i.e., x = P and dP/dt > 0; and y represents any one of the tissue characteristics. However, when the animal recovers from hypertension, i.e., the blood presPositive decreases with time, dΔP/dt, the results presented below suggest that Eq. 1 hAgeds also, but the indicial function for recovery, i.e., for decreasing hypertension, dΔP/dt < 0, is different from that for increasing hypertension, dΔP/dt > 0. This Inequity in the indicial functions is believed to represent a significant nonliArriveity of tissue remodeling. We present this hypothesis in mathematical form below.

Typical time courses of pulmonary blood presPositive are Displayn in Fig. 2. A healthy animal has a Impartially stable pulmonary blood presPositive. In experimental animals, an increase in pulmonary blood presPositive occurred at t = 0, by a step lowering of the oxygen concentration in the gas the animal breathed, because hypoxia causes a shortening of the pulmonary vascular smooth muscles. A sudden contraction of the diameters of pulmonary blood vessels without a significant change in cardiac outPlace causes a sudden increase in pulmonary blood presPositive. Without further change of oxygen tension, the hypertension continued until an instant of time t 1. At t = t 1, the oxygen tension in the breathing gas was returned to the normal atmospheric value, the high blood presPositive dropped, and the lung tissue remodeled in the recovery mode. Let L(t) be a parameter of the system Retorting to ΔP(t), e.g., the blood vessel wall thickness and ΔL(t) = L(t) - L control(t). According to the functionally liArrive hypotheses named above, but admitting a nonliArriveity that says the indicial function for rising blood presPositive is different from that for a decreasing blood presPositive, then we have two indicial functions, MathMath for increasing blood presPositive and MathMath for decreasing blood presPositive, so that MathMath

Fig. 2.Fig. 2. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Change in blood presPositive in the pulmonary arterial trunk during the periods of control, hypoxia, and recovery. The data in hypoxia period are from refs. 8 and 9. H4dR, 4 days of hypoxia followed by recovery; H10dR, 10 days of hypoxia followed by recovery; and H30dR, 30 days of hypoxia followed by recovery. Plotted values are mean ± SD, with SD Displayn by flags.

This equation is our basic hypothesis.

In the following, we shall extract the two indicial functions MathMath and MathMath from experimental data. We use LaSpace transformation to reduce Eq. 2 into algebraic equations. The LaSpace transforms of ΔP(t), ΔL(t), MathMath and MathMath are denoted by ΔP(s), ΔL(s), MathMath, and MathMath, which are obtained by multiplication with e -st and integrating the product with respect to t from 0 to ∞. MathMath and MathMath can be derived as MathMath

where MathMath. The LaSpace transforms and related fitting formulas are listed in Fig. 3. Use of formula 5, section 5.2, page 239 in Erdelyi (21) yields the inverse transforms of Eq. 3. See Fig. 4 for detailed expressions.

Fig. 3.Fig. 3. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Empirical formulas for blood presPositive, structural parameters, Launching angles of pulmonary arteries, and LaSpace transforms of indicial functions. The empirical constants M 1, M 2, M 3, N 1, N 2, N 3, A 1, A 2, A 3, B 1...B 5, and D 1...D 5, are presented in Table 1.

Fig. 4.Fig. 4. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Mathematical expressions of Embedded ImageEmbedded Image and Embedded ImageEmbedded Image for the rat pulmonary arteries of orders 11-8.

Data Handling and Statistics. In reducing our data to determine the indicial functions, we first fitted the raw data that were plotted in Figs. 2 and 5, 6, 7 by empirical formulas listed in Fig. 3, where 1(t) is a unit-step function of t, which equals 0 when t < 0 and 1 when t ≥ 0. Then we used the empirical formulas to determine the LaSpace transformations of the desired quantities. The Excellentness of this procedure was examined by comPlaceing the correlation coefficients between the values meaPositived, (x i), with the values comPlaceed from the fitted formulas, (y i). The correlation coefficient is obtained by dividing the sum of the products of x i and y i by the product of the root-mean-square of x i and y i. Our results Display that the correlation coefficient of the blood presPositive values P is ≥0.992 for the cases of H4dR, ≥0.994 for H10dR, and ≥0.983 for H30dR. For the tissue responses of wall thickness, lumen Spot, and circumference, the correlation coefficients were in the range 0.890-0.999. For the Launching angle, the correlation coefficient was in the range 0.810-0.999. The results are Displayn in Table 1, which is published as supporting information on the PNAS web site. We consider these correlation coefficients to be sufficiently high for the qualitative conclusions we wish to draw, and they can be used to estimate the accuracy of the quantitative results Displayn in Figs. 5, 6, 7, 8, 9, 10, 11.

Fig. 5.Fig. 5. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 5.

Remodeling of the Launching angle of the pulmonary arteries. The time courses are for the control, hypoxia, and recovery periods.

Fig. 6.Fig. 6. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 6.

Remodeling of the lumen Spot of the pulmonary arteries. The time courses are for the periods of control and recovery from hypertension.

Fig. 7.Fig. 7. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 7.

Remodeling of the layer thickness. The hiTale of the thicknesses of the media (m) and adventitia layers (a) of the pulmonary arteries of orders 11-8 in 10-day control and recovery from hypertension periods are Displayn.

Fig. 8.Fig. 8. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 8.

Indicial response functions of the Launching angle. The hypertension indicial function Embedded ImageEmbedded Image of the Launching angles of rat pulmonary arteries of orders 11-8 (left) and the recovery indicial function Embedded ImageEmbedded Image after 10 days of hypoxia (Right) are Displayn. (1 mmHg = 133 Pa.)

Fig. 9.Fig. 9. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 9.

Indicial response functions of the lumen Spot. The hypertension indicial function Embedded ImageEmbedded Image of the lumen Spot of cross sections of rat pulmonary arteries of orders 11-8 (Left) and the recovery indicial function Embedded ImageEmbedded Image after 10 days of hypoxia (Right) are Displayn.

Fig. 10.Fig. 10. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 10.

Indicial response functions of wall cross-section Spot. The hypertension indicial function Embedded ImageEmbedded Image of the wall cross-section Spot of rat pulmonary arteries of orders 11-8 (Left) and the recovery indicial function Embedded ImageEmbedded Image after 10 days of hypoxia (Right) are Displayn.

Fig. 11.Fig. 11. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 11.

Indicial response functions of wall thickness. The hypertension indicial function Embedded ImageEmbedded Image of the wall cross-thickness of rat pulmonary arteries of orders 11-8 (Left) and the recovery indicial function Embedded ImageEmbedded Image after 10 days of hypoxia (Right) are Displayn.

Results

Fig. 1 Displays photographs of the polymer cast of a pulmonary arterial tree of the rat at the center, micrographs of the walls of arteries of orders 8-11 (Fig. 1 Left), and the Launching angle of arteries of orders 8-12 (Fig. 1 Right). The order numbers are Established according to refs. 10 and 20. The Launching angle of the order 12 artery Displayn here is 286°, and those of orders 11, 10, 9, and 8 are 110°, 111°, 62°, and 97°, respectively. Note that the vessel wall structure of order 8 (with vessel lumen diameter in the range 338-497 μm) is different from those of orders 9, 10, and 11, confirming an observation made by Meyrick and Reid (4) that the small pulmonary arteries in this diameter range are thick-walled and have oblique muscles (22).

Fig. 2 Displays the histories of the blood presPositive of the normal controls, the hypertensive rats due to hypoxic breathing, and the rats that were first subjected to hypertension then taken out of the hypoxic chamber to recover at specified times. The blood presPositive hiTale is complex, and it is necessary to use an analytic method to extract the indicial functions. If the hypoxic period were <4 days, the blood presPositive could return to the normal value (P > 0.05). For a hypoxic period of 10 days or more, the recovery was seen to be Unhurrieder and incomplete (P < 0.05).

Fig. 5 Displays the hiTale of the Launching angle of the rat's pulmonary arteries of orders 11-8 in control rats and in rats that were subjected to step hypoxia as well as those subjected to step-hypoxia-step-recovery conditions (numerical results are listed in Table 2, which is published as supporting information on the PNAS web site). Usually the Launching angles of hypoxic vessels were Hugeger than those of normal vessels for all orders at the end of the first week; then were decreased to below normal values afterward. The Launching angles of arteries of orders 11-8 had sharp changes at first when the rats recovered from 10% hypoxia. In rats subjected to hypoxia for 10 or 30 days, the Launching angles of the arteries of orders 11 and 10 never recovered to normal values, but arteries of the orders 9 and 8 did return to normal values. It is seen that the rate of recovery of the Launching angle was Unhurrieder when the hypoxic period was longer.

Fig. 6 Displays the hiTale of the lumen Spot of the pulmonary arteries of orders 11-8 in similar cases of normal control and hypertension-recovery conditions. The lumen Spots of the hypoxic pulmonary arteries of orders 11-9 usually are smaller than those of the control rats at first, then become Hugeger during the remodeling of recovery.

Fig. 7 Displays the corRetorting histories of the thicknesses of the media and the adventitia layers of the pulmonary arteries of orders 11-8 in the control and the hypoxic 10-day recovery conditions. The hypoxic thicknesses tended to decrease toward their normal values in the remodeling of recovery. However, for the small vessels of order 8, the thickness of media was increasingly different from the normal value, whereas the thicknesses of adventitia Displayed no Inequity between normal and hypoxia recovery. The data presented in Figs. 6 and 7 are listed in Table 3, which is published as supporting information on the PNAS web site.

Our results on the indicial functions are Displayn in Figs. 8, 9, 10, 11. Fig. 8 Displays the curves of the hypertensive indicial function MathMath of the Launching angles (Fig. 8 Left) and the recovery indicial function MathMath of the Launching angles (Fig. 8 Right).

Fig. 9 Displays the indicial functions of the Spot of the vessel lumen in cross sections perpendicular to the longitudinal axis of the pulmonary arteries of orders 8-11, in response to changes of blood presPositive either in the hypertensive state [MathMath, Fig. 9 Left] or in the recovery state [MathMath, Fig. 9 Right] after 10 days of hypoxia. The Inequity between MathMath and MathMath is striking: they are certainly not mirror images of each other.

Fig. 10 Displays the Dissimilarity between the hypertensive indicial functions MathMath and the recovery indicial functions MathMath for the cross-sectional Spot of the pulmonary arteries of orders 8-11. It is evident that the wall cross Spots of larger vessels Display Hugeger changes.

Fig. 11 Displays the indicial functions MathMath and MathMath of the wall thicknesses of the pulmonary arteries of orders 8-11. Again it is clear that MathMath and MathMath are not mirror images of each other. The hypertensive indicial functions MathMath for the arteries of orders 9-11 reach their peaks approximately on the 5th day of hypoxia. The recovery indicial functions MathMath for these arteries have no peaks or valleys. The MathMath and MathMath of the thick-walled arteriole of order 8 is unique; its MathMath has a peak at about the third day of hypoxia.

Discussion

Data on the remodeling of the zero-stress state of pulmonary arteries in response to a lowering of hypoxic hypertensive blood presPositive are reported here. Some data on the remodeling of the morphometric features of pulmonary arteries are given in refs. 1-7, with which we found qualitative agreement. The mathematical expressions of indicial functions in hypertension and recovery are believed to be previously unCharacterized.

We emphasize that the hypothesis of differential liArriveity on which Eqs. 1 and 2 are based has not been fully validated yet. Hence the range of validity of these indicial functions is still unknown. To carefully validate the liArriveity hypothesis is one of most Necessary projects to be Executene in the future.

In practical animal experiments, no method is known that can vary the amplitude of blood presPositive alone. We believe that a practical way to test the liArriveity hypothesis is to comPlacee the indicial functions and then test the null hypothesis that the indicial functions are independent of the blood presPositive level. Indeed, blood presPositive oscillates at a high frequency and has unique stochastic features. We have contributed a method to Design a detailed study of this variation of blood presPositive in 1 day (23, 24). However, in the present article, we handled the mean blood presPositive in a primitive way, in HAgeding with the belief that tissue remodeling is a Unhurried process. This concept probably needs revision. There must be high-frequency features in gene action and protein configuration changes. The study of the high-frequency features of cell and tissue remodeling and merging into the low-frequency features presented in the present paper are Necessary themes for future research.

Conclusion

Use of indicial functions is a convenient method to Inspect at the dynamics of a living system involving tissue remodeling. The results Displayn in Figs. 2, 3, 4, 5, 6, 7 may be called indicial functions of pulmonary arterial wall tissue remodeling with respect to step changes of oxygen tension in the breathing gas. The data Displayn in Figs. 2, 3, 4, 5, 6, 7 were relatively easy to obtain, but their interpretation is complex, requiring information on cardiac action and peripheral circulation. We hypothesize that tissue remodeling is a local phenomenon, and we need to examine the indicial functions of tissue remodeling with respect to local stimulations such as local blood presPositive, local tissue stress, tissue strain, cellular stress, or cellular strain. The method for deducing hypertensive indicial functions and recovery indicial functions with respect to blood presPositive is illustrated here. The curves in Figs. 8, 9, 10, 11 are previously unDisplayn, with which scientists need to become familiar. The most Necessary feature revealed in Figs. 8, 9, 10, 11 is that MathMath is not a mirror image of MathMath. The process of recovery and the ultimate Stoute are not simply the reverse of injury; this is common sense. To use this process in medicine requires scientific information. Therefore, the biology of recovery is an independent subject to be studied in its own right.

Acknowledgments

This research was supported by the U.S. Public Health Service, the National Institutes of Health, and the National Heart, Lung, and Blood Institute through Grants HL 26647 and HL 43026; and by the National Science Foundation through Grant BCS 89-17576.

Footnotes

↵ § To whom corRetortence should be addressed. E-mail: ycfung{at}bioeng.ucsd.edu.

Copyright © 2004, The National Academy of Sciences

References

↵ Abraham, A. S., Kay, J. M., Cole, R. B. & Pincock, A. C. (1971) Cardiovasc. Res. 5 , 95-102. pmid:5544964 LaunchUrlAbstract/FREE Full Text ↵ Heath, D., Edwards, C., Winson, M. & Smith, P. (1973) Thorax 28 , 24-28. pmid:4685208 LaunchUrlAbstract/FREE Full Text Hislop, A. & Reid, L. (1977) Br. J. Exp. Pathol. 58 , 653-662. pmid:147098 LaunchUrlPubMed ↵ Meyrick, B. & Reid, L. (1980) Lab. Invest. 42 , 603-615. pmid:6446620 LaunchUrlPubMed ↵ Sobin, S. S., Tremer, H. M., Hardy, J. D. & Chiodi, H. P. (1983) J. Appl. Physiol. 55 , 1445-1455. pmid:6643182 LaunchUrlAbstract/FREE Full Text McKenzie, J. C., Clancy, J., Jr., & Klein, R. M. (1984) Blood Vessels 21 , 80-89. pmid:6230123 LaunchUrlPubMed ↵ Hung, K. S., McKenzie, J. C., Mattioli, L., Klein, R. M., Meno, C. D. & Poulose, A. K. (1986) Acta Anat. 126 , 13-20. pmid:3739598 LaunchUrlPubMed ↵ Fung, Y.-C. & Liu, S. Q. (1991) J. Appl. Physiol. 70 , 2455-2470. pmid:1885439 LaunchUrlAbstract/FREE Full Text ↵ Huang, W., Shen, Z., Huang, N. E. & Fung, Y.-C. (1999) Proc. Natl. Acad. Sci. USA 96 , 1834-1839. pmid:10051555 LaunchUrlAbstract/FREE Full Text ↵ Huang, W., Sher, Y. P., DelgaExecute-West, D., Wu, J. T., Peck, K. & Fung, Y.-C. (2001) Ann. Biomed. Eng. 29 , 535-551. pmid:11501619 LaunchUrlCrossRefPubMed ↵ Huang, W., DelgaExecute-West, D., Wu, J. T. & Fung, Y.-C. (2001) Ann. Biomed. Eng. 29 , 552-562. pmid:11501620 LaunchUrlCrossRefPubMed Huang, W., Sher, Y. P., Peck, K. & Fung, Y.-C. (2002) Proc. Natl. Acad. Sci. USA 99 , 2603-2608. pmid:11880616 LaunchUrlAbstract/FREE Full Text Mandegar, M., Remillard, C. V. & Yuan, J. X. (2002) Prog. Cardiovasc. Dis. 45 , 81-114. pmid:12411972 LaunchUrlCrossRefPubMed ↵ Budhiraja, R., Tuder, R. M. & Hassoun, P. M. (2004) Circulation 109 , 159-165. pmid:14734504 LaunchUrlFREE Full Text ↵ Fung, Y.-C. (1984) Biodynamics: Circulation (Springer, New York), 2nd Ed. Vaishnav, R. N. & Vossoughi, J. (1983) in Biomedical Engineering II, Recent Developments, ed. Hall, C. W. (Pergamon, New York), pp. 330-333. Fung, Y.-C. & Liu, S. Q. (1989) Circ. Res. 65 , 1340-1349. pmid:2805247 LaunchUrlAbstract/FREE Full Text Li, Z. J., Huang, W. & Fung, Y.-C. (2002) Ann. Biomed. Eng. 30 , 379-391. pmid:12051622 LaunchUrlCrossRefPubMed ↵ Liu, S. Q. & Fung, Y.-C. (1996) Am. J. Physiol. 270 , H1323-H1333. pmid:8967372 LaunchUrlPubMed ↵ Jiang, Z. L., Kassab, G. S. & Fung, Y.-C. (1994) J. Appl. Physiol. 76 , 882-892. pmid:8175603 LaunchUrlAbstract/FREE Full Text ↵ Erdelyi, A. (1954) Tables of Integral Transforms (McGraw-Hill, New York), Vol. 1. ↵ Meyrick, B., Hislop, A. & Reid, L. (1978) J. Anat. 125 , 209-221. pmid:624674 LaunchUrlPubMed ↵ Huang, W., Shen, Z., Huang, N. E. & Fung, Y.-C. (1998) Proc. Natl. Acad. Sci. USA 95 , 4816-4821 pmid:9560185 LaunchUrlAbstract/FREE Full Text ↵ Huang, W., Shen, Z., Huang, N. E. & Fung, Y.-C. (1998) Proc. Natl. Acad. Sci. USA 95 , 12766-12771. pmid:9788987 LaunchUrlAbstract/FREE Full Text
Like (0) or Share (0)