## Abstract

The goal of this study is to quantitatively describe the remodeling of the zero-stress state of the femoral artery in flow overload. Increased blood flow, approximately as a unit step change, was imposed on the femoral artery by making an arteriovenous (a-v) fistula with the epigastric vein. The a-v fistula was created in the right leg of 36 rats, which were divided equally into six groups (2 days and 1, 2, 4, 8, and 12 wk after the fistula). The vessels in the left leg were used as controls without operative trauma. The in vivo blood pressure, flow, and femoral outer diameter and the in vitro zero-stress state geometry were measured. The in vivo shear rate at the endothelial surface increased approximately as a step function by ∼83%, after 2 days, compared with the control artery. The arterial luminal and wall area significantly increased postsurgically from 0.15 ± 0.02 and 0.22 ± 0.02 mm^{2} to 0.28 ± 0.04 and 0.31 ± 0.05 mm^{2}, respectively, after 12 wk. The wall thickness did not change significantly over time (*P* > 0.1). The opening angle decreased to 82 ± 4.2 degrees postsurgically when compared with controls (102 ± 4.4) after 12 wk and correlated linearly with the thickness-to-radius ratio. Histological analysis revealed vascular smooth muscle cell growth. The remodeling data are expressed mathematically in terms of indicial functions, i.e., change of a particular feature of a blood vessel in response to a unit step change of blood flow. The indicial function approach provides a quantitative description of the remodeling process in the blood vessel wall.

- residual stress
- opening angle
- flow-dependent dilation
- arteriovenous fistula
- indicial response functions

residual strain is defined as the strain in the no-load state (where external forces are zero) in reference to the zero-stress state. It is well known that residual strain exists in blood vessels and that the arterial configuration at zero-stress state is an open sector (5,31). The open sector can be characterized by an opening angle, which is a measure of the residual strain in the arterial wall (20). It has been shown that the residual strain reduces the stress concentration at the inner portion of the arterial wall at physiological loading (1). It was further pointed out that a change of the opening angle is a result of nonuniform growth of dimension of the tissues in the vessel wall (8).

There is no doubt that significant remodeling of the arterial blood vessel occurs in response to flow overload. Although there are many ways to study the structural and mechanical remodeling of the vessels, we chose residual strain as the most relevant quantitative aspect of remodeling because it is a measure of the nonuniformity of growth or resorption in different parts of the blood vessel wall (8). If growth or resorption were uniform in the vessel wall, then the structural change will cause no change in stress and strain in the blood vessel. On the other hand, if one part of the vessel wall outgrows the rest (or resorbs locally), then it will compress the remainder and cause internal stress. The internal stress is hence a measure of the unequal growth or nonuniform remodeling. This internal stress is called residual stress. Measurements of the residual strain yield a simple index for the mismatch of hypertrophy and hyperplasia in different parts of the vessel. Recent studies have demonstrated that the residual strain changes reflect the nonuniformity of growth and remodeling in hypertension, diabetes, aging, and smoke exposure (16, 19, 20, 25). To our knowledge, the effect of flow overload on the residual strain has not been studied. Therefore, the first goal of the present study is to determine the remodeling of the zero-stress state of the femoral artery in response to flow overload. An approximate step increase in arterial blood flow was created in rat femoral artery by an arteriovenous (a-v) fistula. The arterial morphology and opening angle were measured, and the residual strains at the inner and outer wall surfaces were computed at the no-load state. Furthermore, the shear rate at the inner wall and the average circumferential stress and midwall strain were computed in the in vivo state.

A systematic approach to describe the remodeling process is to record the course of changes that take place in various features of the blood vessel after a step change of input variable. Response to a step function is called the indicial response function (IRF), i.e., change of a particular feature of a blood vessel (e.g., wall thickness, opening angle, strain, stress, and so forth) in response to a unit step change of input variable (e.g., flow). If the system is linear, then a convolution of the IRF yields the response to any arbitrary input function within the same limits. This approach was first proposed by Fung (personal communication) and was later used by Liu and Fung (18) to obtain the IRF of arterial remodeling in response to changes in blood pressure. The IRFs of arterial remodeling in response to changes in blood flow have not been determined. Hence, the second goal of the present study is to obtain the IRF of various features of blood vessel in response to flow overload.

## MATERIALS AND METHODS

### Indicial Response Experiments

Ideally, indicial responses should be measured in one vessel. This is not possible, however, if histological measurements have to be made at different stages of tissue remodeling in different animals. Furthermore, there is no method to change blood flow as a strictly mathematical step function. In practice, one has to use many animals with approximate steps of flow and then analyze the results by methods discussed below.

#### Experimental design.

Thirty-six female Wistar rats weighing 236 ± 36 g (means ± SD) were used in this study. The rats first underwent an end-to-side anastomosis of the epigastric vein to the femoral artery in their right leg. The epigastric venous bypass increased blood flow in the femoral artery. The left leg was used as a control without surgical trauma. The rats were randomly divided into six chronic groups that survived 2 days and 1, 2, 4, 8, and 12 wk after the a-v fistula.

The experiments complied with the animal welfare regulations formulated by the Danish National Society for Medical Research and the *Guide for the Care and Use of Laboratory Animals*.

#### Surgical procedure.

Anesthesia was induced and maintained with Hyponorm (0.3 ml/kg im) and Dormicum (0.4 ml/kg ip). Microsurgery was performed under sterile conditions with the use of a Zeiss operating microscope. The femoral artery and vein and the epigastric vein were dissected along their emergence from the fat pad to the junction with the femoral vein. The femoral artery was then dissected out of its sheath over its entire length. The adventitia of the femoral artery was thoroughly removed over an area twice as long and twice as wide as the arteriotomy. Two cuts were made from opposite directions at 45 degrees to the vessel such that they met exactly and formed a mouth “V.” The epigastric vein was cut at a sufficiently distal position to reach the femoral artery. The epigastric venous lumen was washed by heparinized saline solution (50 EU/ml). The end of the distal epigastric vein was sutured to the femoral arterial side with the use of a 10/0 microvascular monofilament nylon suture (S & T Marketing). The skin incision was closed in one layer.

#### In vivo measurements.

At a scheduled time, the original incisions were reopened under a standard anesthetic regime. The outer diameter of the femoral artery in both legs was videotaped under a stereomicroscope (Zeiss, Stemi 2000-C) and later measured with the use of an image analysis system (Optimas). Short segments of the femoral vessels of both legs were isolated, and the arterial blood flow rates were measured with the use of an ultrasound flowmeter (T206 small animal blood flowmeter, Transonic System). The systemic arterial blood pressure was recorded from the left carotid artery (Cardio-Med System). We did not routinely measure arterial pressure in the femoral artery to prevent damage to the short femoral artery. In pilot experiments, however, it was verified that the pressure in the femoral artery is similar to that in the carotid artery.

The rats were killed with an overdose of anesthesia. The vessels were immediately excised and placed in an organ bath containing Krebs solution (with EGTA, 100 mg/l) aerated with 95% O_{2}-5% CO_{2} at room temperature. Adjacent tissue was carefully removed with the aid of a stereomicroscope.

#### No-load and zero-stress state of the femoral artery.

Three to four segments ∼0.2 mm in length were cut from the proximal femoral artery near the site of the anastomosis. The cross section of the segmental rings in the no-load state (zero pressure) was videotaped. The luminal area, wall area, inner and outer wall circumferential length, and wall thickness were measured with the use of an image analysis system. The measurements were averaged over the three to four rings and used for the subsequent analysis. The rings in the no-load state were cut radially at the anterior position, which was marked with ink. The radial cut causes the ring to open up into a sector, which releases the residual stress and has been shown to represent the zero-stress state (5). The cross sections of sectors were videotaped 30 min after the radial cut to allow the vessel creep to subside. The circumferential lengths of the inner and outer surfaces were measured. The opening angle was defined in accordance with Fung (5), i.e., the angle between two radii joining the midpoint of the arc of the inner wall of the vessel to the tips of the sector. All femoral arterial sectors were fixed in formalin for histological preparation.

#### Histological preparations.

The vascular specimens were fixed in formalin over 24 h and embedded in paraffin. Five-micrometer-thick sections were cut and stained with hematoxylin and eosin and elastin-Trichrom stain. The smooth muscle, collagen, and elastin in the arterial wall were distinguished with three different colors in the elastin-Trichrom stain. Cross-sectional wall area, vessel thickness, and smooth muscle cell count per area were determined from the 8-bit BMP color images.

#### Biomechanical analysis.

Analysis of deformation requires calculations of strain. The circumferential strain and stress of the arteries were determined under the assumptions that *1*) the material in the artery is homogenous, and *2*) the vessel shape is cylindrical. The circumferential stretch ratio of the artery (λ_{θ}) can be computed with respect to the zero-state state according to the following equation
Equation 1where *l*
_{θ} is the circumferential length at the no-load or in vivo state, and *L*
_{θ} is the circumferential length at the zero-stress state. In large deformation, one should use Green's strain, defined as
Equation 2where *E*
_{θ} is the circumferential strain in either the no-load (residual strain) or loaded state (in vivo strain) corresponding to the circumference (*l*
_{θ}) at no-load or in vivo states, respectively, in reference to the circumference (*L*
_{θ}) at zero-stress state. Residual strains were computed at the inner and outer wall surfaces, whereas in vivo strain was calculated at the midwall. The midwall circumferential length of the artery was calculated as the average between the inner and outer circumferences.

At an equilibrium condition, the average circumferential stress in an arterial wall at the in vivo pressure can be computed with an assumption that the shape of the artery is cylindrical as follows
Equation 3where ς_{θ} is the circumferential stress, P is the inflation pressure, *r*
_{i} is the arterial inner radius, and *h* is the wall thickness at the in vivo pressure. With an assumption that material in the artery is incompressible, the inner radius of the artery at the in vivo pressure can be computed on the basis of the measured outer diameters at the in vivo and no-load states and the stretch ratio in the longitudinal direction as given by the incompressibility condition
Equation 4where *r*
_{o} and *r*
_{i}are the outer and inner radii at the in vivo state, respectively, and*Ao* and λ_{z} are the cross-sectional area at the no-load state and axial stretch ratio, respectively.

The shear rate due to blood flow on the inner wall of the artery is computed by the equation for laminar flow (γ˙) in circular cylinders
Equation 5where *r*
_{i} is internal radius of blood vessels, and Q is the volumetric blood flow rate. Under the condition of high shear rate, blood viscosity is approximately constant. The wall shear stress is the product of the wall shear rate and the viscosity of blood.

#### Statistical analysis.

The results are expressed as means ± SE. The time course of the changes in the artery after the creation of the a-v fistula and differences between groups were examined with a two-way ANOVA. Student's *t*-test was also used to detect possible differences between groups of data. The results were regarded as significant if *P* < 0.05.

### Analysis of Indicial Response Experiments

Because the experiments are done by approximate step change of blood flow, we have to transform the results to those of exact steps by computation. This can be done if the system is linear with respect to the amplitude of the disturbance about a homeostatic state. Assuming linearity, this can be done as follows. Let *L*(*t*) be a parameter of the system, e.g., the wall thickness, the circumference, the opening angle, and so forth; *t* is time. Let *F*
_{LQ}(t) be the IRF of*L*(t) − *L*(0) to a unit step change of ΔQ(*t*) = ΔQ(0)**
H
**(

*t*), where

**(**

*H**t*) is the heavyside step function, the value of which is zero when

*t*< 0, one when

*t*> 0, and one-half when

*t*= 0. If the flow perturbations consist of a jump from 0 to ΔQ(0) at

*t*= 0 and a continuous function ΔQ(

*t*) that has a finite derivative at

*t*> 0, then under the linearity assumption (see Fung, Ref. 7), the following convolution integral applies Equation 6To determine the IRF, we used the method of Laplace transformation. The Laplace transform of ΔQ(

*t*) is Δ̅Q̅(

*s*), defined by the integral Equation 7Similarly, the Laplace transforms of Δ

*L*(

*t*) and

*F*

_{LQ}(

*t*) are obtained by multiplication with

*e*

^{−st}and integration of the product from zero to infinity to obtainΔ̅L̅ ̅(

*s*) and ̅F̅ ̅L̅Q̅ ̅(

*s*). The Laplace transformation of

*Eq. 6*is Equation 8Our experiments show a blood flow history that is equal to the homeostatic flow plus a step and a perturbation Equation 9then Equation 10aand Equation 10bwhere

*A*,

*A*

_{1},

*A*

_{2}, and

*A*

_{3}are empirical constants determined from curve fitting of experimental data. The response of the various parameters measured can be expressed in one of the following three forms Equation 11a Equation 11b Equation 11cwhere

*B*,

*b*,

*B*

_{1},

*b*

_{1},

*B*

_{2}, and

*b*

_{2}and

*C*

_{1},

*C*

_{2}, and

*C*

_{3}are empirical constants determined from curve fitting of experimental data. The Laplace transform of the input function, ΔQ(

*t*), is Equation 12The Laplace transformation of the output quantities are Equation 13a Equation 13b Equation 13cThe Laplace transform of the IRF, ̅F̅ ̅L̅Q̅ ̅(

*s*), can be determined for each of the cases by substituting

*Eq. 12*and the respective

*Eq. 13*into

*Eq. 8*as shown in the . The inverse transformation of ̅F̅ ̅L̅Q̅ ̅(

*s*) can then be computed to obtain the IRF of remodeling for the various vessel parameters (see ).

## RESULTS

### In Vivo Data

The rats experienced small weight loss 2 days after the a-v fistula and gained weight during the remaining study period (*P* < 0.01), as shown in Fig.1
*A*. They gained ∼30 g (≈12% of body wt) during the 3-mo period.

The systemic arterial blood pressure measured in the carotid artery is shown in Fig. 1
*B*. The arterial pressure did not change postoperatively (*P* > 0.05). The a-v fistula created a significant increase in pressure drop along the femoral artery because of the decrease in downstream pressure at the site of the anastomosis. Hence, the blood flow rate in the anastomosed femoral artery increased to a value three to four times larger than that of the control leg 2 days after the a-v fistula (*P* < 0.01) and did not significantly change during the remaining period (*P* > 0.5) (Fig. 1
*C*). The greatest increase in blood flow occurred at 2 wk after the fistula. The flow-induced remodeling of the in vivo outer diameter of the femoral artery is shown in Fig. 1
*D*.

### No-Load and Zero-Stress State Data

The inner and outer wall circumferences at no-load and zero-stress states are shown in Fig. 2, *A*and *B*, respectively. Increased circumference indicates arterial dilation. Both the luminal and wall area in the no-load state increased postoperatively (*P* < 0.001) (Fig.2, *C* and *D*, respectively). This was further evidenced by the histological findings as will be discussed (see*Histological Data*). The wall thickness did not differ between the control and fistula groups (*P* > 0.1). There was also no statistically significant increase in wall thickness with time in control arteries (*P* > 0.1) (Fig. 2
*E*). The thickness-to-radius ratio, however, decreased significantly with time compared with the control group (*P* < 0.05), as shown in Fig.2
*F*. This was due to an increase in lumen radius. All data of control arteries (Fig. 2, *A*–*F*) showed no statistically significant variation with time (*P* > 0.5). With the exception of wall thickness (*P* > 0.5), all other morphometric parameters (Fig. 2, *A*–*F*) showed significant changes after the creation of the a-v fistula (*P* < 0.001).

The temporal change of the opening angle is shown in Fig.3
*A*. The opening angle did not show significant change in the control group (*P* > 0.1). In the a-v fistula group, the opening angle was unchanged after 4 wk and then subsequently decreased (*P*< 0.01) compared with the control group. The difference between fistula groups (at 8 and 12 wk) and control groups was statistically significant (*P* < 0.05). The arterial residual strain at both the inner and outer surface is presented in Fig. 3
*B*. It can be seen that the outer strain is tensile, whereas the inner strain is compressive for both the control and fistula groups. The average outer residual strain becomes more tensile in the 1- and 2-wk groups, whereas the average inner residual strain becomes less compressive in the 4-, 8-, and 12-wk fistula groups compared with the control. The temporal variations of the inner and outer residual strains in the control and fistula groups were not found to be statistically significant (*P* > 0.1). Figure 3
*C* shows a positive correlation between the opening angle and the wall thickness-to-radius ratio (*r *= 0.906, *P* < 0.05).

### Biomechanical Data

The incompressibility condition expressed in *Eq. 4
* was used to compute the inner radius of the femoral artery, as shown in Fig. 4
*A*. It can be seen that the inner radius in the femoral artery increased from an average value of 0.33 to 0.40 mm in 12 wk after the creation of the fistula. We also computed various biomechanical parameters including wall shear rate, midwall circumferential strain, and mean stress, as shown in Fig. 4,*B*–*D*, respectively. After the creation of the a-v fistula, shear rate, midwall strain, and mean stress significantly increased initially and decreased thereafter (*P* < 0.001).

### Histological Data

In control arteries, the tunica media is composed of an average of four to five smooth muscle cell layers. It is bounded by a heavily stained, single-layered internal and external elastic lamina. There are several layers of continuous elastic fibers throughout the tunica media. The tunica intima consists of a confluent endothelial monolayer. Figure 5 shows a photomicrograph of a right femoral artery during the progression of flow overload. The relationship between the cross-sectional area of arterial media in the zero-stress state and the duration of the a-v fistula for the two groups is shown in Fig.6
*A*. The change in the medial area of the control artery throughout the 12-wk period was not statistically significant. At 4 wk after fistula, the femoral medial area was increased compared with control. The increase in medial cross-sectional area was due to both net smooth muscle cell (SMC) proliferation and cell hypertrophy. Figure 6
*B* shows the SMC proliferation during the progression of the fistula. The change in the SMC count (the total number of SMC in the medial wall of the open sector, i.e., in the zero-stress state) of the control artery was not statistically significant in the 12-wk period. After 4 wk of flow overload, the SMC count was significantly larger in the fistula group compared with the control group. The large increase in medial cross-sectional area, however, was mainly due to SMC hypertrophy. Small increases in intimal thickness were observed in some parts of some arteries.

### IRF Data

The constants *A*, *A*
_{1},*A*
_{2}, and *A*
_{3} in *Eq.9
*, characterizing the change in blood flow, were found to have values of 18.6 ml/min, 9.38 ml/min, 0.214 ml · min^{−1} · wk^{−1}, and −0.00288 ml · min^{−1} · wk^{−2}, respectively, as determined by a least square fit of the flow data (*R*
^{2} = 0.982). The exponential function,*Eq. 11a
*, was used to fit the experimental data for inner and outer circumferences, inner wall area and thickness-to-radius ratio in the no-load state, inner circumference at the zero-stress state, and internal diameter in the in vivo state. The least square fit constants*B* and *b* along with the correlation coefficient are listed in Table 1 for the various parameters. Similarly, a biphasic function, *Eq. 11b
*, was used to fit the experimental data on the inner and outer strain in the no-load state and flow shear rate and the midwall strain in the in vivo state. The empirical constants are given in Table 1. Finally, a cubic function, *Eq. 11c
*, was used to fit the data on the outer circumference and opening angle in the zero-stress state and the wall area in the no-load state with the corresponding constants listed in Table 1.

The curves showing the time history of various parameters (e.g., Figs.2-4 and 6) are not indicial curves, because the flow change, although having an approximate step increase at *t* = 0, did not remain constant at *t* > 0. To simplify the interpretation, we need to extract the IRFs from these curves. Figures7-9 are the IRFs computed from the experimental data shown in Table 1 with the formulas given in the
(*Eqs. 20, 24
*, and *
27
* for*cases I*, *II*, and *III*, respectively). The IRFs of inner and outer vessel circumferences in the no-load and zero-stress states are shown in Fig. 7, *A* and *B*, respectively. Figure 7
*C* shows the indicial response of the opening angle. The histological data on the area of media and SMC count were also extracted in the form of IRFs as shown in Fig.8, *A* and *B*, respectively. Finally, the various biomechanical parameters (radius, shear rate, circumferential strain, and stress) were also expressed in terms of IRFs as shown in Fig. 9,*A*–*D*.

## DISCUSSION

### Flow-Induced Remodeling of Blood Vessels

The first author to clarify the relationship between blood flow and blood vessel remodeling was Thoma (30), who in 1892 observed that in chick embryos, certain pathways of most rapid blood flow increased in caliber and length. Thoma observed that the growth of blood vessel lumen depends on the flow, and the wall thickness depends on the tension in the wall. Schretzenmayr (26) confirmed Thoma's observations in 1933. More recent studies have shown that increased blood flow induces blood vessel dilatation even in small muscular arteries (15, 29). Flow-induced dilation is found to be influenced by local endothelial cell function (15,28). It is now accepted that shear stress acts through the endothelium to regulate both acute vessel tone and chronic remodeling of blood vessel (2). The endothelium acts as a complex mechanical signal-transduction interface between the flowing blood and the vessel wall (2, 23).

### Uniform Shear Hypothesis

The constant wall shear rate hypothesis implies that the volumetric flow rate is proportional to the cube of the vessel radius, assuming a laminar, steady-state, and incompressible Newtonian flow through a rigid cylindrical vessel (13). Hence, to maintain a constant wall shear rate, the cube of the radius must increase in proportion to the increase in blood flow. Blood vessels can accommodate such a change in vessel radius at two levels: acutely through vasoactive mechanisms (flow-dependent constriction or dilation) and chronically by adjusting vascular caliber (12, 32).

A number of studies have previously shown a normalization of wall shear strain after a significant increase in blood flow (10-12,32). Kamiya and Togawa (12) created an a-v fistula between the common carotid artery and the external jugular vein in dogs and studied the remodeling 6–8 mo postoperatively. They found a normalization of shear rate to within 15% despite a fourfold increase in blood flow. Additional a-v fistula studies of the iliac artery of the monkey and rabbit (21, 32) and radial artery of humans with end-stage renal disease (10) also showed a normalization of shear rate by remodeling of vascular caliber. In the present study, we increased the blood flow in the rat femoral artery by creating an epigastric venous-to-femoral arterial fistula, which subsequently increased the wall shear rate or stress (if viscosity is constant) by ∼83% after 2 days. The shear rate decreased to within 27% of the control value, however, after a 12-wk period. The differences in shear rate at 12 wk were not statistically significant, as shown in Fig. 4
*B*.

### Remodeling of the Zero-Stress State

Mechanical factors have been proposed to regulate growth and remodeling of biological tissue (6). Since Fung (5) and Vaishnav and Vossoughi (31) found that residual stress exists in the arterial wall when external loads are absent (no-load state), the zero-stress state has been used as a reference state for mechanical analysis. Furthermore, it is best to measure the structural components of the vessel wall at the zero-stress state, because in this state the morphology and sizes of the cells and extracellular matrix are not distorted by stress and strain. Otherwise, there is the complication of deformation due to internal stress (6). Residual stress can be altered by many factors, including tissue growth, remodeling (changes in material properties), and acute geometrical changes (22). Experimental evidence has shown that the opening angle of an artery increases during hypertension, which indicates an increase in the residual strain (9, 20). The decrease in the opening angle in response to flow overload shown in this study suggests that the loading pattern is an important determinant of growth and remodeling of the zero-stress state.

Experimental and theoretical evidence suggests that geometric remodeling alters the residual strain in the vessel wall (22). In a theoretical analysis, Rodriguez et al. (24) predicted that concentric hypertrophy, which increases the wall thickness-to-radius ratio, increases the opening angle and hence the residual stress. They also showed that eccentric hypertrophy characterized by a decrease in the wall thickness-to-radius ratio may induce a reversed transmural residual stress gradient. Omens et al. (22) provided experimental evidence for this prediction in ventricular remodeling during cardiac postnatal growth in rats. Their study showed that eccentric hypertrophy decreased the opening angle and residual stress in the circumferential direction. Those results in the heart are consistent with our flow-overload remodeling in the femoral artery. This is in contrast to concentric remodeling induced by hypertension, where the opening angle and the residual stress are increased. In hypertension, it is proposed that the increase in residual stress makes the transmural stress and strain distributions more uniform (20). Omens et al. (22) suggested that the decrease in opening angle with a decrease in wall thickness-to-radius ratio in response to flow overload occurs for purely geometric reasons, because the vessel becomes a thin-walled cylinder having a more uniform transmural stress distribution independent of residual stress.

In the present study, the decrease in opening angle is consistent with Fung's (8) hypothesis of nonuniform remodeling; i.e., if the inner wall grows more than the outer wall, the opening angle will be increased, whereas if the outer wall grows more than the inner wall, the opening angle will be decreased. Figure 7 shows that in flow overload, the outer wall grows more than the inner wall, and hence the opening angle is decreased.

### Remodeling of Circumferential Stress and Strain

An increase in wall shear stress causes circumferential dilation, which increases the circumferential stress and strain in the vessel wall. This implies that circumferential stress and strain are additional factors that may play a role in arterial wall remodeling in flow overload. Observation on vein grafts provides evidence to support this hypothesis. Dobrin et al. (3) constrained the vein grafts circumferentially by partially enclosing the grafts in a cuff. The vein grafts were exposed to arterial pressure and flow. The part of the graft located in the cuff could not dilate, whereas the other part could dilate. The two parts were exposed to the same arterial pressure, but the circumferential stress and strain were different. Their results showed that the increases in the vessel medial thickness and diameter were best associated with increased wall stress and strain in the circumferential direction. Other observations from vein grafts suggest the same conclusion (14, 27). Hence, circumferential deformation may play a role as a regulatory factor in arterial remodeling caused by flow-induced dilation.

The previous studies on vascular remodeling in response to physical stress (3, 10-12, 21, 32) did not assess the circumferential strain because the reference state, zero-stress state, for the evaluation of strain was not investigated. One of the goals of the present study was to characterize the remodeling of the zero-stress state and to use the results to compute the strain. In our a-v fistula model, the midwall strain and circumferential stress were increased by ∼53 and 17%, respectively, after 2 days. The vascular remodeling occurred in such a way as to normalize the midwall strain and circumferential stress of the vessel wall, which were found to be within 22 and 12% of their respective control values at the 12-wk period. These differences were not statistically significant, as shown in Fig. 4, *C* and *D*. In fact, the differences between the fistula and control groups were not statistically significant beyond 4 wk and 8 wk for the mean stress and midwall strain, respectively.

The physical factors (wall shear rate, circumferential stress, and midwall strain) that stimulate vascular remodeling may be expressed in terms of the IRFs for the three biomechanical parameters as shown in Fig. 9, *B*–*D*. To compare the relative magnitude of the three biomechanical IRFs, we normalized the quantities with respect to their initial values as shown in Fig.10. It can be seen that the two major biomechanical stimuli for remodeling are wall shear strain and midwall strain. The midwall circumferential strain has a shorter rise time and a shorter decay time than the wall shear rate. This suggests that the circumferential strain may be an important initial stimulus for remodeling.

### IRFs

A systematic approach to describe the remodeling process is to extract the IRFs as given by *Eqs. 20
*, *
24
*, and*
27
*. The use of IRFs simplifies the interpretation of data and greatly increases the potential of using experimental data for prediction of the outcome of future experiments under an arbitrary course of stimulation. It also allows the reduction of complicated sets of experimental data into definitive statements in terms of IRFs. Finally, it presents a quantitative way to verify the basic hypothesis of linearity. This is a powerful engineering approach that provides a rigorous method to study the function of blood vessels by properly formulated boundary-value problems and makes possible a wide variety of applications to the study of tissue engineering.

### Critique of Methods

The surgical trauma (including mechanical dissection, compression, ischemia, barometric and osmotic changes, and so forth) involved in the construction of an a-v fistula may stimulate vascular remodeling and hyperplasia. Because a sham operation was not done on the control leg, this raises the issue of whether any of the biomechanical changes are a result of surgical trauma. Although we did not fully address this issue, we did obtain data that suggest that the effect of surgical dissection is small. In five animals, both the left and right femoral arteries were dissected in preparation for the a-v fistula. The fistula, however, was only created in the right femoral artery. Hemodynamic and morphological data were obtained at 8 wk (2 animals) and 12 wk (3 animals) after the initial surgery. We found no statistically significant differences between the dissected left femoral artery and the corresponding nondissected control in the 8- and 12-wk groups.

The equation used to compute the shear rate, *Eq. 5
*, is based on the assumption that blood flow has a Poiseuille profile. To examine this hypothesis, at the site where the morphometric measurements were made, we must examine the architecture of the vessel. The femoral artery arises from the external iliac artery. Approximately 3 mm from the external iliac bifurcation, the femoral artery gives rise to a Morphy branch, which is 0.07 mm in diameter. The measurements were made at 1 mm downstream of the Morphy branch. The anastomosis was ∼3 mm downstream of the site of measurements. The issue here is as follows: Does the entry flow into the femoral artery become fully developed at the site of measurements? In other words, what is the inlet length (which is defined as the distance through which the velocity profile becomes approximately parabolic)? Lew and Fung (16) have previously shown that the entrance length (*L*
_{e}) is approximately given by *L*
_{e} = 0.08*R*
_{e}
*D*, where*R*
_{e} and *D* are the Reynolds number and diameter of the vessel, respectively. This expression applies for Reynolds numbers in the range of 50–100. The Reynolds number is given by *R*
_{e} = (4ρQ)/(πμ*D*), where ρ, μ, and Q are the density, viscosity, and flow rate of blood, respectively. The range of Reynolds numbers was found to be 48–120 for the range of flow rates and diameters (5.0–18.2 ml/min and 0.60–0.80 mm, respectively). Hence, we obtain an entrance length of 2.3–7.6 mm. Because the diameter of the Morphy branch is relatively small, there is an effective length of 4 mm from the external iliac artery to the site of measurements. Hence, the velocity profile should be nearly parabolic, and *Eq. 5
* is a reasonable approximation.

### Future Studies

*Equations 20
*, *
24
*, and *
27
*are based on the linearity assumption of the response function*L*(*t*) with respect to the amplitude of the flow step. A system is linear if *L*(*t*) is proportional to ΔQ(0) in *Eq. 6
*. Otherwise, the system is nonlinear. The linearity of the system can be tested in two ways. First, one can directly test the relationship between*L*(*t*) and ΔQ(0) by varying the magnitude of the step function ΔQ(0) and measuring the corresponding values of *L*(*t*). Experimentally, the flow rate through the fistula can be varied by constriction of the epigastric vein. Second, validation of linearity can be done by performing a new perturbation, predicting the results according to*Eq. 6
*, and comparing the predication with measured results. For example, one can impose a flow history by constriction of the epigastric vein with an ameroid. The ameroid will slowly occlude the epigastric vein and hence yield a monotonic decrease in flow after the initial rise. The predictions of *Eq. 6
* can be compared with the experimental results of the ameroid experiment to determine the limits of validity of the linearity hypothesis.

If a system is nonlinear, but a small change in ΔQ(0) results in a small change in *L*(*t*), then *Eq.6
* can still be used. If a small change in ΔQ(0) produces a large change in *L*(*t*), then the system is grossly nonlinear and *Eq. 6
* cannot be used. In this case, a record of the way the indicial function depends on ΔQ(0) is the best way to quantitatively express the nonlinearity; i.e., *F*
_{LQ} is not only a function of *t* but is also a function of the magnitude of Q. Modifying the IRFs in this way, one can still use *Eq. 6
*, which now becomes nonlinear. The range of agreement between theory and experiment will correspond to the range of linearity. The goal of future experiments is to find the upper and lower limit to linearity. The limits are related to the borderline between physiology and pathology.

## Acknowledgments

We thank the Institute of Experimental Clinical Research; the NOVO NORDISK Centre of Growth and Regeneration; the Karen Elise Jensens Foundation; and the National Heart, Lung, and Blood Institute Grant 5-R29-HL-55554 for financial support. G. S. Kassab is the recipient of the National Institutes of Health First Award.

## Appendix

#### Case I.

The Laplace transform of indicial function,
̅F̅
̅L̅Q̅
̅(*s*), can be determined for *case I* by substituting *Eqs. 12
* and *
13a
* into *Eq. 8
* to yield

Equation 14

*Equation 14
* can be rewritten as
Equation 15where
Hence, the Laplace transform of the IRF has the form
Equation 16To obtain the inverse Laplace transform of *Eq. 16
*, it is desirable to express it as a sum of polynomial terms using partial decomposition. Hence, *Eq. 16
* can be expressed in the form
Equation 17The use of partial decomposition yields the following three equations for the three unknowns, *X*
_{1},*X*
_{2}, and *X*
_{3}
Equation 18a
Equation 18b
Equation 18c
*Equation 18* is a system of 3 × 3 linear equations, the solution of which is
Equation 19a
Equation 19b
Equation 19cConsequently, the inverse Laplace transform of *Eq. 15
*gives rise to an IRF of the form (Erdelyi, Ref. 4)
Equation 20
Because the constants *B*, *b*, α, and β are given by experimental results (Table 1),*F*
_{LQ}(*t*) can be derived mathematically from the experimental data by using *Eq. 20
*. More complex formulas needed to handle the experimental data are considered in*cases II* and *III*.

#### Case II.

Similarly, *Eqs. 12
* and *
13b
* can be substituted into *Eq. 8
* to yield

Equation 21
Solving for the Laplace transform of the IRF, we obtained
Equation 22
*Equation 22
* can also be expressed as a sum of polynomial terms using partial fraction decomposition in the form
Equation 23
The coefficients of the various terms of the polynomials on the*left* and *right* sides of *Eq. 23
* can be equated to yield a system of five equations for the five unknowns (*X*
_{1}…*X*
_{5}). In matrix form, we have

The system of linear equations is solved numerically for the unknowns *X*
_{1,}
*X*
_{2}… *X*
_{5} for a given set of constants: *B*
_{1}, *B*
_{2},*b*
_{1}, *b*
_{2}, α, and β as given in Table 1. Hence, the solution for the IRF of *Eq. 22
*has the form

Equation 24

#### Case III.

Finally, to obtain the IRF for *case III*, we similarly begin by combining *Eqs. 12
* and *
13c
* into *Eq.8
* to obtain

Equation 25or
Equation 26Again, *Eq. 26
* can be expressed as a sum of simple polynomial expressions, the inverse Laplace transform of which can be evaluated to obtain
Equation 27

The constants *C*
_{1},*C*
_{2}, and *C*
_{3} are listed in Table 1. The constants α and β depend on the constants*A*, *A*
_{2}, and *A*
_{3}as described above.

## Footnotes

Address for reprint requests and other correspondence: G. S. Kassab, Dept. of Bioengineering, Univ. of California at San Diego, 9500 Gilman Dr., La Jolla, CA 92093-0412 (E-mail:kassab{at}bioeng.ucsd.edu).

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- Copyright © 2001 the American Physiological Society