Abstract

Observation of the post-kidney transplant renograms show an increase in the velocity of the blood flow in the iliac arteries on the transplanted side compared with the side without transplant. A hypothesis for explaining the above observation was made and based on this observation we devised a glass model with side branches modeling the aorta and iliac arteries. Water was made to flow in the model with a constant pressure, the volume of water collected per thirty seconds was measured and the findings tabulated for multiple trials. The observations show an increased volume of water collected on the side with side branches open. Later a theoretical electric-circuit model was developed, the observations made were found to be agreeing with the water in glass model. In the electric model the branching tubes as resistors. The equivalent resistance (Eq) of resistors connected in parallel is lesser than the individual resistance (R). The theoretical explanation by electric circuit model and the practical fluid flow studies supports the observations made on the renograms. Thus, it can be concluded that probably the low resistance that the transplanted kidney offers to blood flow on the side the graft is attached made the flow ahead with its counter part in iliac and femoral arteries.

Introduction

The behaviour of blood flow in the post-renal transplant patients are monitored by renograms and other modalities like colour Doppler Imaging to ensure a normal functioning of the kidney after transplantation. The normal transplant kidney has a low resistance arterial vascular bed characterized by streamlined systolic flow and continuous forward flow in diastole on Pulsed Doppler. Normal main renal artery flow velocity ranges between 20 and 52 cm/sec with a mean of 32 cm/sec.1

For kidney failure patients the renogram, before transplant shows the isotope appearing in the lower extremity arteries i.e. right and left side at the same time on serial frames. However after a renal transplant the isotope renograms shows the isotope to travel a greater distance in the Iliac and femoral arteries on the kidney transplanted side when compared to its counterpart on the other side, as determined in serial frames of the renogram. The position of the kidneys are shown in Fig. 1.

Aims and Objectives

This study was designed to assess a phenomenon observed on renograms of post

Fig. 1 :Position of the kidneys (Posterior view, spine removed). Position of the transplanted kidney (front view-schematic).
Fig. 2a :Renogram pretransplant.	Fig. 2b :Renogram posttransplant. The isotope carrying blood reaches and travels a greater distance on the right side when compared to the left (arrow marked). The transplanted kidney is connected to the right iliac vessels in this case.

renal transplant patients and to verify the fluid dynamic explanation hypothesized.

Material and Methods

Renograms performed at K.J. Hospital for patients who underwent renal transplants were studied. Pre and Post renal transplant renograms of this group was analyzed to assess the level of isotope carrying blood on serial frames (Fig. 2).

Based on the observation made, a glass model stimulating the aorta and its bifurcation in the human being was devised at our centre and experiments were conducted using water as fluid for the purpose of assessing flow in the glass model at constant pressure.

In our glass model the system is a steady flow state under a constant pressure head, the walls of the tube being parallel as shown in the diagram (Fig. 3). The glass model is connected to uniform bored rubber tubes at the ends and water is made to flow under a constant pressure head. The out coming water

from the distal end of the tubes i.e., B and C is collected for 30 seconds and the volume is measured. By opening the side branches one following the other the procedure is repeated until all the side branches are opened. For each opening of the branches five trials were carried out similar to the procedure described above and the readings were averaged and tabulated (Table 1). Our observations show an increase in the volume of the water at the side (C) when the side branches are all open.

Increased flow volume through fixed diameter tubes is possible in the system only by increased flow rate i.e., the velocity of flow.

Comparison of the Increment of the Flow in Tube-B with Tube C

Trial - 1 : All the side branches closed (3.48%) difference in the flow possibly due to system error.

Trial - 2,3,4 and 5 : One of the side branches opened serially (4.75% increase in the flow compared with trial - 1).

Trial - 6 : All the side branches opened (7.81% increase in the flow compared with trial- 1 and 2.45% increase in the flow compared with trial 2,3,4 and 5).

The side branches of the glass model represents the transplanted kidneys run off from the aorta in the invivo state.

The Theoretical Model Based on Electrical Circuits

The electrical analogy of the flow path is shown in the following figures. Where, RA, RB, RC and RD are the flow resistance of the corresponding branches.

From Fig. 4a, we can say if pipe diameter is equal for branch B and C then the flow rate through Branch B and C is also same for an equal pressure drop. In the second case the experiment was carried out with opening of sub-branches. This situation is shown in Fig. 4b. The equivalent flow resistance in branch C can be estimated from following equation:

Table 1 : Flow in the tubes (ml/30 S)
Fig. 4a : Flow resistance diagram, when all sub-branches are closed.	Fig. 4b : Flow resistance diagram, when one or more sub-branches are opened.

REq = (RC X RD)/(RC + RD)

From the above equation it can be seen that REq is always less than RC or RD, this means that, by opening one or more sub-branch of branch C, we are basically reducing the equivalent flow path resistance which in turn allows more total flow in branch C and the sub-branch.

Results and Discussion

The blood flow in the human system is pulsatile, its frequency being about 72 beats/min. The flow is laminar and axial involving elastic vessels with narrowing branches at one end of the vasculature. Blood is a complex fluid compared to water. About half of the volume of blood is blood cells. These cells are elastic in nature and hence cell geometry is time dependent.2 This means that blood has to be considered as mix of solid (elastic) with liquid whereas water is much more straightforward fluid for analysis.3 The presence of these blood cells makes blood have in a way that's different from the way water would. However for simplicity, water and a steady flow state under constant pressure through glass tube were used in the model.

Inspite of the above limitations, the glass model experiments shows evidence and the electrical circuit mode offers explanations consistent with the observation made on the renograms. We can conclude that the transplanted kidney creates a low resistance on the transplanted slide of the iliac artery circulation causing larger volume of blood to flow on that side. Thus the radioisotope carrying blood flows faster and reaches a greater distance on the transplanted side when compared to its counterpart on the other side as observed on the renogram.

Acknowledgement

We wish to thank Professor Dr. Placid Rodriguez, formerly Director, Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam and Prof. Vaidyanathan and his team, Department of Engineering IGCAR for their valuable explanation by giving the theoretical circuit example and suggestions for using Computational Fluid Dynamics to arrive at resolved solutions. We are also thankful to the Director, Central Leather Research Institute (CLRI) for his help in making the glass model.

We also thank K.J. Research Foundation Laboratory/Mr. MP Vikraman, Chief X-ray Technologist, Department of Radiology, K.J. Hospital for their timely help through out the study.

References

• Charles V, Zwirewich MD. "Renal transplant imaging and intervention. Practical Aspects - 2".
• Himeno R. "Blood flow simulation toward actual application at hospital", The 5th Asian Computational Fluid Dynamics, Busan, Korea. 2003: 27-30.
• Ken-ichi Tsubota, Shigeo Wada, Hiroki Kamada, et al. "A particle method for blood flow simulation - Application to flowing red blood cells and platelets". J Earth Simulator 2006; 5 : 2-7.

*Research Scholar; **Dean of Faculty; ***Chief Physician; +Founder and Director, KJ Hospital Research and Postgraduate Centre, K.J. Research Foundation, Chennai 600 084.