Biological model of calcium oxalate stones

At present, there are two main models to describe the initiation of calcium oxalate stones in the urine. Finlayson and Reid named it free particle, fixed particle. The theory of free particulate type suggests that the initiating factor of stone formation is due to the supersaturation of calcium oxalate (CaOx) in the tube fluid, and once calcium oxalate is nucleated in the urine, it can grow and accumulate as it passes through other parts of the kidney. If this process occurs quickly enough, some key crystalline particles will form and reach a large enough size to stay in some narrowing of the renal tubule before being squeezed in the collecting duct, and once the particles remain they will continue to grow until the stone is formed, and certain clinical symptoms are produced.

In 1978, Finlayson and Reid used a simplified mathematical model of the kidney, which concluded that this process could not occur in the short time that urine passed through the kidney. According to their model, they believe that a single crystal does not have enough time to form large enough particles to stay before being extruded in the collecting tube, and that the formation of small CaOx nuclei is due to excessive supersaturation of CaOx in the tubular fluid, due to crystallization or infection, or other causes of cell necrosis caused by damage to some sites of renal epithelial cells, and this damage point becomes the adhesion point of the crystalline nucleus precipitated by excessive supersaturation of CaOx in the tubular fluid, which they call the fixed particle model of stone formation.

In the following two decades, the theory of fixed particle model has not been challenged, and during this time, the theory of fixed particle model has become the basis for researchers to study the nucleation and growth of CaOx crystallization in various kidney cell cultures. In 1994, Kok and Khan re-evaluated Finlayson and Reid's model of stone formation and corrected some errors in the original model, and Kok and Khan combined some new evidence on tubular diameter and urinary flow rate through the tubules. Based on these modified models, they concluded that, in principle, they agreed with Finlayson and Reid's findings, but they also believed that in certain cases, when CaOx crystals begin to move in the kidney, they can adhesion as they pass through the tubules elsewhere, and that the free particle model is also feasible. Kok and Khan's argument is essentially a combination of a free particle model and a fixed particle model. In 2002, Robertson re-examined the models of Finlayson and Reid, Kok and Khan, and gave a presentation at FASEB on "The Biological System of Calcium Oxalate", in which Robertson added some new features to his model and modified the findings of the former, and compared the data from the three models in his paper.

1. Comparison of oxalate stone models

All three models are based on known glomerular filtration rate and water and ion reabsorption processes in the kidney. The renal tubular system is divided into 5 parts: the proximal convoluted tubule, the descending and ascending branches of the medullary loop, the distal convoluted tubule and the collecting duct. The pattern of the kidneys is based on a dendritic system, with a total of about 2.5×106 glomeruli in the two kidneys, and the flow of fluid in the kidneys is calculated by the crossing of isotonic water and solute across the proximal tubule.

Finlayson and Reidr's model is the earliest prototype and is relatively simple because they do not consider that the length of the loop of the kidney is not the same, and the length of the loop depends on the depth at which it enters the medulla. The biggest difference between the three models is the calculation of the transfer time through the manifold, with the Finlayson and Reidr models taking longer than the other two models, and this is due to the slower flow rate due to the wider diameter of the manifold in the Finlayson and Reidr models. Recent studies have shown that fluid transport times are shorter in the Kok and Robertison models, which are measured in non-obstructive kidneys at approximately 3 to 4 minutes.

In such a short period of time, the single crystal grows large enough to remain in the narrowing part of the kidney. This is because the calcium oxalate crystal growth process is slow, normally, up to 1-2 μm/min. As Kok and Khan point out, if crystallization adhesion can occur, the particles can form large enough to stay during the time of kidney transport. Although the adhesion of calcium oxalate can be seen in the stone model of fresh urine, as is In the Kok and Khan and Robertson models, the formation of adherent single crystals in the 3-4 min time of fluid from the glomeruli to the binding duct is greater than expected, and even if the Finlayson model is correct, the transport time in the kidney is only increased to 10-11 min, which is still not enough for a single crystal to grow large enough to stay in the collecting duct in their model of a certain width, which is possible for the free particle model, and there are other mechanisms involved in delaying the time it takes for nucleated material to pass through the kidney, so that the single crystal grows large enough to stay in the kidney transport time of 3-4 min。

Second, factors that can delay the time it takes for crystals to pass through the kidneys

Two factors are thought to delay the passage of nucleated substances through the kidney, both of which are based on a more complex system of kidney structure than the previous models. The laminar flow of the previous models is considered to be a straight, uncomplicated geometry of the tubules, which was originally described by Schulz and Schneider, by modeling what is known to be a complex tubular system in which fluid dynamics have potential "non-flow zones" (similar to those in a wide river) in the kidneys, where little or no fluid passes through, For example, at the bend of the pipe or at the junction of two sections of the manifold, these low-flow areas provide a good area for crystallization and facilitate the growth of crystallization until small stones form in this area. Either they flow with the urine, stay in the "non-flowing zone" of the next tube, or are squeezed into the pelvis calyceal system, which has a different internal geometry between the stone model kidney and the normal kidney, and they can affect the further growth of crystals from the kidney.

The second factor that allows the crystalline particles to stay within the tubules is that the tubules are compressed to show a V- or Z-shaped bend. Although the flow rate of the liquid in the compressed tubule is increased, the same particles can stay in the compressed tubule or at least be delayed as it passes through these narrow points. A good study by Graves illustrates this problem. The models of Finlayson and Reidr, Kok and Khan ignore both factors, and it is difficult to describe largely unknown processes with mathematical models.

三、Robertson草酸钙结石模型的其它特征

A recent reconsideration of the models of Finlayson and Reidr, Kok and Khan has led Robertson to include three fluid dynamics factors in the model he studied, which can be analyzed mathematically.

(1) Fluid resistance close to the pipe wall

Schulz et al. point out that some other factors are related to the passage of particles through the tubule, in a laminar flow system, the flow velocity of the liquid is maximum at the central axis of the lumen, the flow velocity is small when approaching the pipe wall, and the flow velocity is zero at the surface of the pipe wall, which means that the velocity of the crystalline particles passing through the tube close to the pipe wall is slower than the flow velocity in the middle of the lumen, from the central axis of the tubule with an inner radius of R radius, and the flow velocity at a certain distance r has the following equation:

Vr=Vr=0×(R2-r2)/R2

Vr is the velocity of flow over a certain distance r; Vr=0 is the velocity at the central axis: R is the inner radius of the tubule.

From the equation, it can be calculated that when r is close to 0.9×R, it can significantly affect the flow time of the liquid within a specific tube length. When r=0.9×R, the time of the cost is about 5 times longer than that at the central axis of the lumen, about 10 times when r=0.95×R, and about 25 times when r=0.98×R. As a result, the transport time of particles in the liquid through specific tubular fragments is significantly delayed near the tube wall due to the delayed effect of the tube wall on the liquid.

(2) Particle resistance near the pipe wall

In addition to the blockage of the flow of liquids by the wall of the tube, the wall of the tube also has a blocking effect on the flow of particulate components in any liquid. When the diameter of the particles is particularly suitable for the diameter in the tube, the speed at which the particles flow in the liquid can be increased, which can be expressed by the following formula:

Vp=Vfluid flow//(1+2.1×p/R)

Vp is the velocity of the particles, Vfluid flow is the velocity of the liquid, P is the radius of the particles, and R is the inner diameter of the tube

(3) When the renal tubule walks upward, the gravity in the tubule acts

At any given time, the direction in which the collecting ducts of the kidneys travel depends on where the papillae are located. In humans, the direction of walking of the collecting duct is standard, with many nipples located at the lower pole of the kidney, thus including the upward walking collecting duct. If there are crystals in a collecting tube that is walking upwards, then these crystals are subjected to gravity, which provides a reaction force for the crystals to run through the tube, causing the crystals to pass through the tube for a longer period of time, so that the individual crystals grow (or stick). The speed at which the pellets run can be expressed by the following formula:

Vg=0.0000374×p2/(1+2.1×p/R)

Vg is the speed at which gravity causes the particle to run downward, P is the radius of the particle, and R is the inner diameter of the tube. This formula is an evolution of the second formula.

With the exception of the bending action proposed by Schulz and Schneider and the kinks action pointed out by Gravers, all of these factors are included in Robertson's model.

Fourth, the factors that delay the crystallization of calcium oxalate stones in the renal tubule through time

According to several studies of percutaneous renal puncture, water is reattracted within the proximal tubule, the descending loop of the medullary loop, and the collecting duct. Calcium is reabsorbed in the proximal tubule, ascending loop of the medullary loop, and distal tubule, and oxalate is reabsorbed in the initial segment of the proximal tubule but resecreted at the apt end of the proximal tubule. When calcium oxalate passes through the renal tubules, the tube fluid is in a relatively supersaturated state (RSS), and in the range of RSS, the value 1 represents the solubility of the solution in relation to salt, the value <1 represents the solution in the unsaturated state, and the value >1 represents the solution in the supersaturated state. If the RSS is much greater than 1 (the value is around 14), the solution reaches a point called FP, at which the calcium oxalate can precipitate automatically in a short period of time. It is important to note that FP is inversely proportional to the formation time of calcium oxalate. When the FP value is 14, the generation time of calcium oxalate is 5min. When the FP value is 12, the generation time of calcium oxalate is 1h. At an FP value of 10, calcium oxalate needs to be produced for 4 days.

In other words, the longer the residence time of the supersaturated solution, the lower the automatic precipitation point of crystallization in the crystalline particle solution. This means that if the flow of the liquid is delayed, crystallization may be nucleated in a lower supersaturated solution near the wall of the tubule. Robertson's model shows that crystals close to the tube wall (r=0.98*R) can be delayed until they grow large enough to lodge or fall into the urine. Under these conditions, the particles can be delayed from a few minutes to 60 minutes at a rate of 1-2 μm/min, allowing the individual crystals to grow large enough to be able to stay at some points in the tubule and form nuclei of tiny stones, and if adhesion can also occur, the time for the stones to stagnate is shorter, and the larger particles that form are more likely to stay or fall from the upward tubule so that they reach a large enough size to prevent being squeezed out of the tubule in the short term.

Under normal dietary control, the concentration of oxalate in the urine is very low enough to form crystals in the descending branch of the medullary loop, and when the supersaturated liquid in the tube reaches crystallization nucleation, any crystals formed end up only at the end of the collecting duct. However, the crystals formed are smaller and can be easily expelled from the collecting tube. Even after oxalic acid loading is achieved, the normal group has sufficient levels of supersaturation to crystallize in the collecting ducts, and there is a small increase in the number of crystals in patients with stones, because of the high urinary oxalate concentration, calcium oxalate nucleation can occur in the descending segment of the loop and in the collecting duct, which can be delayed in the descending loop of the loop during the described process and lead to the formation and aggregation of large individual crystals. These phenomena can occur in the urine of patients with a normal diet, but especially in patients with oxalic acid loading for stones. Due to the rapid growth of crystals in the supersaturated solution, some oxalic acid is depleted, and the level of supersaturated solution in the collecting tube becomes low, resulting in a decrease in the formation of small crystals.

It is clear from the three models that when crystals pass through the tubules, the individual crystals do not grow large enough to stay in the collecting ducts unless they are delayed. In non-obstructive kidneys, the delay takes about 3-4 minutes. Finlayson suggests that initiation of crystallization nucleus formation occurs at some point of damage on the renal epithelial cells, where crystallization is delayed. Finlayson concluded that the "fixed particle" model was the initiating cause of calcium oxalate stone formation. Based on current data on the size of crystals and the time of transport through the kidney. Kok and Khan and Hess and Kok hypothesized that there were other factors that could cause some crystals to form large enough to remain in the tubules during the transport time of the urine through the tubules, allowing calcium oxalate crystals to adhere and aggregate. From their model, they concluded that as long as there is an adhesion aggregation of crystals, the mechanism of the free particle model may exist. Aggregation of crystals has been observed in fresh, warm urine of patients with recurrent calcium oxalate stones, which supports the free particle model. However, this cannot occur during the normal time when urine passes through the kidneys, because during this time, the size of the single crystals cannot form large enough to form aggregates.

In the latest model, Robertson's model includes several hydrodynamic factors that have been overlooked in previous models: (1) at the central axis of the tubule, the speed of the crystallization operation is different from that of the liquid flow rate, (2) the crystallization is slower than that at the axis of the tube, (3) if the calcium oxalate crystallization is in a sufficiently supersaturated state, the crystallization operation is delayed for 2-3 hours due to various factors, and the crystallization can continue to grow (4) at the long medullary loop of the tubule, the crystallization is more likely to be delayed, and (5) the crystallization may stop or even fall into the ascending segment of the renal tubule。

Robertson's model suggests that (a) calcium oxalate crystals can begin to form in the medullary loop descending branch and collecting duct, depending on the predominant concentration of oxalic acid, (b) the concentration of oxalic acid in the tube increases slightly, and crystals form only in the collecting duct and the crystals formed are smaller, and (c) when the concentration of oxalic acid in the tube fluid increases significantly, crystals can form in two places, the larger nucleated crystals at the medullary loop descending branch, and the smaller nucleated crystals at the collecting place. As a result of high calcium and high concentrations of oxalic acid in the urine, patients with stones have the likelihood of increased crystal formation and the formation of larger crystals and aggregations of crystals associated with calcium oxalate stones. If these factors occur in human kidneys, the free particulate model may be an initiating mechanism for calcium oxalate stone formation.