## Abstract

An analytical Population Balance Equation model is developed and used to assess the risk of critical renal stone formation for astronauts during future space missions. The model uses the renal biochemical profile of the subject as input and predicts the steady-state size distribution of the nucleating, growing, and agglomerating calcium oxalate crystals during their transit through the kidney. The model is verified through comparison with published results of several crystallization experiments. Numerical results indicate that the model is successful in clearly distinguishing between 1-G normal and 1-G recurrent stone-former subjects based solely on their published 24-h urine biochemical profiles. Numerical case studies further show that the predicted renal calculi size distribution for a microgravity astronaut is closer to that of a recurrent stone former on Earth rather than to a normal subject in 1 G. This interestingly implies that the increase in renal stone risk level in microgravity is relatively more significant for a normal person than a stone former. However, numerical predictions still underscore that the stone-former subject carries by far the highest absolute risk of critical stone formation during space travel.

- nephrolithiasis
- gravity
- weightlessness
- crystal nucleation
- crystal growth
- agglomeration

it is estimated that around 5% of the US population will have a clinically significant renal stone incident in their lifetime. With the first evidence of kidney stones in humans found in an Egyptian mummy at E1 Amrah dating back to 4800 BC (29). it seems that mankind has been struggling with nephrolithiasis for centuries. In the advent of the space age, the risk of astronauts developing kidney stones has also become a serious issue for the National Aeronautics and Space Administration (NASA), especially since the duration of future space expeditions will be substantially increasing.

The concern seems to be justified since a recent survey of renal stone formation in US astronauts has revealed 14 recorded episodes (23). Several incidents occurred in the preflight period (*n* = 5), while other episodes (*n* = 9) were in the postflight phase. The time period for the onset of symptomatic stone formation following return to Earth ranged from 9 to 120 mo after landing. Six of the nine postflight episodes occurred after 1994, which corresponded with the extension of the shuttle missions to 12 days.

Renal stones come in different types, and the formation of a specific stone type usually depends upon the existence of particular risk factors that are often revealed by the subject's 24-h urine biochemical analysis. Nevertheless, the most common renal stone and a main component in stones of mixed composition is calcium oxalate (CaOx). It accounts for ∼85% of the clinical occurrences of nephrolithiasis. Four of the six kidney stones of known composition developed by the astronauts have also been CaOx (30).

Space travel has a profound impact on nephron biochemistry. Due to bone atrophy (6, 23, 30) and lower urine volumes caused by dehydration (35, 37), concentration of both calcium and oxalate and hence the urinary supersaturation of CaOx can become significantly elevated. There are other contributing factors such as the astronauts' high protein and sodium level that in addition to inducing higher calcium and pH levels can also provide nidi that may serve as nucleation sites, thus promoting CaOx precipitation (37, 41). The higher pH levels will also lower the concentration of citrate, one of the primary inhibitors of CaOx growth and agglomeration (38, 41). In short, alterations in the renal biochemistry of astronauts tend to provide favorable conditions for increased nucleation, growth, and agglomeration of CaOx crystals in the renal system.

Because of the considerable danger that a clinically significant renal stone incident can pose to the astronauts' health and to the success of future missions, NASA has focused on using Probabilistic Risk Assessment (PRA) models (31) to assess and mitigate the risks associated with development of renal colic for various mission scenarios. In view of the scarcity of environmentally relevant data and absence of a statistically meaningful clinical subject pool, NASA is further relying on computational results from deterministic models to inform the PRA analyses with renal calculi size distributions that occur in space due to microgravity-induced renal biochemical alterations. In this paper, we address this need by presenting a deterministic system model for nucleation, growth, and agglomeration of CaOx crystals in the kidney using the rigorous framework of the Population Balance Equation (PBE) (24). One of the significant advantage of this system or “lumped” model is its amenability to analytic solutions based on two simplifying but acceptable assumptions. As a result, it can provide fast solutions to accommodate the numerous Monte Carlo-generated parametric simulations that are required in the future by the PRA analyses.

CaOx stone development has been subject to various theoretical considerations, both in isolation (7, 20–22) and in the context of its passage through the renal system (3, 4, 9, 12, 27). Finlayson (3) was the first to consider the ducts of Bellini and the pelvis as a system of continuous crystallizers in series and applied a mixed-suspension mixed-product removal (MSMPR) analysis similar to the ones employed extensively in chemical reactor engineering to predict the size distributions for a population of nucleating and growing crystals. Subsequently, Finlayson and Reid (4) and later Kavanagh (9) concluded that for the calculi to become large enough to cause a blockage, some kind of particle fixation or retention such as wall adhesion must take place. This became known as the fixed-particle concept (4). Finlayson's analysis, however, considered only nucleation and growth and neglected the important effects of agglomeration.

Roughly two decades after Finlyason's fixed-particle concept, Kok and Khan (12) used more appropriate data for the tubular dimensions and accounted for particle agglomeration, albeit, in a quite simplified manner, to show that tubular blockage may be possible even without particle fixation when agglomeration is taken into account. This has come to be known the free-particle concept. Robertson (27) complemented the previous theoretical treatments of the problem by considering the effects of tube wall, fluid drag, and gravity and concluded that there is still a possibility that wall drag and gravity may delay the passage of stones traveling close to the tubule walls to an extent that they would grow large enough to cause an obstruction.

Although the precise mechanisms for kidney stone formation and growth are still not well understood, recent studies by Kim et al. (10) and Evan et al. (2) using advanced endoscopic imaging and comprehensive physiological biopsies suggest three possible free/fixed pathways to stone formation: *1*) nucleation and growth on the Randall plaque deposits in the papillae; *2*) growth after adherence to a possibly injured section of collecting or Bellini ducts; and *3*) homogeneous nucleation and growth in the urine during passage through the nephron. The Randall plaque overgrowth mechanism seems to be the most common pathway for CaOx stone development in idiopathic stone formers.

In the present work, we have assumed that the kidney can be represented as a MSMPR continuous crystallizer and a rigorous model for nucleation, growth, and agglomeration of CaOx renal calculi is developed based on the PBE. In part 1 of our study presented here, the model is used to assess the potential risks of renal stone development during space travel by predicting the population and size distributions of renal calculi for normal and stone-forming subjects on Earth and for astronauts after short- and long-duration space missions. In part II of our study presented in an accompanying paper (8), the model will be used to perform a comparative assessment of the efficacy of several possible dietary countermeasures in reducing the risk of renal stone development during space travel. The main inputs to the model for these simulations are the published urine biochemical data representing each of the subjects.

It should be also noted that the risk assessment presented here is based on comparative comparisons of the CaOx crystal aggregate size distributions as predicted by the PBE model for the four different subject biochemistries and not based on prediction of a given clinical stone occurrence brought about through nucleation and growth of the aggregate in a specific location in the kidney such as on a Randall plaque surface or on an injured section of the nephron. As such, the terms “stone,” “crystal,” “particle,” and “renal calculi” are interchangeably used throughout the paper to represent the CaOx crystal aggregates that formed in the kidney through the simultaneous nucleation, growth, and agglomeration mechanisms.

## MATERIALS AND METHODS

The steady-state length-based population density distribution, *n*(*D*), for the nucleating, growing, and agglomerating renal calculi that are carried by the urinary flow through the kidney, as represented by a MSMPR crystallizer, can be described by the steady-state PBE (5, 24) as follows:
(1)

In the above equation, *D* is the crystal diameter, τ is the time constant for urinary flow passage through the kidney given by τ = *V*/*Q*, where *V* is the effective conduit volume, *Q* is the volumetric urinary flow rate, and *β* is the agglomeration kernel that is assumed to be a constant and independent of diameter in the present analysis. The linear growth rate, *G*_{D}, has also been assumed to be constant and independent of diameter.

The first term in the left-hand side of *Eq. 1* represents passage (loss) of CaOx crystal population, *n*(*D*), through the nephron by the urinary flow, and the second term accounts for loss of the particles of a given size due to growth. The first and second terms on the right-hand side of *Eq. 1* respectively account for birth (gain) and death (loss) in the population of crystals of a given size due to agglomeration.

In writing *Eq. 1*, it has been inherently assumed that nucleation occurs at size zero. Thus *Eq. 1* is subject to the initial condition:
(2)

Here, *B*^{o} is the nucleation rate defined by an empirical equation as:
(3)

where *K*_{b} is the nucleation rate constant for which we use a fitted value for CaOx based on a range of measured nucleation rates reported in the literature (1, 3, 40).

*Equation 1* is subject to two important simplifying assumptions. The first is that the growth rate is considered independent of crystal diameter. The physical mechanisms that govern nucleation and crystal growth are relatively well known and have been subject to a comprehensive monograph by Nielsen and Christoffersen (22). General treatments of the kinetics of crystal precipitation from binary electrolytes have been presented by Nielsen (21) and Nancollas (20), where the latter has devoted special attention to crystallization of CaOx in relation to urinary stone formation. For CaOx precipitation, the reaction rate is very small compared with the transport rate especially for smaller stone sizes. Kassemi et al. (7) have described the interaction between species transport and reaction kinetics and its effect on the growth rate. They have shown that it is justified to assume that the growth is limited by the slow reaction rate of the ionic salt for crystal diameters roughly <1 mm. Under these circumstances, the ionic activities of calcium and oxalate at the surface of the crystal approach their bulk urinary values and thus the growth rate becomes independent of crystal size. It can be expressed as a second-order reaction (7, 21) in terms of the bulk urinary concentration, *C*_{ca,|∞} and *C*_{ox,|∞}, and activity coefficient, *f2*, as:
(4)

Here, *K*_{g} and *K*_{so} are, respectively, the linear growth rate constant and the equilibrium solubility product of calcium oxalate and *S* and *RS* are, respectively, the supersaturation and relative supersaturation of CaOx in urine.

The second important assumption in this paper is with regard to the agglomeration term in *Eq. 1*. Here, following the work of Liao and Hulburt (14) and Liu and Thompson (15), we have adopted a more simplified formulation of the agglomeration term whereupon the length of the particles are the conserved dimension rather than their volume. The great advantage of this simplification as will be presented later is that it renders *Eq. 1* amenable to an analytic solution that results in fast computations, an important objective of the present work for future incorporation into the PRA analysis. Liu and Thompson have stated that while the length-preserving assumption is not strictly true for agglomeration of deformable spherical particles, it can be of acceptable validity for agglomeration of rigid particles, especially in the case of end-to-end adhesion of nonspherical or longitudinal crystal aggregates that are representative of CaOx renal calculi. Even under these more forgiving conditions, care has to be taken as to how the model is used. Liu and Thompson have carefully examined the analytic solution generated using this assumption and have established a parametric mathematical criterion that ensures its validity. As will be shown later, this criterion essentially imposes a restriction on residence time through the kidney that we have carefully adhered to for generating the solutions presented here.

Finally, as the crystals nucleate and grow, the ionic concentrations of calcium and oxalate in the urine deplete. The amount of depletion can be considerable, especially for a stone former, thus limiting or reducing the resulting steady-state growth rate. The depletion effect is incorporated into the model by performing a mass balance that results in a simple algebraic equation for determining the steady-state (depleted) concentrations of calcium and oxalate in the kidney: (5)

The above conservation equation simply states that the change in the steady-state ionic concentrations of calcium and oxalate from its initial/inlet or pregrowth and prenucleation value is due to the uptake (depletion) rate of the ionic species by the population of nucleating and growing crystals.

*Equation 1* subject to initial condition (*Eq. 2*), together with growth and nucleation relationships (*Eqs. 3* and *4*) and conservation (*Eq. 5*), constitute a closed set of equations that can be solved iteratively to predict the size distribution of renal calculi for different biochemical input parameters of the problem.

### Solution Methodology

An analytic solution to *Eqs. 1* and *2* has been provided by Liao and Hulbert (14) in terms of a modified first- order Bessel function of the first kind, *I*(*x*). This is accomplished by transforming the original equations in to the Laplace domain, then using the meaningful root of the resulting algebraic equation and transforming back to the real domain through an inversion relationship provided by Roberts and Kaufman (25). The solution presented in dimensionless parameters is as follows:
(6)

Tavare et al. (32) have successfully used the approximate (length conserving) solution presented by *Eq. 6* to fit experimental data for a MSMPR crystallizer to estimate the rates of the associated kinetics processes. Later, Hounslow (5) used numerical solutions generated by his more exact (volume conserving) mathematical model to demonstrate that Tavare's estimations underpredicted the loss of crystals by aggregation. They attributed this underprediction to the length-conserving approximation of *Eq. 1*. Liu and Thompson (15) performed a careful mathematical analysis of *Eq. 6*, defining a parametric range beyond which the solution becomes inaccurate due to the simplification of length preservation in the agglomeration birth term and established a mathematical criterion for its validity. For the renal stone problem at hand, the Liu and Thompson criterion is conveniently expressed as a restriction on the residence time in the nephron that can be expressed in terms of the parameters of the problem as,
(7)

where *N*_{T} is the total number of particles defined by
(8)

To verify and validate the model, we have used the analytic solution provided by *Eq. 6* subject to restrictions imposed by *Eq. 7* to successfully predict the measured population density vs. diameter data of two MSMPR experiments for crystallization of nickel ammonium sulfate (32) and CaOx (39). The model also predicted the data provided by a simple in vitro CaOx renal calculi-related experiment by Finlayson (3) quite well. The verification of the present model by the three above-mentioned experimental results are shown in Fig. 1, *A*–*C*.

## RESULTS AND DISCUSSION

Weightlessness has a profound effect on the renal biochemistries of the astronauts. To investigate and quantify the impact of the changes in nephron biochemistry on the risk of critical stone development during space travel, four different case studies were constructed based on published ground-based (1 G) and space (microgravity) renal biochemical data as shown in Table 1 for *1*) a normal subject at 1 G; *2*) a stone former at 1 G; *3*) an astronaut in microgravity; and *4*) a stone former in microgravity. The primary renal biochemical data for each of these cases that consist of calcium, oxalate, citrate, and magnesium concentrations together with daily urine volumetric flow rate as determined from published 24-h urine biochemical assays are also included in this table.

The first case considered here represents a normal subject on Earth. The concentrations of the primary urine constituents for this case as included in Table 1 were derived from 24-h excretion values considered to be safely in a risk-free range according to typical 24-h laboratory assays and stone profiles used by the Mineral Metabolism Laboratory at University of Texas Southwestern Medical Center (34). The biochemical data for the second subject representing a recurrent stone former on Earth were obtained from published values by Robertson et al. (26) and used by Laube et al. (13) in their assessment of the magnitude of urinary constituents' depletion for a typical stone-forming subject. Whitson et al. (36) have reported the 24-h urine excretion rates obtained from 86 astronauts on the day of landing. These renal data form the basis of the third subject designated as a “microgravity astronaut.” Finally, the last subject represents a microgravity stone former and was constructed using the long-duration flight data provided by Whitson et al. (38) as extracted from the 24-h postflight (return + 0–2) urine samples obtained from astronauts who were subject to the placebo test. In this case, the SDs associated with the respective excretion rates were used to construct a hypothetical “worst case” microgravity scenario as given by the entries for this case in Table 1.

The urinary calcium and oxalate concentrations in the kidney are adjusted along the nephron by a complex regulatory mechanism that controls the secretion-reabsorption process. This can specifically result in the buildup of calcium and oxalate in the collecting ducts as a result of major water reabsorption. Unfortunately, one of the important limitations of working with the 0D system-level model presented here is its inability to accommodate such spatial variations. Of course, the common practice in the system-level modeling approach is to use a weighted or volume-integrated mean concentration. However, this still requires knowledge of the initial concentration and its variation along the nephron. Since this data were not available for the four subject case studies described above, we had two alternatives: either to use arbitrarily estimated values for the initial concentration and its variation based on intuition or to use the measured excreted values as an acceptable representation for the four subjects in our comparative study. Since the former had the danger of introducing additional unqualified uncertainty in our results without much gain, the latter approach was adopted as it was deemed to be more reasonable in view of the comparative nature of our relative risk assessment.

According to the context explained above, for each subject case study, the main inputs to the model are the measured excreted concentration of calcium and oxalate. Using these concentrations, the relative supersaturation for CaOx for each case can be calculated as defined by *Eq. 4*. However, to use this equation, the values of the free ionic calcium and oxalate activities are needed, which vary from case to case based on the respective urinary compositions. In this work, the urinary relative supersaturation was estimated using the methodology and monograms provided by Marshal and Robertson (16). It must be emphasized that that Marshal-Robertson estimation is based on a predesignated assumption that the free ionic concentration of calcium and oxalate are, respectively, 55 and 80% of their total concentrations; these are the ions that are free to bind to each other to form CaOx. The rest of calcium and oxalate ions are assumed to be bound with other constituents of urine such as citrate and magnesium to form soluble complexes. This is a fair approximation for urine's indirect inhibition potential and serves well for the comparative case studies among the subjects in Table 1, where there is little difference between the magnesium and citrate concentrations among the different subjects. Recently, Rodgers et al. (28) used the speciation code JESS (17) to investigate the nature and extent of solution complexes and salt precipitations in the urine. JESS computations are solely based on thermodynamic principles, and kinetic factors that limit growth rates are not considered (28). Moreover, JESS calculations do not account for agglomeration effects that play a crucial role in increasing the stone size as will be shown in the current study.

The other inputs to the model are the volumetric urine flow rate, *Q*, that determines the urine residence time, *τ*, in the nephron, and the kinetic rate constants *K*_{b}, *K*_{g}, and, *β* that, respectively, determine the rates of the nucleation, growth, and agglomeration mechanisms.

The nominal values of *K*_{b}, *K*_{g}, and *β* for CaOx as provided by Zauner and Jones (39) are included in Table 2 but do not reflect direct urinary inhibition. Since citrate is an important natural inhibitor in urine, the value of these rate constants have to be adjusted for the citrate content of the urine. This direct inhibitory effect of citrate on the rate constants is accounted for in the present work through adjustment of the rate constants based on the published data in the literature as discussed in detail in our companion paper (8). Here, the inhibited rate constants for each subject as adjusted based on their reported 24-h urinary citrate concentrations (see Table 1) are included in Table 2.

For comparative assessments of risk among the four subjects, we have to establish a criterion for designating a “maximum” predicted aggregate diameter from the computed size distributions. Naturally, as the diameter of the aggregate increases, its corresponding population or number density decreases. Here, we have chosen the diameter corresponding to a number density of, *N*_{i} = 1, number per milliliter, as a meaningful representation of the “maximum” diameter for each simulation. It is also helpful to designate a relative particle size for a potentially “minimum-risk” condition. Robertson (27) used a relative characteristic particle diameter of ∼200 μm for a relatively risk-free condition. Robertson's residence time for passage through the nephron section is ∼4–5 min (27). However, in our simulations, we have adopted a nominal residence time of ∼10 min for passage through the whole kidney based on a similar estimate used by Finlayson (3). Notwithstanding, in the discussion of results that follows, we have chosen to err on the side of caution by retaining Robertson's relative diameter for safe passage as a conservative criterion for our risk assessment. However, we emphasize that this designation was adopted in our analysis for the purpose of comparative assessment of relative risk between the four subjects based solely on their respective CaOx aggregate size distributions and is not associated with any absolute physiological measure or criterion for retention of stones in a particular section of the nephron.

### Comparison Between Normal and Stone-Former Subjects in 1 G and Microgravity

Comparisons between the predicted renal stone developments for the four subjects included in Table 1 are presented in Fig. 2, *A* and *B*. The cumulative number densities
(9)

for these subjects are plotted against the respective crystal diameters using a log-linear scale in Fig. 2*A*. This scale is useful for delineating the trend of how the size at peak population (most common size) changes from a normal subject to a stone former and from a ground-based subject to the astronaut in microgravity. A skewed bell distribution is prevalent for all four cases as expected. It is encouraging, as evident from a comparison between respective size distributions in Fig. 2, that the model can easily distinguish between a normal person and a stone former in 1 G based solely on their respective urine biochemistries as input. It is also interesting to observe in Fig. 2*A* that on a relative comparative basis, the renal calculi size distribution of the microgravity astronaut resembles more closely the shape and magnitude of a recurrent stone former's distribution on Earth rather than those of a normal subject in 1 G. This is mainly caused by the shift in the astronauts' renal biochemistry as a result of exposure to microgravity and space environment. Table 3 indicates that the peak population of stones occurs at a diameter of 0.25 μm for a 1-G normal subject and moves consistently to larger stone sizes of 2.8, 5, and 6.3 μm, respectively, for the microgravity astronaut and 1-G and microgravity stone-former cases. Consequently, both the peak population density and the diameter at which the peak population occurs are the largest for the hypothetical worst-case scenario of a stone former in microgravity.

Figure 2*B* displays the length-based population density, *n* (*D*), as a function of crystal aggregate diameters on linear-log scale for the same case study. One of the advantages of the length-based population density is that unlike, *N*_{i}, it is independent of the particle size range so that results from simulation using different predesignated stone ranges can be compared with each other in a meaningful way. The linear-log scale used in this plot is also conducive to making a comparative assessment of the largest renal stone size that occurs for each of the subjects.

The length-based population density drops sharply as sizes increases for all the four subjects, as revealed in Fig. 2*B*. The fastest rate of decline takes place for the 1-G normal subject. Again, the behavior of the microgravity astronaut is comparatively closest to the 1-G recurrent stone former rather than the 1-G normal subject. We see from Fig. 2*B* and Table 3 that the maximum CaOx aggregate size that is predicted for the astronaut in microgravity (∼143 μm) is about one-fifth the size of that predicted for a stone former on Earth (∼686 μm). Although the maximum predicted stone size for the microgravity astronaut is below our risk-free criterion of 200 μm in diameter, it is still alarmingly close to the limit. A summary of the predicted maximum renal calculi sizes for the four subjects is presented in Table 3, where it is clearly evident that the normal subject on Earth is risk free, the microgravity astronaut is below but close to the critical range, and the subjects with stone-forming biochemistries in both 1 G and microgravity can produce stones on the order of 1 mm in diameter, clearly in the risk region. In considering these results, the reader is cautioned that there is a substantial sensitivity in the analytic predictions to the large uncertainties in the published values of the kinetic rate constants that are important inputs to the model, as will be quantified and discussed later. In this light, the trends and relative comparisons delineated by the numerical results in Fig. 2 *A* and *B* are more reliable than the predicted absolute values.

### Effects of Depletion

One of the other interesting phenomena revealed in this study is the effect of calcium and oxalate depletion. When growth and nucleation occur in the nephron, calcium and oxalate ions are consumed by the precipitation of CaOx, leading to a decrease in their bulk urinary concentrations and a reduction in the urinary supersaturation. This is a phenomenon that has been discussed in previous articles (13) but never addressed through rigorous mathematical modeling. A comparison between the predicted population density distributions for the four subjects with depletion included (dashed line) or omitted (solid line) in the analysis is also included in Fig. 2, *A* and *B*. The plots show that the depletion effect can be quite significant for the stone-forming subjects who exhibit high growth and nucleation rates due to their elevated supersaturations. By the same token, depletion does not seem to be an important factor for the low to moderate urinary supersaturation levels and growth/nucleation rates prevalent among normal-G and microgravity astronaut subjects. It is interesting to note that for the stone formers, both on Earth and microgravity, depletion acts as a protective negative feedback mechanism without which their particle size distributions would be even more drastically shifted toward larger diameters.

In considering these results, the reader is cautioned that this analysis does not consider the regulatory mechanism which plays an important role in determining the final excreted calcium concentration values. Thus how much of the removed calcium concentration by the nucleation-growth process may be subsequently compensated for by adjustments made by the regulatory mechanisms is not known or revealed by the model. Nevertheless, the above results still underscore the fact that the depletion effect is an important factor to be considered, especially, for a stone former. In this regard, as also pointed out by other investigators (13), there is a possibility that the use of urinary supersaturation levels evaluated from the 24-h urine laboratory analysis as the main distinguishing risk factor may have inherent shortcomings. Since the 24-h urine measurements are made distal to the growth and nucleation processes in the nephron, they may not be true representatives of the pregrowth urinary supersaturation levels that are the real measure of the subject's nucleation/growth potential.

### Impact of Agglomeration

Agglomeration can result in a relatively rapid enhancement of the renal calculi sizes in the kidney. To underscore the importance of this mechanism, the previous four-subject case studies were repeated with the contribution of agglomeration absent in the analysis. Comparisons between agglomeration (solid line) and nonagglomeration (dashed line) results for the four subjects as presented in Fig. 3, *A* and *B*, indicate that agglomeration does not make a noticeable difference for the normal 1-G case but has a large influence on the three other subjects. On one hand, as shown in Fig. 3*A*, the effect of agglomeration is to reduce the peak population densities for the microgravity astronaut and the 1-G and microgravity stone-former subjects and shift the respective crystal diameters at peak population densities toward smaller sizes. On the other hand, as revealed in Fig. 3*B*, agglomeration significantly increases the maximum diameter of the renal calculi for the microgravity astronaut and 1-G and microgravity stone formers. For example, in the microgravity astronaut case, agglomeration results in a roughly twofold increase in the maximum diameter. For the stone-formers subjects in 1 G and microgravity, the impact is even more significant, as indicated in Fig. 3*B* and quantified in Table 4.

Both effects of agglomeration are further revealed by comparing the results presented in Tables 3 and 4 for agglomerating and nonagglomerating cases, respectively. Here, it is observed that agglomeration nearly halves the peak population diameters while increasing the maximum aggregate diameter by 2-, 3-, and 5-fold for the microgravity astronaut and 1-G and microgravity stone formers, respectively.

### Sensitivity of the Predictions to Kinetic Rate Constants and Transit Time

As mentioned before, there is a significant degree of uncertainty and scatter in the nominal values of the kinetic rate constants for nucleation, growth, and agglomeration as extracted from the very limited data published in the literature. Quantitative data and information on the impact of the various constituents of urine such as citrate and magnesium on the magnitude of the kinetic rate constants are even more scarce and less reliable. As a result, a parametric analysis is undertaken to indicate the sensitivity of the predicted results to variations in the values of these kinetic constants. This sensitivity analysis is performed for the case of the microgravity astronaut because it is the subject of greatest interest.

The variation of population densities with changes in the value of nucleation rate constant, *K*_{b}, for the case of the microgravity astronaut are presented in Fig. 4, *A* and *B*. An increase in the value of *K*_{b} increases the peak population density and shifts it toward the smaller crystal sizes as indicated in Fig. 4*A* and Table 5. This is to be expected since the number of crystals in the “zero” nucleation size is augmented with increasing *K*_{b} and skews the bell-shaped curve toward the smaller diameters. What is less intuitive is that larger *K*_{b} values also favor strongly the formation of larger stones in the kidney, as evident from Fig. 4. Thus it is interesting to note that the size of the largest stone formed in the nephron is almost as sensitive to the nucleation rate as to growth and agglomeration. The uncertainty in value of the nucleation rate constant can be as large as an order of magnitude. Table 5 indicates that a change of nucleation rate constant from 5 (10)^{6} to 5(10)^{7} no.·m^{−3}·s^{−1} results in a shift of the predicted maximum aggregate size for the microgravity astronaut from 143 to 376 μm.

The impact of variation of the growth rate constant, *K*_{g}, on the CaOx population densities are shown in Fig. 5, *A* and *B*. Figure 5*A* shows that an increase in the value of *K*_{g} shifts the peak of the bell-shaped curve toward larger diameter calculi while preserving the shape of the curve almost intact. This is to be expected since with the assumption of diameter-independent, kinetically (reaction) controlled growth adopted in this analysis, crystals of all sizes grow at the same rate. It is important to note that the growth rate constant has a significant impact on the renal calculi size. Therefore, the sensitivity of the predictions to this constant is quite important. The spread in the value of *K*_{g} available in the published literature is large and again spans over an order of magnitude. For example, the nominal value of *K*_{g} provided by Zauner and Jones (39) at 5.5(10)^{−10} m/s is roughly an order of magnitude larger than the value reported by Meyer and Smith (18) at ∼5.9(10)^{−11} m/s. Moreover, Millan et al. (19) have shown over an order of magnitude change in the value of *K*_{g} with increasing citrate concentration. Figure 5*B* and Table 6 indicate that a similar order of magnitude change in *K*_{g} from 5.9(10)^{−11} to 5.0(10)^{−10} m/s can shift the maximum aggregate size for the microgravity astronaut from a relatively safe value of 176 μm to a relatively risky range of 830 μm. It is apparent that for a precise prediction of a critical stone occurrence using a deterministic model, more accurate and reliable measurements of the nucleation and reaction rates of CaOx in urine will be required. Finally, Fig. 5*A* indicates that with an increase in *K*_{g} and the associated growth rate the effect of calcium and oxalate depletion become more pronounced, causing a decline in the peak stone population above a *K*_{g} value of 10^{−10} m^{3}/s.

The dependence of CaOx population densities on the magnitude of the agglomeration kernel is indicated in Fig. 6, *A* and *B*. In physical terms, each time two particles agglomerate a “death” or elimination occurs in the population densities of the smaller agglomerating renal calculi while a “birth” or addition happens in the population density of the newly formed larger renal calculi. Mathematically, this is represented by the birth and death source terms in *Eq. 1*. Thus, as shown in Fig. 6*B*, an increase in the value of the agglomeration kernel causes a dip in the population densities of the smaller particles and a rise in the population densities of the larger crystals. This results in a corresponding decrease in the peak population density with increasing *β* and a shift toward smaller diameter stones, as shown in Fig. 6*A* and Table 7.

The results presented in Fig. 6 clearly indicate that the impact of agglomeration on the size distiribution of renal calculi is significant. Unfortunately, as was the case with the reaction rate consistent for CaOx, there is also around an order of magnitude scatter in the nominal values of the agglomeration kernel reported in the literature (40). The situation is, again, further exacerbated by the scarcity of data on how the agglomeration constant is altered due to the direct inhibitive effect of certain constituents of urine such as citrate (11). The value of the agglomeration kernel for the microgravity astronaut as included in Table 2 includes an adjustment due to the citrate content of the subject's urine (8, 11). Without the citrate inhibition effect, the magnitude of *β* is given by its nominal value of 2.87(10)^{−14} m^{3}/s in Table 2. This would result in a shift of the maximum stone size from 143 μm for *β* = 2.87(10)^{−14} m^{3}/s to ∼400 μm for *β* = 2.87(10)^{−14} m^{3}/s as indicated in Table 7.

In the case studies presented here, the transit time was computed based on the daily volumetric urinary flow rate of each subject as included in Table 1. As such, there is no distinction between the transit times of urine and renal calculi in the model. However, in reality gravity, wall friction, and possible nucleation and/or adhesion to a specific location such as on a Randall plaque formation or on an injured section of tubule or duct wall (33) can all increase the transit time of the stone appreciably with a profound increase in the risk of renal stone formation.

Urine itself is a complex organic fluid with many constituents that may both promote or inhibit CaOx nucleation and precipitation. Some of the kinetic inhibitory properties of urine have been considered in the present model, and the effects of citrates and pyrophosphates are specifically examined in more detail in the companion paper (8). However, the list of constituents that may act as urinary inhibitors to CaOx precipitation are exhaustive, and data and an understanding of their inhibitive properties and mechanisms are not yet sufficiently complete to make their inclusion into deterministic mathematical models, even on a phenomenological basis, feasible at this point. As mentioned in the beginning, it is our hope that some of the complexities that could not be incorporated into our current deterministic model can be statistically accounted for when the present model is embedded in the framework of PRA analyses to provide a more realistic risk assessment of critical stone occurrence for future mission scenarios.

### Conclusion

In this work, an analytic PBE model was presented to predict the steady-state size distribution of nucleating, growing, and agglomerating renal calculi during their transit through the kidney in 1 G and microgravity based solely on using the renal biochemical profile of the subject as input. The model was verified through comparison with the published results provided by several MSRPP crystallization experiments, including an in vitro CaOx experiment related to renal stone formation. Four subjects were considered based on their published 1 G and microgravity biochemical profiles, namely, 1-G normal, microgravity astronaut, and 1-G recurrent and microgravity stone formers. From the results of a comprehensive case study involving these four subjects, the following assessments were made.

*1*) The model was successful in clearly distinguishing between a 1-G normal and a 1-G recurrent stone former based on their published 24-h urine biochemical profiles.

*2*) The predicted crystal aggregate size distribution for a microgravity astronaut was closer to that of a recurrent stone former on Earth than a normal risk-free subject in 1 G, underscoring the detrimental effect of space-altered renal biochemistries.

*3*) Due to microgravity-related renal biochemical alterations, the increase in risk level for developing renal stones in microgravity was relatively more significant for a normal person going into space than a stone former. However, numerical predictions also clearly underscore that the stone-former subject has still by far the highest absolute risk of critical stone formation during space travel.

*4*) For stone formers both on Earth and in space, depletion of calcium and oxalate is an important factor to be considered. This points to the shortcoming of the relative supersaturation levels determined by the 24-h urine measurements performed distal to the growth process as a definitive measure of the risk.

*5*) Agglomeration was found to be a crucial mechanism for stone size enhancement in both 1 G and microgravity.

There are two main factors that will determine whether a critical stone incident will occur. The first is renal biochemistry that dictates the rate of stone size enhancement due to growth and agglomeration, and the second is the residence time of renal calculi that is determined by their transport through the nephron by the urinary flow. The lag that might occur due to a nonslip boundary condition (in both 1 G and microgravity) or to gravity effects in upward flowing tubules (only in 1 G), or to nucleation and growth on Randall plaque surfaces or on injured sections of the nephron could not be included in the present “lumped” transport analysis. To consider these important transport effects, the PBE renal stone model needs to be coupled to a two-phase CFD model for stone and urine transport through the nephron. While a coupled CFD-PBE analysis was outside the scope of the present work, it is part of our ongoing model development effort.

## GRANTS

We gratefully acknowledge funding support from the Exploration Medical Capabilities Element of NASA's Human Research.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

M.K. provided conception and design of research; M.K. analyzed data; M.K. interpreted results of experiments; M.K. drafted manuscript; M.K. edited and revised manuscript; M.K. approved final version of manuscript; D.A.T. performed experiments; D.A.T. prepared figures.

## ACKNOWLEDGMENTS

Many thanks go to Drs. DeVon Griffin and Jerry Myers from the NASA Glenn Research Center and Dr. Robert Pietrzyk from the NASA Johnson Space Center for constant support and valuable advice.

- Copyright © 2016 the American Physiological Society