Personalized graphical models for plasma concentrations

In our analysis, we constructed a directed graphical model [10]G_k for each patient k. These graphical models represented joint conditional distributions over random variables for concentration levels Y_km, levels of nutrients in diet X_kj, and effects of those nutrients in multiple levels of detail. The graph is illustrated in Fig 2A, including the effects of nutrients in the general (β_kjm), dialysis treatment (g_ljm), and personal levels (b_kjm). Every concentration m = 1, …, M was conditioned with the same j = 1, …, p nutrients and personal treatment type l = 1, …, L; hence the edges of the graph are directed towards the concentration levels Y_km.

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Fig 2. Structure of the graphical model with general, treatment, and personal level nutrient effects.

A) The hierarchical graphical model for effects of nutrients in general, treatment, and personal levels. B) Information flow of the queried nutrients when conditioned with the current diet, personal effects, and concentration limits. The figure is drawn with diagrams.net (v 22.0.6, https://app.diagrams.net).

https://doi.org/10.1371/journal.pone.0291153.g002

The concentration random variables Y_kmi were assumed to follow gamma distribution that allows only positive values and skews to the right, thus allowing occasional values that were considerably above average [15] as follows (1) where α_m was a shape parameter of gamma distribution for concentration m. We used an inverse scale parameterization of gamma distribution where its rate parameter was obtained by dividing this shape parameter with an expected value μ_kmi for the ith observation of patient k. By this formulation, the expected concentration was clearly defined as a linear combination of the amounts of nutrients and other predictors and their effects with (2) that summed the general effect of a nutrient j with the effect variation caused by the dialysis treatment l, and finally, the personal variation within the treatment for patient k. This formed a nested two-level hierarchical model for repeated personal observations and the effect of treatment that every patient undergoes; either hospital, home, or peritoneal dialysis. Random variable X_kji denotes ith observation of predictor j for patient k, and random variables Z_kji denote predictors whose effects were assumed to vary between treatments or patients. In this analysis, all predictors were assumed to vary, and thus X_kji and Z_kji referred to the same variables, but different sets of predictors could be selected here.

Both parameters of gamma distribution are required to be positive, but we needed to keep the estimated effects of nutrients in an additive scale for direct interpretation. The additive scale can cause problems as it allows also negative values. This was solved by transforming the expected value μ_kmi linearly by adding both sides a constant c which was sufficiently large to move the estimation to strictly non-negative values while not affecting the scale of the estimated effects. For parameter α_m, we could apply an exponent link function that enforced non-negative values. It transformed the values to a logarithmic scale, but their direct interpretation was not important for this analysis. The result was a linearly shifted distribution of form (3) where the original distribution Y_kmi in Eq (1) was obtained with reverse transformations.

Estimating nutritional effects with hierarchical seemingly unrelated regressions

The effects of nutrients were estimated by considering the M different concentration distributions in Eq (1) as seemingly unrelated regressions (SUR) [9]. In SURs, there are no direct relationships between the regression models, but their hierarchical effects can be dependent on each other. This allowed us to consider all the concentrations simultaneously even though they are not directly related. The estimation was conducted by formulating all expected value regressions from Eq (2) as one univariate matrix: (4) where the model matrix X comprises all the predictor variables and was created using a Kronecker product, which is a matrix outer product resulting in a block matrix structure. This pattern follows a block-diagonal arrangement, where the same set of predictors is replicated across all M concentration models. Similarly, block matrices Z^(g) and Z^(b) were formed with Kronecker products to repeat, in a block-diagonal pattern, the predictors for treatment and personal effects for each 1, …, L treatment and 1, …, K person. These matrices contain the same data but were organized differently for selecting the relevant input for each patient and their respective treatment when multiplied by the effect vectors β, g, and b.

The treatment and personal effect vectors were drawn from multivariate Gaussian distributions, and , that were centered to general effects β and had variance-covariance matrices Σ_g and Σ_b. The variance-covariance matrices were defined with diagonal matrices and containing standard deviations for the effects as well as correlation matrices C_g and C_b that decompose into triangular Cholesky decomposition matrices L_g and L_b as follows (5)

This model formulation produces the correlation matrices C_g and C_b whose structures contain similar the within-model correlations at diagonal blocks and cross-model correlations at off-diagonal blocks: (6) where matrix blocks D^(m), m = 1, …, M, denote correlations of effects within a concentration m and blocks C^(nm) denote effect correlations between concentrations m and n = 1, …, M. The correlation matrix blocks are transposed across the diagonal. This system is illustrated as a personal graphical model in Fig 2A, which depicts these within-model and cross-model correlations with random variables ρ_ijmn of the graphical model.

The estimated variance-covariance matrices provided all the necessary information for predicting the personal effects also for new patients with previously unseen observations. This approach was employed in the cross-validation predictions of this study. However, in general, it’s important to exercise caution with out-of-sample predictions. The predictions were generated using Eq (4) by placing the new observations in data matrices. Personal effect predictions follow distribution where z is a standard normal distribution that is scaled with estimated Cholesky decomposition and the standard deviation of the effects . In the cross-validation, we utilized the expected values of this predictive distribution as they are the most probable effects to produce the observed concentrations with the given predictors.

Finally, we acknowledged that some nutrients were likely to have collinearly similar effects on concentrations, which could result in inaccurate parameter estimations [16]. To decorrelate the independent variables, a QR decomposition [17] was used to deconstruct the model matrix of common effects X into an orthogonal matrix Q and an upper-triangular matrix R. This deconstruction was further developed into a thin QR decomposition, X = Q*R*, which is equivalent but more computationally efficient, as and were scaled down by the number of observations n. By estimating parameters β*^(m) = R*β^(m), significantly less correlated common effect coefficients β^(m) = R*⁻¹β*^(m) needed to be calculated as the matrix Q is orthogonal with independent columns.

Probabilistic inference with personal generative models

Personalized models were made generative by defining proper prior distributions for all parameters, including the nutrient intakes in each patient’s diet. Scholars recognize [18] that intake levels follow positive and right-skewed distributions, such as log − normal or gamma distributions. However, intake distribution is different than estimating the current level of intake as required in this task. Normal distribution is a better option here, as it captures the variance of the dietary intake observations but reduces the unseen intake levels. We assumed that the mean of observations was the expected value of the intake, and the standard deviation of observations provides uncertainty: (7)

We denote the resulting, fully estimated personalized graph with . Personal estimations of the nutritional effects were obtained at the personal, most detailed, level of the hierarchical model, allowing for fine-grained adjustments that mitigate possible bias inherent in upper levels of the model. In this graph, the concentration estimates were assumed to settle near their observed values when using the current nutrient intakes and the estimated effects of nutrients. This allowed analysis of the personal compositions of the concentrations, and in addition, we could also condition the graph for any of its variables. This probabilistic inference [19] for intake recommendations is illustrated in Fig 2B where the edges connecting to the queried nutrients were now reversed toward them. We denote this subset of r = 1, …, R nutrients that were queried for recommendations with Q_kr and the rest of j = 1, …, J − R nutrients continue to be denoted with X_kj. They maintained their current personalized estimations while the conditioned nutrient variables were defined with new proposal distributions that allowed all possible healthy levels for these nutrients. In this work, we used uniform distributions with nutrient-specific limits, but a more informative distribution could also be used. By denoting this personalized graph with modified variables as , the conditional probability for the intake recommendation can be formulated as (8) where the queried nutrients Q_kr are conditioned with values that cause all plasma concentrations Y_m to reside within their predefined lower and upper bounds and with probability c.

Algorithm for recommendation sampling

Algorithm (1) estimates the conditional recommendation distribution in Eq (8) by acceptance sampling. In summary, the algorithm samples diet proposals from the previously defined variables Q_kr and X_kj, and evaluates the probability of how confidently this diet proposal produces plasma concentrations that reside within the requested limits. The distribution of these levels of confidence forms the personal recommendation for the queried nutrients Q_kr. For clear reporting and systematic comparison between patients, the algorithm returns 2.5%- and 97.5%-quantile values for every recommended nutrient; the example of the full distribution of Eq (8) can be seen in Fig 3.

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Fig 3. Detailed intake recommendation for Patient 36.

The intake recommendation for Patient 36 on the left shows the inferred diet configurations predicted to produce normal ranges of potassium (P-K), phosphate (fP-Pi), and albumin (P-Alb) concentrations. These limits are marked with white areas in the concentration panels on the right. The black point in the middle of the intake plot represents the patient’s current potassium and phosphorous intake producing the current concentrations shown with dashed vertical lines. The colored rectangle shows the limits of intake constrained by the correspondingly-colored concentration predictions at the boundaries of the normal ranges. The figure is plotted with ggplot2 package for R language (v 3.4.1, https://ggplot2.tidyverse.org).

https://doi.org/10.1371/journal.pone.0291153.g003

Algorithm 1: Multivariate Acceptance Sampling for Intake Recommendation

Input:

,   ⊳ Conditioned graphical model including variables Q_r with priors

,   ⊳ Targeted lower and upper limits for concentrations Y_m

l_x,   ⊳ Quantile of intake random variables ( in ) used in estimation

l_β,   ⊳ Quantile of nutrient effects ( in ) used in estimation

S,   ⊳ Number of drawn samples from the queried nutrients

c   ⊳ Targeted confidence level

Output:

  ⊳ Probabilities for queried nutrients reaching the concentration targets

,   ⊳ 95%-quantile of R-dimensional recommendation distribution

⊳ Expected concentration m without the intake of queried nutrients

  ⊳ Maximum probability for reaching target for concentration m

P^max   ⊳ Maximum probability for reaching all the concentration targets

Algorithm RecommendationSampling ( , , , l_x, l_β, S, c):

1 P^max ← 0;

2 ;

3 p ← number of nutrient random variables in

4 -quantile value of in for all j = 1, …, p − R;

5 -quantile value of in for all j = 1, …, p, m = 1, …, M;

6 -parameter of concentration distribution m in ;

7 foreach concentration m in do

8  ;

9  ;

10  ;

11 end

12 for samples s = 1, …, S do

13  ;

14  foreach concentration m in do

15   ;

16   ;

17   ;

18   ;

19  end

20  ;

21  ;

22 end

23 for queried nutrients r = 1, …, R do

24  ;

25  ;

26  ;

27 end

The steps of the Algorithm (1) are explained in detail. To provide transparent reasoning, the algorithm tracks the overall and concentration-specific probabilities of reaching the targets, and these are initialized to 0 in Steps (1) and (2). The algorithm iterates extensively on the nutrient random variables and their effects, and their number in the graph is counted in Step (3). To manage the uncertainty in the potentially wide distributions of these variables, the algorithm uses point estimates in their quantile values. By default, the expected values, in the mean of the distributions, are used, but tail quantiles are also useful in sensitivity analysis. The used quantiles are provided by parameters l_x and l_β and they are used to acquire point estimates of these random variables in Steps (4) and (5).

To optimize the sampling, the loop in Step (6) calculates concentration-specific parameters that remain fixed during the sampling. Step (7) calculates the expected values of concentrations without the effect of the queried nutrients. This provides constant baselines of concentration levels that do not change during the sampling. These constants are used in Steps (8) and (9) to determine the well-defined minimum and maximum limits for the sampled system. The expected values are used instead of the required concentration limits if they are lower than the concentration’s lower limit or higher than the concentration’s upper limit to prevent the sampling from failing.

The main loop in Step (11) draws and evaluates the S number of samples from the queried nutrients. Vectors of R coincidental samples are drawn from Q_r in Step (12); a sampling function that is provided with the queried random variables as well as the lower and upper sampling limits from Steps (8) and (9). The queried nutrients Q_r are assumed to follow suitable proposal distributions for sampling. For actual sampling, we used No-U-Turn sampling (NUTS) [20] algorithm to draw the samples from this constrained posterior. This algorithm is efficient in sampling these multidimensional distributions.

After a diet proposal is sampled, the loop in Step (13) iterates all the concentrations and calculates their expected values in Step (14) as well as the full posterior distribution in Step (15). The posterior distribution is used in Step (16) to evaluate the confidence level for each diet proposal by assessing the amount of cumulative density between the concentration limits. The maximum of the reached confidence level is stored for each concentration in Step (17) for diagnostic purposes.

The algorithm seeks intake proposals that are estimated to take all the concentrations within the normal ranges, between and , with the minimum confidence level c; therefore, a minimum of achieved concentrations’ confidence level is stored for the current proposal in Step (19). Step (20) stores also the maximum overall confidence level for diagnostic purposes.

Finally, in Steps (22)-(25), minimum and maximum recommendations are picked for all the queried nutrients. In this implementation, we report 2.5% and 97.5% quantile values of a subset C_r for the queried distributions that have confidence levels in sampled proposals for all the concentrations. These quantile values are returned as results in and .

Implementation

The probabilistic models for both hierarchical effects of nutrients and personalized recommendations were implemented with the probabilistic programming language Stan [21] and the estimation of the models was completed through Stan’s implementation of No-U-Turn sampling [20]. Personalized graphical models that use the effect estimations from the hierarchical model were constructed with a custom R-code and iGraph R-package [22]. These graphs represent random variables as nodes and their connections as edges. The estimated distributions of random variables are stored in the properties of the nodes, which allowed us to fully execute Bayesian inference over the graph. The inference for personalized recommendations through Algorithm (1) was implemented in a custom R-code that filtered the samples that were drawn from the recommendation model.

Creating personalized models for all patients

Our main interest was to personalize patients’ diets to reach the suggested normal ranges of plasma potassium (P-K), fasting plasma phosphorous (fP-Pi), and plasma albumin (P-Alb) in Table 3. However, seven of 37 patients had missing plasma albumin measurements. To overcome the missing data, we created personalized models for these patients to predict the missing values before the main analysis. The hierarchical model used in Eq (1) learns typical reactions from patients who have all three concentration measurements available and then generalizes the reactions for patients who have missing measurement values. In the final data set, we imputed the predicted albumin concentrations for these patients. We assumed that this data imputation provided a minimal bias in data and allowed us to use all of the available measurements.

The main analysis was conducted by first estimating the system of hierarchical nutrient effect models in Eq (1) with the three concentration targets and then constructing the personalized graphical models for k = 1, …, 37 patients. In these personalized models, patients’ current nutrient intake variables X_kj were estimated by using Eq (7). The collected intake data are described in Table 1. The table shows also the differing intakes of these nutrients within the study. The input data also included personal factors, such as medication, which are described in Table 2. The estimated effects of phosphorous and potassium are provided in Table 4 at all hierarchically modeled levels. The effects of all considered nutrients are presented in S3 Table, which is arranged in decreasing order of between-treatment variations; with this arrangement, the effect of water on plasma albumin is noticeably different between treatments. Table 5 highlights the strongest effects with high individual variation. The table shows also the estimated variations of the effects between dialysis treatments (). It is notable that most varying effects are related to plasma albumin concentration. Further reasoning for the effect predictions is available in S2 and S3 Figs. The correlation plots of dialysis treatments and personalized matrices in these figures form a network of associated effects for the sample population. S1 and S2 Tables provide detailed figures of effects that have the highest correlations in levels of treatment and individual patients.

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Table 4. Effects of potassium and phosphorous between different dialysis treatments and between patients within the same treatment.

https://doi.org/10.1371/journal.pone.0291153.t004

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Table 5. Strongest and personally most varying effects of nutrients and medication.

https://doi.org/10.1371/journal.pone.0291153.t005

Comparing simulated personalized recommendations

Personalized recommendations were acquired by executing Algorithm (1) for all patients k = 1, …, 37, and providing a conditioned personalized graphical model as an input. The conditioning was done similarly to that of Fig 1B in which phosphorous and potassium were separated as the queried variables Q₁ and Q₂, which left the other nutrients, X_j, j = 1, …, 20, unmodified. These queried variables were defined with the following proposal distributions (9) where the upper limits of the proposed intakes reflect maximum intakes in the general population. These limits were borrowed from the FinRavinto study [23] that studies the intake and nutrition of the Finnish adult population, and in which these limits are the maximum intake for 95% of studied subjects (n = 565). For the personalized concentration limits, given with the algorithm parameters and , we used the normal ranges reported by laboratory staff and KDOQI guidelines [14]. The plasma albumin upper limit was personalized by the patient’s age, as laboratory staff suggested. The exact ranges that were used are reported in Table 3. We executed the recommendation sampling with long chains of S = 10, 000 sample draws for each patient. Besides the means of nutrient level and nutrient effect distributions, the recommendation was executed also with their 5% and 95% quantile values for sensitivity analysis [24] of recommendations. The confidence level was also varied with the parameter options c = 90% and c = 80%.

The algorithm resulted in a recommendation with confidence for 22 of 37 patients, and 17 of those patients could exceed the general recommendations of potassium or phosphorous intake. For the rest of the patients, sufficiently confident recommendations could not be reached. When the 5% and 95% tail quantile values (l_x and l_β) were used as estimations of current nutrient levels and the nutrient effect, only a few patients received confident recommendations, indicating that distributions for these variables were overly wide and would need further observations to achieve confidence for clinical use. Fig 4 illustrates the estimated recommendations for each patient; the first two columns display the recommended intake, and the following three columns offer the predicted concentration levels based on these recommendations. The reported intake recommendations were taken from the 5% and 95% quantile values in the estimated two-dimensional recommendation distributions. This reporting method allowed us to compare otherwise uneven recommendation distributions. It demonstrates that the estimated recommendations range from 1 to 5531 mg/d for potassium and 4 to 2527 mg/d for phosphorous with varying personal configurations. These personalized recommendations are considerably different than the general recommendations of 2500 mg/d for potassium and 1000 mg/d for phosphorous [8].

Detailed analysis of personalized recommendations

From all the personalized recommendations in Fig 4, the estimated two-dimensional distribution for Patient 36 is expanded in Fig 3 for a closer analysis. The black dot on the intake recommendation plot indicates that the patient’s current intake of potassium is approximately 2260 mg/d and phosphorous is approximately 860 mg/d. This intake configuration produced the patient’s current plasma concentrations which are indicated by the dashed line at the concentration plots: 3.8 mmol/l plasma potassium, 1.4 mmol/l plasma phosphate, and 40.5 plasma albumin. To determine the personal ranges of intake recommendation, the patient’s two-dimensional intake recommendation distribution offered simulated intake proposals that were predicted to take all three concentrations to normal ranges; the darkening colors indicate the increasing confidence of the estimation. The rectangle over this distribution defines the area where both intakes produce over 90% confidence in achieving the targeted concentrations. This method for defining regions over uneven recommendation distributions was chosen for systematic reporting and comparison between patients. The rectangle in Fig 3 defines the same recommendation of 759–2702 mg/d for potassium and 693–1473 mg/d for phosphorous that was provided to Patient 36 in Fig 4. The rectangle’s limits have matching colors with the predictive distributions of the concentrations. These four distributions are at the limits of concentration normal ranges and provide constraints for the recommendation.

The gray bars in Fig 4 indicate the counterfactual expected values of concentrations where the effects of potassium and phosphorous are totally omitted. This shows any available room in the concentrations to absorb the effects of phosphorous and potassium. For 7 of 22 patients who received confident recommendations, the initial concentrations were already over the upper limits, but the linear combinations of proposed intakes and their effects decreased the final concentrations within the normal range. For the rest of the patients, this initial level concentration was already within the normal range or lower than its lower limit. Especially for plasma albumin, this lower limit was difficult to reach for one-third of the studied patients.

Evaluation of the personal models

Due to the limited number of personal observations, our primary purpose was to model the reactions of the patients in the studied data, but not to make generalizations about the reactions found. We developed several model candidates to explore the mechanisms in data. The models were evaluated with both visual posterior predictive checks [25, 26] and normalized root mean square errors (NRMSE) that were calculated over all concentration predictions for K patients and M concentrations with (10)

The posterior predictive check in Fig 5 compares the distribution of the measured concentrations with the samples drawn from the multivariate model. This check verified that the model is unbiased as the draws from the concentration model are centered with the measured concentrations. The predictions for phosphate and potassium concentrations have very little variance, but for albumin concentration, the prediction variance is higher. Additionally, we ran simulations with the personal graphical models by drawing samples from current diets and predicting personal concentrations. As no conditioning is done on the diets, the concentrations are expected to settle on their measured values. The differences between the simulated concentrations and the measured concentrations are estimated with the normalized root mean square errors (NRMSE) in Table 6. Here we considered an alternative effect model (mv3_cross_single_level) that omits the dialysis treatment as a separate layer in the hierarchical model but this increased the normalized error from 0.003 to 0.078. We also considered the concentrations as separate univariate models (separate_pk_fppi), but it increased the overall normalized error to 0.081 indicating the effect of multivariate modeling. All of these models were estimated with a Bayesian modeling framework Stan [21] with four chains and 3000 sample draws from each chain with a 1000 sample warm-up period. Model diagnostics confirmed that there were no divergent transitions during the sampling. The sampling produced an average effective sample size (ESS) of 580 over all model parameters. The chains are also considered mixed as R-hat convergence diagnostic is 1.01 on average. It is recommended that the effective sample size should be well over 100 and R-hat less than 1.05 [27] for parameters to be trusted.

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Fig 5. Posterior predictive check (PPC) of the final nutritional effect model.

Posterior Predictive Check (PPC) for the final model version: Black lines represent observed concentration levels, while purple lines overlay the predicted concentration levels when all predictor nutrients match observed values. For an unbiased model, the observed and predicted concentrations are expected to be aligned. The figure is plotted with bayesplot package for R language (v 1.10, http://mc-stan.org/bayesplot/).

https://doi.org/10.1371/journal.pone.0291153.g005

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Table 6. Comparison of model structures for personal nutrient effects.

https://doi.org/10.1371/journal.pone.0291153.t006

For external validation, we executed 10-fold cross-validation for the final version (mv3_cross_two_levels) of the model. In this process, we split the dialysis patient data into ten folds with three or four of all 37 patients removed from each fold. Personal reactions were predicted for these removed patients while the rest of the data were used for the model estimation. The prediction was made with Eq (1) by applying the unseen concentration measurements Y_kmi and corresponding nutrition data X_kjm with the assumption that common effects () and dialysis treatments effects () remained as previously modeled. Only the personal variations from the dialysis treatment averages were predicted to random variables where the standard normal distribution z ∼ Normal(0, I) is transformed with the Cholesky decomposition from Eq (5) holding the learned correlations between nutritional effects. The related correlation matrix is illustrated in S3 Fig with the highest correlations highlighted in S2 Table. Multiplying the distribution with adds the estimated standard deviations of personal effects to it. As a result, the prediction seeks to find the most probable combination of effects from distribution given the observations from new patient k. We evaluated also the predicted concentrations with NRMSE and it increased to 0.170 from 0.003 in comparison with the in-sample model. In addition to the smaller amount of data, the increased error can also be explained with unseen patients that react differently than was estimated in the Choslesky matrix of that fold. S5 Fig illustrates the cross-validation folds of patients and compares the modeled and the predicted NRMSE values.

Finally, we conducted a sensitivity analysis [24] of recommendations by creating personal graphs and recommendations also from the cross-validation predictions. The predicted recommendations are shown in S4 Fig. In comparison to the in-sample recommendations of Fig 4, there are five patients whose predicted reactions did not produce recommendations confident enough, and three patients who could have recommendations with the predicted reactions but not with the in-sample model. All of these changes occurred in recommendations that are very close to the target limits already. The aim of the recommendation sensitivity analysis is to ensure that the recommendations are robust and react consistently.

Limitations and reliability of the study

The data collected for this study had certain limitations, including a small sample size of patients (n = 37) and a limited number of observations per patient. Each patient underwent only two dietary interviews and corresponding laboratory analyses of concentrations. The scarcity of data can be attributed to the fragility of renal patients undergoing dialysis, which poses challenges in conducting extensive studies involving this population. The main motivation of this study was to address the malnutrition-induced weakness experienced by these patients, but the subject is complicated to investigate, and thus literature on the nutritional reactions of renal disease patients is also limited. Given the small amount of data, it is not recommended to generalize the findings of this study beyond the studied patients without further research.

However, we showed in Table 6 that including the patients’ grouping in different types of dialysis treatments improved the personal models’ fit considerably, and as a result, the multivariate model repeats the observed concentrations without bias in Fig 5. This makes the expected values of nutritional reactions solid to conclude that there exist personal differences within this patient population, and there is a need for personal inference of nutrition intake. Another source of uncertainty in the inference is the estimation of patients’ current intake levels. In this partial recommendation, only levels of phosphate and potassium were conditioned, while the rest of the diet remained unmodified. For accurate recommendations, the contribution of the unmodified diet should be accurately estimated, and Algorithm (1) brings these contributions out with parameters to support the reasoning.

A major modeling decision was to choose whether the relations between the concentrations should be estimated, or if they should be considered independently. The latter provides a sparse and computationally efficient structure in a Bayesian network [28], but we have demonstrated that with this small amount of data, also multivariate computation can be accomplished. We conclude that if there exist cross-model correlations, then a multi-target recommendation requires a multivariate model of the reactions. In this analysis, sparse cross-model correlations are proven to exist, and including them enhanced the predictions as was seen in the decrease of normalized root mean square error.

Application of the inference method

Our goal was to develop a recommendation method that enables practitioners to transparently follow the inference process. By estimating the nutrient components contributing to plasma concentrations, our method allows for inference regarding the hypothetical scenario where the intake of phosphorus and potassium is completely omitted. These concentration baseline parameters, denoted as , serve as fixed starting points in the recommendation algorithm. Based on these baselines, the recommendation method offers three options for proceeding. In the first two options, the estimated baselines are either above or below the normal ranges of concentrations. If simulating intakes of phosphorus and potassium alone fails to reach the normal ranges, the recommendation is deemed unsuccessful. In the third option, if the baselines already fall within the normal range, it is still necessary to accumulate a sufficient probability mass from the predictive distributions of concentrations within the normal ranges to provide a confident recommendation. As the model gains more information about individual reactions, the predictive distributions are expected to become more tightly concentrated, with their expected values approaching the limits of the normal ranges. Consequently, the method can generate confident recommendations for more permissive diets. This concept is illustrated in Fig 3, where the predictive distributions of plasma potassium push against the boundaries of the normal ranges, thus constraining the corresponding intake recommendation. The breadth of these distributions signifies the uncertainty stemming from personalized estimations of diet and its effects. For reporting purposes, the recommendations presented in this study utilize the expected values of the predicted concentration distributions.

In addition to facilitating comparisons between patients, our method also supports personalized nutritional guidance, as exemplified in Fig 3. This figure depicts the current phosphorus and potassium intakes of Patient 36, along with the corresponding concentration levels. Despite all considered plasma concentrations falling within the normal ranges with the given intake levels, it is crucial to determine the boundaries of intake necessary to maintain concentrations within those ranges. Based on the personal model developed for Patient 36, Algorithm (1) generates a recommendation for potassium intake ranging between 759 and 2702 mg/d, and for phosphorus intake ranging between 693 and 1473 mg/d. Notably, the expected concentration values for a simulated diet without potassium and phosphorus intake already fall within the normal ranges, along with substantial portions of the concentration predictive distributions. By comparing the corresponding colors of the intake recommendation limits and the predictive distributions of concentrations in Fig 3, we can deduce certain relationships. The concentration of plasma potassium (P-K) increases as potassium intake increases, while it also becomes evident that plasma potassium concentration increases with decreasing phosphorus intake. On the other hand, phosphorus intake has minimal impact on plasma phosphate concentration. However, a decrease in phosphorus intake results in an increase in plasma albumin concentration (P-Alb). Together, these reactions define the lower limit of the phosphorus intake recommendation and the upper limit of the potassium intake recommendation. As phosphorus intake increases and potassium intake decreases, the lower limits of the normal ranges for potassium and albumin concentrations are first met. For many patients, the recommendation is constrained by the upper limit of fasting plasma phosphate and the lower limit of plasma albumin. In clinical practice, making choices among different diet options that yield computationally identical concentrations requires nutritional expertise and may vary between patients, depending on individual factors.

Considerations on the personal effects of nutrients

In the partial recommendation presented in this study, only the levels of phosphate and potassium were specifically adjusted, while the remainder of the diet was left unmodified. The factors taken into account for personalized recommendations were the estimated nutritional reactions and the composition of the current diet. The literature on the nutritional reactions of renal disease patients is scarce, but our hierarchical reaction model indicated that incorporating the type of dialysis treatment as a nested level of grouping improved the accuracy of the model. While there are differences in the average reactions among patients receiving different dialysis treatments, individual variations in reactions were also substantial for many factors. Notably, the average effect of water intake on plasma albumin showed the greatest variation among the different dialysis treatments. Patients undergoing hospital hemodialysis exhibited weaker reactions compared to those undergoing home or peritoneal dialysis. This difference is likely attributed to the less frequent occurrence of hospital hemodialysis and the treatment guidance in hospital settings that restricts fluid and sodium intake [7][14]. Our results also revealed weaker sodium reactions in hospital dialysis.

It is worth noting that the nutrient effects on plasma albumin concentration exhibited the greatest variability among patients. However, it is important to interpret this result cautiously, as the posterior predictive check in Fig 5 indicated a high variance in the model predictions for plasma albumin concentration, despite the predictions being unbiased. We attribute this high variance to the fact that plasma albumin concentration cannot be fully predicted based on the nutrient intake. In fact, it is reported that plasma albumin and prealbumin concentrations should not be relied upon as exclusive nutritional markers [29]. These concentrations may decrease in the presence of inflammation, regardless of the underlying nutritional status. Upon treating the malnutrition, the inflammation may subside, leading to an increase in albumin concentration. Polyunsaturated fatty acids (PUFA) are known to have inflammatory properties [30] and can aid in the recovery process. However, the specific effect of PUFA on albumin concentration has been shown to vary among patients. Thus, the composition of albumin concentration is more intricate compared to the other considered concentrations. Lastly, the intakes of both phosphorus and potassium also exhibit individually varying effects on all the concentrations, as presented in Table 4. Addressing this variation is our primary contribution and is reflected in the resulting personalized recommendations.

Future work

Currently, Algorithm (1) requires that all concentration targets must be fully reached before providing a recommendation. However, it is possible to allow patients to partially reach the targets, which would necessitate defining an order of importance among the concentrations. In such a scenario, achieving the most important concentration would be required, while the less important concentrations would need to be reached to the greatest extent possible. In our future work, we intend to personalize the intake recommendations for all nutrients or as many as practically feasible. We have observed that modifications in phosphorus and potassium levels alone were not sufficient for all patients. However, this poses computationally a more challenging problem, as there will be numerous dimensions to consider in the recommendation distribution beyond these two nutrients. Currently, the recommendation algorithm provides parameters that can be used to track the inference process. However, collecting and interpreting the relevant parameters may be a tedious task. Therefore, in our future work, we aim to enhance the interpretability of the recommendations by incorporating an algorithmic explanation alongside them. This will help users better understand the reasoning behind the personalized recommendations and facilitate their implementation in clinical practice.