日本ソーシャルデータサイエンス論文誌
第 5 巻 第 1 号（2021 年 3 月）

原著論文

Simulation of Severity of Diabetic Nephropathy
Using a Markov Chain
Shinya Mizuno†1, Haruka Ohba†1, Tatsuo Yanagawa†2†3, Keiko Koyano†2†3,
Shuhei Iida†2†3, Tokimune Kou†4, Hajime Okuno†4 and Naokazu Yamaki†4
Abstract: In Japan, National healthcare expenditure in 2015 was 42,364.4 billion yen, 3.8% more than the previous year, whichindicates a significant problem. As diabetes becomes severe, it costs a lot for dialysis and medication. As the population withdiabetes is increasing and diabetes is a risk factor causing complications, it is necessary to undertake efforts to ensure that
diabetes does not lead to severe illness. In this study, we construct a simulation with a Markov chain on diabetes, which will
become an increasingly important issue in the future. First, we create state distribution using eGFR and urine protein. The initial
distribution first uses eGFR and urine protein tested values. The final distribution uses the last inspection value existing as data.
We calculate the average inspection period from the data and make it the unit period of the Markov chain. We calculate thetransition probability matrix from the inspection data and observe the state transition by stationary distribution and simulation.
This simulation clarifies the progressive severity of diabetes, making it easier to deal with stages leading to severe illness.
Simulations are categorized according to patient attributes and implemented so that they can be applied in many cases.
Keywords: Markov chain, simulation, transition probability, stationary distribution

1. INTRODUCTION

the same time, the calculation process is complicated, and there
are drawbacks, for example, it is difficult for humans to grasp

In Japan, national healthcare expenditure in 2015 was 42,364.4

the process leading to the result. First, we use statistics to grasp

billion yen, an increase of 3.8% from the previous year [1]. With

the overall trend to basically analyze clinical data. For that, it is

the increase in national medical expenses, the diabetes-affectednecessary to create an environment with a database structure

population and Impaired glucose tolerance in Japan arethat can analyze data. We consider it necessary to clarify theestimated to be about 10 million people [2], which indicates a

temporal trend of each data and to clarify the state transition ofsignificant problem. As diabetes becomes severe, it costs a lot

the patient. Essentially, a Markov chain is assumed as stationary,for dialysis and medication [3]. As the population of diabetes isbut temporal transition is easy to express, and flexible

increasing, and diabetes is also a risk factor causing

expression can be undertaken by linking with simulation.

complications, efforts that do not lead to severe illness are

In this study, we construct a simulation with a Markov chain onnecessary [4].

diabetes, which will become an increasingly important issue in

Various studies have been undertaken on diabetes to date.

the future. In the complications of diabetes, we focus on

Specifically, many studies have been undertaken on clinicalnephropathy this case. The severity of nephropathy is regulated

research [5-10], but research using information science has also

by eGFR and urine protein. So, we first create state distribution

been increasing recently. Statistical approaches to improvement

using eGFR and urine protein. From this state distribution, weof diabetic nephropathy patients have been conducted [11-13].

try to simulate the deterioration of diabetic nephropathy. The

In recent years, research has been undertaken to predict theinitial distribution first uses eGFR and urine protein tested

number of patients in the future using machine learning theory,

values. The final distribution uses the last inspection value

such as neural network [14,15]. Research using Markov chain toexisting as data. We calculate the average inspection period

measure transitions per unit time and to predict the number offrom the data and make it the unit period of the Markov chain.

patients in the future has also been undertaken [16-19]. The

We calculate the transition probability matrix from theapplication of artificial intelligence to the medical field isinspection data and observe the state transition by stationary

increasing, and machine learning is one of the big tools, but at

distribution and simulation. This simulation clarifies the
progressive severity of diabetes, making it easier to deal with

†1 Shizuoka Institute of Science and Technology(Correspondence author: mizuno.shinya@sist.ac.jp)†2 Nerima General Hospital†3 Institute for Healthcare Quality Improvement, Tokyo Healthcare Foundation†4 Research Institute of Systems Planning, Inc.
投稿日：2020 年 12 月 25 日

stages leading to severe illness. Simulations are categorizedaccording to patient attributes and implemented so that they canbe applied in many cases.

採録日：2021 年 3 月 13 日

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第 5 巻 第 1 号（2021 年 3 月）

Fig. 1 Database structure for analysis of severe diabetes
Table 1 Explanation of factors usedstate. When eGFR decreases, urine protein becomes +, diabetic

2. BASIC ANALYSIS OF EACH ELEMENT

nephropathy gets worse, and state 𝐼 shows severe condition.

In order to simulate the severity of diabetes, we first perform abasic analysis of each factor. We also use databases to analyze

Table 2 Classification by eGFR and urine protein

the data. The database structure constructed in this study isshown in Figure 1. The factor and attribute used for thesimulation in this study is shown in Table 1.
In order to distinguish the severity of diabetes in patients, weclassify the data as shown in Table 2 using eGFR and urine
protein. We used a table with fewer states from the original table

2.1 Patient state transition by factor[20]. We reduced the number of states to clarify the analysis, but

We first compare the first and last data of the patient's

adopt many attributes of patients. State 𝐴 indicates a normal

examination according to Table 2. Patients are classified as 𝐴

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第 5 巻 第 1 号（2021 年 3 月）

Fig. 2 State transition of patients classified by eGFR and urinary protein
to 𝐼 by eGFR and urine protein testing. The results are shown

but there is a transition to state 𝐴 , and improvement isin Figure 2 and Appendix A.1. The numerical value on theobserved.

right-hand side of Figure 2 shows the number of subjects of theelement. The top row in Figure 2 represents the state

2.2 Time-series observation of patient's state transition

classification in the first examination result of all patients. The

Next, we observe at the state transition of patients in time

second line shows the final status of all patients. We observe

series. This graph shows how the patient's condition changes

that the proportion of state 𝐴 and 𝐵 decrease according to thefrom 𝐴 to 𝐼 over time. When the state changes from 𝐴 to 𝐼,

period. On the other hand, in condition 𝐼, which is considered to

the value of eGFR is less than 30 and the value of urine protein

be serious diabetes, the ratio increases from 1.6% to 5.51%,

is (+) or more. Figure 3 shows how the value of eGFR decreases.

Increase rate of state 𝐼 is 3.44. The next line is a state transition

Figure 4 shows the change of urinary protein with time series.

when divided by sex. There is no big difference by sex. The next

Figure 5 shows a state transition from state 𝐴 to state 𝐼 overline shows state transitions by age and the influence of age istime. These figures show that patients' conditions do notlarge: as age rises, the severity rate also increases. Next, during

suddenly worsen, but tend to worsen over time. For example,

classification of the state using the initial Mg concentration, thethe decrease rate of eGFR is -18.4 [𝑚𝑙/𝑚𝑖𝑛/1.73𝑚2 /𝑦𝑒𝑎𝑟],state does not become severe when it is in the proper range fromand although it is affected by aging, the decrease is not sharp.

1.8 to 2.4, but the proportion of patients who are out of the

We need to use the patient's state transition as an important

appropriate range leads to severe cases. The next item classifies

signal to prevent patient condition deterioration.

the state by the average Mg concentration value. Similarly,
patients outside the reference value are likely to become
severely ill. When classified by BMI, although the number ofpeople is small, even if the BMI is below the reference value, it
leads to serious illness. In addition, we confirm the statetransition due to the difference in numerical values in HbA 1 c,
LDL, HDL, but we do not obtain a big difference from otherfactors. Finally, when SGLT inhibitors are used, it has a strong
effect on non-severity. The transition to state 𝐼 is very small,ⓒ2021 Japan Social Data Science Society

37日本ソーシャルデータサイエンス論文誌
第 5 巻 第 1 号（2021 年 3 月）
in state 𝐼 is significant. Similarly, when SGLT inhibitors areused, the mother ratio is also tested. In this case, the 𝑝 value is0.01944, and a significant difference is obtained in the ratio of
the number of people in state 𝐼 depending on whether or not
SGLT inhibitors are used.
Table 3 Initial Mg concentration, number of people using SGLT
inhibitors, and number of people in state 𝐼
Fig. 3 Transition of eGFR when the state changes from 𝐴 to 𝐼

3. DIABETES SEVERITY SIMULATION
Here, a simulation is carried out to confirm the state of severe
diabetes. Let {𝑋𝑛 , 𝑛 = 0, 1, 2, ⋯ , } be a stochastic process thattakes on a finite or countable number of possible values, such as
Fig. 4 Transition of protein when the state changes from 𝐴 to 𝐼

the set of non-negative integers {0,1,2, ⋯ }, which describes
clinical unit time. If 𝑋𝑛 = 𝑎 , then the process is said to be instate 𝑎 at time 𝑛 . We suppose that 𝑋𝑛 takes the state{𝑎, 𝑏, 𝑐, ⋯ , 𝑖} defined in Table 2. We suppose that whenever theprocess is in state 𝑎, there is a fixed probability 𝑝𝑎𝑏 that it will
next be in state 𝑏. {𝑋𝑛 } is interpreted as stating that, for a
Markov chain, the conditional distribution of any future state
𝑋𝑛+1, given the past states 𝑋0 , 𝑋1 , ⋯ , 𝑋𝑛−1 and the present
state 𝑋𝑛 , is independent of the past states and depends only on
the present state. The value 𝑝𝑎𝑏 represents the probability thatthe process will, when in state 𝑎, next make a transition to stateFig. 5 Transition of state when the state changes from 𝐴 to 𝐼𝑏 [30].

2.3 Consideration of influence of patient state transition

3.1 Simulation of severe diabetes in all patients

Next, we consider the data related to patient state transitions.

First, the transition probability matrix is calculated using all

From Figure 2, we observe that the transition to state 𝐼 ispatient data. Every time the patient is examined for eGFR or

greatlymagnesium

urinary protein, the state transition is confirmed. If only one ofconcentrations within the standard value and patients taking

the tests is received, we use the latest one for the data notvalues outside the reference value. Therefore, we check whether

inspected. Appendix A.2 is a transition probability matrix when

there is a difference in ratio of the final state 𝐼 depending on

all patient data are used. The average treatment period of thesewhether the initial inspection data value of Mg concentration isdata was 1618 days, the average treatment interval was 49 days,

within the reference value or not. As Table 3 shows, when theand the average treatment number was 33 times. Therefore, we

Mg concentration is 828 people outside the standard value andassume an interval of 49 days for one transition. From this179 people outside the reference value, the number of people intransition probability matrix, the transition probability fromstate 𝐼 at the end is 39 people and 22 people, respectively. In

state 𝐼 to state 𝐼 is the highest value of 0.86123. The transition

this case, when the mother ratio is tested, the 𝑝 value becomesfrom state 𝐻 to state 𝐼 is also 0.10423, which is a very large

0.00023, and from the initial Mg concentration, the result isvalue compared to the other. This indicates that diabetes isobtained that the difference in the ratio of the number of people

difficult to improve if it becomes severe. We also calculate thedifferent

betweenpatients

takingtransition probabilities for each element.

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Table 4 Stationery distribution and square error of each element
3.2 Calculation of stationary distribution of each element

hours transition to state 𝐼. As age increases, the transition rate toHere, we calculate the stationary distribution for each element

state 𝐼 increases. The transition from state 𝐵 is not muchand compare it with the state ratio actually obtained in the data.

different to the transition from state 𝐴, but the transition from

Table 4 shows the difference between the stationary distribution

states 𝐶 and 𝐷 to state 𝐼 greatly increases. Patients who were

and the actual data in that study. The difference from the actual

not in states 𝐴 and 𝐵 in the initial diagnosis need to be careful

data is calculated by subtracting the value and calculating thewith state transition. Furthermore, in the case of patients whose

sum of squares. We can see that the steady distribution tends toinitial Mg concentration is out of the reference value range, thebe lower than the actual data. There is also a tendency for errors

transition to state 𝐼 is very high even if the condition is 𝐴 or

to tend to be large in Mg-related distributions, and we need

𝐵. By contrast, when using SGLT inhibitors, the transition toattention. However, the correlation coefficient between thestate 𝐼 is a very low value. By using SGLT inhibitors, it isstationary distribution of state 𝐼 and the actual data is 0.97,

considered that the transition to severe diabetes can bewhich can be said to well represent the influence of each

prevented.

element. The probability of state 𝐼 in the case of using SGLT
inhibitors is 0.0157, and from the actual data, it is 0.0156,
indicating high accuracy.

3.4 State simulation of aggregation
Next, we try to summarize the state to improve the simulation
accuracy. In the first transition probability matrix, convergence

3.3 Transition from specific stateis also unstable, as shown in left of Figure 6. By aggregating

Next, we make sure how long it takes for diabetes to shift fromstates 𝐴 and 𝐵, the error of the data with the actual data isstate to state 𝐼 over time. In all patients, 1.3% of the patients

reduced by 81%, and the accuracy can be increased as right oftransition to state 𝐼 after 10 unit hours, even in state 𝐴 in the

Figure 6 and Appendix A.3.

first diagnosis from Table 5 of “All data” column. Similarly,
3.1% of patients in 20 unit hours and 3.8% of patients in 30 unit

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Table 5 Time-series probability from a specific state to state 𝐼

4. CLINICAL FEEDBACK AND
APPLICATION

From the analysis of this study, there are also interesting results
in clinical practice. Normally, HbA1c is considered to have a
bad influence on diabetes if it is 5.9 or higher. However, in

In this section, we consider how to feed back the simulation

Appendix A.4, patients with HbA1c smaller than 5.9 tend to get

results of diabetic nephropathy to the clinic.

worse. The number of patients with HbA1c less than 5.9 is small,but the proportion of state 𝐴 is only 18.78%, the proportion of

4.1 Clinical feedback

state 𝐴 is quite small compared to that of HbA1c is more than

From the results of this study, we found that various factors

5.9 or the whole patient. The condition (𝐴 + 𝐵) ratio of patientsaffect the severity of diabetic nephropathy. In particular, it waswith HbA1c less than 5.9 is 52.49%, less than 62.02% of theconfirmed that the influence of the Mg concentration was large.

whole patient. Also, the ratio of state 𝐻 is very large, 7.28%.

Patients whose Mg concentration is out of a reference value that

Patients with HbA1c less than 5.9 are considered to have severe

introduced the proper range from 1.8 to 2.4 for Mg

diabetes already due to other factors. There are the following

concentration in Section 2.1 have a significantly higher

reasons : Patients who have deteriorated just before dialysis may

probability of becoming the state 𝐼 than those in a reference

sometimes become hypoglycemic without diabetes medicine. As

value. The simulation also shows that the patient has a fasterkidney function decreases, doctors often reduce the amount ofrate of symptoms worsening. Patients whose initial Mg

medication or reduce insulin. Therefore, the numerical value ofconcentration is out of a reference value are also distributed

HbA1c decrease [31]. BMI also tends to be severe for patients

except 𝐴 and 𝐵 compared to patients within a reference value,

under 25. It implies that diabetes worsened and the numericaland we need attention to the patient.

value of BMI decreased. We think that this analysis result seemsⓒ2021 Japan Social Data Science Society

40日本ソーシャルデータサイエンス論文誌
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Fig. 6 Convergence of square error, Left : normal, Right : aggregating states 𝐴 and 𝐵
to contain a lot of useful information for clinical doctors.

REFERENCES
4.2 Cooperation with medical system
Currently in Japan there is a framework to conduct medical
cooperation. As an example, there is Tonet [32]. The corefunction of diabetes-associated path is disease management ofpatients. Because the severity prediction function is not thecurrent system, it is possible to support the function using theresults of this study.

5. CONCLUSION
In this study, we constructed a simulation with a Markov chain
on diabetes to prevent severe diabetes. In order to distinguishthe severity of diabetes in patients, we classified the state ofpatients using eGFR and urine protein. First, using each factor,
we showed the distribution of the patient's initial and ending
status, and we examined which factors were more influential.
We confirmed that age, Mg concentration, and use of SGLT
inhibitors have a large influence on the state change of diabetic
nephropathy. Temporal changes were also shown to worsen overtime.
In this study, we considered how changing the deterioration ofdiabetic nephropathy with the passage of time is important, anddeveloped a model using a Markov chain. From the actual data,a transition probability matrix based on each factor and a steady
distribution were calculated. At this stage, the unit of thetransition used the average inspection interval calculated fromthe actual data. By simulation using the transition probability
matrix for each factor, the situation deteriorating to state 𝐼
became clear. In addition, by aggregating states, we reduced
simulation errors and devised measures to improve simulation
accuracy. We hope that we can effectively predict the results ofthe examination and use the model of this study to predictpatient condition deterioration.
As a future task, because we used the simple stateclassification in Table 2, we need to subdivide the state tosimulate detailly. In addition, we think a more realistic
simulation considering an influence between element items.

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A. Appendix
A.1 State transition of patients classified by eGFR and urinary protein

A.2 Transition probability matrix for all patientsⓒ2021 Japan Social Data Science Society

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A.3 Stationary distribution and square error of each element aggregated state 𝐴 and 𝐵

A.4 State distribution when classified by HbA1c and BMIⓒ2021 Japan Social Data Science Society

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