Crystallization Kinetics of Tacrolimus Monohydrate in an Ethanol–Water System

Abstract

Nucleation and growth during the crystallization process are crucial steps that determine the crystal structure, size, morphology, and purity. A thorough understanding of these mechanisms is essential for producing crystalline products with consistent properties. This study investigates the solubility of tacrolimus (FK506) in an ethanol–water system (1:1, v/v) and examines its crystallization kinetics using batch crystallization experiments. Initially, the solubility of FK506 was measured, and classical nucleation theory was employed to analyze the induction period to determine interfacial free energy ($\gamma$) and other nucleation parameters, including the critical nucleus radius ($r^{*}$), critical free energy ($∆G^{*}$), and the molecular count of the critical nucleus ($i^{*}$). Crystallization kinetics under seeded conditions were also measured, and the parameters of the kinetic model were analyzed to understand the effects of process states such as temperature on the crystallization process. The results suggested that increasing temperature and supersaturation promotes nucleation. The surface entropy factor ($f$) indicates that the tacrolimus crystal growth mechanism is a two-dimensional nucleation growth. The growth process follows the particle size-independent growth law proposed by McCabe. The estimated kinetic parameters reveal the effects of supersaturation, temperature, and suspension density on the nucleation and growth rates.

1. Introduction

Tacrolimus, also known as FK506, is a macrolide antibiotic [1]. Since its approval by the FDA in 1994 for the prevention of solid organ transplantation, it has been widely used in the treatment of acute and chronic rejection of liver, kidney, and bone marrow transplants, becoming a first-line drug for the prevention of transplant rejection [2]. Its monohydrate is now widely used as an active pharmaceutical ingredient (API) in clinical practice and has great commercial value. Tacrolimus is a fermentation product that is first fermented in the presence of generating bacteria to obtain a fermentation broth and then extracted using various processes, such as leaching, to obtain a crude product. The crude product is purified and decontaminated to obtain a refined product, which is in the form of an oil and is usually crystallized using a mixed solvent system to obtain a finished product of tacrolimus.

Crystallization is an important step in the purification of an active pharmaceutical ingredient (API) during development. More than 90% of pharmaceutical products are produced through at least one crystallization step [3,4]. In terms of pharmaceutical crystallization processes, the purity, crystal size and shape, and polycrystalline form are often considered as the main critical quality attributes (CQA) that need to be closely monitored and tightly controlled [5,6]. Consequently, monitoring, modeling, and controlling product particle size has become a central focus in quality by design (QbD) to ensure product quality and performance [7]. A large number of studies have been reported on the modelling of crystallization processes. Zhao et al. carried out a detailed study of the kinetics of primary nucleation of benzoic acid in an ethanol–water system and described a relationship between the antisolvent content and nucleation [8]. Zheng and Li carried out data determination and prediction modeling of the induction time of nucleation for cooling crystallization in an inositol–water system [9]. Zhao et al. investigated the kinetics of anti-solvent cooling and crystallization of 50-ribonucleotide and explored the effects of the addition of different antisolvents. The effects of different antisolvent addition rates and cooling rates on MSZW and crystallization kinetics were explored [10]. Kim et al. conducted a study on the modeling of the cooling crystallization process of amorphous species using sucrose and KNO₃ as examples and successfully predicted the CSD of the crystalline product [11]. Nyande et al. proposed a novel data-driven approach for the development of a mathematical model of the crystallization process, where the model agrees well with the data from the experiments, but the form of the determined kinetics is unknown [12]. The study of the crystallization kinetics of tacrolimus helps to control the crystallization behavior and optimize the crystallization process [13]. However, the crystallization kinetics of tacrolimus monohydrate have not been examined.

In this study, the crystallization kinetics of the intermittent crystallization process of tacrolimus monohydrate were investigated and modelled. The main objective of agitated crystallization is to make a homogeneous solution mixture with a consistent morphology of the product [14,15,16]. The nucleation and growth kinetics during batch crystallization were calculated from population balance equations. The induction period of tacrolimus at different temperatures and supersaturation levels was determined experimentally using the turbidimetric method. Nucleation parameters such as interfacial free energy ($\gamma$), critical nuclear radius ($r^{*}$), critical nuclear free energy ($∆G^{*}$), critical nuclear molecule number ($i^{*}$), and nucleation mechanism were calculated. These results help to explain the nucleation and growth process of tacrolimus monohydrate and may provide corresponding theoretical guidance for the crystallization and purification processes.

2. Theory

2.1. Primary Nucleation

Based on classical nucleation theory, Arrhenius proposed a nucleation rate equation [17,18,19,20]:

$$J=Aexp\left(-{\displaystyle \frac{∆G^{*}}{kT}}\right)$$

(1)

where k is the Boltzmann constant, and $∆G^{*}$ is the critical free energy of the nucleus. The term $∆G$ can be expressed as the sum of the surface free energy ($∆G_S$) and the bulk free energy ($∆G_V$) of the clusters in the solid phase:

$$∆G=∆G_S+∆G_V$$

(2)

It is assumed that the nuclei formed by cluster aggregation are spherical. Gibb’s free energy is expressed as follows:

$$∆G=4\pi {{(r}^{*})}^2\gamma +{\displaystyle \frac{4}{3}}\pi {{(r}^{*})}^3∆G_V$$

(3)

where $\gamma$ is the interfacial energy, and $r^{*}$ is the radius of the critical nucleus. When a critical nucleus is formed, the critical free energy satisfies the following equation [21]:

$${\displaystyle \frac{d\left(\mathsf{\Delta }G\right)}{dr}}=8\pi r^{*}\gamma +4\pi (r^{*}{)}^2\mathsf{\Delta }{G}_V=0\to r^{*}=-{\displaystyle \frac{2\gamma }{\mathsf{\Delta }{G}_V}}$$

(4)

Substituting Equation (4) into Equation (3) gives the critical free energy for the formation of nuclei as follows:

$$\mathsf{\Delta }{G}^{*}={\displaystyle \frac{4}{3}}\pi (r^{*}{)}^2\gamma$$

(5)

According to the Gibbs–Thomson formula, we can obtain the following transformation:

$$lnS=ln{\displaystyle \frac{C}{C^{*}}}={\displaystyle \frac{2\gamma V_m}{kT{r}^{*}}}$$

(6)

where $C^{*}$ is the solubility of the solution, $C$ is the actual concentration, and $V_m$ is the molar crystal volume. This leads to the uniform nucleation rate formula, which is expressed as follows:

$$J=Aexp\left[-{\displaystyle \frac{16\pi {\gamma }^3{V}_m^2}{3k^2{T}^2{\left(lnS\right)}^2}}\right]$$

(7)

In classical nucleation theory, the induction period is considered to be inversely proportional to the rate of nucleation [22]:

$$t_{ind}\propto J^{-1}$$

(8)

According to Equations (7) and (8), the relationship between the induction period and supersaturation can be obtained by taking the logarithm of both sides at the same time:

$$ln{t}_{ind}=K+{\displaystyle \frac{16\pi {\gamma }^3{V}_m^2}{3k^3{T}^3{\left(lnS\right)}^2}}$$

(9)

At constant temperature, $ln{t}_{ind}$ and ${1/\left(\mathrm{l}\mathrm{n}S\right)}^2$ should be a linear relationship with slope b:

$$b={\displaystyle \frac{16\pi {\gamma }^3{V}_m^2}{3k^3{T}^3}}$$

(10)

The interfacial energy ($\gamma$) can be obtained from the following equation:

$$\gamma ={\left({\displaystyle \frac{3b{k}^3{T}^3}{16\pi V_m^2}}\right)}^{{\displaystyle \frac{1}{3}}}$$

(11)

The number of critical molecules $i^{*}$ is expressed as follows:

$$i^{*}={\displaystyle \frac{4\pi (r^{*}{)}^3}{3V_m}}$$

(12)

The surface entropy coefficient $f$ is an important measure of the surface roughness of a crystal and can be calculated from the interfacial energy:

$$f={\displaystyle \frac{4V_m^{\frac{2}{3}}\gamma }{kT}}$$

(13)

2.2. Population Balance Equations (PBEs)

During crystallization, population balance equations can be used to describe the change in crystal size distribution in space and time. It is assumed that crystal growth is independent of crystal size and that the solution volume does not change. Neglecting aggregation and fragmentation, the population balance equation can take the following form [23]:

$${\displaystyle \frac{\partial n\left(L,t\right)}{\partial t}}+{\displaystyle \frac{\partial \left[\mathrm{G}n\left(L,t\right)\right]}{\partial L}}=0$$

(14)

where $\mathrm{G}$ is the growth rate, $n$ is the number density, $t$ is the crystallization time, and $L$ is the crystal size. In this study, FBRM was used to characterize the change in particle size of tacrolimus during cooling crystallization [24]. The number density ($n_i$) is the number of crystals in the $\mathsf{\Delta }{L}_i$ region per unit volume of suspension. The number of particles $N_i$ in the ith channel can be approximated as follows:

$$N_i=n_i\times \mathsf{\Delta }{L}_i$$

(15)

In the majority of industrial crystallizers, the main source of nucleation is secondary nucleation [25]. A number of nucleation mechanisms have been proposed in the course of research. The most widely recognized of these are fluid shear stress nucleation and contact nucleation. Fluid shear stress nucleation refers to the presence of shear stress within the fluid boundary layer, which removes particles adsorbed on the crystal, thus forming a new nucleus. Contact nucleation is the process in which a crystal collides with an external object to form a new nucleus [26,27,28]. Due to the complexity of the mechanism, the understanding of secondary nucleation is still incomplete. Moreover, crystal growth is usually controlled by mass transfer as a function of the supersaturation ratio and stirring rate. In this study, an empirical model of secondary nucleation, which is widely used in industry, and a size-independent growth model are considered for the mathematical description [29,30,31]:

$$B=k_b{M}_t^j{{\omega }_r}^i{S}^h$$

(16)

$$G=k_{\mathrm{g}}exp\left(-{\displaystyle \frac{E_a}{RT}}\right)S^{\mathrm{g}}{{\omega }_r}^v$$

(17)

where $k_b$ and $k_{\mathrm{g}}$ are the pre-exponential factors; $i$, $j$, $h$, $\mathrm{g}$, and $v$ are the secondary nucleation and growth parameters; $M_t$ is the suspension density; and ${\omega }_r$ is the stirring speed. A method of moment transformation is introduced to track the CSD($m_j$) at a particular moment to reduce the difficulty of solving the PBE [32]. The defining Equation (19) for the order moment $j$ of the granularity with respect to the origin in the grain number density distribution is expressed as follows:

$$m_j=\underset{0}{\stackrel{\infty }{\int }}n\left(L,t\right)L^{j}dL \left(j=\mathrm{0,1},\mathrm{2,3}\dots \right)$$

(18)

The nucleation rate and growth rate can be calculated from the following equation:

$${\displaystyle \frac{d{m}_0}{dt}}={\displaystyle \frac{d{\int }_0^{\mathrm{\infty }} ndL}{dt}}={\displaystyle \frac{dN}{dt}}=B_0$$

(19)

$${\displaystyle \frac{d{m}_1}{dt}}={\displaystyle \frac{d{\int }_0^{\mathrm{\infty }} nLdL}{dt}}={\displaystyle \frac{d{\lambda }_T}{dt}}=m_{0}G$$

(20)

3. Experiments & Materials

3.1. Materials

Anhydrous ethanol was obtained from Tianjin Yuanli Chemical Co., Ltd. Tianjin, China, with a purity of over than 99.5%. Deionized water was prepared in the laboratory. The solvent system used in the experiment was obtained by mixing ethanol and deionized water in equal volumes. FK506 was acquired from Hebei Huasheng Pharmaceutical Co., Ltd. Shijiazhuang, China and underwent two rounds of purification before use.

3.2. Experimental Procedure

The solubility of FK506 was determined gravimetrically in the range of 273.15 K to 333.15 K at atmospheric pressure. In the experiment, 15 mL of solvent was added to a 20 mL sample bottle, which was then heated or cooled to the desired temperature. An excess of FK506 was added, and the solution was stirred on a magnetic stirrer at 400 rpm for 24 h to achieve solid–liquid equilibrium. Subsequently, 3 mL of saturated supernatant was withdrawn from each vial using a pre-warmed syringe and filtered through an organic membrane filter (0.22 μm). The filtrate was placed in pre-weighed surface dishes and dried in an oven at 323.15 K for 12 h until the solvent had completely evaporated. After cooling to room temperature, the samples were weighed on an analytical balance (ME204/02, Mettler Toledo, Zurich, Switzerland) [33]. To minimize experimental errors, the above procedure was repeated three times, and the average of the three values was used to calculate the solubility results [34]:

$$x_1={\displaystyle \frac{m_1/M_1}{m_1/M_1+m_2/M_2+m_3/M_3}}$$

(21)

where m₁, m₂, and m₃ denote the mass of tacrolimus, water, and ethanol, respectively, and M₁, M₂, and M₃ denote the molar mass of tacrolimus, water, and ethanol, respectively.

The nucleation induction period was determined using a parallel crystallizer (Crystalline, Technobis Alkmaar, The Netherlands) [35]. The supersaturated solution is first prepared with an ethanol–water solution and tacrolimus monohydrate, kept at 5 K above the saturation temperature for 15–30 min to remove fine crystals from the solution. Then, the solution is immediately transferred to a predetermined parallel crystallizer for the experiment. Since the rate of cooling affects the nucleation induction time, the solution is rapidly cooled at 20 K/min to the desired experimental temperature. Therefore, the cooling time is less than 3 min. In addition, taking into account thermal conduction effects, we specify that the solution reaches the experimental temperature 1 min after the system temperature reaches the experimental temperature, which is much shorter than the induction times measured in the table. Due to unavoidable factors, a threshold of 95% light transmission was used in this study to determine the induction time. The stirring speed was set at 400 rpm.

The cooling crystallization experiments were carried out in a 100 mL jacketed crystallizer with the temperature controlled by a thermostatic water bath (CF41, Julabo, Seelbach, German), and the change in the number of crystals during the crystallization process was monitored using a FBRM (G400, Mettler Toledo). Samples were taken every ten minutes during the crystallization process. Changes in the concentration of the solution were monitored gravimetrically, and the mass of the crystals in the system at different moments of the crystallization process was calculated to obtain the nucleation kinetics and growth kinetics. The crystal sizes were graded using standard sieving methods, and the mean chord lengths and equivalent volume diameters of the different size classes were de-defined using FBRM and a particle size analyzer (Malvern Mastersizer Hydro 3000MU Malvern Panalytical, Malvern, UK). FBRM field data were corrected by mathematical correlation.

4. Results and Discussion

4.1. Solubility

The solubility was determined at 293.15~333.15 K in an ethanol–water system. All the determinations were repeated three times, and the average results are shown in Figure 1. It is evident that the solubility of tacrolimus decreases rapidly as the temperature decreases.

A mathematical description of solubility using cubic polynomials for subsequent experimental applications is provided:

$$x_A=a{T}^3+b{T}^2+cT+d$$

(22)

where $a=2.68687\times {10}^{-9}$, $b=2.76384\times {10}^{-7}$, and $c=-0.89442$; $d=0.171612$.

4.2. Primary Nucleation Kinetics of Tacrolimus

4.2.1. Induction Time

The induction period was measured using a parallel crystallizer, and Figure 2 shows the variation in temperature versus transmittance with time for a set of experimental conditions (S = 3.55 and T = 308.15 K). When the transmittance decreases, it indicates the beginning of nucleation in solution.

Figure 3 shows the relationship between supersaturation and induction time. The induction time decreases nonlinearly with increasing supersaturation and temperature, suggesting that these factors play an important role in the crystal nucleation process. The shorter the induction time, the higher the nucleation rate. The main influences on nucleation vary with environmental conditions. With an increase in supersaturation or temperature, more solute molecules aggregate into clusters, increasing the chance of reaching critical nucleation and thus shortening the induction time. As the nucleation temperature increases and the supersaturation decreases, the temperature becomes the key factor accelerating the nucleation process. Under the opposite conditions, supersaturation becomes the key factor. Although supersaturation is the main driver of crystallization, it is not always the controlling factor; supersaturation and nucleation temperature together control the crystallization process.

Figure 4 reflects the relationship between $ln{t}_{ind}$ and ${1/\left(\mathrm{l}\mathrm{n}S\right)}^2$. According to classical nucleation theory, there are usually two regions of homogeneous nucleation and inhomogeneous nucleation on the relationship diagram. Actually, it is difficult to detect homogeneous nucleation from the solution. Therefore, the interface can be estimated directly using Equations (9) and (10) by referring to the methods used in the study of primary nucleation kinetics [8].

4.2.2. Nucleation Parameters

The slope (b) was calculated according to classical nucleation theory and Equation (9). From this, the value of the interfacial free energy at each temperature was determined (

Table 1. Table of the γ and f for homogeneous nucleation of tacrolimus.

T (K)	Slope (b)	$\mathit{\gamma }\left(\mathbf{m}\mathbf{J}\cdot {\mathbf{m}}^{-2}\right)$	$\mathit{f}$	R2
308.15 K	14.53	3.76	3.81	0.9971
303.15 K	15.31	3.77	3.88	0.9937
298.15 K	17.94	3.90	4.08	0.9953

Note: R² is correlation coefficient.

).

The interfacial free energy ($\gamma$) of tacrolimus decreases slightly with increasing temperature, suggesting that higher temperatures favor crystal nucleation. In other words, the increase in temperature promotes the formation of crystal nuclei. The $f$ values recorded at different temperatures ranged between 3 and 5. Therefore, it is presumed that the crystallization of tacrolimus in the ethanol–water system follows a two-dimensional nucleation growth mechanism.

According to the classical nucleation theory, the nucleation parameters $∆G^{*}$, $r^{*}$, and $i^{*}$ are obtained by substituting $\gamma$ into the equations, as shown in Figure 5. When the supersaturation is increased while maintaining the same supersaturation, or the nucleation temperature is increased while maintaining the same temperature, the nucleation parameters decrease, and the nucleation rate ($J$) increases. These results are consistent with classical nucleation theory and emphasize the joint influence of supersaturation and temperature on the nucleation process. Therefore, at a given temperature, increasing the supersaturation increases the nucleation rate and shortens the induction time.

4.3. Cooling Crystallization Kinetics of Tacrolimus

4.3.1. Dynamics Data

In this study, the crystallization process was examined using cooled crystallization with a temperature range of 323.15 K to 298.15 K. Crystal samples were collected every 10 minutes and analyzed for supersaturation and suspension density using the gravimetric method, with the FBRM scanning the crystal chord length distribution every 2 s.

Table 2. Table of experimental arrangement.

No.	T (K)	${\mathit{w}}_{\mathit{r}}$ (rpm)	S	Seed Loaded (%)	ΔT (K/min)
1	323.15	200	1.200	2	0.10
2	323.15	250	1.200	2	0.11
3	323.15	350	1.200	2	0.12

demonstrates the experimental design of this study. Some of the data from the first set of experiments are shown in

Table 3. Part of the data from the kinetic experiments.

No.	T (K)	${\mathit{w}}_{\mathit{r}}$ (rpm)	MT (g/g)	S	B (${\#·\mathbf{m}}^{-3}{\mathbf{s}}^{-1}$)	G ($\mathbf{m}·{\mathbf{s}}^{-1}$)
1-1	323.15	200	$6.50\times {10}^{-3}$	1.200	$1.98\times {10}^6$	$3.31\times {10}^{-8}$
1-2	321.06	200	$6.10\times {10}^{-3}$	1.355	$2.54\times {10}^6$	$3.39\times {10}^{-8}$
1-3	318.98	200	$6.71\times {10}^{-3}$	1.519	$3.28\times {10}^6$	$3.45\times {10}^{-8}$
1-4	316.89	200	$8.32\times {10}^{-3}$	1.652	$4.27\times {10}^6$	$3.49\times {10}^{-8}$
1-5	314.81	200	$1.09\times {10}^{-2}$	1.754	$5.58\times {10}^6$	$3.50\times {10}^{-8}$
1-6	312.72	200	$1.46\times {10}^{-2}$	1.824	$7.32\times {10}^6$	$3.50\times {10}^{-8}$
1-7	310.64	200	$1.92\times {10}^{-2}$	1.863	$9.64\times {10}^6$	$3.47\times {10}^{-8}$
1-8	308.55	200	$2.49\times {10}^{-2}$	1.872	$1.27\times {10}^7$	$3.43\times {10}^{-8}$
1-9	306.47	200	$3.15\times {10}^{-2}$	1.849	$1.69\times {10}^7$	$3.36\times {10}^{-8}$
1-10	304.38	200	$3.92\times {10}^{-2}$	1.796	$2.24\times {10}^7$	$3.29\times {10}^{-8}$
1-11	302.30	200	$4.79\times {10}^{-2}$	1.712	$2.98\times {10}^7$	$3.20\times {10}^{-8}$
1-10	300.21	200	$5.76\times {10}^{-2}$	1.598	$3.98\times {10}^7$	$3.11\times {10}^{-8}$
1-12	298.15	200	$6.82\times {10}^{-2}$	1.455	$5.28\times {10}^7$	$3.01\times {10}^{-8}$

. From Figure 6, it can be seen that there is a linear relationship between the logarithm of the density of the crystals and the particle size. Therefore, the crystal growth rate of tacrolimus seems to be independent of the particle size. In this experiment, we analyzed the growth rate of tacrolimus using a particle size-independent growth model based on McCabe’s ΔL law. The nucleation and growth rates were derived using moment analysis.

4.3.2. Dynamical Equations

The kinetic equations for the cooling crystallization of tacrolimus in ethanol–water can be obtained by substituting the experimentally obtained temperature, concentration, suspension density, rotational speed, and kinetic data into the kinetic model in order to determine the parameters of the equations as shown in Equations (23) and (24). The parameters were fitted using the lsqcurvefit function in MATLAB R2021b.

$$B_0=1.8158\times {10}^2{S}^{0.1583}{Mt}^{1.1933}{{\omega }_r}^{2.9918} \left(R^2=0.9355\right)$$

(23)

$$G=7.8808\times {10}^{-9}\mathrm{ex}\mathrm{p}\left(-{\displaystyle \frac{5472.014}{RT}}\right)S^{0.1204}{{\omega }_r}^{0.6878} (R^2=0.9870)$$

(24)

Table 4. The confidence interval of the estimated parameter.

Parameter	Confidence Interval (95%)
Kinetic modelling of nucleation
kb	(144.1924, 228.6473)
S	(0.0569, 0.2597)
Mt	(1.1821, 1.2046)
${\omega }_r$	(2.9520, 3.0317)
Kinetic modelling of growth
kg	$(6.2583\times {10}^{-9}$$, 9.9239\times {10}^{-9}$)
S	(0.0190, 0.2218)
${\omega }_r$	(0.6479, 0.7276)

shows the calculated confidence intervals for the parameters. From the data analysis, it can be concluded that in the nucleation rate equation, the supersaturation index is 0.1583, and the suspension density index is 1.1933. This indicates that both supersaturation and suspension density have a positive effect on the secondary nucleation rate. It can be concluded that although supersaturation promotes the formation of new nuclei during the crystallization process, nucleation dependent on the suspension density is also important and a non-negligible factor. By analyzing the growth rate equation, the exponential value of supersaturation is 0.1204, which suggests that an increase in supersaturation favors crystal growth. However, comparing the parameters of the nucleation and growth equations, the effect of supersaturation on nucleation is more significant. At high supersaturation, crystal growth will be limited by the sharp increase in nucleation. Decreasing the degree of supersaturation of the solution leads to a more significant decrease in crystal nucleation, thus providing sufficient space for crystal growth. Therefore, if large size crystals are desired, crystallization in a low saturation solution may be an option.

4.3.3. Impact Factors

The effect of suspension density on the nucleation rate of tacrolimus is shown in Figure 7. The nucleation rate also increases with increasing suspension density. The increase in suspended particles in solution exacerbates secondary nucleation. However, the increase in suspended particles also limits the growth of crystal nuclei.

The effect of the supersaturation ratio on the nucleation and growth kinetics of crystals is shown in Figure 8. Increasing the supersaturation ratio enhances both the nucleation and growth rates of tacrolimus. Specifically, the nucleation and growth kinetics are 0.1583 and 0.1204, respectively, indicating a more pronounced effect on nucleation. These data suggest that while higher supersaturation accelerates nucleation, it can inhibit crystal growth. Excessive supersaturation results in high nucleation rates, which creates numerous small crystals with limited space for further growth, leading to non-uniformity and poor crystal quality. Therefore, selecting an appropriate supersaturation ratio is crucial in the industrial crystallization processes to optimize crystal quality.

Figure 9 illustrates the loading of the kinetic model with the experimental data. These models can provide suitable prediction and guidance for the crystallization process, which is helpful and instructive in adjusting the process parameters to obtain better products.

5. Conclusions

In this paper, guided by the classical nucleation theory, we obtained the nucleation parameters of tacrolimus monohydrate by measuring its nucleation induction time and analyzed the effects of supersaturation and temperature on its primary nucleation kinetics. We also determine and model the crystallization kinetics of tacrolimus monohydrate in an ethanol–water system. The results suggested that both temperature and supersaturation positively affect the primary nucleation, and the nucleation induction time decreases with increasing supersaturation. The interfacial free energy decreases with increasing temperature, while the nucleation parameters (critical free energy, critical nuclear radius, and critical number of molecules) decrease. In addition, the nucleation rate gradually increases. The surface entropy factor ranged from 3 to 5, indicating that the nucleation mechanism of tacrolimus monohydrate was characterized by two-dimensional nucleation growth. Using the population equilibrium equation, we developed a crystallization kinetic model and found that the crystal growth of tacrolimus monohydrate is independent of size. The model emphasizes that the effect of supersaturation on the nucleation rate is particularly significant. Finally, we validated the model against experimental results of crystallization under conditions of intermittent stirring and found that the experimental values were in good agreement with the model predictions. These findings deepen our understanding of the nucleation and growth process of tacrolimus monohydrate and may provide valuable theoretical guidance for the crystallization and purification process.