Mathematical solutions in internal dose assessment: A comparison of Python-based differential equation solvers in biokinetic modeling

A biokinetic model describes the uptake, retention, and clearance of incorporated radionuclides in a biological system using mathematical formalisms defined as a function of intake pathways. Due to the complexity of biological systems, the biokinetic models are classified based on the specific intake scenario (Nosske et al 2011, Li 2018, Zhang et al 2021):

• Human Respiratory Tract Model (HRTM): Describes the deposition of inhaled radionuclides in the respiratory tract, mechanical clearance to other regions of the respiratory tract and the alimentary tract, and absorption into blood;
• Human Alimentary Tract Model (HATM): Quantifies the intake of radionuclides by ingestion, absorption into blood, and clearance (excretion) through feces;
• Wound Model: Describes radionuclide uptake, retention, and clearance through wound contamination with subsequent absorption to blood; and
• Systemic Model: Describes the systemic circulation of radionuclides after entering the blood through injection (either accidental injection or radiopharmaceutical applications) and the aforementioned internal exposure pathways.

Prior to the mid-1960s, knowledge of the quantification of internally incorporated radionuclides was limited (Stather 2004). However, the mandate to increase knowledge of radiation monitoring and prevention of undue exposure was improved as a result of the establishment of organizations, among which was the International Commission on Radiological Protection (ICRP). Building on the foundation laid by the first ICRP biokinetic models for assessing radionuclide intake and dose in the body (ICRP 1959), significant advancements have been realized for prospective and retrospective radiological protection over the years. To this effort, the framework of a compartment-based method found much applicability for use in internal dosimetry applications (Sanchez 2005). The compartmental analysis applied to biological systems separates the organs and tissues of the body into several individual compartments, which are connected through the metabolic exchange of materials (i.e. internalized radionuclides) (Vicini and Brill 2008). The system of compartments is mathematically modeled as a system of coupled ordinary differential equations (ODEs). This mathematical concept describes a state of evolution, given that the state variable changes over time. The conformity of the system of compartments to an ODE allows for its applicability in chemical reactions studies, drug kinetics in pharmacology, and metabolic systems studies (Anderson 1983, Aro 1996, Postawa et al 2020).

Moreover, due to the complexity of systems constituting biokinetic models, computational capabilities were harnessed for solving the coupled system and have been pervasively applied in internal dosimetry (Killough and Eckerman 1984, Birchall and James 1989, Sadre Momtaz and Sheikhangafsheh 2021). Several solving methods, algorithms, and computer codes have been developed for solving general kinetic problems, including biokinetics (Dunford and Johnson 1987, French et al 1988, Strenge et al 1990, Eckerman et al 2006, Fell et al 2007, Bertelli et al 2008). However, most of these codes and tools are based on routine programming languages lacking modern dynamic capabilities, despite the flexibility of symbolic representations in coding syntax and dynamic memory. As new operating system versions have evolved and new models have been developed, migrating toolkits across platforms cannot be assumed to be a simple translation and compilation process (Stabin et al 2005). Moreover, open-source toolkits for internal dose assessment are limited, if available, and functionalities are even further limited. Expansions of enhanced and customizable solution tools have necessitated developing compartmental models in Python codified form with an enhanced coupling capability.

The objective of this study was to test the capabilities of different solvers and methods in the modern programming language Python for solving an extensive coupled system of ODEs for biokinetic modeling. Python is an interpreted and dynamic programming language that offers flexibility by supporting an agile development process (Schmitt et al 2022). The choice to explore Python capabilities for solving the biokinetic models was due to the existence of its direct interfacing potential with a multilevel parallel object-oriented framework for an optimized parametric analysis–a promising platform for the interoperation of disparate codes, modules, and tools.

Metabolic models are divided into several compartments, which are solved as systems of ODEs to estimate the biodistribution and retention of the radioactive material in the body (ICRP 1972). However, the coupled system for a given radionuclide and clearance class could comprise several rapidly and slowly varying components resulting in a problem known as stiffness. Stiffness has been identified as a prominent challenge inherent to the complexities of biokinetic modeling. Stiff systems can result in significant numerical instabilities if not solved with the appropriate solvers and methods, where stability implies solutions that do not exhibit oscillating derivatives, inaccurate solutions (including negative solutions) or divergent solutions (a solution that happens to be unusually large positive or large negative), and solutions with amplification factor less than or equal to one over time (indicating growth in error with time as the solution evolves) (Yano et al 2012). This study conducted a careful analysis of standard ODE solvers and solving methods in Python, defining the application regime for Python solvers based on their applicability to biokinetic modeling. As such, this study's framing elaborates on the problem challenge and stiffness quantification.

This study focuses on the methodological approach adopted and recommended by ICRP (1994, 1995, 1996, 2006, 2007, 2015, 2016a, 2016b, 2017) for internal dose assessment by reconstructing the biokinetic models into Python code and solving the system with different Python numerical solvers through practical examples. The radionuclides explored in this study were selected to demonstrate the solving capabilities of Python-based ODE solvers and solving methods, which are ⁶⁰Co (moderate clearing–Type M) and ¹³¹I (fast clearing–Type F). Both ⁶⁰Co (Type M) and ¹³¹I (Type F) were chosen to encapsulate rapidly and slowly varying components to indicate the level of stiffness of the differential equations posed by the biokinetic models. The choice for a moderate and rapid clearing radionuclide is necessary to demonstrate the tolerance level of the solvers for both the HTRM and systemic biokinetic model. These solvers and solving methods can be comprehensively applied and adapted for solving any other element-specific biokinetic modeling problem. Cobalt, in general, can be encountered in various chemical forms, including metal dust, oxides, nitrates, and chlorides (ICRP 2017). Cobalt-60 is of concern for emergency public protection primarily due to its potential for use in radiological dispersal devices, where inhalation is considered the most probable mode of intake from internal exposures (Dewji et al 2013). Conversely, iodine may be encountered in industry in various chemical and physical forms, including vapors and gases, organic compounds, and particulate forms (ICRP 2017). Iodine-131 is associated with nuclear power plant effluent from the release of fission products (DJurović et al 2016, Sulaiman et al 2018) or an improvised nuclear device (Marcus et al 2006, Chen and Tenforde 2010, Pan and Ungar 2012). Practical examples given in this paper for demonstration employed the ICRP HRTM given in Publication 130 (ICRP 2015); however, the fundamental mathematical approach governing the estimation of activity burden in the body remains consistent and could be adopted for other internal exposure scenarios of concern.

The methodology applied in this study first addressed the framework of compartment-based biokinetic models, from which different mathematical solutions of ODEs in Python were compared and analyzed for stability. Both explicit and implicit methods, an optimized implicit method, and an algebraic methodology, such as matrix exponentiation, were compared. The biokinetic solutions obtained using the exploited Python solvers and the solving methods were compared further with the biokinetic retention solutions from the ICRP Occupational Intake of Radionuclides (OIR) Electronic Annex Distribution Set to quantify the agreement with published data ICRP (2015, 2016b, 2017). The solvers with the most stable solutions were then recommended for further development of a robust computational framework.

2.1. ODE form of biokinetic systems

The dynamic behavior of an inhaled radionuclide in the human respiratory tract is described in ICRP Publication 130 (ICRP 2015). The ICRP Publication 130 model is the updated form of the previous HRTM defined in Publication 66 (ICRP 1994). From the time of intake of a radionuclide, the rate of exchange of the material concentration framed as a mathematical description of the evolution of radionuclides in the body is represented in its general form in equation (1) (ICRP 2015, Issa and Serge 2021, Zhang et al 2021):

$$\begin{equation}\frac{{{\text{d}}{A_{i,j}}\left( t \right)}}{{{\text{d}}t}} = \mathop \sum \limits_{\begin{array}{*{20}{c}} {k = 1} \\ {k \ne j} \end{array}}^M {A_{i,k}}{\lambda _{i,k,j}} - {A_{i,j}}\left[ {\mathop \sum \limits_{\begin{array}{*{20}{c}} {k = 1} \\ {k \ne j} \end{array}}^M {\lambda _{i,k,j}} + \lambda _i^{\text{D}}} \right] + \mathop \sum \limits_{k = 1}^{i - 1} {A_{k,j}}{\beta _{k,i}}\lambda _{i\,\,}^{\text{D}}\end{equation} \tag{ 1 }$$

where M is the number of compartments describing the kinetics;

${\lambda _{i,k,j}}$ is the fractional transfer rate of chain member i, from donor compartment j to receiver compartment k in the biokinetic model;

$\lambda _i^{\text{D}}$ is the physical decay constant of decay chain member i; and

${\beta _{k,i}}$ is the fraction of decays of decay chain member k forming i.

${A_{i,j}}\left( t \right)$ is the activity (Bq) of the radionuclide i in compartment j at time t.

Thus, equation (1) serves as the governing equation for radionuclide and its progeny's distribution in the body conforming to a system first-order ordinary differential equation. With the initial conditions for each compartment (${A_{i,j}}\left( 0 \right)$), the system of ODEs can be solved using applicable numerical/algebraic methods. ICRP Publication 130 (2015) further details the framework for utilizing biokinetic models to develop internal dose coefficients.

2.2. ODE solvers and solving methods

First, solvers in the solve_ivp package from Python Scipy scientific library (Christensen et al 2015, Virtanen et al 2020) were used to solve the ODE system. The solve_ivp package is a condensed solver comprising six solvers (table 1) divided into stiff and non-stiff solvers, where LSODA is a solver known for its universal adaptation.

Table 1. Python language solve_ivp ODE solvers with their respective methods (Hagen and Mayorov 2019).

Method	Solver	Recommended choice of ODE problems
Explicit	RK45	Non-stiff
	RK23	Non-stiff
	DOP853	Non-stiff
Implicit	Radau	Stiff
	BDF	Stiff
Universal choice	LSODA	Universal

In general, ODEs can be classified as non-stiff and stiff based on the problem being modeled. Non-stiff problems are problems whereby all the components evolve simultaneously, while stiff problems are classified as problems with rapidly and slowly varying components; these generally arise in biokinetic modeling of the inhalation pathway (Byrne and Hindmarsh 1987, Omale et al 2014). A stiff problem, typically encountered in internal dosimetry, is known to benefit from analytical or semi-analytical approaches such as eigenvalue methods, where analytical expressions are derived numerically (Killough and Eckerman 1984, Birchall and James 1989, Fell et al 2007). The benefit of a numerical method is derived from its versatility and is, therefore, better adapted to tackle non-linearities (Fell et al 2007). The numerical solvers and methods employed in this work are discussed as follows. The solve_ivp function is part of a class-based package of explicit and implicit ODE solvers in SciPy for solving standard initial value problems (Hagen and Mayorov 2019). The function in SciPy can be called by default using scipy.integrate.solve_ivp(fun, t_span, y0, method = 'RK45', t_eval = None, dense_output = False, events = None, vectorized = False, args = None, **options), where

• fun defines the right-hand side of the equations, which has a calling signature as func(t,y);
• t_span defines the time interval of integration (0,tf);
• y0 defines the initial conditions or an array like the initial state of the system, method specify the solving method to implement;
• t_eval defines the time for which the solutions are stored;
• dense_output specifies whether to implement a continuous solution;
• events tracks events that applies termination if that specific event occurs or redirect computation based on the direction of zero crossing;
• vectorized parameter specifies if fun is vectorized;
• args are arguments to pass to user-defined functions; and
• **options include
  • first_step—initial step size;
  • max_step—maximum step size;
  • rtol and atol—relative and absolute tolerances that ensure local error control by keeping the local error estimate less than atol + rtol × abs(y);
  • jac—Jacobian matrix/array;
  • jac_sparsity—defines the sparsity structure of the Jacobian matrix for a finite-difference approximation;
  • lband—defines the bandwidth of the Jacobian for the LSODA method;
  • uband—defines bandwidth of the Jacobian for the LSODA method; and
  • min_step—defines the minimum allowed step size for the LSODA method.

Methods explored in this work were RK45, RK23, DOP853, Radau, BDF, and LSODA.

For each of the biokinetic models, matrices of constant coefficients (sparse matrix in most cases) were defined, and the right-hand side of the ODEs was further expressed as matrix multiplications. The matrix multiplications were then fed into fun in the solve_ivp function. Implementing the default parameters with each of the methods solutions over a 50 year period were obtained for a single acute unit intake of activity in the respiratory tract. The default implementations include the following:

• 1.  
  Step sizes were allowed to be chosen by the algorithm.
• 2.  
  By default, relative and absolute tolerances (rtol and atol) utilized are 10⁻³ and 10⁻⁶, respectively.
• 3.  
  Jacobian was approximated using the finite difference method.

The optional function parameters of solve_ivp, such as rtol, atol, and time step (max_step), were further adjusted to exercise strict error control. The rtol and atol values were reduced to 10⁻¹² and 10⁻¹⁵, respectively, and the function was reevaluated. Based on the stiffness level of the ¹³¹I scenario, the maximum step size was restricted to 0.2, in addition to rtol and atol of 10⁻¹² and 10⁻¹⁵. The label for restricting the step size, in addition to the reduced tolerances, was additionally customized. The biokinetic model problem sets were solved for each case discussed, and the activity retained in each compartment (for the specific radionuclide) was tracked.

2.2.1. Explicit solvers

According to SciPy, the RK45 method, which is the default method for solve_ivp, is an explicit Runge–Kutta method of order 5(4) (Dormand and Prince 1980). The RK45 method controls the error by assuming the accuracy of the fourth-order method, but steps are taken using fifth-order local extrapolation. The method utilizes a quartic interpolation polynomial–a fourth-degree polynomial (Shampine 1986, Hairer and Ostermann 1990) and can be applied to a complex domain–applicable to the other explicit methods investigated. RK23 is another explicit Runge–Kutta method of order 3(2), where in contrast to RK45, the error is controlled by assuming the accuracy of the second-order method and third-order local interpolation method is utilized by the steps taken (Bogacki and Shampine 1989). When computing continuous solutions, RK23 uses a cubic Hermite polynomial for its dense output. The last solely explicit solver/method exploited was DOP853 of order 8, which is a Python implementation of the DOP853 algorithm initially written in FORTRAN (Hairer et al 1993).

2.2.2. Implicit solvers

The next category of solving methods investigated was implicit methods. Radau is an implicit method of the Radau IIA family of order 5 (Wanner and Hairer 1996). To satisfy the collocation condition in the problem domain, Radau employs cubic polynomials for dense output. Additionally, Radau IIA methods are considered a one-step collocation-type method, which is approximated by polynomials for which the stability of a stiff differential equation is tested by utilizing sub-diagonal Padé approximations (Hairer and Wanner 2015). For the BDF method, a multi-step variable order (1–5) based on backward differentiation formula is employed, for which accuracy is enhanced using the numerical differentiation formulas modification (Shampine and Reichelt 1997) and can be applied to complex domains.

2.2.3. Universal solvers

The final method investigated was LSODA. LSODA is a FORTRAN wrapper from ODEPACK, which is equipped with automatic stiffness detection capability. ODEPACK is a collection of FORTRAN solvers for standard initial value ODE problems (Hindmarsh 1983). LSODA is usually considered a universal choice, and its algorithm permits switching between Adams (a method for non-stiff systems) and BDF (a method suitable for stiff systems of equations) (Petzold 1983). Moreover, some problem types may still need to be fully classified or understood. In this case, SciPy recommends using RK45; if it results in unusually many iterations (diverge or fail), the problem under study is probably stiff and thus should be approached with Radau or BDF (Virtanen et al 2020).

2.2.4. Matrix exponential methods

Matrix theory is considered to be very important for finding the solution to a system of linear ODEs (Bellman 1997). A study published by Birchall and James (1989) described a microcomputer algorithm for solving first-order compartmental models using matrix algebra by essentially finding the exponential of the matrix of constant coefficients. The study by Birchall and James (1989) highlighted some difficulties with existing methodologies for matrix exponentiation evaluation, such as the complexity of evaluation through eigenvalue decomposition and for non-symmetric matrices. Earlier publications also made a major assumption for using eigenvalue methods for relatively simple compartmental models (Killough and Eckerman 1984). According to Killough and Eckerman (1984), to solve an initial value problem (IVP) of the form

$$\begin{equation}\frac{{{\text{d}}Y}}{{{\text{d}}t}} = \,AY\end{equation} \tag{ 2 }$$

$$\begin{equation}Y\left( 0 \right) = \,{Y^{\,0}}\end{equation} \tag{ 3 }$$

where $Y$ is a vector of N functions;

$A$ is a constant N × N matrix coefficient; and

${Y^{\,0}}$ is a vector of initial values of $Y$.

It was assumed that the eigenvectors form a linearly independent set. An eigenvalue is a particular set of scalars, which are also known as characteristic roots, connected with a linear system of equations. An eigenvector is a non-zero vector in linear algebra that can only change direction when scaled by a characteristic root known as the corresponding eigenvalue (Hoffman and Kunze 1971). However, the eigenvalue approach breaks down for cases when the system contains degenerate compartments, defined as consecutive compartments with similar transfer rates. A solution to curb this failure is to perturb the model parameters by a small fraction for the perturbed system to have linearly independent eigenvectors (Killough and Eckerman 1984). To avoid problems with eigenvalue approaches for analytical evaluation, this work exploited other approaches for matrix exponential. A unique solution to the problem in equation (2) can be denoted as ${{\text{e}}^{At}}$. This matrix exponential which can be defined in a standardized form by power series denoted in equation (4) (Polig 2001, Moler and Van Loan 2003).

$$\begin{equation}{{\text{e}}^{At}} = \sum\limits_{i = 0}^\infty \frac{{{{\left( {At} \right)}^i}}}{{i!}}.\end{equation} \tag{ 4 }$$

To compute equation (4), Birchall and James (1989) proposed an optimized series expansion approach due to the difficulty encountered in using standard Taylor series expansion. In this study, the matrix exponential method (expm), which uses the Padé approximation in Python language, is employed for analytical insight to quantify the deviation of the numerical methods (Al-Mohy and Higham 2010). For an n × n matrix A, Padé approximation for a simple case as ${{\text{e}}^A}$ can be expressed as (Arioli et al 1996, Sidje 1998, Li et al 2011):

$$\begin{equation}{R_{pq}}\left( A \right) = \frac{{{N_{pq}}\left( A \right)}}{{{D_{pq}}\left( A \right)}}\,\end{equation} \tag{ 5 }$$

where

$$\begin{equation}{N_{pq}}\left( A \right) = \mathop \sum \limits_{j = 0}^p \frac{{\left( {p + q - j} \right)!p!}}{{\left( {p + q} \right)!j!\left( {p - j} \right)!}}{A^j}\,\end{equation} \tag{ 6 }$$

$$\begin{equation}{D_{pq}}\left( A \right) = \mathop \sum \limits_{j = 0}^q \frac{{\left( {p + q - j} \right)!q!}}{{\left( {p + q} \right)!j!\left( {q - j} \right)!}}{\left( { - A} \right)^j}\,\end{equation} \tag{ 7 }$$

where $N$ has a degree of $p$; and $D$ has a degree of $q$. The Padé approximation, in this case, can be referred to as (p,q) Padé approximation. For ${{\text{e}}^{At}}$, expm evaluates the linear system using the analogous form of equation (5) as given in equation (8).

$$\begin{equation}{R_{pq}}\left( {At} \right) = \frac{{{N_{pq}}\left( {At} \right)}}{{{D_{pq}}\left( {At} \right)}}\,.\end{equation} \tag{ 8 }$$

Additionally, the Python language expm function implements a new scaling and squaring algorithm for matrix exponentiation with a variable order that is decided based on the array provided (Al-Mohy and Higham 2010). The benefit of this approach is that it alleviates over-scaling problems by exploiting: (1) the concept of triangularity to ensure that the diagonal and first-off diagonal are computed accurately and (2) a backward error analysis for a refined truncation error bound. The matrix exponential method, derived from matrix theory, is considered advantageous compared to numerical methods because it has been known to provide virtually exact solutions to a system of equations (Ball and Adams 1967). It is worth noting that matrix exponential methods are primarily limited to linear systems of ODEs with some exceptions (Ball and Adams 1967, Moler and Van Loan 2003). Nonetheless, a conscious effort was made to investigate the linear algebraic method in addition to the numerical methods in Python programming language to have a robust and comprehensive computational framework to handle both stability and non-linear problems and have the best efficiency in terms of computational time.

2.3. Case studies: ⁶⁰Co and ¹³¹I ICRP HRTM biokinetic solving methods

According to ICRP Publication 66 (ICRP 1994), the HRTM system is divided into four anatomical regions: the extrathoracic region (ET), comprising of the anterior nose (ET1) and the posterior nasal passages, larynx, pharynx, and mouth (ET2); the bronchial region (BB), consisting of the trachea and bronchi from which deposited material is cleared by ciliary action; the bronchiolar region (bb) consisting of the bronchioles and terminal bronchioles; and lastly the alveolar-interstitial region (AI), consisting of the bronchioles with some alveoli apposed, the alveolar ducts and sacs with their alveoli, and the interstitial connective tissue, with each of these four regions containing lymphatic tissue (LT) or its components. These collectively define the ICRP Publication 66 HRTM. An updated version of the HRTM was reported in the ICRP Publication 130, where the oral passage is no longer included in region ET2 for consistency with the HATM (ICRP 2015). ICRP Publication 130 can be consulted for detailed information regarding the revised HRTM (ICRP 2015).

The ICRP Publication 130 HRTM is coupled to the radionuclide-specific ⁶⁰Co (ICRP 2016b) and ¹³¹I (ICRP 2017) systemic models through absorption into blood (ICRP 2015) and the HATM (ICRP 2006), where the complete coupled compartmental model is illustrated in figures 1–3, respectively. In contrast to the systemic models, the HRTM adds complexity to the coupled system of the HRTM and the systemic models through competing clearance modes (mechanical clearance and clearance due to particle dissolution and uptake).

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Assuming the clearance classes of Type M for ⁶⁰Co and Type F for ¹³¹I, respectively, with the particle size of 5 µm, which is the baseline assumption for occupational exposure, Python explicit and implicit ODE solvers and a Python-based matrix exponentiation solving method were used to solve for the coupled system for a unit intake of activity in the human respiratory tract over a 50 year time period. The unit intake of the activity is partitioned into various regions of the human respiratory tract based on the ICRP Publication 130 HRTM for a normal nose breather. These apportionments were read as non-zero initial conditions (fractional deposition) by the ODE-solving methods. The solution of the set of ODEs was evaluated as a fraction of the activity of the radionuclide in each compartment. Discussion of the stability performance for each method was conducted to test the solvers via an adult occupational exposure selected as the reference model (ICRP 2015).

2.4. Stiffness and stability

For a coupled linear system of equations with constant coefficients matrix, stiffness can be determined by the relative magnitude of the eigenvalues of the coefficients matrix (Anderson 1983). The relative magnitude is also referred to as the stiffness ratio. The larger the stiffness ratio, the higher the stiffness level and the more restrictive the stability criteria. A large stiffness ratio is evident if eigenvalues of the coefficient matrix are negative or have negative real parts and are widely sparse, posing some difficulties when evaluating the system of ODEs (Anderson 1983, Byrne and Hindmarsh 1987). For each of the coefficient matrices, equation (9) is used to measure the model's stiffness.

$$\begin{equation}{\text{Stiffness Ratio }}\left( {{\text{SR}}} \right) = \,\frac{{\left| {{\lambda _L}} \right|}}{{\left| {{\lambda _S}} \right|}}\, \unicode{x2A7E} \,\frac{{{{\text{max}}_i}\left| {{A_{ii}}} \right|}}{{{{\text{min}}_i}\left| {{A_{ii}}} \right|}}\end{equation} \tag{ 9 }$$

where $\left| {{\lambda _L}} \right|$ is the magnitude of the maximum eigenvalue;

$\left| {{\lambda _S}} \right|$ is the magnitude of the minimum eigenvalue;

${\text{ma}}{{\text{x}}_i}\left| {{A_{ii}}} \right|$ is the maximum magnitude value of the diagonal or the loss term of the coefficient matrix; and

${\text{mi}}{{\text{n}}_i}\left| {{A_{ii}}} \right|$ is the minimum magnitude value of the diagonal or the loss term of the coefficient matrix.

Although the stiffness ratio has been a commonly referenced approach for measuring stiffness, it breaks down in the case of real positive eigenvalues, specifically zero eigenvalues resulting in undefine ratio. To obtain useful information on the measure of stiffness for both Type M ⁶⁰Co and Type F ¹³¹I, zero eigenvalues were discarded, and the stiffness ratio was evaluated with the real negative eigenvalues. The stiffness ratio for ⁶⁰Co and ¹³¹I were evaluated as $1.03 \times {10^{17}}$ and $6.024 \times {10^{22}}$, respectively. A system with mixed types of eigenvalues (positive and negative) is essentially an unstable system.

Stability, conversely, is a property that ensures that the error associated with the numerical approximation does not grow and thus ensures that the ODE solution does not diverge over time (Yano et al 2012). For a detailed stability analysis of the initial value problem posed by the biokinetic models beyond the measure of stiffness, a numerical experiment was performed, and a growth factor known as the numerical amplification factor (G) was employed.

$$\begin{equation}{\text{G}} = \,\left| {\frac{{{q_{n + 1}}}}{{{q_n}}}} \right|\, \unicode{x2A7D} \,1\end{equation} \tag{ 10 }$$

where ${q_{n + 1}}$ is the numerical solution for the current time step, and ${q_n}$ is the numerical solution for the previous time step. The solution is classified as absolutely stable if, for all time steps, G is less than or equal to 1 over time (Yano et al 2012, Patera and Yano 2014).

2.5. Computational platform

For an eventual coupling with a multi-level parallel object-oriented framework for advanced parametric analysis, the HRTM biokinetic model was reconstructed and solved in Python for a unit activity of inhaled ⁶⁰Co of Type M and ¹³¹I of Type F. In this study, initial value problem (IVP) solvers and solving methods were leveraged in Python to numerically and algebraically solve the system of ODEs posed by the biokinetic models and to explore the tolerance levels of the solving methods, hence the choice of moderate and fast clearing radionuclides.

Figures 4–9 give the total whole-body retention solutions of the system of coupled ordinary differential equations solved with the following Python solvers: RK45, RK23, DOP853, Radau, BDF, and LSODA for a unit activity of inhaled ⁶⁰Co of Type M, respectively. As shown in figures 4–6, oscillating solutions were observed for RK45, RK23, and DOP853. For Radau, BDF, and LSODA, as shown in figures 7–9, the retention curves for the whole body did not yield oscillatory or divergent solutions.

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The RK45, RK23, DOP853, Radau, BDF, and LSODA solvers were also subjected to a system of ODEs posed by inhaling ¹³¹I of Type F and the solutions for whole body retention are reported in figures 10–15. These results demonstrated that all experimented solvers (both implicit and explicit) for the total body activity retained resulted in significant oscillations for the ¹³¹I scenario. As a result, the optional capabilities for the implicit solvers were further probed by customizing the optional function parameters since the above computations for ⁶⁰Co and ¹³¹I using these implicit methods implemented the default approach. The choice for specifically adapting the parameters for the implicit methods (Radau and BDF) was based on the recommendation by Python SciPy developers proposing the utility of implicit methods for instances where there are unusually many iterations (divergence) of the solution (Virtanen et al 2020). Moreover, explicit solvers utilized several minutes performing many integrations to approximate the solutions, thereby negating further exploration. Customization in this study utilized the optional capability of the implicit solvers to regulate the local error in computing the solution, as well as optimizing the computational time, as detailed in the methodology. Comparable outcomes for these solvers were observed for both the human respiratory tract and whole-body retention, depicted in the graphical data for whole-body retention as a representation of the global stability effect.

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Retention results are given in figures 16–21, from which solutions were obtained utilizing the local error monitoring capabilities and controlling the step size for the numerical integration of Radau and BDF for ⁶⁰Co of Type M and ¹³¹I of Type F, respectively.

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Tables 2 and 3 report the calculation time for each of the solvers used and indicate solvers that resulted in oscillatory or divergent solutions in specific compartments of the biokinetic model problem. It is important to note that despite the apparent oscillations observed in some of the solutions for the solvers for the ¹³¹I scenario, oscillations in ⁶⁰Co were negligible. However, these non-oscillatory solutions do not necessarily indicate stability. Tables 4–7 provide the summary results from the stability analysis of the solvers quantitatively for ⁶⁰Co of Type M and ¹³¹I of Type F, respectively, showing the maximum G, as defined in equation (10). Although the maximum G values of Radau-Customized and BDF-Customized for ⁶⁰Co, as well as of Radau-Customized2 and BDF-Customized2 for ¹³¹I, were slightly greater than 1, a trend was observed over time which further examined the stability metric in this problem framework. Figures 22 and 23 provide an illustration of this trend for Kidneys (⁶⁰Co) and Thyroid (¹³¹I).

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Stiff problems mostly favor solution methods based on eigenvalue methodologies and, thus, algebraic methods. This study made a conscious effort to compute the biokinetic solutions for the radionuclides using a linear algebraic method (expm) in Python, which employs Padé approximation for which figures 24 and 25 represent the retention of whole-body radioactivity burden for ⁶⁰Co Type M and ¹³¹I Type F, respectively. For verification purposes, the retention resulting from this method was then compared with the solutions of the customized Python IVP implicit solvers by finding the percent relative difference between expm and Radau, ((Radau-expm/expm) $\times$ 100) and expm and BDF ((BDF-expm/expm) $\times$ 100) for ⁶⁰Co of Type M and ¹³¹I of Type F. The results for the relative difference are shown in figures 26 and 27.

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As indicated in the methodology, the inhalation scenarios for ⁶⁰Co of Type M and for ¹³¹I of Type F were derived from the ICRP OIR series of publications (ICRP 2015, 2016b, 2017). As a result, the biokinetic retention solutions from OIR Electronic Annex Distribution Set were compared with the solutions obtained with the solvers/solving methods. The percent relative difference (RD %) for total body retention are reported in tables 8 and 9.

Several ODE solvers and solving methods in Python were employed for solving the biokinetic models from inhaled ⁶⁰Co of Type M and ¹³¹I Type F. For the total body retention of inhaled ⁶⁰Co, the examined solvers (figures 4–9) resulted in an unusual iteration except for Radau, BDF, and LSODA, which were not visually observable in the figures. However, careful analysis of the compartmental activity, through a compartment-by-compartment analysis, demonstrated oscillatory solutions in some of the compartments for the solvers except the customized (local error control) Radau solver (table 2), where customized BDF implemented with the same customization of error tolerance as Radau. For ¹³¹I, solutions with both explicit and implicit solvers in their general form resulted in unstable and oscillatory solutions, as shown in figures 10–15. The relative and absolute tolerances (rtol and atol) were reduced to enable local error control for accuracy, and the activity retained in each compartment was computed and plotted for the whole body for ⁶⁰Co and ¹³¹I in figures 16–21, respectively. However, as shown in the plots of figures 18 and 19, there were significant oscillations in the solutions for both Radau and BDF. For ¹³¹I, the relative and absolute tolerance chosen saturated at 10⁻¹⁵ and 10⁻¹², respectively, without any significant effect due to the highly stiff nature of the ¹³¹I coupled HRTM and systemic biokinetics. The resolution implemented entailed restricting the maximum time points to have an upper bound on the error. Figures 20 and 21 depict non-oscillatory (smooth) solutions. The customization confirms the importance of the optional function parameters of the Python solvers where needed and the need to pay particular attention to their usage.

Additionally, as computational time is essential to the development of modeling tools and, thus, internal dosimetry tools, the calculation time of the solvers were tracked, which is reported in tables 2 and 3 for ⁶⁰Co and ¹³¹I, respectively, indicating that the implicit solvers and a versatile choice (LSODA) were computationally efficient for the biokinetic model problem. These calculation times were system-specific and should not be considered absolute. However, the magnitude of solving speed capabilities for each solver was maintained, regardless of the computer-based system in use. In contrast, explicit solvers, such as RK45, RK23, and DOP853, took approximately 17–35 min to solve the biokinetic model problems, for which the longest time spent achieving a solution was for the case of the coupled ¹³¹I HRTM and systemic biokinetic model. Thus, to achieve numerically optimal solutions with limited accuracy for stiff problems, explicit solvers performed many integrations with very small step sizes, which can sometimes be considered unacceptable. Also, color-coded cells under 'specific compartment divergence' in tables 2 and 3 summarize the solvers that resulted in some level of instability (or oscillatory solutions) and those that were stable in solving the inhalation modeling problems.

4.1. Stability evaluation

4.2. Comparison with the ICRP OIR electronic annex distribution set

The capabilities of solving a complex system of coupled ODEs for biokinetic modeling, using solvers in Python, were explored, where implicit methods demonstrated practical applicability for both moderate and fast clearing radionuclides through a stability analysis. Characteristic tolerance levels were explored for the constituent in Python by being subjected to rapidly and slowly varying coupled ODE systems. The outcome of the implicit solvers (Radau and BDF) with the stable solutions was then compared to the solution obtained with a straightforward (albeit complex) algebraic solving method (expm), where both yielded relatively comparable outcomes. The customized implicit solvers (Radau and BDF) and expm also yielded comparable results when compared with the ICRP OIR Electronic Annex Distribution Set for the inhalation scenarios investigated. Consequently, the Python-based implicit ODE solvers and the Python-based algebraic solving methods yielded optimized performance for solving the metabolic problem regarding calculation speed and stability. This study establishes the foundational knowledge to further develop a robust multi-level object-oriented based framework, leveraging the dynamism in Python for improved dose reconstruction in radiation protection and nuclear medicine applications.

Funding for this project is provided by: Department of Defense—Peer Reviewed Medical Research Program (DOD PRMRP) under Award W81XWH-21-1-0984; National Institutes of Health/National Institute of Allergy and Infectious Diseases (NIH/NIAID) Radiation and Nuclear Countermeasures Program under Award 1P01AI165380-01; and Department of Energy/National Nuclear Security Administration (DOE-NNSA) Nuclear Emergency Support Team (NEST) via Sandia National Laboratories under Award DE-NA0003525 SNL21-CM-420.

The authors of this work express their gratitude to Dr Luiz Bertelli for his technical assistance

All data that support the findings of this study are included within the article (and any supplementary information files).