The mathematical models
- NL models for bacterial growth
- ODEs for bacterial growth
- Stochastic models for bacterial growth
- Error functions
NL models for bacterial growth
In this case, we are supposed to know the analytic formula of microbial growth; in particular, we have implemented some models from "Statistical evaluation of mathematical models for microbial growth" and added some piecewise models. They are:
Exponential
\[N(t) = N_0 \cdot e^{\mu \cdot t}\]
where $\mu$ is the growth rate, and $N_0$ is the starting condition.
Gompertz
\[N(t) = N_{\text{max}} \cdot e^{-e^{-\mu \cdot (t - t_{\text{L}})}}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ is the total growth, and $t_{\text{L}}$ is the lag time.
Logistic
\[N(t) = \frac{N_{\text{max}}}{1 + \left( \frac{N_{\text{max}}}{N_0} - 1 \right) \exp\left( - \mu \cdot t \right)}\]
where $\mu$ is the growth rate, $N_0$ is the starting condition, and $N_{\text{max}}$ is the total growth.
Richards model
\[N(t) = \frac{N_{\text{max}}}{[1 + \nu \cdot e^{-\mu \cdot (t - t^{\text{L}})}]^{\frac{1}{\nu}}}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ is the total growth, $t_{\text{L}}$ is the lag time, and $\nu$ is a shape constant.
Weibull
\[N(t) = N_{\text{max}} - (N_{\text{max}} - N_0) \cdot e^{-(\mu \cdot t)^{\nu}}\]
where $\mu$ is the growth rate, $N_0$ is the starting condition, $N_{\text{max}}$ is the total growth, and $\nu$ is a shape constant.
Morgan
\[N(t) = \frac{N_0 \cdot K^{\nu} + N_{\text{max}} \cdot t^{\nu}}{K^{\nu} + t^{\nu}}\]
where $N_0$ is the starting condition, $N_{\text{max}}$ is the total growth, and $K$ and $\nu$ are shape constants.
Bertalanffy
\[N(t) = N_0 + (N_{\text{max}} - N_0) \cdot (1 - e^{-\mu \cdot t})^{\frac{1}{\nu}}\]
where $N_0$ is the starting condition, $N_{\text{max}}$ is the total growth, $\mu$ is the growth rate, and $\nu$ is a shape constant.
Piece-Wise Linear-Logistic
\[N(t) = \begin{cases} N_0, & t < t_{\text{L}} \\ N(t) = \frac{N_{\text{max}}}{1 + \left( \frac{N_{\text{max}}}{N_0} - 1 \right) \exp\left( - \mu \cdot (t - t_{\text{L}}) \right)}, & t_{\text{L}} \leq t \end{cases}\]
where $N_0$ is the starting condition, $N_{\text{max}}$ is the total growth, $\mu$ is the growth rate, and $t_{\text{L}}$ is the lag time.
Piece-wise Exponential-Logistic
\[N(t) = \begin{cases} N_0 \exp{(\mu_0 \cdot t)}, & t < t_{\text{L}} \\ \frac{N_{\text{max}}}{1 + \left( \frac{N_{\text{max}}}{N_0 \exp{(\mu_0 \cdot t_{\text{L}})}} - 1 \right) \exp\left( - \mu \cdot (t - t_{\text{L}}) \right)}, & t_{\text{L}} \leq t \end{cases}\]
where $N_0$ is the starting condition, $N_{\text{max}}$ is the total growth, $\mu$ is the growth rate, $t_{\text{L}}$ is the lag time, and $\mu_0$ is the growth rate during the lag phase.
To call these models use the string present in this table, the parameters will be returned in the same order of this table.
| Model Name | Parameters List | String to call |
|---|---|---|
| Exponential | $N_0, \mu$ | "NL_exponential" |
| Gompertz | $N_{\text{max}}, \mu, t_{\text{L}}$ | "NL_Gompertz" |
| Logistic | $N_{\text{max}}, N_0,\mu$ | "NL_logistic" |
| Richards model | $N_{\text{max}}, \mu,\nu,t_{\text{L}}$ | "NL_Richards" |
| Weibull | $N_{\text{max}}, N_0,\mu,\nu$ | "NL_Weibull" |
| Morgan | $N_{\text{max}}, N_0,K,\nu$ | "NL_Morgan" |
| Bertalanffy | $N_{\text{max}}, N_0,\mu,\nu$ | "NL_Bertalanffy" |
| piece-wise linear-logistic | $N_0, N_{\text{max}},\mu, t_\text{L}$ | "NL_piecewise_lin_logistic" |
| piece-wise exponential-logistic | $N_0, N_{\text{max}},\mu, t_\text{L}, t_\text{L},\mu_0$ | "NL_piecewise_exp_logistic" |
ODEs for bacterial growth
The models implemented in Kinbiont are the following:
- Exponential:
\[\frac{d N(t)}{dt} =\mu N(t)\]
where $\mu$ is the growth rate.
- Hyper Gompertz:
\[\frac{d N(t)}{dt} = \mu \, \log \left( \frac{N_{\text{max}}}{N(t)} \right)^{(1-n)}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth and $n$ a shape constant.
- Hyper Logistic:
\[\frac{d N(t)}{dt} = \frac{\mu}{N_{\text{max}}} \, N(t)^{(1-n)} (N(t) - N_{\text{max}})^{(1+n)}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth and $n$ a shape constant.
- Bertalanffy-Richards:
\[\frac{d N(t)}{dt} = \frac{\mu}{N_\text{max}^n } \cdot \left ( N_\text{max}^n - N^n(t) \right)\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, and $n$ a shape constant.
- Logistic:
\[\frac{d N(t)}{dt} = \mu \left( 1 - \frac{N(t)}{N_{\text{max}}} \right) \, N(t)\]
where $\mu$ is the growth rate, and $N_{\text{max}}$ the total growth.
- Adjusted Logistic:
\[\frac{d N(t)}{dt} = \mu \left( 1 - \left(\frac{N(t)}{N_{\text{max}}}\right) ^n \right) \, N(t)\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth and $n$ a shape constant.
- Gompertz:
\[\frac{d N(t)}{dt} = \mu \, N(t) \, \log \left( \frac{N_{\text{max}}}{N(t)} \right)\]
where $\mu$ is the growth rate, and $N_{\text{max}}$ the total growth.
- Baranyi Richards:
\[\frac{d N(t)}{dt} = \frac{t^n}{t^n + \lambda^n} \, \mu \left( 1 - \frac{N(t)}{N_{\text{max}}} \right) \, N(t)\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $\lambda$ is the lag time and $n$ a shape constant.
- Baranyi Roberts:
\[\frac{d N(t)}{dt} = \frac{t^n}{t^n + \lambda^n} \, \mu \left( 1 - \left( \frac{N(t)}{N_{\text{max}}} \right)^m \right) \, N(t)\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $\lambda$ is the lag time, $n$ and $m$ are shape constants.
- Piece-wise Adjusted Logistic:
\[\frac{d N(t)}{dt} = \begin{cases} \text{const.} \, N(t) & t < t_{\text{L}} \\ \mu \left( 1 - \left( \frac{N(t)}{N_{\text{max}}} \right)^m \right) \, N(t) & t \geq t_{\text{L}} \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $t_\text{L}$ is the lag time, $m$ is shape constant, and $c$ the growth rate during the lag phase (can be 0).
- Triple Piece-wise Adjusted Logistic:
\[\frac{d N(t)}{dt} = \begin{cases} c_1 \cdot N(t) & \text{for } t < t_{\text{L}}, \\ \mu \left( 1 - \left( \frac{N(t)}{N_{\text{max}}} \right)^m \right) \cdot N(t) & \text{for } t_{\text{L}} \leq t < t_{\text{stat}}, \\ c_2 \cdot N(t) & \text{for } t \geq t_{\text{stat}}, \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $t_\text{L}$ is the lag time, $m$ is a shape constant, $c_1$ the growth rate during the lag phase (can be 0), $t_{\text{stat}} $ the time when stationary phase starts, and $c_2$ the growth rate during the stationary phase.
- Triple Piece-wise:
\[\frac{d N(t)}{dt} = \begin{cases} c_1 \cdot N(t) & \text{for } t < t_{\text{L}}, \\ \mu \cdot N(t) & \text{for } t_{\text{L}} \leq t < t_{\text{stat}},\\ c_2 \cdot \left(1 - \log \left( \frac{N(t)}{N_{\text{max}}} \right)\right) & \text{for } t \geq t_{\text{stat}}, \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $t_\text{L}$ is the lag time, $c_1$ the growth rate during the lag phase (can be 0), $t_{\text{stat}} $ the time when stationary phase starts, and $c_2$ the growth rate during the stationary phase.
- Triple Piece-wise Exponential:
\[\frac{d N(t)}{dt} = \begin{cases} c_1 \cdot N(t) & \text{for } t < t_{\text{L}}, \\ \mu \cdot N(t) & \text{for } t_{\text{L}} \leq t < t_{\text{stat}}, \\ c_2 \cdot N(t) & \text{for } t \geq t_{\text{stat}}, \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $t_\text{L}$ is the lag time, $c_1$ the growth rate during the lag phase (can be 0), $t_{\text{stat}} $ the time when stationary phase starts, and $c_2$ the growth rate during the stationary phase.
- Four Piece-wise Exponential:
\[\frac{d N(t)}{dt} = \begin{cases} c_1 \cdot N(t) & \text{for } t < t_1, \\ \mu \cdot N(t) & \text{for } t_1 \leq t < t_2, \\ c_2 \cdot N(t) & \text{for } t_2 \leq t < t_3,\\ c_3 \cdot N(t) & \text{for } t \geq t_3, \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $t_1$ is the lag time, $c_1$ the growth rate during the lag phase (can be 0), $t_2 $ the time when a growth after exponential growths, $c_2$ the growth rate during this phase, $t_3$ the start of stationary phase and, $c_3$ the growth rate during the stationary phase.
- Heterogeneous Population Model (HPM):
\[\begin{cases} N(t) = N_1(t) + N_2(t), \\ \frac{d N_1(t)}{dt} = - r_{\text{L}} \cdot N_1(t), \\ \frac{d N_2(t)}{dt} = r_{\text{L}} \cdot N_1(t) + \mu \cdot N_2(t) \cdot \left(1 - \frac{N_1(t) + N_2(t)}{N_{\text{max}}}\right), \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, and $r_\text{L}$ is the lag rate (i.e. the rate of transition between $N_1(t)$ and $N_2(t)$). Note that these models assume that the cells are in two states: $N_1(t)$ dormant cells (the cells are not able to reproduce because they are in the lag phase) and $N_2(t)$ active cells, which are able to duplicate. At the start, all the cells are assumed in the dormant state (i.e., $N_{1}(start) = OD(start)$, and $N_{2}(start) = 0.0$) .
- Exponential Heterogeneous Population Model:
\[\begin{cases} N(t) = N_1(t) + N_2(t) \\ \frac{d N_1(t)}{dt} = - \text{r}_{\text{L}} \, N_1(t) \\ \frac{d N_2(t)}{dt} = \text{r}_{\text{L}} \, N_1(t) + \mu \, N_2(t) \end{cases}\]
where similarly to the HPM model, $N_1$ and $N_2$ refer to the populations of dormant and active cells, respectively. $\mu$ is the growth rate, and the lag rate $r_\text{L}$ denotes the transition between the $N_1$ and $N_2$ populations. Here, we also assume that all cells are in the dormant state at the start (i.e., $N_{1}(t = 0) = \text{OD}(t = 0)$, and $N_{2}(t = 0) = 0$).
- Adjusted Heterogeneous Population Model:
\[\begin{cases} N(t) = N_1(t) + N_2(t), \\ \frac{d N_1(t)}{dt} = - r_{\text{L}} \cdot N_1(t), \\ \frac{d N_2(t)}{dt} = r_{\text{L}} \cdot N_1(t) + \mu \cdot N_2(t) , \end{cases}\]
where $\mu$ is the growth rate, and $N_{\text{max}}$ the total growth. Note that these models assume that the cells are in two states: $N_1(t)$ dormant cells (the cells are not able to reproduce because they are in the lag phase) and $N_2(t)$ active cells, which are able to duplicate.At the start, all the cells are assumed in the dormant state (i.e., $N_{1}(start) = OD(start)$, and $N_{2}(start) = 0.0$) .
- Heterogeneous Population Model with Inhibition:
\[\begin{cases} N(t) = N_1(t) + N_2(t) \\ \frac{d N_1(t)}{dt} = - r_{\text{L}} \cdot N_1(t) \\ \frac{d N_2(t)}{dt} = r_{\text{L}} \cdot N_1(t) + \mu \cdot N_2(t) \cdot \left(1 - \left(\frac{N_1(t) + N_2(t)}{N_{\text{max}}}\right)^m\right) \end{cases}\]
where $\mu$ is the growth rate, $N_{\text{max}}$ the total growth, $r_\text{L}$ is the lag rate (i.e. the rate of transition between $N_1(t)$ and $N_2(t)$) and $m$ a shape constant. Note that these models assume that the cells are in two states: $N_1(t)$ dormant cells (the cells are not able to reproduce because they are in the lag phase) and $N_2(t)$ active cells, which are able to duplicate.At the start, all the cells are assumed in the dormant state (i.e., $N_{1}(start) = OD(start)$, and $N_{2}(start) = 0.0$) .
- Heterogeneous Population Model with Inhibition and Death:
\[\begin{cases} N(t) = N_1(t) + N_2(t) + N_3(t), \\ \frac{d N_1(t)}{dt} = - r_{\text{L}} \cdot N_1(t), \\ \frac{d N_2(t)}{dt} = r_{\text{L}} \cdot N_1(t) + \mu \cdot N_2(t) - r_{\text{I}} \cdot N_2(t), \\ \frac{d N_3(t)}{dt} = - r_{\text{D}} \cdot N_3(t) + r_{\text{I}} \cdot N_2(t), \end{cases}\]
where $\mu$ is the growth rate, $r_\text{L}$ is the lag rate (i.e. the rate of transition between $N_1(t)$ and $N_2(t)$) , $r_\text{inhibition}$ is the rate of which cell are inhibited (i.e. the rate of transition between $N_2(t)$ and $N_3(t)$), and $r_{\text{D}}$ is the rate of which cell are die.
Note that these models assume that the cells are in three states: $N_1(t)$ dormant cells (the cells are not able to reproduce because they are in the lag phase), $N_2(t)$ active cells, which are able to duplicate, and inactive cells $N_3(t)$ that die at a rate $r_{\text{D}}$. At the start, all the cells are assumed in the dormant state (i.e., $N_{1}(\text{start}) = OD(\text{start})$, $N_{2}(\text{start}) = 0.0$, and $N_{3}(\text{start}) = 0.0$).
- Heterogeneous Population Model with Inhibition, Death and Resistance:
\[\begin{cases} N(t) = N_1(t) + N_2(t) + N_3(t), \\ \frac{d N_1(t)}{dt} = - r_{\text{L}} \cdot N_1(t), \\ \frac{d N_2(t)}{dt} = r_{\text{L}} \cdot N_1(t) + \mu \cdot N_2(t) - r_{\text{I}} \cdot N_2(t), \\ \frac{d N_3(t)}{dt} = - r_{\text{D}} \cdot N_3(t) \left(1 - \frac{N_3(t)}{N_{\text{res}}}\right) + r_{\text{I}} \cdot N_2(t), \quad \text{with} \quad N_3(t) \leq N_{\text{res}} \end{cases}\]
where $\mu$ is the growth rate, $r_\text{L}$ is the lag rate (i.e. the rate of transition between $N_1(t)$ and $N_2(t)$) , $r_\text{inhibition}$ is the rate of which cell are inhibited (i.e. the rate of transition between $N_2(t)$ and $N_3(t)$), $r_{\text{D}}$ is the rate of which cell are die, and $N_{\text{res}}$ it the number of cell that will be inactive but do not die.
To call these models use the string present in this table, the parameters will be returned in the same order of this table.
| Model Name | Parameters | String to call |
|---|---|---|
| Exponential ODE | label_exp, well, model, gr, th_max_gr, emp_max_gr, loss | "exponential" |
| Hyper Gompertz | label_exp, well, model, gr, N_max, shape, th_max_gr, emp_max_gr, loss | "hyper_gompertz" |
| Hyper Logistic | label_exp, well, model, doubling_time, gr, N_max, shape, th_max_gr, emp_max_gr, loss | "hyper_logistic" |
| Von Bertalanffy ODE | label_exp, well, model, alpha, beta, a, b, th_max_gr, emp_max_gr, loss | "ode_von_bertalanffy" |
| Bertalanffy-Richards | label_exp, well, model, gr, N_max, shape, th_max_gr, emp_max_gr, loss | "bertalanffy_richards" |
| Logistic | label_exp, well, model, gr, N_max, th_max_gr, emp_max_gr, loss | "logistic" |
| Adjusted Logistic | label_exp, well, model, gr, N_max, shape, th_max_gr, emp_max_gr, loss | "alogistic" |
| Gompertz | label_exp, well, model, gr, N_max, th_max_gr, emp_max_gr, loss | "gompertz" |
| Baranyi Richards | label_exp, well, model, gr, N_max, lag_time, shape, th_max_gr, emp_max_gr, loss | "baranyi_richards" |
| Baranyi Roberts | label_exp, well, model, gr, N_max, lag_time, shape_1, shape_2, th_max_gr, emp_max_gr, loss | "baranyi_roberts" |
| Piece-wise Adjusted Logistic | label_exp, well, model, gr, N_max, lag, shape, linear_const, th_max_gr, emp_max_gr, loss | "piecewise_adjusted_logistic" |
| Triple Piece-wise Adjusted Logistic | label_exp, well, model, gr, N_max, lag, shape, linear_const, t_stationary, linear_lag, th_max_gr, emp_max_gr, loss | "triple_piecewise_adjusted_logistic" |
| Triple Piece-wise | label_exp, well, model, gr, gr_2, gr_3, lag, t_stationary, th_max_gr, emp_max_gr, loss | "ODE_triple_piecewise" |
| Triple Piece-wise Exponential | label_exp, well, model, gr, gr_2, gr_3, lag, t_stationary, th_max_gr, emp_max_gr, loss | "ODE_triple_piecewise_exponential" |
| Four Piece-wise Exponential | label_exp, well, model, gr, gr_2, gr_3, gr_4, lag, t_decay_gr, t_stationary, th_max_gr, emp_max_gr, loss | "ODE_four_piecewise" |
| Diauxic Piecewise Adjusted Logistic | label_exp, well, model, gr_1, N_max, shape_1, lag, linear_const, t_shift, gr_2, N_max_2, shape_2, end_second_lag, lag_2_gr, th_max_gr, emp_max_gr, loss | "Diauxic_piecewise_adjusted_logistic" |
| Heterogeneous Population Model (HPM McKellar) | label_exp, well, model, gr, exit_lag_rate, N_max, th_max_gr, emp_max_gr, loss | "HPM" |
| Exponential Heterogeneous Population Model (HPM McKellar) | label_exp, well, model, gr, exit_lag_rate, th_max_gr, emp_max_gr, loss | "HPM_exp" |
| Adjusted Heterogeneous Population Model | label_exp, well, model, gr, exit_lag_rate, N_max, shape, th_max_gr, emp_max_gr, loss | "aHPM" |
| Heterogeneous Population Model with Inhibition | label_exp, well, model, gr, exit_lag_rate, inactivation_rate, th_max_gr, emp_max_gr, loss | "HPM_3_inhibition" |
| Heterogeneous Population Model with Inhibition and Death | label_exp, well, model, gr, exit_lag_rate, inactivation_rate, death_rate, th_max_gr, emp_max_gr, loss | "HPM_3_death" |
| Heterogeneous Population Model with Inhibition, Death and Resistance | label_exp, well, model, gr, exit_lag_rate, inactivation_rate, death_rate, n_res, shape, th_max_gr, emp_max_gr, loss | "aHPM_3_death_resistance" |
Stochastic models for bacterial growth
In the stochastic version of the growth models, the growth rate of each population component (denoted as $\mu_i$) is evaluated based on the concentration of the limiting nutrient and then the number of birth event is evaluated with the Poisson approximation. The user is required to specify the starting amount of nutrients and the volume of the solution. Various kinetic growth models are considered. Note that these models can be used only during simulations. In the following, we use $\nu$ to represent the limiting nutrient concentration throughout, $\mu_\text{max}$ denotes the maximum possible growth rate, $k_1$ (for $i=1,2$) is a numerical constant whose specific meaning depends on the model, $N$ indicates the number of present cells, and $N_\text{max}$ is the carrying capacity in the Verhulst model.
- Monod Model:
\[\mu(\nu;k_1,\mu_\text{max}) = \displaystyle{\mu_\text{max} \frac{\nu}{k_1+\nu}}.\]
- Haldane Model:
\[\mu(\nu;k_1,k_2,\mu_\text{max}) = \displaystyle{\mu_\text{max} \frac{\nu}{k_1+\nu +\frac{\nu^2}{k_2}}}.\]
- Blackman Model:
\[\mu(\nu;k_1,\mu_\text{max}) = \displaystyle{\mu_\text{max} \frac{\nu}{k_1}}.\]
- Tessier Model:
\[\mu(\nu;k_1,\mu_\text{max}) = \displaystyle{\mu_\text{max} (1 - e^{k_1\nu })}.\]
- Moser Model:
\[\mu(\nu;k_1,k_2,\mu_\text{max}) = \displaystyle{\mu_\text{max} \frac{\nu^{k_2}}{k_1+\nu^{k_2}}}.\]
- Aiba-Edwards Model:
\[\mu([\text{Nut.}]; k_1, k_2, \mu_\text{max}) = \mu_\text{max} \frac{[\text{Nut.}]}{k_1 + [\text{Nut.}]} e^{-\frac{[\text{Nut.}]}{k_2}}.\]
- Verhulst Model:
\[\mu(N;N_\text{max},\mu_\text{max}) = \displaystyle{\mu_\text{max} \left(1-\frac{N}{N_\text{max}}\right)}.\]
Error functions
The user can choose to use different error functions to perfor the fitting. Each fitting API has its keyword argument to change the loss. The possible options are described in the following section.
In the equations of this list, the notation is the following: $n$ the number of time points $t_i$, $\hat{N}(t_i, \{P\})$ is the proposed numerical solution at time $t_i$ and using the parameters $\{P\}$, and $N(t_i)$ is the data value at $t_i$.
"L2": Minimize the L2 norm of the difference between the numerical solution of the desired model and the given data.
\[\mathcal{L}(\{P\}) = \frac{1}{n} \sum_{i=1}^n \left(N(t_i) - \hat{N}(t_i, \{P\})\right)^2\]
"RE": Minimize the relative error between the solution and data.
\[\mathcal{L}(\{P\}) = \frac{1}{n} \sum_{i=1}^n 0.5 \, \left(1 - \frac{D(t_i)}{\bar{N}(t_i, \{P\})}\right)^2\]
"L2_derivative": Minimize the L2 norm of the difference between the specific growth rate of the numerical solution of the desired model and the corresponding derivatives of the data.
\[\mathcal{L}(\{P\}) = \frac{1}{n} \sum_{i=1}^n \left(\frac{dD(t_i)}{dt} - \frac{d\bar{N}(t_i, \{P\})}{dt}\right)^2\]
"blank_weighted_L2": Minimize a weighted version of the L2 norm, where the difference between the solution and data is weighted based on a distribution obtained from empirical blank data.
\[\mathcal{L}(\{P\}) = \frac{1}{n} \sum_{i=1}^n \left(1 - P(N(t_i) - \hat{N}(t_i, \{P\})|\text{noise})\right) \, \left(N(t_i) - \hat{N}(t_i, \{P\})\right)^2\]
where $P(N(t_i) - \hat{N}(t_i, \{P\})|\text{noise})$ is the probability distribution of the empirical blank data. "L2_log": Minimize the logarithm of the L2 norm of the difference between the numerical solution of the desired model and the given data.
\[\mathcal{L}(\{P\}) = \log\left(\frac{1}{n} \sum_{i=1}^n \left(N(t_i) - \hat{N}(t_i, \{P\})\right)^2\right)\]
"RE_log": Minimize the logarithm of the relative error between the solution and data.
\[\mathcal{L}(\{P\})= \log\left(\frac{1}{n} \sum_{i=1}^n 0.5 \, \left(1 - \frac{D(t_i)}{\bar{N}(t_i, \{P\})}\right)^2\right)\]
"L2_std_blank": Minimize the L2 norm of the difference between the numerical solution of the desired model and the data, normalized by the standard deviation of empirical blank data.
\[\mathcal{L}(\{P\}) = \frac{1}{n} \sum_{i=1}^n \left(\frac{N(t_i) - \hat{N}(t_i, \{P\})}{\text{std\_blank}}\right)^2\]
where $\text{std\_blank}$ is the standard deviation of the empirical blank data.