1. INTRODUCTION:
1.1. IGF-1 hormone secretion, kinetics, and its role in the body:
IGF-1 is an essential peptide hormone also known as somatomedin secreted from the liver and regulates the proliferation, remodeling and repair of muscular, tissue, and skeletal system development1,2. The secretion of IGF-1 in the liver is regulated by growth hormone from the pituitary gland3, but the cause of high IGF-1 levels arises mainly due to liver metastasis, and its risk level on bone cells is not defined yet.
The proposed model-based study of IGF-1 by Lee et al. explained the prime curse causing abnormal bone cell proliferation is due to secretion of IGF-1 hormone and its excessive deposition over the skeletal system4. Many studies proposed that abnormal IGF-1 level causes a risk of breast, prostate, lung, and rectal cancer5. It is also known that high stimulation of the pituitary gland releases a risk amount of growth factors which further causes increase secretion of IGF-1 hormone and ultimately causes liver metastasis4. However, the pathological cause of the high concentration of IGF-1 in bone cancer due to liver metastasis is still unclear. Thus, the high IGF-1 concentration in the liver and skeletal system helps to design the hypothesis to understand the cause of bone cancer due to the increased concentration of IGF-1 level in the condition of liver metastasis by mathematical analysis. Firmly, the compartmental model of IGF-1 was long back designed by Bouroujerdi et al. in 1994 to study its kinetics in the body, and its metabolism. Their study was simulated mathematically by using a linear differential equation, and the results present the binding efficacy of IGF-1 on its receptor, and concentration of unbound, and bound forms of IGF-1 in plasma, interstitial fluid, and also estimated IGF-1 metabolic clearance rate6. With the support of this data, we aimed to design a three-compartment model and explain the risk level of IGF-1 to cause bone cancer mainly due to liver metastasis. Additionally, the role of IGF-1 on human cell proliferation, and its interaction with metabolite cyclo-glycyl-proline along with the action of bound, and unbound IGF-1 over IGF binding protein has been studied by scientist Phillips et al. through a mathematical compartment model, by using the differential equation, and mathematical simulation was done using MATLAB[R2020a]7. The research work conducted by Rieunier et al. explains the therapeutic target to treat bone metastasis through IGF-1R, elaborates the pathophysiological mechanism of bone cancer due to high IGF-1 concentration, and also interprets the pharmacological role of anti-IGF-1R in healing bone cancer8. The other investigational report published by scientists Jyoti et al.9 Explained the application of differential equation in determining the risk of bone cancer due to leukaemia which gives support to our study of bone cancer analysis with a mathematical application using a specific method i.e., Differential equation. Since the secretion of IGF-1 is from the liver, thus based on a review of literature, we propose and design a compartmental model that links the bridge between the problem arising due to liver metastasis and analysis of the risk of bone cancer via increased concentration of IGF-1 hormone secretion. The mathematical modeling in epidemiology proved to estimate the realistic and reliable results comparable to sophisticated lab table work, thus reducing expenses and time for research conducted by Sravanthi et al., 2017; Reddy and Rambabu, 2017,10,11.
The one pre-clinical evidence reported by Subhashini and Ruth represented the anti-cancer activity of the herbal drug Euphorbia hirta L on cell culture of bone cancerous cells MG-63 cells12. Additionally, our review also includes the pre-clinical study results of Ali and Makhou, 2022 reported the beneficial effect of hypolipidemic drugs i.e., statins improved bone cell synthesis and treat the condition of osteoporosis13. Moreover, while designing our research compartment model, the mechanism of osteogenesis is well understood by review given by scientists Ganta et al., 202114 and further the design of our compartment modeling was built using the overview explained by Debnath and Kumar, 202015.
1.2. Model design of IGF-1:
Previous studies could only give the idea of the kinetic of IGF-1 in the body, and their role in body development but none could explain the risk level of IGF-1 in causing bone cancer16. Our study aimed to design the compartment model of the IGF-1 circulatory system, and by mathematical simulation using ordinary differential equation enabled to trace the risk of bone cancer due to liver metastasis. The liver metastasis further causes an increase in IGF-1 hormone which in turn gets deposited over the skeletal system, and more IGF-1 molecules trigger the uncontrolled proliferation of bone cells and thus become the cause of bone cancer8. We aimed to estimate the concentration of IGF-1 levels in the skeletal system to understand the risk factor of IGF-1, and liver metastasis in leading bone cancer.
2. MATERIALS AND METHODS:
2.1. Modeling IGF-1 kinetics:
The Pharmacokinetic compartmental model is sketched to trace the risk of the IGF-1 concentration level causing uncontrolled bone cell proliferation due to liver metastasis. The IGF-1 is synthesized in the liver considered compartment x1 and released in the circulatory system which is compartment x2 further showing its action in the growth, and functioning of the skeletal system by regulating bone cell proliferation and remodeling assumed as compartment x33. The IGF-1 concentration is estimated in compartments x1, x2, and x3 by using mathematical simulation, which in turn enabled us to understand the risk of bone cancer due to liver metastasis inducing high IGF-1 synthesis. The rate constants of IGF-1 taken as ke1 as efflux from the liver, kL from the liver to blood plasma, ke2 efflux from blood plasma and kP from blood plasma to the skeletal system. As there is no efflux from the skeletal system, it is assumed that IGF-1 hormone continues to deposit and accumulate over bone and trigger extensive bone cell proliferation ultimately causing bone cancer.
Figure1: The schematic sketch of IGF-1 synthesize in compartment x1 indicates liver, flow-through plasma as compartment x2, and deposit over the bone as compartment x3.
The investigation is held using an ordinary differential equation and computation simulation is performed using MATLAB (version R2020a).
The LODE’s are
(3)
2.2. The Model Parameters:
The normal parameter values of IGF-1 kinetics in the body are considered from the literature review as follows:
The initial amount of IGF-1 in the liver x01 =
62.5 × 10-2 mg17 ;
The initial amount of IGF-1 in plasma x02= 0 mg
The initial amount of IGF-1 in bone x03= 0 mg
[From the property of kinetic of a matter(drug), when time t=0, in compartment x2 and compartment x3, the amount of IGF-1 should be zero.]
The initial amount of IGF-1 in the liver (In cancer) x01= 77.5 × 10-2 mg17;
Volume of Liver vL=1470 ml3 18,19;
Volume of Plasma vP=3500 ml3 6;
Volume of bone vB=
mm3 20;
Elimination rate from liver 
=0.30 min-1 17;
Rate constant from liver to plasma kL=0.30 min-1 6;
Rate constant from plasma to bones kP=0.138 min-1 6;
Clearance rate from plasma
= 0.064 min-1 6;
The compartment model explains the kinetics of IGF-1 peptide
hormone from liver x1 (the origin of IGF-1 production) towards bones
x3 in the body through plasma x2. The elimination rate constant
from liver and plasma is considered as and , whereas the rate constant of IGF-1 from the
liver to plasma is kL and from plasma to bone is kP. The study
analysis includes estimating the risk level of IGF-1 produced in the condition of
liver metastasis and thus its effect on the skeletal system ultimately results in
causing bone cancer.
3. CALCULATIONS:
3.1. Mathematical application: Ordinary Differential Equation:
The ordinary differential equation of the designed compartmental model is simplified and mathematical simulation is estimated using the high throughput technique tool MATLAB (version R2020a).
The solutions tothe above equations are
(4)
When, t = 0, (5)
Then, 
(6)
From eqns. (4) – (6)
(7)
(8)
When, t = 0, 
(9)
Then, (10)
From eqns. (8) – (10)
(11)
And,
(12)
When, t = 0, 
(13)
(14)
From eq. (12) – (14)
(15)
3.2 Model Parameter Conditions: Considering Cancer:
The cancer state of the liver is the root cause to increase
in the production rate of IGF-1 hormone, and the major role of IGF-1 is to regulate
the development of the skeletal system and help in body metabolism21.
Thus, due to liver cancer, the volume of liver vL increases to 3024ml3,
and the initial mass of IGF-1 x01 raised to 
. Due to the high production of IGF-1, its release
rate from kL and is also considered to vary from high to low
constants i.e., 0.35, 0.30, and 0.24 min-1. Also, the rate constant from
plasma to bone varies kP as 0.1, 0.135, and 0.170 min-1.
Initially, the normal IGF-1 concentration at normal rate constants and volumes has been analysed, then the parameter condition was considered as per the severity of liver cancer, and accordingly IGF-1 concentration analysis in x1, x2, and x3 is done using the MATLAB computing technique.
4. RESULT AND DISCUSSION:
In Figure 2, the parameter conditions taken as normal liver physiology and estimated results will be a normal concentration of IGF-1 which does not affect the skeletal system. Whereas rest of the figures i.e., fig 3,4,5,6, and 7 indicate the condition of liver cancer where only its initial weight x01 and volume vL of liver calculated as per cancerous condition18,19; and the remaining parameter values are considered from low to high range to understand the severity of liver cancer to raise IGF-1 concentration in the skeletal system. Thus, the parameter values are just considered as cases of liver cancer with normal and abnormal kinetic irrespective of changing the initial weight of the cancerous liver (x01 is fixed in all cases of liver cancer).
Figures 2a, 2b, and 2c indicate the concentration of IGF-1 in compartments x1, x2, and x3 respectively considering normal physiology of liver functions, IGF-1 synthesis, and its rate constant.
Figures 2a, 2b, and 2c indicate the concentration of IGF-1 in compartments x1, x2, and x3 respectively considering normal physiology of liver functions. All parameter values are given in 2.2.
Figures 3a, 3b, and 3c show the concentrations of IGF-1
in x1, x2, and x3 considering the condition of
liver metastasis. Due to liver cancer, the volume of the liver taken is 3024 ml3,
and the initial mass of IGF-1 in the liver is 
as its synthesis increases due to liver cancer,
whereas rate constants of IGF-1 are assumed as normal.
The figures 3a, 3b, 3c show the concentrations of IGF-1
in x1, x2, x3 considering the condition of liver
metastasis. All parameter values are given in 2.2 except 
, 
.
For figure 4, the parameter value is considered as an abnormal
increase in liver volume (vL=3500 ml3) and IGF-1 synthesis
(x01=
mg), whereas rate constants of IGF-1 are considered
normal.
The figures 4a, 4b, 4c show the concentrations of IGF-1 in x1, x2, x3 considering the condition of liver metastasis. All parameter values are given in 2.2 except X01 = 64.04 × 10-2, 77.5 × 10-2, 89 × 69 × 10-2, VL = 2500, 3000 3500 ml3.
The figures 5a, 5b, and 5c show the result of IGF-1 concentration
in case of an abnormal decrease in the rate constant of IGF-1 from the liver to
plasma (kL) and the rate of hepatic clearance 
as 0.35, 0.30, and 0.24 min-1 are
considered, and the conditions of liver cancer as explained in figure 2 are also
considered.
The Figures 5a, 5b, 5c shows the concentrations of IGF-1
in x1, x2, x3. All parameter values are given in
2.2 except x01 = 77.5 × 10-2,VL = 3024ml3,
= kL = 0.35,0.30,0.24
The figures 6a, 6b, and 6c estimate the IGF-1 concentrations
in x1, x2, and x3 due to change in the rate constant
from plasma to bone (kP) and elimination rate (
), the condition of liver metastasis of increased
liver volume and initial mass of IGF-1 is also considered.
The figures 6a, 6b, and 6c estimate the IGF-1 concentrations in x1, x2, and x3. All parameter values are given in 2.2 except,
X01 = 77.5 × 10-2, VL = 3024 ml3
Figure 7 considers all the abnormal parameter values of
liver metastasis together and helps to understand the abnormal IGF-1 concentration
in all the compartments x1, x2, and x3, while estimating
three cases are taken to explain the severity of IGF-1 concentration, and among
them, an initial mass of IGF-1 (x01), and liver volume (vL)
is considered constant in all cases as mg, and 3024 ml3.Case 1 of figure
7 is taken as ke1 =kL=0.35, = 0.040, kP=0.1 min-1.
Case 2 considered as = kL= 0.30, =0.060, kP= 0.135 min-1.
Case 3 is taken as =kL=0.24, =0.080, kP=0.170 min-1.
Thus, the varying rate constants = kL, and kp in cases 1, 2, and 3 are considered
as a condition where rate constant ranges from lowest to high value as per literature
review22,23,24.
The result analysis of all three cases indicates the clear outlook of IGF-1 concentration in case of liver cancer and its toxic accumulation over bone shows that the IGF-1 is the prime risk factor that can lead to bone cancer which is causing the abnormal stimulation of bone cell mitosis and thus an abnormal increase in a number of bone cells.
The figures 7a, 7b, and 7c estimate the IGF-1 concentrations in x1, x2, and x3. All parameter values are given in 2.2 except
X01 = 77.5 × 10-2, VL = 3024 ml3
= kL = 0.35,0,30,0.24. 
, and kp 0.1, 0.135, 0.170.
The result analysis considering different conditions
of model parameter value helps to understand the reason of cause of bone cancer
due to IGF-1 concentrations in x1, x2, and x3 compartments.
Figure 7c represents the IGF-1 concentration as a risk factor causing bone cancer
due to liver metastasis where the parameter values of kinetics considered as = kL= 0.24, = 0.080, kP = 0.170. The scientists
Lee and Okos designed a mathematical model which explained the effect of IGF-1 in
regulating bone cell proliferation; they used an ordinary differential equation
to explain the relation between serum IGF-1 level and bone calcium accretion, which
indirectly co-relates the role of calcium in regulating bone cell remodeling and
repair25.
In our previous studies, we could successfully design a mathematical compartment model of biliary circulation, and using fractional ordinary differential equation Caputo model we estimated the variable concentration of bile acid causing various pathology related to the biliary system26, and also designed the mathematical compartment of cholesterol kinetics from liver to heart and using the ordinary differential equation we estimated the toxic level of cholesterol causing heart diseases. Thus, with the present study, we could successfully design a mathematical model of IGF-1 kinetics simplified the equation using an ordinary differential equation, and estimate the toxic level which is a cause of bone cancer.
5. CONCLUSION:
As the results outcomes of our study clearly explain the effect of uncontrolled liver cell proliferation with increased IGF-1 level is a root cause of leading bone cancer. Such analysis may help to understand the concentration of risk level of IGF-1 causing bone cancer, and accordingly suggests deciding the drug, and dosage regimen along with the therapeutic target considering the skeletal system can help to overcome the toxicity of IGF-1. The mathematical method of an ordinary differential equation and simulation is done using MATLAB helped in estimating the concentration of IGF-1 by the virtual assumption of the body system, and IGF-1 kinetics. The future perspective suggests that our study results can help to conduct preclinical studies, and comparative analysis can be made.
6. CONFLICT OF INTEREST:
The authors have no conflicts of interest regarding this investigation.
7. ACKNOWLEDGMENT:
We are thankful to DST-FIST for providing infrastructural facilities at School of Mathematical Sciences, SRTM University, Nanded with the aid of which this research work has been carried out.
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Received on 09.05.2022 Modified on 06.10.2022
Accepted on 25.02.2023 © RJPT All right reserved
Research J. Pharm. and Tech 2023; 16(9):4199-4205.
DOI: 10.52711/0974-360X.2023.00687