Mathematical model of turmeric

In the previous post (link), we saw a sample pharmacokinetic model that would give a mathematical explanation for the concentration of a drug in the blood. In this post, we will see the adaptation of this model for a mathematical model of turmeric.

Turmeric is a plant native to the Indian subcontinent and south-east asia. It is widely used as a medicinal herb in Ayurvedic and other ancient medical sciences. The main ingredient that is thought to be therapeutic is Curcumin (ref). Turmeric in Ayurvedic medications is either applied topically on the skin, or administered orally. We will begin our examination of turmeric by looking at its oral administration. To adapt the math model approach to turmeric, assuming curcumin is the active ingredient, one needs to know how much curcumin turmeric contains, how it is absorbed into the blood, how it distributes to the organs and how it is eliminated.

Each preparation of turmeric is said to have 1-7% of curcumin (ref). A lot of curcumin gets metabolized in the liver and the gut through what is known as the first pass metabolism (ref), which leads to low amount of curcumin in the blood when it is administered orally. About 33% of curcumin that is administered supposedly reaches the blood stream. Maximum concentration in the blood stream generally occurs 1-2 hours postdose and is found to decline within 12 hours. In a study which used an 8 g/day dose daily (ref), there was a peak blood concentration of 1.75 +/- 0.8 uM*. In a phase 1 study of Curcumin as a chemo preventive agent (ref), 25 patients were administered 3 doses of curcumin at 4, 6, and 8 g per day and blood concentrations of curcumin were recorded.

Using some massaging of parameters I described in a previous post (ref), I managed to make a PK model of curcumin in the format of what is generally called a 2 compartment PK model (ref). One fit for the dose of 4 g is shown below where the red line indicates our model prediction and the blue line indicates experimental data.

The parameter choices used for this fit are something I cannot share. But with this model ready, I can predict the curcumin concentration in blood upon a dose of curcumin. If the curcumin concentration in a sample of turmeric is known, one can easily use the model to predict curcumin concentration in blood for a given dose of turmeric.

In the next post, we will study a sample pharmacodynamic model of turmeric.

A sample pharmacokinetic model.

In the previous post (link), I gave one of the reasons why one should prepare a mathematical model - so that we can predict if the drug concentration in the body is within the levels we desire. This kind of modeling is also called pharmacokinetic modeling. In this post, we will explore in further detail what form pharmacokinetic predictions will take.

As we learnt before, there are four key processes a drug undergoes after it is administered to the human body. These are Absorption, Distribution, Metabolism, and Excretion, in short known as (ADME). We will understand the effects of these processes in the context of a very simple mathematical model. Assume for a moment that a patient was given a pill with an active ingredient. On taking a pill, the first event that the pill would undergo is disintegration in the gastrointestinal tract of the subject. The liberated drug ingredients are absorbed by the bloodstream. During this time, the drug concentration rises in the blood stream.

Figure: A typical plasma drug concentration time curve (Ref: link)

As the blood is transported across the body, the drug leaves the blood stream and drug concentration in the blood starts falling. This is the distribution and metabolism phase of the drug life in the body. During this time, as the drug passes through excretory organs such as the kidney or the liver, the drug is also getting eliminated from the blood. However, this process gets more prominent as the drug concentration falls below a certain limit. The figure above shows a typical drug concentration curve in the blood. Also shown are the minimum effective concentration and the maximum safe concentration (minimum toxic concentration) that we saw in the last post (link). Analyzing various parts of the curve gives among others, the following parameters:

ka : Absorption rate constant : - Parameters like these give the rising shape of the curve above.
Q: Intercompartmental clearance: - Parameters like these explain the post absorption phase of the curve. They explain how the drug gets taken up by other organs in the body.
Cl: Clearance: - Parameters like these explain the elimination phase of the curve.
V: Volume of distribution: Parameters like these help give the volume in which the drug is distributed. Depending on the drug, these can range from the volume of the blood in the human body to a range determined by the protein binding of drug in the body.

If C is the concentration of drug in the blood, then we can write the following equations to predict C:

dA/dt = -ka*A
dC/dt = ka*A/Vc - Q*C - Cl*C

where A is amount of drug administered and dA/dt and dC/dt indicate miniscule changes in A and C over miniscule changes in time (for more info on above equations, see link).

There are several other pharmacokinetic parameters of importance and we will see them in the context of the models discussed in the future. Plasma drug curves as the one shown above are supported by rigorous clinical data with drug sampling done at short intervals so as to be create a virtual facsimile of the drug processes. Studying ayurvedic drugs in similar context gets possible if after identification of active ingredients of the drug repeated sampling is done in the blood to assess the blood levels of the drug as it travels through the body. One example is turmeric which is widely used spice in Indian cooking. Curcumin, one of the chemical compounds found in turmeric is currently receiving a lot of focus because of its anti-oxidant and anti-inflammatory properties. In the next post, we will look at a hypothetical model for Curcumin.

Minimum effective (and toxic) concentration of a drug

In the previous post (click), I explained that correctly accounting for the pharmacokinetics of a drug can help explain the concentration of the drug in the blood and in the target organ. The question that obviously follows this point is:

What is the desired concentration of the drug?

As we commonly notice, each drug we take has a certain amount associated with it and a certain frequency of administration. The purpose of this is to raise the concentration of the drug to a level that it effectively reaches the target organ to cause its effect. The minimum amount of drug required for the drug to have its intended effect is called the minimum effective concentration of a drug. While it is possible that higher concentrations of the drug show beneficial effect too, the drug must be at the minimum effective concentration to have its effect. Below this, the drug doesn't show efficacy.

If too little drug is a problem, too much is also a problem. The converse of the minimum effective concentration is called the minimum toxic concentration of a drug. It is the minimum level above which toxic side effects of the drug are observed.

So the desired concentration of the drug is such that it has its desired effect without being toxic. As a result, most drug delivery strategies are designed such that the drug concentration remains between the minimum effective and minimum toxic concentration of a drug. Once we know through experimentation the various body processes that impact the concentration of the drug, we can generate a virtual facsimile of the processes the drug undergoes using computer programs and mathematical equations, a process we know as mathematical modeling. Once a model is ready, countless scenarios of hypothetical and existing drug dosing can be tested and we can check if we are maintaining the drug concentration at the desired level. The presence of a model helps improve the drug development process by minimizing the number of experiments required.

Routine drug development processes in the pharmaceutical industry are increasingly adapting pharmacokinetic modeling to propose drug doses, frequencies, and sometimes novel ways of drug administration to maintain drug concentration at the desired level. Applying pharmacokinetic models in Ayurveda can allow Ayurveda too to evolve at a similar pace as modern medicine. In the next post, we will see a sample pharmacokinetic model, important terms involved and possible applications for such models in the world of Ayurveda.

Why do we need to model an Ayurvedic drug?

In the previous post (click), we mentioned that we intend to mathematically model ayurvedic drugs. But what are the advantages of mathematically modeling ayurvedic drugs? Before we answer that question, we need to understand what are models and why is modeling done.

What are models?

So what are the models that we talk about? This is not about the latest fashion or jazzy cars and bikes. The model we refer to here is a description of a system using mathematical concepts and language. Mathematical modeling is used in a wide array of fields to first describe a system for example, atmospheric changes, strength of a car hood, or even atomic arrangement of a sheet of metal. The description is in terms of physical properties of the systems, for example, their ability to conduct heat, convection of gases, structural rigidity, etc. Then the impact on the system by an agent is incorporated into the model, such as a heat source, a car crash, an impending hurricane, etc. The closeness of the mathematical description of the system we created helps us predict what would happen when the agent impacts the system. In this way, a mathematical model helps us generate ideas the system would undergo even before running an actual test, and it helps us reduce cost of experiments.

Why do we need to model a drug?

So now we know what are models and why is modeling done. Now why do we need to model any drug? Well our body is governed by the same principles of physics that govern the rest of the universe. Our various organ systems have some commonality with our exterior surroundings. Our nervous system is a good conductor of electricity, our bones that provide our bodies rigidity are similar to construction materials of buildings that undergo physical pressure from the environment, our blood vessels are similar to water pipes and bear the impact of the blood they carry, the list is endless. So naturally these organ systems are fit to be mathematically modeled. 

When the drug enters the body, it undergoes four major processes. It is absorbed (A) into the blood stream (or directly delivered into the blood stream), it is distributed (D) to the body organs where the blood carries it, it is metabolized (M) or broken down in the body organs where it reaches, the metabolized end products enter the blood stream again, and then it is excreted (E) via body organs such as the kidneys, liver, or sometimes even the skin. Collectively, these processes are called the ADME of the drug, with one letter standing for each of the four processes. It is also known as pharmacokinetics. Correctly understanding the impact of these processes on the drug can lead us to predict what is the concentration of the drug in the blood, or in a target organ. Mathematical modeling these bodily processes helps us to predict drug concentration in the blood or in a target organ. 

Once we describe what the drug undergoes as it is administered to the body, we then need to take into account the intended impact of the drug. This can take many forms, but in general here is where we look at how the drug alleviates either the symptoms of the disease, or cures the disease. The study of drug action on the body is called pharmacodynamics. Correctly understanding what the drug does to the body or pharmacodynamics of the drug helps us predict how the drug would carry out its intended action. Most mathematical models of a drug take into account pharmacokinetics or pharmacodynamics or both and vary in complexity depending upon the problem question at hand. 

Why do we need to model an Ayurvedic drug?

As we saw in the previous post (click), ayurvedic treatments are similar to modern day drugs, except that the chemical compound of the drug is a naturally occurring one, not the one chemically synthesized in a lab. The same reasons why we would model modern day drugs can be applied to ayurvedic drugs, and these would help us understand the concentration of ayurvedic drugs in the blood stream and how these drugs go about curing the disease. 

Introduction

In the Hindu epic Ramayana, in the great war in Lanka against the demon Ravana, Shri Lakshmana gets injured. To cure him, Shri Rama requests Hanumana to fly to the Himalayas in search of a herb (Sanjivany) on mount Dronagiri in the Himalayas. Unable to locate the exact herb, Hanumana brings the whole mountain to Lanka so that Ayurveda practitioners in Lanka can heal Lakshmana. Through time, Sanjivany has become mythical today and nobody knows for perfection which of today's herbs it refers to.

Ayurveda is a traditional Indian medicine, the practice of which dates back 1000s of years. Ayurveda believes that diseases could be healed by a balance of 'doshas' or elements that make up the body, namely 'vata', 'pitta', and 'kapha' (air, fire, and phlegm) and one of the tools it uses to perform this is administration of herbs as medicine. However, each herb contains multiple chemical compounds which are digested by the body, absorbed in the blood, and transported to the various parts of the body. These processes can be understood using laws of physics and written as mathematical equations, following which they could be simulated on a computer.

In current use, Ayurveda declares a disease cured by observation of disappearance of symptoms rather than an analysis of disease identifiers disappearing. Mathematically describing the outcome of an administration of a herb can help develop an accurate and precise understanding of how an Ayurvedic drug helps cure a disease. In scientific parlance, this process of mathematically describing a physical phenomenon is called modeling, and using the model to predict what might happen in an untested scenario is called a simulation.

This blog and post marks the beginning of a quest to understand Ayurvedic treatments in depth using mathematical equations and laws of nature. As Sanjivany is a popular mythical herb, I chose to call the blog by this name. The purpose of this blog is to generate interest in the unexplored field of mathematical modeling of ayurvedic drugs, entice collaborations, take criticisms, and improve the concepts explained above.